A cross-diffusion system derived from a Fokker-Planck equation with partial averaging

NASA Astrophysics Data System (ADS)

Jüngel, Ansgar; Zamponi, Nicola

2017-02-01

A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.

A two-dimensional kinematic dynamo model of the ionospheric magnetic field at Venus

NASA Technical Reports Server (NTRS)

Cravens, T. E.; Wu, D.; Shinagawa, H.

1990-01-01

The results of a high-resolution, two-dimensional, time dependent, kinematic dynamo model of the ionospheric magnetic field of Venus are presented. Various one-dimensional models are considered and the two-dimensional model is then detailed. In this model, the two-dimensional magnetic induction equation, the magnetic diffusion-convection equation, is numerically solved using specified plasma velocities. Origins of the vertical velocity profile and of the horizontal velocities are discussed. It is argued that the basic features of the vertical magnetic field profile remain unaltered by horizontal flow effects and also that horizontal plasma flow can strongly affect the magnetic field for altitudes above 300 km.

NUMERICAL ANALYSES FOR TREATING DIFFUSION IN SINGLE-, TWO-, AND THREE-PHASE BINARY ALLOY SYSTEMS

NASA Technical Reports Server (NTRS)

Tenney, D. R.

1994-01-01

This package consists of a series of three computer programs for treating one-dimensional transient diffusion problems in single and multiple phase binary alloy systems. An accurate understanding of the diffusion process is important in the development and production of binary alloys. Previous solutions of the diffusion equations were highly restricted in their scope and application. The finite-difference solutions developed for this package are applicable for planar, cylindrical, and spherical geometries with any diffusion-zone size and any continuous variation of the diffusion coefficient with concentration. Special techniques were included to account for differences in modal volumes, initiation and growth of an intermediate phase, disappearance of a phase, and the presence of an initial composition profile in the specimen. In each analysis, an effort was made to achieve good accuracy while minimizing computation time. The solutions to the diffusion equations for single-, two-, and threephase binary alloy systems are numerically calculated by the three programs NAD1, NAD2, and NAD3. NAD1 treats the diffusion between pure metals which belong to a single-phase system. Diffusion in this system is described by a one-dimensional Fick's second law and will result in a continuous composition variation. For computational purposes, Fick's second law is expressed as an explicit second-order finite difference equation. Finite difference calculations are made by choosing the grid spacing small enough to give convergent solutions of acceptable accuracy. NAD2 treats diffusion between pure metals which form a two-phase system. Diffusion in the twophase system is described by two partial differential equations (a Fick's second law for each phase) and an interface-flux-balance equation which describes the location of the interface. Actual interface motion is obtained by a mass conservation procedure. To account for changes in the thicknesses of the two phases as diffusion progresses, a variable grid technique developed by Murray and Landis is employed. These equations are expressed in finite difference form and solved numerically. Program NAD3 treats diffusion between pure metals which form a two-phase system with an intermediate third phase. Diffusion in the three-phase system is described by three partial differential expressions of Fick's second law and two interface-flux-balance equations. As with the two-phase case, a variable grid finite difference is used to numerically solve the diffusion equations. Computation time is minimized without sacrificing solution accuracy by treating the three-phase problem as a two-phase problem when the thickness of the intermediate phase is less than a preset value. Comparisons between these programs and other solutions have shown excellent agreement. The programs are written in FORTRAN IV for batch execution on the CDC 6600 with a central memory requirement of approximately 51K (octal) 60 bit words.

Void Formation during Diffusion - Two-Dimensional Approach

NASA Astrophysics Data System (ADS)

Wierzba, Bartek

2016-06-01

The final set of equations defining the interdiffusion process in solid state is presented. The model is supplemented by vacancy evolution equation. The competition between the Kirkendall shift, backstress effect and vacancy migration is considered. The proper diffusion flux based on the Nernst-Planck formula is proposed. As a result, the comparison of the experimental and calculated evolution of the void formation in the Fe-Pd diffusion couple is shown.

A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation

NASA Astrophysics Data System (ADS)

Tayebi, A.; Shekari, Y.; Heydari, M. H.

2017-07-01

Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection-diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection-diffusion equation (V-OTFA-DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.

Applications of an exponential finite difference technique

DOE Office of Scientific and Technical Information (OSTI.GOV)

Handschuh, R.F.; Keith, T.G. Jr.

1988-07-01

An exponential finite difference scheme first presented by Bhattacharya for one dimensional unsteady heat conduction problems in Cartesian coordinates was extended. The finite difference algorithm developed was used to solve the unsteady diffusion equation in one dimensional cylindrical coordinates and was applied to two and three dimensional conduction problems in Cartesian coordinates. Heat conduction involving variable thermal conductivity was also investigated. The method was used to solve nonlinear partial differential equations in one and two dimensional Cartesian coordinates. Predicted results are compared to exact solutions where available or to results obtained by other numerical methods.

Interface- and discontinuity-aware numerical schemes for plasma 3-T radiation diffusion in two and three dimensions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dai, William W., E-mail: dai@lanl.gov; Scannapieco, Anthony J.

2015-11-01

A set of numerical schemes is developed for two- and three-dimensional time-dependent 3-T radiation diffusion equations in systems involving multi-materials. To resolve sub-cell structure, interface reconstruction is implemented within any cell that has more than one material. Therefore, the system of 3-T radiation diffusion equations is solved on two- and three-dimensional polyhedral meshes. The focus of the development is on the fully coupling between radiation and material, the treatment of nonlinearity in the equations, i.e., in the diffusion terms and source terms, treatment of the discontinuity across cell interfaces in material properties, the formulations for both transient and steady states,more » the property for large time steps, and second order accuracy in both space and time. The discontinuity of material properties between different materials is correctly treated based on the governing physics principle for general polyhedral meshes and full nonlinearity. The treatment is exact for arbitrarily strong discontinuity. The scheme is fully nonlinear for the full nonlinearity in the 3-T diffusion equations. Three temperatures are fully coupled and are updated simultaneously. The scheme is general in two and three dimensions on general polyhedral meshes. The features of the scheme are demonstrated through numerical examples for transient problems and steady states. The effects of some simplifications of numerical schemes are also shown through numerical examples, such as linearization, simple average of diffusion coefficient, and approximate treatment for the coupling between radiation and material.« less

A geometrical multi-scale numerical method for coupled hygro-thermo-mechanical problems in photovoltaic laminates.

PubMed

Lenarda, P; Paggi, M

A comprehensive computational framework based on the finite element method for the simulation of coupled hygro-thermo-mechanical problems in photovoltaic laminates is herein proposed. While the thermo-mechanical problem takes place in the three-dimensional space of the laminate, moisture diffusion occurs in a two-dimensional domain represented by the polymeric layers and by the vertical channel cracks in the solar cells. Therefore, a geometrical multi-scale solution strategy is pursued by solving the partial differential equations governing heat transfer and thermo-elasticity in the three-dimensional space, and the partial differential equation for moisture diffusion in the two dimensional domains. By exploiting a staggered scheme, the thermo-mechanical problem is solved first via a fully implicit solution scheme in space and time, with a specific treatment of the polymeric layers as zero-thickness interfaces whose constitutive response is governed by a novel thermo-visco-elastic cohesive zone model based on fractional calculus. Temperature and relative displacements along the domains where moisture diffusion takes place are then projected to the finite element model of diffusion, coupled with the thermo-mechanical problem by the temperature and crack opening dependent diffusion coefficient. The application of the proposed method to photovoltaic modules pinpoints two important physical aspects: (i) moisture diffusion in humidity freeze tests with a temperature dependent diffusivity is a much slower process than in the case of a constant diffusion coefficient; (ii) channel cracks through Silicon solar cells significantly enhance moisture diffusion and electric degradation, as confirmed by experimental tests.

Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution.

PubMed

Pedron, I T; Mendes, R S; Malacarne, L C; Lenzi, E K

2002-04-01

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.

On the local well-posedness and a Prodi-Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion

NASA Astrophysics Data System (ADS)

Larios, Adam; Pei, Yuan

2017-07-01

We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. To the best of our knowledge, this is the first Prodi-Serrin-type criterion for such a 3D hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-Boussinesq equations. We note that, as a special case, these results include the 3D non-diffusive Boussinesq system and the 3D MHD equations. Moreover, they can be extended without difficulty to include the case of a Coriolis rotational term.

A numerical solution for the diffusion equation in hydrogeologic systems

USGS Publications Warehouse

Ishii, A.L.; Healy, R.W.; Striegl, Robert G.

1989-01-01

The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)

Generalized fourier analyses of the advection-diffusion equation - Part II: two-dimensional domains

NASA Astrophysics Data System (ADS)

Voth, Thomas E.; Martinez, Mario J.; Christon, Mark A.

2004-07-01

Part I of this work presents a detailed multi-methods comparison of the spatial errors associated with the one-dimensional finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. In Part II we extend the analysis to two-dimensional domains and also consider the effects of wave propagation direction and grid aspect ratio on the phase speed, and the discrete and artificial diffusivities. The observed dependence of dispersive and diffusive behaviour on propagation direction makes comparison of methods more difficult relative to the one-dimensional results. For this reason, integrated (over propagation direction and wave number) error and anisotropy metrics are introduced to facilitate comparison among the various methods. With respect to these metrics, the consistent mass Galerkin and consistent mass control-volume finite element methods, and their streamline upwind derivatives, exhibit comparable accuracy, and generally out-perform their lumped mass counterparts and finite-difference based schemes. While this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common mathematical framework. Published in 2004 by John Wiley & Sons, Ltd.

Can phoretic particles swim in two dimensions?

NASA Astrophysics Data System (ADS)

Sondak, David; Hawley, Cory; Heng, Siyu; Vinsonhaler, Rebecca; Lauga, Eric; Thiffeault, Jean-Luc

2016-12-01

Artificial phoretic particles swim using self-generated gradients in chemical species (self-diffusiophoresis) or charges and currents (self-electrophoresis). These particles can be used to study the physics of collective motion in active matter and might have promising applications in bioengineering. In the case of self-diffusiophoresis, the classical physical model relies on a steady solution of the diffusion equation, from which chemical gradients, phoretic flows, and ultimately the swimming velocity may be derived. Motivated by disk-shaped particles in thin films and under confinement, we examine the extension to two dimensions. Because the two-dimensional diffusion equation lacks a steady state with the correct boundary conditions, Laplace transforms must be used to study the long-time behavior of the problem and determine the swimming velocity. For fixed chemical fluxes on the particle surface, we find that the swimming velocity ultimately always decays logarithmically in time. In the case of finite Péclet numbers, we solve the full advection-diffusion equation numerically and show that this decay can be avoided by the particle moving to regions of unconsumed reactant. Finite advection thus regularizes the two-dimensional phoretic problem.

Normal versus anomalous self-diffusion in two-dimensional fluids: memory function approach and generalized asymptotic Einstein relation.

PubMed

Shin, Hyun Kyung; Choi, Bongsik; Talkner, Peter; Lee, Eok Kyun

2014-12-07

Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. At higher densities a range of normal diffusion can be identified. The motion presumably changes to sub-diffusion for even higher densities.

Normal versus anomalous self-diffusion in two-dimensional fluids: Memory function approach and generalized asymptotic Einstein relation

NASA Astrophysics Data System (ADS)

Shin, Hyun Kyung; Choi, Bongsik; Talkner, Peter; Lee, Eok Kyun

2014-12-01

Based on the generalized Langevin equation for the momentum of a Brownian particle a generalized asymptotic Einstein relation is derived. It agrees with the well-known Einstein relation in the case of normal diffusion but continues to hold for sub- and super-diffusive spreading of the Brownian particle's mean square displacement. The generalized asymptotic Einstein relation is used to analyze data obtained from molecular dynamics simulations of a two-dimensional soft disk fluid. We mainly concentrated on medium densities for which we found super-diffusive behavior of a tagged fluid particle. At higher densities a range of normal diffusion can be identified. The motion presumably changes to sub-diffusion for even higher densities.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

PubMed

Singh, Brajesh K; Srivastava, Vineet K

2015-04-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

PubMed Central

Singh, Brajesh K.; Srivastava, Vineet K.

2015-01-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

Mixing Regimes in a Spatially Confined, Two-Dimensional, Supersonic Shear Layer

DTIC Science & Technology

1992-07-31

MODEL ................................... 3 THE MODEL PROBLEMS .............................................. 6 THE ONE-DIMENSIONAL PROBLEM...the effects of the numerical diffusion on the spectrum. Guirguis et al.ś and Farouk et al."’ have studied spatially evolving mixing layers for equal...approximations. Physical and Numerical Model General Formulation We solve the time-dependent, two-dimensional, compressible, Navier-Stokes equations for a

Generalized Fourier analyses of the advection-diffusion equation - Part I: one-dimensional domains

NASA Astrophysics Data System (ADS)

Christon, Mark A.; Martinez, Mario J.; Voth, Thomas E.

2004-07-01

This paper presents a detailed multi-methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi-discretizations of the scalar advection-diffusion equation. The errors are reported in terms of non-dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew-symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov-Galerkin and its control-volume finite element analogue, the streamline upwind control-volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi-discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super-convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behaviour. In Part II of this paper, we consider two-dimensional semi-discretizations of the advection-diffusion equation and also assess the affects of grid-induced anisotropy observed in the non-dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi-methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

Diffusion Characteristics of Upwind Schemes on Unstructured Triangulations

NASA Technical Reports Server (NTRS)

Wood, William A.; Kleb, William L.

1998-01-01

The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and dimensionally-split finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the dimensionally-split finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2-3 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a 20-25 speedup over finite volume.

Reaction-diffusion systems coupled at the boundary and the Morse-Smale property

NASA Astrophysics Data System (ADS)

Broche, Rita de Cássia D. S.; de Oliveira, Luiz Augusto F.

We study an one-dimensional nonlinear reaction-diffusion system coupled on the boundary. Such system comes from modeling problems of temperature distribution on two bars of same length, jointed together, with different diffusion coefficients. We prove the transversality property of unstable and stable manifolds assuming all equilibrium points are hyperbolic. To this end, we write the system as an equation with noncontinuous diffusion coefficient. We then study the nonincreasing property of the number of zeros of a linearized nonautonomous equation as well as the Sturm-Liouville properties of the solutions of a linear elliptic problem.

A third-order computational method for numerical fluxes to guarantee nonnegative difference coefficients for advection-diffusion equations in a semi-conservative form

NASA Astrophysics Data System (ADS)

Sakai, K.; Watabe, D.; Minamidani, T.; Zhang, G. S.

2012-10-01

According to Godunov theorem for numerical calculations of advection equations, there exist no higher-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations in a semi-conservative form, in which there exist two kinds of numerical fluxes at a cell surface and these two fluxes are not always coincident in non-uniform velocity fields. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter. We extend the present method into multi-dimensional equations. Numerical experiments for advection-diffusion equations showed nonoscillatory solutions.

Downstream boundary effects on the frequency of self-excited oscillations in transonic diffuser flows

NASA Astrophysics Data System (ADS)

Hsieh, T.

1986-10-01

Investigation of downstream boundary effects on the frequency of self-excited oscillations in two-dimensional, separated transonic diffuser flows were conducted numerically by solving the compressible, Reynolds-averaged, thin-layer Navier-Stokes equation with two equation turbulence models. It was found that the flow fields are very sensitive to the location of the downstream boundary. Extension of the diffuser downstream boundary significantly reduces the frequency and amplitude of oscillations for pressure, velocity, and shock. The existence of a suction slot in the experimental setpup obscures the physical downstream boundary and therefore presents a difficulty for quantitative comparisons between computation and experiment.

Logarithmic Superdiffusion in Two Dimensional Driven Lattice Gases

NASA Astrophysics Data System (ADS)

Krug, J.; Neiss, R. A.; Schadschneider, A.; Schmidt, J.

2018-03-01

The spreading of density fluctuations in two-dimensional driven diffusive systems is marginally anomalous. Mode coupling theory predicts that the diffusivity in the direction of the drive diverges with time as (ln t)^{2/3} with a prefactor depending on the macroscopic current-density relation and the diffusion tensor of the fluctuating hydrodynamic field equation. Here we present the first numerical verification of this behavior for a particular version of the two-dimensional asymmetric exclusion process. Particles jump strictly asymmetrically along one of the lattice directions and symmetrically along the other, and an anisotropy parameter p governs the ratio between the two rates. Using a novel massively parallel coupling algorithm that strongly reduces the fluctuations in the numerical estimate of the two-point correlation function, we are able to accurately determine the exponent of the logarithmic correction. In addition, the variation of the prefactor with p provides a stringent test of mode coupling theory.

An Exponential Finite Difference Technique for Solving Partial Differential Equations. M.S. Thesis - Toledo Univ., Ohio

NASA Technical Reports Server (NTRS)

Handschuh, Robert F.

1987-01-01

An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that were more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.

exponential finite difference technique for solving partial differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Handschuh, R.F.

1987-01-01

An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that weremore » more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.« less

A two-dimensional, time-dependent model of suspended sediment transport and bed reworking for continental shelves

USGS Publications Warehouse

Harris, C.K.; Wiberg, P.L.

2001-01-01

A two-dimensional, time-dependent solution to the transport equation is formulated to account for advection and diffusion of sediment suspended in the bottom boundary layer of continental shelves. This model utilizes a semi-implicit, upwind-differencing scheme to solve the advection-diffusion equation across a two-dimensional transect that is configured so that one dimension is the vertical, and the other is a horizontal dimension usually aligned perpendicular to shelf bathymetry. The model calculates suspended sediment concentration and flux; and requires as input wave properties, current velocities, sediment size distributions, and hydrodynamic sediment properties. From the calculated two-dimensional suspended sediment fluxes, we quantify the redistribution of shelf sediment, bed erosion, and deposition for several sediment sizes during resuspension events. The two-dimensional, time-dependent approach directly accounts for cross-shelf gradients in bed shear stress and sediment properties, as well as transport that occurs before steady-state suspended sediment concentrations have been attained. By including the vertical dimension in the calculations, we avoid depth-averaging suspended sediment concentrations and fluxes, and directly account for differences in transport rates and directions for fine and coarse sediment in the bottom boundary layer. A flux condition is used as the bottom boundary condition for the transport equation in order to capture time-dependence of the suspended sediment field. Model calculations demonstrate the significance of both time-dependent and spatial terms on transport and depositional patterns on continental shelves. ?? 2001 Elsevier Science Ltd. All rights reserved.

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

DOE Office of Scientific and Technical Information (OSTI.GOV)

Boubendir, Yassine; Mendez, Vicenc; Rotstein, Horacio G.

2010-09-15

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delaymore » time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.« less

From analytical solutions of solute transport equations to multidimensional time-domain random walk (TDRW) algorithms

NASA Astrophysics Data System (ADS)

Bodin, Jacques

2015-03-01

In this study, new multi-dimensional time-domain random walk (TDRW) algorithms are derived from approximate one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) analytical solutions of the advection-dispersion equation and from exact 1-D, 2-D, and 3-D analytical solutions of the pure-diffusion equation. These algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point, which may be incomplete due to 2-D or 3-D transverse dispersion or diffusion. The method is extended to heterogeneous media, represented as a piecewise collection of homogeneous media. The particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries. The accuracy of the multi-dimensional TDRW method is verified against (i) exact analytical solutions of solute transport in homogeneous media and (ii) finite-difference simulations in a synthetic 2-D heterogeneous medium of simple geometry. The results demonstrate that the method is ideally suited to purely diffusive transport and to advection-dispersion transport problems dominated by advection. Conversely, the method is not recommended for highly dispersive transport problems because the accuracy of the advection-dispersion TDRW algorithms degrades rapidly for a low Péclet number, consistent with the accuracy limit of the approximate analytical solutions. The proposed approach provides a unified methodology for deriving multi-dimensional time-domain particle equations and may be applicable to other mathematical transport models, provided that appropriate analytical solutions are available.

Effective electrodiffusion equation for non-uniform nanochannels.

PubMed

Marini Bettolo Marconi, Umberto; Melchionna, Simone; Pagonabarraga, Ignacio

2013-06-28

We derive a one-dimensional formulation of the Planck-Nernst-Poisson equation to describe the dynamics of a symmetric binary electrolyte in channels whose section is nanometric and varies along the axial direction. The approach is in the spirit of the Fick-Jacobs diffusion equation and leads to a system of coupled equations for the partial densities which depends on the charge sitting at the walls in a non-trivial fashion. We consider two kinds of non-uniformities, those due to the spatial variation of charge distribution and those due to the shape variation of the pore and report one- and three-dimensional solutions of the electrokinetic equations.

Derivation of a closed form analytical expression for fluorescence recovery after photo bleaching in the case of continuous bleaching during read out

NASA Astrophysics Data System (ADS)

Endress, E.; Weigelt, S.; Reents, G.; Bayerl, T. M.

2005-01-01

Measurements of very slow diffusive processes in membranes, like the diffusion of integral membrane proteins, by fluorescence recovery after photo bleaching (FRAP) are hampered by bleaching of the probe during the read out of the fluorescence recovery. In the limit of long observation time (very slow diffusion as in the case of large membrane proteins), this bleaching may cause errors to the recovery function and thus provides error-prone diffusion coefficients. In this work we present a new approach to a two-dimensional closed form analytical solution of the reaction-diffusion equation, based on the addition of a dissipative term to the conventional diffusion equation. The calculation was done assuming (i) a Gaussian laser beam profile for bleaching the spot and (ii) that the fluorescence intensity profile emerging from the spot can be approximated by a two-dimensional Gaussian. The detection scheme derived from the analytical solution allows for diffusion measurements without the constraint of observation bleaching. Recovery curves of experimental FRAP data obtained under non-negligible read-out bleaching for native membranes (rabbit endoplasmic reticulum) on a planar solid support showed excellent agreement with the analytical solution and allowed the calculation of the lipid diffusion coefficient.

General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations

NASA Astrophysics Data System (ADS)

Lin, Guoxing

2018-05-01

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. A general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion { 0 < α , β ≤ 2 } based on the fractional derivative, has not been obtained, where α and β are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. The theory and the CTRW simulation agree with each other. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.

Rarefied gas flows through a curved channel: Application of a diffusion-type equation

NASA Astrophysics Data System (ADS)

Aoki, Kazuo; Takata, Shigeru; Tatsumi, Eri; Yoshida, Hiroaki

2010-11-01

Rarefied gas flows through a curved two-dimensional channel, caused by a pressure or a temperature gradient, are investigated numerically by using a macroscopic equation of convection-diffusion type. The equation, which was derived systematically from the Bhatnagar-Gross-Krook model of the Boltzmann equation and diffuse-reflection boundary condition in a previous paper [K. Aoki et al., "A diffusion model for rarefied flows in curved channels," Multiscale Model. Simul. 6, 1281 (2008)], is valid irrespective of the degree of gas rarefaction when the channel width is much shorter than the scale of variations of physical quantities and curvature along the channel. Attention is also paid to a variant of the Knudsen compressor that can produce a pressure raise by the effect of the change of channel curvature and periodic temperature distributions without any help of moving parts. In the process of analysis, the macroscopic equation is (partially) extended to the case of the ellipsoidal-statistical model of the Boltzmann equation.

Reaction-diffusion systems and external morphogen gradients: the two-dimensional case, with an application to skeletal pattern formation.

PubMed

Glimm, Tilmann; Zhang, Jianying; Shen, Yun-Qiu; Newman, Stuart A

2012-03-01

We investigate a reaction-diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction-diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction-diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.

Corrected simulations for one-dimensional diffusion processes with naturally occurring boundaries.

PubMed

Shafiey, Hassan; Gan, Xinjun; Waxman, David

2017-11-01

To simulate a diffusion process, a usual approach is to discretize the time in the associated stochastic differential equation. This is the approach used in the Euler method. In the present work we consider a one-dimensional diffusion process where the terms occurring, within the stochastic differential equation, prevent the process entering a region. The outcome is a naturally occurring boundary (which may be absorbing or reflecting). A complication occurs in a simulation of this situation. The term involving a random variable, within the discretized stochastic differential equation, may take a trajectory across the boundary into a "forbidden region." The naive way of dealing with this problem, which we refer to as the "standard" approach, is simply to reset the trajectory to the boundary, based on the argument that crossing the boundary actually signifies achieving the boundary. In this work we show, within the framework of the Euler method, that such resetting introduces a spurious force into the original diffusion process. This force may have a significant influence on trajectories that come close to a boundary. We propose a corrected numerical scheme, for simulating one-dimensional diffusion processes with naturally occurring boundaries. This involves correcting the standard approach, so that an exact property of the diffusion process is precisely respected. As a consequence, the proposed scheme does not introduce a spurious force into the dynamics. We present numerical test cases, based on exactly soluble one-dimensional problems with one or two boundaries, which suggest that, for a given value of the discrete time step, the proposed scheme leads to substantially more accurate results than the standard approach. Alternatively, the standard approach needs considerably more computation time to obtain a comparable level of accuracy to the proposed scheme, because the standard approach requires a significantly smaller time step.

Corrected simulations for one-dimensional diffusion processes with naturally occurring boundaries

NASA Astrophysics Data System (ADS)

Shafiey, Hassan; Gan, Xinjun; Waxman, David

2017-11-01

To simulate a diffusion process, a usual approach is to discretize the time in the associated stochastic differential equation. This is the approach used in the Euler method. In the present work we consider a one-dimensional diffusion process where the terms occurring, within the stochastic differential equation, prevent the process entering a region. The outcome is a naturally occurring boundary (which may be absorbing or reflecting). A complication occurs in a simulation of this situation. The term involving a random variable, within the discretized stochastic differential equation, may take a trajectory across the boundary into a "forbidden region." The naive way of dealing with this problem, which we refer to as the "standard" approach, is simply to reset the trajectory to the boundary, based on the argument that crossing the boundary actually signifies achieving the boundary. In this work we show, within the framework of the Euler method, that such resetting introduces a spurious force into the original diffusion process. This force may have a significant influence on trajectories that come close to a boundary. We propose a corrected numerical scheme, for simulating one-dimensional diffusion processes with naturally occurring boundaries. This involves correcting the standard approach, so that an exact property of the diffusion process is precisely respected. As a consequence, the proposed scheme does not introduce a spurious force into the dynamics. We present numerical test cases, based on exactly soluble one-dimensional problems with one or two boundaries, which suggest that, for a given value of the discrete time step, the proposed scheme leads to substantially more accurate results than the standard approach. Alternatively, the standard approach needs considerably more computation time to obtain a comparable level of accuracy to the proposed scheme, because the standard approach requires a significantly smaller time step.

Scanning tunneling spectroscopy study of the proximity effect in a disordered two-dimensional metal.

PubMed

Serrier-Garcia, L; Cuevas, J C; Cren, T; Brun, C; Cherkez, V; Debontridder, F; Fokin, D; Bergeret, F S; Roditchev, D

2013-04-12

The proximity effect between a superconductor and a highly diffusive two-dimensional metal is revealed in a scanning tunneling spectroscopy experiment. The in situ elaborated samples consist of superconducting single crystalline Pb islands interconnected by a nonsuperconducting atomically thin disordered Pb wetting layer. In the vicinity of each superconducting island the wetting layer acquires specific tunneling characteristics which reflect the interplay between the proximity-induced superconductivity and the inherent electron correlations of this ultimate diffusive two-dimensional metal. The observed spatial evolution of the tunneling spectra is accounted for theoretically by combining the Usadel equations with the theory of dynamical Coulomb blockade; the relevant length and energy scales are extracted and found in agreement with available experimental data.

Far-field analysis of coupled bulk and boundary layer diffusion toward an ion channel entrance.

PubMed Central

Schumaker, M F; Kentler, C J

1998-01-01

We present a far-field analysis of ion diffusion toward a channel embedded in a membrane with a fixed charge density. The Smoluchowski equation, which represents the 3D problem, is approximated by a system of coupled three- and two-dimensional diffusions. The 2D diffusion models the quasi-two-dimensional diffusion of ions in a boundary layer in which the electrical potential interaction with the membrane surface charge is important. The 3D diffusion models ion transport in the bulk region outside the boundary layer. Analytical expressions for concentration and flux are developed that are accurate far from the channel entrance. These provide boundary conditions for a numerical solution of the problem. Our results are used to calculate far-field ion flows corresponding to experiments of Bell and Miller (Biophys. J. 45:279, 1984). PMID:9591651

A cubic spline approximation for problems in fluid mechanics

NASA Technical Reports Server (NTRS)

Rubin, S. G.; Graves, R. A., Jr.

1975-01-01

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Sparse learning of stochastic dynamical equations

NASA Astrophysics Data System (ADS)

Boninsegna, Lorenzo; Nüske, Feliks; Clementi, Cecilia

2018-06-01

With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastic SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential and the projected dynamics of a two-dimensional diffusion process.

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

NASA Astrophysics Data System (ADS)

Zhang, Yu-Feng; Wu, Li-Xin; Rui, Wen-Juan

2015-05-01

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding (2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation (BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing (2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the (2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the (2+1)-dimensional AKNS equation (also called the Davey-Stewartson hierarchy), a kind of (2+1)-dimensional Schrödinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new (2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the (2+1)-dimensional integrable coupling, which is further reduced to the standard (2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known (1+1)-dimensional AKNS hierarchy, the (1+1)-dimensional nonlinear Schrödinger equation are all special cases of the (2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the (2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated. Supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), the National Natural Science Foundation of China under Grant No. 11371361, the Fundamental Research Funds for the Central Universities (2013XK03), and the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016

Comments on the Diffusive Behavior of Two Upwind Schemes

NASA Technical Reports Server (NTRS)

Wood, William A.; Kleb, William L.

1998-01-01

The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and locally one-dimensional finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2.5 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a speedup of 29 over finite volume.

Multigrid for hypersonic viscous two- and three-dimensional flows

NASA Technical Reports Server (NTRS)

Turkel, E.; Swanson, R. C.; Vatsa, V. N.; White, J. A.

1991-01-01

The use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows is considered. The time dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that removes the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are given for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional flow over a blunt biconic.

An analysis of curvature effects for the control of wall-bounded shear flows

NASA Technical Reports Server (NTRS)

Gatski, T. B.; Savill, A. M.

1989-01-01

The Reynolds stress transport equations are used to predict the effects of simultaneous and sequential combinations of distortions on turbulent boundary layers. The equations are written in general orthogonal curvilinear coordinates, with the curvature terms expressed in terms of the principal radii of curvature of the respective coordinate surfaces. Results are obtained for the cases of two-dimensional and three-dimensional flows in the limit where production and pressure-strain redistribution dominate over diffusion effects.

A 3D model for rain-induced landslides based on molecular dynamics with fractal and fractional water diffusion

NASA Astrophysics Data System (ADS)

Martelloni, Gianluca; Bagnoli, Franco; Guarino, Alessio

2017-09-01

We present a three-dimensional model of rain-induced landslides, based on cohesive spherical particles. The rainwater infiltration into the soil follows either the fractional or the fractal diffusion equations. We analytically solve the fractal partial differential equation (PDE) for diffusion with particular boundary conditions to simulate a rainfall event. We developed a numerical integration scheme for the PDE, compared with the analytical solution. We adapt the fractal diffusion equation obtaining the gravimetric water content that we use as input of a triggering scheme based on Mohr-Coulomb limit-equilibrium criterion. This triggering is then complemented by a standard molecular dynamics algorithm, with an interaction force inspired by the Lennard-Jones potential, to update the positions and velocities of particles. We present our results for homogeneous and heterogeneous systems, i.e., systems composed by particles with same or different radius, respectively. Interestingly, in the heterogeneous case, we observe segregation effects due to the different volume of the particles. Finally, we analyze the parameter sensibility both for the triggering and the propagation phases. Our simulations confirm the results of a previous two-dimensional model and therefore the feasible applicability to real cases.

ANALYTICAL SOLUTIONS OF THE ATMOSPHERIC DIFFUSION EQUATION WITH MULTIPLE SOURCES AND HEIGHT-DEPENDENT WIND SPEED AND EDDY DIFFUSIVITIES. (R825689C072)

EPA Science Inventory

Abstract

Three-dimensional analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities are derived in a systematic fashion. For homogeneous Neumann (total reflection), Dirichlet (total adsorpti...

ANALYTICAL SOLUTIONS OF THE ATMOSPHERIC DIFFUSION EQUATION WITH MULTIPLE SOURCES AND HEIGHT-DEPENDENT WIND SPEED AND EDDY DIFFUSIVITIES. (R825689C048)

EPA Science Inventory

Abstract

Three-dimensional analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities are derived in a systematic fashion. For homogeneous Neumann (total reflection), Dirichlet (total adsorpti...

Laser one-dimensional range profile and the laser two-dimensional range profile of cylinders

NASA Astrophysics Data System (ADS)

Gong, Yanjun; Wang, Mingjun; Gong, Lei

2015-10-01

Laser one-dimensional range profile, that is scattering power from pulse laser scattering of target, is a radar imaging technology. The laser two-dimensional range profile is two-dimensional scattering imaging of pulse laser of target. Laser one-dimensional range profile and laser two-dimensional range profile are called laser range profile(LRP). The laser range profile can reflect the characteristics of the target shape and surface material. These techniques were motivated by applications of laser radar to target discrimination in ballistic missile defense. The radar equation of pulse laser is given in this paper. This paper demonstrates the analytical model of laser range profile of cylinder based on the radar equation of the pulse laser. Simulations results of laser one-dimensional range profiles of some cylinders are given. Laser range profiles of cylinder, whose surface material with diffuse lambertian reflectance, is given in this paper. Laser range profiles of different pulse width of cylinder are given in this paper. The influences of geometric parameters, pulse width, attitude on the range profiles are analyzed.

Comparison between results of solution of Burgers' equation and Laplace's equation by Galerkin and least-square finite element methods

NASA Astrophysics Data System (ADS)

Adib, Arash; Poorveis, Davood; Mehraban, Farid

2018-03-01

In this research, two equations are considered as examples of hyperbolic and elliptic equations. In addition, two finite element methods are applied for solving of these equations. The purpose of this research is the selection of suitable method for solving each of two equations. Burgers' equation is a hyperbolic equation. This equation is a pure advection (without diffusion) equation. This equation is one-dimensional and unsteady. A sudden shock wave is introduced to the model. This wave moves without deformation. In addition, Laplace's equation is an elliptical equation. This equation is steady and two-dimensional. The solution of Laplace's equation in an earth dam is considered. By solution of Laplace's equation, head pressure and the value of seepage in the directions X and Y are calculated in different points of earth dam. At the end, water table is shown in the earth dam. For Burgers' equation, least-square method can show movement of wave with oscillation but Galerkin method can not show it correctly (the best method for solving of the Burgers' equation is discrete space by least-square finite element method and discrete time by forward difference.). For Laplace's equation, Galerkin and least square methods can show water table correctly in earth dam.

Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

NASA Astrophysics Data System (ADS)

Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye

2018-04-01

The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

Influence of two-dimensional hygrothermal gradients on interlaminar stresses near free edges

NASA Technical Reports Server (NTRS)

Farley, G. L.; Herakovich, C. T.

1977-01-01

Interlaminar stresses are determined for mechanical loading, uniform hygrothermal loading, and gradient moisture loading through implementation of a finite element computer code. Nonuniform two-dimensional hygroscopic gradients are obtained from a finite difference solution of the diffusion equation. It is shown that hygroscopic induced stresses can be larger than those resulting from mechanical and thermal loading, and that the distribution of the interlaminar normal stress may be changed significantly in the presence of a two-dimensional moisture gradient in the boundary layer of a composite laminate.

Evaluation of Full Reynolds Stress Turbulence Models in FUN3D

NASA Technical Reports Server (NTRS)

Dudek, Julianne C.; Carlson, Jan-Renee

2017-01-01

Full seven-equation Reynolds stress turbulence models are promising tools for today’s aerospace technology challenges. This paper examines two such models for computing challenging turbulent flows including shock-wave boundary layer interactions, separation and mixing layers. The Wilcox and the SSG/LRR full second-moment Reynolds stress models have been implemented into the FUN3D (Fully Unstructured Navier-Stokes Three Dimensional) unstructured Navier-Stokes code and were evaluated for four problems: a transonic two-dimensional diffuser, a supersonic axisymmetric compression corner, a compressible planar shear layer, and a subsonic axisymmetric jet. Simulation results are compared with experimental data and results computed using the more commonly used Spalart-Allmaras (SA) one-equation and the Menter Shear Stress Transport (SST-V) two-equation turbulence models.

Evaluation of Full Reynolds Stress Turbulence Models in FUN3D

NASA Technical Reports Server (NTRS)

Dudek, Julianne C.; Carlson, Jan-Renee

2017-01-01

Full seven-equation Reynolds stress turbulence models are a relatively new and promising tool for todays aerospace technology challenges. This paper uses two stress-omega full Reynolds stress models to evaluate challenging flows including shock-wave boundary layer interactions, separation and mixing layers. The Wilcox and the SSG/LRR full second-moment Reynolds stress models have been implemented into the FUN3D (Fully Unstructured Navier-Stokes Three Dimensional) unstructured Navier-Stokes code and are evaluated for four problems: a transonic two-dimensional diffuser, a supersonic axisymmetric compression corner, a compressible planar shear layer, and a subsonic axisymmetric jet. Simulation results are compared with experimental data and results using the more commonly used Spalart-Allmaras (SA) one-equation and the Menter Shear Stress Transport (SST-V) two-equation turbulence models.

A Gibbs point field model for the spatial pattern of coronary capillaries

NASA Astrophysics Data System (ADS)

Karch, R.; Neumann, M.; Neumann, F.; Ullrich, R.; Neumüller, J.; Schreiner, W.

2006-09-01

We propose a Gibbs point field model for the pattern of coronary capillaries in transverse histologic sections from human hearts, based on the physiology of oxygen supply from capillaries to tissue. To specify the potential energy function of the Gibbs point field, we draw on an analogy between the equation of steady-state oxygen diffusion from an array of parallel capillaries to the surrounding tissue and Poisson's equation for the electrostatic potential of a two-dimensional distribution of identical point charges. The influence of factors other than diffusion is treated as a thermal disturbance. On this basis, we arrive at the well-known two-dimensional one-component plasma, a system of identical point charges exhibiting a weak (logarithmic) repulsive interaction that is completely characterized by a single dimensionless parameter. By variation of this parameter, the model is able to reproduce many characteristics of real capillary patterns.

Dynamics of Robertson–Walker spacetimes with diffusion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Alho, A., E-mail: aalho@math.ist.utl.pt; Calogero, S., E-mail: calogero@chalmers.se; Machado Ramos, M.P., E-mail: mpr@mct.uminho.pt

2015-03-15

We study the dynamics of spatially homogeneous and isotropic spacetimes containing a fluid undergoing microscopic velocity diffusion in a cosmological scalar field. After deriving a few exact solutions of the equations, we continue by analyzing the qualitative behavior of general solutions. To this purpose we recast the equations in the form of a two dimensional dynamical system and perform a global analysis of the flow. Among the admissible behaviors, we find solutions that are asymptotically de-Sitter both in the past and future time directions and which undergo accelerated expansion at all times.

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations.

PubMed

Sánchez-Garduño, Faustino; Pérez-Velázquez, Judith

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D (0) = 0) and advection-degenerate (at h '(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h ( u ): (1)   h '( u ) is constant k , (2)   h '( u ) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.

Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations

PubMed Central

Sánchez-Garduño, Faustino

2016-01-01

This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (at D(0) = 0) and advection-degenerate (at h′(0) = 0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection term h(u): (1)  h′(u) is constant k, (2)  h′(u) = ku with k > 0, and (3) it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, where k = 0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE. PMID:27689131

Diffusion accessibility as a method for visualizing macromolecular surface geometry.

PubMed

Tsai, Yingssu; Holton, Thomas; Yeates, Todd O

2015-10-01

Important three-dimensional spatial features such as depth and surface concavity can be difficult to convey clearly in the context of two-dimensional images. In the area of macromolecular visualization, the computer graphics technique of ray-tracing can be helpful, but further techniques for emphasizing surface concavity can give clearer perceptions of depth. The notion of diffusion accessibility is well-suited for emphasizing such features of macromolecular surfaces, but a method for calculating diffusion accessibility has not been made widely available. Here we make available a web-based platform that performs the necessary calculation by solving the Laplace equation for steady state diffusion, and produces scripts for visualization that emphasize surface depth by coloring according to diffusion accessibility. The URL is http://services.mbi.ucla.edu/DiffAcc/. © 2015 The Protein Society.

Exploding dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in one and two spatial dimensions. A review and a perspective

NASA Astrophysics Data System (ADS)

Cartes, C.; Descalzi, O.; Brand, H. R.

2014-10-01

We review the work on exploding dissipative solitons in one and two spatial dimensions. Features covered include: the transition from modulated to exploding dissipative solitons, the analogue of the Ruelle-Takens scenario for dissipative solitons, inducing exploding dissipative solitons by noise, two classes of exploding dissipative solitons in two spatial dimensions, diffusing asymmetric exploding dissipative solitons as a model for a two-dimensional extended chaotic system. As a perspective we outline the interaction of exploding dissipative solitons with quasi one-dimensional dissipative solitons, breathing quasi one-dimensional solutions and their possible connection with experimental results on convection, and the occurence of exploding dissipative solitons in reaction-diffusion systems. It is a great pleasure to dedicate this work to our long-time friend Hans (Prof. Dr. Hans Jürgen Herrmann) on the occasion of his 60th birthday.

Mathematical analysis of a sharp-diffuse interfaces model for seawater intrusion

NASA Astrophysics Data System (ADS)

Choquet, C.; Diédhiou, M. M.; Rosier, C.

2015-10-01

We consider a new model mixing sharp and diffuse interface approaches for seawater intrusion phenomena in free aquifers. More precisely, a phase field model is introduced in the boundary conditions on the virtual sharp interfaces. We thus include in the model the existence of diffuse transition zones but we preserve the simplified structure allowing front tracking. The three-dimensional problem then reduces to a two-dimensional model involving a strongly coupled system of partial differential equations of parabolic type describing the evolution of the depths of the two free surfaces, that is the interface between salt- and freshwater and the water table. We prove the existence of a weak solution for the model completed with initial and boundary conditions. We also prove that the depths of the two interfaces satisfy a coupled maximum principle.

Nodal Green’s Function Method Singular Source Term and Burnable Poison Treatment in Hexagonal Geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

A.A. Bingham; R.M. Ferrer; A.M. ougouag

2009-09-01

An accurate and computationally efficient two or three-dimensional neutron diffusion model will be necessary for the development, safety parameters computation, and fuel cycle analysis of a prismatic Very High Temperature Reactor (VHTR) design under Next Generation Nuclear Plant Project (NGNP). For this purpose, an analytical nodal Green’s function solution for the transverse integrated neutron diffusion equation is developed in two and three-dimensional hexagonal geometry. This scheme is incorporated into HEXPEDITE, a code first developed by Fitzpatrick and Ougouag. HEXPEDITE neglects non-physical discontinuity terms that arise in the transverse leakage due to the transverse integration procedure application to hexagonal geometry andmore » cannot account for the effects of burnable poisons across nodal boundaries. The test code being developed for this document accounts for these terms by maintaining an inventory of neutrons by using the nodal balance equation as a constraint of the neutron flux equation. The method developed in this report is intended to restore neutron conservation and increase the accuracy of the code by adding these terms to the transverse integrated flux solution and applying the nodal Green’s function solution to the resulting equation to derive a semi-analytical solution.« less

The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model

NASA Astrophysics Data System (ADS)

Liu, Chang-jiang; Pang, Song; Xu, Qiang; He, Ling; Yang, Shao-peng; Qing, Yun-jie

2018-06-01

The motion of many solid-gas two-phase flows can be described by the Boltzmann equation. In order to simplify the Boltzmann equation, the convective-diffusion term is reserved and the collision term is replaced by the three-dimensional Bharnagar-Gross-Krook (BGK) model. Then the simplified Boltzmann equation is solved by homotopy perturbation method (HPM), and its approximate analytical solution is obtained. Through the analyzing, it is proved that the analytical solution satisfies all the constraint conditions, and its formation is in accord with the formation of the solution that is obtained by traditional Chapman-Enskog method, and the solving process of HPM is much more simple and convenient. This preliminarily shows the effectiveness and rapidness of HPM to solve the Boltzmann equation. The results obtained herein provide some theoretical basis for the further study of dynamic model of solid-gas two-phase flows, such as the sturzstrom of high-speed distant landslide caused by microseism and the sand storm caused by strong breeze.

Electromigration of intergranular voids in metal films for microelectronic interconnects

NASA Astrophysics Data System (ADS)

Averbuch, Amir; Israeli, Moshe; Ravve, Igor

2003-04-01

Voids and cracks often occur in the interconnect lines of microelectronic devices. They increase the resistance of the circuits and may even lead to a fatal failure. Voids may occur inside a single grain, but often they appear on the boundary between two grains. In this work, we model and analyze numerically the migration and evolution of an intergranular void subjected to surface diffusion forces and external voltage applied to the interconnect. The grain-void interface is considered one-dimensional, and the physical formulation of the electromigration and diffusion model results in two coupled fourth-order one-dimensional time-dependent PDEs. The boundary conditions are specified at the triple points, which are common to both neighboring grains and the void. The solution of these equations uses a finite difference scheme in space and a Runge-Kutta integration scheme in time, and is also coupled to the solution of a static Laplace equation describing the voltage distribution throughout the grain. Since the voltage distribution is required only along the interface line, the two-dimensional discretization of the grain interior is not needed, and the static problem is solved by the boundary element method at each time step. The motion of the intergranular void was studied for different ratios between the diffusion and the electric field forces, and for different initial configurations of the void.

Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations

NASA Astrophysics Data System (ADS)

Moghaderi, Hamid; Dehghan, Mehdi; Donatelli, Marco; Mazza, Mariarosa

2017-12-01

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank-Nicolson scheme and the so-called weighted and shifted Grünwald formula. By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate.

Neuronal models in infinite-dimensional spaces and their finite-dimensional projections: Part II.

PubMed

Brzychczy, S; Leszczyński, H; Poznanski, R R

2012-09-01

Application of comparison theorem is used to examine the validitiy of the "lumped parameter assumption" in describing the behavior of solutions of the continuous cable equation U(t) = DU(xx)+f(U) with the discrete cable equation dV(n)/dt = d*(V(n+1) - 2V(n) + V(n-1)) + f(V(n)), where f is a nonlinear functional describing the internal diffusion of electrical potential in single neurons. While the discrete cable equation looks like a finite difference approximation of the continuous cable equation, solutions of the two reveal significantly different behavior which imply that the compartmental models (spiking neurons) are poor quantifiers of neurons, contrary to what is commonly accepted in computational neuroscience.

Computation and visualization of spreading depression based on reaction-diffusion equation with recovery

NASA Astrophysics Data System (ADS)

Ding, Hongxia; Chen, Shangbin; Zeng, Shuai; Zeng, Shaoqun; Liu, Qian; Luo, Qingming

2008-12-01

Spreading depression (SD) shows as propagating suppression of electrical activity, which relates with migraine and focal cerebral ischaemia. The putative mechanism of SD is the reaction-diffusion hypothesis involving potassium ions. In part inspired by optical imaging of two SD waves collision, we aimed to show the merged and large wavefront but not annihilation during collision by experimental and computational study. This paper modified Reggia et al established bistable equation with recovery to compute and visualize SD. Firstly, the media tissue of SD was assumed as one-dimensional continuum. The Crank-Nicholson method was used to solve the modified equations with recovery term. Then, the computation results were extended to two-dimensional space by symmetry. One individual SD was visualized as a concentric wave initiating from the stimulation point. The mergence but not annihilation of two colliding waves of SD was demonstrated. In addition, the dynamics of SD depending on the parameters was studied and presented. The results allied SD with the emerging concepts of volume transmission. This work not only supplied a paradigm to compute and visualize SD but also became a tool to explore the mechanisms of SD.

A Three-Fold Approach to the Heat Equation: Data, Modeling, Numerics

ERIC Educational Resources Information Center

Spayd, Kimberly; Puckett, James

2016-01-01

This article describes our modeling approach to teaching the one-dimensional heat (diffusion) equation in a one-semester undergraduate partial differential equations course. We constructed the apparatus for a demonstration of heat diffusion through a long, thin metal rod with prescribed temperatures at each end. The students observed the physical…

A Two Colorable Fourth Order Compact Difference Scheme and Parallel Iterative Solution of the 3D Convection Diffusion Equation

NASA Technical Reports Server (NTRS)

Zhang, Jun; Ge, Lixin; Kouatchou, Jules

2000-01-01

A new fourth order compact difference scheme for the three dimensional convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it Only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with the Gauss-Seidel type iterative method. This is compared with the known 19 point fourth order compact differenCe scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15 point and the 19 point fourth order compact schemes.

Multilevel Sequential Monte Carlo Samplers for Normalizing Constants

DOE PAGES

Moral, Pierre Del; Jasra, Ajay; Law, Kody J. H.; ...

2017-08-24

This article considers the sequential Monte Carlo (SMC) approximation of ratios of normalizing constants associated to posterior distributions which in principle rely on continuum models. Therefore, the Monte Carlo estimation error and the discrete approximation error must be balanced. A multilevel strategy is utilized to substantially reduce the cost to obtain a given error level in the approximation as compared to standard estimators. Two estimators are considered and relative variance bounds are given. The theoretical results are numerically illustrated for two Bayesian inverse problems arising from elliptic partial differential equations (PDEs). The examples involve the inversion of observations of themore » solution of (i) a 1-dimensional Poisson equation to infer the diffusion coefficient, and (ii) a 2-dimensional Poisson equation to infer the external forcing.« less

Numerical method for solution of systems of non-stationary spatially one-dimensional nonlinear differential equations

NASA Technical Reports Server (NTRS)

Morozov, S. K.; Krasitskiy, O. P.

1978-01-01

A computational scheme and a standard program is proposed for solving systems of nonstationary spatially one-dimensional nonlinear differential equations using Newton's method. The proposed scheme is universal in its applicability and its reduces to a minimum the work of programming. The program is written in the FORTRAN language and can be used without change on electronic computers of type YeS and BESM-6. The standard program described permits the identification of nonstationary (or stationary) solutions to systems of spatially one-dimensional nonlinear (or linear) partial differential equations. The proposed method may be used to solve a series of geophysical problems which take chemical reactions, diffusion, and heat conductivity into account, to evaluate nonstationary thermal fields in two-dimensional structures when in one of the geometrical directions it can take a small number of discrete levels, and to solve problems in nonstationary gas dynamics.

Correlations, soliton modes, and non-Hermitian linear mode transmutation in the one-dimensional noisy Burgers equation.

PubMed

Fogedby, Hans C

2003-08-01

Using the previously developed canonical phase space approach applied to the noisy Burgers equation in one dimension, we discuss in detail the growth morphology in terms of nonlinear soliton modes and superimposed linear modes. We moreover analyze the non-Hermitian character of the linear mode spectrum and the associated dynamical pinning, and mode transmutation from diffusive to propagating behavior induced by the solitons. We discuss the anomalous diffusion of growth modes, switching and pathways, correlations in the multisoliton sector, and in detail the correlations and scaling properties in the two-soliton sector.

Stable dissipative optical vortex clusters by inhomogeneous effective diffusion.

PubMed

Li, Huishan; Lai, Shiquan; Qui, Yunli; Zhu, Xing; Xie, Jianing; Mihalache, Dumitru; He, Yingji

2017-10-30

We numerically show the generation of robust vortex clusters embedded in a two-dimensional beam propagating in a dissipative medium described by the generic cubic-quintic complex Ginzburg-Landau equation with an inhomogeneous effective diffusion term, which is asymmetrical in the two transverse directions and periodically modulated in the longitudinal direction. We show the generation of stable optical vortex clusters for different values of the winding number (topological charge) of the input optical beam. We have found that the number of individual vortex solitons that form the robust vortex cluster is equal to the winding number of the input beam. We have obtained the relationships between the amplitudes and oscillation periods of the inhomogeneous effective diffusion and the cubic gain and diffusion (viscosity) parameters, which depict the regions of existence and stability of vortex clusters. The obtained results offer a method to form robust vortex clusters embedded in two-dimensional optical beams, and we envisage potential applications in the area of structured light.

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise

NASA Astrophysics Data System (ADS)

Nezhadhaghighi, Mohsen Ghasemi

2017-08-01

Here, we present results of numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy-tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of μ -stable Lévy noise. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α . We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ -stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise.

PubMed

Nezhadhaghighi, Mohsen Ghasemi

2017-08-01

Here, we present results of numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy-tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of μ-stable Lévy noise. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α. We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ-stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

The Effect of Multigrid Parameters in a 3D Heat Diffusion Equation

NASA Astrophysics Data System (ADS)

Oliveira, F. De; Franco, S. R.; Pinto, M. A. Villela

2018-02-01

The aim of this paper is to reduce the necessary CPU time to solve the three-dimensional heat diffusion equation using Dirichlet boundary conditions. The finite difference method (FDM) is used to discretize the differential equations with a second-order accuracy central difference scheme (CDS). The algebraic equations systems are solved using the lexicographical and red-black Gauss-Seidel methods, associated with the geometric multigrid method with a correction scheme (CS) and V-cycle. Comparisons are made between two types of restriction: injection and full weighting. The used prolongation process is the trilinear interpolation. This work is concerned with the study of the influence of the smoothing value (v), number of mesh levels (L) and number of unknowns (N) on the CPU time, as well as the analysis of algorithm complexity.

Galactic Cosmic-ray Transport in the Global Heliosphere: A Four-Dimensional Stochastic Model

NASA Astrophysics Data System (ADS)

Florinski, V.

