Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory

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

Mendl, Christian B.; Lu, Jianfeng; Lukkarinen, Jani

We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic theory. The Wigner function serves as a common interface between the microscopic and kinetic level. We demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe the additional quasiconservation of the phonon density, defined via an ensemble average of the related microscopic field variables and exactly conserved by the kinetic equations. On superkinetic time scales, density quasiconservation is lost while energy remainsmore » conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. Furthermore, the precise mechanism remains unknown and is not captured by the Boltzmann-Peierls equations.« less

An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov-Darwin particle-in-cell algorithm

NASA Astrophysics Data System (ADS)

Chen, G.; Chacón, L.

2014-10-01

A recent proof-of-principle study proposes a nonlinear electrostatic implicit particle-in-cell (PIC) algorithm in one dimension (Chen et al., 2011). The algorithm employs a kinetically enslaved Jacobian-free Newton-Krylov (JFNK) method, and conserves energy and charge to numerical round-off. In this study, we generalize the method to electromagnetic simulations in 1D using the Darwin approximation to Maxwell's equations, which avoids radiative noise issues by ordering out the light wave. An implicit, orbit-averaged, time-space-centered finite difference scheme is employed in both the 1D Darwin field equations (in potential form) and the 1D-3V particle orbit equations to produce a discrete system that remains exactly charge- and energy-conserving. Furthermore, enabled by the implicit Darwin equations, exact conservation of the canonical momentum per particle in any ignorable direction is enforced via a suitable scattering rule for the magnetic field. We have developed a simple preconditioner that targets electrostatic waves and skin currents, and allows us to employ time steps O(√{mi /me } c /veT) larger than the explicit CFL. Several 1D numerical experiments demonstrate the accuracy, performance, and conservation properties of the algorithm. In particular, the scheme is shown to be second-order accurate, and CPU speedups of more than three orders of magnitude vs. an explicit Vlasov-Maxwell solver are demonstrated in the "cold" plasma regime (where kλD ≪ 1).

A k-omega-multivariate beta PDF for supersonic combustion

NASA Technical Reports Server (NTRS)

Alexopoulos, G. A.; Baurle, R. A.; Hassan, H. A.

1992-01-01

In an attempt to study the interaction between combustion and turbulence in supersonic flows, an assumed PDF has been employed. This makes it possible to calculate the time average of the chemical source terms that appear in the species conservation equations. In order to determine the averages indicated in an equation, two transport equations, one for the temperature (enthalpy) variance and one for Q, are required. Model equations are formulated for such quantities. The turbulent time scale controls the evolution. An algebraic model similar to that used by Eklund et al was used in an attempt to predict the recent measurements of Cheng et al. Predictions were satisfactory before ignition but were less satisfactory after ignition. One of the reasons for this behavior is the inadequacy of the algebraic turbulence model employed. Because of this, the objective of this work is to develop a k-omega model to remedy the situation.

A fully vectorized numerical solution of the incompressible Navier-Stokes equations. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Patel, N.

1983-01-01

A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.

Energy balance and mass conservation in reduced order models of fluid flows

NASA Astrophysics Data System (ADS)

Mohebujjaman, Muhammad; Rebholz, Leo G.; Xie, Xuping; Iliescu, Traian

2017-10-01

In this paper, we investigate theoretically and computationally the conservation properties of reduced order models (ROMs) for fluid flows. Specifically, we investigate whether the ROMs satisfy the same (or similar) energy balance and mass conservation as those satisfied by the Navier-Stokes equations. All of our theoretical findings are illustrated and tested in numerical simulations of a 2D flow past a circular cylinder at a Reynolds number Re = 100. First, we investigate the ROM energy balance. We show that using the snapshot average for the centering trajectory (which is a popular treatment of nonhomogeneous boundary conditions in ROMs) yields an incorrect energy balance. Then, we propose a new approach, in which we replace the snapshot average with the Stokes extension. Theoretically, the Stokes extension produces an accurate energy balance. Numerically, the Stokes extension yields more accurate results than the standard snapshot average, especially for longer time intervals. Our second contribution centers around ROM mass conservation. We consider ROMs created using two types of finite elements: the standard Taylor-Hood (TH) element, which satisfies the mass conservation weakly, and the Scott-Vogelius (SV) element, which satisfies the mass conservation pointwise. Theoretically, the error estimates for the SV-ROM are sharper than those for the TH-ROM. Numerically, the SV-ROM yields significantly more accurate results, especially for coarser meshes and longer time intervals.

Direct simulation of groundwater age

USGS Publications Warehouse

Goode, Daniel J.

1996-01-01

A new method is proposed to simulate groundwater age directly, by use of an advection-dispersion transport equation with a distributed zero-order source of unit (1) strength, corresponding to the rate of aging. The dependent variable in the governing equation is the mean age, a mass-weighted average age. The governing equation is derived from residence-time-distribution concepts for the case of steady flow. For the more general case of transient flow, a transient governing equation for age is derived from mass-conservation principles applied to conceptual “age mass.” The age mass is the product of the water mass and its age, and age mass is assumed to be conserved during mixing. Boundary conditions include zero age mass flux across all noflow and inflow boundaries and no age mass dispersive flux across outflow boundaries. For transient-flow conditions, the initial distribution of age must be known. The solution of the governing transport equation yields the spatial distribution of the mean groundwater age and includes diffusion, dispersion, mixing, and exchange processes that typically are considered only through tracer-specific solute transport simulation. Traditional methods have relied on advective transport to predict point values of groundwater travel time and age. The proposed method retains the simplicity and tracer-independence of advection-only models, but incorporates the effects of dispersion and mixing on volume-averaged age. Example simulations of age in two idealized regional aquifer systems, one homogeneous and the other layered, demonstrate the agreement between the proposed method and traditional particle-tracking approaches and illustrate use of the proposed method to determine the effects of diffusion, dispersion, and mixing on groundwater age.

A Well-Balanced Path-Integral f-Wave Method for Hyperbolic Problems with Source Terms

PubMed Central

2014-01-01

Systems of hyperbolic partial differential equations with source terms (balance laws) arise in many applications where it is important to compute accurate time-dependent solutions modeling small perturbations of equilibrium solutions in which the source terms balance the hyperbolic part. The f-wave version of the wave-propagation algorithm is one approach, but requires the use of a particular averaged value of the source terms at each cell interface in order to be “well balanced” and exactly maintain steady states. A general approach to choosing this average is developed using the theory of path conservative methods. A scalar advection equation with a decay or growth term is introduced as a model problem for numerical experiments. PMID:24563581

Mesoscale Modeling of LX-17 Under Isentropic Compression

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

Springer, H K; Willey, T M; Friedman, G

Mesoscale simulations of LX-17 incorporating different equilibrium mixture models were used to investigate the unreacted equation-of-state (UEOS) of TATB. Candidate TATB UEOS were calculated using the equilibrium mixture models and benchmarked with mesoscale simulations of isentropic compression experiments (ICE). X-ray computed tomography (XRCT) data provided the basis for initializing the simulations with realistic microstructural details. Three equilibrium mixture models were used in this study. The single constituent with conservation equations (SCCE) model was based on a mass-fraction weighted specific volume and the conservation of mass, momentum, and energy. The single constituent equation-of-state (SCEOS) model was based on a mass-fraction weightedmore » specific volume and the equation-of-state of the constituents. The kinetic energy averaging (KEA) model was based on a mass-fraction weighted particle velocity mixture rule and the conservation equations. The SCEOS model yielded the stiffest TATB EOS (0.121{micro} + 0.4958{micro}{sup 2} + 2.0473{micro}{sup 3}) and, when incorporated in mesoscale simulations of the ICE, demonstrated the best agreement with VISAR velocity data for both specimen thicknesses. The SCCE model yielded a relatively more compliant EOS (0.1999{micro}-0.6967{micro}{sup 2} + 4.9546{micro}{sup 3}) and the KEA model yielded the most compliant EOS (0.1999{micro}-0.6967{micro}{sup 2}+4.9546{micro}{sup 3}) of all the equilibrium mixture models. Mesoscale simulations with the lower density TATB adiabatic EOS data demonstrated the least agreement with VISAR velocity data.« less

On the transition towards slow manifold in shallow-water and 3D Euler equations in a rotating frame

NASA Technical Reports Server (NTRS)

Mahalov, A.

1994-01-01

The long-time, asymptotic state of rotating homogeneous shallow-water equations is investigated. Our analysis is based on long-time averaged rotating shallow-water equations describing interactions of large-scale, horizontal, two-dimensional motions with surface inertial-gravity waves field for a shallow, uniformly rotating fluid layer. These equations are obtained in two steps: first by introducing a Poincare/Kelvin linear propagator directly into classical shallow-water equations, then by averaging. The averaged equations describe interaction of wave fields with large-scale motions on time scales long compared to the time scale 1/f(sub o) introduced by rotation (f(sub o)/2-angular velocity of background rotation). The present analysis is similar to the one presented by Waleffe (1991) for 3D Euler equations in a rotating frame. However, since three-wave interactions in rotating shallow-water equations are forbidden, the final equations describing the asymptotic state are simplified considerably. Special emphasis is given to a new conservation law found in the asymptotic state and decoupling of the dynamics of the divergence free part of the velocity field. The possible rising of a decoupled dynamics in the asymptotic state is also investigated for homogeneous turbulence subjected to a background rotation. In our analysis we use long-time expansion, where the velocity field is decomposed into the 'slow manifold' part (the manifold which is unaffected by the linear 'rapid' effects of rotation or the inertial waves) and a formal 3D disturbance. We derive the physical space version of the long-time averaged equations and consider an invariant, basis-free derivation. This formulation can be used to generalize Waleffe's (1991) helical decomposition to viscous inhomogeneous flows (e.g. problems in cylindrical geometry with no-slip boundary conditions on the cylinder surface and homogeneous in the vertical direction).

Audible thunder characteristic and the relation between peak frequency and lightning parameters

NASA Astrophysics Data System (ADS)

Yuhua, Ouyang; Ping, Yuan

2012-02-01

In recent summers, some natural lightning optical spectra and audible thunder signals were observed. Twelve events on 15 August 2008 are selected as samples since some synchronizing information about them are obtained, such as lightning optical spectra, surface E-field changes, etc. By using digital filter and Fourier transform, thunder frequency spectra in observation location have been calculated. Then the two main propagation effects, finite amplitude propagation and attenuation by air, are calculated. Upon that we take the test thunder frequency spectra and work backward to recalculate the original frequency spectra near generation location. Thunder frequency spectra and the frequency distribution varying with distance are researched. According to the theories on plasma, the channel temperature and electron density are further calculated by transition parameters of lines in lightning optical spectra. Pressure and the average ionization degree of each discharge channel are obtained by using Saha equations, charge conservation equations and particle conservation equations. Moreover, the relationship between the peak frequency of each thunder and channel parameters of the lightning is studied.

On symmetries, conservation laws and exact solutions of the nonlinear Schrödinger-Hirota equation

NASA Astrophysics Data System (ADS)

Akbulut, Arzu; Taşcan, Filiz

2018-04-01

In this paper, conservation laws and exact solution are found for nonlinear Schrödinger-Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger-Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.

Relativistic Kinetic Theory

NASA Astrophysics Data System (ADS)

Vereshchagin, Gregory V.; Aksenov, Alexey G.

2017-02-01

Preface; Acknowledgements; Acronyms and definitions; Introduction; Part I. Theoretical Foundations: 1. Basic concepts; 2. Kinetic equation; 3. Averaging; 4. Conservation laws and equilibrium; 5. Relativistic BBGKY hierarchy; 6. Basic parameters in gases and plasmas; Part II. Numerical Methods: 7. The basics of computational physics; 8. Direct integration of Boltzmann equations; 9. Multidimensional hydrodynamics; Part III. Applications: 10. Wave dispersion in relativistic plasma; 11. Thermalization in relativistic plasma; 12. Kinetics of particles in strong fields; 13. Compton scattering in astrophysics and cosmology; 14. Self-gravitating systems; 15. Neutrinos, gravitational collapse and supernovae; Appendices; Bibliography; Index.

Separation of acoustic waves in isentropic flow perturbations

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

Henke, Christian, E-mail: christian.henke@atlas-elektronik.com

2015-04-15

The present contribution investigates the mechanisms of sound generation and propagation in the case of highly-unsteady flows. Based on the linearisation of the isentropic Navier–Stokes equation around a new pathline-averaged base flow, it is demonstrated for the first time that flow perturbations of a non-uniform flow can be split into acoustic and vorticity modes, with the acoustic modes being independent of the vorticity modes. Therefore, we can propose this acoustic perturbation as a general definition of sound. As a consequence of the splitting result, we conclude that the present acoustic perturbation is propagated by the convective wave equation and fulfilsmore » Lighthill’s acoustic analogy. Moreover, we can define the deviations of the Navier–Stokes equation from the convective wave equation as “true” sound sources. In contrast to other authors, no assumptions on a slowly varying or irrotational flow are necessary. Using a symmetry argument for the conservation laws, an energy conservation result and a generalisation of the sound intensity are provided. - Highlights: • First splitting of non-uniform flows in acoustic and non-acoustic components. • These result leads to a generalisation of sound which is compatible with Lighthill’s acoustic analogy. • A closed equation for the generation and propagation of sound is given.« less

Nongeostrophic theory of zonally averaged circulation. I - Formulation

NASA Technical Reports Server (NTRS)

Tung, Ka Kit

1986-01-01

A nongeostrophic theory of zonally averaged circulation is formulated using the nonlinear primitive equations (mass conservation, thermodynamics, and zonal momentum) on a sphere. The relationship between the mean meridional circulation and diabatic heating rate is studied. Differences between results of nongeostropic theory and the geostrophic formulation concerning the role of eddy forcing of the diabatic circulation and the nonlinear nearly inviscid limit versus the geostrophic limit are discussed. Consideration is given to the Eliassen-Palm flux divergence, the Eliassen-Palm pseudodivergence, the nonacceleration theorem, and the nonlinear nongeostrophic Taylor relationship.

A hybrid Reynolds averaged/PDF closure model for supersonic turbulent combustion

NASA Technical Reports Server (NTRS)

Frankel, Steven H.; Hassan, H. A.; Drummond, J. Philip

1990-01-01

A hybrid Reynolds averaged/assumed pdf approach has been developed and applied to the study of turbulent combustion in a supersonic mixing layer. This approach is used to address the 'laminar-like' treatment of the thermochemical terms that appear in the conservation equations. Calculations were carried out for two experiments involving H2-air supersonic turbulent mixing. Two different forms of the pdf were implemented. In general, the results show modest improvement from previous calculations. Moreover, the results appear to be somewhat independent of the form of the assumed pdf.

Conservation laws and conserved quantities for (1+1)D linearized Boussinesq equations

NASA Astrophysics Data System (ADS)

Carvalho, Cindy; Harley, Charis

2017-05-01

Conservation laws and physical conserved quantities for the (1+1)D linearized Boussinesq equations at a constant water depth are presented. These equations describe incompressible, inviscid, irrotational fluid flow in the form of a non steady solitary wave. A systematic multiplier approach is used to obtain the conservation laws of the system of third order partial differential equations (PDEs) in dimensional form. Physical conserved quantities are derived by integrating the conservation laws in the direction of wave propagation and imposing decaying boundary conditions in the horizontal direction. One of these is a newly discovered conserved quantity which relates to an energy flux density.