2009-04-01

We study galactic cosmic-ray transport in the outer heliosphere and heliosheath using a newly developed transport model based on stochastic integration of the phase-space trajectories of Parker's equation. The model employs backward integration of the diffusion-convection transport equation using Ito calculus and is four-dimensional in space+momentum. We apply the model to the problem of galactic proton transport in the heliosphere during a negative solar minimum. Model results are compared with the Voyager measurements of galactic proton radial gradients and spectra in the heliosheath. We show that the heliosheath is not as efficient in diverting cosmic rays during solar minima as predicted by earlier two-dimensional models.

Numerical investigations of passive scalar transport in Taylor-Couette flows: Counter-rotation effect

NASA Astrophysics Data System (ADS)

Ouazib, Nabila; Salhi, Yacine; Si-Ahmed, El-Khider; Legrand, Jack; Degrez, G.

2017-07-01

Numerical methods for solving convection-diffusion-reaction (CDR) scalar transport equation in three-dimensional flow are used in the present investigation. The flow is confined between two concentric cylinders both the inner cylinder and the outer one are allowed to rotate. Direct numerical simulations (DNS) have been achieved to study the effects of the gravitational and the centrifugal potentials on the stability of incompressible Taylor-Couette flow. The Navier-Stokes equations and the uncoupled convection-diffusion-reaction equation are solved using a spectral development in one direction combined together with a finite element discretization in the two remaining directions. The complexity of the patterns is highlighted. Since, it increases as the rotation rates of the cylinders increase. In addition, the effect of the counter-rotation of the cylinders on the mass transfer is pointed out.

A non-linear dimension reduction methodology for generating data-driven stochastic input models

NASA Astrophysics Data System (ADS)

Ganapathysubramanian, Baskar; Zabaras, Nicholas

2008-06-01

Stochastic analysis of random heterogeneous media (polycrystalline materials, porous media, functionally graded materials) provides information of significance only if realistic input models of the topology and property variations are used. This paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data-driven strategy. Given a set of microstructure realizations (input samples) generated from given statistical information about the medium topology, the framework constructs a reduced-order stochastic representation of the thermal diffusivity. This problem of constructing a low-dimensional stochastic representation of property variations is analogous to the problem of manifold learning and parametric fitting of hyper-surfaces encountered in image processing and psychology. Denote by M the set of microstructures that satisfy the given experimental statistics. A non-linear dimension reduction strategy is utilized to map M to a low-dimensional region, A. We first show that M is a compact manifold embedded in a high-dimensional input space Rn. An isometric mapping F from M to a low-dimensional, compact, connected set A⊂Rd(d≪n) is constructed. Given only a finite set of samples of the data, the methodology uses arguments from graph theory and differential geometry to construct the isometric transformation F:M→A. Asymptotic convergence of the representation of M by A is shown. This mapping F serves as an accurate, low-dimensional, data-driven representation of the property variations. The reduced-order model of the material topology and thermal diffusivity variations is subsequently used as an input in the solution of stochastic partial differential equations that describe the evolution of dependant variables. A sparse grid collocation strategy (Smolyak algorithm) is utilized to solve these stochastic equations efficiently. We showcase the methodology by constructing low-dimensional input stochastic models to represent thermal diffusivity in two-phase microstructures. This model is used in analyzing the effect of topological variations of two-phase microstructures on the evolution of temperature in heat conduction processes.

On the Maxwell-Stefan approach to diffusion: a general resolution in the transient regime for one-dimensional systems.

PubMed

Leonardi, Erminia; Angeli, Celestino

2010-01-14

The diffusion process in a multicomponent system can be formulated in a general form by the generalized Maxwell-Stefan equations. This formulation is able to describe the diffusion process in different systems, such as, for instance, bulk diffusion (in the gas, liquid, and solid phase) and diffusion in microporous materials (membranes, zeolites, nanotubes, etc.). The Maxwell-Stefan equations can be solved analytically (only in special cases) or by numerical approaches. Different numerical strategies have been previously presented, but the number of diffusing species is normally restricted, with only few exceptions, to three in bulk diffusion and to two in microporous systems, unless simplifications of the Maxwell-Stefan equations are considered. In the literature, a large effort has been devoted to the derivation of the analytic expression of the elements of the Fick-like diffusion matrix and therefore to the symbolic inversion of a square matrix with dimensions n x n (n being the number of independent components). This step, which can be easily performed for n = 2 and remains reasonable for n = 3, becomes rapidly very complex in problems with a large number of components. This paper addresses the problem of the numerical resolution of the Maxwell-Stefan equations in the transient regime for a one-dimensional system with a generic number of components, avoiding the definition of the analytic expression of the elements of the Fick-like diffusion matrix. To this aim, two approaches have been implemented in a computational code; the first is the simple finite difference second-order accurate in time Crank-Nicolson scheme for which the full mathematical derivation and the relevant final equations are reported. The second is based on the more accurate backward differentiation formulas, BDF, or Gear's method (Shampine, L. F. ; Gear, C. W. SIAM Rev. 1979, 21, 1.), as implemented in the Livermore solver for ordinary differential equations, LSODE (Hindmarsh, A. C. Serial Fortran Solvers for ODE Initial Value Problems, Technical Report; https://computation.llnl.gov/casc/odepack/odepack_ home.html (2006).). Both methods have been applied to a series of specific problems, such as bulk diffusion of acetone and methanol through stagnant air, uptake of two components on a microporous material in a model system, and permeation across a microporous membrane in model systems, both with the aim to validate the method and to add new information to the comprehension of the peculiar behavior of these systems. The approach is validated by comparison with different published results and with analytic expressions for the steady-state concentration profiles or fluxes in particular systems. The possibility to treat a generic number of components (the limitation being essentially the computational power) is also tested, and results are reported on the permeation of a five component mixture through a membrane in a model system. It is worth noticing that the algorithm here reported can be applied also to the Fick formulation of the diffusion problem with concentration-dependent diffusion coefficients.

Stratified Shear Flows In Pipe Geometries

NASA Astrophysics Data System (ADS)

Harabin, George; Camassa, Roberto; McLaughlin, Richard; UNC Joint Fluids Lab Team Team

2015-11-01

Exact and series solutions to the full Navier-Stokes equations coupled to the advection diffusion equation are investigated in tilted three-dimensional pipe geometries. Analytic techniques for studying the three-dimensional problem provide a means for tackling interesting questions such as the optimal domain for mass transport, and provide new avenues for experimental investigation of diffusion driven flows. Both static and time dependent solutions will be discussed. NSF RTG DMS-0943851, NSF RTG ARC-1025523, NSF DMS-1009750.

Finite Difference Formulation for Prediction of Water Pollution

NASA Astrophysics Data System (ADS)

Johari, Hanani; Rusli, Nursalasawati; Yahya, Zainab

2018-03-01

Water is an important component of the earth. Human being and living organisms are demand for the quality of water. Human activity is one of the causes of the water pollution. The pollution happened give bad effect to the physical and characteristic of water contents. It is not practical to monitor all aspects of water flow and transport distribution. So, in order to help people to access to the polluted area, a prediction of water pollution concentration must be modelled. This study proposed a one-dimensional advection diffusion equation for predicting the water pollution concentration transport. The numerical modelling will be produced in order to predict the transportation of water pollution concentration. In order to approximate the advection diffusion equation, the implicit Crank Nicolson is used. For the purpose of the simulation, the boundary condition and initial condition, the spatial steps and time steps as well as the approximations of the advection diffusion equation have been encoded. The results of one dimensional advection diffusion equation have successfully been used to predict the transportation of water pollution concentration by manipulating the velocity and diffusion parameters.

Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

NASA Astrophysics Data System (ADS)

Chen, Shanzhen; Jiang, Xiaoyun

2012-08-01

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1), the Helmholtz equation (α→0) and the wave equation (α=2) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.

A three-dimensional spin-diffusion model for micromagnetics

PubMed Central

Abert, Claas; Ruggeri, Michele; Bruckner, Florian; Vogler, Christoph; Hrkac, Gino; Praetorius, Dirk; Suess, Dieter

2015-01-01

We solve a time-dependent three-dimensional spin-diffusion model coupled to the Landau-Lifshitz-Gilbert equation numerically. The presented model is validated by comparison to two established spin-torque models: The model of Slonzewski that describes spin-torque in multi-layer structures in the presence of a fixed layer and the model of Zhang and Li that describes current driven domain-wall motion. It is shown that both models are incorporated by the spin-diffusion description, i.e., the nonlocal effects of the Slonzewski model are captured as well as the spin-accumulation due to magnetization gradients as described by the model of Zhang and Li. Moreover, the presented method is able to resolve the time dependency of the spin-accumulation. PMID:26442796

Evaluation of Full Reynolds Stress Turbulence Models in FUN3D

NASA Technical Reports Server (NTRS)

Dudek, Julianne C.; Carlson, Jan-Renee

2017-01-01

Full seven-equation Reynolds stress turbulence models are a relatively new and promising tool for todays aerospace technology challenges. This paper uses two stress-omega full Reynolds stress models to evaluate challenging flows including shock-wave boundary layer interactions, separation and mixing layers. The Wilcox and the SSGLRR full second-moment Reynolds stress models are evaluated for four problems: a transonic two-dimensional diffuser, a supersonic axisymmetric compression corner, a compressible planar shear layer, and a subsonic axisymmetric jet. Simulation results are compared with experimental data and results using the more commonly used Spalart-Allmaras (SA) one-equation and the Menter Shear Stress Transport (SST) two-equation models.

First-Passage Times in d -Dimensional Heterogeneous Media

NASA Astrophysics Data System (ADS)

Vaccario, G.; Antoine, C.; Talbot, J.

2015-12-01

Although there are many theoretical studies of the mean first-passage time (MFPT), most neglect the diffusive heterogeneity of real systems. We present exact analytical expressions for the MFPT and residence times of a pointlike particle diffusing in a spherically symmetric d -dimensional heterogeneous system composed of two concentric media with different diffusion coefficients with an absorbing inner boundary (target) and a reflecting outer boundary. By varying the convention, e.g., Itō, Stratonovich, or isothermal, chosen to interpret the overdamped Langevin equation with multiplicative noise describing the diffusion process, we find different predictions and counterintuitive results for the residence time in the outer region and hence for the MFPT, while the residence time in the inner region is independent of the convention. This convention dependence of residence times and the MFPT could provide insights about the heterogeneous diffusion in a cell or in a tumor, or for animal and insect searches inside their home range.

CRASH: A BLOCK-ADAPTIVE-MESH CODE FOR RADIATIVE SHOCK HYDRODYNAMICS-IMPLEMENTATION AND VERIFICATION

DOE Office of Scientific and Technical Information (OSTI.GOV)

Van der Holst, B.; Toth, G.; Sokolov, I. V.

We describe the Center for Radiative Shock Hydrodynamics (CRASH) code, a block-adaptive-mesh code for multi-material radiation hydrodynamics. The implementation solves the radiation diffusion model with a gray or multi-group method and uses a flux-limited diffusion approximation to recover the free-streaming limit. Electrons and ions are allowed to have different temperatures and we include flux-limited electron heat conduction. The radiation hydrodynamic equations are solved in the Eulerian frame by means of a conservative finite-volume discretization in either one-, two-, or three-dimensional slab geometry or in two-dimensional cylindrical symmetry. An operator-split method is used to solve these equations in three substeps: (1)more » an explicit step of a shock-capturing hydrodynamic solver; (2) a linear advection of the radiation in frequency-logarithm space; and (3) an implicit solution of the stiff radiation diffusion, heat conduction, and energy exchange. We present a suite of verification test problems to demonstrate the accuracy and performance of the algorithms. The applications are for astrophysics and laboratory astrophysics. The CRASH code is an extension of the Block-Adaptive Tree Solarwind Roe Upwind Scheme (BATS-R-US) code with a new radiation transfer and heat conduction library and equation-of-state and multi-group opacity solvers. Both CRASH and BATS-R-US are part of the publicly available Space Weather Modeling Framework.« less

Bracket formulations and energy- and helicity-preserving numerical methods for incompressible two-phase flows

NASA Astrophysics Data System (ADS)

Suzuki, Yukihito

2018-03-01

A diffuse interface model for three-dimensional viscous incompressible two-phase flows is formulated within a bracket formalism using a skew-symmetric Poisson bracket together with a symmetric negative semi-definite dissipative bracket. The budgets of kinetic energy, helicity, and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method. The Cahn-Hilliard and Allen-Cahn equations are employed as diffuse interface models, in which the equalities of densities and viscosities of two different phases are assumed. Numerical experiments on the motion of periodic arrays of tubes and those of droplets have been conducted to examine the properties and usefulness of the proposed method.

Equivalence of Fluctuation Splitting and Finite Volume for One-Dimensional Gas Dynamics

NASA Technical Reports Server (NTRS)

Wood, William A.

1997-01-01

The equivalence of the discretized equations resulting from both fluctuation splitting and finite volume schemes is demonstrated in one dimension. Scalar equations are considered for advection, diffusion, and combined advection/diffusion. Analysis of systems is performed for the Euler and Navier-Stokes equations of gas dynamics. Non-uniform mesh-point distributions are included in the analyses.

Discontinuous Finite Element Quasidiffusion Methods

DOE PAGES

Anistratov, Dmitriy Yurievich; Warsa, James S.

2018-05-21

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

Discontinuous Finite Element Quasidiffusion Methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Anistratov, Dmitriy Yurievich; Warsa, James S.

Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and themore » LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.« less

Fokker-Planck description for the queue dynamics of large tick stocks.

PubMed

Garèche, A; Disdier, G; Kockelkoren, J; Bouchaud, J-P

2013-09-01

Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. "Jump" events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.

Fokker-Planck description for the queue dynamics of large tick stocks

NASA Astrophysics Data System (ADS)

Garèche, A.; Disdier, G.; Kockelkoren, J.; Bouchaud, J.-P.

2013-09-01

Motivated by empirical data, we develop a statistical description of the queue dynamics for large tick assets based on a two-dimensional Fokker-Planck (diffusion) equation. Our description explicitly includes state dependence, i.e., the fact that the drift and diffusion depend on the volume present on both sides of the spread. “Jump” events, corresponding to sudden changes of the best limit price, must also be included as birth-death terms in the Fokker-Planck equation. All quantities involved in the equation can be calibrated using high-frequency data on the best quotes. One of our central findings is that the dynamical process is approximately scale invariant, i.e., the only relevant variable is the ratio of the current volume in the queue to its average value. While the latter shows intraday seasonalities and strong variability across stocks and time periods, the dynamics of the rescaled volumes is universal. In terms of rescaled volumes, we found that the drift has a complex two-dimensional structure, which is a sum of a gradient contribution and a rotational contribution, both stable across stocks and time. This drift term is entirely responsible for the dynamical correlations between the ask queue and the bid queue.

Multi-Dimensional Asymptotically Stable 4th Order Accurate Schemes for the Diffusion Equation

NASA Technical Reports Server (NTRS)

Abarbanel, Saul; Ditkowski, Adi

1996-01-01

An algorithm is presented which solves the multi-dimensional diffusion equation on co mplex shapes to 4th-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions fail.

Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation

NASA Astrophysics Data System (ADS)

Abuasad, Salah; Hashim, Ishak

2018-04-01

In this paper, we present the homotopy decomposition method with a modified definition of beta fractional derivative for the first time to find exact solution of one-dimensional time-fractional diffusion equation. In this method, the solution takes the form of a convergent series with easily computable terms. The exact solution obtained by the proposed method is compared with the exact solution obtained by using fractional variational homotopy perturbation iteration method via a modified Riemann-Liouville derivative.

Comparison of fluid neutral models for one-dimensional plasma edge modeling with a finite volume solution of the Boltzmann equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Horsten, N., E-mail: niels.horsten@kuleuven.be; Baelmans, M.; Dekeyser, W.

2016-01-15

We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assumingmore » equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.« less

Cellular automaton formulation of passive scalar dynamics

NASA Technical Reports Server (NTRS)

Chen, Hudong; Matthaeus, William H.

1987-01-01

Cellular automata modeling of the advection of a passive scalar in a two-dimensional flow is examined in the context of discrete lattice kinetic theory. It is shown that if the passive scalar is represented by tagging or 'coloring' automation particles a passive advection-diffusion equation emerges without use of perturbation expansions. For the specific case of the hydrodynamic lattice gas model of Frisch et al. (1986), the diffusion coefficient is calculated by perturbation.

An approximate analysis of the diffusing flow in a self-controlled heat pipe.

NASA Technical Reports Server (NTRS)

Somogyi, D.; Yen, H. H.

1973-01-01

Constant-density two-dimensional axisymmetric equations are presented for the diffusing flow of a class of self-controlled heat pipes. The analysis is restricted to the vapor space. Condensation of the vapor is related to its mass fraction at the wall by the gas kinetic formula. The Karman-Pohlhausen integral method is applied to obtain approximate solutions. Solutions are presented for a water heat pipe with neon control gas.

Polarization radiation in the planetary atmosphere delimited by a heterogeneous diffusely reflecting surface

NASA Technical Reports Server (NTRS)

Strelkov, S. A.; Sushkevich, T. A.

1983-01-01

Spatial frequency characteristics (SFC) and the scattering functions were studied in the two cases of a uniform horizontal layer with absolutely black bottom, and an isolated layer. The mathematical model for these examples describes the horizontal heterogeneities in a light field with regard to radiation polarization in a three dimensional planar atmosphere, delimited by a heterogeneous surface with diffuse reflection. The perturbation method was used to obtain vector transfer equations which correspond to the linear and nonlinear systems of polarization radiation transfer. The boundary value tasks for the vector transfer equation that is a parametric set and one dimensional are satisfied by the SFC of the nonlinear system, and are expressed through the SFC of linear approximation. As a consequence of the developed theory, formulas were obtained for analytical calculation of albedo in solving the task of dissemination of polarization radiation in the planetary atmosphere with uniform Lambert bottom.

Surface Diffusion in Systems of Interacting Brownian Particles

NASA Astrophysics Data System (ADS)

Mazroui, M'hammed; Boughaleb, Yahia

The paper reviews recent results on diffusive phenomena in two-dimensional periodic potential. Specifically, static and dynamic properties are investigated by calculating different correlation functions. Diffusion process is first studied for one-dimensional system by using the Fokker-Planck equation which is solved numerically by the matrix continued fraction method in the case of bistable potential. The transition from hopping to liquid-like diffusion induced by variation of some parameters is discussed. This study will therefore serve to demonstrate the influence of this form of potential. Further, an analytical approximation for the dc-conductivity is derived for a wide damping range in the framework of the Linear Response Theory. On the basis of this expression, calculations of the ac conductivity of two-dimensional system with Frenkel-Kontorova pair interaction in the intermediate friction regime is performed by using the continued fraction expansion method. The dc-conductivity expression is used to determine the rest of the development. By varying the density of mobile ions we discuss commensurability effects. To get information about the diffusion mechanism, the full width at half maximum λω(q), of the quasi-elastic line of the dynamical structure factor S(q,ω) is computed. The calculations are extended up to large values of q covering several Brillouin zones. The analysis of λω(q) with different parameters shows that the most probable diffusion process in good two-dimensional superionic conductors consists of a competition between a back correlated hopping in one direction and forward correlated hopping in addition to liquid-like motions in the other direction.

Fractional Number Operator and Associated Fractional Diffusion Equations

NASA Astrophysics Data System (ADS)

Rguigui, Hafedh

2018-03-01

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Garcia, Andres

Transport and reaction in zeolites and other porous materials, such as mesoporous silica particles, has been a focus of interest in recent years. This is in part due to the possibility of anomalous transport effects (e.g. single-file diffusion) and its impact in the reaction yield in catalytic processes. Computational simulations are often used to study these complex nonequilibrium systems. Computer simulations using Molecular Dynamics (MD) techniques are prohibitive, so instead coarse grained one-dimensional models with the aid of Kinetic Monte Carlo (KMC) simulations are used. Both techniques can be computationally expensive, both time and resource wise. These coarse-grained systems canmore » be exactly described by a set of coupled stochastic master equations, that describe the reaction-diffusion kinetics of the system. The equations can be written exactly, however, coupling between the equations and terms within the equations make it impossible to solve them exactly; approximations must be made. One of the most common methods to obtain approximate solutions is to use Mean Field (MF) theory. MF treatments yield reasonable results at high ratios of reaction rate k to hop rate h of the particles, but fail completely at low k=h due to the over-estimation of fluxes of particles within the pore. We develop a method to estimate fluxes and intrapore diffusivity in simple one- dimensional reaction-diffusion models at high and low k=h, where the pores are coupled to an equilibrated three-dimensional fluid. We thus successfully describe analytically these simple reaction-diffusion one-dimensional systems. Extensions to models considering behavior with long range steric interactions and wider pores require determination of multiple boundary conditions. We give a prescription to estimate the required parameters for these simulations. For one dimensional systems, if single-file diffusion is relaxed, additional parameters to describe particle exchange have to be introduced. We use Langevin Molecular Dynamics (MD) simulations to assess these parameters.« less

A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements

NASA Astrophysics Data System (ADS)

Qin, Shanlin; Liu, Fawang; Turner, Ian W.

2018-03-01

The consideration of diffusion processes in magnetic resonance imaging (MRI) signal attenuation is classically described by the Bloch-Torrey equation. However, many recent works highlight the distinct deviation in MRI signal decay due to anomalous diffusion, which motivates the fractional order generalization of the Bloch-Torrey equation. In this work, we study the two-dimensional multi-term time and space fractional diffusion equation generalized from the time and space fractional Bloch-Torrey equation. By using the Galerkin finite element method with a structured mesh consisting of rectangular elements to discretize in space and the L1 approximation of the Caputo fractional derivative in time, a fully discrete numerical scheme is derived. A rigorous analysis of stability and error estimation is provided. Numerical experiments in the square and L-shaped domains are performed to give an insight into the efficiency and reliability of our method. Then the scheme is applied to solve the multi-term time and space fractional Bloch-Torrey equation, which shows that the extra time derivative terms impact the relaxation process.

An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sun, Wenjun, E-mail: sun_wenjun@iapcm.ac.cn; Jiang, Song, E-mail: jiang@iapcm.ac.cn; Xu, Kun, E-mail: makxu@ust.hk

The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon's mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction between radiation and material. In the optical thin limit, the photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the gray radiative transfer equations, where the radiation transportmore » equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [14], and is an extension of a recent work [12] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional gray radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon's mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach.« less

Phase transition of traveling waves in bacterial colony pattern

NASA Astrophysics Data System (ADS)

Wakano, Joe Yuichiro; Komoto, Atsushi; Yamaguchi, Yukio

2004-05-01

Depending on the growth condition, bacterial colonies can exhibit different morphologies. Many previous studies have used reaction diffusion equations to reproduce spatial patterns. They have revealed that nonlinear reaction term can produce diverse patterns as well as nonlinear diffusion coefficient. Typical reaction term consists of nutrient consumption, bacterial reproduction, and sporulation. Among them, the functional form of sporulation rate has not been biologically investigated. Here we report experimentally measured sporulation rate. Then, based on the result, a reaction diffusion model is proposed. One-dimensional simulation showed the existence of traveling wave solution. We study the wave form as a function of the initial nutrient concentration and find two distinct types of solution. Moreover, transition between them is very sharp, which is analogous to phase transition. The velocity of traveling wave also shows sharp transition in nonlinear diffusion model, which is consistent with the previous experimental result. The phenomenon can be explained by separatrix in reaction term dynamics. Results of two-dimensional simulation are also shown and discussed.

Time-dependent transport of energetic particles in magnetic turbulence: computer simulations versus analytical theory

NASA Astrophysics Data System (ADS)

Arendt, V.; Shalchi, A.

2018-06-01

We explore numerically the transport of energetic particles in a turbulent magnetic field configuration. A test-particle code is employed to compute running diffusion coefficients as well as particle distribution functions in the different directions of space. Our numerical findings are compared with models commonly used in diffusion theory such as Gaussian distribution functions and solutions of the cosmic ray Fokker-Planck equation. Furthermore, we compare the running diffusion coefficients across the mean magnetic field with solutions obtained from the time-dependent version of the unified non-linear transport theory. In most cases we find that particle distribution functions are indeed of Gaussian form as long as a two-component turbulence model is employed. For turbulence setups with reduced dimensionality, however, the Gaussian distribution can no longer be obtained. It is also shown that the unified non-linear transport theory agrees with simulated perpendicular diffusion coefficients as long as the pure two-dimensional model is excluded.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis.

PubMed

MacDonald, G; Mackenzie, J A; Nolan, M; Insall, R H

2016-03-15

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

The novel implicit LU-SGS parallel iterative method based on the diffusion equation of a nuclear reactor on a GPU cluster

NASA Astrophysics Data System (ADS)

Zhang, Jilin; Sha, Chaoqun; Wu, Yusen; Wan, Jian; Zhou, Li; Ren, Yongjian; Si, Huayou; Yin, Yuyu; Jing, Ya

2017-02-01

GPU not only is used in the field of graphic technology but also has been widely used in areas needing a large number of numerical calculations. In the energy industry, because of low carbon, high energy density, high duration and other characteristics, the development of nuclear energy cannot easily be replaced by other energy sources. Management of core fuel is one of the major areas of concern in a nuclear power plant, and it is directly related to the economic benefits and cost of nuclear power. The large-scale reactor core expansion equation is large and complicated, so the calculation of the diffusion equation is crucial in the core fuel management process. In this paper, we use CUDA programming technology on a GPU cluster to run the LU-SGS parallel iterative calculation against the background of the diffusion equation of the reactor. We divide one-dimensional and two-dimensional mesh into a plurality of domains, with each domain evenly distributed on the GPU blocks. A parallel collision scheme is put forward that defines the virtual boundary of the grid exchange information and data transmission by non-stop collision. Compared with the serial program, the experiment shows that GPU greatly improves the efficiency of program execution and verifies that GPU is playing a much more important role in the field of numerical calculations.

Axi-symmetric generalized thermoelastic diffusion problem with two-temperature and initial stress under fractional order heat conduction

NASA Astrophysics Data System (ADS)

Deswal, Sunita; Kalkal, Kapil Kumar; Sheoran, Sandeep Singh

2016-09-01

A mathematical model of fractional order two-temperature generalized thermoelasticity with diffusion and initial stress is proposed to analyze the transient wave phenomenon in an infinite thermoelastic half-space. The governing equations are derived in cylindrical coordinates for a two dimensional axi-symmetric problem. The analytical solution is procured by employing the Laplace and Hankel transforms for time and space variables respectively. The solutions are investigated in detail for a time dependent heat source. By using numerical inversion method of integral transforms, we obtain the solutions for displacement, stress, temperature and diffusion fields in physical domain. Computations are carried out for copper material and displayed graphically. The effect of fractional order parameter, two-temperature parameter, diffusion, initial stress and time on the different thermoelastic and diffusion fields is analyzed on the basis of analytical and numerical results. Some special cases have also been deduced from the present investigation.

Building 1D resonance broadened quasilinear (RBQ) code for fast ions Alfvénic relaxations

NASA Astrophysics Data System (ADS)

Gorelenkov, Nikolai; Duarte, Vinicius; Berk, Herbert

2016-10-01

The performance of the burning plasma is limited by the confinement of superalfvenic fusion products, e.g. alpha particles, which are capable of resonating with the Alfvénic eigenmodes (AEs). The effect of AEs on fast ions is evaluated using a resonance line broadened diffusion coefficient. The interaction of fast ions and AEs is captured for cases where there are either isolated or overlapping modes. A new code RBQ1D is being built which constructs diffusion coefficients based on realistic eigenfunctions that are determined by the ideal MHD code NOVA. The wave particle interaction can be reduced to one-dimensional dynamics where for the Alfvénic modes typically the particle kinetic energy is nearly constant. Hence to a good approximation the Quasi-Linear (QL) diffusion equation only contains derivatives in the angular momentum. The diffusion equation is then one dimensional that is efficiently solved simultaneously for all particles with the equation for the evolution of the wave angular momentum. The evolution of fast ion constants of motion is governed by the QL diffusion equations which are adapted to find the ion distribution function.

Continuous time anomalous diffusion in a composite medium.

PubMed

Stickler, B A; Schachinger, E

2011-08-01

The one-dimensional continuous time anomalous diffusion in composite media consisting of a finite number of layers in immediate contact is investigated. The diffusion process itself is described with the help of two probability density functions (PDFs), one of which is an arbitrary jump-length PDF, and the other is a long-tailed waiting-time PDF characterized by the waiting-time index β∈(0,1). The former is assumed to be a function of the space coordinate x and the time coordinate t while the latter is a function of x and the time interval. For such an environment a very general form of the diffusion equation is derived which describes the continuous time anomalous diffusion in a composite medium. This result is then specialized to two particular forms of the jump-length PDF, namely the continuous time Lévy flight PDF and the continuous time truncated Lévy flight PDF. In both cases the PDFs are characterized by the Lévy index α∈(0,2) which is regarded to be a function of x and t. It is possible to demonstrate that for particular choices of the indices α and β other equations for anomalous diffusion, well known from the literature, follow immediately. This demonstrates the very general applicability of the derivation and of the resulting fractional differential equation discussed here.

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 1: Analysis description

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Bui, Trong T.

1993-01-01

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the Analysis Description, and presents the equations and solution procedure. The governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models are described in detail.

A diffusion model of protected population on bilocal habitat with generalized resource

NASA Astrophysics Data System (ADS)

Vasilyev, Maxim D.; Trofimtsev, Yuri I.; Vasilyeva, Natalya V.

2017-11-01

A model of population distribution in a two-dimensional area divided by an ecological barrier, i.e. the boundaries of natural reserve, is considered. Distribution of the population is defined by diffusion, directed migrations and areal resource. The exchange of specimens occurs between two parts of the habitat. The mathematical model is presented in the form of a boundary value problem for a system of non-linear parabolic equations with variable parameters of diffusion and growth function. The splitting space variables, sweep method and simple iteration methods were used for the numerical solution of a system. A set of programs was coded in Python. Numerical simulation results for the two-dimensional unsteady non-linear problem are analyzed in detail. The influence of migration flow coefficients and functions of natural birth/death ratio on the distributions of population densities is investigated. The results of the research would allow to describe the conditions of the stable and sustainable existence of populations in bilocal habitat containing the protected and non-protected zones.

An axisymmetric single-path model for gas transport in the conducting airways.

PubMed

Madasu, Srinath; Borhan, All; Ultman, James S

2006-02-01

In conventional one-dimensional single-path models, radially averaged concentration is calculated as a function of time and longitudinal position in the lungs, and coupled convection and diffusion are accounted for with a dispersion coefficient. The axisymmetric single-path model developed in this paper is a two-dimensional model that incorporates convective-diffusion processes in a more fundamental manner by simultaneously solving the Navier-Stokes and continuity equations with the convection-diffusion equation. A single airway path was represented by a series of straight tube segments interconnected by leaky transition regions that provide for flow loss at the airway bifurcations. As a sample application, the model equations were solved by a finite element method to predict the unsteady state dispersion of an inhaled pulse of inert gas along an airway path having dimensions consistent with Weibel's symmetric airway geometry. Assuming steady, incompressible, and laminar flow, a finite element analysis was used to solve for the axisymmetric pressure, velocity and concentration fields. The dispersion calculated from these numerical solutions exhibited good qualitative agreement with the experimental values, but quantitatively was in error by 20%-30% due to the assumption of axial symmetry and the inability of the model to capture the complex recirculatory flows near bifurcations.

Designing herbicide formulation characteristics to maximize efficacy and minimize rice injury in paddy environments.

PubMed

Cryer, S A; Mann, R K; Erhardt-Zabik, S; Keeney, F N; Handy, P R

2001-06-01

Mathematical descriptors, coupled with experimental observations, are used to quantify differential uptake of an experimental herbicide in Japonica and Indica rice (Oryza sativa, non-target) and barnyardgrass (Echinochloa crus-galli, target). Partitioning, degradation, plant uptake and metabolism are described using mass-balance conservation equations in the form of kinetic approximations. Estimated environmental concentrations, governed by the pesticide formulation, are described using superimposed analytical solutions for the one-dimensional diffusion equation in spherical coordinates and by a finite difference representation of the two-dimensional diffusion equation in Cartesian coordinates. Formulation attributes from granules include active ingredient release rates, particle sizes, pesticide loading, and granule spacing. The diffusion model for pesticide transport is coupled with the compartment model to follow the fate and transport of a pesticide from its initial application location to various environmental matrices of interest. Formulation effects, partitioning and degradation in the various environmental matrices, differential plant uptake and metabolism, and dose-response information for plants are accounted for. This novel model provides a mechanism for selecting formulation delivery systems that optimize specific attributes (such as weed control or the therapeutic index) for risk-assessment procedures. In this report we describe how this methodology was used to explore the factors affecting herbicide efficacy and to define an optimal release rate for a granule formulation.

Finite Element Analysis of Poroelastic Composites Undergoing Thermal and Gas Diffusion

NASA Technical Reports Server (NTRS)

Salamon, N. J. (Principal Investigator); Sullivan, Roy M.; Lee, Sunpyo

1995-01-01

A theory for time-dependent thermal and gas diffusion in mechanically time-rate-independent anisotropic poroelastic composites has been developed. This theory advances previous work by the latter two authors by providing for critical transverse shear through a three-dimensional axisymmetric formulation and using it in a new hypothesis for determining the Biot fluid pressure-solid stress coupling factor. The derived governing equations couple material deformation with temperature and internal pore pressure and more strongly couple gas diffusion and heat transfer than the previous theory. Hence the theory accounts for the interactions between conductive heat transfer in the porous body and convective heat carried by the mass flux through the pores. The Bubnov Galerkin finite element method is applied to the governing equations to transform them into a semidiscrete finite element system. A numerical procedure is developed to solve the coupled equations in the space and time domains. The method is used to simulate two high temperature tests involving thermal-chemical decomposition of carbon-phenolic composites. In comparison with measured data, the results are accurate. Moreover unlike previous work, for a single set of poroelastic parameters, they are consistent with two measurements in a restrained thermal growth test.

Shape in Picture: Mathematical Description of Shape in Grey-Level Images

DTIC Science & Technology

1992-09-11

representation is scale-space, derived frrr- the linear isotropic diffusion equation; recently other types of equations have been considered. Multiscale...recognition of dimensions in the general case of an arbitrary denominator is similar to that just explained. 3 Linear Inequalities in the Two-Dimensional...solid region containing all pixels of the space, whose coordinates satisfy a linear inequality. A Um C scspt fr Digital Geometry 41 s a a v--’ -0 7 O

An application of a two-equation model of turbulence to three-dimensional chemically reacting flows

NASA Technical Reports Server (NTRS)

Lee, J.

1994-01-01

A numerical study of three dimensional chemically reacting and non-reacting flowfields is conducted using a two-equation model of turbulence. A generalized flow solver using an implicit Lower-Upper (LU) diagonal decomposition numerical technique and finite-rate chemistry has been coupled with a low-Reynolds number two-equation model of turbulence. This flow solver is then used to study chemically reacting turbulent supersonic flows inside combustors with synergetic fuel injectors. The reacting and non-reacting turbulent combustor solutions obtained are compared with zero-equation turbulence model solutions and with available experimental data. The hydrogen-air chemistry is modeled using a nine-species/eighteen reaction model. A low-Reynolds number k-epsilon model was used to model the effect of turbulence because, in general, the low-Reynolds number k-epsilon models are easier to implement numerically and are far more general than algebraic models. However, low-Reynolds number k-epsilon models require a much finer near-wall grid resolution than high-Reynolds number models to resolve accurately the near-wall physics. This is especially true in complex flowfields, where the stiff nature of the near-wall turbulence must be resolved. Therefore, the limitations imposed by the near-wall characteristics and compressible model corrections need to be evaluated further. The gradient-diffusion hypothesis is used to model the effects of turbulence on the mass diffusion process. The influence of this low-Reynolds number turbulence model on the reacting flowfield predictions was studied parametrically.

Diffusion in random networks: Asymptotic properties, and numerical and engineering approximations

NASA Astrophysics Data System (ADS)

Padrino, Juan C.; Zhang, Duan Z.

2016-11-01

The ensemble phase averaging technique is applied to model mass transport by diffusion in random networks. The system consists of an ensemble of random networks, where each network is made of a set of pockets connected by tortuous channels. Inside a channel, we assume that fluid transport is governed by the one-dimensional diffusion equation. Mass balance leads to an integro-differential equation for the pores mass density. The so-called dual porosity model is found to be equivalent to the leading order approximation of the integration kernel when the diffusion time scale inside the channels is small compared to the macroscopic time scale. As a test problem, we consider the one-dimensional mass diffusion in a semi-infinite domain, whose solution is sought numerically. Because of the required time to establish the linear concentration profile inside a channel, for early times the similarity variable is xt- 1 / 4 rather than xt- 1 / 2 as in the traditional theory. This early time sub-diffusive similarity can be explained by random walk theory through the network. In addition, by applying concepts of fractional calculus, we show that, for small time, the governing equation reduces to a fractional diffusion equation with known solution. We recast this solution in terms of special functions easier to compute. Comparison of the numerical and exact solutions shows excellent agreement.

Formulation, Implementation and Validation of a Two-Fluid model in a Fuel Cell CFD Code

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jain, Kunal; Cole, J. Vernon; Kumar, Sanjiv

2008-12-01

Water management is one of the main challenges in PEM Fuel Cells. While water is essential for membrane electrical conductivity, excess liquid water leads to flooding of catalyst layers. Despite the fact that accurate prediction of two-phase transport is key for optimal water management, understanding of the two-phase transport in fuel cells is relatively poor. Wang et. al. have studied the two-phase transport in the channel and diffusion layer separately using a multiphase mixture model. The model fails to accurately predict saturation values for high humidity inlet streams. Nguyen et. al. developed a two-dimensional, two-phase, isothermal, isobaric, steady state modelmore » of the catalyst and gas diffusion layers. The model neglects any liquid in the channel. Djilali et. al. developed a three-dimensional two-phase multicomponent model. The model is an improvement over previous models, but neglects drag between the liquid and the gas phases in the channel. In this work, we present a comprehensive two-fluid model relevant to fuel cells. Models for two-phase transport through Channel, Gas Diffusion Layer (GDL) and Channel-GDL interface, are discussed. In the channel, the gas and liquid pressures are assumed to be same. The surface tension effects in the channel are incorporated using the continuum surface force (CSF) model. The force at the surface is expressed as a volumetric body force and added as a source to the momentum equation. In the GDL, the gas and liquid are assumed to be at different pressures. The difference in the pressures (capillary pressure) is calculated using an empirical correlations. At the Channel-GDL interface, the wall adhesion affects need to be taken into account. SIMPLE-type methods recast the continuity equation into a pressure-correction equation, the solution of which then provides corrections for velocities and pressures. However, in the two-fluid model, the presence of two phasic continuity equations gives more freedom and more complications. A general approach would be to form a mixture continuity equation by linearly combining the phasic continuity equations using appropriate weighting factors. Analogous to mixture equation for pressure correction, a difference equation is used for the volume/phase fraction by taking the difference between the phasic continuity equations. The relative advantages of the above mentioned algorithmic variants for computing pressure correction and volume fractions are discussed and quantitatively assessed. Preliminary model validation is done for each component of the fuel cell. The two-phase transport in the channel is validated using empirical correlations. Transport in the GDL is validated against results obtained from LBM and VOF simulation techniques. The Channel-GDL interface transport will be validated against experiment and empirical correlation of droplet detachment at the interface.« less

Extrapolation techniques applied to matrix methods in neutron diffusion problems

NASA Technical Reports Server (NTRS)

Mccready, Robert R

1956-01-01

A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.

An adaptive grid algorithm for one-dimensional nonlinear equations

NASA Technical Reports Server (NTRS)

Gutierrez, William E.; Hills, Richard G.

1990-01-01

Richards' equation, which models the flow of liquid through unsaturated porous media, is highly nonlinear and difficult to solve. Step gradients in the field variables require the use of fine grids and small time step sizes. The numerical instabilities caused by the nonlinearities often require the use of iterative methods such as Picard or Newton interation. These difficulties result in large CPU requirements in solving Richards equation. With this in mind, adaptive and multigrid methods are investigated for use with nonlinear equations such as Richards' equation. Attention is focused on one-dimensional transient problems. To investigate the use of multigrid and adaptive grid methods, a series of problems are studied. First, a multigrid program is developed and used to solve an ordinary differential equation, demonstrating the efficiency with which low and high frequency errors are smoothed out. The multigrid algorithm and an adaptive grid algorithm is used to solve one-dimensional transient partial differential equations, such as the diffusive and convective-diffusion equations. The performance of these programs are compared to that of the Gauss-Seidel and tridiagonal methods. The adaptive and multigrid schemes outperformed the Gauss-Seidel algorithm, but were not as fast as the tridiagonal method. The adaptive grid scheme solved the problems slightly faster than the multigrid method. To solve nonlinear problems, Picard iterations are introduced into the adaptive grid and tridiagonal methods. Burgers' equation is used as a test problem for the two algorithms. Both methods obtain solutions of comparable accuracy for similar time increments. For the Burgers' equation, the adaptive grid method finds the solution approximately three times faster than the tridiagonal method. Finally, both schemes are used to solve the water content formulation of the Richards' equation. For this problem, the adaptive grid method obtains a more accurate solution in fewer work units and less computation time than required by the tridiagonal method. The performance of the adaptive grid method tends to degrade as the solution process proceeds in time, but still remains faster than the tridiagonal scheme.

A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

USGS Publications Warehouse

Smith, Peter E.

2006-01-01

A semi-implicit, finite-difference method for the numerical solution of the three-dimensional equations for circulation in estuaries is presented and tested. The method uses a three-time-level, leapfrog-trapezoidal scheme that is essentially second-order accurate in the spatial and temporal numerical approximations. The three-time-level scheme is shown to be preferred over a two-time-level scheme, especially for problems with strong nonlinearities. The stability of the semi-implicit scheme is free from any time-step limitation related to the terms describing vertical diffusion and the propagation of the surface gravity waves. The scheme does not rely on any form of vertical/horizontal mode-splitting to treat the vertical diffusion implicitly. At each time step, the numerical method uses a double-sweep method to transform a large number of small tridiagonal equation systems and then uses the preconditioned conjugate-gradient method to solve a single, large, five-diagonal equation system for the water surface elevation. The governing equations for the multi-level scheme are prepared in a conservative form by integrating them over the height of each horizontal layer. The layer-integrated volumetric transports replace velocities as the dependent variables so that the depth-integrated continuity equation that is used in the solution for the water surface elevation is linear. Volumetric transports are computed explicitly from the momentum equations. The resulting method is mass conservative, efficient, and numerically accurate.

Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation.

PubMed

Berry, Hugues

2002-10-01

Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes.

Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation.

PubMed Central

Berry, Hugues

2002-01-01

Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes. PMID:12324410

Semi-implicit integration factor methods on sparse grids for high-dimensional systems

NASA Astrophysics Data System (ADS)

Wang, Dongyong; Chen, Weitao; Nie, Qing

2015-07-01

Numerical methods for partial differential equations in high-dimensional spaces are often limited by the curse of dimensionality. Though the sparse grid technique, based on a one-dimensional hierarchical basis through tensor products, is popular for handling challenges such as those associated with spatial discretization, the stability conditions on time step size due to temporal discretization, such as those associated with high-order derivatives in space and stiff reactions, remain. Here, we incorporate the sparse grids with the implicit integration factor method (IIF) that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions. We combine IIF, in which the reaction is treated implicitly and the diffusion is treated explicitly and exactly, with various sparse grid techniques based on the finite element and finite difference methods and a multi-level combination approach. The overall method is found to be efficient in terms of both storage and computational time for solving a wide range of PDEs in high dimensions. In particular, the IIF with the sparse grid combination technique is flexible and effective in solving systems that may include cross-derivatives and non-constant diffusion coefficients. Extensive numerical simulations in both linear and nonlinear systems in high dimensions, along with applications of diffusive logistic equations and Fokker-Planck equations, demonstrate the accuracy, efficiency, and robustness of the new methods, indicating potential broad applications of the sparse grid-based integration factor method.

Photon migration in non-scattering tissue and the effects on image reconstruction

NASA Astrophysics Data System (ADS)

Dehghani, H.; Delpy, D. T.; Arridge, S. R.

1999-12-01

Photon propagation in tissue can be calculated using the relationship described by the transport equation. For scattering tissue this relationship is often simplified and expressed in terms of the diffusion approximation. This approximation, however, is not valid for non-scattering regions, for example cerebrospinal fluid (CSF) below the skull. This study looks at the effects of a thin clear layer in a simple model representing the head and examines its effect on image reconstruction. Specifically, boundary photon intensities (total number of photons exiting at a point on the boundary due to a source input at another point on the boundary) are calculated using the transport equation and compared with data calculated using the diffusion approximation for both non-scattering and scattering regions. The effect of non-scattering regions on the calculated boundary photon intensities is presented together with the advantages and restrictions of the transport code used. Reconstructed images are then presented where the forward problem is solved using the transport equation for a simple two-dimensional system containing a non-scattering ring and the inverse problem is solved using the diffusion approximation to the transport equation.

A non-linear dimension reduction methodology for generating data-driven stochastic input models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ganapathysubramanian, Baskar; Zabaras, Nicholas

Stochastic analysis of random heterogeneous media (polycrystalline materials, porous media, functionally graded materials) provides information of significance only if realistic input models of the topology and property variations are used. This paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data-driven strategy. Given a set of microstructure realizations (input samples) generated from given statistical information about the medium topology, the framework constructs a reduced-order stochastic representation of the thermal diffusivity. This problem of constructing a low-dimensional stochastic representation of property variations is analogous to the problem ofmore » manifold learning and parametric fitting of hyper-surfaces encountered in image processing and psychology. Denote by M the set of microstructures that satisfy the given experimental statistics. A non-linear dimension reduction strategy is utilized to map M to a low-dimensional region, A. We first show that M is a compact manifold embedded in a high-dimensional input space R{sup n}. An isometric mapping F from M to a low-dimensional, compact, connected set A is contained in R{sup d}(d<

Flow regimes for fluid injection into a confined porous medium

DOE PAGES

Zheng, Zhong; Guo, Bo; Christov, Ivan C.; ...

2015-02-24

We report theoretical and numerical studies of the flow behaviour when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection, a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid flow is mainly driven by the injection, and the governingmore » equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous. In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated.« less

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 3: Programmer's reference

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Bui, Trong T.

1993-01-01

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.

The 2.5-dimensional equivalent sources method for directly exposed and shielded urban canyons.

PubMed

Hornikx, Maarten; Forssén, Jens

2007-11-01

When a domain in outdoor acoustics is invariant in one direction, an inverse Fourier transform can be used to transform solutions of the two-dimensional Helmholtz equation to a solution of the three-dimensional Helmholtz equation for arbitrary source and observer positions, thereby reducing the computational costs. This previously published approach [D. Duhamel, J. Sound Vib. 197, 547-571 (1996)] is called a 2.5-dimensional method and has here been extended to the urban geometry of parallel canyons, thereby using the equivalent sources method to generate the two-dimensional solutions. No atmospheric effects are considered. To keep the error arising from the transform small, two-dimensional solutions with a very fine frequency resolution are necessary due to the multiple reflections in the canyons. Using the transform, the solution for an incoherent line source can be obtained much more efficiently than by using the three-dimensional solution. It is shown that the use of a coherent line source for shielded urban canyon observer positions leads mostly to an overprediction of levels and can yield erroneous results for noise abatement schemes. Moreover, the importance of multiple facade reflections in shielded urban areas is emphasized by vehicle pass-by calculations, where cases with absorptive and diffusive surfaces have been modeled.

Surface evolution in bare bamboo-type metal lines under diffusion and electric field effects

NASA Astrophysics Data System (ADS)

Averbuch, Amir; Israeli, Moshe; Nathan, Menachem; Ravve, Igor

2003-07-01

Irregularities such as voids and cracks often occur in bamboo-type metal lines of microelectronic interconnects. They increase the resistance of the circuits, and may even lead to a fatal failure. In this work, we analyze numerically the electromigration of an unpassivated bamboo-type line with pre-existing irregularities in its top surface (also called a grain-void interface). The bamboo line is subjected to surface diffusion forces and external electric fields. Under these forces, initial defects may either heal or become worse. The grain-void interface is considered to be one-dimensional, and the physical formulation of an electromigration and diffusion model results in two coupled, fourth order, one-dimensional time-dependent PDEs, with the boundary conditions imposed at the electrode points and at the triple point, which belongs to two neighboring grains and the void. These equations are discretized by finite differences on a regular grid in space, and by a Runge-Kutta integration scheme in time, and solved simultaneously with a static Laplace equation describing the voltage distribution throughout each grain, when the substrate conductivity is neglected. Since the voltage distribution is required only along an interface line, the two-dimensional discretization of the grain interior is not needed, and the static problem is solved by the boundary element method at each time step. The motion of the interface line is studied for different ratios between diffusion and electric field forces, and for different initial configurations of the grain-void interface. We study plain and tilted contour lines, considering positive and negative tilts with respect to the external electric field, a stepped contour with field lines entering or exiting the 'step', and a number of modifications of the classical Mullins problem of thermal grooving. We also consider a two-grain Mullins problem with a normal and tilted boundary between the grains, examining positive and negative tilts.

Analytical modeling of operating characteristics of premixing-prevaporizing fuel-air mixing passages. Volume 1: Analysis and results

NASA Technical Reports Server (NTRS)

Anderson, O. L.; Chiappetta, L. M.; Edwards, D. E.; Mcvey, J. B.

1982-01-01

A model for predicting the distribution of liquid fuel droplets and fuel vapor in premixing-prevaporizing fuel-air mixing passages of the direct injection type is reported. This model consists of three computer programs; a calculation of the two dimensional or axisymmetric air flow field neglecting the effects of fuel; a calculation of the three dimensional fuel droplet trajectories and evaporation rates in a known, moving air flow; a calculation of fuel vapor diffusing into a moving three dimensional air flow with source terms dependent on the droplet evaporation rates. The fuel droplets are treated as individual particle classes each satisfying Newton's law, a heat transfer, and a mass transfer equation. This fuel droplet model treats multicomponent fuels and incorporates the physics required for the treatment of elastic droplet collisions, droplet shattering, droplet coalescence and droplet wall interactions. The vapor diffusion calculation treats three dimensional, gas phase, turbulent diffusion processes. The analysis includes a model for the autoignition of the fuel air mixture based upon the rate of formation of an important intermediate chemical species during the preignition period.

High-accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear Schrödinger-type equations

NASA Astrophysics Data System (ADS)

Khawaja, U. Al; Al-Refai, M.; Shchedrin, Gavriil; Carr, Lincoln D.

2018-06-01

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations, including the fractional nonlinear Schrödinger equation and the fractional nonlinear diffusion equation. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. In the specific case of the fractional nonlinear Schrödinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.

Characterization of chaotic electroconvection near flat electrodes under oscillatory voltages

NASA Astrophysics Data System (ADS)

Kim, Jeonglae; Davidson, Scott; Mani, Ali

2017-11-01

Onset of hydrodynamic instability and chaotic electroconvection in aqueous systems are studied by directly solving the two-dimensional coupled Poisson-Nernst-Planck and Navier-Stokes equations. An aqueous binary electrolyte is bounded by two planar electrodes where time-harmonic voltage is applied at a constant oscillation frequency. The governing equations are solved using a fully-conservative second-order-accurate finite volume discretization and a second-order implicit Euler time advancement. At a sufficiently high amplitude of applied voltage, the system exhibits chaotic behaviors involving strong hydrodynamic mixing and enhanced electroconvection. The system responses are characterized as a function of oscillation frequency, voltage magnitude, and the ratio of diffusivities of two ion species. Our results indicate that electroconvection is most enhanced for frequencies on the order of inverse system RC time scale. We will discuss the dependence of this optimal frequency on the asymmetry of the diffusion coefficients of ionic species. Supported by the Stanford's Precourt Institute.

Unity and diversity in mixing: Stretching, diffusion, breakup, and aggregation in chaotic flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ottino, J.M.

1991-05-01

Experiments and theory have produced a reasonably good qualitative understanding of the evolution of chaotic mixing of passive tracers, especially in two-dimensional time-periodic flow fields. Such an understanding forms a fabric for the evolution of breakup, aggregation, and diffusion-controlled reactions in more complex flows. These systems can be viewed as a population of microstructures'' whose behavior is dictated by iterations of a chaotic flow; microstructures break, diffuse, and aggregate, causing the population to evolve in space and time. This paper presents simple physical models for such processes. Self-similarity is common to all the problems; examples arise in the context ofmore » the distribution of stretchings within chaotic flows, in the asymptotic evolution of diffusion-reaction processes at striation thickness scales, in the equilibrium distribution of drop sizes generated upon mixing of immiscible fluids, in the equations describing mean-field kinetics of coagulation, in the sequence of actions necessary for the destruction of islands in two-dimensional flow, and in the fractal structure of clusters produced upon aggregation in chaotic flows.« less

Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation With Small Viscosity

NASA Astrophysics Data System (ADS)

Beck, Margaret; Wayne, C. Eugene

2009-01-01

The large-time behavior of solutions to the Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted L^2 space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this metastable manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.

Bioheat model evaluations of laser effects on tissues: role of water evaporation and diffusion

NASA Astrophysics Data System (ADS)

Nagulapally, Deepthi; Joshi, Ravi P.; Thomas, Robert J.

2011-03-01

A two-dimensional, time-dependent bioheat model is applied to evaluate changes in temperature and water content in tissues subjected to laser irradiation. Our approach takes account of liquid-to-vapor phase changes and a simple diffusive flow of water within the biotissue. An energy balance equation considers blood perfusion, metabolic heat generation, laser absorption, and water evaporation. The model also accounts for the water dependence of tissue properties (both thermal and optical), and variations in blood perfusion rates based on local tissue injury. Our calculations show that water diffusion would reduce the local temperature increases and hot spots in comparison to simple models that ignore the role of water in the overall thermal and mass transport. Also, the reduced suppression of perfusion rates due to tissue heating and damage with water diffusion affect the necrotic depth. Two-dimensional results for the dynamic temperature, water content, and damage distributions will be presented for skin simulations. It is argued that reduction in temperature gradients due to water diffusion would mitigate local refractive index variations, and hence influence the phenomenon of thermal lensing. Finally, simple quantitative evaluations of pressure increases within the tissue due to laser absorption are presented.

UFO: A THREE-DIMENSIONAL NEUTRON DIFFUSION CODE FOR THE IBM 704

DOE Office of Scientific and Technical Information (OSTI.GOV)

Auerbach, E.H.; Jewett, J.P.; Ketchum, M.A.

A description of UFO, a code for the solution of the fewgroup neutron diffusion equation in three-dimensional Cartesian coordinates on the IBM 704, is given. An accelerated Liebmann flux iteration scheme is used, and optimum parameters can be calculated by the code whenever they are required. The theory and operation of the program are discussed. (auth)

Molecular Dynamic Simulation of Water Vapor and Determination of Diffusion Characteristics in the Pore

NASA Astrophysics Data System (ADS)

Nikonov, Eduard G.; Pavluš, Miron; Popovičová, Mária

2018-02-01

One of the varieties of pores, often found in natural or artificial building materials, are the so-called blind pores of dead-end or saccate type. Three-dimensional model of such kind of pore has been developed in this work. This model has been used for simulation of water vapor interaction with individual pore by molecular dynamics in combination with the diffusion equation method. Special investigations have been done to find dependencies between thermostats implementations and conservation of thermodynamic and statistical values of water vapor - pore system. The two types of evolution of water - pore system have been investigated: drying and wetting of the pore. Full research of diffusion coefficient, diffusion velocity and other diffusion parameters has been made.

Incorporating convection into one-dimensional solute redistribution during crystal growth from the melt I. The steady-state solution

NASA Astrophysics Data System (ADS)

Yen, C. T.; Tiller, W. A.

1992-03-01

A one-dimensional mathematical analysis is made of the redistribution of solute which occurs during crystal growth from a convected melt. In this analysis, the important contribution from lateral melt convection to one-dimensional solute redistribution analysis is taken into consideration via an annihilation/creation term in the one-dimensional solute transport equation. Calculations of solute redistribution under steady-state conditions have been carried out analytically. It is found that this new solute redistribution model overcomes several weaknesses that occur when applying the Burton, Prim and Slichter solute segregation equation (1953) in real melt growth situations. It is also found that, with this correction, the diffusion coefficients for solute's in liquid silicon are now found to be in the same range as other liquid metal diffusion coefficients.

A nonlinear equation for ionic diffusion in a strong binary electrolyte

PubMed Central

Ghosal, Sandip; Chen, Zhen

2010-01-01

The problem of the one-dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson–Nernst–Planck (PNP) system, consists of a diffusion equation for each species augmented by transport owing to a self-consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important problems in physics, such as electron and hole diffusion across semiconductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here, we derive a more general theory by exploiting the ratio of the Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear partial differential equation that provides a better approximation than the classical linear equation for ambipolar diffusion, but reduces to it in the appropriate limit. PMID:21818176

Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain

NASA Astrophysics Data System (ADS)

Chae, Myeongju; Choi, Kyudong; Kang, Kyungkeun; Lee, Jihoon

2018-07-01

We consider a simplified model of tumor angiogenesis, described by a Keller-Segel equation on the two dimensional domain (x , y) ∈ R ×Sλ where Sλ is the circle of perimeter λ. It is known that the system allows planar traveling wave solutions of an invading type. In case that λ is sufficiently small, we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted Sobolev space. When chemical diffusion is present, it can be shown that the system is linearly stable. Lastly, we prove that any solution with our front condition eventually becomes planar under certain regularity conditions.

Fractional-calculus diffusion equation

PubMed Central

2010-01-01

Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677

Macroscopic Modeling of a One-Dimensional Electrochemical Cell using the Poisson-Nernst-Planck Equations

NASA Astrophysics Data System (ADS)

Yan, David

This thesis presents the one-dimensional equations, numerical method and simulations of a model to characterize the dynamical operation of an electrochemical cell. This model extends the current state-of-the art in that it accounts, in a primitive way, for the physics of the electrolyte/electrode interface and incorporates diffuse-charge dynamics, temperature coupling, surface coverage, and polarization phenomena. The one-dimensional equations account for a system with one or two mobile ions of opposite charge, and the electrode reaction we consider (when one is needed) is a one-electron electrodeposition reaction. Though the modeled system is far from representing a realistic electrochemical device, our results show a range of dynamics and behaviors which have not been observed previously, and explore the numerical challenges required when adding more complexity to a model. Furthermore, the basic transport equations (which are developed in three spatial dimensions) can in future accomodate the inclusion of additional physics, and coupling to more complex boundary conditions that incorporate two-dimensional surface phenomena and multi-rate reactions. In the model, the Poisson-Nernst-Planck equations are used to model diffusion and electromigration in an electrolyte, and the generalized Frumkin-Butler-Volmer equation is used to model reaction kinetics at electrodes. An energy balance equation is derived and coupled to the diffusion-migration equation. The model also includes dielectric polarization effects by introducing different values of the dielectric permittivity in different regions of the bulk, as well as accounting for surface coverage effects due to adsorption, and finite size "crowding", or steric effects. Advection effects are not modeled but could in future be incorporated. In order to solve the coupled PDE's, we use a variable step size second order scheme in time and finite differencing in space. Numerical tests are performed on a simplified system and the scheme's stability and convergence properties are discussed. While evaluating different methods for discretizing the coupled flux boundary condition, we discover a thresholding behaviour in the adaptive time stepper, and perform additional tests to investigate it. Finally, a method based on ghost points is chosen for its favorable numerical properties compared to the alternatives. With this method, we are able to run simulations with a large range of parameters, including any value of the nondimensionalized Debye length epsilon. The numerical code is first used to run simulations to explore the effects of polarization, surface coverage, and temperature. The code is also used to perform frequency sweeps of input signals in order to mimic impedance spectroscopy experiments. Finally, in Chapter 5, we use our model to apply ramped voltages to electrochemical systems, and show theoretical and simulated current-voltage curves for liquid and solid thin films, cells with blocking (polarized) electrodes, and electrolytes with background charge. Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. No general theory of linear-sweep voltammetry is available, however, for unsupported electrolytes and for other situations where diffuse charge effects play a role. We show theoretical and simulated current-voltage curves for liquid and solid thin films, cells with blocking electrodes, and membranes with fixed background charge. The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, but capacitive charging of the double layers is also studied, for early time transients at reactive electrodes and for non-reactive blocking electrodes. The final chapter highlights the role of diffuse charge in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

Two-dimensional description of surface-bounded exospheres with application to the migration of water molecules on the Moon

NASA Astrophysics Data System (ADS)

Schorghofer, Norbert

2015-05-01

On the Moon, water molecules and other volatiles are thought to migrate along ballistic trajectories. Here, this migration process is described in terms of a two-dimensional partial differential equation for the surface concentration, based on the probability distribution of thermal ballistic hops. A random-walk model, a corresponding diffusion coefficient, and a continuum description are provided. In other words, a surface-bounded exosphere is described purely in terms of quantities on the surface, which can provide computational and conceptual advantages. The derived continuum equation can be used to calculate the steady-state distribution of the surface concentration of volatile water molecules. An analytic steady-state solution is obtained for an equatorial ring; it reveals the width and mass of the pileup of molecules at the morning terminator.

Application of Central Upwind Scheme for Solving Special Relativistic Hydrodynamic Equations

PubMed Central

Yousaf, Muhammad; Ghaffar, Tayabia; Qamar, Shamsul

2015-01-01

The accurate modeling of various features in high energy astrophysical scenarios requires the solution of the Einstein equations together with those of special relativistic hydrodynamics (SRHD). Such models are more complicated than the non-relativistic ones due to the nonlinear relations between the conserved and state variables. A high-resolution shock-capturing central upwind scheme is implemented to solve the given set of equations. The proposed technique uses the precise information of local propagation speeds to avoid the excessive numerical diffusion. The second order accuracy of the scheme is obtained with the use of MUSCL-type initial reconstruction and Runge-Kutta time stepping method. After a discussion of the equations solved and of the techniques employed, a series of one and two-dimensional test problems are carried out. To validate the method and assess its accuracy, the staggered central and the kinetic flux-vector splitting schemes are also applied to the same model. The scheme is robust and efficient. Its results are comparable to those obtained from the sophisticated algorithms, even in the case of highly relativistic two-dimensional test problems. PMID:26070067

Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction

NASA Astrophysics Data System (ADS)

Sid, S.; Terrapon, V. E.; Dubief, Y.

2018-02-01

The goal of the present study is threefold: (i) to demonstrate the two-dimensional nature of the elasto-inertial instability in elasto-inertial turbulence (EIT), (ii) to identify the role of the bidimensional instability in three-dimensional EIT flows, and (iii) to establish the role of the small elastic scales in the mechanism of self-sustained EIT. Direct numerical simulations of viscoelastic fluid flows are performed in both two- and three-dimensional straight periodic channels using the Peterlin finitely extensible nonlinear elastic model (FENE-P). The Reynolds number is set to Reτ=85 , which is subcritical for two-dimensional flows but beyond the transition for three-dimensional ones. The polymer properties selected correspond to those of typical dilute polymer solutions, and two moderate Weissenberg numbers, Wiτ=40 ,100 , are considered. The simulation results show that sustained turbulence can be observed in two-dimensional subcritical flows, confirming the existence of a bidimensional elasto-inertial instability. The same type of instability is also observed in three-dimensional simulations where both Newtonian and elasto-inertial turbulent structures coexist. Depending on the Wi number, one type of structure can dominate and drive the flow. For large Wi values, the elasto-inertial instability tends to prevail over the Newtonian turbulence. This statement is supported by (i) the absence of typical Newtonian near-wall vortices and (ii) strong similarities between two- and three-dimensional flows when considering larger Wi numbers. The role of small elastic scales is investigated by introducing global artificial diffusion (GAD) in the hyperbolic transport equation for polymers. The aim is to measure how the flow reacts when the smallest elastic scales are progressively filtered out. The study results show that the introduction of large polymer diffusion in the system strongly damps a significant part of the elastic scales that are necessary to feed turbulence, eventually leading to flow laminarization. A sufficiently high Schmidt number (weakly diffusive polymers) is necessary to allow self-sustained turbulence to settle. Although EIT can withstand a low amount of diffusion and remains in a nonlaminar chaotic state, adding a finite amount of GAD in the system can have an impact on the dynamics and lead to important quantitative changes, even for Schmidt numbers as large as 102. The use of GAD should therefore be avoided in viscoelastic flow simulations.

A parallel algorithm for the two-dimensional time fractional diffusion equation with implicit difference method.

PubMed

Gong, Chunye; Bao, Weimin; Tang, Guojian; Jiang, Yuewen; Liu, Jie

2014-01-01

It is very time consuming to solve fractional differential equations. The computational complexity of two-dimensional fractional differential equation (2D-TFDE) with iterative implicit finite difference method is O(M(x)M(y)N(2)). In this paper, we present a parallel algorithm for 2D-TFDE and give an in-depth discussion about this algorithm. A task distribution model and data layout with virtual boundary are designed for this parallel algorithm. The experimental results show that the parallel algorithm compares well with the exact solution. The parallel algorithm on single Intel Xeon X5540 CPU runs 3.16-4.17 times faster than the serial algorithm on single CPU core. The parallel efficiency of 81 processes is up to 88.24% compared with 9 processes on a distributed memory cluster system. We do think that the parallel computing technology will become a very basic method for the computational intensive fractional applications in the near future.

Numerical solutions of Navier-Stokes equations for compressible turbulent two/three dimensional flows in terminal shock region of an inlet/diffuser

NASA Technical Reports Server (NTRS)

Liu, N. S.; Shamroth, S. J.; Mcdonald, H.

1983-01-01

The multidimensional ensemble averaged compressible time dependent Navier Stokes equations in conjunction with mixing length turbulence model and shock capturing technique were used to study the terminal shock type of flows in various flight regimes occurring in a diffuser/inlet model. The numerical scheme for solving the governing equations is based on a linearized block implicit approach and the following high Reynolds number calculations were carried out: (1) 2 D, steady, subsonic; (2) 2 D, steady, transonic with normal shock; (3) 2 D, steady, supersonic with terminal shock; (4) 2 D, transient process of shock development and (5) 3 D, steady, transonic with normal shock. The numerical results obtained for the 2 D and 3 D transonic shocked flows were compared with corresponding experimental data; the calculated wall static pressure distributions agree well with the measured data.

Multivariate space - time analysis of PRE-STORM precipitation

NASA Technical Reports Server (NTRS)

Polyak, Ilya; North, Gerald R.; Valdes, Juan B.

1994-01-01

This paper presents the methodologies and results of the multivariate modeling and two-dimensional spectral and correlation analysis of PRE-STORM rainfall gauge data. Estimated parameters of the models for the specific spatial averages clearly indicate the eastward and southeastward wave propagation of rainfall fluctuations. A relationship between the coefficients of the diffusion equation and the parameters of the stochastic model of rainfall fluctuations is derived that leads directly to the exclusive use of rainfall data to estimate advection speed (about 12 m/s) as well as other coefficients of the diffusion equation of the corresponding fields. The statistical methodology developed here can be used for confirmation of physical models by comparison of the corresponding second-moment statistics of the observed and simulated data, for generating multiple samples of any size, for solving the inverse problem of the hydrodynamic equations, and for application in some other areas of meteorological and climatological data analysis and modeling.

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK

PubMed Central

2014-01-01

Background Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system’s set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This “code-based” approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution. Results As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations. Conclusions The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts. PMID:24725437

Simulations of pattern dynamics for reaction-diffusion systems via SIMULINK.

PubMed

Wang, Kaier; Steyn-Ross, Moira L; Steyn-Ross, D Alistair; Wilson, Marcus T; Sleigh, Jamie W; Shiraishi, Yoichi

2014-04-11

Investigation of the nonlinear pattern dynamics of a reaction-diffusion system almost always requires numerical solution of the system's set of defining differential equations. Traditionally, this would be done by selecting an appropriate differential equation solver from a library of such solvers, then writing computer codes (in a programming language such as C or Matlab) to access the selected solver and display the integrated results as a function of space and time. This "code-based" approach is flexible and powerful, but requires a certain level of programming sophistication. A modern alternative is to use a graphical programming interface such as Simulink to construct a data-flow diagram by assembling and linking appropriate code blocks drawn from a library. The result is a visual representation of the inter-relationships between the state variables whose output can be made completely equivalent to the code-based solution. As a tutorial introduction, we first demonstrate application of the Simulink data-flow technique to the classical van der Pol nonlinear oscillator, and compare Matlab and Simulink coding approaches to solving the van der Pol ordinary differential equations. We then show how to introduce space (in one and two dimensions) by solving numerically the partial differential equations for two different reaction-diffusion systems: the well-known Brusselator chemical reactor, and a continuum model for a two-dimensional sheet of human cortex whose neurons are linked by both chemical and electrical (diffusive) synapses. We compare the relative performances of the Matlab and Simulink implementations. The pattern simulations by Simulink are in good agreement with theoretical predictions. Compared with traditional coding approaches, the Simulink block-diagram paradigm reduces the time and programming burden required to implement a solution for reaction-diffusion systems of equations. Construction of the block-diagram does not require high-level programming skills, and the graphical interface lends itself to easy modification and use by non-experts.

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2006-01-01

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

Diffusion model to describe osteogenesis within a porous titanium scaffold.

PubMed

Schmitt, M; Allena, R; Schouman, T; Frasca, S; Collombet, J M; Holy, X; Rouch, P

2016-01-01

In this study, we develop a two-dimensional finite element model, which is derived from an animal experiment and allows simulating osteogenesis within a porous titanium scaffold implanted in ewe's hemi-mandible during 12 weeks. The cell activity is described through diffusion equations and regulated by the stress state of the structure. We compare our model to (i) histological observations and (ii) experimental data obtained from a mechanical test done on sacrificed animal. We show that our mechano-biological approach provides consistent numerical results and constitutes a useful tool to predict osteogenesis pattern.

Three-dimensional stochastic modeling of radiation belts in adiabatic invariant coordinates

NASA Astrophysics Data System (ADS)

Zheng, Liheng; Chan, Anthony A.; Albert, Jay M.; Elkington, Scot R.; Koller, Josef; Horne, Richard B.; Glauert, Sarah A.; Meredith, Nigel P.

2014-09-01

A 3-D model for solving the radiation belt diffusion equation in adiabatic invariant coordinates has been developed and tested. The model, named Radbelt Electron Model, obtains a probabilistic solution by solving a set of Itô stochastic differential equations that are mathematically equivalent to the diffusion equation. This method is capable of solving diffusion equations with a full 3-D diffusion tensor, including the radial-local cross diffusion components. The correct form of the boundary condition at equatorial pitch angle α0=90° is also derived. The model is applied to a simulation of the October 2002 storm event. At α0 near 90°, our results are quantitatively consistent with GPS observations of phase space density (PSD) increases, suggesting dominance of radial diffusion; at smaller α0, the observed PSD increases are overestimated by the model, possibly due to the α0-independent radial diffusion coefficients, or to insufficient electron loss in the model, or both. Statistical analysis of the stochastic processes provides further insights into the diffusion processes, showing distinctive electron source distributions with and without local acceleration.

Diffusion in random networks

DOE PAGES

Zhang, Duan Z.; Padrino, Juan C.

2017-06-01

The ensemble averaging technique is applied to model mass transport by diffusion in random networks. The system consists of an ensemble of random networks, where each network is made of pockets connected by tortuous channels. Inside a channel, fluid transport is assumed to be governed by the one-dimensional diffusion equation. Mass balance leads to an integro-differential equation for the pocket mass density. The so-called dual-porosity model is found to be equivalent to the leading order approximation of the integration kernel when the diffusion time scale inside the channels is small compared to the macroscopic time scale. As a test problem,more » we consider the one-dimensional mass diffusion in a semi-infinite domain. Because of the required time to establish the linear concentration profile inside a channel, for early times the similarity variable is xt $-$1/4 rather than xt $-$1/2 as in the traditional theory. We found this early time similarity can be explained by random walk theory through the network.« less

Solution of a cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula

NASA Astrophysics Data System (ADS)

Remizov, I. D.

2012-07-01

The Cauchy problem for a class of diffusion equations in a Hilbert space is studied. It is proved that the Cauchy problem in well posed in the class of uniform limits of infinitely smooth bounded cylindrical functions on the Hilbert space, and the solution is presented in the form of the so-called Feynman formula, i.e., a limit of multiple integrals against a gaussian measure as the multiplicity tends to infinity. It is also proved that the solution of the Cauchy problem depends continuously on the diffusion coefficient. A process reducing an approximate solution of an infinite-dimensional diffusion equation to finding a multiple integral of a real function of finitely many real variables is indicated.

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 2: User's guide

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Bui, Trong T.

1993-01-01

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the User's Guide, and describes the program's features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.

A GENERALIZED TWO-COMPONENT MODEL OF SOLAR WIND TURBULENCE AND AB INITIO DIFFUSION MEAN-FREE PATHS AND DRIFT LENGTHSCALES OF COSMIC RAYS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wiengarten, T.; Fichtner, H.; Kleimann, J.

2016-12-10

We extend a two-component model for the evolution of fluctuations in the solar wind plasma so that it is fully three-dimensional (3D) and also coupled self-consistently to the large-scale magnetohydrodynamic equations describing the background solar wind. The two classes of fluctuations considered are a high-frequency parallel-propagating wave-like piece and a low-frequency quasi-two-dimensional component. For both components, the nonlinear dynamics is dominanted by quasi-perpendicular spectral cascades of energy. Driving of the fluctuations by, for example, velocity shear and pickup ions is included. Numerical solutions to the new model are obtained using the Cronos framework, and validated against previous simpler models. Comparing results frommore » the new model with spacecraft measurements, we find improved agreement relative to earlier models that employ prescribed background solar wind fields. Finally, the new results for the wave-like and quasi-two-dimensional fluctuations are used to calculate ab initio diffusion mean-free paths and drift lengthscales for the transport of cosmic rays in the turbulent solar wind.« less

Guiding brine shrimp through mazes by solving reaction diffusion equations

NASA Astrophysics Data System (ADS)

Singal, Krishma; Fenton, Flavio

Excitable systems driven by reaction diffusion equations have been shown to not only find solutions to mazes but to also to find the shortest path between the beginning and the end of the maze. In this talk we describe how we can use the Fitzhugh-Nagumo model, a generic model for excitable media, to solve a maze by varying the basin of attraction of its two fixed points. We demonstrate how two dimensional mazes are solved numerically using a Java Applet and then accelerated to run in real time by using graphic processors (GPUs). An application of this work is shown by guiding phototactic brine shrimp through a maze solved by the algorithm. Once the path is obtained, an Arduino directs the shrimp through the maze using lights from LEDs placed at the floor of the Maze. This method running in real time could be eventually used for guiding robots and cars through traffic.

Diffusion Monte Carlo study of strongly interacting two-dimensional Fermi gases

DOE PAGES

Galea, Alexander; Dawkins, Hillary; Gandolfi, Stefano; ...

2016-02-01

Ultracold atomic Fermi gases have been a popular topic of research, with attention being paid recently to two-dimensional (2D) gases. In this work, we perform T=0 ab initio diffusion Monte Carlo calculations for a strongly interacting two-component Fermi gas confined to two dimensions. We first go over finite-size systems and the connection to the thermodynamic limit. After that, we illustrate pertinent 2D scattering physics and properties of the wave function. We then show energy results for the strong-coupling crossover, in between the Bose-Einstein condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes. Our energy results for the BEC-BCS crossover are parametrized to producemore » an equation of state, which is used to determine Tan's contact. We carry out a detailed comparison with other microscopic results. Lastly, we calculate the pairing gap for a range of interaction strengths in the strong coupling regime, following from variationally optimized many-body wave functions.« less

Analysis of sound propagation in ducts using the wave envelope concept

NASA Technical Reports Server (NTRS)

Baumeister, K. J.

1974-01-01

A finite difference formulation is presented for sound propagation in a rectangular two-dimensional duct without steady flow for plane wave input. Before the difference equations are formulated, the governing Helmholtz equation is first transformed to a form whose solution does not oscillate along the length of the duct. This transformation reduces the required number of grid points by an order of magnitude, and the number of grid points becomes independent of the sound frequency. Physically, the transformed pressure represents the amplitude of the conventional sound wave. Example solutions are presented for sound propagation in a one-dimensional straight hard-wall duct and in a two-dimensional straight soft-wall duct without steady flow. The numerical solutions show evidence of the existence along the duct wall of a developing acoustic pressure diffusion boundary layer which is similar in nature to the conventional viscous flow boundary layer. In order to better illustrate this concept, the wave equation and boundary conditions are written such that the frequency no longer appears explicitly in them. The frequency effects in duct propagation can be visualized solely as an expansion and stretching of the suppressor duct.

Numerical Solution of Time-Dependent Problems with a Fractional-Power Elliptic Operator

NASA Astrophysics Data System (ADS)

Vabishchevich, P. N.

2018-03-01

A time-dependent problem in a bounded domain for a fractional diffusion equation is considered. The first-order evolution equation involves a fractional-power second-order elliptic operator with Robin boundary conditions. A finite-element spatial approximation with an additive approximation of the operator of the problem is used. The time approximation is based on a vector scheme. The transition to a new time level is ensured by solving a sequence of standard elliptic boundary value problems. Numerical results obtained for a two-dimensional model problem are presented.

On Hilbert-Schmidt norm convergence of Galerkin approximation for operator Riccati equations

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1988-01-01

An abstract approximation framework for the solution of operator algebraic Riccati equations is developed. The approach taken is based on a formulation of the Riccati equation as an abstract nonlinear operator equation on the space of Hilbert-Schmidt operators. Hilbert-Schmidt norm convergence of solutions to generic finite dimensional Galerkin approximations to the Riccati equation to the solution of the original infinite dimensional problem is argued. The application of the general theory is illustrated via an operator Riccati equation arising in the linear-quadratic design of an optimal feedback control law for a 1-D heat/diffusion equation. Numerical results demonstrating the convergence of the associated Hilbert-Schmidt kernels are included.

Three dimensional flow field inside compressor rotor, including blade boundary layers

NASA Technical Reports Server (NTRS)

Galmes, J. M.; Pouagere, M.; Lakshminarayana, B.

1982-01-01

The Reynolds stress equation, pressure strain correlation, and dissipative terms and diffusion are discussed in relation to turbulence modelling using the Reynolds stress model. Algebraic modeling of Reynolds stresses and calculation of the boundary layer over an axial cylinder are examined with regards to the kinetic energy model for turbulence modelling. The numerical analysis of blade and hub wall boundary layers, and an experimental study of rotor blade boundary layer in an axial flow compressor rotor are discussed. The Patankar-Spalding numerical method for two dimensional boundary layers is included.

Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach

NASA Astrophysics Data System (ADS)

Kim, Changho; Nonaka, Andy; Bell, John B.; Garcia, Alejandro L.; Donev, Aleksandar

2017-03-01

We develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuating hydrodynamics (FHD). For hydrodynamic systems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poisson fluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamic fluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. By comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.

Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach

DOE PAGES

Kim, Changho; Nonaka, Andy; Bell, John B.; ...

2017-03-24

Here, we develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuating hydrodynamics (FHD). For hydrodynamic systems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poisson fluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules,more » to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamic fluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. Furthermore, by comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.« less

A new boundary scheme for simulation of gas flow in kerogen pores with considering surface diffusion effect

NASA Astrophysics Data System (ADS)

Wang, Lingquan; Zeng, Zhong; Zhang, Liangqi; Qiao, Long; Zhang, Yi; Lu, Yiyu

2018-04-01

Navier-Stokes (NS) equations with no-slip boundary conditions fail to realistically describe micro-flows with considering nanoscale phenomena. Particularly, in kerogen pores, slip-flow and surface diffusion are important. In this study, we propose a new slip boundary scheme for the lattice Boltzmann (LB) method through the non-equilibrium extrapolation scheme to simulate the slip-flow considering surface diffusion effect. Meanwhile, the second-order slip velocity can be taken into account. The predicted characteristics in a two-dimensional micro-flow, including slip-velocity, velocity distribution along the flow direction with/without surface diffusion are present. The results in this study are compared with available analytical and reference results, and good agreements are achieved.

A direct numerical method for predicting concentration profiles in a turbulent boundary layer over a flat plate. M.S. Thesis

NASA Technical Reports Server (NTRS)

Dow, J. W.

1972-01-01

A numerical solution of the turbulent mass transport equation utilizing the concept of eddy diffusivity is presented as an efficient method of investigating turbulent mass transport in boundary layer type flows. A FORTRAN computer program is used to study the two-dimensional diffusion of ammonia, from a line source on the surface, into a turbulent boundary layer over a flat plate. The results of the numerical solution are compared with experimental data to verify the results of the solution. Several other solutions to diffusion problems are presented to illustrate the versatility of the computer program and to provide some insight into the problem of mass diffusion as a whole.

Diffusion Forecasting Model with Basis Functions from QR-Decomposition

NASA Astrophysics Data System (ADS)

Harlim, John; Yang, Haizhao

2018-06-01

The diffusion forecasting is a nonparametric approach that provably solves the Fokker-Planck PDE corresponding to Itô diffusion without knowing the underlying equation. The key idea of this method is to approximate the solution of the Fokker-Planck equation with a discrete representation of the shift (Koopman) operator on a set of basis functions generated via the diffusion maps algorithm. While the choice of these basis functions is provably optimal under appropriate conditions, computing these basis functions is quite expensive since it requires the eigendecomposition of an N× N diffusion matrix, where N denotes the data size and could be very large. For large-scale forecasting problems, only a few leading eigenvectors are computationally achievable. To overcome this computational bottleneck, a new set of basis functions constructed by orthonormalizing selected columns of the diffusion matrix and its leading eigenvectors is proposed. This computation can be carried out efficiently via the unpivoted Householder QR factorization. The efficiency and effectiveness of the proposed algorithm will be shown in both deterministically chaotic and stochastic dynamical systems; in the former case, the superiority of the proposed basis functions over purely eigenvectors is significant, while in the latter case forecasting accuracy is improved relative to using a purely small number of eigenvectors. Supporting arguments will be provided on three- and six-dimensional chaotic ODEs, a three-dimensional SDE that mimics turbulent systems, and also on the two spatial modes associated with the boreal winter Madden-Julian Oscillation obtained from applying the Nonlinear Laplacian Spectral Analysis on the measured Outgoing Longwave Radiation.

Diffusion Forecasting Model with Basis Functions from QR-Decomposition

NASA Astrophysics Data System (ADS)

Harlim, John; Yang, Haizhao

2017-12-01

The diffusion forecasting is a nonparametric approach that provably solves the Fokker-Planck PDE corresponding to Itô diffusion without knowing the underlying equation. The key idea of this method is to approximate the solution of the Fokker-Planck equation with a discrete representation of the shift (Koopman) operator on a set of basis functions generated via the diffusion maps algorithm. While the choice of these basis functions is provably optimal under appropriate conditions, computing these basis functions is quite expensive since it requires the eigendecomposition of an N× N diffusion matrix, where N denotes the data size and could be very large. For large-scale forecasting problems, only a few leading eigenvectors are computationally achievable. To overcome this computational bottleneck, a new set of basis functions constructed by orthonormalizing selected columns of the diffusion matrix and its leading eigenvectors is proposed. This computation can be carried out efficiently via the unpivoted Householder QR factorization. The efficiency and effectiveness of the proposed algorithm will be shown in both deterministically chaotic and stochastic dynamical systems; in the former case, the superiority of the proposed basis functions over purely eigenvectors is significant, while in the latter case forecasting accuracy is improved relative to using a purely small number of eigenvectors. Supporting arguments will be provided on three- and six-dimensional chaotic ODEs, a three-dimensional SDE that mimics turbulent systems, and also on the two spatial modes associated with the boreal winter Madden-Julian Oscillation obtained from applying the Nonlinear Laplacian Spectral Analysis on the measured Outgoing Longwave Radiation.

Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations.

PubMed

Feng, Bao-Feng; Malomed, Boris A; Kawahara, Takuji

2002-11-01

We present a two-dimensional (2D) generalization of the stabilized Kuramoto-Sivashinsky system, based on the Kadomtsev-Petviashvili (KP) equation including dissipation of the generic [Newell-Whitehead-Segel (NWS)] type and gain. The system directly applies to the description of gravity-capillary waves on the surface of a liquid layer flowing down an inclined plane, with a surfactant diffusing along the layer's surface. Actually, the model is quite general, offering a simple way to stabilize nonlinear media, combining the weakly 2D dispersion of the KP type with gain and NWS dissipation. Other applications are internal waves in multilayer fluids flowing down an inclined plane, double-front flames in gaseous mixtures, etc. Parallel to this weakly 2D model, we also introduce and study a semiphenomenological one, whose dissipative terms are isotropic, rather than of the NWS type, in order to check if qualitative results are sensitive to the exact form of the lossy terms. The models include an additional linear equation of the advection-diffusion type, linearly coupled to the main KP-NWS equation. The extra equation provides for stability of the zero background in the system, thus opening a way for the existence of stable localized pulses. We focus on the most interesting case, when the dispersive part of the system is of the KP-I type, which corresponds, e.g., to capillary waves, and makes the existence of completely localized 2D pulses possible. Treating the losses and gain as small perturbations and making use of the balance equation for the field momentum, we find that the equilibrium between the gain and losses may select two steady-state solitons from their continuous family existing in the absence of the dissipative terms (the latter family is found in an exact analytical form, and is numerically demonstrated to be stable). The selected soliton with the larger amplitude is expected to be stable. Direct simulations completely corroborate the analytical predictions, for both the physical and phenomenological models.

A Comparison of Some Difference Schemes for a Parabolic Problem of Zero-Coupon Bond Pricing

NASA Astrophysics Data System (ADS)

Chernogorova, Tatiana; Vulkov, Lubin

2009-11-01

This paper describes a comparison of some numerical methods for solving a convection-diffusion equation subjected by dynamical boundary conditions which arises in the zero-coupon bond pricing. The one-dimensional convection-diffusion equation is solved by using difference schemes with weights including standard difference schemes as the monotone Samarskii's scheme, FTCS and Crank-Nicolson methods. The schemes are free of spurious oscillations and satisfy the positivity and maximum principle as demanded for the financial and diffusive solution. Numerical results are compared with analytical solutions.

Diffusive and localization behavior of electromagnetic waves in a two-dimensional random medium

NASA Astrophysics Data System (ADS)

Wang, Ken Kang-Hsin; Ye, Zhen

2003-10-01

In this paper, we discuss the transport phenomena of electromagnetic waves in a two-dimensional random system which is composed of arrays of electrical dipoles, following the model presented earlier by Erdogan et al. [J. Opt. Soc. Am. B 10, 391 (1993)]. A set of self-consistent equations is presented, accounting for the multiple scattering in the system, and is then solved numerically. A strong localization regime is discovered in the frequency domain. The transport properties within, near the edge of, and nearly outside the localization regime are investigated for different parameters such as filling factor and system size. The results show that within the localization regime, waves are trapped near the transmitting source. Meanwhile, the diffusive waves follow an intuitive but expected picture. That is, they increase with traveling path as more and more random scattering incurs, followed by a saturation, then start to decay exponentially when the travelling path is large enough, signifying the localization effect. For the cases where the frequencies are near the boundary of or outside the localization regime, the results of diffusive waves are compared with the diffusion approximation, showing less encouraging agreement as in other systems [Asatryan et al., Phys. Rev. E 67, 036605 (2003)].

Computation for Electromigration in Interconnects of Microelectronic Devices

NASA Astrophysics Data System (ADS)

Averbuch, Amir; Israeli, Moshe; Ravve, Igor; Yavneh, Irad

2001-03-01

Reliability and performance of microelectronic devices depend to a large extent on the resistance of interconnect lines. Voids and cracks may occur in the interconnects, causing a severe increase in the total resistance and even open circuits. In this work we analyze void motion and evolution due to surface diffusion effects and applied external voltage. The interconnects under consideration are three-dimensional (sandwich) constructs made of a very thin metal film of possibly variable thickness attached to a substrate of nonvanishing conductance. A two-dimensional level set approach was applied to study the dynamics of the moving (assumed one-dimensional) boundary of a void in the metal film. The level set formulation of an electromigration and diffusion model results in a fourth-order nonlinear (two-dimensional) time-dependent PDE. This equation was discretized by finite differences on a regular grid in space and a Runge-Kutta integration scheme in time, and solved simultaneously with a second-order static elliptic PDE describing the electric potential distribution throughout the interconnect line. The well-posed three-dimensional problem for the potential was approximated via singular perturbations, in the limit of small aspect ratio, by a two-dimensional elliptic equation with variable coefficients describing the combined local conductivity of metal and substrate (which is allowed to vary in time and space). The difference scheme for the elliptic PDE was solved by a multigrid technique at each time step. Motion of voids in both weak and strong electric fields was examined, and different initial void configurations were considered, including circles, ellipses, polygons with rounded corners, a butterfly, and long grooves. Analysis of the void behavior and its influence on the resistance gives the circuit designer a tool for choosing the proper parameters of an interconnect (width-to-length ratio, properties of the line material, conductivity of the underlayer, etc.).

Fick's second law transformed: one path to cloaking in mass diffusion.

PubMed

Guenneau, S; Puvirajesinghe, T M

2013-06-06

Here, we adapt the concept of transformational thermodynamics, whereby the flux of temperature is controlled via anisotropic heterogeneous diffusivity, for the diffusion and transport of mass concentration. The n-dimensional, time-dependent, anisotropic heterogeneous Fick's equation is considered, which is a parabolic partial differential equation also applicable to heat diffusion, when convection occurs, for example, in fluids. This theory is illustrated with finite-element computations for a liposome particle surrounded by a cylindrical multi-layered cloak in a water-based environment, and for a spherical multi-layered cloak consisting of layers of fluid with an isotropic homogeneous diffusivity, deduced from an effective medium approach. Initial potential applications could be sought in bioengineering.

Study on low intensity aeration oxygenation model and optimization for shallow water

NASA Astrophysics Data System (ADS)

Chen, Xiao; Ding, Zhibin; Ding, Jian; Wang, Yi

2018-02-01

Aeration/oxygenation is an effective measure to improve self-purification capacity in shallow water treatment while high energy consumption, high noise and expensive management refrain the development and the application of this process. Based on two-film theory, the theoretical model of the three-dimensional partial differential equation of aeration in shallow water is established. In order to simplify the equation, the basic assumptions of gas-liquid mass transfer in vertical direction and concentration diffusion in horizontal direction are proposed based on engineering practice and are tested by the simulation results of gas holdup which are obtained by simulating the gas-liquid two-phase flow in aeration tank under low-intensity condition. Based on the basic assumptions and the theory of shallow permeability, the model of three-dimensional partial differential equations is simplified and the calculation model of low-intensity aeration oxygenation is obtained. The model is verified through comparing the aeration experiment. Conclusions as follows: (1)The calculation model of gas-liquid mass transfer in vertical direction and concentration diffusion in horizontal direction can reflect the process of aeration well; (2) Under low-intensity conditions, the long-term aeration and oxygenation is theoretically feasible to enhance the self-purification capacity of water bodies; (3) In the case of the same total aeration intensity, the effect of multipoint distributed aeration on the diffusion of oxygen concentration in the horizontal direction is obvious; (4) In the shallow water treatment, reducing the volume of aeration equipment with the methods of miniaturization, array, low-intensity, mobilization to overcome the high energy consumption, large size, noise and other problems can provide a good reference.

Vectorized and multitasked solution of the few-group neutron diffusion equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zee, S.K.; Turinsky, P.J.; Shayer, Z.

1989-03-01

A numerical algorithm with parallelism was used to solve the two-group, multidimensional neutron diffusion equations on computers characterized by shared memory, vector pipeline, and multi-CPU architecture features. Specifically, solutions were obtained on the Cray X/MP-48, the IBM-3090 with vector facilities, and the FPS-164. The material-centered mesh finite difference method approximation and outer-inner iteration method were employed. Parallelism was introduced in the inner iterations using the cyclic line successive overrelaxation iterative method and solving in parallel across lines. The outer iterations were completed using the Chebyshev semi-iterative method that allows parallelism to be introduced in both space and energy groups. Formore » the three-dimensional model, power, soluble boron, and transient fission product feedbacks were included. Concentrating on the pressurized water reactor (PWR), the thermal-hydraulic calculation of moderator density assumed single-phase flow and a closed flow channel, allowing parallelism to be introduced in the solution across the radial plane. Using a pinwise detail, quarter-core model of a typical PWR in cycle 1, for the two-dimensional model without feedback the measured million floating point operations per second (MFLOPS)/vector speedups were 83/11.7. 18/2.2, and 2.4/5.6 on the Cray, IBM, and FPS without multitasking, respectively. Lower performance was observed with a coarser mesh, i.e., shorter vector length, due to vector pipeline start-up. For an 18 x 18 x 30 (x-y-z) three-dimensional model with feedback of the same core, MFLOPS/vector speedups of --61/6.7 and an execution time of 0.8 CPU seconds on the Cray without multitasking were measured. Finally, using two CPUs and the vector pipelines of the Cray, a multitasking efficiency of 81% was noted for the three-dimensional model.« less

Adaptive macro finite elements for the numerical solution of monodomain equations in cardiac electrophysiology.

PubMed

Heidenreich, Elvio A; Ferrero, José M; Doblaré, Manuel; Rodríguez, José F

2010-07-01

Many problems in biology and engineering are governed by anisotropic reaction-diffusion equations with a very rapidly varying reaction term. This usually implies the use of very fine meshes and small time steps in order to accurately capture the propagating wave while avoiding the appearance of spurious oscillations in the wave front. This work develops a family of macro finite elements amenable for solving anisotropic reaction-diffusion equations with stiff reactive terms. The developed elements are incorporated on a semi-implicit algorithm based on operator splitting that includes adaptive time stepping for handling the stiff reactive term. A linear system is solved on each time step to update the transmembrane potential, whereas the remaining ordinary differential equations are solved uncoupled. The method allows solving the linear system on a coarser mesh thanks to the static condensation of the internal degrees of freedom (DOF) of the macroelements while maintaining the accuracy of the finer mesh. The method and algorithm have been implemented in parallel. The accuracy of the method has been tested on two- and three-dimensional examples demonstrating excellent behavior when compared to standard linear elements. The better performance and scalability of different macro finite elements against standard finite elements have been demonstrated in the simulation of a human heart and a heterogeneous two-dimensional problem with reentrant activity. Results have shown a reduction of up to four times in computational cost for the macro finite elements with respect to equivalent (same number of DOF) standard linear finite elements as well as good scalability properties.

Isostable reduction with applications to time-dependent partial differential equations.

PubMed

Wilson, Dan; Moehlis, Jeff

2016-07-01

Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.

Cross-stream diffusion under pressure-driven flow in microchannels with arbitrary aspect ratios: a phase diagram study using a three-dimensional analytical model

PubMed Central

Song, Hongjun; Wang, Yi; Pant, Kapil

2011-01-01

This article presents a three-dimensional analytical model to investigate cross-stream diffusion transport in rectangular microchannels with arbitrary aspect ratios under pressure-driven flow. The Fourier series solution to the three-dimensional convection–diffusion equation is obtained using a double integral transformation method and associated eigensystem calculation. A phase diagram derived from the dimensional analysis is presented to thoroughly interrogate the characteristics in various transport regimes and examine the validity of the model. The analytical model is verified against both experimental and numerical models in terms of the concentration profile, diffusion scaling law, and mixing efficiency with excellent agreement (with <0.5% relative error). Quantitative comparison against other prior analytical models in extensive parameter space is also performed, which demonstrates that the present model accommodates much broader transport regimes with significantly enhanced applicability. PMID:22247719

Cross-stream diffusion under pressure-driven flow in microchannels with arbitrary aspect ratios: a phase diagram study using a three-dimensional analytical model.

PubMed

Song, Hongjun; Wang, Yi; Pant, Kapil

2012-01-01

This article presents a three-dimensional analytical model to investigate cross-stream diffusion transport in rectangular microchannels with arbitrary aspect ratios under pressure-driven flow. The Fourier series solution to the three-dimensional convection-diffusion equation is obtained using a double integral transformation method and associated eigensystem calculation. A phase diagram derived from the dimensional analysis is presented to thoroughly interrogate the characteristics in various transport regimes and examine the validity of the model. The analytical model is verified against both experimental and numerical models in terms of the concentration profile, diffusion scaling law, and mixing efficiency with excellent agreement (with <0.5% relative error). Quantitative comparison against other prior analytical models in extensive parameter space is also performed, which demonstrates that the present model accommodates much broader transport regimes with significantly enhanced applicability.

Effects of the approximations of light propagation on quantitative photoacoustic tomography using two-dimensional photon diffusion equation and linearization

NASA Astrophysics Data System (ADS)

Okawa, Shinpei; Hirasawa, Takeshi; Kushibiki, Toshihiro; Ishihara, Miya

2017-12-01

Quantitative photoacoustic tomography (QPAT) employing a light propagation model will play an important role in medical diagnoses by quantifying the concentration of hemoglobin or a contrast agent. However, QPAT by the light propagation model with the three-dimensional (3D) radiative transfer equation (RTE) requires a huge computational load in the iterative forward calculations involved in the updating process to reconstruct the absorption coefficient. The approximations of the light propagation improve the efficiency of the image reconstruction for the QPAT. In this study, we compared the 3D/two-dimensional (2D) photon diffusion equation (PDE) approximating 3D RTE with the Monte Carlo simulation based on 3D RTE. Then, the errors in a 2D PDE-based linearized image reconstruction caused by the approximations were quantitatively demonstrated and discussed in the numerical simulations. It was clearly observed that the approximations affected the reconstructed absorption coefficient. The 2D PDE-based linearized algorithm succeeded in the image reconstruction of the region with a large absorption coefficient in the 3D phantom. The value reconstructed in the phantom experiment agreed with that in the numerical simulation, so that it was validated that the numerical simulation of the image reconstruction predicted the relationship between the true absorption coefficient of the target in the 3D medium and the reconstructed value with the 2D PDE-based linearized algorithm. Moreover, the the true absorption coefficient in 3D medium was estimated from the 2D reconstructed image on the basis of the prediction by the numerical simulation. The estimation was successful in the phantom experiment, although some limitations were revealed.

A one-dimensional heat-transport model for conduit flow in karst aquifers

USGS Publications Warehouse

Long, Andrew J.; Gilcrease, P.C.

2009-01-01

A one-dimensional heat-transport model for conduit flow in karst aquifers is presented as an alternative to two or three-dimensional distributed-parameter models, which are data intensive and require knowledge of conduit locations. This model can be applied for cases where water temperature in a well or spring receives all or part of its water from a phreatic conduit. Heat transport in the conduit is simulated by using a physically-based heat-transport equation that accounts for inflow of diffuse flow from smaller openings and fissures in the surrounding aquifer during periods of low recharge. Additional diffuse flow that is within the zone of influence of the well or spring but has not interacted with the conduit is accounted for with a binary mixing equation to proportion these different water sources. The estimation of this proportion through inverse modeling is useful for the assessment of contaminant vulnerability and well-head or spring protection. The model was applied to 7 months of continuous temperature data for a sinking stream that recharges a conduit and a pumped well open to the Madison aquifer in western South Dakota. The simulated conduit-flow fraction to the well ranged from 2% to 31% of total flow, and simulated conduit velocity ranged from 44 to 353 m/d.

Multidimensional flamelet-generated manifolds for partially premixed combustion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nguyen, Phuc-Danh; Vervisch, Luc; Subramanian, Vallinayagam

2010-01-15

Flamelet-generated manifolds have been restricted so far to premixed or diffusion flame archetypes, even though the resulting tables have been applied to nonpremixed and partially premixed flame simulations. By using a projection of the full set of mass conservation species balance equations into a restricted subset of the composition space, unsteady multidimensional flamelet governing equations are derived from first principles, under given hypotheses. During the projection, as in usual one-dimensional flamelets, the tangential strain rate of scalar isosurfaces is expressed in the form of the scalar dissipation rates of the control parameters of the multidimensional flamelet-generated manifold (MFM), which ismore » tested in its five-dimensional form for partially premixed combustion, with two composition space directions and three scalar dissipation rates. It is shown that strain-rate-induced effects can hardly be fully neglected in chemistry tabulation of partially premixed combustion, because of fluxes across iso-equivalence-ratio and iso-progress-of-reaction surfaces. This is illustrated by comparing the 5D flamelet-generated manifold with one-dimensional premixed flame and unsteady strained diffusion flame composition space trajectories. The formal links between the asymptotic behavior of MFM and stratified flame, weakly varying partially premixed front, triple-flame, premixed and nonpremixed edge flames are also evidenced. (author)« less

Border-crossing model for the diffusive coarsening of two-dimensional and quasi-two-dimensional wet foams

NASA Astrophysics Data System (ADS)

Schimming, C. D.; Durian, D. J.

2017-09-01

For dry foams, the transport of gas from small high-pressure bubbles to large low-pressure bubbles is dominated by diffusion across the thin soap films separating neighboring bubbles. For wetter foams, the film areas become smaller as the Plateau borders and vertices inflate with liquid. So-called "border-blocking" models can explain some features of wet-foam coarsening based on the presumption that the inflated borders totally block the gas flux; however, this approximation dramatically fails in the wet or unjamming limit where the bubbles become close-packed spheres and coarsening proceeds even though there are no films. Here, we account for the ever-present border-crossing flux by a new length scale defined by the average gradient of gas concentration inside the borders. We compute that it is proportional to the geometric average of film and border thicknesses, and we verify this scaling by numerical solution of the diffusion equation. We similarly consider transport across inflated vertices and surface Plateau borders in quasi-two-dimensional foams. And we show how the d A /d t =K0(n -6 ) von Neumann law is modified by the appearance of terms that depend on bubble size and shape as well as the concentration gradient length scales. Finally, we use the modified von Neumann law to compute the growth rate of the average bubble area, which is not constant.

Analytic model for ring pattern formation by bacterial swarmers

NASA Astrophysics Data System (ADS)

Arouh, Scott

2001-03-01

We analyze a model proposed by Medvedev, Kaper, and Kopell (the MKK model) for ring formation in two-dimensional bacterial colonies of Proteus mirabilis. We correct the model to formally include a feature crucial of the ring generation mechanism: a bacterial density threshold to the nonlinear diffusivity of the MKK model. We numerically integrate the model equations, and observe the logarithmic profiles of the bacterial densities near the front. These lead us to define a consolidation front distinct from the colony radius. We find that this consolidation front propagates outward toward the colony radius with a nearly constant velocity. We then implement the corrected MKK equations in two dimensions and compare our results with biological experiment. Our numerical results indicate that the two-dimensional corrected MKK model yields smooth (rather than branched) rings, and that colliding colonies merge if grown in phase but not if grown out of phase. We also introduce a model, based on coupling the MKK model to a nutrient field, for simulating experimentally observed branched rings.

Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation

NASA Astrophysics Data System (ADS)

Kamiya, Ryo; Kanki, Masataka; Mase, Takafumi; Tokihiro, Tetsuji

2017-01-01

We introduce a so-called coprimeness-preserving non-integrable extension to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equations defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence.

Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations

NASA Astrophysics Data System (ADS)

Fouxon, Itzhak; Denisov, Sergey; Zaburdaev, Vasily; Barkai, Eli

2017-04-01

We consider super-diffusive Lévy walks in d≥slant 2 dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called ‘infinite density’, describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d  >  1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.

The development of a three-dimensional partially elliptic flow computer program for combustor research

NASA Technical Reports Server (NTRS)

Pan, Y. S.

1978-01-01

A three dimensional, partially elliptic, computer program was developed. Without requiring three dimensional computer storage locations for all flow variables, the partially elliptic program is capable of predicting three dimensional combustor flow fields with large downstream effects. The program requires only slight increase of computer storage over the parabolic flow program from which it was developed. A finite difference formulation for a three dimensional, fully elliptic, turbulent, reacting, flow field was derived. Because of the negligible diffusion effects in the main flow direction in a supersonic combustor, the set of finite-difference equations can be reduced to a partially elliptic form. Only the pressure field was governed by an elliptic equation and requires three dimensional storage; all other dependent variables are governed by parabolic equations. A numerical procedure which combines a marching integration scheme with an iterative scheme for solving the elliptic pressure was adopted.

TEMPEST. Transient 3-D Thermal-Hydraulic

DOE Office of Scientific and Technical Information (OSTI.GOV)

Eyler, L.L.

TEMPEST is a transient, three-dimensional, hydrothermal program that is designed to analyze a range of coupled fluid dynamic and heat transfer systems of particular interest to the Fast Breeder Reactor (FBR) thermal-hydraulic design community. The full three-dimensional, time-dependent equations of motion, continuity, and heat transport are solved for either laminar or turbulent fluid flow, including heat diffusion and generation in both solid and liquid materials. The equations governing mass, momentum, and energy conservation for incompressible flows and small density variations (Boussinesq approximation) are solved using finite-difference techniques. Analyses may be conducted in either cylindrical or Cartesian coordinate systems. Turbulence ismore » treated using a two-equation model. Two auxiliary plotting programs, SEQUEL and MANPLOT, for use with TEMPEST output are included. SEQUEL may be operated in batch or interactive mode; it generates data required for vector plots, contour plots of scalar quantities, line plots, grid and boundary plots, and time-history plots. MANPLOT reads the SEQUEL-generated data and creates the hardcopy plots. TEMPEST can be a valuable hydrothermal design analysis tool in areas outside the intended FBR thermal-hydraulic design community.« less

Application of the method of lines for solutions of the Navier-Stokes equations using a nonuniform grid distribution

NASA Technical Reports Server (NTRS)

Abolhassani, J. S.; Tiwari, S. N.

1983-01-01

The feasibility of the method of lines for solutions of physical problems requiring nonuniform grid distributions is investigated. To attain this, it is also necessary to investigate the stiffness characteristics of the pertinent equations. For specific applications, the governing equations considered are those for viscous, incompressible, two dimensional and axisymmetric flows. These equations are transformed from the physical domain having a variable mesh to a computational domain with a uniform mesh. The two governing partial differential equations are the vorticity and stream function equations. The method of lines is used to solve the vorticity equation and the successive over relaxation technique is used to solve the stream function equation. The method is applied to three laminar flow problems: the flow in ducts, curved-wall diffusers, and a driven cavity. Results obtained for different flow conditions are in good agreement with available analytical and numerical solutions. The viability and validity of the method of lines are demonstrated by its application to Navier-Stokes equations in the physical domain having a variable mesh.

Forward and back diffusion through argillaceous formations

NASA Astrophysics Data System (ADS)

Yang, Minjune; Annable, Michael D.; Jawitz, James W.

2017-05-01

The exchange of solutes between aquifers and lower-permeability argillaceous formations is of considerable interest for solute and contaminant fate and transport. We present a synthesis of analytical solutions for solute diffusion between aquifers and single aquitard systems, validated in well-controlled experiments, and applied to several data sets from laboratory and field-scale problems with diffusion time and length scales ranging from 10-2 to 108 years and 10-2 to 102 m. One-dimensional diffusion models were applied using the method of images to consider the general cases of a finite aquitard bounded by two aquifers at the top and bottom, or a semiinfinite aquitard bounded by an aquifer. The simpler semiinfinite equations are appropriate for all domains with dimensionless relative diffusion length, ZD < 0.7. At dimensionless length scales above this threshold, application of semiinfinite equations to aquitards of finite thickness leads to increasing errors and solutions based on the method of images are required. Measured resident solute concentration profiles in aquitards and flux-averaged solute concentrations in surrounding aquifers were accurately modeled by appropriately accounting for generalized dynamic aquifer-aquitard boundary conditions, including concentration gradient reversals. Dimensionless diffusion length scales were used to illustrate the transferability of these relatively simple models to physical systems with dimensions that spanned 10 orders of magnitude. The results of this study offer guidance on the application of a simplified analytical approach to environmentally important layered problems with one or two diffusion interfaces.