Triangle based TVD schemes for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Durlofsky, Louis J.; Osher, Stanley; Engquist, Bjorn

1990-01-01

A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.

A multi-dimensional nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell (PIC) algorithm

NASA Astrophysics Data System (ADS)

Chen, Guangye; Chacón, Luis; CoCoMans Team

2014-10-01

For decades, the Vlasov-Darwin model has been recognized to be attractive for PIC simulations (to avoid radiative noise issues) in non-radiative electromagnetic regimes. However, the Darwin model results in elliptic field equations that renders explicit time integration unconditionally unstable. Improving on linearly implicit schemes, fully implicit PIC algorithms for both electrostatic and electromagnetic regimes, with exact discrete energy and charge conservation properties, have been recently developed in 1D. This study builds on these recent algorithms to develop an implicit, orbit-averaged, time-space-centered finite difference scheme for the particle-field equations in multiple dimensions. The algorithm conserves energy, charge, and canonical-momentum exactly, even with grid packing. A simple fluid preconditioner allows efficient use of large timesteps, O (√{mi/me}c/veT) larger than the explicit CFL. We demonstrate the accuracy and efficiency properties of the of the algorithm with various numerical experiments in 2D3V.

Cropping management using color and color infrared aerial photographs

NASA Technical Reports Server (NTRS)

Morgan, K. M.; Morris-Jones, D. R.; Lee, G. B.; Kiefer, R. W.

1979-01-01

The Universal Soil Loss Equation (USLE) is a widely accepted tool for erosion prediction and conservation planning. Solving this equation yields the long-term average annual soil loss that can be expected from rill and inter-rill erosion. In this study, manual interpretation of color and color infrared 70 mm photography at the scale of 1:60,000 is used to determine the cropping management factor in the USLE. Accurate information was collected about plowing practices and crop residue cover (unharvested vegetation) for the winter season on agricultural land in Pheasant Branch Creek watershed in Dane County, Wisconsin.

Evaluation of a research circulation control airfoil using Navier-Stokes methods

NASA Technical Reports Server (NTRS)

Shrewsbury, George D.

1987-01-01

The compressible Reynolds time averaged Navier-Stokes equations were used to obtain solutions for flows about a two dimensional circulation control airfoil. The governing equations were written in conservation form for a body-fitted coordinate system and solved using an Alternating Direction Implicit (ADI) procedure. A modified algebraic eddy viscosity model was used to define the turbulent characteristics of the flow, including the wall jet flow over the Coanda surface at the trailing edge. Numerical results are compared to experimental data obtained for a research circulation control airfoil geometry. Excellent agreement with the experimental results was obtained.

Scaling and scale invariance of conservation laws in Reynolds transport theorem framework

NASA Astrophysics Data System (ADS)

Haltas, Ismail; Ulusoy, Suleyman

2015-07-01

Scale invariance is the case where the solution of a physical process at a specified time-space scale can be linearly related to the solution of the processes at another time-space scale. Recent studies investigated the scale invariance conditions of hydrodynamic processes by applying the one-parameter Lie scaling transformations to the governing equations of the processes. Scale invariance of a physical process is usually achieved under certain conditions on the scaling ratios of the variables and parameters involved in the process. The foundational axioms of hydrodynamics are the conservation laws, namely, conservation of mass, conservation of linear momentum, and conservation of energy from continuum mechanics. They are formulated using the Reynolds transport theorem. Conventionally, Reynolds transport theorem formulates the conservation equations in integral form. Yet, differential form of the conservation equations can also be derived for an infinitesimal control volume. In the formulation of the governing equation of a process, one or more than one of the conservation laws and, some times, a constitutive relation are combined together. Differential forms of the conservation equations are used in the governing partial differential equation of the processes. Therefore, differential conservation equations constitute the fundamentals of the governing equations of the hydrodynamic processes. Applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework instead of applying to the governing partial differential equations may lead to more fundamental conclusions on the scaling and scale invariance of the hydrodynamic processes. This study will investigate the scaling behavior and scale invariance conditions of the hydrodynamic processes by applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework.

On the conservation laws of Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

San, Sait; Yaşar, Emrullah

2015-05-01

In this study, the nonlocal conservation theorem and multiplier approach are performed on the 1 + 1 dimensional Derrida-Lebowitz-Speer-Spohn (DLSS) equation which arises in quantum semi conductor theory. We obtain local conservation laws by using the both methods. Furthermore by utilizing the relationship between conservation laws and Lie point symmetries, the DLSS equation is reduced to third order ordinary differential equation.

Filtering of non-linear instabilities

NASA Technical Reports Server (NTRS)

Khosla, P. K.; Rubin, S. G.

1978-01-01

For Courant numbers larger than one and cell Reynolds numbers larger than two, oscillations and in some cases instabilities are typically found with implicit numerical solutions of the fluid dynamics equations. This behavior has sometimes been associated with the loss of diagonal dominance of the coefficient matrix. It is shown that these problems can be related to the choice of the spatial differences, with the resulting instability related to aliasing or nonlinear interaction. Appropriate filtering can reduce the intensity of these oscillations and possibly eliminate the instability. These filtering procedures are equivalent to a weighted average of conservation and nonconservation differencing. The entire spectrum of filtered equations retains a three point character as well as second order spatial accuracy. Burgers equation was considered as a model.

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

Cheviakov, Alexei F., E-mail: chevaikov@math.usask.ca

Partial differential equations of the form divN=0, N{sub t}+curl M=0 involving two vector functions in R{sup 3} depending on t, x, y, z appear in different physical contexts, including the vorticity formulation of fluid dynamics, magnetohydrodynamics (MHD) equations, and Maxwell's equations. It is shown that these equations possess an infinite family of local divergence-type conservation laws involving arbitrary functions of space and time. Moreover, it is demonstrated that the equations of interest have a rather special structure of a lower-degree (degree two) conservation law in R{sup 4}(t,x,y,z). The corresponding potential system has a clear physical meaning. For the Maxwell's equations,more » it gives rise to the scalar electric and the vector magnetic potentials; for the vorticity equations of fluid dynamics, the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables; for MHD equations, the potential equations yield a generalization of the Galas-Bogoyavlenskij potential that describes magnetic surfaces of ideal MHD equilibria. The lower-degree conservation law is further shown to yield curl-type conservation laws and determined potential equations in certain lower-dimensional settings. Examples of new nonlocal conservation laws, including an infinite family of nonlocal material conservation laws of ideal time-dependent MHD equations in 2+1 dimensions, are presented.« less

Consistent three-equation model for thin films

NASA Astrophysics Data System (ADS)

Richard, Gael; Gisclon, Marguerite; Ruyer-Quil, Christian; Vila, Jean-Paul

2017-11-01

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.

Conservation form of the equations of fluid dynamics in general nonsteady coordinates

NASA Astrophysics Data System (ADS)

Zhang, H.; Camarero, R.; Kahawita, R.

1985-11-01

Many of the differential equations arising in fluid dynamics may be stated in conservation-law form. A number of investigations have been conducted with the aim to derive the conservation-law form of the Navier-Stokes equations in general nonsteady coordinate systems. The present note has the objective to illustrate a mathematical methodology with which such forms of the equations may be derived in an easier and more general fashion. For numerical applications, the scalar form of the equations is eventually provided. Attention is given to the conservation form of equations in curvilinear coordinates and numerical considerations.

Conservation properties of numerical integration methods for systems of ordinary differential equations

NASA Technical Reports Server (NTRS)

Rosenbaum, J. S.

1976-01-01

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

Boundary Conditions for Infinite Conservation Laws

NASA Astrophysics Data System (ADS)

Rosenhaus, V.; Bruzón, M. S.; Gandarias, M. L.

2016-12-01

Regular soliton equations (KdV, sine-Gordon, NLS) are known to possess infinite sets of local conservation laws. Some other classes of nonlinear PDE possess infinite-dimensional symmetries parametrized by arbitrary functions of independent or dependent variables; among them are Zabolotskaya-Khokhlov, Kadomtsev-Petviashvili, Davey-Stewartson equations and Born-Infeld equation. Boundary conditions were shown to play an important role for the existence of local conservation laws associated with infinite-dimensional symmetries. In this paper, we analyze boundary conditions for the infinite conserved densities of regular soliton equations: KdV, potential KdV, Sine-Gordon equation, and nonlinear Schrödinger equation, and compare them with boundary conditions for the conserved densities obtained from infinite-dimensional symmetries with arbitrary functions of independent and dependent variables.

Lie symmetries and conservation laws for the time fractional Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Rui, Wenjuan; Zhang, Xiangzhi

2016-05-01

This paper investigates the invariance properties of the time fractional Derrida-Lebowitz-Speer-Spohn (FDLSS) equation with Riemann-Liouville derivative. By using the Lie group analysis method of fractional differential equations, we derive Lie symmetries for the FDLSS equation. In a particular case of scaling transformations, we transform the FDLSS equation into a nonlinear ordinary fractional differential equation. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

Conservation-form equations of unsteady open-channel flow

USGS Publications Warehouse

Lai, C.; Baltzer, R.A.; Schaffranek, R.W.

2002-01-01

The unsteady open-channel flow equations are typically expressed in a variety of forms due to the imposition of differing assumptions, use of varied dependent variables, and inclusion of different source/sink terms. Questions often arise as to whether a particular equation set is expressed in a form consistent with the conservation-law definition. The concept of conservation form is developed to clarify the meaning mathematically. Six sets of unsteady-flow equations typically used in engineering practice are presented and their conservation properties are identified and discussed. Results of the theoretical development and analysis of the equations are substantiated in a set of numerical experiments conducted using alternate equation forms. Findings of these analytical and numerical efforts demonstrate that the choice of dependent variable is the fundamental factor determining the nature of the conservation properties of any particular equation form.

Gyrokinetic equations and full f solution method based on Dirac's constrained Hamiltonian and inverse Kruskal iteration

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

Heikkinen, J. A.; Nora, M.

2011-02-15

Gyrokinetic equations of motion, Poisson equation, and energy and momentum conservation laws are derived based on the reduced-phase-space Lagrangian and inverse Kruskal iteration introduced by Pfirsch and Correa-Restrepo [J. Plasma Phys. 70, 719 (2004)]. This formalism, together with the choice of the adiabatic invariant J= as one of the averaging coordinates in phase space, provides an alternative to the standard gyrokinetics. Within second order in gyrokinetic parameter, the new equations do not show explicit ponderomotivelike or polarizationlike terms. Pullback of particle information with an iterated gyrophase and field dependent gyroradius function from the gyrocenter position defined by gyroaveraged coordinates allowsmore » direct numerical integration of the gyrokinetic equations in particle simulation of the field and particles with full distribution function. As an example, gyrokinetic systems with polarization drift either present or absent in the equations of motion are considered.« less

Self-adjusting grid methods for one-dimensional hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Harten, A.; Hyman, J. M.

1983-01-01

The automatic adjustment of a grid which follows the dynamics of the numerical solution of hyperbolic conservation laws is given. The grid motion is determined by averaging the local characteristic velocities of the equations with respect to the amplitudes of the signals. The resulting algorithm is a simple extension of many currently popular Godunov-type methods. Computer codes using one of these methods can be easily modified to add the moving mesh as an option. Numerical examples are given that illustrate the improved accuracy of Godunov's and Roe's methods on a self-adjusting mesh. Previously announced in STAR as N83-15008

Two-dimensional coupled mathematical modeling of fluvial processes with intense sediment transport and rapid bed evolution

NASA Astrophysics Data System (ADS)

Yue, Zhiyuan; Cao, Zhixian; Li, Xin; Che, Tao

2008-09-01

Alluvial rivers may experience intense sediment transport and rapid bed evolution under a high flow regime, for which traditional decoupled mathematical river models based on simplified conservation equations are not applicable. A two-dimensional coupled mathematical model is presented, which is generally applicable to the fluvial processes with either intense or weak sediment transport. The governing equations of the model comprise the complete shallow water hydrodynamic equations closed with Manning roughness for boundary resistance and empirical relationships for sediment exchange with the erodible bed. The second-order Total-Variation-Diminishing version of the Weighted-Average-Flux method, along with the HLLC approximate Riemann Solver, is adapted to solve the governing equations, which can properly resolve shock waves and contact discontinuities. The model is applied to the pilot study of the flooding due to a sudden outburst of a real glacial-lake.

Traveltime and longitudinal dispersion in Illinois streams

USGS Publications Warehouse

Graf, Julia B.

1986-01-01

Twenty-seven measurements of traveltime and longitudinal dispersion in 10 Illinois streams made from 1975 to 1982 provide data needed for estimating traveltime of peak concentration of a conservative solute, traveltime of the leading edge of a solute cloud, peak concentration resulting from injection of a given quantity of solute, and passage time of solute past a given point on a stream. These four variables can be estimated graphically for each stream from distance of travel and either discharge at the downstream end of the reach or flow-duration frequency. From equations developed from field measurements, the traveltime and dispersion characteristics also can be estimated for other unregulated streams in Illinois that have drainage areas less than about 1,500 square miles. For unmeasured streams, traveltime of peak concentration and of the leading edge of the cloud are related to discharge at the downstream end of the reach and to distance of travel. For both measured and unmeasured streams, peak concentration and passage time are best estimated from the relation of each to traveltime. In measured streams, dispersion efficiency is greater than that predicted by Fickian diffusion theory. The rate of decrease in peak concentration with traveltime is about equal to the rate of increase in passage time. Average velocity in a stream reach, given by the velocity of the center of solute mass in that reach, can be estimated from an equation developed from measured values. The equation relates average reach velocity to discharge at the downstream end of the reach. Average reach velocities computed for 9 of the 10 streams from available equations that are based on hydraulic-geometry relations are high relative to measured values. The estimating equation developed from measured velocities provides estimates of average reach velocity that are closer to measured velocities than are those computed using equations developed from hydraulic-geometry relations.

Some aspects of steam-water flow simulation in geothermal wells

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

Shulyupin, Alexander N.

1996-01-24

Actual aspects of steam-water simulation in geothermal wells are considered: necessary quality of a simulator, flow regimes, mass conservation equation, momentum conservation equation, energy conservation equation and condition equations. Shortcomings of traditional hydraulic approach are noted. Main questions of simulator development by the hydraulic approach are considered. New possibilities of a simulation with the structure approach employment are noted.

On a new class of completely integrable nonlinear wave equations. I. Infinitely many conservation laws

NASA Astrophysics Data System (ADS)

Nutku, Y.

1985-06-01

We point out a class of nonlinear wave equations which admit infinitely many conserved quantities. These equations are characterized by a pair of exact one-forms. The implication that they are closed gives rise to equations, the characteristics and Riemann invariants of which are readily obtained. The construction of the conservation laws requires the solution of a linear second-order equation which can be reduced to canonical form using the Riemann invariants. The hodograph transformation results in a similar linear equation. We discuss also the symplectic structure and Bäcklund transformations associated with these equations.