Some remarks on relativistic diffusion and the spectral dimension criterion

NASA Astrophysics Data System (ADS)

Muniz, C. R.; Cunha, M. S.; Filho, R. N. Costa; Bezerra, V. B.

2015-01-01

The spectral dimension ds for high energies is calculated using the Relativistic Schrödinger Equation Analytically Continued (RSEAC) instead of the so-called Telegraph's equation (TE), in both ultraviolet (UV) and infrared (IR) regimens. Regarding the TE, the recent literature presents difficulties related to its stochastic derivation and interpretation, advocating the use of the RSEAC to properly describe the relativistic diffusion phenomena. Taking into account that the Lorentz symmetry is broken in UV regime at Lifshitz point, we show that there exists a degeneracy in very high energies, meaning that both the RSEAC and TE correctly describe the diffusion processes at these energy scales, at least under the spectral dimension criterion. In fact, both the equations yield the same result, namely, ds=2 , a dimensional reduction that is compatible with several theories of quantum gravity. This result is reached even when one takes into account a cosmological model, as for example, the de Sitter universe. On the other hand, in the IR regimen, such degeneracy is lifted in favor of the approach via TE, due to the fact that only this equation provides the correct value for ds, which is equal to the actual number of spacetime dimensions, i.e., ds=4 , while RSEAC yields ds=3 , so that a diffusing particle described by this method experiences a three-dimensional spacetime.

Spatiotemporal pattern formation in a prey-predator model under environmental driving forces

NASA Astrophysics Data System (ADS)

Sirohi, Anuj Kumar; Banerjee, Malay; Chakraborti, Anirban

2015-09-01

Many existing studies on pattern formation in the reaction-diffusion systems rely on deterministic models. However, environmental noise is often a major factor which leads to significant changes in the spatiotemporal dynamics. In this paper, we focus on the spatiotemporal patterns produced by the predator-prey model with ratio-dependent functional response and density dependent death rate of predator. We get the reaction-diffusion equations incorporating the self-diffusion terms, corresponding to random movement of the individuals within two dimensional habitats, into the growth equations for the prey and predator population. In order to have the noise added model, small amplitude heterogeneous perturbations to the linear intrinsic growth rates are introduced using uncorrelated Gaussian white noise terms. For the noise added system, we then observe spatial patterns for the parameter values lying outside the Turing instability region. With thorough numerical simulations we characterize the patterns corresponding to Turing and Turing-Hopf domain and study their dependence on different system parameters like noise-intensity, etc.

Numberical Solution to Transient Heat Flow Problems

ERIC Educational Resources Information Center

Kobiske, Ronald A.; Hock, Jeffrey L.

1973-01-01

Discusses the reduction of the one- and three-dimensional diffusion equation to the difference equation and its stability, convergence, and heat-flow applications under different boundary conditions. Indicates the usefulness of this presentation for beginning students of physics and engineering as well as college teachers. (CC)

An improved design method and experimental performance of two dimensional curved wall diffusers

NASA Technical Reports Server (NTRS)

Yang, T.; Hudson, W. G.; El-Nashar, A. M.

1972-01-01

A computer design program was developed to incorporate the suction slots in solving the potential flow equations with prescribed boundary conditions. Using the contour generated from this program two Griffith diffusers were tested having area ratios AR = 3 and 4. The inlet Reynolds number ranged from 600,000 to 7 million. It was found that the slot suction required for metastable operation depends on the sidewall suction applied. Values of slot suction of 8% of the inlet flow rate was required for AR = 4 with metastable condition, provided that enough sidewall suction was applied. For AR = 3, the values of slot suction was about 25% lower than those required for AR = 4. For nearly all unseparated test runs, the effectiveness was 100% and the exit flow was uniform. In addition to the Griffith diffusers, dump and cusp diffusers of comparable area ratios were built and tested. The results obtained from these diffusers were compared with those of the Griffith diffusers. Flow separation occurred in all test runs with the dump and cusp diffusers.

Calculation of the flow field in supersonic mixed-compression inlets at angle of attack using the three-dimensional method of characteristics with discrete shock wave fitting

NASA Technical Reports Server (NTRS)

Vadyak, J.; Hoffman, J. D.

1978-01-01

The influence of molecular transport is included in the computation by treating viscous and thermal diffusion terms in the governing partial differential equations as correction terms in the method of characteristics scheme. The development of a production type computer program is reported which is capable of calculating the flow field in a variety of axisymmetric mixed-compression aircraft inlets. The results agreed well with those produced by the two-dimensional method characteristics when axisymmetric flow fields are computed. For three-dimensional flow fields, the results agree well with experimental data except in regions of high viscous interaction and boundary layer removal.

Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion

NASA Astrophysics Data System (ADS)

Sánchez-Vizuet, Tonatiuh; Cerfon, Antoine J.

2018-02-01

We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo-spectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials called Maxwell polynomials. Taking a one-dimensional equation describing energy diffusion due to Fokker-Planck collisions with a Maxwell-Boltzmann background distribution as the test bench for the performance of the scheme, we find that Maxwell based discretizations outperform other commonly used schemes in most situations, often by orders of magnitude. This provides a strong motivation for their use in high-dimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a non-modal time stepping instability in their most straightforward implementation, so that special care must be given to the discrete representation of the linear operators in order to benefit from the advantages provided by Maxwell polynomials.

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Bui, Trong T.

1993-01-01

A computer code called Proteus 3D has been developed to solve the three dimensional, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort has been to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation have been emphasized. The governing equations are solved in generalized non-orthogonal body-fitted coordinates by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the Analysis Description, and presents the equations and solution procedure. It describes in detail the governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models.

Magnetic Field Line Random Walk in Arbitrarily Stretched Isotropic Turbulence

NASA Astrophysics Data System (ADS)

Wongpan, P.; Ruffolo, D.; Matthaeus, W. H.; Rowlands, G.

2006-12-01

Many types of space and laboratory plasmas involve turbulent fluctuations with an approximately uniform mean magnetic field B_0, and the field line random walk plays an important role in guiding particle motions. Much of the relevant literature concerns isotropic turbulence, and has mostly been perturbative, i.e., for small fluctuations, or based on numerical simulations for specific conditions. On the other hand, solar wind turbulence is apparently anisotropic, and has been modeled as a sum of idealized two-dimensional and one dimensional (slab) components, but with the deficiency of containing no oblique wave vectors. In the present work, we address the above issues with non-perturbative analytic calculations of diffusive field line random walks for unpolarized, arbitrarily stretched isotropic turbulence, including the limits of nearly one-dimensional (highly stretched) and nearly two-dimensional (highly squashed) turbulence. We develop implicit analytic formulae for the diffusion coefficients D_x and D_z, two coupled integral equations in which D_x and D_z appear inside 3-dimensional integrals over all k-space, are solved numerically with the aid of Mathematica routines for specific cases. We can vary the parameters B0 and β, the stretching along z for constant turbulent energy. Furthermore, we obtain analytic closed-form solutions in all extreme cases. We obtain 0.54 < D_z/D_x < 2, indicating an approximately isotropic random walk even for very anisotropic (unpolarized) turbulence, a surprising result. For a given β, the diffusion coefficient vs. B0 can be described by a Padé approximant. We find quasilinear behavior at high B0 and percolative behavior at low B_0. Partially supported by a Sritrangthong Scholarship from the Faculty of Science, Mahidol University; the Thailand Research Fund; NASA Grant NNG05GG83G; and Thailand's Commission for Higher Education.

Two-dimensional evolution equation of finite-amplitude internal gravity waves in a uniformly stratified fluid

PubMed

Kataoka; Tsutahara; Akuzawa

2000-02-14

We derive a fully nonlinear evolution equation that can describe the two-dimensional motion of finite-amplitude long internal waves in a uniformly stratified three-dimensional fluid of finite depth. The derived equation is the two-dimensional counterpart of the evolution equation obtained by Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)]. In the small-amplitude limit, our equation is reduced to the celebrated Kadomtsev-Petviashvili equation.

Bulk diffusion in a kinetically constrained lattice gas

NASA Astrophysics Data System (ADS)

Arita, Chikashi; Krapivsky, P. L.; Mallick, Kirone

2018-03-01

In the hydrodynamic regime, the evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient encapsulating all microscopic details of the dynamics. This diffusion coefficient is, in principle, determined by a Green-Kubo formula. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be computed except when a lattice gas additionally satisfies the gradient condition. We develop a procedure to systematically obtain analytical approximations for the diffusion coefficient for non-gradient lattice gases with known equilibrium. The method relies on a variational formula found by Varadhan and Spohn which is a version of the Green-Kubo formula particularly suitable for diffusive lattice gases. Restricting the variational formula to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We apply this approach to a kinetically constrained non-gradient lattice gas in two dimensions, viz. to the Kob-Andersen model on the square lattice.

Asymptotic Analysis of Time-Dependent Neutron Transport Coupled with Isotopic Depletion and Radioactive Decay

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brantley, P S

2006-09-27

We describe an asymptotic analysis of the coupled nonlinear system of equations describing time-dependent three-dimensional monoenergetic neutron transport and isotopic depletion and radioactive decay. The classic asymptotic diffusion scaling of Larsen and Keller [1], along with a consistent small scaling of the terms describing the radioactive decay of isotopes, is applied to this coupled nonlinear system of equations in a medium of specified initial isotopic composition. The analysis demonstrates that to leading order the neutron transport equation limits to the standard time-dependent neutron diffusion equation with macroscopic cross sections whose number densities are determined by the standard system of ordinarymore » differential equations, the so-called Bateman equations, describing the temporal evolution of the nuclide number densities.« less

Non-equilibrium radiation from viscous chemically reacting two-phase exhaust plumes

NASA Technical Reports Server (NTRS)

Penny, M. M.; Smith, S. D.; Mikatarian, R. R.; Ring, L. R.; Anderson, P. G.

1976-01-01

A knowledge of the structure of the rocket exhaust plumes is necessary to solve problems involving plume signatures, base heating, plume/surface interactions, etc. An algorithm is presented which treats the viscous flow of multiphase chemically reacting fluids in a two-dimensional or axisymmetric supersonic flow field. The gas-particle flow solution is fully coupled with the chemical kinetics calculated using an implicit scheme to calculate chemical production rates. Viscous effects include chemical species diffusion with the viscosity coefficient calculated using a two-equation turbulent kinetic energy model.

Indispensable finite time corrections for Fokker-Planck equations from time series data.

PubMed

Ragwitz, M; Kantz, H

2001-12-17

The reconstruction of Fokker-Planck equations from observed time series data suffers strongly from finite sampling rates. We show that previously published results are degraded considerably by such effects. We present correction terms which yield a robust estimation of the diffusion terms, together with a novel method for one-dimensional problems. We apply these methods to time series data of local surface wind velocities, where the dependence of the diffusion constant on the state variable shows a different behavior than previously suggested.

The modified semi-discrete two-dimensional Toda lattice with self-consistent sources

NASA Astrophysics Data System (ADS)

Gegenhasi

2017-07-01

In this paper, we derive the Grammian determinant solutions to the modified semi-discrete two-dimensional Toda lattice equation, and then construct the semi-discrete two-dimensional Toda lattice equation with self-consistent sources via source generation procedure. The algebraic structure of the resulting coupled modified differential-difference equation is clarified by presenting its Grammian determinant solutions and Casorati determinant solutions. As an application of the Grammian determinant and Casorati determinant solution, the explicit one-soliton and two-soliton solution of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources are given. We also construct another form of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources which is the Bäcklund transformation for the semi-discrete two-dimensional Toda lattice equation with self-consistent sources.

Stabilized finite element methods to simulate the conductances of ion channels

NASA Astrophysics Data System (ADS)

Tu, Bin; Xie, Yan; Zhang, Linbo; Lu, Benzhuo

2015-03-01

We have previously developed a finite element simulator, ichannel, to simulate ion transport through three-dimensional ion channel systems via solving the Poisson-Nernst-Planck equations (PNP) and Size-modified Poisson-Nernst-Planck equations (SMPNP), and succeeded in simulating some ion channel systems. However, the iterative solution between the coupled Poisson equation and the Nernst-Planck equations has difficulty converging for some large systems. One reason we found is that the NP equations are advection-dominated diffusion equations, which causes troubles in the usual FE solution. The stabilized schemes have been applied to compute fluids flow in various research fields. However, they have not been studied in the simulation of ion transport through three-dimensional models based on experimentally determined ion channel structures. In this paper, two stabilized techniques, the SUPG and the Pseudo Residual-Free Bubble function (PRFB) are introduced to enhance the numerical robustness and convergence performance of the finite element algorithm in ichannel. The conductances of the voltage dependent anion channel (VDAC) and the anthrax toxin protective antigen pore (PA) are simulated to validate the stabilization techniques. Those two stabilized schemes give reasonable results for the two proteins, with decent agreement with both experimental data and Brownian dynamics (BD) simulations. For a variety of numerical tests, it is found that the simulator effectively avoids previous numerical instability after introducing the stabilization methods. Comparison based on our test data set between the two stabilized schemes indicates both SUPG and PRFB have similar performance (the latter is slightly more accurate and stable), while SUPG is relatively more convenient to implement.

Generalization of one-dimensional solute transport: A stochastic-convective flow conceptualization

NASA Astrophysics Data System (ADS)

Simmons, C. S.

1986-04-01

A stochastic-convective representation of one-dimensional solute transport is derived. It is shown to conceptually encompass solutions of the conventional convection-dispersion equation. This stochastic approach, however, does not rely on the assumption that dispersive flux satisfies Fick's diffusion law. Observable values of solute concentration and flux, which together satisfy a conservation equation, are expressed as expectations over a flow velocity ensemble, representing the inherent random processess that govern dispersion. Solute concentration is determined by a Lagrangian pdf for random spatial displacements, while flux is determined by an equivalent Eulerian pdf for random travel times. A condition for such equivalence is derived for steady nonuniform flow, and it is proven that both Lagrangian and Eulerian pdfs are required to account for specified initial and boundary conditions on a global scale. Furthermore, simplified modeling of transport is justified by proving that an ensemble of effectively constant velocities always exists that constitutes an equivalent representation. An example of how a two-dimensional transport problem can be reduced to a single-dimensional stochastic viewpoint is also presented to further clarify concepts.

Role of Rayleigh numbers on characteristics of double diffusive salt fingers

NASA Astrophysics Data System (ADS)

Rehman, F.; Singh, O. P.

2018-05-01

Double diffusion convection, driven by two constituents of the fluid with different molecular diffusivity, is widely applied in oceanography and large number of other fields like astrophysics, geology, chemistry and metallurgy. In case of ocean, heat (T) and salinity (S) are the two components with varying diffusivity, where heat diffuses hundred times faster than salt. Component (T) stabilizes the system whereas components (S) destabilizes the system with overall density remains stable and forms the rising and sinking fingers known as salt fingers. Recent observations suggest that salt finger characteristics such as growth rates, wavenumber, and fluxes are strongly depending on the Rayleigh numbers as major driving force. In this paper, we corroborate this observation with the help of experiments, numerical simulations and linear theory. An eigenvalue expression for growth rate is derived from the linearized governing equations with explicit dependence on Rayleigh numbers, density stability ratio, Prandtl number and diffusivity ratio. Expressions for fastest growing fingers are also derived as a function various non-dimensional parameter. The predicted results corroborate well with the data reported from the field measurements, experiments and numerical simulations.

Pattern formation for NO+N H3 on Pt(100): Two-dimensional numerical results

NASA Astrophysics Data System (ADS)

Uecker, Hannes

2005-01-01

The Lombardo-Fink-Imbihl model of the NO+NH3 reaction on a Pt(100) surface consists of seven coupled ordinary differential equations (ODE) and shows stable relaxation oscillations with sharp transitions in the relevant temperature range. Here we study numerically the effect of coupling of these oscillators by surface diffusion in two dimensions. We find different types of patterns, in particular phase clusters and standing waves. In models of related surface reactions such clustered solutions are known to exist only under a global coupling through the gas phase. This global coupling is replaced here by relatively fast diffusion of two variables which are kinetically slaved in the ODE. We also compare our simulations with experimental results and discuss some shortcomings of the model.

An incompressible two-dimensional multiphase particle-in-cell model for dense particle flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Snider, D.M.; O`Rourke, P.J.; Andrews, M.J.

1997-06-01

A two-dimensional, incompressible, multiphase particle-in-cell (MP-PIC) method is presented for dense particle flows. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Difficulties associated with calculating interparticle interactions for dense particle flows with volume fractions above 5% have been eliminated by mapping particle properties to a Eulerian grid and then mapping back computed stress tensors to particle positions. This approach utilizes the best of Eulerian/Eulerian continuum models and Eulerian/Lagrangian discrete models. The solution scheme allows for distributions of types, sizes, and density of particles,more » with no numerical diffusion from the Lagrangian particle calculations. The computational method is implicit with respect to pressure, velocity, and volume fraction in the continuum solution thus avoiding courant limits on computational time advancement. MP-PIC simulations are compared with one-dimensional problems that have analytical solutions and with two-dimensional problems for which there are experimental data.« less

Modelling Detailed-Chemistry Effects on Turbulent Diffusion Flames using a Parallel Solution-Adaptive Scheme

NASA Astrophysics Data System (ADS)

Jha, Pradeep Kumar

Capturing the effects of detailed-chemistry on turbulent combustion processes is a central challenge faced by the numerical combustion community. However, the inherent complexity and non-linear nature of both turbulence and chemistry require that combustion models rely heavily on engineering approximations to remain computationally tractable. This thesis proposes a computationally efficient algorithm for modelling detailed-chemistry effects in turbulent diffusion flames and numerically predicting the associated flame properties. The cornerstone of this combustion modelling tool is the use of parallel Adaptive Mesh Refinement (AMR) scheme with the recently proposed Flame Prolongation of Intrinsic low-dimensional manifold (FPI) tabulated-chemistry approach for modelling complex chemistry. The effect of turbulence on the mean chemistry is incorporated using a Presumed Conditional Moment (PCM) approach based on a beta-probability density function (PDF). The two-equation k-w turbulence model is used for modelling the effects of the unresolved turbulence on the mean flow field. The finite-rate of methane-air combustion is represented here by using the GRI-Mech 3.0 scheme. This detailed mechanism is used to build the FPI tables. A state of the art numerical scheme based on a parallel block-based solution-adaptive algorithm has been developed to solve the Favre-averaged Navier-Stokes (FANS) and other governing partial-differential equations using a second-order accurate, fully-coupled finite-volume formulation on body-fitted, multi-block, quadrilateral/hexahedral mesh for two-dimensional and three-dimensional flow geometries, respectively. A standard fourth-order Runge-Kutta time-marching scheme is used for time-accurate temporal discretizations. Numerical predictions of three different diffusion flames configurations are considered in the present work: a laminar counter-flow flame; a laminar co-flow diffusion flame; and a Sydney bluff-body turbulent reacting flow. Comparisons are made between the predicted results of the present FPI scheme and Steady Laminar Flamelet Model (SLFM) approach for diffusion flames. The effects of grid resolution on the predicted overall flame solutions are also assessed. Other non-reacting flows have also been considered to further validate other aspects of the numerical scheme. The present schemes predict results which are in good agreement with published experimental results and reduces the computational cost involved in modelling turbulent diffusion flames significantly, both in terms of storage and processing time.

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 3: Programmer's reference

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Bui, Trong T.

1993-01-01

A computer code called Proteus 3D was developed to solve the three-dimensional, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 2: User's guide

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Bui, Trong T.

1993-01-01

A computer code called Proteus 3D was developed to solve the three-dimensional, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This User's Guide describes the program's features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.

Quantitative Examination of Corrosion Damage by Means of Thermal Response Measurements

NASA Technical Reports Server (NTRS)

Rajic, Nik

1998-01-01

Two computational methods are presented that enable a characterization of corrosion damage to be performed from thermal response measurements derived from a standard flash thermographic inspection. The first is based upon a one dimensional analytical solution to the heat diffusion equation and presumes the lateral extent of damage is large compared to the residual structural thickness, such that lateral heat diffusion effects can be considered insignificant. The second proposed method, based on a finite element optimization scheme, addresses the more general case where these conditions are not met. Results from an experimental application are given to illustrate the precision, robustness and practical efficacy of both methods.

a Bounded Finite-Difference Discretization of a Two-Dimensional Diffusion Equation with Logistic Nonlinear Reaction

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.

In the present manuscript, we introduce a finite-difference scheme to approximate solutions of the two-dimensional version of Fisher's equation from population dynamics, which is a model for which the existence of traveling-wave fronts bounded within (0,1) is a well-known fact. The method presented here is a nonstandard technique which, in the linear regime, approximates the solutions of the original model with a consistency of second order in space and first order in time. The theory of M-matrices is employed here in order to elucidate conditions under which the method is able to preserve the positivity and the boundedness of solutions. In fact, our main result establishes relatively flexible conditions under which the preservation of the positivity and the boundedness of new approximations is guaranteed. Some simulations of the propagation of a traveling-wave solution confirm the analytical results derived in this work; moreover, the experiments evince a good agreement between the numerical result and the analytical solutions.

Premixed flames in closed cylindrical tubes

NASA Astrophysics Data System (ADS)

Metzener, Philippe; Matalon, Moshe

2001-09-01

We consider the propagation of a premixed flame, as a two-dimensional sheet separating unburned gas from burned products, in a closed cylindrical tube. A nonlinear evolution equation, that describes the motion of the flame front as a function of its mean position, is derived. The equation contains a destabilizing term that results from the gas motion induced by thermal expansion and has a memory term associated with vorticity generation. Numerical solutions of this equation indicate that, when diffusion is stabilizing, the flame evolves into a non-planar form whose shape, and its associated symmetry properties, are determined by the Markstein parameter, and by the initial data. In particular, we observe the development of convex axisymmetric or non-axisymmetric flames, tulip flames and cellular flames.

Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model

NASA Astrophysics Data System (ADS)

Dehghan, Mehdi; Mohammadi, Vahid

2017-03-01

As is said in [27], the tumor-growth model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models [27]. Simulations of this practical model using numerical methods can be applied for evaluating it. The present paper investigates the solution of the tumor growth model with meshless techniques. Meshless methods are applied based on the collocation technique which employ multiquadrics (MQ) radial basis function (RBFs) and generalized moving least squares (GMLS) procedures. The main advantages of these choices come back to the natural behavior of meshless approaches. As well as, a method based on meshless approach can be applied easily for finding the solution of partial differential equations in high-dimension using any distributions of points on regular and irregular domains. The present paper involves a time-dependent system of partial differential equations that describes four-species tumor growth model. To overcome the time variable, two procedures will be used. One of them is a semi-implicit finite difference method based on Crank-Nicolson scheme and another one is based on explicit Runge-Kutta time integration. The first case gives a linear system of algebraic equations which will be solved at each time-step. The second case will be efficient but conditionally stable. The obtained numerical results are reported to confirm the ability of these techniques for solving the two and three-dimensional tumor-growth equations.

Multigrid approaches to non-linear diffusion problems on unstructured meshes

NASA Technical Reports Server (NTRS)

Mavriplis, Dimitri J.; Bushnell, Dennis M. (Technical Monitor)

2001-01-01

The efficiency of three multigrid methods for solving highly non-linear diffusion problems on two-dimensional unstructured meshes is examined. The three multigrid methods differ mainly in the manner in which the nonlinearities of the governing equations are handled. These comprise a non-linear full approximation storage (FAS) multigrid method which is used to solve the non-linear equations directly, a linear multigrid method which is used to solve the linear system arising from a Newton linearization of the non-linear system, and a hybrid scheme which is based on a non-linear FAS multigrid scheme, but employs a linear solver on each level as a smoother. Results indicate that all methods are equally effective at converging the non-linear residual in a given number of grid sweeps, but that the linear solver is more efficient in cpu time due to the lower cost of linear versus non-linear grid sweeps.

Numerical Simulation of Hydrogen Air Supersonic Coaxial Jet

NASA Astrophysics Data System (ADS)

Dharavath, Malsur; Manna, Pulinbehari; Chakraborty, Debasis

2017-10-01

In the present study, the turbulent structure of coaxial supersonic H2-air jet is explored numerically by solving three dimensional RANS equations along with two equation k-ɛ turbulence model. Grid independence of the solution is demonstrated by estimating the error distribution using Grid Convergence Index. Distributions of flow parameters in different planes are analyzed to explain the mixing and combustion characteristics of high speed coaxial jets. The flow field is seen mostly diffusive in nature and hydrogen diffusion is confined to core region of the jet. Both single step laminar finite rate chemistry and turbulent reacting calculation employing EDM combustion model are performed to find the effect of turbulence-chemistry interaction in the flow field. Laminar reaction predicts higher H2 mol fraction compared to turbulent reaction because of lower reaction rate caused by turbulence chemistry interaction. Profiles of major species and temperature match well with experimental data at different axial locations; although, the computed profiles show a narrower shape in the far field region. These results demonstrate that standard two equation class turbulence model with single step kinetics based turbulence chemistry interaction can describe H2-air reaction adequately in high speed flows.

Methods for Solving Gas Damping Problems in Perforated Microstructures Using a 2D Finite-Element Solver

PubMed Central

Veijola, Timo; Råback, Peter

2007-01-01

We present a straightforward method to solve gas damping problems for perforated structures in two dimensions (2D) utilising a Perforation Profile Reynolds (PPR) solver. The PPR equation is an extended Reynolds equation that includes additional terms modelling the leakage flow through the perforations, and variable diffusivity and compressibility profiles. The solution method consists of two phases: 1) determination of the specific admittance profile and relative diffusivity (and relative compressibility) profiles due to the perforation, and 2) solution of the PPR equation with a FEM solver in 2D. Rarefied gas corrections in the slip-flow region are also included. Analytic profiles for circular and square holes with slip conditions are presented in the paper. To verify the method, square perforated dampers with 16–64 holes were simulated with a three-dimensional (3D) Navier-Stokes solver, a homogenised extended Reynolds solver, and a 2D PPR solver. Cases for both translational (in normal to the surfaces) and torsional motion were simulated. The presented method extends the region of accurate simulation of perforated structures to cases where the homogenisation method is inaccurate and the full 3D Navier-Stokes simulation is too time-consuming.

A three-dimensional, finite element model for coastal and estuarine circulation

USGS Publications Warehouse

Walters, R.A.

1992-01-01

This paper describes the development and application of a three-dimensional model for coastal and estuarine circulation. The model uses a harmonic expansion in time and a finite element discretization in space. All nonlinear terms are retained, including quadratic bottom stress, advection and wave transport (continuity nonlinearity). The equations are solved as a global and a local problem, where the global problem is the solution of the wave equation formulation of the shallow water equations, and the local problem is the solution of the momentum equation for the vertical velocity profile. These equations are coupled to the advection-diffusion equation for salt so that density gradient forcing is included in the momentum equations. The model is applied to a study of Delaware Bay, U.S.A., where salinity intrusion is the primary focus. ?? 1991.

A variable turbulent Prandtl and Schmidt number model study for scramjet applications

NASA Astrophysics Data System (ADS)

Keistler, Patrick

A turbulence model that allows for the calculation of the variable turbulent Prandtl (Prt) and Schmidt (Sct) numbers as part of the solution is presented. The model also accounts for the interactions between turbulence and chemistry by modeling the corresponding terms. Four equations are added to the baseline k-zeta turbulence model: two equations for enthalpy variance and its dissipation rate to calculate the turbulent diffusivity, and two equations for the concentrations variance and its dissipation rate to calculate the turbulent diffusion coefficient. The underlying turbulence model already accounts for compressibility effects. The variable Prt /Sct turbulence model is validated and tuned by simulating a wide variety of experiments. Included in the experiments are two-dimensional, axisymmetric, and three-dimensional mixing and combustion cases. The combustion cases involved either hydrogen and air, or hydrogen, ethylene, and air. Two chemical kinetic models are employed for each of these situations. For the hydrogen and air cases, a seven species/seven reaction model where the reaction rates are temperature dependent and a nine species/nineteen reaction model where the reaction rates are dependent on both pressure and temperature are used. For the cases involving ethylene, a 15 species/44 reaction reduced model that is both pressure and temperature dependent is used, along with a 22 species/18 global reaction reduced model that makes use of the quasi-steady-state approximation. In general, fair to good agreement is indicated for all simulated experiments. The turbulence/chemistry interaction terms are found to have a significant impact on flame location for the two-dimensional combustion case, with excellent experimental agreement when the terms are included. In most cases, the hydrogen chemical mechanisms behave nearly identically, but for one case, the pressure dependent model would not auto-ignite at the same conditions as the experiment and the other chemical model. The model was artificially ignited in that case. For the cases involving ethylene combustion, the chemical model has a profound impact on the flame size, shape, and ignition location. However, without quantitative experimental data, it is difficult to determine which one is more suitable for this particular application.

The coupling between flame surface dynamics and species mass conservation in premixed turbulent combustion

NASA Technical Reports Server (NTRS)

Trouve, A.; Veynante, D.; Bray, K. N. C.; Mantel, T.

1994-01-01

Current flamelot models based on a description of the flame surface dynamics require the closure of two inter-related equations: a transport equation for the mean reaction progress variable, (tilde)c, and a transport equation for the flame surface density, Sigma. The coupling between these two equations is investigated using direct numerical simulations (DNS) with emphasis on the correlation between the turbulent fluxes of (tilde)c, bar(pu''c''), and Sigma, (u'')(sub S)Sigma. Two different DNS databases are used in the present work: a database developed at CTR by A. Trouve and a database developed by C. J. Rutland using a different code. Both databases correspond to statistically one-dimensional premixed flames in isotropic turbulent flow. The run parameters, however, are significantly different, and the two databases correspond to different combustion regimes. It is found that in all simulated flames, the correlation between bar(pu''c'') and (u'')(sub S)Sigma is always strong. The sign, however, of the turbulent flux of (tilde)c or Sigma with respect to the mean gradients, delta(tilde)c/delta(x) or delta(Sigma)/delta(x), is case-dependent. The CTR database is found to exhibit gradient turbulent transport of (tilde)c and Sigma, whereas the Rutland DNS features counter-gradient diffusion. The two databases are analyzed and compared using various tools (a local analysis of the flow field near the flame, a classical analysis of the conservation equation for (tilde)(u''c''), and a thin flame theoretical analysis). A mechanism is then proposed to explain the discrepancies between the two databases and a preliminary simple criterion is derived to predict the occurrence of gradient/counter-gradient turbulent diffusion.

An accurate two-dimensional LBIC solution for bipolar transistors

NASA Astrophysics Data System (ADS)

Benarab, A.; Baudrand, H.; Lescure, M.; Boucher, J.

1988-05-01

A complete solution of the diffusion problem of carriers generated by a located light beam in the emitter and base region of a bipolar structure is presented. Green's function method and moment method are used to solve the 2-D diffusion equation in these regions. From the Green's functions solution of these equations, the light beam induced currents (LBIC) in the different junctions of the structure due to an extended generation represented by a rectangular light spot; are thus decided. The equations of these currents depend both on the parameters which characterise the structure, surface states, dimensions of the emitter and the base region, and the characteristics of the light spot, that is to say, the width and the wavelength. Curves illustrating the variation of the various LBIC in the base region junctions as a function of the impact point of the light beam ( x0) for different values of these parameters are discussed. In particular, the study of the base-emitter currents when the light beam is swept right across the sample illustrates clearly a good geometrical definition of the emitter region up to base end of the emitter-base space-charge areas and a "whirl" lateral diffusion beneath this region, (i.e. the diffusion of the generated carriers near the surface towards the horizontal base-emitter junction and those created beneath this junction towards the lateral (B-E) junctions).

Analytical solutions of the one-dimensional advection-dispersion solute transport equation subject to time-dependent boundary conditions

USDA-ARS?s Scientific Manuscript database

Analytical solutions of the advection-dispersion solute transport equation remain useful for a large number of applications in science and engineering. In this paper we extend the Duhamel theorem, originally established for diffusion type problems, to the case of advective-dispersive transport subj...

Two-dimensional numerical model for the high electron mobility transistor

NASA Astrophysics Data System (ADS)

Loret, Dany

1987-11-01

A two-dimensional numerical drift-diffusion model for the High Electron Mobility Transistor (HEMT) is presented. Special attention is paid to the modeling of the current flow over the heterojunction. A finite difference scheme is used to solve the equations, and a variable mesh spacing was implemented to cope with the strong variations of functions near the heterojunction. Simulation results are compared to experimental data for a 0.7 μm gate length device. Small-signal transconductances and cut-off frequency obtained from the 2-D model agree well with the experimental values from S-parameter measurements. It is shown that the numerical models give good insight into device behaviour, including important parasitic effects such as electron injection into the bulk GaAs.

Trends and Techniques for Space Base Electronics

NASA Technical Reports Server (NTRS)

Trotter, J. D.; Wade, T. E.; Gassaway, J. D.

1979-01-01

Simulations of various phosphorus and boron diffusions in SOS were completed and a sputtering system, furnaces, and photolithography related equipment were set up. Double layer metal experiments initially utilized wet chemistry techniques. By incorporating ultrasonic etching of the vias, premetal cleaning a modified buffered HF, phosphorus doped vapox, and extended sintering, yields of 98% were obtained using the standard test pattern. A two dimensional modeling program was written for simulating short channel MOSFETs with nonuniform substrate doping. A key simplifying assumption used is that the majority carriers can be represented by a sheet charge at the silicon dioxide silicon interface. Although the program is incomplete, the two dimensional Poisson equation for the potential distribution was achieved. The status of other Z-D MOSFET simulation programs is summarized.

An axisymmetric non-hydrostatic model for double-diffusive water systems

NASA Astrophysics Data System (ADS)

Hilgersom, Koen; Zijlema, Marcel; van de Giesen, Nick

2018-02-01

The three-dimensional (3-D) modelling of water systems involving double-diffusive processes is challenging due to the large computation times required to solve the flow and transport of constituents. In 3-D systems that approach axisymmetry around a central location, computation times can be reduced by applying a 2-D axisymmetric model set-up. This article applies the Reynolds-averaged Navier-Stokes equations described in cylindrical coordinates and integrates them to guarantee mass and momentum conservation. The discretized equations are presented in a way that a Cartesian finite-volume model can be easily extended to the developed framework, which is demonstrated by the implementation into a non-hydrostatic free-surface flow model. This model employs temperature- and salinity-dependent densities, molecular diffusivities, and kinematic viscosity. One quantitative case study, based on an analytical solution derived for the radial expansion of a dense water layer, and two qualitative case studies demonstrate a good behaviour of the model for seepage inflows with contrasting salinities and temperatures. Four case studies with respect to double-diffusive processes in a stratified water body demonstrate that turbulent flows are not yet correctly modelled near the interfaces and that an advanced turbulence model is required.

Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow

NASA Astrophysics Data System (ADS)

Zheng, Lin; Zheng, Song; Zhai, Qinglan

2016-02-01

In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.

Analysis and design of numerical schemes for gas dynamics 1: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence

NASA Technical Reports Server (NTRS)

Jameson, Antony

1994-01-01

The theory of non-oscillatory scalar schemes is developed in this paper in terms of the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multi-dimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for one-dimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multi-dimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.

Lee wave breaking region: the map of instability development scenarios

NASA Astrophysics Data System (ADS)

Yakovenko, S. N.

2017-10-01

Numerical study of a stably stratified flow above the two-dimensional cosine-shaped obstacle has been performed by DNS and LES. These methods were implemented to solve the three-dimensional Navier-Stokes equations in the Boussinesq approximation, together with by the scalar diffusion equation. The results of scanning in the wide ranges of physical parameters (Reynolds and Prandtl/Schmidt numbers relating to laboratory experiment cases and atmospheric or oceanic situations) are presented for instability and turbulence development scenarios in the overturning internal lee waves. The latter is generated by the obstacle in a flow with the constant inflow values of velocity and stable density gradient. Evolution of lee-wave breaking is explored by visualization of velocity and scalar (density) fields, and the analysis of spectra. Based on the numerical simulation results, the power-law dependence on Reynolds number is demonstrated for the wavelength of the most unstable perturbation.

Generalized time-dependent Schrödinger equation in two dimensions under constraints

NASA Astrophysics Data System (ADS)

Sandev, Trifce; Petreska, Irina; Lenzi, Ervin K.

2018-01-01

We investigate a generalized two-dimensional time-dependent Schrödinger equation on a comb with a memory kernel. A Dirac delta term is introduced in the Schrödinger equation so that the quantum motion along the x-direction is constrained at y = 0. The wave function is analyzed by using Green's function approach for several forms of the memory kernel, which are of particular interest. Closed form solutions for the cases of Dirac delta and power-law memory kernels in terms of Fox H-function, as well as for a distributed order memory kernel, are obtained. Further, a nonlocal term is also introduced and investigated analytically. It is shown that the solution for such a case can be represented in terms of infinite series in Fox H-functions. Green's functions for each of the considered cases are analyzed and plotted for the most representative ones. Anomalous diffusion signatures are evident from the presence of the power-law tails. The normalized Green's functions obtained in this work are of broader interest, as they are an important ingredient for further calculations and analyses of some interesting effects in the transport properties in low-dimensional heterogeneous media.

Theoretical examination of effective oxygen diffusion coefficient and electrical conductivity of polymer electrolyte fuel cell porous components

NASA Astrophysics Data System (ADS)

Inoue, Gen; Yokoyama, Kouji; Ooyama, Junpei; Terao, Takeshi; Tokunaga, Tomomi; Kubo, Norio; Kawase, Motoaki

2016-09-01

The reduction of oxygen transfer resistance through porous components consisting of a gas diffusion layer (GDL), microporous layer (MPL), and catalyst layer (CL) is very important to reduce the cost and improve the performance of a PEFC system. This study involves a systematic examination of the relationship between the oxygen transfer resistance of the actual porous components and their three-dimensional structure by direct measurement with FIB-SEM and X-ray CT. Numerical simulations were carried out to model the properties of oxygen transport. Moreover, based on the model structure and theoretical equations, an approach to the design of new structures is proposed. In the case of the GDL, the binder was found to obstruct gas diffusion with a negative effect on performance. The relative diffusion coefficient of the MPL is almost equal to that of the model structure of particle packing. However, that of CL is an order of magnitude less than those of the other two components. Furthermore, an equation expressing the relative diffusion coefficient of each component can be obtained with the function of porosity. The electrical conductivity of MPL, which is lower than that of the carbon black packing, is considered to depend on the contact resistance.

Analytical approximations for spatial stochastic gene expression in single cells and tissues

PubMed Central

Smith, Stephen; Cianci, Claudia; Grima, Ramon

2016-01-01

Gene expression occurs in an environment in which both stochastic and diffusive effects are significant. Spatial stochastic simulations are computationally expensive compared with their deterministic counterparts, and hence little is currently known of the significance of intrinsic noise in a spatial setting. Starting from the reaction–diffusion master equation (RDME) describing stochastic reaction–diffusion processes, we here derive expressions for the approximate steady-state mean concentrations which are explicit functions of the dimensionality of space, rate constants and diffusion coefficients. The expressions have a simple closed form when the system consists of one effective species. These formulae show that, even for spatially homogeneous systems, mean concentrations can depend on diffusion coefficients: this contradicts the predictions of deterministic reaction–diffusion processes, thus highlighting the importance of intrinsic noise. We confirm our theory by comparison with stochastic simulations, using the RDME and Brownian dynamics, of two models of stochastic and spatial gene expression in single cells and tissues. PMID:27146686

Summer Work Experience: Determining Methane Combustion Mechanisms and Sub-Scale Diffuser Properties for Space Transporation System Engine Testing

NASA Technical Reports Server (NTRS)

Williams, Powtawche N.

1998-01-01

To assess engine performance during the testing of Space Shuttle Main Engines (SSMEs), the design of an optimal altitude diffuser is studied for future Space Transportation Systems (STS). For other Space Transportation Systems, rocket propellant using kerosene is also studied. Methane and dodecane have similar reaction schemes as kerosene, and are used to simulate kerosene combustion processes at various temperatures. The equations for the methane combustion mechanism at high temperature are given, and engine combustion is simulated on the General Aerodynamic Simulation Program (GASP). The successful design of an altitude diffuser depends on the study of a sub-scaled diffuser model tested through two-dimensional (2-D) flow-techniques. Subroutines given calculate the static temperature and pressure at each Mach number within the diffuser flow. Implementing these subroutines into program code for the properties of 2-D compressible fluid flow determines all fluid characteristics, and will be used in the development of an optimal diffuser design.

Solution of the Burnett equations for hypersonic flows near the continuum limit

NASA Technical Reports Server (NTRS)

Imlay, Scott T.

1992-01-01

The INCA code, a three-dimensional Navier-Stokes code for analysis of hypersonic flowfields, was modified to analyze the lower reaches of the continuum transition regime, where the Navier-Stokes equations become inaccurate and Monte Carlo methods become too computationally expensive. The two-dimensional Burnett equations and the three-dimensional rotational energy transport equation were added to the code and one- and two-dimensional calculations were performed. For the structure of normal shock waves, the Burnett equations give consistently better results than Navier-Stokes equations and compare reasonably well with Monte Carlo methods. For two-dimensional flow of Nitrogen past a circular cylinder the Burnett equations predict the total drag reasonably well. Care must be taken, however, not to exceed the range of validity of the Burnett equations.

Spline-Based Parameter Estimation Techniques for Two-Dimensional Convection and Diffusion Equations.

DTIC Science & Technology

1986-07-01

brassicae ) were related at a point adjacent to and downwind from a cabbage ( brassica ) crop (9]. Although Wright [161 had rejected anemotaxis as a...tunnel experiments by Coaker and Smith [71 indicated that female E. brassicae do fly upwind in the presence of brassica odor. To resolve this issue Hawkes...sought to calculate dispersal rates of E. Brassicae released from a point exposed to brassica odor. When recapture data suggested random dispersal

Numerical calculation on a two-step subdiffusion behavior of lateral protein movement in plasma membranes

NASA Astrophysics Data System (ADS)

Sumi, Tomonari; Okumoto, Atsushi; Goto, Hitoshi; Sekino, Hideo

2017-10-01

A two-step subdiffusion behavior of lateral movement of transmembrane proteins in plasma membranes has been observed by using single-molecule experiments. A nested double-compartment model where large compartments are divided into several smaller ones has been proposed in order to explain this observation. These compartments are considered to be delimited by membrane-skeleton "fences" and membrane-protein "pickets" bound to the fences. We perform numerical simulations of a master equation using a simple two-dimensional lattice model to investigate the heterogeneous diffusion dynamics behavior of transmembrane proteins within plasma membranes. We show that the experimentally observed two-step subdiffusion process can be described using fence and picket models combined with decreased local diffusivity of transmembrane proteins in the vicinity of the pickets. This allows us to explain the two-step subdiffusion behavior without explicitly introducing nested double compartments.

From quantum stochastic differential equations to Gisin-Percival state diffusion

NASA Astrophysics Data System (ADS)

Parthasarathy, K. R.; Usha Devi, A. R.

2017-08-01

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ (L2(R+ ) ⊗(Cn⊕Cn ) ) and the Hilbert space L2(μ ) , where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B (t ) ,t ≥0 } , we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

Environmental solid particle effects on compressor cascade performance

NASA Technical Reports Server (NTRS)

Tabakoff, W.; Balan, C.

1982-01-01

The effect of suspended solid particles on the performance of the compressor cascade was investigated experimentally in a specially built cascade tunnel, using quartz sand particles. The cascades were made of NACA 65(10)10 airfoils. Three cascades were tested, one accelerating cascade and two diffusing cascades. The theoretical analysis assumes inviscid and incompressible two dimensional flow. The momentum exchange between the fluid and the particle is accounted for by the interphase force terms in the fluid momentum equation. The modified fluid phase momentum equations and the continuity equation are reduced to the conventional stream function vorticity formulation. The method treats the fluid phase in the Eulerian system and the particle phase in Lagrangian system. The experimental results indicate a small increase in the blade surface static pressures, while the theoretical results indicate a small decrease. The theoretical analysis, also predicts the loss in total pressure associated with the particulate flow through the cascade.

Simulation of 2-dimensional viscous flow through cascades using a semi-elliptic analysis and hybrid C-H grids

NASA Technical Reports Server (NTRS)

Ramamurti, R.; Ghia, U.; Ghia, K. N.

1988-01-01

A semi-elliptic formulation, termed the interacting parabolized Navier-Stokes (IPNS) formulation, is developed for the analysis of a class of subsonic viscous flows for which streamwise diffusion is neglible but which are significantly influenced by upstream interactions. The IPNS equations are obtained from the Navier-Stokes equations by dropping the streamwise viscous-diffusion terms but retaining upstream influence via the streamwise pressure-gradient. A two-step alternating-direction-explicit numerical scheme is developed to solve these equations. The quasi-linearization and discretization of the equations are carefully examined so that no artificial viscosity is added externally to the scheme. Also, solutions to compressible as well as nearly compressible flows are obtained without any modification either in the analysis or in the solution process. The procedure is applied to constricted channels and cascade passages formed by airfoils of various shapes. These geometries are represented using numerically generated curilinear boundary-oriented coordinates forming an H-grid. A hybrid C-H grid, more appropriate for cascade of airfoils with rounded leading edges, was also developed. Satisfactory results are obtained for flows through cascades of Joukowski airfoils.

Roles of Diffusion Dynamics in Stem Cell Signaling and Three-Dimensional Tissue Development.

PubMed

McMurtrey, Richard J

2017-09-15

Recent advancements in the ability to construct three-dimensional (3D) tissues and organoids from stem cells and biomaterials have not only opened abundant new research avenues in disease modeling and regenerative medicine but also have ignited investigation into important aspects of molecular diffusion in 3D cellular architectures. This article describes fundamental mechanics of diffusion with equations for modeling these dynamic processes under a variety of scenarios in 3D cellular tissue constructs. The effects of these diffusion processes and resultant concentration gradients are described in the context of the major molecular signaling pathways in stem cells that both mediate and are influenced by gas and nutrient concentrations, including how diffusion phenomena can affect stem cell state, cell differentiation, and metabolic states of the cell. The application of these diffusion models and pathways is of vital importance for future studies of developmental processes, disease modeling, and tissue regeneration.

Simulations of spray autoignition and flame establishment with two-dimensional CMC

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wright, Y.M.; Boulouchos, K.; De Paola, G.

2005-12-01

The unsteady two-dimensional conditional moment closure (CMC) model with first-order closure of the chemistry and supplied with standard models for the conditional convection and turbulent diffusion terms has been interfaced with a commercial engine CFD code and analyzed with two numerical methods, an 'exact' calculation with the method of lines and a faster fractional-step method. The aim was to examine the sensitivity of the predictions to the operator splitting errors and to identify the extent to which spatial transport terms are important for spray autoignition problems. Despite the underlying simplifications, solution of the full CMC equations allows a single modelmore » to be used for the autoignition, flame propagation ('premixed mode'), and diffusion flame mode of diesel combustion, which makes CMC a good candidate model for practical engine calculations. It was found that (i) the conditional averages have significant spatial gradients before ignition and during the premixed mode and (ii) that the inclusion of physical-space transport affects the calculation of the autoignition delay time, both of which suggest that volume-averaged CMC approaches may be inappropriate for diesel-like problems. A balance of terms in the CMC equation before and after autoignition shows the relative magnitude of spatial transport and allows conjectures on the structure of the premixed phase of diesel combustion. Very good agreement with available experimental data is found concerning ignition delays and the effect of background air turbulence on them.« less

Magnetoconvection dynamics in a stratified layer. 1: Two-dimensional simulations and visualization

NASA Astrophysics Data System (ADS)

Lantz, Steven R.; Sudan, R. N.

1995-03-01

To gain insight in the problem of fluid convection below solar photosphere, time-dependent magnetohydrodynamic convection is studied by numerical simulation to the magneto-anelastic equations, a model appropiate for low Mach numbers. Numerical solutions to the equations are generated on a two-dimensional Cartesian mesh by a finite-difference, predictor-corrector algorithm. The thermodynamic properties of the fluid are held constant at the rigid, stress-free top and bottom boundaries of the computational box, while lateral boundaries are treated as periodic. In most runs the background polytropic fluid configuration is held fixed at Rayleigh number R = 5.44 times the critical value, Prandtl number P = 1.8, and aspect ratio a = 1, while the magnetic parameters are allowed to vary. The resulting dynamical behavior is shown to be strongly influenced by a horizontal magnetic field which is imposed at the bottom boundary. As the field strength increases from zero, an initially unsteady 'single-roll' state, featuring complex time dependence is replaced by a steady 'traveling-wave tilted state; then, an oscillatory or 'sloshing' state; then, a steady two-poll state with no tilting; and finally, a stationary state. Because the magnetic field is matched onto a potential field at the top boundary, it can penetrate into the nonconducting region above. By varying a magnetic diffusivity, the concentrations of weak magnetic fields at the top of these flows can be shown to be explainable in terms of an advection-diffusion balance.

Boundary Layer Model for Air Pollutant Concentrations Due to Highway Traffic

ERIC Educational Resources Information Center

Ragland, Kenneth W.; Peirce, J. Jeffrey

1975-01-01

A numerical solution of the three-dimensional steady-state diffusion equation for a finite width line source is presented. The wind speed and eddy diffusivity as a function of height above the roadway are obtained. Normalized ground level and elevated concentrations near a highway are obtained for winds perpendicular, parallel, and at 45 degrees.…

On the transition from the Ginzburg-Landau equation to the extended Fisher-Kolmogorov equation

NASA Astrophysics Data System (ADS)

Rottschäfer, Vivi; Doelman, Arjen

1998-07-01

The Ginzburg-Landau (GL) equation ‘generically’ describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains. In this paper we consider the transition from this generic situation to a degenerate (co-dimension 2) case in which the GL approach is no longer valid. Instead of studying a general underlying model problem, we consider a two-dimensional system of coupled reaction-diffusion equations in one spatial dimension. We show that near the degeneration the behaviour of small perturbations is governed by the extended Fisher-Kolmogorov (eFK) equation (at leading order). The relation between the GL-equation and the eFK-equation is quite subtle, but can be analysed in detail. The main goal of this paper is to study this relation, which we do asymptotically. The asymptotic analysis is compared to numerical simulations of the full reaction-diffusion system. As one approaches the co-dimension 2 point, we observe that the stable stationary periodic patterns predicted by the GL-equation evolve towards various different families of stable, stationary (but not necessarily periodic) so-called ‘multi-bump’ solutions. In the literature, these multi-bump patterns are shown to exist as solutions of the eFK-equation, but there is no proof of the asymptotic stability of these solutions. Our results suggest that these multi-bump patterns can also be asymptotically stable in large classes of model problems.

Similarity solutions of some two-space-dimensional nonlinear wave evolution equations

NASA Technical Reports Server (NTRS)

Redekopp, L. G.

1980-01-01

Similarity reductions of the two-space-dimensional versions of the Korteweg-de Vries, modified Korteweg-de Vries, Benjamin-Davis-Ono, and nonlinear Schroedinger equations are presented, and some solutions of the reduced equations are discussed. Exact dispersive solutions of the two-dimensional Korteweg-de Vries equation are obtained, and the similarity solution of this equation is shown to be reducible to the second Painleve transcendent.

Methods for characterizing subsurface volatile contaminants using in-situ sensors

DOEpatents

Ho, Clifford K [Albuquerque, NM

2006-02-21

An inverse analysis method for characterizing diffusion of vapor from an underground source of volatile contaminant using data taken by an in-situ sensor. The method uses one-dimensional solutions to the diffusion equation in Cartesian, cylindrical, or spherical coordinates for isotropic and homogenous media. If the effective vapor diffusion coefficient is known, then the distance from the source to the in-situ sensor can be estimated by comparing the shape of the predicted time-dependent vapor concentration response curve to the measured response curve. Alternatively, if the source distance is known, then the effective vapor diffusion coefficient can be estimated using the same inverse analysis method. A triangulation technique can be used with multiple sensors to locate the source in two or three dimensions. The in-situ sensor can contain one or more chemiresistor elements housed in a waterproof enclosure with a gas permeable membrane.

a Cell Vertex Algorithm for the Incompressible Navier-Stokes Equations on Non-Orthogonal Grids

NASA Astrophysics Data System (ADS)

Jessee, J. P.; Fiveland, W. A.

1996-08-01

The steady, incompressible Navier-Stokes (N-S) equations are discretized using a cell vertex, finite volume method. Quadrilateral and hexahedral meshes are used to represent two- and three-dimensional geometries respectively. The dependent variables include the Cartesian components of velocity and pressure. Advective fluxes are calculated using bounded, high-resolution schemes with a deferred correction procedure to maintain a compact stencil. This treatment insures bounded, non-oscillatory solutions while maintaining low numerical diffusion. The mass and momentum equations are solved with the projection method on a non-staggered grid. The coupling of the pressure and velocity fields is achieved using the Rhie and Chow interpolation scheme modified to provide solutions independent of time steps or relaxation factors. An algebraic multigrid solver is used for the solution of the implicit, linearized equations.A number of test cases are anlaysed and presented. The standard benchmark cases include a lid-driven cavity, flow through a gradual expansion and laminar flow in a three-dimensional curved duct. Predictions are compared with data, results of other workers and with predictions from a structured, cell-centred, control volume algorithm whenever applicable. Sensitivity of results to the advection differencing scheme is investigated by applying a number of higher-order flux limiters: the MINMOD, MUSCL, OSHER, CLAM and SMART schemes. As expected, studies indicate that higher-order schemes largely mitigate the diffusion effects of first-order schemes but also shown no clear preference among the higher-order schemes themselves with respect to accuracy. The effect of the deferred correction procedure on global convergence is discussed.

Parallel SOR methods with a parabolic-diffusion acceleration technique for solving an unstructured-grid Poisson equation on 3D arbitrary geometries

NASA Astrophysics Data System (ADS)

Zapata, M. A. Uh; Van Bang, D. Pham; Nguyen, K. D.

2016-05-01

This paper presents a parallel algorithm for the finite-volume discretisation of the Poisson equation on three-dimensional arbitrary geometries. The proposed method is formulated by using a 2D horizontal block domain decomposition and interprocessor data communication techniques with message passing interface. The horizontal unstructured-grid cells are reordered according to the neighbouring relations and decomposed into blocks using a load-balanced distribution to give all processors an equal amount of elements. In this algorithm, two parallel successive over-relaxation methods are presented: a multi-colour ordering technique for unstructured grids based on distributed memory and a block method using reordering index following similar ideas of the partitioning for structured grids. In all cases, the parallel algorithms are implemented with a combination of an acceleration iterative solver. This solver is based on a parabolic-diffusion equation introduced to obtain faster solutions of the linear systems arising from the discretisation. Numerical results are given to evaluate the performances of the methods showing speedups better than linear.

1D Resonance line Broadened Quasilinear (RBQ1D) code for fast ion Alfvenic relaxations and its validations

NASA Astrophysics Data System (ADS)

Gorelenkov, Nikolai; Duarte, Vinicius; Podesta, Mario

2017-10-01

The performance of the burning plasma can be limited by the requirements to confine the superalfvenic fusion products which are capable of resonating with the Alfvénic eigenmodes (AEs). The effect of AEs on fast ions is evaluated using the quasi-linear approach [Berk et al., Ph.Plasmas'96] generalized for this problem recently [Duarte et al., Ph.D.'17]. The generalization involves the resonance line broadened interaction regions with the diffusion coefficient prescribed to find the evolution of the velocity distribution function. The baseline eigenmode structures are found using the NOVA-K code perturbatively [Gorelenkov et al., Ph.Plasmas'99]. A RBQ1D code allowing the diffusion in radial direction is presented here. The wave particle interaction can be reduced to one-dimensional dynamics where for the Alfvénic modes typically the particle kinetic energy is nearly constant. Hence to a good approximation the Quasi-Linear (QL) diffusion equation only contains derivatives in the angular momentum. The diffusion equation is then one dimensional that is efficiently solved simultaneously for all particles with the equation for the evolution of the wave angular momentum. The RBQ1D is validated against recent DIIID results [Collins et al., PRL'16]. Supported by the US Department of Energy under DE-AC02-09CH11466.

Solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili dynamic equation in dust-acoustic plasmas

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-09-01

Nonlinear two-dimensional Kadomtsev-Petviashvili (KP) equation governs the behaviour of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions. By using the reductive perturbation method, the two-dimensional dust-acoustic solitary waves (DASWs) in unmagnetized cold plasma consisting of dust fluid, ions and electrons lead to a KP equation. We derived the solitary travelling wave solutions of the two-dimensional nonlinear KP equation by implementing sech-tanh, sinh-cosh, extended direct algebraic and fraction direct algebraic methods. We found the electrostatic field potential and electric field in the form travelling wave solutions for two-dimensional nonlinear KP equation. The solutions for the KP equation obtained by using these methods can be demonstrated precisely and efficiency. As an illustration, we used the readymade package of Mathematica program 10.1 to solve the original problem. These solutions are in good agreement with the analytical one.

Chaotic dynamics and diffusion in a piecewise linear equation

NASA Astrophysics Data System (ADS)

Shahrear, Pabel; Glass, Leon; Edwards, Rod

2015-03-01

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

An idealised study of the effects of small scales on chemistry in a two-dimensional turbulent flow.

NASA Astrophysics Data System (ADS)

Chaalal, F. Ait; Bartello, P.; Bourqui, M.

2009-04-01

The non-linear nature of stratospheric chemical reactions makes them sensitive to mixing and diffusion. Most stratospheric Climate-Chemistry Models neglect the effects of sub-grid flow structures on chemistry. Several previous studies have pointed out that such unresolved small scales could significantly affect the chemistry. However this problem has not been thoroughly studied from a theoretical point of view. To fulfill this gap, we investigate the interactions between advection, diffusion and chemistry for a simple bimolecular reaction between two initially unmixed reactants, within the framework of two-dimensional isotropic and homogenous turbulence. This is a highly simplified representation of quasi-isentropic mixing in the stratosphere. Our goal here is to describe and understand how the production rate of the product species is affected by the size of the smallest scales of the tracer field, as determined by the tracer diffusion coefficient ΰ. The spatial average of the prognostic equation for the product's concentration involves the covariance of the reactants. The time evolution of this covariance depends in turn on a dissipative term, and on second and third order chemical terms. The set of equations is not closed and any finite resolution model would need a parameterization of the dissipation and a closure hypothesis on the chemical terms. To study these terms, we perform ensembles of direct numerical simulations using a pseudo-spectral two-dimensional periodic model. The ensembles span different initial conditions of the flow and different tracer diffusion coefficients ΰ. Our results show a strong dependence of the total production on the diffusion coefficient. This production scales like ΰp(t) , where p(t) is a positive decreasing function of time. This scaling is very similar to the one found by Tan et al. (1998) for atmospheric flows on the deactivation of chlorine by nitrogen oxide at the southern edge of the winter time polar vortex. Furthermore, the time derivative of the reactants' covariance is found to be only very weakly dependent on the chemical reaction rate, for both slow and fast chemistries compared to the advection. The variations of the dissipation and of the chemical terms with the reaction speed compensate each other. As a consequence, the calculation of the product's concentration using the covariance of the dissipation without chemistry is a good approximation of the effect of diffusion with chemistry. Reference Tan, DGH; Haynes, PH; MacKenzie, AR; et al., Effects of fluid-dynamical stirring and mixing on the deactivation of stratospheric chlorine, Journal of Geophysical Research-Atmospheres, Volume: 103 Issue: D1 Pages: 1585-1605 (1998).

Fourier analysis of finite element preconditioned collocation schemes

NASA Technical Reports Server (NTRS)

Deville, Michel O.; Mund, Ernest H.

1990-01-01

The spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes is investigated. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the tranverse direction) of the 2-D Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.

Simulation of propagation of the HPM in the low-pressure argon plasma

NASA Astrophysics Data System (ADS)

Zhigang, LI; Zhongcai, YUAN; Jiachun, WANG; Jiaming, SHI

2018-02-01

The propagation of the high-power microwave (HPM) with a frequency of 6 GHz in the low-pressure argon plasma was studied by the method of fluid approximation. The two-dimensional transmission model was built based on the wave equation, the electron drift-diffusion equations and the heavy species transport equations, which were solved by means of COMSOL Multiphysics software. The simulation results showed that the propagation characteristic of the HPM was closely related to the average electron density of the plasma. The attenuation of the transmitted wave increased nonlinearly with the electron density. Specifically, the growth of the attenuation slowed down as the electron density increased uniformly. In addition, the concrete transmission process of the HPM wave in the low-pressure argon plasma was given.

A deterministic particle method for one-dimensional reaction-diffusion equations

NASA Technical Reports Server (NTRS)

Mascagni, Michael

1995-01-01

We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.

A two-dimensional lattice equation as an extension of the Heideman-Hogan recurrence

NASA Astrophysics Data System (ADS)

Kamiya, Ryo; Kanki, Masataka; Mase, Takafumi; Tokihiro, Tetsuji

2018-03-01

We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somos-like recurrences, and introduce one of its two dimensional extensions. The two dimensional lattice equation we present is linearizable in both directions, and has the Laurent and the coprimeness properties. Moreover, its reduction produces a generalized family of the Heideman-Hogan recurrence. Higher order examples of two dimensional linearizable lattice equations related to the Dana Scott recurrence are also discussed.

Hybrid diffusion-P3 equation in N-layered turbid media: steady-state domain.

PubMed

Shi, Zhenzhi; Zhao, Huijuan; Xu, Kexin

2011-10-01

This paper discusses light propagation in N-layered turbid media. The hybrid diffusion-P3 equation is solved for an N-layered finite or infinite turbid medium in the steady-state domain for one point source using the extrapolated boundary condition. The Fourier transform formalism is applied to derive the analytical solutions of the fluence rate in Fourier space. Two inverse Fourier transform methods are developed to calculate the fluence rate in real space. In addition, the solutions of the hybrid diffusion-P3 equation are compared to the solutions of the diffusion equation and the Monte Carlo simulation. For the case of small absorption coefficients, the solutions of the N-layered diffusion equation and hybrid diffusion-P3 equation are almost equivalent and are in agreement with the Monte Carlo simulation. For the case of large absorption coefficients, the model of the hybrid diffusion-P3 equation is more precise than that of the diffusion equation. In conclusion, the model of the hybrid diffusion-P3 equation can replace the diffusion equation for modeling light propagation in the N-layered turbid media for a wide range of absorption coefficients.

A Review of the Ginzburg-Syrovatskii's Galactic Cosmic-Ray Propagation Model and its Leaky-Box Limit

NASA Technical Reports Server (NTRS)

Barghouty, A. F.

2012-01-01

Phenomenological models of galactic cosmic-ray propagation are based on a diffusion equation known as the Ginzburg-Syrovatskii s equation, or variants (or limits) of this equation. Its one-dimensional limit in a homogeneous volume, known as the leaky-box limit or model, is sketched here. The justification, utility, limitations, and a typical numerical implementation of the leaky-box model are examined in some detail.

Selection theory of free dendritic growth in a potential flow.

PubMed

von Kurnatowski, Martin; Grillenbeck, Thomas; Kassner, Klaus

2013-04-01

The Kruskal-Segur approach to selection theory in diffusion-limited or Laplacian growth is extended via combination with the Zauderer decomposition scheme. This way nonlinear bulk equations become tractable. To demonstrate the method, we apply it to two-dimensional crystal growth in a potential flow. We omit the simplifying approximations used in a preliminary calculation for the same system [Fischaleck, Kassner, Europhys. Lett. 81, 54004 (2008)], thus exhibiting the capability of the method to extend mathematical rigor to more complex problems than hitherto accessible.

The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations

NASA Technical Reports Server (NTRS)

Schlesinger, R. E.; Johnson, D. R.; Uccellini, L. W.

1983-01-01

In the present investigation, a one-dimensional linearized analysis is used to determine the effect of Asselin's (1972) time filter on both the computational stability and phase error of numerical solutions for the shallow water wave equations, in cases with diffusion but without rotation. An attempt has been made to establish the approximate optimal values of the filtering parameter nu for each of the 'lagged', Dufort-Frankel, and Crank-Nicholson diffusion schemes, suppressing the computational wave mode without materially altering the physical wave mode. It is determined that in the presence of diffusion, the optimum filter length depends on whether waves are undergoing significant propagation. When moderate propagation is present, with or without diffusion, the Asselin filter has little effect on the spatial phase lag of the physical mode for the leapfrog advection scheme of the three diffusion schemes considered.

Nonequilibrium generalised Langevin equation for the calculation of heat transport properties in model 1D atomic chains coupled to two 3D thermal baths.

PubMed

Ness, H; Stella, L; Lorenz, C D; Kantorovich, L

2017-04-28

We use a generalised Langevin equation scheme to study the thermal transport of low dimensional systems. In this approach, the central classical region is connected to two realistic thermal baths kept at two different temperatures [H. Ness et al., Phys. Rev. B 93, 174303 (2016)]. We consider model Al systems, i.e., one-dimensional atomic chains connected to three-dimensional baths. The thermal transport properties are studied as a function of the chain length N and the temperature difference ΔT between the baths. We calculate the transport properties both in the linear response regime and in the non-linear regime. Two different laws are obtained for the linear conductance versus the length of the chains. For large temperatures (T≳500 K) and temperature differences (ΔT≳500 K), the chains, with N>18 atoms, present a diffusive transport regime with the presence of a temperature gradient across the system. For lower temperatures (T≲500 K) and temperature differences (ΔT≲400 K), a regime similar to the ballistic regime is observed. Such a ballistic-like regime is also obtained for shorter chains (N≤15). Our detailed analysis suggests that the behaviour at higher temperatures and temperature differences is mainly due to anharmonic effects within the long chains.

Rossby wave activity in a two-dimensional model - Closure for wave driving and meridional eddy diffusivity

NASA Technical Reports Server (NTRS)

Hitchman, Matthew H.; Brasseur, Guy

1988-01-01

A parameterization of the effects of Rossby waves in the middle atmosphere is proposed for use in two-dimensional models. By adding an equation for conservation of Rossby wave activity, closure is obtained for the meridional eddy fluxes and body force due to Rossby waves. Rossby wave activity is produced in a climatological fashion at the tropopause, is advected by a group velocity which is determined solely by model zonal winds, and is absorbed where it converges. Absorption of Rossby wave activity causes both an easterly torque and an irreversible mixing of potential vorticity, represented by the meridional eddy diffusivity, K(yy). The distribution of Rossby wave driving determines the distribution of K(yy), which is applied to all of the chemical constituents. This provides a self-consistent coupling of the wave activity with the winds, tracer distributions and the radiative field. Typical winter stratospheric values for K(yy) of 2 million sq m/sec are obtained. Poleward tracer advection is enhanced and meridional tracer gradients are reduced where Rossby wave activity is absorbed in the model.

Horizontal density-gradient effects on simulation of flow and transport in the Potomac Estuary

USGS Publications Warehouse

Schaffranek, Raymond W.; Baltzer, Robert A.; ,

1990-01-01

A two-dimensional, depth-integrated, hydrodynamic/transport model of the Potomac Estuary between Indian Head and Morgantown, Md., has been extended to include treatment of baroclinic forcing due to horizontal density gradients. The finite-difference model numerically integrates equations of mass and momentum conservation in conjunction with a transport equation for heat, salt, and constituent fluxes. Lateral and longitudinal density gradients are determined from salinity distributions computed from the convection-diffusion equation and an equation of state that expresses density as a function of temperature and salinity; thus, the hydrodynamic and transport computations are directly coupled. Horizontal density variations are shown to contribute significantly to momentum fluxes determined in the hydrodynamic computation. These fluxes lead to enchanced tidal pumping, and consequently greater dispersion, as is evidenced by numerical simulations. Density gradient effects on tidal propagation and transport behavior are discussed and demonstrated.

Reynolds-Stress Budgets in an Impinging Shock Wave/Boundary-Layer Interaction

NASA Technical Reports Server (NTRS)

Vyas, Manan A.; Yoder, Dennis A.; Gaitonde, Datta V.

2018-01-01

Implicit large-eddy simulation (ILES) of a shock wave/boundary-layer interaction (SBLI) was performed. Comparisons with experimental data showed a sensitivity of the current prediction to the modeling of the sidewalls. This was found to be common among various computational studies in the literature where periodic boundary conditions were used in the spanwise direction, as was the case in the present work. Thus, although the experiment was quasi-two-dimensional, the present simulation was determined to be two-dimensional. Quantities present in the exact equation of the Reynolds-stress transport, i.e., production, molecular diffusion, turbulent transport, pressure diffusion, pressure strain, dissipation, and turbulent mass flux were calculated. Reynolds-stress budgets were compared with past large-eddy simulation and direct numerical simulation datasets in the undisturbed portion of the turbulent boundary layer to validate the current approach. The budgets in SBLI showed the growth in the production term for the primary normal stress and energy transfer mechanism was led by the pressure strain term in the secondary normal stresses. The pressure diffusion term, commonly assumed as negligible by turbulence model developers, was shown to be small but non-zero in the normal stress budgets, however it played a key role in the primary shear stress budget.

Surge dynamics coupled to pore-pressure evolution in debris flows

USGS Publications Warehouse

Savage, S.B.; Iverson, R.M.; ,

2003-01-01

Temporally and spatially varying pore-fluid pressures exert strong controls on debris-flow motion by mediating internal and basal friction at grain contacts. We analyze these effects by deriving a one-dimensional model of pore-pressure diffusion explicitly coupled to changes in debris-flow thickness. The new pore-pressure equation is combined with Iverson's (1997) extension of the depth-averaged Savage-Hutter (1989, 1991) granular avalanche equations to predict motion of unsteady debris-flow surges with evolving pore-pressure distributions. Computational results illustrate the profound effects of pore-pressure diffusivities on debris-flow surge depths and velocities. ?? 2003 Millpress,.

Hybrid asymptotic-numerical approach for estimating first-passage-time densities of the two-dimensional narrow capture problem.

PubMed

Lindsay, A E; Spoonmore, R T; Tzou, J C

2016-10-01

A hybrid asymptotic-numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with a reflecting boundary. As motivation for this study, we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean. This implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined. To solve the underlying diffusion equation, the method of Laplace transforms is used to obtain an elliptic problem of modified Helmholtz type. In the limit of vanishing trap sizes, each trap is represented as a Dirac point source that permits the solution of the transform equation to be represented as a superposition of Helmholtz Green's functions. Using this solution, we construct asymptotic short-time solutions of the first-passage-time density, which captures peaks associated with rapid capture by the absorbing traps. When numerical evaluation of the Helmholtz Green's function is employed followed by numerical inversion of the Laplace transform, the method reproduces the density for larger times. We demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time-dependent solution of the diffusion equation and discrete particle simulations. In particular, we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged. The hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes.

Dynamic screening in a two-species asymmetric exclusion process

NASA Astrophysics Data System (ADS)

Kim, Kyung Hyuk; den Nijs, Marcel

2007-08-01

The dynamic scaling properties of the one-dimensional Burgers equation are expected to change with the inclusion of additional conserved degrees of freedom. We study this by means of one-dimensional (1D) driven lattice gas models that conserve both mass and momentum. The most elementary version of this is the Arndt-Heinzel-Rittenberg (AHR) process, which is usually presented as a two-species diffusion process, with particles of opposite charge hopping in opposite directions and with a variable passing probability. From the hydrodynamics perspective this can be viewed as two coupled Burgers equations, with the number of positive and negative momentum quanta individually conserved. We determine the dynamic scaling dimension of the AHR process from the time evolution of the two-point correlation functions, and find numerically that the dynamic critical exponent is consistent with simple Kardar-Parisi-Zhang- (KPZ) type scaling. We establish that this is the result of perfect screening of fluctuations in the stationary state. The two-point correlations decay exponentially in our simulations and in such a manner that in terms of quasiparticles, fluctuations fully screen each other at coarse grained length scales. We prove this screening rigorously using the analytic matrix product structure of the stationary state. The proof suggests the existence of a topological invariant. The process remains in the KPZ universality class but only in the sense of a factorization, as (KPZ)2 . The two Burgers equations decouple at large length scales due to the perfect screening.

Fractional Dynamics of Single File Diffusion in Dusty Plasma Ring

NASA Astrophysics Data System (ADS)

Muniandy, S. V.; Chew, W. X.; Asgari, H.; Wong, C. S.; Lim, S. C.

2011-11-01

Single file diffusion (SFD) refers to the constrained motion of particles in quasi-one-dimensional channel such that the particles are unable to pass each other. Possible SFD of charged dust confined in biharmonic annular potential well with screened Coulomb interaction is investigated. Transition from normal diffusion to anomalous sub-diffusion behaviors is observed. Deviation from SFD's mean square displacement scaling behavior of 1/2-exponent may occur in strongly interacting systems. A phenomenological model based on fractional Langevin equation is proposed to account for the anomalous SFD behavior in dusty plasma ring.

Monte-Carlo simulation of a stochastic differential equation

NASA Astrophysics Data System (ADS)

Arif, ULLAH; Majid, KHAN; M, KAMRAN; R, KHAN; Zhengmao, SHENG

2017-12-01

For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient, Monte Carlo (MC) techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem. Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.

Structure of two-dimensional solitons in the context of a generalized Kadomtsev-Petviashvili equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Abramyan, L.A.; Stepanyants, Yu.A.

1988-04-01

The structure of steady-state two-dimensional solutions of the soliton type with quadratic and cubic nonlinearities and power-law dispersion is analyzed numerically. It is shown that steadily coupled two-dimensional multisolitons can exist for positive dispersion in a broad class of equations, which generalize the Kadomtsev-Petviashvili equation.

Computations of the three-dimensional flow and heat transfer within a coolant passage of a radial turbine blade

NASA Technical Reports Server (NTRS)

Shih, T. I.-P.; Roelke, R. J.; Steinthorsson, E.

1991-01-01

A numerical code is developed for computing three-dimensional, turbulent, compressible flow within coolant passages of turbine blades. The code is based on a formulation of the compressible Navier-Stokes equations in a rotating frame of reference in which the velocity dependent variable is specified with respect to the rotating frame instead of the inertial frame. The algorithm employed to obtain solutions to the governing equation is a finite-volume LU algorithm that allows convection, source, as well as diffusion terms to be treated implicitly. In this study, all convection terms are upwind differenced by using flux-vector splitting, and all diffusion terms are centrally differenced. This paper describes the formulation and algorithm employed in the code. Some computed solutions for the flow within a coolant passage of a radial turbine are also presented.

Double-Diffusive Convection in Rotational Shear

DTIC Science & Technology

2015-03-01

salt finger development is 0 and 0Z ZT S> > . The model uses the Boussinesq equations of motion with the linear equations of state, are expressed in...reference density from the Boussinesq approximation. ( )top bottom Z T T T H − = (2.2) The resultant non-dimensionalized equations for the model are...S T k k t = to determine how the system evolved during the simulation. B. VERSIONS OF THE BASIC MODEL This research was based on four separate

An Ab Initio and Kinetic Monte Carlo Simulation Study of Lithium Ion Diffusion on Graphene

PubMed Central

Zhong, Kehua; Yang, Yanmin; Xu, Guigui; Zhang, Jian-Min; Huang, Zhigao

2017-01-01

The Li+ diffusion coefficients in Li+-adsorbed graphene systems were determined by combining first-principle calculations based on density functional theory with Kinetic Monte Carlo simulations. The calculated results indicate that the interactions between Li ions have a very important influence on lithium diffusion. Based on energy barriers directly obtained from first-principle calculations for single-Li+ and two-Li+ adsorbed systems, a new equation predicting energy barriers with more than two Li ions was deduced. Furthermore, it is found that the temperature dependence of Li+ diffusion coefficients fits well to the Arrhenius equation, rather than meeting the equation from electrochemical impedance spectroscopy applied to estimate experimental diffusion coefficients. Moreover, the calculated results also reveal that Li+ concentration dependence of diffusion coefficients roughly fits to the equation from electrochemical impedance spectroscopy in a low concentration region; however, it seriously deviates from the equation in a high concentration region. So, the equation from electrochemical impedance spectroscopy technique could not be simply used to estimate the Li+ diffusion coefficient for all Li+-adsorbed graphene systems with various Li+ concentrations. Our work suggests that interactions between Li ions, and among Li ion and host atoms will influence the Li+ diffusion, which determines that the Li+ intercalation dependence of Li+ diffusion coefficient should be changed and complex. PMID:28773122

Numerical simulation of the hydrodynamical combustion to strange quark matter

NASA Astrophysics Data System (ADS)

Niebergal, Brian; Ouyed, Rachid; Jaikumar, Prashanth

2010-12-01

We present results from a numerical solution to the burning of neutron matter inside a cold neutron star into stable u,d,s quark matter. Our method solves hydrodynamical flow equations in one dimension with neutrino emission from weak equilibrating reactions, and strange quark diffusion across the burning front. We also include entropy change from heat released in forming the stable quark phase. Our numerical results suggest burning front laminar speeds of 0.002-0.04 times the speed of light, much faster than previous estimates derived using only a reactive-diffusive description. Analytic solutions to hydrodynamical jump conditions with a temperature-dependent equation of state agree very well with our numerical findings for fluid velocities. The most important effect of neutrino cooling is that the conversion front stalls at lower density (below ≈2 times saturation density). In a two-dimensional setting, such rapid speeds and neutrino cooling may allow for a flame wrinkle instability to develop, possibly leading to detonation.

Brownian dynamics of confined rigid bodies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Delong, Steven; Balboa Usabiaga, Florencio; Donev, Aleksandar, E-mail: donev@courant.nyu.edu

2015-10-14

We introduce numerical methods for simulating the diffusive motion of rigid bodies of arbitrary shape immersed in a viscous fluid. We parameterize the orientation of the bodies using normalized quaternions, which are numerically robust, space efficient, and easy to accumulate. We construct a system of overdamped Langevin equations in the quaternion representation that accounts for hydrodynamic effects, preserves the unit-norm constraint on the quaternion, and is time reversible with respect to the Gibbs-Boltzmann distribution at equilibrium. We introduce two schemes for temporal integration of the overdamped Langevin equations of motion, one based on the Fixman midpoint method and the othermore » based on a random finite difference approach, both of which ensure that the correct stochastic drift term is captured in a computationally efficient way. We study several examples of rigid colloidal particles diffusing near a no-slip boundary and demonstrate the importance of the choice of tracking point on the measured translational mean square displacement (MSD). We examine the average short-time as well as the long-time quasi-two-dimensional diffusion coefficient of a rigid particle sedimented near a bottom wall due to gravity. For several particle shapes, we find a choice of tracking point that makes the MSD essentially linear with time, allowing us to estimate the long-time diffusion coefficient efficiently using a Monte Carlo method. However, in general, such a special choice of tracking point does not exist, and numerical techniques for simulating long trajectories, such as the ones we introduce here, are necessary to study diffusion on long time scales.« less

Influence investigation of a void region on modeling light propagation in a heterogeneous medium.

PubMed

Yang, Defu; Chen, Xueli; Ren, Shenghan; Qu, Xiaochao; Tian, Jie; Liang, Jimin

2013-01-20

A void region exists in some biological tissues, and previous studies have shown that inaccurate images would be obtained if it were not processed. A hybrid radiosity-diffusion method (HRDM) that couples the radiosity theory and the diffusion equation has been proposed to deal with the void problem and has been well demonstrated in two-dimensional and three-dimensional (3D) simple models. However, the extent of the impact of the void region on the accuracy of modeling light propagation has not been investigated. In this paper, we first implemented and verified the HRDM in 3D models, including both the regular geometries and a digital mouse model, and then investigated the influences of the void region on modeling light propagation in a heterogeneous medium. Our investigation results show that the influence of the region can be neglected when the size of the void is less than a certain range, and other cases must be taken into account.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Broda, Jill Terese

The neutron flux across the nuclear reactor core is of interest to reactor designers and others. The diffusion equation, an integro-differential equation in space and energy, is commonly used to determine the flux level. However, the solution of a simplified version of this equation when automated is very time consuming. Since the flux level changes with time, in general, this calculation must be made repeatedly. Therefore solution techniques that speed the calculation while maintaining accuracy are desirable. One factor that contributes to the solution time is the spatial flux shape approximation used. It is common practice to use the samemore » order flux shape approximation in each energy group even though this method may not be the most efficient. The one-dimensional, two-energy group diffusion equation was solved, for the node average flux and core k-effective, using two sets of spatial shape approximations for each of three reactor types. A fourth-order approximation in both energy groups forms the first set of approximations used. The second set used combines a second-order approximation with a fourth-order approximation in energy group two. Comparison of the results from the two approximation sets show that the use of a different order spatial flux shape approximation results in considerable loss in accuracy for the pressurized water reactor modeled. However, the loss in accuracy is small for the heavy water and graphite reactors modeled. The use of different order approximations in each energy group produces mixed results. Further investigation into the accuracy and computing time is required before any quantitative advantage of the use of the second-order approximation in energy group one and the fourth-order approximation in energy group two can be determined.« less

A numerical study of three-dimensional flame propagation over thin solids in purely forced concurrent flow including gas-phase radiation

NASA Astrophysics Data System (ADS)

Feier, Ioan I., Jr.

The effect of flame radiation on concurrent-flow flame spread over a thin solid sample of finite width in a low-speed wind tunnel is modeled using three-dimensional full Navier-Stokes equations and three-dimensional flame radiation transfer equations. The formulation includes the conservation of mass, momentum, energy, and species: fuel vapor, oxygen, carbon dioxide and water vapor. The SN discrete ordinates method is used to solve the radiation transfer equation with a mean absorption coefficient kappa = Ckappa p, where kappap is the Planck mean absorption coefficient of the gas mixture. The varying parameter C has a value between 0 and 1; C represents the strength of flame radiation. In addition, the solid fuel absorptivity alpha is varied to ascertain the effect of flame radiation heat feedback to the solid. The flow tunnel modeled has a dimension of 10x10x30 cm, the solid fuel has a width of 6-cm with two 1-cm inert strips as edges. Incoming forced flow velocity (5 cm/s) of 21% oxygen is assumed. For comparison with the three-dimensional results, corresponding two-dimensional computations are also performed. Detailed spatial flame profiles, solid surface profiles, and heat fluxes are presented. Increasing the flame radiation strength decreases the flame length. Although flame radiation provides an additional heat transfer mechanism to preheat the solid, it is insufficient to offset the decreased convective heating due to the shorter flame; the net effect is a slower spread rate. The percentage of unreacted fuel vapor that escapes from the flame is under 2%. It is theorized that some of the pyrolyzed fuel vapor diffuses sideway and reacts at the flame edges. A radiative energy balance is analyzed also. Flame radiative feedback to the solid plays a more important role in two-dimensional flames. With high solid fuel absorptivity, a peak in the flame spread rate occurs at an intermediate value of flame radiation strength---due to the competition between two mechanisms: gas-radiation heat loss weakening the flame and the radiative feedback boosting the solid pyrolysis. Two-dimensional calculations suggest that a larger percentage of unreacted fuel vapor can escape from the flame when the flame radiation strength is high.

Quantum Hamilton equations of motion for bound states of one-dimensional quantum systems

NASA Astrophysics Data System (ADS)

Köppe, J.; Patzold, M.; Grecksch, W.; Paul, W.

2018-06-01

On the basis of Nelson's stochastic mechanics derivation of the Schrödinger equation, a formal mathematical structure of non-relativistic quantum mechanics equivalent to the one in classical analytical mechanics has been established in the literature. We recently were able to augment this structure by deriving quantum Hamilton equations of motion by finding the Nash equilibrium of a stochastic optimal control problem, which is the generalization of Hamilton's principle of classical mechanics to quantum systems. We showed that these equations allow a description and numerical determination of the ground state of quantum problems without using the Schrödinger equation. We extend this approach here to deliver the complete discrete energy spectrum and related eigenfunctions for bound states of one-dimensional stationary quantum systems. We exemplify this analytically for the one-dimensional harmonic oscillator and numerically by analyzing a quartic double-well potential, a model of broad importance in many areas of physics. We furthermore point out a relation between the tunnel splitting of such models and mean first passage time concepts applied to Nelson's diffusion paths in the ground state.

Selection of intracellular calcium patterns in a model with clustered Ca2+ release channels

NASA Astrophysics Data System (ADS)

Shuai, J. W.; Jung, P.

2003-03-01

A two-dimensional model is proposed for intracellular Ca2+ waves, which incorporates both the discrete nature of Ca2+ release sites in the endoplasmic reticulum membrane and the stochastic dynamics of the clustered inositol 1,4,5-triphosphate (IP3) receptors. Depending on the Ca2+ diffusion coefficient and concentration of IP3, various spontaneous Ca2+ patterns, such as calcium puffs, local waves, abortive waves, global oscillation, and tide waves, can be observed. We further investigate the speed of the global waves as a function of the IP3 concentration and the Ca2+ diffusion coefficient and under what conditions the spatially averaged Ca2+ response can be described by a simple set of ordinary differential equations.

Basis adaptation and domain decomposition for steady partial differential equations with random coefficients

DOE PAGES

Tipireddy, R.; Stinis, P.; Tartakovsky, A. M.

2017-09-04

In this paper, we present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate low-dimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support ourmore » construction with numerical experiments for a steady-state diffusion equation with a random spatially dependent coefficient. Lastly, our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.« less

Basis adaptation and domain decomposition for steady-state partial differential equations with random coefficients

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tipireddy, R.; Stinis, P.; Tartakovsky, A. M.

We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate low-dimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support our construction with numericalmore » experiments for a steady-state diffusion equation with a random spatially dependent coefficient. Our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.« less

Whitham modulation theory for (2  +  1)-dimensional equations of Kadomtsev–Petviashvili type

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Biondini, Gino; Rumanov, Igor

2018-05-01

Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev–Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the two-dimensional Benjamin–Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich–Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.

An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher-KPP equation

NASA Astrophysics Data System (ADS)

Shapovalov, A. V.; Trifonov, A. Yu.

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher-KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

Renormalized charge in a two-dimensional model of colloidal suspension from hypernetted chain approach.

PubMed

Camargo, Manuel; Téllez, Gabriel

2008-04-07

The renormalized charge of a simple two-dimensional model of colloidal suspension was determined by solving the hypernetted chain approximation and Ornstein-Zernike equations. At the infinite dilution limit, the asymptotic behavior of the correlation functions is used to define the effective interactions between the components of the system and these effective interactions were compared to those derived from the Poisson-Boltzmann theory. The results we obtained show that, in contrast to the mean-field theory, the renormalized charge does not saturate, but exhibits a maximum value and then decays monotonically as the bare charge increases. The results also suggest that beyond the counterion layer near to the macroion surface, the ionic cloud is not a diffuse layer which can be handled by means of the linearized theory, as the two-state model claims, but a more complex structure is settled by the correlations between microions.

Generalized fractional diffusion equations for accelerating subdiffusion and truncated Lévy flights

NASA Astrophysics Data System (ADS)

Chechkin, A. V.; Gonchar, V. Yu.; Gorenflo, R.; Korabel, N.; Sokolov, I. M.

2008-08-01

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. For processes lacking such scaling the corresponding description may be given by diffusion equations with fractional derivatives of distributed order. Such equations were introduced in A. V. Chechkin, R. Gorenflo, and I. Sokolov [Phys. Rev. E 66, 046129 (2002)] for the description of the processes getting more anomalous in the course of time (decelerating subdiffusion and accelerating superdiffusion). Here we discuss the properties of diffusion equations with fractional derivatives of the distributed order for the description of anomalous relaxation and diffusion phenomena getting less anomalous in the course of time, which we call, respectively, accelerating subdiffusion and decelerating superdiffusion. For the former process, by taking a relatively simple particular example with two fixed anomalous diffusion exponents we show that the proposed equation effectively describes the subdiffusion phenomenon with diffusion exponent varying in time. For the latter process we demonstrate by a particular example how the power-law truncated Lévy stable distribution evolves in time to the distribution with power-law asymptotics and Gaussian shape in the central part. The special case of two different orders is characteristic for the general situation in which the extreme orders dominate the asymptotics.

Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-01-01

The propagation of three-dimensional nonlinear irrotational flow of an inviscid and incompressible fluid of the long waves in dispersive shallow-water approximation is analyzed. The problem formulation of the long waves in dispersive shallow-water approximation lead to fifth-order Kadomtsev-Petviashvili (KP) dynamical equation by applying the reductive perturbation theory. By using an extended auxiliary equation method, the solitary travelling-wave solutions of the two-dimensional nonlinear fifth-order KP dynamical equation are derived. An analytical as well as a numerical solution of the two-dimensional nonlinear KP equation are obtained and analyzed with the effects of external pressure flow.

Validation of a numerical method for interface-resolving simulation of multicomponent gas-liquid mass transfer and evaluation of multicomponent diffusion models

NASA Astrophysics Data System (ADS)

Woo, Mino; Wörner, Martin; Tischer, Steffen; Deutschmann, Olaf

2018-03-01

The multicomponent model and the effective diffusivity model are well established diffusion models for numerical simulation of single-phase flows consisting of several components but are seldom used for two-phase flows so far. In this paper, a specific numerical model for interfacial mass transfer by means of a continuous single-field concentration formulation is combined with the multicomponent model and effective diffusivity model and is validated for multicomponent mass transfer. For this purpose, several test cases for one-dimensional physical or reactive mass transfer of ternary mixtures are considered. The numerical results are compared with analytical or numerical solutions of the Maxell-Stefan equations and/or experimental data. The composition-dependent elements of the diffusivity matrix of the multicomponent and effective diffusivity model are found to substantially differ for non-dilute conditions. The species mole fraction or concentration profiles computed with both diffusion models are, however, for all test cases very similar and in good agreement with the analytical/numerical solutions or measurements. For practical computations, the effective diffusivity model is recommended due to its simplicity and lower computational costs.

Correction to verdonck and tuerlinckx (2014).

PubMed

2015-01-01

Reports an error in "The Ising Decision Maker: A binary stochastic network for choice response time" by Stijn Verdonck and Francis Tuerlinckx (Psychological Review, 2014[Jul], Vol 121[3], 422-462). An inaccurate assumption in Appendix B (provided in the erratum) led to an oversimplified result in Equation 18 (the diffusion equations associated with the microscopically defined dynamics). The authors sincerely thank Rani Moran for making them aware of the problem. Only the expression of the diffusion coefficient D is incorrect, and should be changed, as indicated in the erratum. Both the cause of the problem and the solution are also explained in the erratum. (The following abstract of the original article appeared in record 2014-31650-006.) The Ising Decision Maker (IDM) is a new formal model for speeded two-choice decision making derived from the stochastic Hopfield network or dynamic Ising model. On a microscopic level, it consists of 2 pools of binary stochastic neurons with pairwise interactions. Inside each pool, neurons excite each other, whereas between pools, neurons inhibit each other. The perceptual input is represented by an external excitatory field. Using methods from statistical mechanics, the high-dimensional network of neurons (microscopic level) is reduced to a two-dimensional stochastic process, describing the evolution of the mean neural activity per pool (macroscopic level). The IDM can be seen as an abstract, analytically tractable multiple attractor network model of information accumulation. In this article, the properties of the IDM are studied, the relations to existing models are discussed, and it is shown that the most important basic aspects of two-choice response time data can be reproduced. In addition, the IDM is shown to predict a variety of observed psychophysical relations such as Piéron's law, the van der Molen-Keuss effect, and Weber's law. Using Bayesian methods, the model is fitted to both simulated and real data, and its performance is compared to the Ratcliff diffusion model. (PsycINFO Database Record (c) 2015 APA, all rights reserved).

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

NASA Astrophysics Data System (ADS)

de Almeida, Valmor F.

2017-07-01

A phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equation and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.

Viscous, resistive MHD stability computed by spectral techniques

NASA Technical Reports Server (NTRS)

Dahlburg, R. B.; Zang, T. A.; Montgomery, D.; Hussaini, M. Y.

1983-01-01

Expansions in Chebyshev polynomials are used to study the linear stability of one dimensional magnetohydrodynamic (MHD) quasi-equilibria, in the presence of finite resistivity and viscosity. The method is modeled on the one used by Orszag in accurate computation of solutions of the Orr-Sommerfeld equation. Two Reynolds like numbers involving Alfven speeds, length scales, kinematic viscosity, and magnetic diffusivity govern the stability boundaries, which are determined by the geometric mean of the two Reynolds like numbers. Marginal stability curves, growth rates versus Reynolds like numbers, and growth rates versus parallel wave numbers are exhibited. A numerical result which appears general is that instability was found to be associated with inflection points in the current profile, though no general analytical proof has emerged. It is possible that nonlinear subcritical three dimensional instabilities may exist, similar to those in Poiseuille and Couette flow.

Thermoelectric transport in two-dimensional giant Rashba systems

NASA Astrophysics Data System (ADS)

Xiao, Cong; Li, Dingping; Ma, Zhongshui; Niu, Qian

Thermoelectric transport in strongly spin-orbit coupled two-dimensional Rashba systems is studied using the analytical solution of the linearized Boltzmann equation. To highlight the effects of inter-band scattering, we assume point-like potential impurities, and obtain the band-and energy-dependent transport relaxation times. Unconventional transport behaviors arise when the Fermi level lies near or below the band crossing point (BCP), such as the non-Drude electrical conducivity below the BCP, the failure of the standard Mott relation linking the Peltier coefficient to the electrical conductivity near the BCP, the enhancement of diffusion thermopower and figure of merit below the BCP, the zero-field Hall coefficient which is not inversely proportional to and not a monotonic function of the carrier density, the enhanced Nernst coefficient below the BCP, and the enhanced current-induced spin-polarization efficiency.

A ray tracing model for leaf bidirectional scattering studies

NASA Technical Reports Server (NTRS)

Brakke, T. W.; Smith, J. A.

1987-01-01

A leaf is modeled as a deterministic two-dimensional structure consisting of a network of circular arcs designed to represent the internal morphology of major species. The path of an individual ray through the leaf is computed using geometric optics. At each intersection of the ray with an arc, the specular reflected and transmitted rays are calculated according to the Snell and Fresnel equations. Diffuse scattering is treated according to Lambert's law. Absorption is also permitted but requires a detailed knowledge of the spectral attenuation coefficients. An ensemble of initial rays are chosen for each incident direction with the initial intersection points on the leaf surface selected randomly. The final equilibrium state after all interactions then yields the leaf bidirectional reflectance and transmittance distributions. The model also yields the internal two dimensional light gradient profile of the leaf.

An Investigation of Wave Propagations in Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Hu, Fang Q.

2004-01-01

Analysis of the discontinuous Galerkin method has been carried out for one- and two-dimensional system of hyperbolic equations. Analytical, as well as numerical, properties of wave propagation in a DGM scheme are derived and verified with direct numerical simulations. In addition to a systematic examination of the dissipation and dispersion errors, behaviours of a DG scheme at an interface of two different grid topologies are also studied. Under the same framework, a quantitative discrete analysis of various artificial boundary conditions is also conducted. Progress has been made in numerical boundary condition treatment that is closely related to the application of DGM in aeroacoustics problems. Finally, Fourier analysis of DGM for the Convective diffusion equation has also be studied in connection with the application of DG schemes for the Navier-Stokes equations. This research has resulted in five(5) publications, plus one additional manuscript in preparation, four(4) conference presentations, and three(3) departmental seminars, as summarized in part II. Abstracts of papers are given in part 111 of this report.

Numerical simulation of rarefied gas flow through a slit

NASA Technical Reports Server (NTRS)

Keith, Theo G., Jr.; Jeng, Duen-Ren; De Witt, Kenneth J.; Chung, Chan-Hong

1990-01-01

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas from one reservoir to another through a two-dimensional slit. The cases considered are for hard vacuum downstream pressure, finite pressure ratios, and isobaric pressure with thermal diffusion, which are not well established in spite of the simplicity of the flow field. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, three kinds of collision sampling techniques, the time counter (TC) method, the null collision (NC) method, and the no time counter (NTC) method, are used.

A stratospheric aerosol model with perturbations induced by the space shuttle particulate effluents

NASA Technical Reports Server (NTRS)

Rosen, J. M.; Hofmann, D. J.

1977-01-01

A one dimensional steady state stratospheric aerosol model is developed that considers the subsequent perturbations caused by including the expected space shuttle particulate effluents. Two approaches to the basic modeling effort were made: in one, enough simplifying assumptions were introduced so that a more or less exact solution to the descriptive equations could be obtained; in the other approach very few simplifications were made and a computer technique was used to solve the equations. The most complex form of the model contains the effects of sedimentation, diffusion, particle growth and coagulation. Results of the perturbation calculations show that there will probably be an immeasurably small increase in the stratospheric aerosol concentration for particles larger than about 0.15 micrometer radius.

Subdiffusion in Membrane Permeation of Small Molecules.

PubMed

Chipot, Christophe; Comer, Jeffrey

2016-11-02

Within the solubility-diffusion model of passive membrane permeation of small molecules, translocation of the permeant across the biological membrane is traditionally assumed to obey the Smoluchowski diffusion equation, which is germane for classical diffusion on an inhomogeneous free-energy and diffusivity landscape. This equation, however, cannot accommodate subdiffusive regimes, which have long been recognized in lipid bilayer dynamics, notably in the lateral diffusion of individual lipids. Through extensive biased and unbiased molecular dynamics simulations, we show that one-dimensional translocation of methanol across a pure lipid membrane remains subdiffusive on timescales approaching typical permeation times. Analysis of permeant motion within the lipid bilayer reveals that, in the absence of a net force, the mean squared displacement depends on time as t 0.7 , in stark contrast with the conventional model, which assumes a strictly linear dependence. We further show that an alternate model using a fractional-derivative generalization of the Smoluchowski equation provides a rigorous framework for describing the motion of the permeant molecule on the pico- to nanosecond timescale. The observed subdiffusive behavior appears to emerge from a crossover between small-scale rattling of the permeant around its present position in the membrane and larger-scale displacements precipitated by the formation of transient voids.

Exact solutions and conservation laws of the system of two-dimensional viscous Burgers equations

NASA Astrophysics Data System (ADS)

Abdulwahhab, Muhammad Alim

2016-10-01

Fluid turbulence is one of the phenomena that has been studied extensively for many decades. Due to its huge practical importance in fluid dynamics, various models have been developed to capture both the indispensable physical quality and the mathematical structure of turbulent fluid flow. Among the prominent equations used for gaining in-depth insight of fluid turbulence is the two-dimensional Burgers equations. Its solutions have been studied by researchers through various methods, most of which are numerical. Being a simplified form of the two-dimensional Navier-Stokes equations and its wide range of applicability in various fields of science and engineering, development of computationally efficient methods for the solution of the two-dimensional Burgers equations is still an active field of research. In this study, Lie symmetry method is used to perform detailed analysis on the system of two-dimensional Burgers equations. Optimal system of one-dimensional subalgebras up to conjugacy is derived and used to obtain distinct exact solutions. These solutions not only help in understanding the physical effects of the model problem but also, can serve as benchmarks for constructing algorithms and validation of numerical solutions of the system of Burgers equations under consideration at finite Reynolds numbers. Independent and nontrivial conserved vectors are also constructed.

Rigorous results for the minimal speed of Kolmogorov-Petrovskii-Piscounov monotonic fronts with a cutoffa)

NASA Astrophysics Data System (ADS)

Benguria, Rafael D.; Depassier, M. Cristina; Loss, Michael

2012-12-01

We study the effect of a cutoff on the speed of pulled fronts of the one-dimensional reaction diffusion equation. To accomplish this, we first use variational techniques to prove the existence of a heteroclinic orbit in phase space for traveling wave solutions of the corresponding reaction diffusion equation under conditions that include discontinuous reaction profiles. This existence result allows us to prove rigorous upper and lower bounds on the minimal speed of monotonic fronts in terms of the cut-off parameter ɛ. From these bounds we estimate the range of validity of the Brunet-Derrida formula for a general class of reaction terms.

A computer model for one-dimensional mass and energy transport in and around chemically reacting particles, including complex gas-phase chemistry, multicomponent molecular diffusion, surface evaporation, and heterogeneous reaction

NASA Technical Reports Server (NTRS)

Cho, S. Y.; Yetter, R. A.; Dryer, F. L.

1992-01-01

Various chemically reacting flow problems highlighting chemical and physical fundamentals rather than flow geometry are presently investigated by means of a comprehensive mathematical model that incorporates multicomponent molecular diffusion, complex chemistry, and heterogeneous processes, in the interest of obtaining sensitivity-related information. The sensitivity equations were decoupled from those of the model, and then integrated one time-step behind the integration of the model equations, and analytical Jacobian matrices were applied to improve the accuracy of sensitivity coefficients that are calculated together with model solutions.

Analysis of the discontinuous Galerkin method applied to the European option pricing problem

NASA Astrophysics Data System (ADS)

Hozman, J.

2013-12-01

In this paper we deal with a numerical solution of a one-dimensional Black-Scholes partial differential equation, an important scalar nonstationary linear convection-diffusion-reaction equation describing the pricing of European vanilla options. We present a derivation of the numerical scheme based on the space semidiscretization of the model problem by the discontinuous Galerkin method with nonsymmetric stabilization of diffusion terms and with the interior and boundary penalty. The main attention is paid to the investigation of a priori error estimates for the proposed scheme. The appended numerical experiments illustrate the theoretical results and the potency of the method, consequently.

Development and application of a three dimensional numerical model for predicting pollutant and sediment transport using an Eulerian-Lagrangian marker particle technique

NASA Technical Reports Server (NTRS)

Pavish, D. L.; Spaulding, M. L.

1977-01-01

A computer coded Lagrangian marker particle in Eulerian finite difference cell solution to the three dimensional incompressible mass transport equation, Water Advective Particle in Cell Technique, WAPIC, was developed, verified against analytic solutions, and subsequently applied in the prediction of long term transport of a suspended sediment cloud resulting from an instantaneous dredge spoil release. Numerical results from WAPIC were verified against analytic solutions to the three dimensional incompressible mass transport equation for turbulent diffusion and advection of Gaussian dye releases in unbounded uniform and uniformly sheared uni-directional flow, and for steady-uniform plug channel flow. WAPIC was utilized to simulate an analytic solution for non-equilibrium sediment dropout from an initially vertically uniform particle distribution in one dimensional turbulent channel flow.

Steady motion of skyrmions and domains walls under diffusive spin torques

NASA Astrophysics Data System (ADS)

Elías, Ricardo Gabriel; Vidal-Silva, Nicolas; Manchon, Aurélien

2017-03-01

We explore the role of the spin diffusion of conducting electrons in two-dimensional magnetic textures (domain walls and skyrmions) with spatial variation of the order of the spin precession length λex. The effect of diffusion reflects in four additional torques that are third order in spatial derivatives of magnetization and bilinear in λex and in the nonadiabatic parameter β'. In order to study the dynamics of the solitons when these diffusive torques are present, we derive the Thiele equation in the limit of steady motion and we compare the results with the nondiffusive limit. When considering a homogenous current these torques increase the longitudinal velocity of transverse domain walls of width Δ by a factor (λex/Δ)2(α/3), α being the magnetic damping constant. In the case of single skyrmions with core radius r0these new contributions tend to increase the Magnus effect in an amount proportional to (λex/r0) 2(1 +2 α β') .

Numerical Analysis of Transient Temperature Response of Soap Film

NASA Astrophysics Data System (ADS)

Tanaka, Seiichi; Tatesaku, Akihiro; Dantsuka, Yuki; Fujiwara, Seiji; Kunimine, Kanji

2015-11-01

Measurements of thermophysical properties of thin liquid films are important to understand interfacial phenomena due to film structures composed of amphiphilic molecules in soap film, phospholipid bilayer of biological cell and emulsion. A transient hot-wire technique for liquid films less than 1 \\upmu m thick such as soap film has been proposed to measure the thermal conductivity and diffusivity simultaneously. Two-dimensional heat conduction equations for a solid cylinder with a liquid film have been solved numerically. The temperature of a thin wire with liquid film increases steeply with its own heat generation. The feasibility of this technique is verified through numerical experiments for various thermal conductivities, diffusivities, and film thicknesses. Calculated results indicate that the increase in the volumetric average temperature of the thin wire sufficiently varies with the change of thermal conductivity and diffusivity of the soap film. Therefore, the temperature characteristics could be utilized to evaluate both the thermal conductivity and diffusivity using the Gauss-Newton method.

A radially resolved kinetic model for nonlocal electron ripple diffusion losses in tokamaks

DOE Office of Scientific and Technical Information (OSTI.GOV)

Robertson, Scott

A relatively simple radially resolved kinetic model is applied to the ripple diffusion problem for electrons in tokamaks. The distribution function f(r,v) is defined on a two-dimensional grid, where r is the radial coordinate and v is the velocity coordinate. Particle transport in the radial direction is from ripple and banana diffusion and transport in the velocity direction is described by the Fokker-Planck equation. Particles and energy are replaced by source functions that are adjusted to maintain a constant central density and temperature. The relaxed profiles of f(r,v) show that the electron distribution function at the wall contains suprathermal electronsmore » that have diffused from the interior that enhance ripple transport. The transport at the periphery is therefore nonlocal. The energy replacement times from the computational model are near to the experimental replacement times for tokamak discharges in the compilation by Pfeiffer and Waltz [Nucl. Fusion 19, 51 (1979)].« less

Repeated-cascade theory of strong turbulence in a magnetized plasma

NASA Technical Reports Server (NTRS)

Tchen, C. M.

1976-01-01

A two-dimensional Navier-Stokes equation of vorticity in fluid turbulence is used to model drift turbulence in a plasma with a strong constant magnetic field and a constant mean density gradient. The nonlinear eddy diffusivity is described by a time-integrated Lagrangian correlation of velocities, and the repeated-cascade method is employed to choose the rank accounting for nearest-neighbor interactions, to calculate the Lagrangian correlation, and to close the correlation hierarchy. As a result, the diffusivity becomes dependent on the plasma's induced diffusion and is represented by a memory chain that is cut off by similarity and inertial randomization. Spectral laws relating the kinetic-energy spectrum to the -5, -5/2, -3, and -11 powers of wavenumber are derived for the velocity subranges of production, approach to inertia, inertia, and dissipation, respectively. It is found that the diffusivity is proportional to some inverse power of the magnetic field, that power being 1, 2/3, 5/6, and 2, respectively, for the four velocity subranges.

A parallel algorithm for nonlinear convection-diffusion equations

NASA Technical Reports Server (NTRS)

Scroggs, Jeffrey S.

1990-01-01

A parallel algorithm for the efficient solution of nonlinear time-dependent convection-diffusion equations with small parameter on the diffusion term is presented. The method is based on a physically motivated domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. The method is suitable for the solution of problems arising in the simulation of fluid dynamics. Experimental results for a nonlinear equation in two-dimensions are presented.

AN IMMERSED BOUNDARY METHOD FOR COMPLEX INCOMPRESSIBLE FLOWS

EPA Science Inventory

An immersed boundary method for time-dependant, three- dimensional, incompressible flows is presented in this paper. The incompressible Navier-Stokes equations are discretized using a low-diffusion flux splitting method for the inviscid fluxes and a second order central differenc...