The Finite-Surface Method for incompressible flow: a step beyond staggered grid

NASA Astrophysics Data System (ADS)

Hokpunna, Arpiruk; Misaka, Takashi; Obayashi, Shigeru

2017-11-01

We present a newly developed higher-order finite surface method for the incompressible Navier-Stokes equations (NSE). This method defines the velocities as a surface-averaged value on the surfaces of the pressure cells. Consequently, the mass conservation on the pressure cells becomes an exact equation. The only things left to approximate is the momentum equation and the pressure at the new time step. At certain conditions, the exact mass conservation enables the explicit n-th order accurate NSE solver to be used with the pressure treatment that is two or four order less accurate without loosing the apparent convergence rate. This feature was not possible with finite volume of finite difference methods. We use Fourier analysis with a model spectrum to determine the condition and found that the range covers standard boundary layer flows. The formal convergence and the performance of the proposed scheme is compared with a sixth-order finite volume method. Finally, the accuracy and performance of the method is evaluated in turbulent channel flows. This work is partially funded by a research colloaboration from IFS, Tohoku university and ASEAN+3 funding scheme from CMUIC, Chiang Mai University.

A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm

DOE PAGES

Chen, G.; Chacón, L.

2015-08-11

For decades, the Vlasov–Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. We explore a fully implicit PIC algorithm for the Vlasov–Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space–time-centered, employing particle sub-cycling and orbit-averaging. This algorithm conserves total energy, local charge,more » canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. Finally, we demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D–3V.« less

Conservation laws and symmetries of a generalized Kawahara equation

NASA Astrophysics Data System (ADS)

Gandarias, Maria Luz; Rosa, Maria; Recio, Elena; Anco, Stephen

2017-06-01

The generalized Kawahara equation ut = a(t)uxxxxx + b(t)uxxx + c(t) f (u)ux appears in many physical applications. A complete classification of low-order conservation laws and point symmetries is obtained for this equation, which includes as a special case the usual Kawahara equation ut = αuux + βu2ux + γuxxx + μuxxxxx. A general connection between conservation laws and symmetries for the generalized Kawahara equation is derived through the Hamiltonian structure of this equation and its relationship to Noether's theorem using a potential formulation.

Evaluation and modification of five techniques for estimating stormwater runoff for watersheds in west-central Florida

USGS Publications Warehouse

Trommer, J.T.; Loper, J.E.; Hammett, K.M.

1996-01-01

Several traditional techniques have been used for estimating stormwater runoff from ungaged watersheds. Applying these techniques to water- sheds in west-central Florida requires that some of the empirical relationships be extrapolated beyond tested ranges. As a result, there is uncertainty as to the accuracy of these estimates. Sixty-six storms occurring in 15 west-central Florida watersheds were initially modeled using the Rational Method, the U.S. Geological Survey Regional Regression Equations, the Natural Resources Conservation Service TR-20 model, the U.S. Army Corps of Engineers Hydrologic Engineering Center-1 model, and the Environmental Protection Agency Storm Water Management Model. The techniques were applied according to the guidelines specified in the user manuals or standard engineering textbooks as though no field data were available and the selection of input parameters was not influenced by observed data. Computed estimates were compared with observed runoff to evaluate the accuracy of the techniques. One watershed was eliminated from further evaluation when it was determined that the area contributing runoff to the stream varies with the amount and intensity of rainfall. Therefore, further evaluation and modification of the input parameters were made for only 62 storms in 14 watersheds. Runoff ranged from 1.4 to 99.3 percent percent of rainfall. The average runoff for all watersheds included in this study was about 36 percent of rainfall. The average runoff for the urban, natural, and mixed land-use watersheds was about 41, 27, and 29 percent, respectively. Initial estimates of peak discharge using the rational method produced average watershed errors that ranged from an underestimation of 50.4 percent to an overestimation of 767 percent. The coefficient of runoff ranged from 0.20 to 0.60. Calibration of the technique produced average errors that ranged from an underestimation of 3.3 percent to an overestimation of 1.5 percent. The average calibrated coefficient of runoff for each watershed ranged from 0.02 to 0.72. The average values of the coefficient of runoff necessary to calibrate the urban, natural, and mixed land-use watersheds were 0.39, 0.16, and 0.08, respectively. The U.S. Geological Survey regional regression equations for determining peak discharge produced errors that ranged from an underestimation of 87.3 percent to an over- estimation of 1,140 percent. The regression equations for determining runoff volume produced errors that ranged from an underestimation of 95.6 percent to an overestimation of 324 percent. Regression equations developed from data used for this study produced errors that ranged between an underestimation of 82.8 percent and an over- estimation of 328 percent for peak discharge, and from an underestimation of 71.2 percent to an overestimation of 241 percent for runoff volume. Use of the equations developed for west-central Florida streams produced average errors for each type of watershed that were lower than errors associated with use of the U.S. Geological Survey equations. Initial estimates of peak discharges and runoff volumes using the Natural Resources Conservation Service TR-20 model, produced average errors of 44.6 and 42.7 percent respectively, for all the watersheds. Curve numbers and times of concentration were adjusted to match estimated and observed peak discharges and runoff volumes. The average change in the curve number for all the watersheds was a decrease of 2.8 percent. The average change in the time of concentration was an increase of 59.2 percent. The shape of the input dimensionless unit hydrograph also had to be adjusted to match the shape and peak time of the estimated and observed flood hydrographs. Peak rate factors for the modified input dimensionless unit hydrographs ranged from 162 to 454. The mean errors for peak discharges and runoff volumes were reduced to 18.9 and 19.5 percent, respectively, using the average calibrated input parameters for ea

Optical display for radar sensing

NASA Astrophysics Data System (ADS)

Szu, Harold; Hsu, Charles; Willey, Jefferson; Landa, Joseph; Hsieh, Minder; Larsen, Louis V.; Krzywicki, Alan T.; Tran, Binh Q.; Hoekstra, Philip; Dillard, John T.; Krapels, Keith A.; Wardlaw, Michael; Chu, Kai-Dee

2015-05-01

Boltzmann headstone S = kB Log W turns out to be the Rosette stone for Greek physics translation optical display of the microwave sensing hieroglyphics. The LHS is the molecular entropy S measuring the degree of uniformity scattering off the sensing cross sections. The RHS is the inverse relationship (equation) predicting the Planck radiation spectral distribution parameterized by the Kelvin temperature T. Use is made of the conservation energy law of the heat capacity of Reservoir (RV) change T Δ S = -ΔE equals to the internal energy change of black box (bb) subsystem. Moreover, an irreversible thermodynamics Δ S > 0 for collision mixing toward totally larger uniformity of heat death, asserted by Boltzmann, that derived the so-called Maxwell-Boltzmann canonical probability. Given the zero boundary condition black box, Planck solved a discrete standing wave eigenstates (equation). Together with the canonical partition function (equation) an average ensemble average of all possible internal energy yielded the celebrated Planck radiation spectral (equation) where the density of states (equation). In summary, given the multispectral sensing data (equation), we applied Lagrange Constraint Neural Network (LCNN) to solve the Blind Sources Separation (BSS) for a set of equivalent bb target temperatures. From the measurements of specific value, slopes and shapes we can fit a set of Kelvin temperatures T's for each bb targets. As a result, we could apply the analytical continuation for each entropy sources along the temperature-unique Planck spectral curves always toward the RGB color temperature display for any sensing probing frequency.

[The study on the characteristics and particle densities of lightning discharge plasma].

PubMed

Wang, Jie; Yuan, Ping; Zhang, Hua-ming; Shen, Xiao-zhi

2008-09-01

According to the wavelengths, relative intensities and transition parameters of lines in cloud-to-ground lightning spectra obtained by a slit-less spectrograph in Qinghai province and Xizang municipality, and by theoretical calculations of plasma, the average temperature and electron density for individual lightning discharge channel were calculated, and then, using Saha equations, electric charge conservation equations and particle conservation equations, the particle densities of every ionized-state, the mass density, pressure and the average ionization degree were obtained. Moreover, the average ionization degree and characteristics of particle distributions in each lightning discharge channel were analyzed. Local thermodynamic equilibrium and an optically thin emitting gas were assumed in the calculations. The result shows that the characteristics of lightning discharge plasma have strong relationships with lightning intensities. For a certain return stroke channel, both temperatures and electron densities of different positions show tiny trend of falling away with increasing height along the discharge channel. Lightning channels are almost completely ionized, and the first ionized particles occupy the main station while N II has the highest particle density. On the other hand, the relative concentrations of N II and O II are near a constant in lightning channels with different intensities. Generally speaking, the more intense the lightning discharge, the higher are the values of channel temperature, electron density and relative concentrations of highly ionized particles, but the lower the concentration of the neutral atoms. After considering the Coulomb interactions between positive and negative particles in the calculations, the results of ionization energies decrease, and the particle densities of atoms and first ionized ions become low while high-ionized ions become high. At a temperature of 28000 K, the pressure of the discharge channel due to electrons, atoms and ions is about 10 atmospheric pressure, and it changes for different lightning stroke with different intensity. The mass density of channel is lower and changes from 0.01 to 0.1 compared to the mass density of air at standard temperature and pressure (STP).

Efficient Low Dissipative High Order Schemes for Multiscale MHD Flows, I: Basic Theory

NASA Technical Reports Server (NTRS)

Sjoegreen, Bjoern; Yee, H. C.

2003-01-01

The objective of this paper is to extend our recently developed highly parallelizable nonlinear stable high order schemes for complex multiscale hydrodynamic applications to the viscous MHD equations. These schemes employed multiresolution wavelets as adaptive numerical dissipation controls t o limit the amount of and to aid the selection and/or blending of the appropriate types of dissipation to be used. The new scheme is formulated for both the conservative and non-conservative form of the MHD equations in curvilinear grids. The four advantages of the present approach over existing MHD schemes reported in the open literature are as follows. First, the scheme is constructed for long-time integrations of shock/turbulence/combustion MHD flows. Available schemes are too diffusive for long-time integrations and/or turbulence/combustion problems. Second, unlike exist- ing schemes for the conservative MHD equations which suffer from ill-conditioned eigen- decompositions, the present scheme makes use of a well-conditioned eigen-decomposition obtained from a minor modification of the eigenvectors of the non-conservative MHD equations t o solve the conservative form of the MHD equations. Third, this approach of using the non-conservative eigensystem when solving the conservative equations also works well in the context of standard shock-capturing schemes for the MHD equations. Fourth, a new approach to minimize the numerical error of the divergence-free magnetic condition for high order schemes is introduced. Numerical experiments with typical MHD model problems revealed the applicability of the newly developed schemes for the MHD equations.

The reality of artificial viscosity

DOE PAGES

Margolin, L. G.

2018-02-24

Artificial viscosity is used in the computer simulation of high Reynolds number flows and is one of the oldest numerical artifices. In this work, I will describe the origin and the interpretation of artificial viscosity as a physical phenomenon. The basis of this interpretation is the finite scale theory, which describes the evolution of integral averages of the fluid solution over finite (length) scales. I will outline the derivation of finite scale Navier–Stokes equations and highlight the particular properties of the equations that depend on the finite scales. Those properties include enslavement, inviscid dissipation, and a law concerning the partitionmore » of total flux of conserved quantities into advective and diffusive components.« less

HYDRODYNAMIC SIMULATION OF THE UPPER POTOMAC ESTUARY.

USGS Publications Warehouse

Schaffranck, Raymond W.

1986-01-01

Hydrodynamics of the upper extent of the Potomac Estuary between Indian Head and Morgantown, Md. , are simulated using a two-dimensional model. The model computes water-surface elevations and depth-averaged velocities by numerically integrating finite-difference forms of the equations of mass and momentum conservation using the alternating direction implicit method. The fundamental, non-linear, unsteady-flow equations, upon which the model is formulated, include additional terms to account for Coriolis acceleration and meteorological influences. Preliminary model/prototype data comparisons show agreement to within 9% for tidal flow volumes and phase differences within the measured-data-recording interval. Use of the model to investigate the hydrodynamics and certain aspects of transport within this Potomac Estuary reach is demonstrated. Refs.

Introduction to COFFE: The Next-Generation HPCMP CREATE-AV CFD Solver

NASA Technical Reports Server (NTRS)

Glasby, Ryan S.; Erwin, J. Taylor; Stefanski, Douglas L.; Allmaras, Steven R.; Galbraith, Marshall C.; Anderson, W. Kyle; Nichols, Robert H.

2016-01-01

HPCMP CREATE-AV Conservative Field Finite Element (COFFE) is a modular, extensible, robust numerical solver for the Navier-Stokes equations that invokes modularity and extensibility from its first principles. COFFE implores a flexible, class-based hierarchy that provides a modular approach consisting of discretization, physics, parallelization, and linear algebra components. These components are developed with modern software engineering principles to ensure ease of uptake from a user's or developer's perspective. The Streamwise Upwind/Petrov-Galerkin (SU/PG) method is utilized to discretize the compressible Reynolds-Averaged Navier-Stokes (RANS) equations tightly coupled with a variety of turbulence models. The mathematics and the philosophy of the methodology that makes up COFFE are presented.

The reality of artificial viscosity

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

Margolin, L. G.

Artificial viscosity is used in the computer simulation of high Reynolds number flows and is one of the oldest numerical artifices. In this work, I will describe the origin and the interpretation of artificial viscosity as a physical phenomenon. The basis of this interpretation is the finite scale theory, which describes the evolution of integral averages of the fluid solution over finite (length) scales. I will outline the derivation of finite scale Navier–Stokes equations and highlight the particular properties of the equations that depend on the finite scales. Those properties include enslavement, inviscid dissipation, and a law concerning the partitionmore » of total flux of conserved quantities into advective and diffusive components.« less

Experimental investigation and numerical modelling of positive corona discharge: ozone generation

NASA Astrophysics Data System (ADS)

Yanallah, K; Pontiga, F; Fernández-Rueda, A; Castellanos, A

2009-03-01

The spatial distribution of the species generated in a wire-cylinder positive corona discharge in pure oxygen has been computed using a plasma chemistry model that includes the most significant reactions between electrons, ions, atoms and molecules. The plasma chemistry model is included in the continuity equations of each species, which are coupled with Poisson's equation for the electric field and the energy conservation equation for the gas temperature. The current-voltage characteristic measured in the experiments has been used as an input data to the numerical simulation. The numerical model is able to reproduce the basic structure of the positive corona discharge and highlights the importance of Joule heating on ozone generation. The average ozone density has been computed as a function of current intensity and compared with the experimental measurements of ozone concentration determined by UV absorption spectroscopy.

Amplitude equation for under water sand-ripples in one dimension.

NASA Astrophysics Data System (ADS)

Schnipper, Teis; Mertens, Keith; Ellegaard, Clive; Bohr, Tomas

2007-11-01

Sand-ripples under oscillatory water flow form periodic patterns with wave lengths primarily controlled by the amplitude d of the water motion. We present an amplitude equation for sand-ripples in one spatial dimension which captures the formation of the ripples as well as secondary bifurcations observed when the amplitude d is suddenly varied. The equation has the form [ ht=- ɛ(h-h)+((hx)^2-1)hxx- hxxxx+ δ((hx)^2)xx] which, due to the first term, is neither completely local (it has long-range coupling through the average height h) nor has local sand conservation. We discuss why this is reasonable and how this term (with ɛ˜d-2) stops the coarsening process at a finite wavelength proportional to d. We compare our numerical results with experimental observations in a narrow channel.

Kinetic energy equations for the average-passage equation system

NASA Technical Reports Server (NTRS)

Johnson, Richard W.; Adamczyk, John J.

1989-01-01

Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.

Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham-Broer-Kaup-Like Equations

NASA Astrophysics Data System (ADS)

Wang, Xiu-Bin; Tian, Shou-Fu; Qin, Chun-Yan; Zhang, Tian-Tian

2017-03-01

In this article, a generalised Whitham-Broer-Kaup-Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham-Broer-Kaup-Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.

Numerical Prediction of Periodic Vortex Shedding in Subsonic and Transonic Turbine Cascade Flows

NASA Astrophysics Data System (ADS)

Mensink, C.

1996-05-01

Periodic vortex shedding at the trailing edge of a turbine cascade has been investigated numerically for a subsonic and a transonic cascade flow. The numerical investigation was carried out by a finite volume multiblock code, solving the 2D compressible Reynolds-averaged Navier-Stokes equations on a set of non-overlapping grid blocks that are connected in a conservative way. Comparisons are made with experimental results previously obtained by Sieverding and Heinemann.

Thermal Modeling for the Next Generation of Radiofrequency Exposure Limits: Commentary.

PubMed

Foster, Kenneth R; Ziskin, Marvin C; Balzano, Quirino

2017-07-01

This commentary evaluates two sets of guidelines for human exposure to radiofrequency (RF) energy, focusing on the frequency range above the "transition" frequency at 3-10 GHz where the guidelines change their basic restrictions from specific absorption rate to incident power density, through the end of the RF band at 300 GHz. The analysis is based on a simple thermal model based on Pennes' bioheat equation (BHTE) (Pennes 1948) assuming purely surface heating; an Appendix provides more details about the model and its range of applicability. This analysis suggests that present limits are highly conservative relative to their stated goals of limiting temperature increase in tissue. As applied to transmitting devices used against the body, they are much more conservative than product safety standards for touch temperature for personal electronics equipment that are used in contact with the body. Provisions in the current guidelines for "averaging time" and "averaging area" are not consistent with scaling characteristics of the bioheat equation and should be refined. The authors suggest the need for additional limits on fluence for protection against brief, high intensity pulses at millimeter wave frequencies. This commentary considers only thermal hazards, which form the basis of the current guidelines, and excludes considerations of reported "non-thermal" effects of exposure that would have to be evaluated in the process of updating the guidelines.

First-passage dynamics of linear stochastic interface models: numerical simulations and entropic repulsion effect

NASA Astrophysics Data System (ADS)

Gross, Markus

2018-03-01

A fluctuating interfacial profile in one dimension is studied via Langevin simulations of the Edwards–Wilkinson equation with non-conserved noise and the Mullins–Herring equation with conserved noise. The profile is subject to either periodic or Dirichlet (no-flux) boundary conditions. We determine the noise-driven time-evolution of the profile between an initially flat configuration and the instant at which the profile reaches a given height M for the first time. The shape of the averaged profile agrees well with the prediction of weak-noise theory (WNT), which describes the most-likely trajectory to a fixed first-passage time. Furthermore, in agreement with WNT, on average the profile approaches the height M algebraically in time, with an exponent that is essentially independent of the boundary conditions. However, the actual value of the dynamic exponent turns out to be significantly smaller than predicted by WNT. This ‘renormalization’ of the exponent is explained in terms of the entropic repulsion exerted by the impenetrable boundary on the fluctuations of the profile around its most-likely path. The entropic repulsion mechanism is analyzed in detail for a single (fractional) Brownian walker, which describes the anomalous diffusion of a tagged monomer of the interface as it approaches the absorbing boundary. The present study sheds light on the accuracy and the limitations of the weak-noise approximation for the description of the full first-passage dynamics.

Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

NASA Astrophysics Data System (ADS)

Rasin, Alexander G.

2010-06-01

The application of the Gardner method for the generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely Bäcklund transformations and initial conservation laws, follows from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for the generation of symmetries is developed and applied to the discrete Korteweg-de Vries equation. It can also be applied to all the other ABS equations.

Equivalence transformations and conservation laws for a generalized variable-coefficient Gardner equation

NASA Astrophysics Data System (ADS)

de la Rosa, R.; Gandarias, M. L.; Bruzón, M. S.

2016-11-01

In this paper we study the generalized variable-coefficient Gardner equations of the form ut + A(t) unux + C(t) u2nux + B(t) uxxx + Q(t) u = 0 . This class broadens out many other equations previously considered: Johnpillai and Khalique (2010), Molati and Ramollo (2012) and Vaneeva et al. (2015). The use of the equivalence group of this class allows us to perform an exhaustive study and a simple and clear formulation of the results. Some conservation laws are derived for the nonlinearly self-adjoint equations by using a general theorem on conservation laws. We also construct conservation laws by applying the multipliers method.

Properties of finite difference models of non-linear conservative oscillators

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1988-01-01

Finite-difference (FD) approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically. A generalized formulation of the Duffing and modified Duffing equations is derived and analyzed using several FD techniques, and it is concluded that, although it is always possible to contstruct FD models of conservative oscillators which are themselves conservative, caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation.

Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

2018-06-01

In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov's nonlocal conservation method is applied to construct Cls for time fractional CDGDK.

Momentum conserving Brownian dynamics propagator for complex soft matter fluids

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

Padding, J. T.; Briels, W. J.

2014-12-28

We present a Galilean invariant, momentum conserving first order Brownian dynamics scheme for coarse-grained simulations of highly frictional soft matter systems. Friction forces are taken to be with respect to moving background material. The motion of the background material is described by locally averaged velocities in the neighborhood of the dissolved coarse coordinates. The velocity variables are updated by a momentum conserving scheme. The properties of the stochastic updates are derived through the Chapman-Kolmogorov and Fokker-Planck equations for the evolution of the probability distribution of coarse-grained position and velocity variables, by requiring the equilibrium distribution to be a stationary solution.more » We test our new scheme on concentrated star polymer solutions and find that the transverse current and velocity time auto-correlation functions behave as expected from hydrodynamics. In particular, the velocity auto-correlation functions display a long time tail in complete agreement with hydrodynamics.« less

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.

Adjoint-Based Methodology for Time-Dependent Optimization

NASA Technical Reports Server (NTRS)

Yamaleev, N. K.; Diskin, B.; Nielsen, E. J.

2008-01-01

This paper presents a discrete adjoint method for a broad class of time-dependent optimization problems. The time-dependent adjoint equations are derived in terms of the discrete residual of an arbitrary finite volume scheme which approximates unsteady conservation law equations. Although only the 2-D unsteady Euler equations are considered in the present analysis, this time-dependent adjoint method is applicable to the 3-D unsteady Reynolds-averaged Navier-Stokes equations with minor modifications. The discrete adjoint operators involving the derivatives of the discrete residual and the cost functional with respect to the flow variables are computed using a complex-variable approach, which provides discrete consistency and drastically reduces the implementation and debugging cycle. The implementation of the time-dependent adjoint method is validated by comparing the sensitivity derivative with that obtained by forward mode differentiation. Our numerical results show that O(10) optimization iterations of the steepest descent method are needed to reduce the objective functional by 3-6 orders of magnitude for test problems considered.

Traveling wave solutions and conservation laws for nonlinear evolution equation

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

2018-02-01

In this work, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation. We acquire new types of traveling wave solutions for the governing equation. We show that the equation is nonlinear self-adjoint by obtaining suitable substitution. Therefore, we construct conservation laws for the equation using new conservation theorem. The obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium. The constraint condition for the existence of solitons is stated. Some three dimensional figures for some of the acquired results are illustrated.

A Novel Blast-mitigation Concept for Light Tactical Vehicles

DTIC Science & Technology

2013-01-01

analysis which utilizes the mass and energy (but not linear momentum ) conservation equations is provided. It should be noted that the identical final...results could be obtained using an analogous analysis which combines the mass and the linear momentum conservation equations. For a calorically...governing mass, linear momentum and energy conservation and heat conduction equations are solved within ABAQUS/ Explicit with a second-order accurate

7 CFR 610.13 - Equations for predicting soil loss due to wind erosion.

Code of Federal Regulations, 2014 CFR

2014-01-01

... 7 Agriculture 6 2014-01-01 2014-01-01 false Equations for predicting soil loss due to wind erosion... RESOURCES CONSERVATION SERVICE, DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.13 Equations for predicting soil loss due to wind erosion. (a) The...

7 CFR 610.13 - Equations for predicting soil loss due to wind erosion.

Code of Federal Regulations, 2010 CFR

2010-01-01

... 7 Agriculture 6 2010-01-01 2010-01-01 false Equations for predicting soil loss due to wind erosion... RESOURCES CONSERVATION SERVICE, DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.13 Equations for predicting soil loss due to wind erosion. (a) The...

7 CFR 610.13 - Equations for predicting soil loss due to wind erosion.

Code of Federal Regulations, 2012 CFR

2012-01-01

... 7 Agriculture 6 2012-01-01 2012-01-01 false Equations for predicting soil loss due to wind erosion... RESOURCES CONSERVATION SERVICE, DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.13 Equations for predicting soil loss due to wind erosion. (a) The...

7 CFR 610.13 - Equations for predicting soil loss due to wind erosion.

Code of Federal Regulations, 2011 CFR

2011-01-01

... 7 Agriculture 6 2011-01-01 2011-01-01 false Equations for predicting soil loss due to wind erosion... RESOURCES CONSERVATION SERVICE, DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.13 Equations for predicting soil loss due to wind erosion. (a) The...

7 CFR 610.13 - Equations for predicting soil loss due to wind erosion.

Code of Federal Regulations, 2013 CFR

2013-01-01

... 7 Agriculture 6 2013-01-01 2013-01-01 false Equations for predicting soil loss due to wind erosion... RESOURCES CONSERVATION SERVICE, DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.13 Equations for predicting soil loss due to wind erosion. (a) The...

Conservative mixing, competitive mixing and their applications

NASA Astrophysics Data System (ADS)

Klimenko, A. Y.

2010-12-01

In many of the models applied to simulations of turbulent transport and turbulent combustion, the mixing between particles is used to reflect the influence of the continuous diffusion terms in the transport equations. Stochastic particles with properties and mixing can be used not only for simulating turbulent combustion, but also for modeling a large spectrum of physical phenomena. Traditional mixing, which is commonly used in the modeling of turbulent reacting flows, is conservative: the total amount of scalar is (or should be) preserved during a mixing event. It is worthwhile, however, to consider a more general mixing that does not possess these conservative properties; hence, our consideration lies beyond traditional mixing. In non-conservative mixing, the particle post-mixing average becomes biased towards one of the particles participating in mixing. The extreme form of non-conservative mixing can be called competitive mixing or competition: after a mixing event, the loser particle simply receives the properties of the winner particle. Particles with non-conservative mixing can be used to emulate various phenomena involving competition. In particular, we investigate cyclic behavior that can be attributed to complex competing systems. We show that the localness and intransitivity of competitive mixing are linked to the cyclic behavior.

Computation of Laminar and Turbulent Flow in 90-Degree Square-Duct and Pipe Bends Using the Navier-Stokes Equations

DTIC Science & Technology

1982-04-01

R.M. and Warming, R.F.: An Implicit Finite - Difference Algorithm for Hyperbolic Systems in Conservation Law Form. Journal of Computational Physics...Quincy Street C-40) Arlington, VA 22217 D 82 05-.10 I0, S4CURITY CLASSIFICATION OF THIS ’E(Wha, Doae Entotwed) Slength scale. Six different flow cases...forces upstream have produced a non-zero velocity gradient normal to the plane of curvature. Fluid with above (/below) average nioiiiei.tuili migrates

Numerical solution of the unsteady Navier-Stokes equation

NASA Technical Reports Server (NTRS)

Osher, Stanley J.; Engquist, Bjoern

1985-01-01

The construction and the analysis of nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws are discussed. These schemes share many desirable properties with total variation diminishing schemes, but TVD schemes have at most first-order accuracy, in the sense of truncation error, at extrema of the solution. In this paper a uniformly second-order approximation is constructed, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time. This is achieved via a nonoscillatory piecewise linear reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell.

On conservation laws for a generalized Boussinesq equation

NASA Astrophysics Data System (ADS)

Anco, S.; Rosa, M.; Gandarias, M. L.

2017-07-01

In this work, we study a Boussinesq equation with a strong damping term from the point of view of the Lie theory. By using the low order conservation laws we apply the conservation laws multiplier method to the associated potential systems.

A single-scattering correction for the seismo-acoustic parabolic equation.

PubMed

Collins, Michael D

2012-04-01

An efficient single-scattering correction that does not require iterations is derived and tested for the seismo-acoustic parabolic equation. The approach is applicable to problems involving gradual range dependence in a waveguide with fluid and solid layers, including the key case of a sloping fluid-solid interface. The single-scattering correction is asymptotically equivalent to a special case of a single-scattering correction for problems that only have solid layers [Küsel et al., J. Acoust. Soc. Am. 121, 808-813 (2007)]. The single-scattering correction has a simple interpretation (conservation of interface conditions in an average sense) that facilitated its generalization to problems involving fluid layers. Promising results are obtained for problems in which the ocean bottom interface has a small slope.

Travelling wave solutions and conservation laws for the Korteweg-de Vries-Bejamin-Bona-Mahony equation

NASA Astrophysics Data System (ADS)

Simbanefayi, Innocent; Khalique, Chaudry Masood

2018-03-01

In this work we study the Korteweg-de Vries-Benjamin-Bona-Mahony (KdV-BBM) equation, which describes the two-way propagation of waves. Using Lie symmetry method together with Jacobi elliptic function expansion and Kudryashov methods we construct its travelling wave solutions. Also, we derive conservation laws of the KdV-BBM equation using the variational derivative approach. In this method, we begin by computing second-order multipliers for the KdV-BBM equation followed by a derivation of the respective conservation laws for each multiplier.

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.

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.

Lie symmetry analysis, explicit solutions and conservation laws for the space-time fractional nonlinear evolution equations

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa; Baleanu, Dumitru

2018-04-01

This paper studies the symmetry analysis, explicit solutions, convergence analysis, and conservation laws (Cls) for two different space-time fractional nonlinear evolution equations with Riemann-Liouville (RL) derivative. The governing equations are reduced to nonlinear ordinary differential equation (ODE) of fractional order using their Lie point symmetries. In the reduced equations, the derivative is in Erdelyi-Kober (EK) sense, power series technique is applied to derive an explicit solutions for the reduced fractional ODEs. The convergence of the obtained power series solutions is also presented. Moreover, the new conservation theorem and the generalization of the Noether operators are developed to construct the nonlocal Cls for the equations . Some interesting figures for the obtained explicit solutions are presented.

Variational symmetries, conserved quantities and identities for several equations of mathematical physics

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

Donchev, Veliko, E-mail: velikod@ie.bas.bg

2014-03-15

We find variational symmetries, conserved quantities and identities for several equations: envelope equation, Böcher equation, the propagation of sound waves with losses, flow of a gas with losses, and the nonlinear Schrödinger equation with losses or gains, and an electro-magnetic interaction. Most of these equations do not have a variational description with the classical variational principle and we find such a description with the generalized variational principle of Herglotz.

Aortic valve orifice equation independent of valvular flow intervals: application to aortic valve area computation in aortic stenosis and comparison with the Gorlin formula.