Cattaneo-Christov double-diffusion theory for three-dimensional flow of viscoelastic nanofluid with the effect of heat generation/absorption

NASA Astrophysics Data System (ADS)

Hayat, Tasawar; Qayyum, Sajid; Shehzad, Sabir Ali; Alsaedi, Ahmed

2018-03-01

The present research article focuses on three-dimensional flow of viscoelastic(second grade) nanofluid in the presence of Cattaneo-Christov double-diffusion theory. Flow caused is due to stretching sheet. Characteristics of heat transfer are interpreted by considering the heat generation/absorption. Nanofluid theory comprises of Brownian motion and thermophoresis. Cattaneo-Christov double-diffusion theory is introduced in the energy and concentration expressions. Such diffusions are developed as a part of formulating the thermal and solutal relaxation times framework. Suitable variables are implemented for the conversion of partial differential systems into a sets of ordinary differential equations. The transformed expressions have been explored through homotopic algorithm. Behavior of sundry variables on the velocities, temperature and concentration are scrutinized graphically. Numerical values of skin friction coefficients are also calculated and examined. Here thermal field enhances for heat generation parameter while reverse situation is noticed for heat absorption parameter.

On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics

NASA Astrophysics Data System (ADS)

Motsepa, Tanki; Masood Khalique, Chaudry

2018-05-01

In this paper, we study a (2+1) dimensional KdV-mKdV equation, which models many physical phenomena of mathematical physics. This equation has two integral terms in it. By an appropriate substitution, we convert this equation into two partial differential equations, which do not have integral terms and are equivalent to the original equation. We then work with the system of two equations and obtain its exact travelling wave solutions in form of Jacobi elliptic functions. Furthermore, we employ the multiplier method to construct conservation laws for the system. Finally, we revert the results obtained into the original variables of the (2+1) dimensional KdV-mKdV equation.

Geometrical approach to tumor growth.

PubMed

Escudero, Carlos

2006-08-01

Tumor growth has a number of features in common with a physical process known as molecular beam epitaxy. Both growth processes are characterized by the constraint of growth development to the body border, and surface diffusion of cells and particles at the growing edge. However, tumor growth implies an approximate spherical symmetry that makes necessary a geometrical treatment of the growth equations. The basic model was introduced in a former paper [C. Escudero, Phys. Rev. E 73, 020902(R) (2006)], and in the present work we extend our analysis and try to shed light on the possible geometrical principles that drive tumor growth. We present two-dimensional models that reproduce the experimental observations, and analyze the unexplored three-dimensional case, for which interesting conclusions on tumor growth are derived.

Fluctuations and symmetries in two-dimensional active gels.

PubMed

Sarkar, N; Basu, A

2011-04-01

Motivated by the unique physical properties of biological active matter, e.g., cytoskeletal dynamics in eukaryotic cells, we set up effective two-dimensional (2d) coarse-grained hydrodynamic equations for the dynamics of thin active gels with polar or nematic symmetries. We use the well-known three-dimensional (3d) descriptions (K. Kruse et al., Eur. Phys. J. E 16, 5 (2005); A. Basu et al., Eur. Phys. J. E 27, 149 (2008)) for thin active-gel samples confined between parallel plates with appropriate boundary conditions to derive the effective 2d constitutive relations between appropriate thermodynamic fluxes and generalised forces for small deviations from equilibrium. We consider three distinct cases, characterised by spatial symmetries and boundary conditions, and show how such considerations dictate the structure of the constitutive relations. We use these to study the linear instabilities, calculate the correlation functions and the diffusion constant of a small tagged particle, and elucidate their dependences on the activity or nonequilibrium drive.

[Effects of several low-molecular-weight organic acids on the release kinetic of endosulfan from red soil].

PubMed

Zhao, Zhen-hua; Wu, Yu; Jiang, Xin; Xia, Li-ling; Ni, Li-xiao

2009-10-15

The kinetic release behaviors of a-endosulfan from red soil with three kinds of low-molecular-weight organic acids (LMWOA: oxalate, tartrate and citrate) solution and water leaching were investigated by kinetic device designed by ourselves and batch method. The results show that: the release percentage of endosulfan from red soil by tartrate and citrate solution (10 mmol/L) can increase by 7%-18% more than that by distilled water and oxalate solution, especially for tartrate solution. There is no significant difference between distilled water and oxalate solution for the release percentage of endosulfan (p > 0.05). There are two stages of quick and slow for the release of endosulfan from red soil, and the leaching speed is quicker especially for the initial 200 mL leaching solution. When using distilled water or oxalate solution as leaching solution, the best equations that described the kinetic release behavior of endosulfan from red soil were parabola diffuse equation and double constant equation, and weren't the apparent first dynamics equation that represented the simple surface diffusion mechanism. The kinetic release behavior of endosulfan in tartrate or citrate leaching system can be described by Elovich equation (R2 > 0.99, p < 0.0001), it implied that the simple surface diffusion mechanism is not the primary factor that effected the release of endosulfan, which three-dimensional molecule structure is complex, from red soil in aqueous phase leaching systems, and it maybe related to the outward diffuse mechanism from soil particle, activation and deactivation function of soil particles surface, the dissolution of soil mineral surface and structure change of inherent organic matter that coating onto the soil mineral surface induced by LMW organic acid. It suggested that the tartrate and citrate induced the complication of the release mechanisms of the pesticides from red soil.

Homogenization of Large-Scale Movement Models in Ecology

USGS Publications Warehouse

Garlick, M.J.; Powell, J.A.; Hooten, M.B.; McFarlane, L.R.

2011-01-01

A difficulty in using diffusion models to predict large scale animal population dispersal is that individuals move differently based on local information (as opposed to gradients) in differing habitat types. This can be accommodated by using ecological diffusion. However, real environments are often spatially complex, limiting application of a direct approach. Homogenization for partial differential equations has long been applied to Fickian diffusion (in which average individual movement is organized along gradients of habitat and population density). We derive a homogenization procedure for ecological diffusion and apply it to a simple model for chronic wasting disease in mule deer. Homogenization allows us to determine the impact of small scale (10-100 m) habitat variability on large scale (10-100 km) movement. The procedure generates asymptotic equations for solutions on the large scale with parameters defined by small-scale variation. The simplicity of this homogenization procedure is striking when compared to the multi-dimensional homogenization procedure for Fickian diffusion,and the method will be equally straightforward for more complex models. ?? 2010 Society for Mathematical Biology.

One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

NASA Astrophysics Data System (ADS)

Sakaguchi, Hidetsugu; Ishibashi, Kazuya

2018-06-01

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable ϕ. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.

Steady MHD free convection heat and mass transfer flow about a vertical porous surface with thermal diffusion and induced magnetic field

NASA Astrophysics Data System (ADS)

Touhid Hossain, M. M.; Afruz-Zaman, Md.; Rahman, Fouzia; Hossain, M. Arif

2013-09-01

In this study the thermal diffusion effect on the steady laminar free convection flow and heat transfer of viscous incompressible MHD electrically conducting fluid above a vertical porous surface is considered under the influence of an induced magnetic field. The governing non-dimensional equations relevant to the problem, containing the partial differential equations, are transformed by usual similarity transformations into a system of coupled non-linear ordinary differential equations and will be solved analytically by using the perturbation technique. On introducing the non-dimensional concept and applying Boussinesq's approximation, the solutions for velocity field, temperature distribution and induced magnetic field to the second order approximations are obtained for large suction with different selected values of the established dimensionless parameters. The influences of these various establish parameters on the velocity and temperature fields and on the induced magnetic fields are exhibited under certain assumptions and are studied graphically in the present analysis. It is observed that the effects of thermal-diffusion and large suction have great importance on the velocity, temperature and induced magnetic fields and mass concentration for several fluids considered, so that their effects should be taken into account with other useful parameters associated. It is also found that the dimensionless Prandtl number, Grashof number, Modified Grashof number and magnetic parameter have an appreciable influence on the concerned independent variables.

Three New (2+1)-dimensional Integrable Systems and Some Related Darboux Transformations

NASA Astrophysics Data System (ADS)

Guo, Xiu-Rong

2016-06-01

We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1, then under the framework of zero curvature equations we generate two (2+1)-dimensional integrable hierarchies, including the (2+1)-dimensional shallow water wave (SWW) hierarchy and the (2+1)-dimensional Kaup-Newell (KN) hierarchy. Through reduction of the (2+1)-dimensional hierarchies, we get a (2+1)-dimensional SWW equation and a (2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the (2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the (2+1)-dimensional KN equation could be deduced. Finally, with the help of the spatial spectral matrix of SWW hierarchy, we generate a (2+1) heat equation and a (2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang-Mills equations. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Shandong Provincial Natural Science Foundation of China under Grant Nos. ZR2012AQ011, ZR2013AL016, ZR2015EM042, National Social Science Foundation of China under Grant No. 13BJY026, the Development of Science and Technology Project under Grant No. 2015NS1048 and A Project of Shandong Province Higher Educational Science and Technology Program under Grant No. J14LI58

Computation of three-dimensional three-phase flow of carbon dioxide using a high-order WENO scheme

NASA Astrophysics Data System (ADS)

Gjennestad, Magnus Aa.; Gruber, Andrea; Lervåg, Karl Yngve; Johansen, Øyvind; Ervik, Åsmund; Hammer, Morten; Munkejord, Svend Tollak

2017-11-01

We have developed a high-order numerical method for the 3D simulation of viscous and inviscid multiphase flow described by a homogeneous equilibrium model and a general equation of state. Here we focus on single-phase, two-phase (gas-liquid or gas-solid) and three-phase (gas-liquid-solid) flow of CO2 whose thermodynamic properties are calculated using the Span-Wagner reference equation of state. The governing equations are spatially discretized on a uniform Cartesian grid using the finite-volume method with a fifth-order weighted essentially non-oscillatory (WENO) scheme and the robust first-order centered (FORCE) flux. The solution is integrated in time using a third-order strong-stability-preserving Runge-Kutta method. We demonstrate close to fifth-order convergence for advection-diffusion and for smooth single- and two-phase flows. Quantitative agreement with experimental data is obtained for a direct numerical simulation of an air jet flowing from a rectangular nozzle. Quantitative agreement is also obtained for the shape and dimensions of the barrel shock in two highly underexpanded CO2 jets.

Global dynamics of a nonlocal delayed reaction-diffusion equation on a half plane

NASA Astrophysics Data System (ADS)

Hu, Wenjie; Duan, Yueliang

2018-04-01

We consider a delayed reaction-diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson's blowfly equation and the diffusive Mackey-Glass equation.

Numerical study of the flow in a three-dimensional thermally driven cavity

NASA Astrophysics Data System (ADS)

Rauwoens, Pieter; Vierendeels, Jan; Merci, Bart

2008-06-01

Solutions for the fully compressible Navier-Stokes equations are presented for the flow and temperature fields in a cubic cavity with large horizontal temperature differences. The ideal-gas approximation for air is assumed and viscosity is computed using Sutherland's law. The three-dimensional case forms an extension of previous studies performed on a two-dimensional square cavity. The influence of imposed boundary conditions in the third dimension is investigated as a numerical experiment. Comparison is made between convergence rates in case of periodic and free-slip boundary conditions. Results with no-slip boundary conditions are presented as well. The effect of the Rayleigh number is studied. Results are computed using a finite volume method on a structured, collocated grid. An explicit third-order discretization for the convective part and an implicit central discretization for the acoustic part and for the diffusive part are used. To stabilize the scheme an artificial dissipation term for the pressure and the temperature is introduced. The discrete equations are solved using a time-marching method with restrictions on the timestep corresponding to the explicit parts of the solver. Multigrid is used as acceleration technique.

Effects of Drift-Shell Splitting by Chorus Waves on Radiation Belt Electrons

NASA Astrophysics Data System (ADS)

Chan, A. A.; Zheng, L.; O'Brien, T. P., III; Tu, W.; Cunningham, G.; Elkington, S. R.; Albert, J.

2015-12-01

Drift shell splitting in the radiation belts breaks all three adiabatic invariants of charged particle motion via pitch angle scattering, and produces new diffusion terms that fully populate the diffusion tensor in the Fokker-Planck equation. Based on the stochastic differential equation method, the Radbelt Electron Model (REM) simulation code allows us to solve such a fully three-dimensional Fokker-Planck equation, and to elucidate the sources and transport mechanisms behind the phase space density variations. REM has been used to perform simulations with an empirical initial phase space density followed by a seed electron injection, with a Tsyganenko 1989 magnetic field model, and with chorus wave and ULF wave diffusion models. Our simulation results show that adding drift shell splitting changes the phase space location of the source to smaller L shells, which typically reduces local electron energization (compared to neglecting drift-shell splitting effects). Simulation results with and without drift-shell splitting effects are compared with Van Allen Probe measurements.

Taylor dispersion in two-dimensional bacterial turbulence

NASA Astrophysics Data System (ADS)

Huang, Yongxiang; Ou, Wenyu; Chen, Ming; Lu, Zhiming; Jiang, Nan; Liu, Yulu; Qiu, Xiang; Zhou, Quan

2017-05-01

In this work, single particle dispersion was analyzed for a bacterial turbulence by retrieving the virtual Lagrangian trajectory via numerical integration of the Lagrangian equation. High-order displacement functions were calculated for cases with and without mean velocity effect. The two-regime power-law behavior for short and long time evolutions was identified experimentally, which was separated by the Lagrangian integral time. For the case with the mean velocity effect, the experimental Hurst numbers were determined to be 0.94 and 0.97 for short and long time evolutions, respectively. For the case without the mean velocity effect, the values were 0.88 and 0.58. Moreover, very weak intermittency correction was detected. All measured Hurst numbers were above 1/2, the value of the normal diffusion, which verifies the super-diffusion behavior of living fluid. This behavior increases the efficiency of bacteria to obtain food.

The solution of non-linear hyperbolic equation systems by the finite element method

NASA Technical Reports Server (NTRS)

Loehner, R.; Morgan, K.; Zienkiewicz, O. C.

1984-01-01

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

A moving mesh finite difference method for equilibrium radiation diffusion equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yang, Xiaobo, E-mail: xwindyb@126.com; Huang, Weizhang, E-mail: whuang@ku.edu; Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn

2015-10-01

An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativitymore » of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.« less

Non-linear instability analysis of the two-dimensional Navier-Stokes equation: The Taylor-Green vortex problem

NASA Astrophysics Data System (ADS)

Sengupta, Tapan K.; Sharma, Nidhi; Sengupta, Aditi

2018-05-01

An enstrophy-based non-linear instability analysis of the Navier-Stokes equation for two-dimensional (2D) flows is presented here, using the Taylor-Green vortex (TGV) problem as an example. This problem admits a time-dependent analytical solution as the base flow, whose instability is traced here. The numerical study of the evolution of the Taylor-Green vortices shows that the flow becomes turbulent, but an explanation for this transition has not been advanced so far. The deviation of the numerical solution from the analytical solution is studied here using a high accuracy compact scheme on a non-uniform grid (NUC6), with the fourth-order Runge-Kutta method. The stream function-vorticity (ψ, ω) formulation of the governing equations is solved here in a periodic square domain with four vortices at t = 0. Simulations performed at different Reynolds numbers reveal that numerical errors in computations induce a breakdown of symmetry and simultaneous fragmentation of vortices. It is shown that the actual physical instability is triggered by the growth of disturbances and is explained by the evolution of disturbance mechanical energy and enstrophy. The disturbance evolution equations have been traced by looking at (a) disturbance mechanical energy of the Navier-Stokes equation, as described in the work of Sengupta et al., "Vortex-induced instability of an incompressible wall-bounded shear layer," J. Fluid Mech. 493, 277-286 (2003), and (b) the creation of rotationality via the enstrophy transport equation in the work of Sengupta et al., "Diffusion in inhomogeneous flows: Unique equilibrium state in an internal flow," Comput. Fluids 88, 440-451 (2013).

Two-Dimensional Array Beam Scanning Via Externally and Mutually Injection Locked Coupled Oscillators

NASA Technical Reports Server (NTRS)

Pogorzelski, Ronald J.

2000-01-01

Some years ago, Stephan proposed an approach to one dimensional (linear) phased array beam steering which requires only a single phase shifter. This involves the use of a linear array of voltage-controlled electronic oscillators coupled to nearest neighbors. The oscillators are mutually injection locked by controlling their coupling and tuning appropriately. Stephan's approach consists of deriving two signals from a master oscillator, one signal phase shifted with respect to the other by means of a single phase shifter. These two signals are injected into the end oscillators of the array. The result is a linear phase progression across the oscillator array. Thus, if radiating elements are connected to each oscillator and spaced uniformly along a line, they will radiate a beam at an angle to that line determined by the phase gradient which is, in turn, determined by the phase difference between the injection signals.The beam direction is therefore controlled by adjusting this phase difference. Recently, Pogorzelski and York presented a formulation which facilitates theoretical analysis of the above beam steering technique. This was subsequently applied by Pogorzelski in analysis of two dimensional beam steering using perimeter detuning of a coupled oscillator array. The formulation is based on a continuum model in which the oscillator phases are represented by a continuous function satisfying a partial differential equation of diffusion type. This equation can be solved via the Laplace transform and the resulting solution exhibits the dynamic behavior of the array as the beam is steered. Stephan's beam steering technique can be similarly generalized to two-dimensional arrays in which the beam control signals are applied to the oscillators on the perimeter of the array. In this paper the continuum model for this two-dimensional case is developed and the dynamic solution for the corresponding aperture phase function is obtained. The corresponding behavior of the resulting far-zone radiation pattern is displayed as well.

Infinite Conservation Laws, Continuous Symmetries and Invariant Solutions of Some Discrete Integrable Equations

NASA Astrophysics Data System (ADS)

Zhang, Yu-Feng; Zhang, Xiang-Zhi; Dong, Huan-He

2017-12-01

Two new shift operators are introduced for which a few differential-difference equations are generated by applying the R-matrix method. These equations can be reduced to the standard Toda lattice equation and (1+1)-dimensional and (2+1)-dimensional Toda-type equations which have important applications in hydrodynamics, plasma physics, and so on. Based on these consequences, we deduce the Hamiltonian structures of two discrete systems. Finally, we obtain some new infinite conservation laws of two discrete equations and employ Lie-point transformation group to obtain some continuous symmetries and part of invariant solutions for the (1+1) and (2+1)-dimensional Toda-type equations. Supported by the Fundamental Research Funds for the Central University under Grant No. 2017XKZD11

Multiple-scattering coefficients and absorption controlled diffusive processes

NASA Astrophysics Data System (ADS)

Godoy, Salvador; García-Colín, L. S.; Micenmacher, Victor

1999-11-01

Multiple-scattering transmission and reflection coefficients (T,R) are introduced in addition to the diffusion coefficient D for the description of ballistic diffusion in the presence of absorption. For 1D (one-dimensional) systems, the measurement of only one between T and D imposes restrictions on the possible values of the other. If D is measured, then T is bounded between the Landauer and Lambert-Beer equations. Measurements of both (T,D) imply the theoretical knowledge of the microscopic absorption Σa and scattering rΣs cross sections.

Hydrodynamics of confined colloidal fluids in two dimensions

NASA Astrophysics Data System (ADS)

Sané, Jimaan; Padding, Johan T.; Louis, Ard A.

2009-05-01

We apply a hybrid molecular dynamics and mesoscopic simulation technique to study the dynamics of two-dimensional colloidal disks in confined geometries. We calculate the velocity autocorrelation functions and observe the predicted t-1 long-time hydrodynamic tail that characterizes unconfined fluids, as well as more complex oscillating behavior and negative tails for strongly confined geometries. Because the t-1 tail of the velocity autocorrelation function is cut off for longer times in finite systems, the related diffusion coefficient does not diverge but instead depends logarithmically on the overall size of the system. The Langevin equation gives a poor approximation to the velocity autocorrelation function at both short and long times.

Multiexponential models of (1+1)-dimensional dilaton gravity and Toda-Liouville integrable models

NASA Astrophysics Data System (ADS)

de Alfaro, V.; Filippov, A. T.

2010-01-01

We study general properties of a class of two-dimensional dilaton gravity (DG) theories with potentials containing several exponential terms. We isolate and thoroughly study a subclass of such theories in which the equations of motion reduce to Toda and Liouville equations. We show that the equation parameters must satisfy a certain constraint, which we find and solve for the most general multiexponential model. It follows from the constraint that integrable Toda equations in DG theories generally cannot appear without accompanying Liouville equations. The most difficult problem in the two-dimensional Toda-Liouville (TL) DG is to solve the energy and momentum constraints. We discuss this problem using the simplest examples and identify the main obstacles to solving it analytically. We then consider a subclass of integrable two-dimensional theories where scalar matter fields satisfy the Toda equations and the two-dimensional metric is trivial. We consider the simplest case in some detail. In this example, we show how to obtain the general solution. We also show how to simply derive wavelike solutions of general TL systems. In the DG theory, these solutions describe nonlinear waves coupled to gravity and also static states and cosmologies. For static states and cosmologies, we propose and study a more general one-dimensional TL model typically emerging in one-dimensional reductions of higher-dimensional gravity and supergravity theories. We especially attend to making the analytic structure of the solutions of the Toda equations as simple and transparent as possible.

Active ideal sedimentation: exact two-dimensional steady states.

PubMed

Hermann, Sophie; Schmidt, Matthias

2018-02-28

We consider an ideal gas of active Brownian particles that undergo self-propelled motion and both translational and rotational diffusion under the influence of gravity. We solve analytically the corresponding Smoluchowski equation in two space dimensions for steady states. The resulting one-body density is given as a series, where each term is a product of an orientation-dependent Mathieu function and a height-dependent exponential. A lower hard wall is implemented as a no-flux boundary condition. Numerical evaluation of the suitably truncated analytical solution shows the formation of two different spatial regimes upon increasing Peclet number. These regimes differ in their mean particle orientation and in their variation of the orientation-averaged density with height.

A Non Local Electron Heat Transport Model for Multi-Dimensional Fluid Codes

NASA Astrophysics Data System (ADS)

Schurtz, Guy

2000-10-01

Apparent inhibition of thermal heat flow is one of the most ancient problems in computational Inertial Fusion and flux-limited Spitzer-Harm conduction has been a mainstay in multi-dimensional hydrodynamic codes for more than 25 years. Theoretical investigation of the problem indicates that heat transport in laser produced plasmas has to be considered as a non local process. Various authors contributed to the non local theory and proposed convolution formulas designed for practical implementation in one-dimensional fluid codes. Though the theory, confirmed by kinetic calculations, actually predicts a reduced heat flux, it fails to explain the very small limiters required in two-dimensional simulations. Fokker-Planck simulations by Epperlein, Rickard and Bell [PRL 61, 2453 (1988)] demonstrated that non local effects could lead to a strong reduction of heat flow in two dimensions, even in situations where a one-dimensional analysis suggests that the heat flow is nearly classical. We developed at CEA/DAM a non local electron heat transport model suitable for implementation in our two-dimensional radiation hydrodynamic code FCI2. This model may be envisionned as the first step of an iterative solution of the Fokker-Planck equations; it takes the mathematical form of multigroup diffusion equations, the solution of which yields both the heat flux and the departure of the electron distribution function to the Maxwellian. Although direct implementation of the model is straightforward, formal solutions of it can be expressed in convolution form, exhibiting a three-dimensional tensor propagator. Reduction to one dimension retrieves the original formula of Luciani, Mora and Virmont [PRL 51, 1664 (1983)]. Intense magnetic fields may be generated by thermal effects in laser targets; these fields, as well as non local effects, will inhibit electron conduction. We present simulations where both effects are taken into account and shortly discuss the coupling strategy between them.

Directly measuring of thermal pulse transfer in one-dimensional highly aligned carbon nanotubes

PubMed Central

Zhang, Guang; Liu, Changhong; Fan, Shoushan

2013-01-01

Using a simple and precise instrument system, we directly measured the thermo-physical properties of one-dimensional highly aligned carbon nanotubes (CNTs). A kind of CNT-based macroscopic materials named super aligned carbon nanotube (SACNT) buckypapers was measured in our experiment. We defined a new one-dimensional parameter, the “thermal transfer speed” to characterize the thermal damping mechanisms in the SACNT buckypapers. Our results indicated that the SACNT buckypapers with different densities have obviously different thermal transfer speeds. Furthermore, we found that the thermal transfer speed of high-density SACNT buckypapers may have an obvious damping factor along the CNTs aligned direction. The anisotropic thermal diffusivities of SACNT buckypapers could be calculated by the thermal transfer speeds. The thermal diffusivities obviously increase as the buckypaper-density increases. For parallel SACNT buckypapers, the thermal diffusivity could be as high as 562.2 ± 55.4 mm2/s. The thermal conductivities of these SACNT buckypapers were also calculated by the equation k = Cpαρ. PMID:23989589

Directly measuring of thermal pulse transfer in one-dimensional highly aligned carbon nanotubes.

PubMed

Zhang, Guang; Liu, Changhong; Fan, Shoushan

2013-01-01

Using a simple and precise instrument system, we directly measured the thermo-physical properties of one-dimensional highly aligned carbon nanotubes (CNTs). A kind of CNT-based macroscopic materials named super aligned carbon nanotube (SACNT) buckypapers was measured in our experiment. We defined a new one-dimensional parameter, the "thermal transfer speed" to characterize the thermal damping mechanisms in the SACNT buckypapers. Our results indicated that the SACNT buckypapers with different densities have obviously different thermal transfer speeds. Furthermore, we found that the thermal transfer speed of high-density SACNT buckypapers may have an obvious damping factor along the CNTs aligned direction. The anisotropic thermal diffusivities of SACNT buckypapers could be calculated by the thermal transfer speeds. The thermal diffusivities obviously increase as the buckypaper-density increases. For parallel SACNT buckypapers, the thermal diffusivity could be as high as 562.2 ± 55.4 mm(2)/s. The thermal conductivities of these SACNT buckypapers were also calculated by the equation k = Cpαρ.

Multiscale solutions of radiative heat transfer by the discrete unified gas kinetic scheme

NASA Astrophysics Data System (ADS)

Luo, Xiao-Ping; Wang, Cun-Hai; Zhang, Yong; Yi, Hong-Liang; Tan, He-Ping

2018-06-01

The radiative transfer equation (RTE) has two asymptotic regimes characterized by the optical thickness, namely, optically thin and optically thick regimes. In the optically thin regime, a ballistic or kinetic transport is dominant. In the optically thick regime, energy transport is totally dominated by multiple collisions between photons; that is, the photons propagate by means of diffusion. To obtain convergent solutions to the RTE, conventional numerical schemes have a strong dependence on the number of spatial grids, which leads to a serious computational inefficiency in the regime where the diffusion is predominant. In this work, a discrete unified gas kinetic scheme (DUGKS) is developed to predict radiative heat transfer in participating media. Numerical performances of the DUGKS are compared in detail with conventional methods through three cases including one-dimensional transient radiative heat transfer, two-dimensional steady radiative heat transfer, and three-dimensional multiscale radiative heat transfer. Due to the asymptotic preserving property, the present method with relatively coarse grids gives accurate and reliable numerical solutions for large, small, and in-between values of optical thickness, and, especially in the optically thick regime, the DUGKS demonstrates a pronounced computational efficiency advantage over the conventional numerical models. In addition, the DUGKS has a promising potential in the study of multiscale radiative heat transfer inside the participating medium with a transition from optically thin to optically thick regimes.

Stochastic modeling of experimental chaotic time series.

PubMed

Stemler, Thomas; Werner, Johannes P; Benner, Hartmut; Just, Wolfram

2007-01-26

Methods developed recently to obtain stochastic models of low-dimensional chaotic systems are tested in electronic circuit experiments. We demonstrate that reliable drift and diffusion coefficients can be obtained even when no excessive time scale separation occurs. Crisis induced intermittent motion can be described in terms of a stochastic model showing tunneling which is dominated by state space dependent diffusion. Analytical solutions of the corresponding Fokker-Planck equation are in excellent agreement with experimental data.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tipireddy, R.; Stinis, P.; Tartakovsky, A. M.

In this paper, we present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate low-dimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support ourmore » construction with numerical experiments for a steady-state diffusion equation with a random spatially dependent coefficient. Lastly, our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.« less

Small particle transport across turbulent nonisothermal boundary layers

NASA Technical Reports Server (NTRS)

Rosner, D. E.; Fernandez De La Mora, J.

1982-01-01

The interaction between turbulent diffusion, Brownian diffusion, and particle thermophoresis in the limit of vanishing particle inertial effects is quantitatively modeled for applications in gas turbines. The model is initiated with consideration of the particle phase mass conservation equation for a two-dimensional boundary layer, including the thermophoretic flux term directed toward the cold wall. A formalism of a turbulent flow near a flat plate in a heat transfer problem is adopted, and variable property effects are neglected. Attention is given to the limit of very large Schmidt numbers and the particle concentration depletion outside of the Brownian sublayer. It is concluded that, in the parameter range of interest, thermophoresis augments the high Schmidt number mass-transfer coefficient by a factor equal to the product of the outer sink and the thermophoretic suction.

First passage times for multiple particles with reversible target-binding kinetics

NASA Astrophysics Data System (ADS)

Grebenkov, Denis S.

2017-10-01

We investigate the first passage problem for multiple particles that diffuse towards a target, partially adsorb there, and then desorb after a finite exponentially distributed residence time. We search for the first time when m particles undergoing such reversible target-binding kinetics are found simultaneously on the target that may trigger an irreversible chemical reaction or a biophysical event. Even if the particles are independent, the finite residence time on the target yields an intricate temporal coupling between particles. We compute analytically the mean first passage time (MFPT) for two independent particles by mapping the original problem to higher-dimensional surface-mediated diffusion and solving the coupled partial differential equations. The respective effects of the adsorption and desorption rates on the MFPT are revealed and discussed.

Self diffusion of interacting membrane proteins.

PubMed Central

Abney, J R; Scalettar, B A; Owicki, J C

1989-01-01

A two-dimensional version of the generalized Smoluchowski equation is used to analyze the time (or distance) dependent self diffusion of interacting membrane proteins in concentrated membrane systems. This equation provides a well established starting point for descriptions of the diffusion of particles that interact through both direct and hydrodynamic forces; in this initial work only the effects of direct interactions are explicitly considered. Data describing diffusion in the presence of hard-core repulsions, soft repulsions, and soft repulsions with weak attractions are presented. The effect that interactions have on the self-diffusion coefficient of a real protein molecule from mouse liver gap junctions is also calculated. The results indicate that self diffusion is always inhibited by direct interactions; this observation is interpreted in terms of the caging that will exist at finite protein concentration. It is also noted that, over small distance scales, the diffusion coefficient is determined entirely by the very strong Brownian forces; therefore, as a function of displacement the self-diffusion coefficient decays (rapidly) from its value at infinite dilution to its steady-state interaction-averaged value. The steady-state self-diffusion coefficient describes motion over distance scales that range from approximately 10 nm to cellular dimensions and is the quantity measured in fluorescence recovery after photobleaching experiments. The short-ranged behavior of the diffusion coefficient is important on the interparticle-distance scale and may therefore influence the rate at which nearest-neighbor collisional processes take place. The hard-disk theoretical results presented here are in excellent agreement with lattice Monte-Carlo results obtained by other workers. The concentration dependence of experimentally measured diffusion coefficients of antibody-hapten complexes bound to the membrane surface is consistent with that predicted by the theory. The variation in experimental diffusion coefficients of integral membrane proteins is greater than that predicted by the theory, and may also reflect protein-induced perturbations in membrane viscosity. PMID:2720077

Intermode light diffusion in multimode optical waveguides with rough surfaces.

PubMed

Stepanov, S; Chaikina, E I; Leskova, T A; Méndez, E R

2005-06-01

A theoretical analysis of incoherent intermode light power diffusion in multimode dielectric waveguides with rough (corrugated) surfaces is presented. The correlation length a of the surface-profile variations is assumed to be sufficiently large (a less less than lambda/2pi) to permit light scattering into the outer space only from the modes close to the critical angles of propagation and yet sufficiently small (a less less than d, where d is the average width of the waveguide) to permit direct interaction between a given mode and a large number of neighboring ones. The cases of a one-dimensional (1D) slab waveguide and a two-dimensional cylindrical waveguide (optical fiber) are analyzed, and we find that in both cases the partial differential equations that govern the evolution of the angular light power profile propagating along the waveguide are 1D and of the diffusion type. However, whereas in the former case the effective conductivity coefficient proves to be linearly dependent on the transverse-mode wave number, in the latter one the linear dependence is for the effective diffusion coefficient. The theoretical predictions are in reasonable agreement with experimental results for the intermode power diffusion in multimode (700 x 700) optical fibers with etched surfaces. The characteristic length of dispersion of a narrow angular power profile evaluated from the correlation length and standard deviation of heights of the surface profile proved to be in good agreement with the experimentally observed changes in the output angular power profiles.

A two-dimensional numerical simulation of a supersonic, chemically reacting mixing layer

NASA Technical Reports Server (NTRS)

Drummond, J. Philip

1988-01-01

Research has been undertaken to achieve an improved understanding of physical phenomena present when a supersonic flow undergoes chemical reaction. A detailed understanding of supersonic reacting flows is necessary to successfully develop advanced propulsion systems now planned for use late in this century and beyond. In order to explore such flows, a study was begun to create appropriate physical models for describing supersonic combustion, and to develop accurate and efficient numerical techniques for solving the governing equations that result from these models. From this work, two computer programs were written to study reacting flows. Both programs were constructed to consider the multicomponent diffusion and convection of important chemical species, the finite rate reaction of these species, and the resulting interaction of the fluid mechanics and the chemistry. The first program employed a finite difference scheme for integrating the governing equations, whereas the second used a hybrid Chebyshev pseudospectral technique for improved accuracy.

A computational study of the flowfield surrounding the Aeroassist Flight Experiment vehicle

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.; Greene, Francis A.

1987-01-01

A symmetric total variation diminishing (STVD) algorithm has been applied to the solution of the three-dimensional hypersonic flowfield surrounding the Aeroassist Flight Experiment (AFE) vehicle. Both perfect-gas and chemical nonequilibrium models have been used. The perfect-gas flows were computed at two different Reynolds numbers, including a flight trajectory point at maximum dynamic pressure, and on two different grids. Procedures for coupling the solution of the species continuity equations with the Navier-Stokes equations in the presence of chemical nonequilibrium are reviewed and tested on the forebody of the AFE and on the complete flowfield assuming noncatalytic wall and no species diffusion. Problems with the STVD algorithm unique to flows with variable thermodynamic properties (real gas) are identified and algorithm modifications are suggested. A potential heating problem caused by strong flow impingement on the nozzle lip in the near wake at 0-deg angle of attack has been identified.

Effect of Wind Velocity on Flame Spread in Microgravity

NASA Technical Reports Server (NTRS)

Prasad, Kuldeep; Olson, Sandra L.; Nakamura, Yuji; Fujita, Osamu; Nishizawa, Katsuhiro; Ito, Kenichi; Kashiwagi, Takashi; Simons, Stephen N. (Technical Monitor)

2002-01-01

A three-dimensional, time-dependent model is developed describing ignition and subsequent transition to flame spread over a thermally thin cellulosic sheet heated by external radiation in a microgravity environment. A low Mach number approximation to the Navier Stokes equations with global reaction rate equations describing combustion in the gas phase and the condensed phase is numerically solved. The effects of a slow external wind (1-20 cm/s) on flame transition are studied in an atmosphere of 35% oxygen concentration. The ignition is initiated at the center part of the sample by generating a line-shape flame along the width of the sample. The calculated results are compared with data obtained in the 10s drop tower. Numerical results exhibit flame quenching at a wind speed of 1.0 cm/s, two localized flames propagating upstream along the sample edges at 1.5 cm/s, a single line-shape flame front at 5.0 cm/s, three flames structure observed at 10.0 cm/s (consisting of a single line-shape flame propagating upstream and two localized flames propagating downstream along sample edges) and followed by two line-shape flames (one propagating upstream and another propagating downstream) at 20.0 cm/s. These observations qualitatively compare with experimental data. Three-dimensional visualization of the observed flame complex, fuel concentration contours, oxygen and reaction rate isosurfaces, convective and diffusive mass flux are used to obtain a detailed understanding of the controlling mechanism, Physical arguments based on lateral diffusive flux of oxygen, fuel depletion, oxygen shadow of the flame and heat release rate are constructed to explain the various observed flame shapes.

Influence of heat conducting substrates on explosive crystallization in thin layers

NASA Astrophysics Data System (ADS)

Schneider, Wilhelm

2017-09-01

Crystallization in a thin, initially amorphous layer is considered. The layer is in thermal contact with a substrate of very large dimensions. The energy equation of the layer contains source and sink terms. The source term is due to liberation of latent heat in the crystallization process, while the sink term is due to conduction of heat into the substrate. To determine the latter, the heat diffusion equation for the substrate is solved by applying Duhamel's integral. Thus, the energy equation of the layer becomes a heat diffusion equation with a time integral as an additional term. The latter term indicates that the heat loss due to the substrate depends on the history of the process. To complete the set of equations, the crystallization process is described by a rate equation for the degree of crystallization. The governing equations are then transformed to a moving co-ordinate system in order to analyze crystallization waves that propagate with invariant properties. Dual solutions are found by an asymptotic expansion for large activation energies of molecular diffusion. By introducing suitable variables, the results can be presented in a universal form that comprises the influence of all non-dimensional parameters that govern the process. Of particular interest for applications is the prediction of a critical heat loss parameter for the existence of crystallization waves with invariant properties.

Anomalously Fast Diffusion of Targeted Carbon Nanotubes in Cellular Spheroids.

PubMed

Wang, Yichun; Bahng, Joong Hwan; Che, Quantong; Han, Jishu; Kotov, Nicholas A

2015-08-25

Understanding transport of carbon nanotubes (CNTs) and other nanocarriers within tissues is essential for biomedical imaging and drug delivery using these carriers. Compared to traditional cell cultures in animal studies, three-dimensional tissue replicas approach the complexity of the actual organs and enable high temporal and spatial resolution of the carrier permeation. We investigated diffusional transport of CNTs in highly uniform spheroids of hepatocellular carcinoma and found that apparent diffusion coefficients of CNTs in these tissue replicas are anomalously high and comparable to diffusion rates of similarly charged molecules with molecular weights 10000× lower. Moreover, diffusivity of CNTs in tissues is enhanced after functionalization with transforming growth factor β1. This unexpected trend contradicts predictions of the Stokes-Einstein equation and previously obtained empirical dependences of diffusivity on molecular mass for permeants in gas, liquid, solid or gel. It is attributed to the planar diffusion (gliding) of CNTs along cellular membranes reducing effective dimensionality of diffusional space. These findings indicate that nanotubes and potentially similar nanostructures are capable of fast and deep permeation into the tissue, which is often difficult to realize with anticancer agents.

A pyramid scheme for three-dimensional diffusion equations on polyhedral meshes

NASA Astrophysics Data System (ADS)

Wang, Shuai; Hang, Xudeng; Yuan, Guangwei

2017-12-01

In this paper, a new cell-centered finite volume scheme is proposed for three-dimensional diffusion equations on polyhedral meshes, which is called as pyramid scheme (P-scheme). The scheme is designed for polyhedral cells with nonplanar cell-faces. The normal flux on a nonplanar cell-face is discretized on a planar face, which is determined by a simple optimization procedure. The resulted discrete form of the normal flux involves only cell-centered and cell-vertex unknowns, and is free from face-centered unknowns. In the case of hexahedral meshes with skewed nonplanar cell-faces, a quite simple expression is obtained for the discrete normal flux. Compared with the second order accurate O-scheme [31], the P-scheme is more robust and the discretization cost is reduced remarkably. Numerical results are presented to show the performance of the P-scheme on various kinds of distorted meshes. In particular, the P-scheme is shown to be second order accurate.

Effects of 3 dimensional crystal geometry and orientation on 1D and 2D time-scale determinations of magmatic processes using olivine and orthopyroxene

NASA Astrophysics Data System (ADS)

Shea, Thomas; Krimer, Daniel; Costa, Fidel; Hammer, Julia

2014-05-01

One of the achievements in recent years in volcanology is the determination of time-scales of magmatic processes via diffusion in minerals and its addition to the petrologists' and volcanologists' toolbox. The method typically requires one-dimensional modeling of randomly cut crystals from two-dimensional thin sections. Here we address the question whether using 1D (traverse) or 2D (surface) datasets exploited from randomly cut 3D crystals introduces a bias or dispersion in the time-scales estimated, and how this error can be improved or eliminated. Computational simulations were performed using a concentration-dependent, finite-difference solution to the diffusion equation in 3D. The starting numerical models involved simple geometries (spheres, parallelepipeds), Mg/Fe zoning patterns (either normal or reverse), and isotropic diffusion coefficients. Subsequent models progressively incorporated more complexity, 3D olivines possessing representative polyhedral morphologies, diffusion anisotropy along the different crystallographic axes, and more intricate core-rim zoning patterns. Sections and profiles used to compare 1, 2 and 3D diffusion models were selected to be (1) parallel to the crystal axes, (2) randomly oriented but passing through the olivine center, or (3) randomly oriented and sectioned. Results show that time-scales estimated on randomly cut traverses (1D) or surfaces (2D) can be widely distributed around the actual durations of 3D diffusion (~0.2 to 10 times the true diffusion time). The magnitude over- or underestimations of duration are a complex combination of the geometry of the crystal, the zoning pattern, the orientation of the cuts with respect to the crystallographic axes, and the degree of diffusion anisotropy. Errors on estimated time-scales retrieved from such models may thus be significant. Drastic reductions in the uncertainty of calculated diffusion times can be obtained by following some simple guidelines during the course of data collection (i.e. selection of crystals and concentration profiles, acquisition of crystallographic orientation data), thus allowing derivation of robust time-scales.

Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping

2016-10-01

Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.

Theory of Stochastic Laplacian Growth

NASA Astrophysics Data System (ADS)

Alekseev, Oleg; Mineev-Weinstein, Mark

2017-07-01

We generalize the diffusion-limited aggregation by issuing many randomly-walking particles, which stick to a cluster at the discrete time unit providing its growth. Using simple combinatorial arguments we determine probabilities of different growth scenarios and prove that the most probable evolution is governed by the deterministic Laplacian growth equation. A potential-theoretical analysis of the growth probabilities reveals connections with the tau-function of the integrable dispersionless limit of the two-dimensional Toda hierarchy, normal matrix ensembles, and the two-dimensional Dyson gas confined in a non-uniform magnetic field. We introduce the time-dependent Hamiltonian, which generates transitions between different classes of equivalence of closed curves, and prove the Hamiltonian structure of the interface dynamics. Finally, we propose a relation between probabilities of growth scenarios and the semi-classical limit of certain correlation functions of "light" exponential operators in the Liouville conformal field theory on a pseudosphere.

Prediction of stream volatilization coefficients

USGS Publications Warehouse

Rathbun, Ronald E.

1990-01-01

Equations are developed for predicting the liquid-film and gas-film reference-substance parameters for quantifying volatilization of organic solutes from streams. Molecular weight and molecular-diffusion coefficients of the solute are used as correlating parameters. Equations for predicting molecular-diffusion coefficients of organic solutes in water and air are developed, with molecular weight and molal volume as parameters. Mean absolute errors of prediction for diffusion coefficients in water are 9.97% for the molecular-weight equation, 6.45% for the molal-volume equation. The mean absolute error for the diffusion coefficient in air is 5.79% for the molal-volume equation. Molecular weight is not a satisfactory correlating parameter for diffusion in air because two equations are necessary to describe the values in the data set. The best predictive equation for the liquid-film reference-substance parameter has a mean absolute error of 5.74%, with molal volume as the correlating parameter. The best equation for the gas-film parameter has a mean absolute error of 7.80%, with molecular weight as the correlating parameter.

Conformal dynamics of precursors to fracture

NASA Astrophysics Data System (ADS)

Barra, F.; Herrera, M.; Procaccia, I.

2003-09-01

An exact integro-differential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2-dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.

The role of fractional time-derivative operators on anomalous diffusion

NASA Astrophysics Data System (ADS)

Tateishi, Angel A.; Ribeiro, Haroldo V.; Lenzi, Ervin K.

2017-10-01

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Generalized Landauer equation: Absorption-controlled diffusion processes

NASA Astrophysics Data System (ADS)

Godoy, Salvador; García-Colín, L. S.; Micenmacher, Victor

1999-05-01

The exact expression of the one-dimensional Boltzmann multiple-scattering coefficients, for the passage of particles through a slab of a given material, is obtained in terms of the single-scattering cross section of the material, including absorption. The remarkable feature of the result is that for multiple scattering in a metal, free from absorption, one recovers the well-known Landauer result for conduction electrons. In the case of particles, such as neutrons, moving through a weak absorbing media, Landuer's formula is modified due to the absorption cross section. For photons, in a strong absorbing media, one recovers the Lambert-Beer equation. In this latter case one may therefore speak of absorption-controlled diffusive processes.

Noise-induced drift in two-dimensional anisotropic systems

NASA Astrophysics Data System (ADS)

Farago, Oded

2017-10-01

We study the isothermal Brownian dynamics of a particle in a system with spatially varying diffusivity. Due to the heterogeneity of the system, the particle's mean displacement does not vanish even if it does not experience any physical force. This phenomenon has been termed "noise-induced drift," and has been extensively studied for one-dimensional systems. Here, we examine the noise-induced drift in a two-dimensional anisotropic system, characterized by a symmetric diffusion tensor with unequal diagonal elements. A general expression for the mean displacement vector is derived and presented as a sum of two vectors, depicting two distinct drifting effects. The first vector describes the tendency of the particle to drift toward the high diffusivity side in each orthogonal principal diffusion direction. This is a generalization of the well-known expression for the noise-induced drift in one-dimensional systems. The second vector represents a novel drifting effect, not found in one-dimensional systems, originating from the spatial rotation in the directions of the principal axes. The validity of the derived expressions is verified by using Langevin dynamics simulations. As a specific example, we consider the relative diffusion of two transmembrane proteins, and demonstrate that the average distance between them increases at a surprisingly fast rate of several tens of micrometers per second.

Numerical study of chemically reacting viscous flow relevant to pulsed detonation engines

NASA Astrophysics Data System (ADS)

Yi, Tae-Hyeong

2005-11-01

A computational fluid dynamics code for two-dimensional, multi-species, laminar Navier-Stokes equations is developed to simulate a recently proposed engine concept for a pulsed detonation based propulsion system and to investigate the feasibility of the engine of the concept. The governing equations that include transport phenomena such as viscosity, thermal conduction and diffusion are coupled with chemical reactions. The gas is assumed to be thermally perfect and in chemically non-equilibrium. The stiffness due to coupling the fluid dynamics and the chemical kinetics is properly taken care of by using a time-operator splitting method and a variable coefficient ordinary differential equation solver. A second-order Roe scheme with a minmod limiter is explicitly used for space descretization, while a second-order, two-step Runge-Kutta method is used for time descretization. In space integration, a finite volume method and a cell-centered scheme are employed. The first-order derivatives in the equations of transport properties are discretized by a central differencing with Green's theorem. Detailed chemistry is involved in this study. Two chemical reaction mechanisms are extracted from GRI-Mech, which are forty elementary reactions with thirteen species for a hydrogen-air mixture and twenty-seven reactions with eight species for a hydrogen-oxygen mixture. The code is ported to a high-performance parallel machine with Message-Passing Interface. Code validation is performed with chemical kinetic modeling for a stoichiometric hydrogen-air mixture, an one-dimensional detonation tube, a two-dimensional, inviscid flow over a wedge and a viscous flow over a flat plate. Detonation is initiated using a numerically simulated arc-ignition or shock-induced ignition system. Various freestream conditions are utilized to study the propagation of the detonation in the proposed concept of the engine. Investigation of the detonation propagation is performed for a pulsed detonation rocket and a supersonic combustion chamber. For a pulsed detonation rocket case, the detonation tube is embedded in a mixing chamber where an initiator is added to the main detonation chamber. Propagating detonation waves in a supersonic combustion chamber is investigated for one- and two-dimensional cases. The detonation initiated by an arc and a shock wave is studied in the inviscid and viscous flow, respectively. Various features including a detonation-shock interaction, a detonation diffraction, a base flow and a vortex are observed.

A simple inertial formulation of the shallow water equations for efficient two-dimensional flood inundation modelling

NASA Astrophysics Data System (ADS)

Bates, Paul D.; Horritt, Matthew S.; Fewtrell, Timothy J.

2010-06-01

SummaryThis paper describes the development of a new set of equations derived from 1D shallow water theory for use in 2D storage cell inundation models where flows in the x and y Cartesian directions are decoupled. The new equation set is designed to be solved explicitly at very low computational cost, and is here tested against a suite of four test cases of increasing complexity. In each case the predicted water depths compare favourably to analytical solutions or to simulation results from the diffusive storage cell code of Hunter et al. (2005). For the most complex test involving the fine spatial resolution simulation of flow in a topographically complex urban area the Root Mean Squared Difference between the new formulation and the model of Hunter et al. is ˜1 cm. However, unlike diffusive storage cell codes where the stable time step scales with (1/Δ x) 2, the new equation set developed here represents shallow water wave propagation and so the stability is controlled by the Courant-Freidrichs-Lewy condition such that the stable time step instead scales with 1/Δ x. This allows use of a stable time step that is 1-3 orders of magnitude greater for typical cell sizes than that possible with diffusive storage cell models and results in commensurate reductions in model run times. For the tests reported in this paper the maximum speed up achieved over a diffusive storage cell model was 1120×, although the actual value seen will depend on model resolution and water surface gradient. Solutions using the new equation set are shown to be grid-independent for the conditions considered and to have an intuitively correct sensitivity to friction, however small instabilities and increased errors on predicted depth were noted when Manning's n = 0.01. The new equations are likely to find widespread application in many types of flood inundation modelling and should provide a useful additional tool, alongside more established model formulations, for a variety of flood risk management studies.

Alignment dynamics of diffusive scalar gradient in a two-dimensional model flow

NASA Astrophysics Data System (ADS)

Gonzalez, M.

2018-04-01

The Lagrangian two-dimensional approach of scalar gradient kinematics is revisited accounting for molecular diffusion. Numerical simulations are performed in an analytic, parameterized model flow, which enables considering different regimes of scalar gradient dynamics. Attention is especially focused on the influence of molecular diffusion on Lagrangian statistical orientations and on the dynamics of scalar gradient alignment.

Double diffusivity model under stochastic forcing

NASA Astrophysics Data System (ADS)

Chattopadhyay, Amit K.; Aifantis, Elias C.

2017-05-01

The "double diffusivity" model was proposed in the late 1970s, and reworked in the early 1980s, as a continuum counterpart to existing discrete models of diffusion corresponding to high diffusivity paths, such as grain boundaries and dislocation lines. It was later rejuvenated in the 1990s to interpret experimental results on diffusion in polycrystalline and nanocrystalline specimens where grain boundaries and triple grain boundary junctions act as high diffusivity paths. Technically, the model pans out as a system of coupled Fick-type diffusion equations to represent "regular" and "high" diffusivity paths with "source terms" accounting for the mass exchange between the two paths. The model remit was extended by analogy to describe flow in porous media with double porosity, as well as to model heat conduction in media with two nonequilibrium local temperature baths, e.g., ion and electron baths. Uncoupling of the two partial differential equations leads to a higher-ordered diffusion equation, solutions of which could be obtained in terms of classical diffusion equation solutions. Similar equations could also be derived within an "internal length" gradient (ILG) mechanics formulation applied to diffusion problems, i.e., by introducing nonlocal effects, together with inertia and viscosity, in a mechanics based formulation of diffusion theory. While being remarkably successful in studies related to various aspects of transport in inhomogeneous media with deterministic microstructures and nanostructures, its implications in the presence of stochasticity have not yet been considered. This issue becomes particularly important in the case of diffusion in nanopolycrystals whose deterministic ILG-based theoretical calculations predict a relaxation time that is only about one-tenth of the actual experimentally verified time scale. This article provides the "missing link" in this estimation by adding a vital element in the ILG structure, that of stochasticity, that takes into account all boundary layer fluctuations. Our stochastic-ILG diffusion calculation confirms rapprochement between theory and experiment, thereby benchmarking a new generation of gradient-based continuum models that conform closer to real-life fluctuating environments.

Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems

NASA Astrophysics Data System (ADS)

Ge, Yongbin; Cao, Fujun

2011-05-01

In this paper, a multigrid method based on the high order compact (HOC) difference scheme on nonuniform grids, which has been proposed by Kalita et al. [J.C. Kalita, A.K. Dass, D.C. Dalal, A transformation-free HOC scheme for steady convection-diffusion on non-uniform grids, Int. J. Numer. Methods Fluids 44 (2004) 33-53], is proposed to solve the two-dimensional (2D) convection diffusion equation. The HOC scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids, consequently, the multigrid method is brand-new for solving the discrete system arising from the difference equation on nonuniform grids. The corresponding multigrid projection and interpolation operators are constructed by the area ratio. Some boundary layer and local singularity problems are used to demonstrate the superiority of the present method. Numerical results show that the multigrid method with the HOC scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems. The present method is also applied to solve the 2D incompressible Navier-Stokes equations using the stream function-vorticity formulation and the numerical solutions of the lid-driven cavity flow problem are obtained and compared with solutions available in the literature.

A Robust and Efficient Method for Steady State Patterns in Reaction-Diffusion Systems

PubMed Central

Lo, Wing-Cheong; Chen, Long; Wang, Ming; Nie, Qing

2012-01-01

An inhomogeneous steady state pattern of nonlinear reaction-diffusion equations with no-flux boundary conditions is usually computed by solving the corresponding time-dependent reaction-diffusion equations using temporal schemes. Nonlinear solvers (e.g., Newton’s method) take less CPU time in direct computation for the steady state; however, their convergence is sensitive to the initial guess, often leading to divergence or convergence to spatially homogeneous solution. Systematically numerical exploration of spatial patterns of reaction-diffusion equations under different parameter regimes requires that the numerical method be efficient and robust to initial condition or initial guess, with better likelihood of convergence to an inhomogeneous pattern. Here, a new approach that combines the advantages of temporal schemes in robustness and Newton’s method in fast convergence in solving steady states of reaction-diffusion equations is proposed. In particular, an adaptive implicit Euler with inexact solver (AIIE) method is found to be much more efficient than temporal schemes and more robust in convergence than typical nonlinear solvers (e.g., Newton’s method) in finding the inhomogeneous pattern. Application of this new approach to two reaction-diffusion equations in one, two, and three spatial dimensions, along with direct comparisons to several other existing methods, demonstrates that AIIE is a more desirable method for searching inhomogeneous spatial patterns of reaction-diffusion equations in a large parameter space. PMID:22773849

Current Fluctuations in One Dimensional Diffusive Systems with a Step Initial Density Profile

NASA Astrophysics Data System (ADS)

Derrida, Bernard; Gerschenfeld, Antoine

2009-12-01

We show how to apply the macroscopic fluctuation theory (MFT) of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim to study the current fluctuations of diffusive systems with a step initial condition. We argue that one has to distinguish between two ways of averaging (the annealed and the quenched cases) depending on whether we let the initial condition fluctuate or not. Although the initial condition is not a steady state, the distribution of the current satisfies a symmetry very reminiscent of the fluctuation theorem. We show how the equations of the MFT can be solved in the case of non-interacting particles. The symmetry of these equations can be used to deduce the distribution of the current for several other models, from its knowledge (Derrida and Gerschenfeld in J. Stat. Phys. 136, 1-15, 2009) for the symmetric simple exclusion process. In the range where the integrated current Qt˜sqrt{t} , we show that the non-Gaussian decay exp [- Q {/t 3}/ t] of the distribution of Q t is generic.

Self-diffusion in a stochastically heated two-dimensional dusty plasma

NASA Astrophysics Data System (ADS)

Sheridan, T. E.

2016-09-01

Diffusion in a two-dimensional dusty plasma liquid (i.e., a Yukawa liquid) is studied experimentally. The dusty plasma liquid is heated stochastically by a surrounding three-dimensional toroidal dusty plasma gas which acts as a thermal reservoir. The measured dust velocity distribution functions are isotropic Maxwellians, giving a well-defined kinetic temperature. The mean-square displacement for dust particles is found to increase linearly with time, indicating normal diffusion. The measured diffusion coefficients increase approximately linearly with temperature. The effective collision rate is dominated by collective dust-dust interactions rather than neutral gas drag, and is comparable to the dusty-plasma frequency.

Multi-dimensional multi-species modeling of transient electrodeposition in LIGA microfabrication.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Evans, Gregory Herbert; Chen, Ken Shuang

2004-06-01

This report documents the efforts and accomplishments of the LIGA electrodeposition modeling project which was headed by the ASCI Materials and Physics Modeling Program. A multi-dimensional framework based on GOMA was developed for modeling time-dependent diffusion and migration of multiple charged species in a dilute electrolyte solution with reduction electro-chemical reactions on moving deposition surfaces. By combining the species mass conservation equations with the electroneutrality constraint, a Poisson equation that explicitly describes the electrolyte potential was derived. The set of coupled, nonlinear equations governing species transport, electric potential, velocity, hydrodynamic pressure, and mesh motion were solved in GOMA, using themore » finite-element method and a fully-coupled implicit solution scheme via Newton's method. By treating the finite-element mesh as a pseudo solid with an arbitrary Lagrangian-Eulerian formulation and by repeatedly performing re-meshing with CUBIT and re-mapping with MAPVAR, the moving deposition surfaces were tracked explicitly from start of deposition until the trenches were filled with metal, thus enabling the computation of local current densities that potentially influence the microstructure and frictional/mechanical properties of the deposit. The multi-dimensional, multi-species, transient computational framework was demonstrated in case studies of two-dimensional nickel electrodeposition in single and multiple trenches, without and with bath stirring or forced flow. Effects of buoyancy-induced convection on deposition were also investigated. To further illustrate its utility, the framework was employed to simulate deposition in microscreen-based LIGA molds. Lastly, future needs for modeling LIGA electrodeposition are discussed.« less

Finite-Dimensional Representations for Controlled Diffusions with Delay

DOE Office of Scientific and Technical Information (OSTI.GOV)

Federico, Salvatore, E-mail: salvatore.federico@unimi.it; Tankov, Peter, E-mail: tankov@math.univ-paris-diderot.fr

2015-02-15

We study stochastic delay differential equations (SDDE) where the coefficients depend on the moving averages of the state process. As a first contribution, we provide sufficient conditions under which the solution of the SDDE and a linear path functional of it admit a finite-dimensional Markovian representation. As a second contribution, we show how approximate finite-dimensional Markovian representations may be constructed when these conditions are not satisfied, and provide an estimate of the error corresponding to these approximations. These results are applied to optimal control and optimal stopping problems for stochastic systems with delay.

Numerical solution of special ultra-relativistic Euler equations using central upwind scheme

NASA Astrophysics Data System (ADS)

Ghaffar, Tayabia; Yousaf, Muhammad; Qamar, Shamsul

2018-06-01

This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.

Kadomtsev−Petviashvili equation for a flow of highly nonisothermal collisionless plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Movsesyants, Yu. B., E-mail: yumovsesyants@gmail.com; Rukhadze, A. A., E-mail: rukh@fpl.gpi.ru; Tyuryukanov, P. M.

2016-01-15

It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev−Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

Kadomtsev-Petviashvili equation for a flow of highly nonisothermal collisionless plasma

NASA Astrophysics Data System (ADS)

Movsesyants, Yu. B.; Rukhadze, A. A.; Tyuryukanov, P. M.

2016-01-01

It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev-Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

Calculation method for steady-state pollutant concentration in mixing zones considering variable lateral diffusion coefficient.

PubMed

Wu, Wen; Wu, Zhouhu; Song, Zhiwen

2017-07-01

Prediction of the pollutant mixing zone (PMZ) near the discharge outfall in Huangshaxi shows large error when using the methods based on the constant lateral diffusion assumption. The discrepancy is due to the lack of consideration of the diffusion coefficient variation. The variable lateral diffusion coefficient is proposed to be a function of the longitudinal distance from the outfall. Analytical solution of the two-dimensional advection-diffusion equation of a pollutant is derived and discussed. Formulas to characterize the geometry of the PMZ are derived based on this solution, and a standard curve describing the boundary of the PMZ is obtained by proper choices of the normalization scales. The change of PMZ topology due to the variable diffusion coefficient is then discussed using these formulas. The criterion of assuming the lateral diffusion coefficient to be constant without large error in PMZ geometry is found. It is also demonstrated how to use these analytical formulas in the inverse problems including estimating the lateral diffusion coefficient in rivers by convenient measurements, and determining the maximum allowable discharge load based on the limitations of the geometrical scales of the PMZ. Finally, applications of the obtained formulas to onsite PMZ measurements in Huangshaxi present excellent agreement.

Development of electromagnetic cascades in the atmosphere including the Landau-Pomeranchuk-Migdal effect

NASA Technical Reports Server (NTRS)

Streitmatter, R. E.; Stephens, S. A.

1985-01-01

Numerical solutions have been obtained for the one-dimensional atmospheric electromagnetic cascade diffusion equations, including the Landau-Pomeranchuk-Migdal and screening effects. Spectra produced by primary gamma rays of various energies are given at a number of deths in the atmosphere.

Quantitative diffusion and swelling kinetic measurements using large-angle interferometric refractometry.

PubMed

Saunders, John E; Chen, Hao; Brauer, Chris; Clayton, McGregor; Chen, Weijian; Barnes, Jack A; Loock, Hans-Peter

2015-12-07

The uptake and release of sorbates into films and coatings is typically accompanied by changes of the films' refractive index and thickness. We provide a comprehensive model to calculate the concentration of the sorbate from the average refractive index and the film thickness, and validate the model experimentally. The mass fraction of the analyte partitioned into a film is described quantitatively by the Lorentz-Lorenz equation and the Clausius-Mosotti equation. To validate the model, the uptake kinetics of water and other solvents into SU-8 films (d = 40-45 μm) were explored. Large-angle interferometric refractometry measurements can be used to characterize films that are between 15 μm to 150 μm thick and, Fourier analysis, is used to determine independently the thickness, the average refractive index and the refractive index at the film-substrate interface at one-second time intervals. From these values the mass fraction of water in SU-8 was calculated. The kinetics were best described by two independent uptake processes having different rates. Each process followed one-dimensional Fickian diffusion kinetics with diffusion coefficients for water into SU-8 photoresist film of 5.67 × 10(-9) cm(2) s(-1) and 61.2 × 10(-9) cm(2) s(-1).

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form ismore » based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less

A flexible nonlinear diffusion acceleration method for the S N transport equations discretized with discontinuous finite elements

DOE PAGES

Schunert, Sebastian; Wang, Yaqi; Gleicher, Frederick; ...

2017-02-21

This paper presents a flexible nonlinear diffusion acceleration (NDA) method that discretizes both the S N transport equation and the diffusion equation using the discontinuous finite element method (DFEM). The method is flexible in that the diffusion equation can be discretized on a coarser mesh with the only restriction that it is nested within the transport mesh and the FEM shape function orders of the two equations can be different. The consistency of the transport and diffusion solutions at convergence is defined by using a projection operator mapping the transport into the diffusion FEM space. The diffusion weak form ismore » based on the modified incomplete interior penalty (MIP) diffusion DFEM discretization that is extended by volumetric drift, interior face, and boundary closure terms. In contrast to commonly used coarse mesh finite difference (CMFD) methods, the presented NDA method uses a full FEM discretized diffusion equation for acceleration. Suitable projection and prolongation operators arise naturally from the FEM framework. Via Fourier analysis and numerical experiments for a one-group, fixed source problem the following properties of the NDA method are established for structured quadrilateral meshes: (1) the presented method is unconditionally stable and effective in the presence of mild material heterogeneities if the same mesh and identical shape functions either of the bilinear or biquadratic type are used, (2) the NDA method remains unconditionally stable in the presence of strong heterogeneities, (3) the NDA method with bilinear elements extends the range of effectiveness and stability by a factor of two when compared to CMFD if a coarser diffusion mesh is selected. In addition, the method is tested for solving the C5G7 multigroup, eigenvalue problem using coarse and fine mesh acceleration. Finally, while NDA does not offer an advantage over CMFD for fine mesh acceleration, it reduces the iteration count required for convergence by almost a factor of two in the case of coarse mesh acceleration.« less

Three Dimensional Compressible Turbulent Flow Computations for a Diffusing S-Duct With/Without Vortex Generators

NASA Technical Reports Server (NTRS)

Cho, Soo-Yong; Greber, Isaac

1994-01-01

Numerical investigations on a diffusing S-duct with/without vortex generators and a straight duct with vortex generators are presented. The investigation consists of solving the full three-dimensional unsteady compressible mass averaged Navier-Stokes equations. An implicit finite volume lower-upper time marching code (RPLUS3D) has been employed and modified. A three-dimensional Baldwin-Lomax turbulence model has been modified in conjunction with the flow physics. A model for the analysis of vortex generators in a fully viscous subsonic internal flow is evaluated. A vortical structure for modeling the shed vortex is used as a source term in the computation domain. The injected vortex paths in the straight duct are compared with the analysis by two kinds of prediction models. The flow structure by the vortex generators are investigated along the duct. Computed results of the flow in a circular diffusing S-duct provide an understanding of the flow structure within a typical engine inlet system. These are compared with the experimental wall static-pressure, static- and total-pressure field, and secondary velocity profiles. Additionally, boundary layer thickness, skin friction values, and velocity profiles in wall coordinates are presented. In order to investigate the effect of vortex generators, various vortex strengths are examined in this study. The total-pressure recovery and distortion coefficients are obtained at the exit of the S-duct. The numerical results clearly depict the interaction between the low velocity flow by the flow separation and the injected vortices.

Uniform high order spectral methods for one and two dimensional Euler equations

NASA Technical Reports Server (NTRS)

Cai, Wei; Shu, Chi-Wang

1991-01-01

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.

Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system

NASA Astrophysics Data System (ADS)

Gambino, G.; Lombardo, M. C.; Sammartino, M.

2018-01-01

In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (sub T mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of sub T -mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.

Convective drying of osmo-dehydrated apple slices: kinetics and spatial behavior of effective mass diffusivity and moisture content

NASA Astrophysics Data System (ADS)

de Farias Aires, Juarez Everton; da Silva, Wilton Pereira; de Almeida Farias Aires, Kalina Lígia Cavalcante; da Silva Júnior, Aluízio Freire; da Silva e Silva, Cleide Maria Diniz Pereira

2018-04-01

The main objective of this study is the presentation of a numerical model of liquid diffusion for the description of the convective drying of apple slices submitted to pretreatment of osmotic dehydration able of predicting the spatial distribution of effective mass diffusivity values in apple slabs. Two models that use numerical solutions of the two-dimensional diffusion equation in Cartesian coordinates with the boundary condition of third kind were proposed to describe drying. The first one does not consider the shrinkage of the product and assumes that the process parameters remain constant along the convective drying. The second one considers the shrinkage of the product and assumes that the effective mass diffusivity of water varies according to the local value of the water content in the apple samples. Process parameters were estimated from experimental data through an optimizer coupled to the numerical solutions. The osmotic pretreatment did not reduce the drying time in relation to the fresh fruits when the drying temperature was equal to 40 °C. The use of the temperature of 60 °C led to a reduction in the drying time. The model that considers the variations in the dimensions of the product and the variation in the effective mass diffusivity proved to be more adequate to describe the process.

Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

PubMed

Vlad, Marcel Ovidiu; Ross, John

2002-12-01

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

Transformation diffusion reconstruction of three-dimensional histology volumes from two-dimensional image stacks.

PubMed

Casero, Ramón; Siedlecka, Urszula; Jones, Elizabeth S; Gruscheski, Lena; Gibb, Matthew; Schneider, Jürgen E; Kohl, Peter; Grau, Vicente

2017-05-01

Traditional histology is the gold standard for tissue studies, but it is intrinsically reliant on two-dimensional (2D) images. Study of volumetric tissue samples such as whole hearts produces a stack of misaligned and distorted 2D images that need to be reconstructed to recover a congruent volume with the original sample's shape. In this paper, we develop a mathematical framework called Transformation Diffusion (TD) for stack alignment refinement as a solution to the heat diffusion equation. This general framework does not require contour segmentation, is independent of the registration method used, and is trivially parallelizable. After the first stack sweep, we also replace registration operations by operations in the space of transformations, several orders of magnitude faster and less memory-consuming. Implementing TD with operations in the space of transformations produces our Transformation Diffusion Reconstruction (TDR) algorithm, applicable to general transformations that are closed under inversion and composition. In particular, we provide formulas for translation and affine transformations. We also propose an Approximated TDR (ATDR) algorithm that extends the same principles to tensor-product B-spline transformations. Using TDR and ATDR, we reconstruct a full mouse heart at pixel size 0.92µm×0.92µm, cut 10µm thick, spaced 20µm (84G). Our algorithms employ only local information from transformations between neighboring slices, but the TD framework allows theoretical analysis of the refinement as applying a global Gaussian low-pass filter to the unknown stack misalignments. We also show that reconstruction without an external reference produces large shape artifacts in a cardiac specimen while still optimizing slice-to-slice alignment. To overcome this problem, we use a pre-cutting blockface imaging process previously developed by our group that takes advantage of Brewster's angle and a polarizer to capture the outline of only the topmost layer of wax in the block containing embedded tissue for histological sectioning. Copyright © 2017 The Authors. Published by Elsevier B.V. All rights reserved.

Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation

NASA Astrophysics Data System (ADS)

Mihalache, D.; Mazilu, D.; Lederer, F.; Leblond, H.; Malomed, B. A.

2008-03-01

We present generic outcomes of collisions between stable solitons with intrinsic vorticity S=1 or S=2 in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity, for the axially symmetric configuration. An essential ingredient of the complex Ginzburg-Landau equation is an effective transverse diffusivity (which is known in models of laser cavities), as vortex solitons cannot be stable without it. For the sake of comparison, results are also included for fundamental three-dimensional solitons, with S=0 . Depending on the collision momentum, χ , three generic outcomes are identified: merger of the solitons into a single one, at small χ ; quasielastic interaction, at large χ ; and creation of an extra soliton, in an intermediate region. In addition to the final outcomes, we also highlight noteworthy features of the transient dynamics.

Modeling electrokinetic flows by consistent implicit incompressible smoothed particle hydrodynamics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pan, Wenxiao; Kim, Kyungjoo; Perego, Mauro

2017-04-01

We present an efficient implicit incompressible smoothed particle hydrodynamics (I2SPH) discretization of Navier-Stokes, Poisson-Boltzmann, and advection-diffusion equations subject to Dirichlet or Robin boundary conditions. It is applied to model various two and three dimensional electrokinetic flows in simple or complex geometries. The I2SPH's accuracy and convergence are examined via comparison with analytical solutions, grid-based numerical solutions, or empirical models. The new method provides a framework to explore broader applications of SPH in microfluidics and complex fluids with charged objects, such as colloids and biomolecules, in arbitrary complex geometries.

Existence, uniqueness and regularity of a time-periodic probability density distribution arising in a sedimentation-diffusion problem

NASA Technical Reports Server (NTRS)

Nitsche, Ludwig C.; Nitsche, Johannes M.; Brenner, Howard

1988-01-01

The sedimentation and diffusion of a nonneutrally buoyant Brownian particle in vertical fluid-filled cylinder of finite length which is instantaneously inverted at regular intervals are investigated analytically. A one-dimensional convective-diffusive equation is derived to describe the temporal and spatial evolution of the probability density; a periodicity condition is formulated; the applicability of Fredholm theory is established; and the parameter-space regions are determined within which the existence and uniqueness of solutions are guaranteed. Numerical results for sample problems are presented graphically and briefly characterized.

Pressure-based high-order TVD methodology for dynamic stall control

NASA Astrophysics Data System (ADS)

Yang, H. Q.; Przekwas, A. J.

1992-01-01

The quantitative prediction of the dynamics of separating unsteady flows, such as dynamic stall, is of crucial importance. This six-month SBIR Phase 1 study has developed several new pressure-based methodologies for solving 3D Navier-Stokes equations in both stationary and moving (body-comforting) coordinates. The present pressure-based algorithm is equally efficient for low speed incompressible flows and high speed compressible flows. The discretization of convective terms by the presently developed high-order TVD schemes requires no artificial dissipation and can properly resolve the concentrated vortices in the wing-body with minimum numerical diffusion. It is demonstrated that the proposed Newton's iteration technique not only increases the convergence rate but also strongly couples the iteration between pressure and velocities. The proposed hyperbolization of the pressure correction equation is shown to increase the solver's efficiency. The above proposed methodologies were implemented in an existing CFD code, REFLEQS. The modified code was used to simulate both static and dynamic stalls on two- and three-dimensional wing-body configurations. Three-dimensional effect and flow physics are discussed.

A stabilized Runge–Kutta–Legendre method for explicit super-time-stepping of parabolic and mixed equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.

2014-01-15

Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge–Kutta-like time-steps to advance the parabolic terms by a time-step that is s{sup 2} times larger than amore » single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge–Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems – a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful in parabolic problems with variable diffusion coefficients. This includes variable coefficient parabolic equations that might give rise to skew symmetric terms. The RKC1 and RKC2 schemes do not share this convex monotonicity preserving property. One-dimensional and two-dimensional von Neumann stability analyses of RKC1, RKC2, RKL1 and RKL2 are also presented, showing that the latter two have some advantages. The paper includes several details to facilitate implementation. A detailed accuracy analysis is presented to show that the methods reach their design accuracies. A stringent set of test problems is also presented. To demonstrate the robustness and versatility of our methods, we show their successful operation on problems involving linear and non-linear heat conduction and viscosity, resistive magnetohydrodynamics, ambipolar diffusion dominated magnetohydrodynamics, level set methods and flux limited radiation diffusion. In a prior paper (Meyer, Balsara and Aslam 2012 [36]) we have also presented an extensive test-suite showing that the RKL2 method works robustly in the presence of shocks in an anisotropically conducting, magnetized plasma.« less

A stabilized Runge-Kutta-Legendre method for explicit super-time-stepping of parabolic and mixed equations

NASA Astrophysics Data System (ADS)

Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.

2014-01-01

Parabolic partial differential equations appear in several physical problems, including problems that have a dominant hyperbolic part coupled to a sub-dominant parabolic component. Explicit methods for their solution are easy to implement but have very restrictive time step constraints. Implicit solution methods can be unconditionally stable but have the disadvantage of being computationally costly or difficult to implement. Super-time-stepping methods for treating parabolic terms in mixed type partial differential equations occupy an intermediate position. In such methods each superstep takes “s” explicit Runge-Kutta-like time-steps to advance the parabolic terms by a time-step that is s2 times larger than a single explicit time-step. The expanded stability is usually obtained by mapping the short recursion relation of the explicit Runge-Kutta scheme to the recursion relation of some well-known, stable polynomial. Prior work has built temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Chebyshev polynomials. Since their stability is based on the boundedness of the Chebyshev polynomials, these methods have been called RKC1 and RKC2. In this work we build temporally first- and second-order accurate super-time-stepping methods around the recursion relation associated with Legendre polynomials. We call these methods RKL1 and RKL2. The RKL1 method is first-order accurate in time; the RKL2 method is second-order accurate in time. We verify that the newly-designed RKL1 and RKL2 schemes have a very desirable monotonicity preserving property for one-dimensional problems - a solution that is monotone at the beginning of a time step retains that property at the end of that time step. It is shown that RKL1 and RKL2 methods are stable for all values of the diffusion coefficient up to the maximum value. We call this a convex monotonicity preserving property and show by examples that it is very useful in parabolic problems with variable diffusion coefficients. This includes variable coefficient parabolic equations that might give rise to skew symmetric terms. The RKC1 and RKC2 schemes do not share this convex monotonicity preserving property. One-dimensional and two-dimensional von Neumann stability analyses of RKC1, RKC2, RKL1 and RKL2 are also presented, showing that the latter two have some advantages. The paper includes several details to facilitate implementation. A detailed accuracy analysis is presented to show that the methods reach their design accuracies. A stringent set of test problems is also presented. To demonstrate the robustness and versatility of our methods, we show their successful operation on problems involving linear and non-linear heat conduction and viscosity, resistive magnetohydrodynamics, ambipolar diffusion dominated magnetohydrodynamics, level set methods and flux limited radiation diffusion. In a prior paper (Meyer, Balsara and Aslam 2012 [36]) we have also presented an extensive test-suite showing that the RKL2 method works robustly in the presence of shocks in an anisotropically conducting, magnetized plasma.

Proxy-equation paradigm: A strategy for massively parallel asynchronous computations

NASA Astrophysics Data System (ADS)

Mittal, Ankita; Girimaji, Sharath

2017-09-01

Massively parallel simulations of transport equation systems call for a paradigm change in algorithm development to achieve efficient scalability. Traditional approaches require time synchronization of processing elements (PEs), which severely restricts scalability. Relaxing synchronization requirement introduces error and slows down convergence. In this paper, we propose and develop a novel "proxy equation" concept for a general transport equation that (i) tolerates asynchrony with minimal added error, (ii) preserves convergence order and thus, (iii) expected to scale efficiently on massively parallel machines. The central idea is to modify a priori the transport equation at the PE boundaries to offset asynchrony errors. Proof-of-concept computations are performed using a one-dimensional advection (convection) diffusion equation. The results demonstrate the promise and advantages of the present strategy.

Multi-Component Diffusion with Application To Computational Aerothermodynamics

NASA Technical Reports Server (NTRS)

Sutton, Kenneth; Gnoffo, Peter A.

1998-01-01

The accuracy and complexity of solving multicomponent gaseous diffusion using the detailed multicomponent equations, the Stefan-Maxwell equations, and two commonly used approximate equations have been examined in a two part study. Part I examined the equations in a basic study with specified inputs in which the results are applicable for many applications. Part II addressed the application of the equations in the Langley Aerothermodynamic Upwind Relaxation Algorithm (LAURA) computational code for high-speed entries in Earth's atmosphere. The results showed that the presented iterative scheme for solving the Stefan-Maxwell equations is an accurate and effective method as compared with solutions of the detailed equations. In general, good accuracy with the approximate equations cannot be guaranteed for a species or all species in a multi-component mixture. 'Corrected' forms of the approximate equations that ensured the diffusion mass fluxes sum to zero, as required, were more accurate than the uncorrected forms. Good accuracy, as compared with the Stefan- Maxwell results, were obtained with the 'corrected' approximate equations in defining the heating rates for the three Earth entries considered in Part II.

A direct method of extracting surface recombination velocity from an electron beam induced current line scan

NASA Astrophysics Data System (ADS)

Ong, Vincent K. S.

1998-04-01

The extraction of diffusion length and surface recombination velocity in a semiconductor with the use of an electron beam induced current line scan has traditionally been done by fitting the line scan into complicated theoretical equations. It was recently shown that a much simpler equation is sufficient for the extraction of diffusion length. The linearization coefficient is the only variable that is needed to be adjusted in the curve fitting process. However, complicated equations are still necessary for the extraction of surface recombination velocity. It is shown in this article that it is indeed possible to extract surface recombination velocity with a simple equation, using only one variable, the linearization coefficient. An intuitive feel for the reason behind the method was discussed. The accuracy of the method was verified with the use of three-dimensional computer simulation, and was found to be even slightly better than that of the best existing method.

Numerical Modeling of HgCdTe Solidification: Effects of Phase Diagram, Double-Diffusion Convection and Microgravity Level

NASA Technical Reports Server (NTRS)

Bune, Andris V.; Gillies, Donald C.; Lehoczky, Sandor L.

1997-01-01

Melt convection, along with species diffusion and segregation on the solidification interface are the primary factors responsible for species redistribution during HgCdTe crystal growth from the melt. As no direct information about convection velocity is available, numerical modeling is a logical approach to estimate convection. Furthermore influence of microgravity level, double-diffusion and material properties should be taken into account. In the present study, HgCdTe is considered as a binary alloy with melting temperature available from a phase diagram. The numerical model of convection and solidification of binary alloy is based on the general equations of heat and mass transfer in two-dimensional region. Mathematical modeling of binary alloy solidification is still a challenging numericial problem. A Rigorous mathematical approach to this problem is available only when convection is not considered at all. The proposed numerical model was developed using the finite element code FIDAP. In the present study, the numerical model is used to consider thermal, solutal convection and a double diffusion source of mass transport.

Characteristic power spectrum of diffusive interface dynamics in the two-dimensional Ising model

NASA Astrophysics Data System (ADS)

Masumoto, Yusuke; Takesue, Shinji

2018-05-01

We investigate properties of the diffusive motion of an interface in the two-dimensional Ising model in equilibrium or nonequilibrium situations. We focused on the relation between the power spectrum of a time sequence of spins and diffusive motion of an interface which was already clarified in one-dimensional systems with a nonequilibrium phase transition like the asymmetric simple exclusion process. It is clarified that the interface motion is a diffusion process with a drift force toward the higher-temperature side when the system is in contact with heat reservoirs at different temperatures and heat transfers through the system. Effects of the width of the interface are also discussed.

A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh–Taylor instability

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tartakovsky, Alexandre M.; Meakin, Paul

2005-08-10

A numerical model based on smoothed particle hydrodynamics (SPH) has been developed and used to simulate the classical two-dimensional Rayleigh–Taylor instability and three-dimensional miscible flow in fracture apertures with complex geometries. To model miscible flow fluid particles with variable, composition dependent, masses were used. By basing the SPH equations on the particle number density artificial surface tension effects were avoided. The simulation results for the growth of a single perturbation driven by the Rayleigh – Taylor instability compare well with numerical results obtained by Fournier et al., and the growth of a perturbation with time can be represented quite wellmore » by a second-degree polynomial, in accord with the linear stability analysis of Duff et al. The dispersion coefficient found from SPH simulation of flow and diffusion in an ideal fracture was in excellent agreement with the value predicted by the theory of Taylor and Aris. The simulations of miscible flow in fracture apertures can be used to determination dispersion coefficients for transport in fractured media - a parameter used in large-scale simulations of contaminant transport.« less

Review of literature on the finite-element solution of the equations of two-dimensional surface-water flow in the horizontal plane

USGS Publications Warehouse

Lee, Jonathan K.; Froehlich, David C.

1987-01-01

Published literature on the application of the finite-element method to solving the equations of two-dimensional surface-water flow in the horizontal plane is reviewed in this report. The finite-element method is ideally suited to modeling two-dimensional flow over complex topography with spatially variable resistance. A two-dimensional finite-element surface-water flow model with depth and vertically averaged velocity components as dependent variables allows the user great flexibility in defining geometric features such as the boundaries of a water body, channels, islands, dikes, and embankments. The following topics are reviewed in this report: alternative formulations of the equations of two-dimensional surface-water flow in the horizontal plane; basic concepts of the finite-element method; discretization of the flow domain and representation of the dependent flow variables; treatment of boundary conditions; discretization of the time domain; methods for modeling bottom, surface, and lateral stresses; approaches to solving systems of nonlinear equations; techniques for solving systems of linear equations; finite-element alternatives to Galerkin's method of weighted residuals; techniques of model validation; and preparation of model input data. References are listed in the final chapter.

Bistability in the chemical master equation for dual phosphorylation cycles.

PubMed

Bazzani, Armando; Castellani, Gastone C; Giampieri, Enrico; Remondini, Daniel; Cooper, Leon N

2012-06-21

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.

Two-dimensional numerical simulation of boron diffusion for pyramidally textured silicon

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ma, Fa-Jun, E-mail: Fajun.Ma@nus.edu.sg; Duttagupta, Shubham; Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, 117576

2014-11-14

Multidimensional numerical simulation of boron diffusion is of great relevance for the improvement of industrial n-type crystalline silicon wafer solar cells. However, surface passivation of boron diffused area is typically studied in one dimension on planar lifetime samples. This approach neglects the effects of the solar cell pyramidal texture on the boron doping process and resulting doping profile. In this work, we present a theoretical study using a two-dimensional surface morphology for pyramidally textured samples. The boron diffusivity and segregation coefficient between oxide and silicon in simulation are determined by reproducing measured one-dimensional boron depth profiles prepared using different boronmore » diffusion recipes on planar samples. The established parameters are subsequently used to simulate the boron diffusion process on textured samples. The simulated junction depth is found to agree quantitatively well with electron beam induced current measurements. Finally, chemical passivation on planar and textured samples is compared in device simulation. Particularly, a two-dimensional approach is adopted for textured samples to evaluate chemical passivation. The intrinsic emitter saturation current density, which is only related to Auger and radiative recombination, is also simulated for both planar and textured samples. The differences between planar and textured samples are discussed.« less

Diffusive transport in the presence of stochastically gated absorption

NASA Astrophysics Data System (ADS)

Bressloff, Paul C.; Karamched, Bhargav R.; Lawley, Sean D.; Levien, Ethan

2017-08-01

We analyze a population of Brownian particles moving in a spatially uniform environment with stochastically gated absorption. The state of the environment at time t is represented by a discrete stochastic variable k (t )∈{0 ,1 } such that the rate of absorption is γ [1 -k (t )] , with γ a positive constant. The variable k (t ) evolves according to a two-state Markov chain. We focus on how stochastic gating affects the attenuation of particle absorption with distance from a localized source in a one-dimensional domain. In the static case (no gating), the steady-state attenuation is given by an exponential with length constant √{D /γ }, where D is the diffusivity. We show that gating leads to slower, nonexponential attenuation. We also explore statistical correlations between particles due to the fact that they all diffuse in the same switching environment. Such correlations can be determined in terms of moments of the solution to a corresponding stochastic Fokker-Planck equation.

Transport equations in an enzymatic glucose fuel cell

NASA Astrophysics Data System (ADS)

Jariwala, Soham; Krishnamurthy, Balaji

2018-01-01

A mathematical model is developed to study the effects of convective flux and operating temperature on the performance of an enzymatic glucose fuel cell with a membrane. The model assumes isothermal operating conditions and constant feed rate of glucose. The glucose fuel cell domain is divided into five sections, with governing equations describing transport characteristics in each region, namely - anode diffusion layer, anode catalyst layer (enzyme layer), membrane, cathode catalyst layer and cathode diffusion layer. The mass transport is assumed to be one-dimensional and the governing equations are solved numerically. The effects flow rate of glucose feed on the performance of the fuel cell are studied as it contributes significantly to the convective flux. The effects of operating temperature on the performance of a glucose fuel cell are also modeled. The cell performances are compared using cell polarization curves, which were found compliant with experimental observations.

2ND EF Conference in Turbulent Heat Transfer, Manchester, UK 1998. Volume 1

DTIC Science & Technology

1998-06-01

study the effect of Pr on statistical properties characterizing the scalar field. Turbulent diffusivities are presented for Pr=0.1-2400. A time scale, r...can be defined from the kinetic energy, fc, and the dissipation of turbulence, e, where r = -. The observed influence of Pr (0.05-10) on a time ...dimensional, time -dependent Navier-Stokes equa- tion in a skew-symmetric form and the advection- diffusion equation. du = (u x w) - VIE - Pxex

Elimination of numerical diffusion in 1 - phase and 2 - phase flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rajamaeki, M.

1997-07-01

The new hydraulics solution method PLIM (Piecewise Linear Interpolation Method) is capable of avoiding the excessive errors, numerical diffusion and also numerical dispersion. The hydraulics solver CFDPLIM uses PLIM and solves the time-dependent one-dimensional flow equations in network geometry. An example is given for 1-phase flow in the case when thermal-hydraulics and reactor kinetics are strongly coupled. Another example concerns oscillations in 2-phase flow. Both the example computations are not possible with conventional methods.

Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

NASA Astrophysics Data System (ADS)

Laslier, Benoît; Toninelli, Fabio Lucio

2018-03-01

We study a reversible continuous-time Markov dynamics of a discrete (2 + 1)-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the {L× L} torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in Luby et al. (SIAM J Comput 31:167-192, 2001): in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2 n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time (Luby et al. 2001) and a certain one-dimensional projection of the dynamics is described by the heat equation (Wilson in Ann Appl Probab 14:274-325, 2004). In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as L\\to∞ to the solution of a non-linear parabolic PDE. The initial profile is assumed to be C 2 differentiable and to contain no "frozen region". The explicit form of the PDE was recently conjectured (Laslier and Toninelli in Ann Henri Poincaré Theor Math Phys 18:2007-2043, 2017) on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg-Landau model (Funaki and Spohn in Commun Math Phys 85:1-36, 1997; Nishikawa in Commun Math Phys 127:205-227, 2003), here the mobility coefficient turns out to be a non-trivial function of the interface slope.

On a model of three-dimensional bursting and its parallel implementation

NASA Astrophysics Data System (ADS)

Tabik, S.; Romero, L. F.; Garzón, E. M.; Ramos, J. I.

2008-04-01

A mathematical model for the simulation of three-dimensional bursting phenomena and its parallel implementation are presented. The model consists of four nonlinearly coupled partial differential equations that include fast and slow variables, and exhibits bursting in the absence of diffusion. The differential equations have been discretized by means of a second-order accurate in both space and time, linearly-implicit finite difference method in equally-spaced grids. The resulting system of linear algebraic equations at each time level has been solved by means of the Preconditioned Conjugate Gradient (PCG) method. Three different parallel implementations of the proposed mathematical model have been developed; two of these implementations, i.e., the MPI and the PETSc codes, are based on a message passing paradigm, while the third one, i.e., the OpenMP code, is based on a shared space address paradigm. These three implementations are evaluated on two current high performance parallel architectures, i.e., a dual-processor cluster and a Shared Distributed Memory (SDM) system. A novel representation of the results that emphasizes the most relevant factors that affect the performance of the paralled implementations, is proposed. The comparative analysis of the computational results shows that the MPI and the OpenMP implementations are about twice more efficient than the PETSc code on the SDM system. It is also shown that, for the conditions reported here, the nonlinear dynamics of the three-dimensional bursting phenomena exhibits three stages characterized by asynchronous, synchronous and then asynchronous oscillations, before a quiescent state is reached. It is also shown that the fast system reaches steady state in much less time than the slow variables.

A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic simulations of neutral gas flows

NASA Astrophysics Data System (ADS)

Güçlü, Y.; Hitchon, W. N. G.

2012-04-01

The term 'Convected Scheme' (CS) refers to a family of algorithms, most usually applied to the solution of Boltzmann's equation, which uses a method of characteristics in an integral form to project an initial cell forward to a group of final cells. As such the CS is a 'forward-trajectory' semi-Lagrangian scheme. For multi-dimensional simulations of neutral gas flows, the cell-centered version of this semi-Lagrangian (CCSL) scheme has advantages over other options due to its implementation simplicity, low memory requirements, and easier treatment of boundary conditions. The main drawback of the CCSL-CS to date has been its high numerical diffusion in physical space, because of the 2nd order remapping that takes place at the end of each time step. By means of a modified equation analysis, it is shown that a high order estimate of the remapping error can be obtained a priori, and a small correction to the final position of the cells can be applied upon remapping, in order to achieve full compensation of this error. The resulting scheme is 4th order accurate in space while retaining the desirable properties of the CS: it is conservative and positivity-preserving, and the overall algorithm complexity is not appreciably increased. Two monotone (i.e. non-oscillating) versions of the fourth order CCSL-CS are also presented: one uses a common flux-limiter approach; the other uses a non-polynomial reconstruction to evaluate the derivatives of the density function. The method is illustrated in simple one- and two-dimensional examples, and a fully 3D solution of the Boltzmann equation describing expansion of a gas into vacuum through a cylindrical tube.

Calibration of a thin metal foil for infrared imaging video bolometer to estimate the spatial variation of thermal diffusivity using a photo-thermal technique.

PubMed

Pandya, Shwetang N; Peterson, Byron J; Sano, Ryuichi; Mukai, Kiyofumi; Drapiko, Evgeny A; Alekseyev, Andrey G; Akiyama, Tsuyoshi; Itomi, Muneji; Watanabe, Takashi

2014-05-01

A thin metal foil is used as a broad band radiation absorber for the InfraRed imaging Video Bolometer (IRVB), which is a vital diagnostic for studying three-dimensional radiation structures from high temperature plasmas in the Large Helical Device. The two-dimensional (2D) heat diffusion equation of the foil needs to be solved numerically to estimate the radiation falling on the foil through a pinhole geometry. The thermal, physical, and optical properties of the metal foil are among the inputs to the code besides the spatiotemporal variation of temperature, for reliable estimation of the exhaust power from the plasma illuminating the foil. The foil being very thin and of considerable size, non-uniformities in these properties need to be determined by suitable calibration procedures. The graphite spray used for increasing the surface emissivity also contributes to a change in the thermal properties. This paper discusses the application of the thermographic technique for determining the spatial variation of the effective in-plane thermal diffusivity of the thin metal foil and graphite composite. The paper also discusses the advantages of this technique in the light of limitations and drawbacks presented by other calibration techniques being practiced currently. The technique is initially applied to a material of known thickness and thermal properties for validation and finally to thin foils of gold and platinum both with two different thicknesses. It is observed that the effect of the graphite layer on the estimation of the thermal diffusivity becomes more pronounced for thinner foils and the measured values are approximately 2.5-3 times lower than the literature values. It is also observed that the percentage reduction in thermal diffusivity due to the coating is lower for high thermal diffusivity materials such as gold. This fact may also explain, albeit partially, the higher sensitivity of the platinum foil as compared to gold.

Calibration of a thin metal foil for infrared imaging video bolometer to estimate the spatial variation of thermal diffusivity using a photo-thermal technique

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pandya, Shwetang N., E-mail: pandya.shwetang@LHD.nifs.ac.jp; Sano, Ryuichi; Peterson, Byron J.

A thin metal foil is used as a broad band radiation absorber for the InfraRed imaging Video Bolometer (IRVB), which is a vital diagnostic for studying three-dimensional radiation structures from high temperature plasmas in the Large Helical Device. The two-dimensional (2D) heat diffusion equation of the foil needs to be solved numerically to estimate the radiation falling on the foil through a pinhole geometry. The thermal, physical, and optical properties of the metal foil are among the inputs to the code besides the spatiotemporal variation of temperature, for reliable estimation of the exhaust power from the plasma illuminating the foil.more » The foil being very thin and of considerable size, non-uniformities in these properties need to be determined by suitable calibration procedures. The graphite spray used for increasing the surface emissivity also contributes to a change in the thermal properties. This paper discusses the application of the thermographic technique for determining the spatial variation of the effective in-plane thermal diffusivity of the thin metal foil and graphite composite. The paper also discusses the advantages of this technique in the light of limitations and drawbacks presented by other calibration techniques being practiced currently. The technique is initially applied to a material of known thickness and thermal properties for validation and finally to thin foils of gold and platinum both with two different thicknesses. It is observed that the effect of the graphite layer on the estimation of the thermal diffusivity becomes more pronounced for thinner foils and the measured values are approximately 2.5–3 times lower than the literature values. It is also observed that the percentage reduction in thermal diffusivity due to the coating is lower for high thermal diffusivity materials such as gold. This fact may also explain, albeit partially, the higher sensitivity of the platinum foil as compared to gold.« less

Modeling persistence of motion in a crowded environment: The diffusive limit of excluding velocity-jump processes

NASA Astrophysics Data System (ADS)

Gavagnin, Enrico; Yates, Christian A.

2018-03-01

Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this behavior is an agent-based velocity-jump process. In the absence of agent-agent interaction, the mean-field continuum limit of the agent-based model (ABM) gives rise to the well known hyperbolic telegraph equation. When agent-agent interaction is included in the ABM, a strictly advective system of partial differential equations (PDEs) can be derived at the population level. However, no diffusive limit of the ABM has been obtained from such a model. Connecting the microscopic behavior of the ABM to a diffusive macroscopic description is desirable, since it allows the exploration of a wider range of scenarios and establishes a direct connection with commonly used statistical tools of movement analysis. In order to connect the ABM at the population level to a diffusive PDE at the population level, we consider a generalization of the agent-based velocity-jump process on a two-dimensional lattice with three forms of agent interaction. This generalization allows us to take a diffusive limit and obtain a faithful population-level description. We investigate the properties of the model at both the individual and population levels and we elucidate some of the models' key characteristic features. In particular, we show an intrinsic anisotropy inherent to the models and we find evidence of a spontaneous form of aggregation at both the micro- and macroscales.

Thermal radiation and mass transfer effects on unsteady MHD free convection flow past a vertical oscillating plate

NASA Astrophysics Data System (ADS)

Rana, B. M. Jewel; Ahmed, Rubel; Ahmmed, S. F.

2017-06-01

Unsteady MHD free convection flow past a vertical porous plate in porous medium with radiation, diffusion thermo, thermal diffusion and heat source are analyzed. The governing non-linear, partial differential equations are transformed into dimensionless by using non-dimensional quantities. Then the resultant dimensionless equations are solved numerically by applying an efficient, accurate and conditionally stable finite difference scheme of explicit type with the help of a computer programming language Compaq Visual Fortran. The stability and convergence analysis has been carried out to establish the effect of velocity, temperature, concentration, skin friction, Nusselt number, Sherwood number, stream lines and isotherms line. Finally, the effects of various parameters are presented graphically and discussed qualitatively.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

NASA Astrophysics Data System (ADS)

Wang, Yan; Zhang, Yufeng; Zhang, Xiangzhi

2016-09-01

We first introduced a linear stationary equation with a quadratic operator in ∂x and ∂y, then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Analytic study of solutions for a (3 + 1) -dimensional generalized KP equation

NASA Astrophysics Data System (ADS)

Gao, Hui; Cheng, Wenguang; Xu, Tianzhou; Wang, Gangwei

2018-03-01

The (3 + 1) -dimensional generalized KP (gKP) equation is an important nonlinear partial differential equation in theoretical and mathematical physics which can be used to describe nonlinear wave motion. Through the Hirota bilinear method, one-solition, two-solition and N-solition solutions are derived via symbolic computation. Two classes of lump solutions, rationally localized in all directions in space, to the dimensionally reduced cases in (2 + 1)-dimensions, are constructed by using a direct method based on the Hirota bilinear form of the equation. It implies that we can derive the lump solutions of the reduced gKP equation from positive quadratic function solutions to the aforementioned bilinear equation. Meanwhile, we get interaction solutions between a lump and a kink of the gKP equation. The lump appears from a kink and is swallowed by it with the change of time. This work offers a possibility which can enrich the variety of the dynamical features of solutions for higher-dimensional nonlinear evolution equations.

Statistical mechanics of shell models for two-dimensional turbulence

NASA Astrophysics Data System (ADS)

Aurell, E.; Boffetta, G.; Crisanti, A.; Frick, P.; Paladin, G.; Vulpiani, A.

1994-12-01

We study shell models that conserve the analogs of energy and enstrophy and hence are designed to mimic fluid turbulence in two-dimensions (2D). The main result is that the observed state is well described as a formal statistical equilibrium, closely analogous to the approach to two-dimensional ideal hydrodynamics of Onsager [Nuovo Cimento Suppl. 6, 279 (1949)], Hopf [J. Rat. Mech. Anal. 1, 87 (1952)], and Lee [Q. Appl. Math. 10, 69 (1952)]. In the presence of forcing and dissipation we observe a forward flux of enstrophy and a backward flux of energy. These fluxes can be understood as mean diffusive drifts from a source to two sinks in a system which is close to local equilibrium with Lagrange multipliers (``shell temperatures'') changing slowly with scale. This is clear evidence that the simplest shell models are not adequate to reproduce the main features of two-dimensional turbulence. The dimensional predictions on the power spectra from a supposed forward cascade of enstrophy and from one branch of the formal statistical equilibrium coincide in these shell models in contrast to the corresponding predictions for the Navier-Stokes and Euler equations in 2D. This coincidence has previously led to the mistaken conclusion that shell models exhibit a forward cascade of enstrophy. We also study the dynamical properties of the models and the growth of perturbations.

The effect of coupled transport phenomena in the Opalinus Clay and implications for radionuclide transport

NASA Astrophysics Data System (ADS)

Soler, Josep M.

2001-12-01

In this study, the potential effects of coupled transport phenomena on radionuclide transport in the vicinity of a repository for vitrified high-level radioactive waste (HLW) and spent nuclear fuel (SF) hosted by the Opalinus Clay in Switzerland, at times equal to or greater than the expected lifetime of the waste canisters (about 1000 years), are addressed. The solute fluxes associated with advection, chemical diffusion, thermal and chemical osmosis, hyperfiltration and thermal diffusion have been incorporated into a simple one-dimensional transport equation. The analytical solution of this equation, with appropriate parameters, shows that thermal osmosis is the only coupled transport mechanism that could, on its own, have a strong effect on repository performance. Based on the results from the analytical model, two-dimensional finite-difference models incorporating advection and thermal osmosis, and taking conservation of fluid mass into account, have been formulated. The results show that, under the conditions in the vicinity of the repository at the time scales of interest, and due to the constraints imposed by conservation of fluid mass, the advective component of flow will oppose and cancel the thermal-osmotic component. The overall conclusion is that coupled phenomena will only have a very minor impact on radionuclide transport in the Opalinus Clay, in terms of fluid and solute fluxes, at least under the conditions prevailing at times equal to or greater than the expected lifetime of the waste canisters (about 1000 years).

Diffusion Influenced Adsorption Kinetics.

PubMed

Miura, Toshiaki; Seki, Kazuhiko

2015-08-27

When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir-Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir-Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integro-differential equation using the Grünwald-Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage.

Application of the order-of-magnitude analysis to a fourth-order RANS closure for simulating a 2D boundary layer

NASA Astrophysics Data System (ADS)

Poroseva, Svetlana V.

2013-11-01

Simulations of turbulent boundary-layer flows are usually conducted using a set of the simplified Reynolds-Averaged Navier-Stokes (RANS) equations obtained by order-of-magnitude analysis (OMA) of the original RANS equations. The resultant equations for the mean-velocity components are closed using the Boussinesq approximation for the Reynolds stresses. In this study OMA is applied to the fourth-order RANS (FORANS) set of equations. The FORANS equations are chosen as they can be closed on the level of the 5th-order correlations without using unknown model coefficients, i.e. no turbulent diffusion modeling is required. New models for the 2nd-, 3rd- and 4th-order velocity-pressure gradient correlations are derived for the current FORANS equations. This set of FORANS equations and models are analyzed for the case of two-dimensional mean flow. The equations include familiar transport terms for the mean-velocity components along with algebraic expressions for velocity correlations of different orders specific to the FORANS approach. Flat plate DNS data (Spalart, 1988) are used to verify these expressions and the areas of the OMA applicability within the boundary layer. The material is based upon work supported by NASA under award NNX12AJ61A.

Arbitrarily high-order time-stepping schemes based on the operator spectrum theory for high-dimensional nonlinear Klein-Gordon equations

NASA Astrophysics Data System (ADS)

Liu, Changying; Wu, Xinyuan

2017-07-01

In this paper we explore arbitrarily high-order Lagrange collocation-type time-stepping schemes for effectively solving high-dimensional nonlinear Klein-Gordon equations with different boundary conditions. We begin with one-dimensional periodic boundary problems and first formulate an abstract ordinary differential equation (ODE) on a suitable infinity-dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula which is essential for the derivation of our arbitrarily high-order Lagrange collocation-type time-stepping schemes for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix under some suitable smoothness assumptions. With regard to the two dimensional Dirichlet or Neumann boundary problems, our new time-stepping schemes coupled with discrete Fast Sine / Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein-Gordon equations effectively. All essential features of the methodology are present in one-dimensional and two-dimensional cases, although the schemes to be analysed lend themselves with equal to higher-dimensional case. The numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear Klein-Gordon equations in the literature.

Recognition of Equations Using a Two-Dimensional Stochastic Context-Free Grammar

NASA Astrophysics Data System (ADS)

Chou, Philip A.

1989-11-01

We propose using two-dimensional stochastic context-free grammars for image recognition, in a manner analogous to using hidden Markov models for speech recognition. The value of the approach is demonstrated in a system that recognizes printed, noisy equations. The system uses a two-dimensional probabilistic version of the Cocke-Younger-Kasami parsing algorithm to find the most likely parse of the observed image, and then traverses the corresponding parse tree in accordance with translation formats associated with each production rule, to produce eqn I troff commands for the imaged equation. In addition, it uses two-dimensional versions of the Inside/Outside and Baum re-estimation algorithms for learning the parameters of the grammar from a training set of examples. Parsing the image of a simple noisy equation currently takes about one second of cpu time on an Alliant FX/80.

Experimental dynamic response of a two-dimensional, Mach 2.7, mixed compression inlet

NASA Technical Reports Server (NTRS)

Baumbick, R. J.; Neiner, G. H.; Cole, G. L.

1972-01-01

A test program was conducted on a two-dimensional supersonic inlet. Internal disturbances in diffuser exit mass flow were produced by oscillating overboard bypass doors. Open-loop dynamic responses of shock position, throat exit and diffuser exit static pressures are presented. The steady-state and dynamic coupling between ducts were also obtained. The experimental results from the two-dimensional inlet are compared to results from a similar size axisymmetric inlet and also to a transfer function synthesis program.

Mixing of a passive scalar in isotropic and sheared homogeneous turbulence

NASA Technical Reports Server (NTRS)

Shirani, E.; Ferziger, J. H.; Reynolds, W. C.

1981-01-01

In order to calculate the velocity and scalar fields, the three dimensional, time-dependent equations of motion and the diffusion equation were solved numerically. The following cases were treated: isotropic, homogeneous turbulence with decay of a passive scalar; and homogeneous turbulent shear flow with a passive scalar whose mean varies linearly in the spanwise direction. The solutions were obtained at relatively low Reynolds numbers so that all of the turbulent scales could be resolved without modeling. Turbulent statistics such as integral length scales, Taylor microscales, Kolmogorov length scale, one- and two-point correlations of velocity-velocity and velocity-scalar, turbulent Prandtl/Schmidt number, r.m.s. values of velocities, the scalar quantity and pressure, skewness, decay rates, and decay exponents were calculated. The results are compared with the available expermental results, and good agreement is obtained.

The effect of dissipative inhomogeneous medium on the statistics of the wave intensity

NASA Technical Reports Server (NTRS)

Saatchi, Sasan S.

1993-01-01

One of the main theoretical points in the theory of wave propagation in random medium is the derivation of closed form equations to describe the statistics of the propagating waves. In particular, in one dimensional problems, the closed form representation of the multiple scattering effects is important since it contributes in understanding such problems like wave localization, backscattering enhancement, and intensity fluctuations. In this the propagation of plane waves in a layer of one-dimensional dissipative random medium is considered. The medium is modeled by a complex permittivity whose real part is a constant representing the absorption. The one dimensional problem is mathematically equivalent to the analysis of a transmission line with randomly perturbed distributed parameters and a single mode lossy waveguide and the results can be used to study the propagation of radio waves through atmosphere and the remote sensing of geophysical media. It is assumed the scattering medium consists of an ensemble of one-dimensional point scatterers randomly positioned in a layer of thickness L with diffuse boundaries. A Poisson impulse process with density lambda is used to model the position of scatterers in the medium. By employing the Markov properties of this process an exact closed form equation of Kolmogorov-Feller type was obtained for the probability density of the reflection coefficient. This equation was solved by combining two limiting cases: (1) when the density of scatterers is small; and (2) when the medium is weakly dissipative. A two variable perturbation method for small lambda was used to obtain solutions valid for thick layers. These solutions are then asymptotically evaluated for small dissipation. To show the effect of dissipation, the mean and fluctuations of the reflected power are obtained. The results were compared with a lossy homogeneous medium and with a lossless inhomogeneous medium and the regions where the effect of absorption is not essential were discussed.