PubMed

Seitz, W; Oppenheimer, L; McIlroy, M; Nelson, D; Operschall, J

1986-12-01

An orifice equation is derived relating the effective aortic valve area, A, the average aortic valve pressure gradient, dP, the stroke volume, SV, and the heart frequency, FH, through considerations of momentum conservation across the aortic valve. This leads to a formula consistent with Newton's second law of motion. The form of the new equation is A = (7.5 X 10(-5)) SV FH2/Pd, where A, VS, FH and Pd are expressed in cm2, ml, s-1 and mmHg, respectively. Aortic valve areas computed with the new orifice equation are found to correlate with those computed by the Gorlin formula in conditions of resting haemodynamic states at a level of r = 0.86, SE = 0.25 cm2, N = 120. The results suggest that the new formula may be considered as an independent orifice equation having a similar domain of validity as the Gorlin formula. The new equation offers the possibility of deriving additional useful haemodynamic relationships through combination with established cardiological formulas and applying it in a noninvasive Doppler ultrasonic or echocardiographic context.

Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny-Lin equation

NASA Astrophysics Data System (ADS)

Rashidi, Saeede; Hejazi, S. Reza

This paper investigates the invariance properties of the time fractional Benny-Lin equation with Riemann-Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto-Sivashinsky equation and Navier-Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny-Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

Nonlinear Conservation Laws and Finite Volume Methods

NASA Astrophysics Data System (ADS)

Leveque, Randall J.

Introduction Software Notation Classification of Differential Equations Derivation of Conservation Laws The Euler Equations of Gas Dynamics Dissipative Fluxes Source Terms Radiative Transfer and Isothermal Equations Multi-dimensional Conservation Laws The Shock Tube Problem Mathematical Theory of Hyperbolic Systems Scalar Equations Linear Hyperbolic Systems Nonlinear Systems The Riemann Problem for the Euler Equations Numerical Methods in One Dimension Finite Difference Theory Finite Volume Methods Importance of Conservation Form - Incorrect Shock Speeds Numerical Flux Functions Godunov's Method Approximate Riemann Solvers High-Resolution Methods Other Approaches Boundary Conditions Source Terms and Fractional Steps Unsplit Methods Fractional Step Methods General Formulation of Fractional Step Methods Stiff Source Terms Quasi-stationary Flow and Gravity Multi-dimensional Problems Dimensional Splitting Multi-dimensional Finite Volume Methods Grids and Adaptive Refinement Computational Difficulties Low-Density Flows Discrete Shocks and Viscous Profiles Start-Up Errors Wall Heating Slow-Moving Shocks Grid Orientation Effects Grid-Aligned Shocks Magnetohydrodynamics The MHD Equations One-Dimensional MHD Solving the Riemann Problem Nonstrict Hyperbolicity Stiffness The Divergence of B Riemann Problems in Multi-dimensional MHD Staggered Grids The 8-Wave Riemann Solver Relativistic Hydrodynamics Conservation Laws in Spacetime The Continuity Equation The 4-Momentum of a Particle The Stress-Energy Tensor Finite Volume Methods Multi-dimensional Relativistic Flow Gravitation and General Relativity References

Numerical Modeling of Saturated Boiling in a Heated Tube

NASA Technical Reports Server (NTRS)

Majumdar, Alok; LeClair, Andre; Hartwig, Jason

2017-01-01

This paper describes a mathematical formulation and numerical solution of boiling in a heated tube. The mathematical formulation involves a discretization of the tube into a flow network consisting of fluid nodes and branches and a thermal network consisting of solid nodes and conductors. In the fluid network, the mass, momentum and energy conservation equations are solved and in the thermal network, the energy conservation equation of solids is solved. A pressure-based, finite-volume formulation has been used to solve the equations in the fluid network. The system of equations is solved by a hybrid numerical scheme which solves the mass and momentum conservation equations by a simultaneous Newton-Raphson method and the energy conservation equation by a successive substitution method. The fluid network and thermal network are coupled through heat transfer between the solid and fluid nodes which is computed by Chen's correlation of saturated boiling heat transfer. The computer model is developed using the Generalized Fluid System Simulation Program and the numerical predictions are compared with test data.

Hyperbolic conservation laws and numerical methods

NASA Technical Reports Server (NTRS)

Leveque, Randall J.

1990-01-01

The mathematical structure of hyperbolic systems and the scalar equation case of conservation laws are discussed. Linear, nonlinear systems and the Riemann problem for the Euler equations are also studied. The numerical methods for conservation laws are presented in a nonstandard manner which leads to large time steps generalizations and computations on irregular grids. The solution of conservation laws with stiff source terms is examined.

Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg–de Vries equation

PubMed Central

Bridges, Thomas J.

2016-01-01

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically. PMID:28119546

Diffusion Processes Satisfying a Conservation Law Constraint

DOE PAGES

Bakosi, J.; Ristorcelli, J. R.

2014-03-04

We investigate coupled stochastic differential equations governing N non-negative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires that a set of fluctuating variables be non-negative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the non-negativity and the unit-sum conservation law constraint are satisfied as the variables evolve in time. We investigate the consequencesmore » of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.« less

Diffusion Processes Satisfying a Conservation Law Constraint

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

Bakosi, J.; Ristorcelli, J. R.

We investigate coupled stochastic differential equations governing N non-negative continuous random variables that satisfy a conservation principle. In various fields a conservation law requires that a set of fluctuating variables be non-negative and (if appropriately normalized) sum to one. As a result, any stochastic differential equation model to be realizable must not produce events outside of the allowed sample space. We develop a set of constraints on the drift and diffusion terms of such stochastic models to ensure that both the non-negativity and the unit-sum conservation law constraint are satisfied as the variables evolve in time. We investigate the consequencesmore » of the developed constraints on the Fokker-Planck equation, the associated system of stochastic differential equations, and the evolution equations of the first four moments of the probability density function. We show that random variables, satisfying a conservation law constraint, represented by stochastic diffusion processes, must have diffusion terms that are coupled and nonlinear. The set of constraints developed enables the development of statistical representations of fluctuating variables satisfying a conservation law. We exemplify the results with the bivariate beta process and the multivariate Wright-Fisher, Dirichlet, and Lochner’s generalized Dirichlet processes.« less

Multi-Hamiltonian structure of equations of hydrodynamic type

NASA Astrophysics Data System (ADS)

Gümral, H.; Nutku, Y.

1990-11-01

The discussion of the Hamiltonian structure of two-component equations of hydrodynamic type is completed by presenting the Hamiltonian operators for Euler's equation governing the motion of plane sound waves of finite amplitude and another quasilinear second-order wave equation. There exists a doubly infinite family of conserved Hamiltonians for the equations of gas dynamics that degenerate into one, namely, the Benney sequence, for shallow-water waves. Infinite sequences of conserved quantities for these equations are also presented. In the case of multicomponent equations of hydrodynamic type, it is shown, that Kodama's generalization of the shallow-water equations admits bi-Hamiltonian structure.

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

Dumbser, Michael, E-mail: michael.dumbser@unitn.it; Balsara, Dinshaw S., E-mail: dbalsara@nd.edu

In this paper a new, simple and universal formulation of the HLLEM Riemann solver (RS) is proposed that works for general conservative and non-conservative systems of hyperbolic equations. For non-conservative PDE, a path-conservative formulation of the HLLEM RS is presented for the first time in this paper. The HLLEM Riemann solver is built on top of a novel and very robust path-conservative HLL method. It thus naturally inherits the positivity properties and the entropy enforcement of the underlying HLL scheme. However, with just the slight additional cost of evaluating eigenvectors and eigenvalues of intermediate characteristic fields, we can represent linearlymore » degenerate intermediate waves with a minimum of smearing. For conservative systems, our paper provides the easiest and most seamless path for taking a pre-existing HLL RS and quickly and effortlessly converting it to a RS that provides improved results, comparable with those of an HLLC, HLLD, Osher or Roe-type RS. This is done with minimal additional computational complexity, making our variant of the HLLEM RS also a very fast RS that can accurately represent linearly degenerate discontinuities. Our present HLLEM RS also transparently extends these advantages to non-conservative systems. For shallow water-type systems, the resulting method is proven to be well-balanced. Several test problems are presented for shallow water-type equations and two-phase flow models, as well as for gas dynamics with real equation of state, magnetohydrodynamics (MHD & RMHD), and nonlinear elasticity. Since our new formulation accommodates multiple intermediate waves and has a broader applicability than the original HLLEM method, it could alternatively be called the HLLI Riemann solver, where the “I” stands for the intermediate characteristic fields that can be accounted for. -- Highlights: •New simple and general path-conservative formulation of the HLLEM Riemann solver. •Application to general conservative and non-conservative hyperbolic systems. •Inclusion of sub-structure and resolution of intermediate characteristic fields. •Well-balanced for single- and two-layer shallow water equations and multi-phase flows. •Euler equations with real equation of state, MHD equations, nonlinear elasticity.« less

Breakdown of the conservative potential equation

NASA Technical Reports Server (NTRS)

Salas, M. D.; Gumbert, C. R.

1986-01-01

The conservative full-potential equation is used to study transonic flow over five airfoil sections. The results of the study indicate that once shock are present in the flow, the qualitative approximation is different from that observed with the Euler equations. The difference in behavior of the potential eventually leads to multiple solutions.

Cubication of Conservative Nonlinear Oscillators

ERIC Educational Resources Information Center

Belendez, Augusto; Alvarez, Mariela L.; Fernandez, Elena; Pascual, Immaculada

2009-01-01

A cubication procedure of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force, and this allows us to approximate the original nonlinear differential equation by a Duffing equation in which the coefficients for the linear…

Explicit wave action conservation for water waves on vertically sheared flows

NASA Astrophysics Data System (ADS)

Quinn, Brenda; Toledo, Yaron; Shrira, Victor

2016-04-01

Water waves almost always propagate on currents with a vertical structure such as currents directed towards the beach accompanied by an under-current directed back toward the deep sea or wind-induced currents which change magnitude with depth due to viscosity effects. On larger scales they also change their direction due to the Coriolis force as described by the Ekman spiral. This implies that the existing wave models, which assume vertically-averaged currents, is an approximation which is far from realistic. In recent years, ocean circulation models have significantly improved with the capability to model vertically-sheared current profiles in contrast with the earlier vertically-averaged current profiles. Further advancements have coupled wave action models to circulation models to relate the mutual effects between the two types of motion. Restricting wave models to vertically-averaged non-turbulent current profiles is obviously problematic in these cases and the primary goal of this work is to derive and examine a general wave action equation which accounts for these shortcoming. The formulation of the wave action conservation equation is made explicit by following the work of Voronovich (1976) and using known asymptotic solutions of the boundary value problem which exploit the smallness of the current magnitude compared to the wave phase velocity and/or its vertical shear and curvature. The adopted approximations are shown to be sufficient for most of the conceivable applications. This provides correction terms to the group velocity and wave action definition accounting for the shear effects, which are fitting for application to operational wave models. In the limit of vanishing current shear, the new formulation reduces to the commonly used Bretherton & Garrett (1968) no-shear wave action equation where the invariant is calculated with the current magnitude taken at the free surface. It is shown that in realistic oceanic conditions, the neglect of the vertical structure of the currents in wave modelling which is currently universal, might lead to significant errors in wave amplitude and the predicted wave ray paths. An extension of the work toward the more complex case of turbulent currents will also be discussed.

General Navier–Stokes-like momentum and mass-energy equations

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

Monreal, Jorge, E-mail: jmonreal@mail.usf.edu

2015-03-15

A new system of general Navier–Stokes-like equations is proposed to model electromagnetic flow utilizing analogues of hydrodynamic conservation equations. Such equations are intended to provide a different perspective and, potentially, a better understanding of electromagnetic mass, energy and momentum behaviour. Under such a new framework additional insights into electromagnetism could be gained. To that end, we propose a system of momentum and mass-energy conservation equations coupled through both momentum density and velocity vectors.

On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure

NASA Astrophysics Data System (ADS)

Nutku, Y.

1987-11-01

The multi-Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained for these equations. Starting from a (primary) Hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi-Hamiltonian structure are obtained and it is shown that the second Hamiltonian operator can be constructed solely through a knowledge of the first Hamiltonian function. The recursion operator which first appears at the level of bi-Hamiltonian structure gives rise to an infinite sequence of conserved Hamiltonians. It is found that in general there exist two different infinite sequences of conserved quantities for these equations. The recursion relation defining higher Hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the (k+1)st Hamiltonian operator which depends on the kth Hamiltonian function. The infinite sequence of conserved Hamiltonians are common to all the higher Hamiltonian structures. The equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri-Hamiltonian structure with two distinct infinite sets of conserved quantities. The isothermal case of γ=1 is an exceptional one that requires separate treatment. This corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to Poisson's equation in nonlinear acoustics.

A Space-Time Conservation Element and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes

NASA Technical Reports Server (NTRS)

Zhang, Zeng-Chan; Yu, S. T. John; Chang, Sin-Chung; Jorgenson, Philip (Technical Monitor)

2001-01-01

In this paper, we report a version of the Space-Time Conservation Element and Solution Element (CE/SE) Method in which the 2D and 3D unsteady Euler equations are simulated using structured or unstructured quadrilateral and hexahedral meshes, respectively. In the present method, mesh values of flow variables and their spatial derivatives are treated as independent unknowns to be solved for. At each mesh point, the value of a flow variable is obtained by imposing a flux conservation condition. On the other hand, the spatial derivatives are evaluated using a finite-difference/weighted-average procedure. Note that the present extension retains many key advantages of the original CE/SE method which uses triangular and tetrahedral meshes, respectively, for its 2D and 3D applications. These advantages include efficient parallel computing ease of implementing non-reflecting boundary conditions, high-fidelity resolution of shocks and waves, and a genuinely multidimensional formulation without using a dimensional-splitting approach. In particular, because Riemann solvers, the cornerstones of the Godunov-type upwind schemes, are not needed to capture shocks, the computational logic of the present method is considerably simpler. To demonstrate the capability of the present method, numerical results are presented for several benchmark problems including oblique shock reflection, supersonic flow over a wedge, and a 3D detonation flow.

Transonic Navier-Stokes solutions of three-dimensional afterbody flows

NASA Technical Reports Server (NTRS)

Compton, William B., III; Thomas, James L.; Abeyounis, William K.; Mason, Mary L.

1989-01-01

The performance of a three-dimensional Navier-Stokes solution technique in predicting the transonic flow past a nonaxisymmetric nozzle was investigated. The investigation was conducted at free-stream Mach numbers ranging from 0.60 to 0.94 and an angle of attack of 0 degrees. The numerical solution procedure employs the three-dimensional, unsteady, Reynolds-averaged Navier-Stokes equations written in strong conservation form, a thin layer assumption, and the Baldwin-Lomax turbulence model. The equations are solved by using the finite-volume principle in conjunction with an approximately factored upwind-biased numerical algorithm. In the numerical procedure, the jet exhaust is represented by a solid sting. Wind-tunnel data with the jet exhaust simulated by high pressure air were also obtained to compare with the numerical calculations.

Anisotropic opinion dynamics

NASA Astrophysics Data System (ADS)

Neirotti, Juan

2016-07-01

We consider the process of opinion formation in a society of interacting agents, where there is a set B of socially accepted rules. In this scenario, we observed that agents, represented by simple feed-forward, adaptive neural networks, may have a conservative attitude (mostly in agreement with B ) or liberal attitude (mostly in agreement with neighboring agents) depending on how much their opinions are influenced by their peers. The topology of the network representing the interaction of the society's members is determined by a graph, where the agents' properties are defined over the vertexes and the interagent interactions are defined over the bonds. The adaptability of the agents allows us to model the formation of opinions as an online learning process, where agents learn continuously as new information becomes available to the whole society (online learning). Through the application of statistical mechanics techniques we deduced a set of differential equations describing the dynamics of the system. We observed that by slowly varying the average peer influence in such a way that the agents attitude changes from conservative to liberal and back, the average social opinion develops a hysteresis cycle. Such hysteretic behavior disappears when the variance of the social influence distribution is large enough. In all the cases studied, the change from conservative to liberal behavior is characterized by the emergence of conservative clusters, i.e., a closed knitted set of society members that follow a leader who agrees with the social status quo when the rule B is challenged.

Geometry of Conservation Laws for a Class of Parabolic Partial Differential Equations

NASA Astrophysics Data System (ADS)

Clelland, Jeanne Nielsen

1996-08-01

I consider the problem of computing the space of conservation laws for a second-order, parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system {cal I} on a 12 -manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of {cal I} modulo the so -called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is:. Theorem. Any conservation law for a second-order, parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal {cal I} on the original 12-manifold M. I show that if a nontrivial conservation law exists, then {cal I} has a deprolongation to an equivalent system {cal J} on a 7-manifold N, and any conservation law for {cal I} can be expressed as a closed 3-form on N which lies in {cal J}. Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA(u _{xx}u_{yy}-u_sp {xy}{2}) + B_1u_{xx }+2B_2u_{xy} +B_3u_ {yy}+C=0crwhere A, B_{i}, C are functions of x, y, t, u, u_{x}, u _{y}, u_{t}. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation u_{t} = {1over2}e ^{-u}(u_{xx}+u_ {yy})has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.

Amplitude equation and long-range interactions in underwater sand ripples in one dimension

NASA Astrophysics Data System (ADS)

Schnipper, Teis; Mertens, Keith; Ellegaard, Clive; Bohr, Tomas

2008-10-01

We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form ht=-γ(h- hmacr )+((hx)2-1)hxx-hxxxx+δ((hx)2)xx which, due to the first term, is neither completely local (it has long-range coupling through the average height hmacr ) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large δ (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.

Dispersion in tidally averaged transport equation

USGS Publications Warehouse

Cheng, R.T.; Casulli, V.

1992-01-01

A general governing inter-tidal transport equation for conservative solutes has been derived without invoking the weakly nonlinear approximation. The governing inter-tidal transport equation is a convection-dispersion equation in which the convective velocity is a mean Lagrangian residual current, and the inter-tidal dispersion coefficient is defined by a dispersion patch. When the weakly nonlinear condition is violated, the physical significance of the Stokes' drift, as used in tidal dynamics, becomes questionable. For nonlinear problems, analytical solutions for the mean Lagrangian residual current and for the inter-tidal dispersion coefficient do not exist, they must be determined numerically. A rectangular tidal inlet with a constriction is used in the first example. The solutions of the residual currents and the computed properties of the inter-tidal dispersion coefficient are used to illuminate the mechanisms of the inter-tidal transport processes. Then, the present formulation is tested in a geometrically complex tidal estuary – San Francisco Bay, California. The computed inter-tidal dispersion coefficients are in the range between 5×104 and 5×106 cm2/sec., which are consistent with the values reported in the literature

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the two-dimensional or axisymmetric, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 1 is the Analysis Description, and 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.

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.

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.

Numerical solution of the Saint-Venant equations by an efficient hybrid finite-volume/finite-difference method

NASA Astrophysics Data System (ADS)

Lai, Wencong; Khan, Abdul A.

2018-04-01

A computationally efficient hybrid finite-volume/finite-difference method is proposed for the numerical solution of Saint-Venant equations in one-dimensional open channel flows. The method adopts a mass-conservative finite volume discretization for the continuity equation and a semi-implicit finite difference discretization for the dynamic-wave momentum equation. The spatial discretization of the convective flux term in the momentum equation employs an upwind scheme and the water-surface gradient term is discretized using three different schemes. The performance of the numerical method is investigated in terms of efficiency and accuracy using various examples, including steady flow over a bump, dam-break flow over wet and dry downstream channels, wetting and drying in a parabolic bowl, and dam-break floods in laboratory physical models. Numerical solutions from the hybrid method are compared with solutions from a finite volume method along with analytic solutions or experimental measurements. Comparisons demonstrates that the hybrid method is efficient, accurate, and robust in modeling various flow scenarios, including subcritical, supercritical, and transcritical flows. In this method, the QUICK scheme for the surface slope discretization is more accurate and less diffusive than the center difference and the weighted average schemes.

Nonlocal conservation laws of the constant astigmatism equation

NASA Astrophysics Data System (ADS)

Hlaváč, Adam; Marvan, Michal

2017-03-01

For the constant astigmatism equation, we construct a system of nonlocal conservation laws (an abelian covering) closed under the reciprocal transformations. The corresponding potentials are functionally independent modulo a Wronskian type relation.

On the Conservation of Cross Helicity and Wave Action in Solar-wind Models with Non-WKB Alfvén Wave Reflection

NASA Astrophysics Data System (ADS)

Chandran, Benjamin D. G.; Perez, Jean C.; Verscharen, Daniel; Klein, Kristopher G.; Mallet, Alfred

2015-09-01

The interaction between Alfvén-wave turbulence and the background solar wind affects the cross helicity (\\int {d}3x {\\boldsymbol{v}}\\cdot {\\boldsymbol{B}}) in two ways. Non-WKB reflection converts outward-propagating Alfvén waves into inward-propagating Alfvén waves and vice versa, and the turbulence transfers momentum to the background flow. When both effects are accounted for, the total cross helicity is conserved. In the special case that the background density and flow speed are independent of time, the equations of cross-helicity conservation and total-energy conservation can be combined to recover a well-known equation derived by Heinemann and Olbert that has been interpreted as a non-WKB generalization of wave-action conservation. This latter equation (in contrast to cross-helicity and energy conservation) does not hold when the background varies in time.

Extension of lattice Boltzmann flux solver for simulation of compressible multi-component flows

NASA Astrophysics Data System (ADS)

Yang, Li-Ming; Shu, Chang; Yang, Wen-Ming; Wang, Yan

2018-05-01

The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers for solving compressible fluid flow problems, is extended to simulate compressible multi-component flows in this work. To solve the two-phase gas-liquid problems, the model equations with stiffened gas equation of state are adopted. In this model, two additional non-conservative equations are introduced to represent the material interfaces, apart from the classical Euler equations. We first convert the interface equations into the full conservative form by applying the mass equation. After that, we calculate the numerical fluxes of the classical Euler equations by the existing LBFS and the numerical fluxes of the interface equations by the passive scalar approach. Once all the numerical fluxes at the cell interface are obtained, the conservative variables at cell centers can be updated by marching the equations in time and the material interfaces can be identified via the distributions of the additional variables. The numerical accuracy and stability of present scheme are validated by its application to several compressible multi-component fluid flow problems.

A comparative study of the nonuniqueness problem of the potential equation

NASA Technical Reports Server (NTRS)

Salas, M. D.; Jameson, A.; Melnik, R. E.

1985-01-01

The nonuniqueness problem occurring at transonic speeds with the conservative potential equation is investigated numerically. The study indicates that the problem is not an inviscid phenomenon, but results from approximate treatment of shock waves inherent in the conservative potential model. A new bound on the limit of validity of the conservative potential model is proposed.

Braneworld gravity within non-conservative gravitational theory

NASA Astrophysics Data System (ADS)

Fabris, J. C.; Caramês, Thiago R. P.; da Silva, J. M. Hoff

2018-05-01

We investigate the braneworld gravity starting from the non-conservative gravitational field equations in a five-dimensional bulk. The approach is based on the Gauss-Codazzi formalism along with the study of the braneworld consistency conditions. The effective gravitational equations on the brane are obtained and the constraint leading to a brane energy-momentum conservation is analyzed.

A semi-Lagrangian approach to the shallow water equation

NASA Technical Reports Server (NTRS)

Bates, J. R.; Mccormick, Stephen F.; Ruge, John; Sholl, David S.; Yavneh, Irad

1993-01-01

We present a formulation of the shallow water equations that emphasizes the conservation of potential vorticity. A locally conservative semi-Lagrangian time-stepping scheme is developed, which leads to a system of three coupled PDE's to be solved at each time level. We describe a smoothing analysis of these equations, on which an effective multigrid solver is constructed. Some results from applying this solver to the static version of these equations are presented.

Kinetics of Polydomain Ordering at Second-Order Phase Transitions (by the Example of the AuCu3 Alloy)

NASA Astrophysics Data System (ADS)

Feldman, E. P.; Stefanovich, L. I.; Gumennyk, K. V.

2008-08-01

Kinetics of polydomain spinodal ordering is studied in alloys of AuCu3 type. We introduce four non-conserved long-range order parameters whose sum, however, is conserved and, using the statistical approach, follow the temporal evolution of their random spatial distribution after a rapid temperature quench. A system of nonlinear differential equations for correlators of second and third order is derived. Asymptotical analysis of this system allows to investigate the scaling regime, which develops on the late stages of evolution and to extract additional information concerning the rate of decrease of the specific volume of disordered regions and the rate of decrease of the average thickness of antiphase boundaries. Comparison of these results to experimental data is given. The quench below the spinodal and the onset of long-range order may be separated by the incubation time, whose origin is different from that in first-order phase transitions. Numerical integration of equations for correlators shows also, that it is possible to prepare a sample in such a way that its further evolution will go with formation of transient kinetically slowed polydomain structures different from the final L12 structure.

Conservational PDF Equations of Turbulence

NASA Technical Reports Server (NTRS)

Shih, Tsan-Hsing; Liu, Nan-Suey

2010-01-01

Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application

Discretization and Preconditioning Algorithms for the Euler and Navier-Stokes Equations on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Bart, Timothy J.; Kutler, Paul (Technical Monitor)

1998-01-01

Chapter 1 briefly reviews several related topics associated with the symmetrization of systems of conservation laws and quasi-conservation laws: (1) Basic Entropy Symmetrization Theory; (2) Symmetrization and eigenvector scaling; (3) Symmetrization of the compressible Navier-Stokes equations; and (4) Symmetrization of the quasi-conservative form of the magnetohydrodynamic (MHD) equations. Chapter 2 describes one of the best known tools employed in the study of differential equations, the maximum principle: any function f(x) which satisfies the inequality f(double prime)>0 on the interval [a,b] attains its maximum value at one of the endpoints on the interval. Chapter three examines the upwind finite volume schemes for scalar and system conservation laws. The basic tasks in the upwind finite volume approach have already been presented: reconstruction, flux evaluation, and evolution. By far, the most difficult task in this process is the reconstruction step.

Electromagnetic momentum and the energy–momentum tensor in a linear medium with magnetic and dielectric properties

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

Crenshaw, Michael E., E-mail: michael.e.crenshaw4.civ@mail.mil

2014-04-15

In a continuum setting, the energy–momentum tensor embodies the relations between conservation of energy, conservation of linear momentum, and conservation of angular momentum. The well-defined total energy and the well-defined total momentum in a thermodynamically closed system with complete equations of motion are used to construct the total energy–momentum tensor for a stationary simple linear material with both magnetic and dielectric properties illuminated by a quasimonochromatic pulse of light through a gradient-index antireflection coating. The perplexing issues surrounding the Abraham and Minkowski momentums are bypassed by working entirely with conservation principles, the total energy, and the total momentum. We derivemore » electromagnetic continuity equations and equations of motion for the macroscopic fields based on the material four-divergence of the traceless, symmetric total energy–momentum tensor. We identify contradictions between the macroscopic Maxwell equations and the continuum form of the conservation principles. We resolve the contradictions, which are the actual fundamental issues underlying the Abraham–Minkowski controversy, by constructing a unified version of continuum electrodynamics that is based on establishing consistency between the three-dimensional Maxwell equations for macroscopic fields, the electromagnetic continuity equations, the four-divergence of the total energy–momentum tensor, and a four-dimensional tensor formulation of electrodynamics for macroscopic fields in a simple linear medium.« less

Vorticity and symplecticity in multi-symplectic, Lagrangian gas dynamics

NASA Astrophysics Data System (ADS)

Webb, G. M.; Anco, S. C.

2016-02-01

The Lagrangian, multi-dimensional, ideal, compressible gas dynamic equations are written in a multi-symplectic form, in which the Lagrangian fluid labels, m i (the Lagrangian mass coordinates) and time t are the independent variables, and in which the Eulerian position of the fluid element {x}={x}({m},t) and the entropy S=S({m},t) are the dependent variables. Constraints in the variational principle are incorporated by means of Lagrange multipliers. The constraints are: the entropy advection equation S t = 0, the Lagrangian map equation {{x}}t={u} where {u} is the fluid velocity, and the mass continuity equation which has the form J=τ where J={det}({x}{ij}) is the Jacobian of the Lagrangian map in which {x}{ij}=\\partial {x}i/\\partial {m}j and τ =1/ρ is the specific volume of the gas. The internal energy per unit volume of the gas \\varepsilon =\\varepsilon (ρ ,S) corresponds to a non-barotropic gas. The Lagrangian is used to define multi-momenta, and to develop de Donder-Weyl Hamiltonian equations. The de Donder-Weyl equations are cast in a multi-symplectic form. The pullback conservation laws and the symplecticity conservation laws are obtained. One class of symplecticity conservation laws give rise to vorticity and potential vorticity type conservation laws, and another class of symplecticity laws are related to derivatives of the Lagrangian energy conservation law with respect to the Lagrangian mass coordinates m i . We show that the vorticity-symplecticity laws can be derived by a Lie dragging method, and also by using Noether’s second theorem and a fluid relabelling symmetry which is a divergence symmetry of the action. We obtain the Cartan-Poincaré form describing the equations and we discuss a set of differential forms representing the equation system.

Consistent description of kinetic equation with triangle anomaly

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

Pu Shi; Gao Jianhua; Wang Qun

2011-05-01

We provide a consistent description of the kinetic equation with a triangle anomaly which is compatible with the entropy principle of the second law of thermodynamics and the charge/energy-momentum conservation equations. In general an anomalous source term is necessary to ensure that the equations for the charge and energy-momentum conservation are satisfied and that the correction terms of distribution functions are compatible to these equations. The constraining equations from the entropy principle are derived for the anomaly-induced leading order corrections to the particle distribution functions. The correction terms can be determined for the minimum number of unknown coefficients in onemore » charge and two charge cases by solving the constraining equations.« less

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.

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

Ghosh, Debojyoti; Constantinescu, Emil M.