An interaction algorithm for prediction of mean and fluctuating velocities in two-dimensional aerodynamic wake flows

NASA Technical Reports Server (NTRS)

Baker, A. J.; Orzechowski, J. A.

1980-01-01

A theoretical analysis is presented yielding sets of partial differential equations for determination of turbulent aerodynamic flowfields in the vicinity of an airfoil trailing edge. A four phase interaction algorithm is derived to complete the analysis. Following input, the first computational phase is an elementary viscous corrected two dimensional potential flow solution yielding an estimate of the inviscid-flow induced pressure distribution. Phase C involves solution of the turbulent two dimensional boundary layer equations over the trailing edge, with transition to a two dimensional parabolic Navier-Stokes equation system describing the near-wake merging of the upper and lower surface boundary layers. An iteration provides refinement of the potential flow induced pressure coupling to the viscous flow solutions. The final phase is a complete two dimensional Navier-Stokes analysis of the wake flow in the vicinity of a blunt-bases airfoil. A finite element numerical algorithm is presented which is applicable to solution of all partial differential equation sets of inviscid-viscous aerodynamic interaction algorithm. Numerical results are discussed.

Investigating axial diffusion in cylindrical pores using confocal single-particle fluorescence correlation spectroscopy.

PubMed

Chen, Fang; Neupane, Bhanu; Li, Peiyuan; Su, Wei; Wang, Gufeng

2016-08-01

We explored the feasibility of using confocal fluorescence correlation spectroscopy to study small nanoparticle diffusion in hundred-nanometer-sized cylindrical pores. By modeling single particle diffusion in tube-like confined three-dimensional space aligned parallel to the confocal optical axis, we showed that two diffusion dynamics can be observed in both original intensity traces and the autocorrelation functions (ACFs): the confined two-dimensional lateral diffusion and the unconfined one-dimensional (1D) axial diffusion. The separation of the axial and confined lateral diffusion dynamics provides an opportunity to study diffusions in different dimensions separately. We further experimentally studied 45 nm carboxylated polystyrene particles diffusing in 300 nm alumina pores. The experimental data showed consistency with the simulation. To extract the accurate axial diffusion coefficient, we found that a 1D diffusion model with a Lorentzian axial collection profile needs to be used to analyze the experimental ACFs. The diffusion of the 45 nm nanoparticles in polyethyleneglycol-passivated 300 nm pores slowed down by a factor of ∼2, which can be satisfactorily explained by hydrodynamic frictions. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

NASA Astrophysics Data System (ADS)

Patrone, Paul N.; Einstein, T. L.; Margetis, Dionisios

2010-12-01

We study analytically and numerically a one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal. Our goal is to formulate and validate analytical techniques for approximately solving systems of coupled nonlinear stochastic differential equations (SDEs) governing fluctuations in surface motion. In our analytical approach, the starting point is the Burton-Cabrera-Frank (BCF) model by which step motion is driven by diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounts for entropic and nearest-neighbor elastic-dipole interactions. By including Gaussian white noise to the equations of motion for terrace widths, we formulate large systems of SDEs under different choices of diffusion coefficients for the noise. We simplify this description via (i) perturbation theory and linearization of the step interactions and, alternatively, (ii) a mean-field (MF) approximation whereby widths of adjacent terraces are replaced by a self-consistent field but nonlinearities in step interactions are retained. We derive simplified formulas for the time-dependent terrace-width distribution (TWD) and its steady-state limit. Our MF analytical predictions for the TWD compare favorably with kinetic Monte Carlo simulations under the addition of a suitably conservative white noise in the BCF equations.

A compressible Navier-Stokes solver with two-equation and Reynolds stress turbulence closure models

NASA Technical Reports Server (NTRS)

Morrison, Joseph H.

1992-01-01

This report outlines the development of a general purpose aerodynamic solver for compressible turbulent flows. Turbulent closure is achieved using either two equation or Reynolds stress transportation equations. The applicable equation set consists of Favre-averaged conservation equations for the mass, momentum and total energy, and transport equations for the turbulent stresses and turbulent dissipation rate. In order to develop a scheme with good shock capturing capabilities, good accuracy and general geometric capabilities, a multi-block cell centered finite volume approach is used. Viscous fluxes are discretized using a finite volume representation of a central difference operator and the source terms are treated as an integral over the control volume. The methodology is validated by testing the algorithm on both two and three dimensional flows. Both the two equation and Reynolds stress models are used on a two dimensional 10 degree compression ramp at Mach 3, and the two equation model is used on the three dimensional flow over a cone at angle of attack at Mach 3.5. With the development of this algorithm, it is now possible to compute complex, compressible high speed flow fields using both two equation and Reynolds stress turbulent closure models, with the capability of eventually evaluating their predictive performance.

Lax representations for matrix short pulse equations

NASA Astrophysics Data System (ADS)

Popowicz, Z.

2017-10-01

The Lax representation for different matrix generalizations of Short Pulse Equations (SPEs) is considered. The four-dimensional Lax representations of four-component Matsuno, Feng, and Dimakis-Müller-Hoissen-Matsuno equations are obtained. The four-component Feng system is defined by generalization of the two-dimensional Lax representation to the four-component case. This system reduces to the original Feng equation, to the two-component Matsuno equation, or to the Yao-Zang equation. The three-component version of the Feng equation is presented. The four-component version of the Matsuno equation with its Lax representation is given. This equation reduces the new two-component Feng system. The two-component Dimakis-Müller-Hoissen-Matsuno equations are generalized to the four-parameter family of the four-component SPE. The bi-Hamiltonian structure of this generalization, for special values of parameters, is defined. This four-component SPE in special cases reduces to the new two-component SPE.

An asymptotic induced numerical method for the convection-diffusion-reaction equation

NASA Technical Reports Server (NTRS)

Scroggs, Jeffrey S.; Sorensen, Danny C.

1988-01-01

A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run time results demonstrate the viability of the method.

Feasibility of High Energy Lasers for Interdiction Activities

DTIC Science & Technology

2017-12-01

2.3.2 Power in the Bucket Another parameter we will use in this study is the power-in-the-bucket. The “bucket” is defined as the area on the target we...the heat diffusion equation for a one -dimensional case (where the x-direction is into the target) and assuming a semi-infinite slab of material. The... studied and modeled. One of the approaches to describe these interactions is by making a one -dimensional mathematical model assuming [8]: 1. A semi

Vapor-liquid equilibrium and equation of state of two-dimensional fluids from a discrete perturbation theory

NASA Astrophysics Data System (ADS)

Trejos, Víctor M.; Santos, Andrés; Gámez, Francisco

2018-05-01

The interest in the description of the properties of fluids of restricted dimensionality is growing for theoretical and practical reasons. In this work, we have firstly developed an analytical expression for the Helmholtz free energy of the two-dimensional square-well fluid in the Barker-Henderson framework. This equation of state is based on an approximate analytical radial distribution function for d-dimensional hard-sphere fluids (1 ≤ d ≤ 3) and is validated against existing and new simulation results. The so-obtained equation of state is implemented in a discrete perturbation theory able to account for general potential shapes. The prototypical Lennard-Jones and Yukawa fluids are tested in its two-dimensional version against available and new simulation data with semiquantitative agreement.

Two-Dimensional Transport Studies for the Composition and Structure of the Io Plasma Torus

NASA Technical Reports Server (NTRS)

Smyth, William H.

2003-01-01

The overall objective of this project is to investigate the roles of local and spatially extended plasma sources created by Io, plasma torus chemistry, and plasma convective and diffusive transport in producing the long-lived S(+), S(++) and O(+) radial ribbon structures of the plasma torus, their System III longitude and local-time asymmetries, their energy sources and their possible time variability. To accomplish this objective, two-dimensional [radial (L) and System III longitude] plasma transport equations for the flux-tube plasma content and energy content will be solved that include the convective motions for both the east-west electric field and co-rotational velocity-lag profile near Io s orbit, radial diffusion, and the spacetime dependent flux-tube production and loss created by both neutral-plasma and plasma-ion reaction chemistry in the plasma torus. For neutral-plasma chemistry, the project will for the first time undertake the calculation of realistic three-dimensional, spatially-extended, and time-varying contributions to the flux-tube ion-production and loss that are produced by Io's corona and extended neutral clouds. The unknown two-dimensional spatial nature of diffusion in the plasma transport will be isolated and better defined in the investigation by the collective consideration of the foregoing different physical processes. For energy transport, the energy flow from hot pickup ions (and a new electron source) to thermal ions and electrons will be included in investigating the System III longitude and local-time temperature asymmetries in the plasma torus. The research is central to the scope of the NASA Sun-Earth Connection Roadmap in Quest II Campaign 4 "Comparative Planetary Space Environments" by addressing key questions for understanding the magnetosphere of planets with high rotation rates and large internal plasma sources and, in addition, is of considerable importance to the NASA Solar System Exploration Science Theme. In this regard, Jupiter is the most extreme example with its rapid rotation and with its inner Galilean satellite Io providing the dominant plasma source for the magnetosphere.

Assessing the inherent uncertainty of one-dimensional diffusions

NASA Astrophysics Data System (ADS)

Eliazar, Iddo; Cohen, Morrel H.

2013-01-01

In this paper we assess the inherent uncertainty of one-dimensional diffusion processes via a stochasticity classification which provides an à la Mandelbrot categorization into five states of uncertainty: infra-mild, mild, borderline, wild, and ultra-wild. Two settings are considered. (i) Stopped diffusions: the diffusion initiates from a high level and is stopped once it first reaches a low level; in this setting we analyze the inherent uncertainty of the diffusion's maximal exceedance above its initial high level. (ii) Stationary diffusions: the diffusion is in dynamical statistical equilibrium; in this setting we analyze the inherent uncertainty of the diffusion's equilibrium level. In both settings general closed-form analytic results are established, and their application is exemplified by stock prices in the stopped-diffusions setting, and by interest rates in the stationary-diffusions setting. These results provide a highly implementable decision-making tool for the classification of uncertainty in the context of one-dimensional diffusions.

Two-dimensional integrating matrices on rectangular grids. [solving differential equations associated with rotating structures

NASA Technical Reports Server (NTRS)

Lakin, W. D.

1981-01-01

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

The structure of mass-loading shocks. [interaction of solar wind with cometary coma or local interstellar medium using two-fluid model

NASA Technical Reports Server (NTRS)

Zank, G. P.; Khabibrakhmanov, I. KH.; Story, T.

1993-01-01

A new two-fluid model which describes mass loading in the solar wind (e.g., the interaction of the solar wind with a cometary coma or the local interstellar medium) is presented. The self-consistent back-reaction of the mass-loaded ions is included through their effective scattering in low-frequency MHD turbulence and the invocation of a diffusive approximation. Such an approximation has the advantage of introducing self-consistent dissipation coefficients into the governing equations, thereby facilitating the investigation of the internal structure of shocks in mass-loading environments. To illustrate the utility of the new model, we consider the structure of cometary shocks in the hypersonic one-dimensional limit, finding that the incoming solar wind is slowed by both mass loading and the development of a large cometary ion pressure gradient. The shock is broadened and smoothed by the cometary ions with a thickness of the order of the cometary ion diffusion scale.

Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barajas-Solano, David A.; Tartakovsky, Alexandre M.

We present a cumulative density function (CDF) method for the probabilistic analysis of $d$-dimensional advection-dominated reactive transport in heterogeneous media. We employ a probabilistic approach in which epistemic uncertainty on the spatial heterogeneity of Darcy-scale transport coefficients is modeled in terms of random fields with given correlation structures. Our proposed CDF method employs a modified Large-Eddy-Diffusivity (LED) approach to close and localize the nonlocal equations governing the one-point PDF and CDF of the concentration field, resulting in a $(d + 1)$ dimensional PDE. Compared to the classsical LED localization, the proposed modified LED localization explicitly accounts for the mean-field advectivemore » dynamics over the phase space of the PDF and CDF. To illustrate the accuracy of the proposed closure, we apply our CDF method to one-dimensional single-species reactive transport with uncertain, heterogeneous advection velocities and reaction rates modeled as random fields.« less

Averaging of random walks and shift-invariant measures on a Hilbert space

NASA Astrophysics Data System (ADS)

Sakbaev, V. Zh.

2017-06-01

We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.

Application of kinetic flux vector splitting scheme for solving multi-dimensional hydrodynamical models of semiconductor devices

NASA Astrophysics Data System (ADS)

Nisar, Ubaid Ahmed; Ashraf, Waqas; Qamar, Shamsul

In this article, one and two-dimensional hydrodynamical models of semiconductor devices are numerically investigated. The models treat the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behavior of electron flow in semiconductor devices. Mathematically, the governing equations form a convection-diffusion type system with a right hand side describing the relaxation effects and interaction with a self consistent electric field. The proposed numerical scheme is a splitting scheme based on the kinetic flux-vector splitting (KFVS) method for the hyperbolic step, and a semi-implicit Runge-Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as low field mobility, device length, lattice temperature and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equations. A two dimensional simulation is also performed by KFVS method for a MESFET device, producing results in good agreement with those obtained by NT-central scheme.

Particle-size segregation and diffusive remixing in shallow granular avalanches

NASA Astrophysics Data System (ADS)

Gray, J. M. N. T.; Chugunov, V. A.

2006-12-01

Segregation and mixing of dissimilar grains is a problem in many industrial and pharmaceutical processes, as well as in hazardous geophysical flows, where the size-distribution can have a major impact on the local rheology and the overall run-out. In this paper, a simple binary mixture theory is used to formulate a model for particle-size segregation and diffusive remixing of large and small particles in shallow gravity-driven free-surface flows. This builds on a recent theory for the process of kinetic sieving, which is the dominant mechanism for segregation in granular avalanches provided the density-ratio and the size-ratio of the particles are not too large. The resulting nonlinear parabolic segregation remixing equation reduces to a quasi-linear hyperbolic equation in the no-remixing limit. It assumes that the bulk velocity is incompressible and that the bulk pressure is lithostatic, making it compatible with most theories used to compute the motion of shallow granular free-surface flows. In steady-state, the segregation remixing equation reduces to a logistic type equation and the ‘S’-shaped solutions are in very good agreement with existing particle dynamics simulations for both size and density segregation. Laterally uniform time-dependent solutions are constructed by mapping the segregation remixing equation to Burgers equation and using the Cole Hopf transformation to linearize the problem. It is then shown how solutions for arbitrary initial conditions can be constructed using standard methods. Three examples are investigated in which the initial concentration is (i) homogeneous, (ii) reverse graded with the coarse grains above the fines, and, (iii) normally graded with the fines above the coarse grains. Time-dependent two-dimensional solutions are also constructed for plug-flow in a semi-infinite chute.

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

NASA Astrophysics Data System (ADS)

Frank, T. D.

2008-02-01

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.

Rogue waves in the two dimensional nonlocal nonlinear Schrödinger equation and nonlocal Klein-Gordon equation.

PubMed

Liu, Wei; Zhang, Jing; Li, Xiliang

2018-01-01

In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota's bilinear method. Like the nonlocal NLS equation, these solutions may have singularities. However, by suitable constraints of parameters, nonsingular breather solutions are generated. Besides, by taking a long wave limit of these obtained soliton solutions, rogue wave solutions and semi-rational solutions are derived. For the two dimensional NLS equation, rogue wave solutions are line rogue waves, which arise from a constant background with a line profile and then disappear into the same background. The semi-rational solutions shows intriguing dynamical behaviours: line rogue wave and line breather arise from a constant background together and then disappear into the constant background again uniformly. For the coupled nonlocal Klein-Gordon equation, rogue waves are localized in both space and time, semi-rational solutions are composed of rogue waves, breathers and periodic line waves. These solutions are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides.

Rogue waves in the two dimensional nonlocal nonlinear Schrödinger equation and nonlocal Klein-Gordon equation

PubMed Central

Zhang, Jing; Li, Xiliang

2018-01-01

In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. Solitons and periodic line waves as exact solutions of these two nonlocal equations are derived by employing the Hirota’s bilinear method. Like the nonlocal NLS equation, these solutions may have singularities. However, by suitable constraints of parameters, nonsingular breather solutions are generated. Besides, by taking a long wave limit of these obtained soliton solutions, rogue wave solutions and semi-rational solutions are derived. For the two dimensional NLS equation, rogue wave solutions are line rogue waves, which arise from a constant background with a line profile and then disappear into the same background. The semi-rational solutions shows intriguing dynamical behaviours: line rogue wave and line breather arise from a constant background together and then disappear into the constant background again uniformly. For the coupled nonlocal Klein-Gordon equation, rogue waves are localized in both space and time, semi-rational solutions are composed of rogue waves, breathers and periodic line waves. These solutions are demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides. PMID:29432495

A numerical solution for two-dimensional Fredholm integral equations of the second kind with kernels of the logarithmic potential form

NASA Technical Reports Server (NTRS)

Gabrielsen, R. E.; Uenal, A.

1981-01-01

Two dimensional Fredholm integral equations with logarithmic potential kernels are numerically solved. The explicit consequence of these solutions to their true solutions is demonstrated. The results are based on a previous work in which numerical solutions were obtained for Fredholm integral equations of the second kind with continuous kernels.

Classifying bilinear differential equations by linear superposition principle

NASA Astrophysics Data System (ADS)

Zhang, Lijun; Khalique, Chaudry Masood; Ma, Wen-Xiu

2016-09-01

In this paper, we investigate the linear superposition principle of exponential traveling waves to construct a sub-class of N-wave solutions of Hirota bilinear equations. A necessary and sufficient condition for Hirota bilinear equations possessing this specific sub-class of N-wave solutions is presented. We apply this result to find N-wave solutions to the (2+1)-dimensional KP equation, a (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation, a (3+1)-dimensional generalized BKP equation and the (2+1)-dimensional BKP equation. The inverse question, i.e., constructing Hirota Bilinear equations possessing N-wave solutions, is considered and a refined 3-step algorithm is proposed. As examples, we construct two very general kinds of Hirota bilinear equations of order 4 possessing N-wave solutions among which one satisfies dispersion relation and another does not satisfy dispersion relation.

Mathematical modeling of microbially induced crown corrosion in wastewater collection systems and laboratory investigation and modeling of sulfuric acid corrosion of concrete

NASA Astrophysics Data System (ADS)

Jahani, Fereidoun

In the model for microbially induced crown corrosion, the diffusion of sulfide inside the concrete pores, its biological conversion to sulfuric acid, and the corrosion of calcium carbonate aggregates are represented. The corrosion front is modeled as a moving boundary. The location of the interface between the corrosion layer and the concrete is determined as part of the solution to the model equations. This model consisted of a system of one dimensional reaction-diffusion equations coupled to an equation describing the movement of the corrosion front. The equations were solved numerically using finite element Galerkin approximation. The concentration profiles of sulfide in the air and the liquid phases, the pH as a function of concrete depth, and the position of the corrosion front. A new equation for the corrosion rate was also derived. A more specific model for the degradation of a concrete specimen exposed to a sulfuric acid solution was also studied. In this model, diffusion of hydrogen ions and their reaction with alkaline components of concrete were expressed using Fick's Law of diffusion. The model equations described the moving boundary, the dissolution rate of alkaline components in the concrete, volume increase of sulfuric acid solution over the concrete specimen, and the boundary conditions on the surface of the concrete. An apparatus was designed and experiments were performed to measure pH changes on the surface of concrete. The data were used to calculate the dissolution rate of the concrete and, with the model, to determine the diffusion rate of sulfuric acid in the corrosion layer and corrosion layer thickness. Electrochemical Impedance Spectroscopy (EIS) was used to study the corrosion rate of iron pins embedded in the concrete sample. The open circuit potential (OCP) determined the onset of corrosion on the surface of the pins. Visual observation of the corrosion layer thickness was in good agreement with the simulation results.

Nonlocal Reformulations of Water and Internal Waves and Asymptotic Reductions

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.

2009-09-01

Nonlocal reformulations of the classical equations of water waves and two ideal fluids separated by a free interface, bounded above by either a rigid lid or a free surface, are obtained. The kinematic equations may be written in terms of integral equations with a free parameter. By expressing the pressure, or Bernoulli, equation in terms of the surface/interface variables, a closed system is obtained. An advantage of this formulation, referred to as the nonlocal spectral (NSP) formulation, is that the vertical component is eliminated, thus reducing the dimensionality and fixing the domain in which the equations are posed. The NSP equations and the Dirichlet-Neumann operators associated with the water wave or two-fluid equations can be related to each other and the Dirichlet-Neumann series can be obtained from the NSP equations. Important asymptotic reductions obtained from the two-fluid nonlocal system include the generalizations of the Benney-Luke and Kadomtsev-Petviashvili (KP) equations, referred to as intermediate-long wave (ILW) generalizations. These 2+1 dimensional equations possess lump type solutions. In the water wave problem high-order asymptotic series are obtained for two and three dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known hyperbolic secant squared solution of the KdV equation; in three dimensions, the first term is the rational lump solution of the KP equation.

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

NASA Astrophysics Data System (ADS)

Fitt, A. D.; Mason, D. P.; Moss, E. A.

2007-11-01

The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.

Discrete effect on the halfway bounce-back boundary condition of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations.

PubMed

Cui, Shuqi; Hong, Ning; Shi, Baochang; Chai, Zhenhua

2016-04-01

In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches.

PubMed

Xu, Zhenli; Ma, Manman; Liu, Pei

2014-07-01

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Documentation of the Surface-Water Routing (SWR1) Process for modeling surface-water flow with the U.S. Geological Survey Modular Ground-Water Model (MODFLOW-2005)

USGS Publications Warehouse

Hughes, Joseph D.; Langevin, Christian D.; Chartier, Kevin L.; White, Jeremy T.

2012-01-01

A flexible Surface-Water Routing (SWR1) Process that solves the continuity equation for one-dimensional and two-dimensional surface-water flow routing has been developed for the U.S. Geological Survey three-dimensional groundwater model, MODFLOW-2005. Simple level- and tilted-pool reservoir routing and a diffusive-wave approximation of the Saint-Venant equations have been implemented. Both methods can be implemented in the same model and the solution method can be simplified to represent constant-stage elements that are functionally equivalent to the standard MODFLOW River or Drain Package boundary conditions. A generic approach has been used to represent surface-water features (reaches) and allows implementation of a variety of geometric forms. One-dimensional geometric forms include rectangular, trapezoidal, and irregular cross section reaches to simulate one-dimensional surface-water features, such as canals and streams. Two-dimensional geometric forms include reaches defined using specified stage-volume-area-perimeter (SVAP) tables and reaches covering entire finite-difference grid cells to simulate two-dimensional surface-water features, such as wetlands and lakes. Specified SVAP tables can be used to represent reaches that are smaller than the finite-difference grid cell (for example, isolated lakes), or reaches that cannot be represented accurately using the defined top of the model. Specified lateral flows (which can represent point and distributed flows) and stage-dependent rainfall and evaporation can be applied to each reach. The SWR1 Process can be used with the MODFLOW Unsaturated Zone Flow (UZF1) Package to permit dynamic simulation of runoff from the land surface to specified reaches. Surface-water/groundwater interactions in the SWR1 Process are mathematically defined to be a function of the difference between simulated stages and groundwater levels, and the specific form of the reach conductance equation used in each reach. Conductance can be specified directly or calculated as a function of the simulated wetted perimeter and defined reach bed hydraulic properties, or as a weighted combination of both reach bed hydraulic properties and horizontal hydraulic conductivity. Each reach can be explicitly coupled to a single specific groundwater-model layer or coupled to multiple groundwater-model layers based on the reach geometry and groundwater-model layer elevations in the row and column containing the reach. Surface-water flow between reservoirs is simulated using control structures. Surface-water flow between reaches, simulated by the diffusive-wave approximation, can also be simulated using control structures. A variety of control structures have been included in the SWR1 Process and include (1) excess-volume structures, (2) uncontrolled-discharge structures, (3) pumps, (4) defined stage-discharge relations, (5) culverts, (6) fixed- or movable-crest weirs, and (7) fixed or operable gated spillways. Multiple control structures can be implemented in individual reaches and are treated as composite flow structures. Solution of the continuity equation at the reach-group scale (a single reach or a user-defined collection of individual reaches) is achieved using exact Newton methods with direct solution methods or exact and inexact Newton methods with Krylov sub-space methods. Newton methods have been used in the SWR1 Process because of their ability to solve nonlinear problems. Multiple SWR1 time steps can be simulated for each MODFLOW time step, and a simple adaptive time-step algorithm, based on user-specified rainfall, stage, flow, or convergence constraints, has been implemented to better resolve surface-water response. A simple linear- or sigmoid-depth scaling approach also has been implemented to account for increased bed roughness at small surface-water depths and to increase numerical stability. A line-search algorithm also has been included to improve the quality of the Newton-step upgrade vector, if possible. The SWR1 Process has been benchmarked against one- and two-dimensional numerical solutions from existing one- and two-dimensional numerical codes that solve the dynamic-wave approximation of the Saint-Venant equations. Two-dimensional solutions test the ability of the SWR1 Process to simulate the response of a surface-water system to (1) steady flow conditions for an inclined surface (solution of Manning's equation), and (2) transient inflow and rainfall for an inclined surface. The one-dimensional solution tests the ability of the SWR1 Process to simulate a looped network with multiple upstream inflows and several control structures. The SWR1 Process also has been compared to a level-pool reservoir solution. A synthetic test problem was developed to evaluate a number of different SWR1 solution options and simulate surface-water/groundwater interaction. The solution approach used in the SWR1 Process may not be applicable for all surface-water/groundwater problems. The SWR1 Process is best suited for modeling long-term changes (days to years) in surface-water and groundwater flow. Use of the SWR1 Process is not recommended for modeling the transient exchange of water between streams and aquifers when local and convective acceleration and other secondary effects (for example, wind and Coriolis forces) are substantial. Dam break evaluations and two-dimensional evaluations of spatially extensive domains are examples where acceleration terms and secondary effects would be significant, respectively.

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

NASA Astrophysics Data System (ADS)

Abd-El-Malek, Mina B.; Kassem, Magda M.; Meky, Mohammed L. M.

2002-03-01

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skin-like membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.

Short-time dynamics of monomers and dimers in quasi-two-dimensional colloidal mixtures.

PubMed

Sarmiento-Gómez, Erick; Villanueva-Valencia, José Ramón; Herrera-Velarde, Salvador; Ruiz-Santoyo, José Arturo; Santana-Solano, Jesús; Arauz-Lara, José Luis; Castañeda-Priego, Ramón

2016-07-01

We report on the short-time dynamics in colloidal mixtures made up of monomers and dimers highly confined between two glass plates. At low concentrations, the experimental measurements of colloidal motion agree well with the solution of the Navier-Stokes equation at low Reynolds numbers; the latter takes into account the increase in the drag force on a colloidal particle due to wall-particle hydrodynamic forces. More importantly, we find that the ratio of the short-time diffusion coefficient of the monomer and that of the center of mass of the dimmer is almost independent of both the dimer molar fraction, x_{d}, and the total packing fraction, ϕ, up to ϕ≈0.5. At higher concentrations, this ratio displays a small but systematic increase. A similar physical scenario is observed for the ratio between the parallel and the perpendicular components of the short-time diffusion coefficients of the dimer. This dynamical behavior is corroborated by means of molecular dynamics computer simulations that include explicitly the particle-particle hydrodynamic forces induced by the solvent. Our results suggest that the effects of colloid-colloid hydrodynamic interactions on the short-time diffusion coefficients are almost identical and factorable in both species.

Diffusion with resetting inside a circle

NASA Astrophysics Data System (ADS)

Chatterjee, Abhinava; Christou, Christos; Schadschneider, Andreas

2018-06-01

We study the Brownian motion of a particle in a bounded circular two-dimensional domain in search for a stationary target on the boundary of the domain. The process switches between two modes: one where it performs a two-dimensional diffusion inside the circle and one where it diffuses along the one-dimensional boundary. During the process, the Brownian particle resets to its initial position with a constant rate r . The Fokker-Planck formalism allows us to calculate the mean time to absorption (MTA) as well as the optimal resetting rate for which the MTA is minimized. From the derived analytical results the parameter regions where resetting reduces the search time can be specified. We also provide a numerical method for the verification of our results.

Verification and Validation of a Coordinate Transformation Method in Axisymmetric Transient Magnetics.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ashcraft, C. Chace; Niederhaus, John Henry; Robinson, Allen C.

We present a verification and validation analysis of a coordinate-transformation-based numerical solution method for the two-dimensional axisymmetric magnetic diffusion equation, implemented in the finite-element simulation code ALEGRA. The transformation, suggested by Melissen and Simkin, yields an equation set perfectly suited for linear finite elements and for problems with large jumps in material conductivity near the axis. The verification analysis examines transient magnetic diffusion in a rod or wire in a very low conductivity background by first deriving an approximate analytic solution using perturbation theory. This approach for generating a reference solution is shown to be not fully satisfactory. A specializedmore » approach for manufacturing an exact solution is then used to demonstrate second-order convergence under spatial refinement and tem- poral refinement. For this new implementation, a significant improvement relative to previously available formulations is observed. Benefits in accuracy for computed current density and Joule heating are also demonstrated. The validation analysis examines the circuit-driven explosion of a copper wire using resistive magnetohydrodynamics modeling, in comparison to experimental tests. The new implementation matches the accuracy of the existing formulation, with both formulations capturing the experimental burst time and action to within approximately 2%.« less

Stochastic modelling of intermittency.

PubMed

Stemler, Thomas; Werner, Johannes P; Benner, Hartmut; Just, Wolfram

2010-01-13

Recently, methods have been developed to model low-dimensional chaotic systems in terms of stochastic differential equations. We tested such methods in an electronic circuit experiment. We aimed to obtain reliable drift and diffusion coefficients even without a pronounced time-scale separation of the chaotic dynamics. By comparing the analytical solutions of the corresponding Fokker-Planck equation with experimental data, we show here that crisis-induced intermittency can be described in terms of a stochastic model which is dominated by state-space-dependent diffusion. Further on, we demonstrate and discuss some limits of these modelling approaches using numerical simulations. This enables us to state a criterion that can be used to decide whether a stochastic model will capture the essential features of a given time series. This journal is © 2010 The Royal Society

HARDI DATA DENOISING USING VECTORIAL TOTAL VARIATION AND LOGARITHMIC BARRIER

PubMed Central

Kim, Yunho; Thompson, Paul M.; Vese, Luminita A.

2010-01-01

In this work, we wish to denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical brain imaging. Diffusion imaging is a relatively new and powerful method to measure the three-dimensional profile of water diffusion at each point in the brain. These images can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI data is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. Unfortunately, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter pre-selected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. Here we propose two variational methods to denoise HARDI data. The first one directly denoises the collected data S, while the second one denoises the so-called sADC (spherical Apparent Diffusion Coefficient), a field of radial functions derived from the data. These two quantities are related by an equation of the form S = SSexp (−b · sADC) (in the noise-free case). By applying these two different models, we will be able to determine which quantity will most accurately preserve data structure after denoising. The theoretical analysis of the proposed models is presented, together with experimental results and comparisons for denoising synthetic and real HARDI data. PMID:20802839

Modeling electrokinetic flows by consistent implicit incompressible smoothed particle hydrodynamics

DOE PAGES

Pan, Wenxiao; Kim, Kyungjoo; Perego, Mauro; ...

2017-01-03

In this paper, we present a consistent implicit incompressible smoothed particle hydrodynamics (I 2SPH) discretization of Navier–Stokes, Poisson–Boltzmann, and advection–diffusion equations subject to Dirichlet or Robin boundary conditions. It is applied to model various two and three dimensional electrokinetic flows in simple or complex geometries. The accuracy and convergence of the consistent I 2SPH are examined via comparison with analytical solutions, grid-based numerical solutions, or empirical models. Lastly, the new method provides a framework to explore broader applications of SPH in microfluidics and complex fluids with charged objects, such as colloids and biomolecules, in arbitrary complex geometries.

A coarse-grid projection method for accelerating incompressible flow computations

NASA Astrophysics Data System (ADS)

San, Omer; Staples, Anne E.

2013-01-01

We present a coarse-grid projection (CGP) method for accelerating incompressible flow computations, which is applicable to methods involving Poisson equations as incompressibility constraints. The CGP methodology is a modular approach that facilitates data transfer with simple interpolations and uses black-box solvers for the Poisson and advection-diffusion equations in the flow solver. After solving the Poisson equation on a coarsened grid, an interpolation scheme is used to obtain the fine data for subsequent time stepping on the full grid. A particular version of the method is applied here to the vorticity-stream function, primitive variable, and vorticity-velocity formulations of incompressible Navier-Stokes equations. We compute several benchmark flow problems on two-dimensional Cartesian and non-Cartesian grids, as well as a three-dimensional flow problem. The method is found to accelerate these computations while retaining a level of accuracy close to that of the fine resolution field, which is significantly better than the accuracy obtained for a similar computation performed solely using a coarse grid. A linear acceleration rate is obtained for all the cases we consider due to the linear-cost elliptic Poisson solver used, with reduction factors in computational time between 2 and 42. The computational savings are larger when a suboptimal Poisson solver is used. We also find that the computational savings increase with increasing distortion ratio on non-Cartesian grids, making the CGP method a useful tool for accelerating generalized curvilinear incompressible flow solvers.

The twin cell model and its excellence in determining the glass transition temperature of thin film metallic glass

NASA Astrophysics Data System (ADS)

Kanjilal, Baishali; Iram, Samreen; Das, Atreyee; Chakrabarti, Haimanti

2018-05-01

This work reports a novel two dimensional approach to the theoretical computation of the glass transition temperature in simple hypothetical icosahedral packed structures based on Thin Film metallic glasses using liquid state theories in the realm of transport properties. The model starts from Navier-Stokes equation and evaluates the statistical average velocity of each different species of atom under the condition of ensemble equality to compute diffusion lengths and the diffusion coefficients as a function of temperature. The additional correction brought in is that of the limited states due to tethering of one nodule vis -a-vis the others. The movement of the molecules use our Twin Cell Model a typical model pertinent for modeling chain motions. A temperature viscosity correction by Cohen and Grest is included through the temperature dependence of the relaxation times for glass formers.

A self-organizing Lagrangian particle method for adaptive-resolution advection-diffusion simulations

NASA Astrophysics Data System (ADS)

Reboux, Sylvain; Schrader, Birte; Sbalzarini, Ivo F.

2012-05-01

We present a novel adaptive-resolution particle method for continuous parabolic problems. In this method, particles self-organize in order to adapt to local resolution requirements. This is achieved by pseudo forces that are designed so as to guarantee that the solution is always well sampled and that no holes or clusters develop in the particle distribution. The particle sizes are locally adapted to the length scale of the solution. Differential operators are consistently evaluated on the evolving set of irregularly distributed particles of varying sizes using discretization-corrected operators. The method does not rely on any global transforms or mapping functions. After presenting the method and its error analysis, we demonstrate its capabilities and limitations on a set of two- and three-dimensional benchmark problems. These include advection-diffusion, the Burgers equation, the Buckley-Leverett five-spot problem, and curvature-driven level-set surface refinement.

Diffusion of massive particles around an Abelian-Higgs string

NASA Astrophysics Data System (ADS)

Saha, Abhisek; Sanyal, Soma

2018-03-01

We study the diffusion of massive particles in the space time of an Abelian Higgs string. The particles in the early universe plasma execute Brownian motion. This motion of the particles is modeled as a two dimensional random walk in the plane of the Abelian Higgs string. The particles move randomly in the space time of the string according to their geodesic equations. We observe that for certain values of their energy and angular momentum, an overdensity of particles is observed close to the string. We find that the string parameters determine the distribution of the particles. We make an estimate of the density fluctuation generated around the string as a function of the deficit angle. Though the thickness of the string is small, the length is large and the overdensity close to the string may have cosmological consequences in the early universe.

Chaos based video encryption using maps and Ikeda time delay system

NASA Astrophysics Data System (ADS)

Valli, D.; Ganesan, K.

2017-12-01

Chaos based cryptosystems are an efficient method to deal with improved speed and highly secured multimedia encryption because of its elegant features, such as randomness, mixing, ergodicity, sensitivity to initial conditions and control parameters. In this paper, two chaos based cryptosystems are proposed: one is the higher-dimensional 12D chaotic map and the other is based on the Ikeda delay differential equation (DDE) suitable for designing a real-time secure symmetric video encryption scheme. These encryption schemes employ a substitution box (S-box) to diffuse the relationship between pixels of plain video and cipher video along with the diffusion of current input pixel with the previous cipher pixel, called cipher block chaining (CBC). The proposed method enhances the robustness against statistical, differential and chosen/known plain text attacks. Detailed analysis is carried out in this paper to demonstrate the security and uniqueness of the proposed scheme.

Causal electric charge diffusion and balance functions in relativistic heavy-ion collisions

NASA Astrophysics Data System (ADS)

Kapusta, Joseph I.; Plumberg, Christopher

2018-01-01

We study the propagation and diffusion of electric charge fluctuations in high-energy heavy-ion collisions using the Cattaneo form for the dissipative part of the electric current. As opposed to the ordinary diffusion equation this form limits the speed at which charge can propagate. Including the noise term in the current, which arises uniquely from the fluctuation-dissipation theorem, we calculate the balance functions for charged hadrons in a simple 1+1-dimensional Bjorken hydrodynamical model. Limiting the speed of propagation of charge fluctuations increases the height and reduces the width of these balance functions when plotted versus rapidity. We also estimate the numerical value of the associated diffusion time constant from anti-de Sitter-space/conformal-field theory.

Evolution of a magnetic flux tube in two-dimensional penetrative convection

NASA Technical Reports Server (NTRS)

Jennings, R. L.; Brandenburg, A.; Nordlund, A.; Stein, R. F.

1992-01-01

Highly supercritical compressible convection is simulated in a two-dimensional domain in which the upper half is unstable to convection while the lower half is stably stratified. This configuration is an idealization of the layers near the base of the solar convection zone. Once the turbulent flow is well developed, a toroidal magnetic field B sub tor is introduced to the stable layer. The field's evolution is governed by an advection-diffusion-type equation, and the Lorentz force does not significantly affect the flow. After many turnover times the field is stratified such that the absolute value of B sub tor/rho is approximately constant in the convective layer, where rho is density, while in the stable layer this ratio decreases linearly with depth. Consequently most of the magnetic flux is stored in the overshoot layer. The inclusion of rotation leads to travelling waves which transport magnetic flux latitudinally in a manner reminiscent of the migrations seen during the solar cycle.

Use of upscaled elevation and surface roughness data in two-dimensional surface water models

USGS Publications Warehouse

Hughes, J.D.; Decker, J.D.; Langevin, C.D.

2011-01-01

In this paper, we present an approach that uses a combination of cell-block- and cell-face-averaging of high-resolution cell elevation and roughness data to upscale hydraulic parameters and accurately simulate surface water flow in relatively low-resolution numerical models. The method developed allows channelized features that preferentially connect large-scale grid cells at cell interfaces to be represented in models where these features are significantly smaller than the selected grid size. The developed upscaling approach has been implemented in a two-dimensional finite difference model that solves a diffusive wave approximation of the depth-integrated shallow surface water equations using preconditioned Newton–Krylov methods. Computational results are presented to show the effectiveness of the mixed cell-block and cell-face averaging upscaling approach in maintaining model accuracy, reducing model run-times, and how decreased grid resolution affects errors. Application examples demonstrate that sub-grid roughness coefficient variations have a larger effect on simulated error than sub-grid elevation variations.

A three-dimensional turbulent compressible flow model for ejector and fluted mixers

NASA Technical Reports Server (NTRS)

Rushmore, W. L.; Zelazny, S. W.

1978-01-01

A three dimensional finite element computer code was developed to analyze ejector and axisymmetric fluted mixer systems whose flow fields are not significantly influenced by streamwise diffusion effects. A two equation turbulence model was used to make comparisons between theory and data for various flow fields which are components of the ejector system, i.e., (1) turbulent boundary layer in a duct; (2) rectangular nozzle (free jet); (3) axisymmetric nozzle (free jet); (4) hypermixing nozzle (free jet); and (5) plane wall jet. Likewise, comparisons of the code with analytical results and/or other numerical solutions were made for components of the axisymmetric fluted mixer system. These included: (1) developing pipe flow; (2) developing flow in an annular pipe; (3) developing flow in an axisymmetric pipe with conical center body and no fluting and (4) developing fluted pipe flow. Finally, two demonstration cases are presented which show the code's ability to analyze both the ejector and axisymmetric fluted mixers.

Comparison of analytical and experimental performance of a wind-tunnel diffuser section

NASA Technical Reports Server (NTRS)

Shyne, R. J.; Moore, R. D.; Boldman, D. R.

1986-01-01

Wind tunnel diffuser performance is evaluated by comparing experimental data with analytical results predicted by an one-dimensional integration procedure with skin friction coefficient, a two-dimensional interactive boundary layer procedure for analyzing conical diffusers, and a two-dimensional, integral, compressible laminar and turbulent boundary layer code. Pressure, temperature, and velocity data for a 3.25 deg equivalent cone half-angle diffuser (37.3 in., 94.742 cm outlet diameter) was obtained from the one-tenth scale Altitude Wind Tunnel modeling program at the NASA Lewis Research Center. The comparison is performed at Mach numbers of 0.162 (Re = 3.097x19(6)), 0.326 (Re = 6.2737x19(6)), and 0.363 (Re = 7.0129x10(6)). The Reynolds numbers are all based on an inlet diffuser diameter of 32.4 in., 82.296 cm, and reasonable quantitative agreement was obtained between the experimental data and computational codes.

An accessible four-dimensional treatment of Maxwell's equations in terms of differential forms

NASA Astrophysics Data System (ADS)

Sá, Lucas

2017-03-01

Maxwell’s equations are derived in terms of differential forms in the four-dimensional Minkowski representation, starting from the three-dimensional vector calculus differential version of these equations. Introducing all the mathematical and physical concepts needed (including the tool of differential forms), using only knowledge of elementary vector calculus and the local vector version of Maxwell’s equations, the equations are reduced to a simple and elegant set of two equations for a unified quantity, the electromagnetic field. The treatment should be accessible for students taking a first course on electromagnetism.

Hot-electron thermocouple and the diffusion thermopower of two-dimensional electrons in GaAs.

PubMed

Chickering, W E; Eisenstein, J P; Reno, J L

2009-07-24

A simple hot-electron thermocouple is realized in a two-dimensional electron system (2DES) and used to measure the diffusion thermopower of the 2DES at zero magnetic field. This hot-electron technique, which requires no micron-scale patterning of the 2DES, is much less sensitive than conventional methods to phonon-drag effects. Our thermopower results are in good agreement with the Mott formula for diffusion thermopower for temperatures up to T approximately 2 K.

Wave and pseudo-diffusion equations from squeezed states

NASA Technical Reports Server (NTRS)

Daboul, Jamil

1993-01-01

We show that the probability distributions P(sub n)(q,p;y) := the absolute value squared of (n(p,q;y), which are obtained from squeezed states, obey an interesting partial differential equation, to which we give two intuitive interpretations: as a wave equation in one space dimension; and as a pseudo-diffusion equation. We also study the corresponding Wehrl entropies S(sub n)(y), and we show that they have minima at zero squeezing, y = 0.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

NASA Astrophysics Data System (ADS)

Di Nucci, Carmine

2018-05-01

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

PubMed

Ablowitz, Mark; Biondini, Gino; Wang, Qiao

2017-09-01

Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hydrodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

Oxygen concentration dependence of silicon oxide dynamical properties

NASA Astrophysics Data System (ADS)

Yajima, Yuji; Shiraishi, Kenji; Endoh, Tetsuo; Kageshima, Hiroyuki

2018-06-01

To understand oxidation in three-dimensional silicon, dynamic characteristics of a SiO x system with various stoichiometries were investigated. The calculated results show that the self-diffusion coefficient increases as oxygen density decreases, and the increase is large when the temperature is low. It also shows that the self-diffusion coefficient saturates, when the number of removed oxygen atoms is sufficiently large. Then, approximate analytical equations are derived from the calculated results, and the previously reported expression is confirmed in the extremely low-SiO-density range.

Bubble statistics in aged wet foams and the Fokker-Planck equation

NASA Astrophysics Data System (ADS)

Zimnyakov, D. A.; Yuvchenko, S. A.; Tzyipin, D. V.; Samorodina, T. V.

2018-04-01

Results of the experimental study of changes in the bubble size statistics during aging of wet foams are discussed. It is proposed that the evolution of the bubble radii distributions can be described in terms of the one dimensional Fokker- Planck equation. The empirical distributions of the bubble radii exhibit a self-similarity of their shapes and can be transformed to a time-independent form using the radius renormalization. Analysis of obtained data allows us to suggest that the drift term of the Fokker-Planck equation dominates in comparison with the diffusion term in the case of aging of isolated quasi-stable wet foams.

Diffusion with Varying Drag; the Runaway Problem.

NASA Astrophysics Data System (ADS)

Rollins, David Kenneth

We study the motion of electrons in an ionized plasma of electrons and ions in an external electric field. A probability distribution function describes the electron motion and is a solution of a Fokker-Planck equation. In zero field, the solution approaches an equilibrium Maxwellian. For arbitrarily small field, electrons overcome the diffusive effects and are freely accelerated by the field. This is the electron runaway phenomenon. We treat the electric field as a small perturbation. We consider various diffusion coefficients for the one dimensional problem and determine the runaway current as a function of the field strength. Diffusion coefficients, non-zero on a finite interval are examined. Some non-trivial cases of these can be solved exactly in terms of known special functions. The more realistic case where the diffusion coefficient decays with velocity are then considered. To determine the runaway current, the equivalent Schrodinger eigenvalue problem is analysed. The smallest eigenvalue is shown to be equal to the runaway current. Using asymptotic matching a solution can be constructed which is then used to evaluate the runaway current. The runaway current is exponentially small as a function of field strength. This method is used to extract results from the three dimensional problem.

Hall-Effect Thruster Simulations with 2-D Electron Transport and Hydrodynamic Ions

NASA Technical Reports Server (NTRS)

Mikellides, Ioannis G.; Katz, Ira; Hofer, Richard H.; Goebel, Dan M.

2009-01-01

A computational approach that has been used extensively in the last two decades for Hall thruster simulations is to solve a diffusion equation and energy conservation law for the electrons in a direction that is perpendicular to the magnetic field, and use discrete-particle methods for the heavy species. This "hybrid" approach has allowed for the capture of bulk plasma phenomena inside these thrusters within reasonable computational times. Regions of the thruster with complex magnetic field arrangements (such as those near eroded walls and magnets) and/or reduced Hall parameter (such as those near the anode and the cathode plume) challenge the validity of the quasi-one-dimensional assumption for the electrons. This paper reports on the development of a computer code that solves numerically the 2-D axisymmetric vector form of Ohm's law, with no assumptions regarding the rate of electron transport in the parallel and perpendicular directions. The numerical challenges related to the large disparity of the transport coefficients in the two directions are met by solving the equations in a computational mesh that is aligned with the magnetic field. The fully-2D approach allows for a large physical domain that extends more than five times the thruster channel length in the axial direction, and encompasses the cathode boundary. Ions are treated as an isothermal, cold (relative to the electrons) fluid, accounting for charge-exchange and multiple-ionization collisions in the momentum equations. A first series of simulations of two Hall thrusters, namely the BPT-4000 and a 6-kW laboratory thruster, quantifies the significance of ion diffusion in the anode region and the importance of the extended physical domain on studies related to the impact of the transport coefficients on the electron flow field.

THE IMPLICIT CONTRIBUTION OF SLAB MODES TO THE PERPENDICULAR DIFFUSION COEFFICIENT OF PARTICLES INTERACTING WITH TWO-COMPONENT TURBULENCE

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shalchi, A., E-mail: andreasm4@yahoo.com

2016-10-20

We explore the transport of energetic particles in two-component turbulence in which the stochastic magnetic field is assumed to be a superposition of slab and two-dimensional modes. It is known that in magnetostatic slab turbulence, the motion of particles across the mean magnetic field is subdiffusive. If a two-dimensional component is added, diffusion is recovered. It was also shown before that in two-component turbulence, the slab modes do not explicitly contribute to the perpendicular diffusion coefficient. In the current paper, the implicit contribution of slab modes is explored and it is shown that this contribution leads to a reduction ofmore » the perpendicular diffusion coefficient. This effect improves the agreement between simulations and analytical theory. Furthermore, the obtained results are relevant for investigations of diffusive shock acceleration.« less

A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions

PubMed Central

Chevalier, Michael W.; El-Samad, Hana

2014-01-01

Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-to-cell variability even in clonal populations. Stochastic biochemical networks have been traditionally modeled as continuous-time discrete-state Markov processes whose probability density functions evolve according to a chemical master equation (CME). In diffusion reaction systems on membranes, the Markov formalism, which assumes constant reaction propensities is not directly appropriate. This is because the instantaneous propensity for a diffusion reaction to occur depends on the creation times of the molecules involved. In this work, we develop a chemical master equation for systems of this type. While this new CME is computationally intractable, we make rational dimensional reductions to form an approximate equation, whose moments are also derived and are shown to yield efficient, accurate results. This new framework forms a more general approach than the Markov CME and expands upon the realm of possible stochastic biochemical systems that can be efficiently modeled. PMID:25481130

A master equation and moment approach for biochemical systems with creation-time-dependent bimolecular rate functions

NASA Astrophysics Data System (ADS)

Chevalier, Michael W.; El-Samad, Hana

2014-12-01

Noise and stochasticity are fundamental to biology and derive from the very nature of biochemical reactions where thermal motion of molecules translates into randomness in the sequence and timing of reactions. This randomness leads to cell-to-cell variability even in clonal populations. Stochastic biochemical networks have been traditionally modeled as continuous-time discrete-state Markov processes whose probability density functions evolve according to a chemical master equation (CME). In diffusion reaction systems on membranes, the Markov formalism, which assumes constant reaction propensities is not directly appropriate. This is because the instantaneous propensity for a diffusion reaction to occur depends on the creation times of the molecules involved. In this work, we develop a chemical master equation for systems of this type. While this new CME is computationally intractable, we make rational dimensional reductions to form an approximate equation, whose moments are also derived and are shown to yield efficient, accurate results. This new framework forms a more general approach than the Markov CME and expands upon the realm of possible stochastic biochemical systems that can be efficiently modeled.

Two-boundary grid generation for the solution of the three dimensional compressible Navier-Stokes equations. Ph.D. Thesis - Old Dominion Univ.

NASA Technical Reports Server (NTRS)

Smith, R. E.

1981-01-01

A grid generation technique called the two boundary technique is developed and applied for the solution of the three dimensional Navier-Stokes equations. The Navier-Stokes equations are transformed from a cartesian coordinate system to a computational coordinate system, and the grid generation technique provides the Jacobian matrix describing the transformation. The two boundary technique is based on algebraically defining two distinct boundaries of a flow domain and the distribution of the grid is achieved by applying functions to the uniform computational grid which redistribute the computational independent variables and consequently concentrate or disperse the grid points in the physical domain. The Navier-Stokes equations are solved using a MacCormack time-split technique. Grids and supersonic laminar flow solutions are obtained for a family of three dimensional corners and two spike-nosed bodies.

Capture zone area distributions for nucleation and growth of islands during submonolayer deposition

DOE Office of Scientific and Technical Information (OSTI.GOV)

Han, Yong; Li, Maozhi; Evans, James W.

2016-12-07

A fundamental evolution equation is developed to describe the distribution of areas of capture zones (CZs) associated with islands formed by homogeneous nucleation and growth during submonolayer deposition on perfect flat surfaces. This equation involves various quantities which characterize subtle spatial aspects of the nucleation process. These quantities in turn depend on the complex stochastic geometry of the CZ tessellation of the surface, and their detailed form determines the CZ area distribution (CZD) including its asymptotic features. For small CZ areas, behavior of the CZD reflects the critical island size, i. For large CZ areas, it may reflect the probabilitymore » for nucleation near such large CZs. Predictions are compared with kinetic Monte Carlo simulation data for models with two-dimensional compact islands with i = 1 (irreversible island formation by diffusing adatom pairs) and i = 0 (adatoms spontaneously convert to stable nuclei, e.g., by exchange with the substrate).« less

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

DOE Office of Scientific and Technical Information (OSTI.GOV)

de Almeida, Valmor F.

In this work, a phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equationmore » and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.« less

An iterative phase-space explicit discontinuous Galerkin method for stellar radiative transfer in extended atmospheres

DOE PAGES

de Almeida, Valmor F.

2017-04-19

In this work, a phase-space discontinuous Galerkin (PSDG) method is presented for the solution of stellar radiative transfer problems. It allows for greater adaptivity than competing methods without sacrificing generality. The method is extensively tested on a spherically symmetric, static, inverse-power-law scattering atmosphere. Results for different sizes of atmospheres and intensities of scattering agreed with asymptotic values. The exponentially decaying behavior of the radiative field in the diffusive-transparent transition region, and the forward peaking behavior at the surface of extended atmospheres were accurately captured. The integrodifferential equation of radiation transfer is solved iteratively by alternating between the radiative pressure equationmore » and the original equation with the integral term treated as an energy density source term. In each iteration, the equations are solved via an explicit, flux-conserving, discontinuous Galerkin method. Finite elements are ordered in wave fronts perpendicular to the characteristic curves so that elemental linear algebraic systems are solved quickly by sweeping the phase space element by element. Two implementations of a diffusive boundary condition at the origin are demonstrated wherein the finite discontinuity in the radiation intensity is accurately captured by the proposed method. This allows for a consistent mechanism to preserve photon luminosity. The method was proved to be robust and fast, and a case is made for the adequacy of parallel processing. In addition to classical two-dimensional plots, results of normalized radiation intensity were mapped onto a log-polar surface exhibiting all distinguishing features of the problem studied.« less