The numerical simulation of meso-, convective-, and microscale atmospheric flows requires the solution of the Euler or the Navier-Stokes equations. Nonhydrostatic weather prediction algorithms often solve the equations in terms of derived quantities such as Exner pressure and potential temperature (and are thus not conservative) and/or as perturbations to the hydrostatically balanced equilibrium state. This paper presents a well-balanced, conservative finite difference formulation for the Euler equations with a gravitational source term, where the governing equations are solved as conservation laws for mass, momentum, and energy. Preservation of the hydrostatic balance to machine precision by the discretized equations is essentialmore » because atmospheric phenomena are often small perturbations to this balance. The proposed algorithm uses the weighted essentially nonoscillatory and compact-reconstruction weighted essentially nonoscillatory schemes for spatial discretization that yields high-order accurate solutions for smooth flows and is essentially nonoscillatory across strong gradients; however, the well-balanced formulation may be used with other conservative finite difference methods. The performance of the algorithm is demonstrated on test problems as well as benchmark atmospheric flow problems, and the results are verified with those in the literature.« less

The fundamental equation of eddy covariance and its application in flux measurements

Treesearch

Lianhong Gu; William J. Massman; Ray Leuning; Stephen G. Pallardy; Tilden Meyers; Paul J. Hanson; Jeffery S. Riggs; Kevin P. Hosman; Bai Yang

2012-01-01

A fundamental equation of eddy covariance (FQEC) is derived that allows the net ecosystem exchange (NEE) Ns of a specified atmospheric constituent s to be measured with the constraint of conservation of any other atmospheric constituent (e.g. N2, argon, or dry air). It is shown that if the condition [equation, see PDF] is true, the conservation of mass can be applied...

LETTER TO THE EDITOR: Bicomplexes and conservation laws in non-Abelian Toda models

NASA Astrophysics Data System (ADS)

Gueuvoghlanian, E. P.

2001-08-01

A bicomplex structure is associated with the Leznov-Saveliev equation of integrable models. The linear problem associated with the zero-curvature condition is derived in terms of the bicomplex linear equation. The explicit example of a non-Abelian conformal affine Toda model is discussed in detail and its conservation laws are derived from the zero-curvature representation of its equation of motion.

Fradkin-Bacry-Ruegg-Souriau perihelion vector for Gorringe-Leach equations

NASA Astrophysics Data System (ADS)

Grandati, Yves; Bérard, Alain; Mohrbach, Hervé

2010-02-01

We show that every generalized Gorringe-Leach equation admits an associated Fradkin-Bacry-Ruegg-Souriau’s vector which, in general, is only a piecewise conserved quantity. In the case of dualizable generalized Gorringe-Leach equations, which include the case of conservative motions in central power law potentials, the image sets of the FBRS vectors for dual classes are dual images of each other.

Evolution of wealth in a non-conservative economy driven by local Nash equilibria.

PubMed

Degond, Pierre; Liu, Jian-Guo; Ringhofer, Christian

2014-11-13

We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed in Degond et al. (2014 J. Stat. Phys. 154, 751-780 (doi:10.1007/s10955-013-0888-4)). The model considers a system of rational agents interacting in a game-theoretical framework. This evolution drives the dynamics of the agents in both wealth and economic configuration variables. The cost function is chosen to represent a risk-averse strategy of each agent. That is, the agent is more likely to interact with the market, the more predictable the market, and therefore the smaller its individual risk. This yields a kinetic equation for an effective single particle agent density with a Nash equilibrium serving as the local thermodynamic equilibrium. We consider a regime of scale separation where the large-scale dynamics is given by a hydrodynamic closure with this local equilibrium. A class of generalized collision invariants is developed to overcome the difficulty of the non-conservative property in the hydrodynamic closure derivation of the large-scale dynamics for the evolution of wealth distribution. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse Gamma distribution, which has been previously considered in the literature, as a local equilibrium for particular choices of the cost function. © 2014 The Author(s) Published by the Royal Society. All rights reserved.

BMS3 invariant fluid dynamics at null infinity

NASA Astrophysics Data System (ADS)

Penna, Robert F.

2018-02-01

We revisit the boundary dynamics of asymptotically flat, three dimensional gravity. The boundary is governed by a momentum conservation equation and an energy conservation equation, which we interpret as fluid equations, following the membrane paradigm. We reformulate the boundary’s equations of motion as Hamiltonian flow on the dual of an infinite-dimensional, semi-direct product Lie algebra equipped with a Lie–Poisson bracket. This gives the analogue for boundary fluid dynamics of the Marsden–Ratiu–Weinstein formulation of the compressible Euler equations on a manifold, M, as Hamiltonian flow on the dual of the Lie algebra of \

Path length differencing and energy conservation of the S[sub N] Boltzmann/Spencer-Lewis equation

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

Filippone, W.L.; Monahan, S.P.

It is shown that the S[sub N] Boltzmann/Spencer-Lewis equations conserve energy locally if and only if they satisfy particle balance and diamond differencing is used in path length. In contrast, the spatial differencing schemes have no bearing on the energy balance. Energy is conserved globally if it is conserved locally and the multigroup cross sections are energy conserving. Although the coupled electron-photon cross sections generated by CEPXS conserve particles and charge, they do not precisely conserve energy. It is demonstrated that these cross sections can be adjusted such that particles, charge, and energy are conserved. Finally, since a conventional negativemore » flux fixup destroys energy balance when applied to path legend, a modified fixup scheme that does not is presented.« less

On the Maxwellian distribution, symmetric form, and entropy conservation for the Euler equations

NASA Technical Reports Server (NTRS)

Deshpande, S. M.

1986-01-01

The Euler equations of gas dynamics have some very interesting properties in that the flux vector is a homogeneous function of the unknowns and the equations can be cast in symmetric hyperbolic form and satisfy the entropy conservation. The Euler equations are the moments of the Boltzmann equation of the kinetic theory of gases when the velocity distribution function is a Maxwellian. The present paper shows the relationship between the symmetrizability and the Maxwellian velocity distribution. The entropy conservation is in terms of the H-function, which is a slight modification of the H-function first introduced by Boltzmann in his famous H-theorem. In view of the H-theorem, it is suggested that the development of total H-diminishing (THD) numerical methods may be more profitable than the usual total variation diminishing (TVD) methods for obtaining wiggle-free solutions.

Modeling of electron cyclotron resonance discharges

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

Meyyappan, M.; Govindan, T.R.

The current trend in plasma processing is the development of high density plasma sources to achieve high deposition and etch rates, uniformity over large ares, and low wafer damage. Here, is a simple model to predict the spatially-averaged plasma characteristics of electron cyclotron resonance (ECR) reactors is presented. The model consists of global conservation equations for species concentration, electron density and energy. A gas energy balance is used to predict the neutral temperature self-consistently. The model is demonstrated for an ECR argon discharge. The predicted behavior of the discharge as a function of system variables agrees well with experimental observations.

Hamiltonian structures for systems of hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Olver, Peter J.; Nutku, Yavuz

1988-07-01

The bi-Hamiltonian structure for a large class of one-dimensional hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one-dimensional elastic media, and the Born-Infeld equation from nonlinear electrodynamics, is exhibited. For polytropic gas dynamics, these results lead to a quadri-Hamiltonian structure. New higher-order entropy-flux pairs (conservation laws) and higher-order symmetries are exhibited.

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

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

McHugh, P.R.; Ramshaw, J.D.

MAGMA is a FORTRAN computer code designed to viscous flow in in situ vitrification melt pools. It models three-dimensional, incompressible, viscous flow and heat transfer. The momentum equation is coupled to the temperature field through the buoyancy force terms arising from the Boussinesq approximation. All fluid properties, except density, are assumed variable. Density is assumed constant except in the buoyancy force terms in the momentum equation. A simple melting model based on the enthalpy method allows the study of the melt front progression and latent heat effects. An indirect addressing scheme used in the numerical solution of the momentum equationmore » voids unnecessary calculations in cells devoid of liquid. Two-dimensional calculations can be performed using either rectangular or cylindrical coordinates, while three-dimensional calculations use rectangular coordinates. All derivatives are approximated by finite differences. The incompressible Navier-Stokes equations are solved using a new fully implicit iterative technique, while the energy equation is differenced explicitly in time. Spatial derivatives are written in conservative form using a uniform, rectangular, staggered mesh based on the marker and cell placement of variables. Convective terms are differenced using a weighted average of centered and donor cell differencing to ensure numerical stability. Complete descriptions of MAGMA governing equations, numerics, code structure, and code verification are provided. 14 refs.« less

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.

Cardiac valve orifice equation independent of valvular flow intervals: application to mitral valve area computation in mitral stenosis and comparison with the Gorlin formula and direct anatomical measurements.

PubMed

Seitz, W; Marino, P; Zanolla, L; Buonanno, C; McIlroy, M; Spiel, M

1984-11-01

An orifice equation is developed which relates the effective mitral valve area (A), the average mitral valve pressure gradient (dP), the cardiac output (Q) and the heart frequency (f) through considerations of momentum conservation across the mitral valve. The form of the new equation is A = (4.75 X 10(-5)Qf/dP, where A, Q, and dP are expressed in cm2, ml X min-1 and mmHg respectively. Mitral valve areas computed with the new orifice formula are found to correlate with those computed by the Gorlin formula in conditions of equilibrium associated with the resting state at a level of r = 0.95, SE = 0.15 cm2, with autopsy measurements at a level of r = 0.85, SE = 0.18 cm2 and with direct anatomical measurements of excised valves at a level of r = 0.78, SE = 0.41 cm2. The results suggest that the new formula may be considered as an independent orifice equation enjoying a similar domain of validity as the Gorlin formula. The new equation offers the possibility of deriving additional useful haemodynamic relationships when used in combination with established cardiological formulas.

Higher-order jump conditions for conservation laws

NASA Astrophysics Data System (ADS)

Oksuzoglu, Hakan

2018-04-01

The hyperbolic conservation laws admit discontinuous solutions where the solution variables can have finite jumps in space and time. The jump conditions for conservation laws are expressed in terms of the speed of the discontinuity and the state variables on both sides. An example from the Gas Dynamics is the Rankine-Hugoniot conditions for the shock speed. Here, we provide an expression for the acceleration of the discontinuity in terms of the state variables and their spatial derivatives on both sides. We derive a jump condition for the shock acceleration. Using this general expression, we show how to obtain explicit shock acceleration formulas for nonlinear hyperbolic conservation laws. We start with the Burgers' equation and check the derived formula with an analytical solution. We next derive formulas for the Shallow Water Equations and the Euler Equations of Gas Dynamics. We will verify our formulas for the Euler Equations using an exact solution for the spherically symmetric blast wave problem. In addition, we discuss the potential use of these formulas for the implementation of shock fitting methods.

How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation

ERIC Educational Resources Information Center

Heras, Jose A.

2009-01-01

The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.

Similarity considerations and conservation laws for magneto-static atmospheres

NASA Technical Reports Server (NTRS)

Webb, G. M.

1986-01-01

The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential. Similarity solutions of the elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field. The solutions are obtained from a consideration of the invariance group of the elliptic equation. The importance of symmetries of the elliptic equation also appears in the determination of conservation laws. It turns out that the elliptic equation can be written as a variational principle, and the symmetries of the variational functional lead (via Noether's theorem) to conservation laws for the equation. As an example of the application of the similarity solutions, a model magnetostatic atmosphere is constructed in which the current density J is proportional to the cube of the magnetic potential, and falls off exponentially with distance vertical to the base, with an 'e-folding' distance equal to the gravitational scale height. The solutions show the interplay between the gravitational force, the J x B force (B, magnetic field induction) and the gas pressure gradient.

Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state

NASA Astrophysics Data System (ADS)

Lee, Bok Jik; Toro, Eleuterio F.; Castro, Cristóbal E.; Nikiforakis, Nikolaos

2013-08-01

For the numerical simulation of detonation of condensed phase explosives, a complex equation of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Cochran-Chan (C-C) EOS, are widely used. However, when a conservative scheme is used for solving the Euler equations with such equations of state, a spurious solution across the contact discontinuity, a well known phenomenon in multi-fluid systems, arises even for single materials. In this work, we develop a generalised Osher-type scheme in an adaptive primitive-conservative framework to overcome the aforementioned difficulties. Resulting numerical solutions are compared with the exact solutions and with the numerical solutions from the Godunov method in conjunction with the exact Riemann solver for the Euler equations with Mie-Grüneisen form of equations of state, such as the JWL and the C-C equations of state. The adaptive scheme is extended to second order and its empirical convergence rates are presented, verifying second order accuracy for smooth solutions. Through a suite of several tests problems in one and two space dimensions we illustrate the failure of conservative schemes and the capability of the methods of this paper to overcome the difficulties.

Using Improved Equation of State to Model Simultaneous Nucleation and Bubble Growth in Thermoplastic Foams

NASA Astrophysics Data System (ADS)

Khan, Irfan; Costeux, Stephane; Adrian, David; Cristancho, Diego

2013-11-01

Due to environmental regulations carbon-dioxide (CO2) is increasingly being used to replace traditional blowing agents in thermoplastic foams. CO2 is dissolved in the polymer matrix under supercritical conditions. In order to predict the effect of process parameters on foam properties using numerical modeling, the P-V-T relationship of the blowing agents should accurately be represented at the supercritical state. Previous studies in the area of foam modeling have all used ideal gas equation of state to predict the behavior of the blowing agent. In this work the Peng-Robinson equation of state is being used to model the blowing agent during its diffusion into the growing bubble. The model is based on the popular ``Influence Volume Approach,'' which assumes a growing boundary layer with depleted blowing agent surrounds each bubble. Classical nucleation theory is used to predict the rate of nucleation of bubbles. By solving the mass balance, momentum balance and species conservation equations for each bubble, the model is capable of predicting average bubble size, bubble size distribution and bulk porosity. The effect of the improved model on the bubble growth and foam properties are discussed.

Two-dimensional Euler and Navier-Stokes Time accurate simulations of fan rotor flows

NASA Technical Reports Server (NTRS)

Boretti, A. A.

1990-01-01

Two numerical methods are presented which describe the unsteady flow field in the blade-to-blade plane of an axial fan rotor. These methods solve the compressible, time-dependent, Euler and the compressible, turbulent, time-dependent, Navier-Stokes conservation equations for mass, momentum, and energy. The Navier-Stokes equations are written in Favre-averaged form and are closed with an approximate two-equation turbulence model with low Reynolds number and compressibility effects included. The unsteady aerodynamic component is obtained by superposing inflow or outflow unsteadiness to the steady conditions through time-dependent boundary conditions. The integration in space is performed by using a finite volume scheme, and the integration in time is performed by using k-stage Runge-Kutta schemes, k = 2,5. The numerical integration algorithm allows the reduction of the computational cost of an unsteady simulation involving high frequency disturbances in both CPU time and memory requirements. Less than 200 sec of CPU time are required to advance the Euler equations in a computational grid made up of about 2000 grid during 10,000 time steps on a CRAY Y-MP computer, with a required memory of less than 0.3 megawords.

AKNS hierarchy, Darboux transformation and conservation laws of the 1D nonautonomous nonlinear Schroedinger equations

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

Zhao Dun; Center for Interdisciplinary Studies, Lanzhou University, Lanzhou 730000; Zhang Yujuan

2011-04-15

By constructing nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, we investigate the nonautonomous nonlinear Schroedinger (NLS) equations which have been used to describe the Feshbach resonance management in matter-wave solitons in Bose-Einstein condensate and the dispersion and nonlinearity managements for optical solitons. It is found that these equations are some special cases of a new integrable model of nonlocal nonautonomous NLS equations. Based on the Lax pairs, the Darboux transformation and conservation laws are explored. It is shown that the local external potentials would break down the classical infinite number of conservation laws. The result indicates that the integrability of the nonautonomous NLSmore » systems may be nontrivial in comparison to the conventional concept of integrability in the canonical case.« less

Finite elements and finite differences for transonic flow calculations

NASA Technical Reports Server (NTRS)

Hafez, M. M.; Murman, E. M.; Wellford, L. C.

1978-01-01

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Mass conservation: 1-D open channel flow equations

USGS Publications Warehouse

DeLong, Lewis L.

1989-01-01

Unsteady flow simulation in natural rivers is often complicated by meandering channels of compound section. Hydraulic properties and the length of the wetted channel may vary significantly as a meandering river inundates its adjacent floodplain. The one-dimensional, unsteady, open-channel flow equations can be extended to simulate floods in channels of compound section. It will be shown that equations derived from the addition of differential equations individually describing flow in main and overbank channels do not in general conserve mass when overbank and main channels are of different lengths.

Uniqueness of Mass-Conserving Self-similar Solutions to Smoluchowski's Coagulation Equation with Inverse Power Law Kernels

NASA Astrophysics Data System (ADS)

Laurençot, Philippe

2018-03-01

Uniqueness of mass-conserving self-similar solutions to Smoluchowski's coagulation equation is shown when the coagulation kernel K is given by K(x,x_*)=2(x x_*)^{-α } , (x,x_*)\\in (0,∞)^2 , for some α >0.

Impact of Periodic Unsteadiness on Performance and Heat Load in Axial Flow Turbomachines

NASA Technical Reports Server (NTRS)

Sharma, Om P.; Stetson, Gary M.; Daniels, William A,; Greitzer, Edward M.; Blair, Michael F.; Dring, Robert P.

1997-01-01

Results of an analytical and experimental investigation, directed at the understanding of the impact of periodic unsteadiness on the time-averaged flows in axial flow turbomachines, are presented. Analysis of available experimental data, from a large-scale rotating rig (LSRR) (low speed rig), shows that in the time-averaged axisymmetric equations the magnitude of the terms representing the effect of periodic unsteadiness (deterministic stresses) are as large or larger than those due to random unsteadiness (turbulence). Numerical experiments, conducted to highlight physical mechanisms associated with the migration of combustor generated hot-streaks in turbine rotors, indicated that the effect can be simulated by accounting for deterministic stress like terms in the time-averaged mass and energy conservation equations. The experimental portion of this program shows that the aerodynamic loss for the second stator in a 1-1/2 stage turbine are influenced by the axial spacing between the second stator leading edge and the rotor trailing edge. However, the axial spacing has little impact on the heat transfer coefficient. These performance changes are believed to be associated with the change in deterministic stress at the inlet to the second stator. Data were also acquired to quantify the impact of indexing the first stator relative to the second stator. For the range of parameters examined, this effect was found to be of the same order as the effect of axial spacing.

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

NASA Astrophysics Data System (ADS)

LeMesurier, Brenton

2012-01-01

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics. Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives. This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.

Dynamic stability of spinning pretwisted beams subjected to axial random forces

NASA Astrophysics Data System (ADS)

Young, T. H.; Gau, C. Y.

2003-11-01

This paper studies the dynamic stability of a pretwisted cantilever beam spinning along its longitudinal axis and subjected to an axial random force at the free end. The axial force is assumed as the sum of a constant force and a random process with a zero mean. Due to this axial force, the beam may experience parametric random instability. In this work, the finite element method is first applied to yield discretized system equations. The stochastic averaging method is then adopted to obtain Ito's equations for the response amplitudes of the system. Finally the mean-square stability criterion is utilized to determine the stability condition of the system. Numerical results show that the stability boundary of the system converges as the first three modes are taken into calculation. Before the convergence is reached, the stability condition predicted is not conservative enough.

Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Meza, L. E. Arroyo; Dutra, A. de Souza; Hott, M. B.; Roy, P.

2015-01-01

By using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrödinger equation (NLSE) with cubic and quintic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) coefficients. We obtain a class of wide localized exact solutions of NLSE in the presence of a number of non-Hermitian parity-time (PT )-symmetric external potentials, which are constituted by a mixing of external potentials and source or drain terms. The exact solutions found here can be applied to theoretical studies of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. We show that, even in the presence of gain or loss terms, stable solutions can be found and that the PT symmetry is an important feature to guarantee the conservation of the average energy of the system.

A Numerical Model Study of Nocturnal Drainage Flows with Strong Wind and Temperature Gradients.

NASA Astrophysics Data System (ADS)

Yamada, T.; Bunker, S.

1989-07-01

A second-moment turbulence-closure model described in Yamada and Bunker is used to simulate nocturnal drainage flows observed during the 1984 ASCOT field expedition in Brush Creek, Colorado. In order to simulate the observed strong wind directional shear and temperature gradients, two modifications are added to the model. The strong wind directional shear was maintained by introducing a `nudging' term in the equation of motion to guide the modeled winds in the layers above the ridge top toward the observed wind direction. The second modification was accomplished by reformulating the conservation equation for the potential temperature in such a way that only the deviation from the horizontally averaged value was prognostically computed.The vegetation distribution used in this study is undoubtedly crude. Nevertheless, the present simulation suggests that tall tree canopy can play an important role in producing inhomogeneous wind distribution, particularly in the levels below the canopy top.

Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids II: Extension to Two Dimensional Scalar Equation

NASA Technical Reports Server (NTRS)

Wang, Z. J.; Liu, Yen; Kwak, Dochan (Technical Monitor)

2002-01-01

The framework for constructing a high-order, conservative Spectral (Finite) Volume (SV) method is presented for two-dimensional scalar hyperbolic conservation laws on unstructured triangular grids. Each triangular grid cell forms a spectral volume (SV), and the SV is further subdivided into polygonal control volumes (CVs) to supported high-order data reconstructions. Cell-averaged solutions from these CVs are used to reconstruct a high order polynomial approximation in the SV. Each CV is then updated independently with a Godunov-type finite volume method and a high-order Runge-Kutta time integration scheme. A universal reconstruction is obtained by partitioning all SVs in a geometrically similar manner. The convergence of the SV method is shown to depend on how a SV is partitioned. A criterion based on the Lebesgue constant has been developed and used successfully to determine the quality of various partitions. Symmetric, stable, and convergent linear, quadratic, and cubic SVs have been obtained, and many different types of partitions have been evaluated. The SV method is tested for both linear and non-linear model problems with and without discontinuities.

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

NASA Astrophysics Data System (ADS)

Lee, D.; Palha, A.; Gerritsma, M.

2018-03-01

A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

Generalized Maxwell equations and charge conservation censorship

NASA Astrophysics Data System (ADS)

Modanese, G.

2017-02-01

The Aharonov-Bohm electrodynamics is a generalization of Maxwell theory with reduced gauge invariance. It allows to couple the electromagnetic field to a charge which is not locally conserved, and has an additional degree of freedom, the scalar field S = ∂αAα, usually interpreted as a longitudinal wave component. By reformulating the theory in a compact Lagrangian formalism, we are able to eliminate S explicitly from the dynamics and we obtain generalized Maxwell equation with interesting properties: they give ∂μFμν as the (conserved) sum of the (possibly non-conserved) physical current density jν, and a “secondary” current density iν which is a nonlocal function of jν. This implies that any non-conservation of jν is effectively “censored” by the observable field Fμν, and yet it may have real physical consequences. We give examples of stationary solutions which display these properties. Possible applications are to systems where local charge conservation is violated due to anomalies of the Adler-Bell-Jackiw (ABJ) kind or to macroscopic quantum tunnelling with currents which do not satisfy a local continuity equation.

Three-Dimensional Navier-Stokes Method with Two-Equation Turbulence Models for Efficient Numerical Simulation of Hypersonic Flows

NASA Technical Reports Server (NTRS)

Bardina, J. E.

1994-01-01

A new computational efficient 3-D compressible Reynolds-averaged implicit Navier-Stokes method with advanced two equation turbulence models for high speed flows is presented. All convective terms are modeled using an entropy satisfying higher-order Total Variation Diminishing (TVD) scheme based on implicit upwind flux-difference split approximations and arithmetic averaging procedure of primitive variables. This method combines the best features of data management and computational efficiency of space marching procedures with the generality and stability of time dependent Navier-Stokes procedures to solve flows with mixed supersonic and subsonic zones, including streamwise separated flows. Its robust stability derives from a combination of conservative implicit upwind flux-difference splitting with Roe's property U to provide accurate shock capturing capability that non-conservative schemes do not guarantee, alternating symmetric Gauss-Seidel 'method of planes' relaxation procedure coupled with a three-dimensional two-factor diagonal-dominant approximate factorization scheme, TVD flux limiters of higher-order flux differences satisfying realizability, and well-posed characteristic-based implicit boundary-point a'pproximations consistent with the local characteristics domain of dependence. The efficiency of the method is highly increased with Newton Raphson acceleration which allows convergence in essentially one forward sweep for supersonic flows. The method is verified by comparing with experiment and other Navier-Stokes methods. Here, results of adiabatic and cooled flat plate flows, compression corner flow, and 3-D hypersonic shock-wave/turbulent boundary layer interaction flows are presented. The robust 3-D method achieves a better computational efficiency of at least one order of magnitude over the CNS Navier-Stokes code. It provides cost-effective aerodynamic predictions in agreement with experiment, and the capability of predicting complex flow structures in complex geometries with good accuracy.

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

NASA Astrophysics Data System (ADS)

Doha, Eid H.; Bhrawy, Ali H.; Abdelkawy, Mohammed A.

2014-09-01

In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

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

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the 2-D or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating-direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 3 is the Programmer's Reference, and describes the program structure, the FORTRAN variables stored in common blocks, and the details of each subprogram.

Constitutive equations for an electroactive polymer

NASA Astrophysics Data System (ADS)

Tixier, Mireille; Pouget, Joël

2016-07-01

Ionic electroactive polymers can be used as sensors or actuators. For this purpose, a thin film of polyelectrolyte is saturated with a solvent and sandwiched between two platinum electrodes. The solvent causes a complete dissociation of the polymer and the release of small cations. The application of an electric field across the thickness results in the bending of the strip and vice versa. The material is modeled by a two-phase continuous medium. The solid phase, constituted by the polymer backbone inlaid with anions, is depicted as a deformable porous media. The liquid phase is composed of the free cations and the solvent (usually water). We used a coarse grain model. The conservation laws of this system have been established in a previous work. The entropy balance law and the thermodynamic relations are first written for each phase and then for the complete material using a statistical average technique and the material derivative concept. One deduces the entropy production. Identifying generalized forces and fluxes provides the constitutive equations of the whole system: the stress-strain relations which satisfy a Kelvin-Voigt model, generalized Fourier's and Darcy's laws and the Nernst-Planck equation.

A minimum entropy principle in the gas dynamics equations

NASA Technical Reports Server (NTRS)

Tadmor, E.

1986-01-01

Let u(x bar,t) be a weak solution of the Euler equations, governing the inviscid polytropic gas dynamics; in addition, u(x bar, t) is assumed to respect the usual entropy conditions connected with the conservative Euler equations. We show that such entropy solutions of the gas dynamics equations satisfy a minimum entropy principle, namely, that the spatial minimum of their specific entropy, (Ess inf s(u(x,t)))/x, is an increasing function of time. This principle equally applies to discrete approximations of the Euler equations such as the Godunov-type and Lax-Friedrichs schemes. Our derivation of this minimum principle makes use of the fact that there is a family of generalized entrophy functions connected with the conservative Euler equations.

Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains

NASA Astrophysics Data System (ADS)

Przedborski, Michelle; Anco, Stephen C.

2017-09-01

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

Entropy-conservative spatial discretization of the multidimensional quasi-gasdynamic system of equations

NASA Astrophysics Data System (ADS)

Zlotnik, A. A.

2017-04-01

The multidimensional quasi-gasdynamic system written in the form of mass, momentum, and total energy balance equations for a perfect polytropic gas with allowance for a body force and a heat source is considered. A new conservative symmetric spatial discretization of these equations on a nonuniform rectangular grid is constructed (with the basic unknown functions—density, velocity, and temperature—defined on a common grid and with fluxes and viscous stresses defined on staggered grids). Primary attention is given to the analysis of entropy behavior: the discretization is specially constructed so that the total entropy does not decrease. This is achieved via a substantial revision of the standard discretization and applying numerous original features. A simplification of the constructed discretization serves as a conservative discretization with nondecreasing total entropy for the simpler quasi-hydrodynamic system of equations. In the absence of regularizing terms, the results also hold for the Navier-Stokes equations of a viscous compressible heat-conducting gas.

Restoration of the contact surface in FORCE-type centred schemes I: Homogeneous two-dimensional shallow water equations

NASA Astrophysics Data System (ADS)

Canestrelli, Alberto; Toro, Eleuterio F.

2012-10-01

Recently, the FORCE centred scheme for conservative hyperbolic multi-dimensional systems has been introduced in [34] and has been applied to Euler and relativistic MHD equations, solved on unstructured meshes. In this work we propose a modification of the FORCE scheme, named FORCE-Contact, that provides improved resolution of contact and shear waves. This paper presents the technique in full detail as applied to the two-dimensional homogeneous shallow water equations. The improvements due to the new method are particularly evident when an additional equation is solved for a tracer, since the modified scheme exactly resolves isolated and steady contact discontinuities. The improvement is considerable also for slowly moving contact discontinuities, for shear waves and for steady states in meandering channels. For these types of flow fields, the numerical results provided by the new FORCE-Contact scheme are comparable with, and sometimes better than, the results obtained from upwind schemes, such as Roes scheme for example. In a companion paper, a similar approach to restoring the missing contact wave and preserving well-balanced properties for non-conservative one- and two-layer shallow water equations is introduced. However, the procedure is general and it is in principle applicable to other multidimensional hyperbolic systems in conservative and non-conservative form, such as the Euler equations for compressible gas dynamics.

A kinetic equation with kinetic entropy functions for scalar conservation laws

NASA Technical Reports Server (NTRS)

Perthame, Benoit; Tadmor, Eitan

1990-01-01

A nonlinear kinetic equation is constructed and proved to be well-adapted to describe general multidimensional scalar conservation laws. In particular, it is proved to be well-posed uniformly in epsilon - the microscopic scale. It is also shown that the proposed kinetic equation is equipped with a family of kinetic entropy functions - analogous to Boltzmann's microscopic H-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, it is proved by both - BV compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a continuum limit, as it converges strongly with epsilon below 0 to the unique entropy solution of the corresponding conservation law.

Scale-Invariant Forms of Conservation Equations in Reactive Fields and a Modified Hydro-Thermo-Diffusive Theory of Laminar Flames

NASA Technical Reports Server (NTRS)

Sohrab, Siavash H.; Piltch, Nancy (Technical Monitor)

2000-01-01

A scale-invariant model of statistical mechanics is applied to present invariant forms of mass, energy, linear, and angular momentum conservation equations in reactive fields. The resulting conservation equations at molecular-dynamic scale are solved by the method of large activation energy asymptotics to describe the hydro-thermo-diffusive structure of laminar premixed flames. The predicted temperature and velocity profiles are in agreement with the observations. Also, with realistic physico-chemical properties and chemical-kinetic parameters for a single-step overall combustion of stoichiometric methane-air premixed flame, the laminar flame propagation velocity of 42.1 cm/s is calculated in agreement with the experimental value.

Turbulent fluid motion IV-averages, Reynolds decomposition, and the closure problem

NASA Technical Reports Server (NTRS)

Deissler, Robert G.

1992-01-01

Ensemble, time, and space averages as applied to turbulent quantities are discussed, and pertinent properties of the averages are obtained. Those properties, together with Reynolds decomposition, are used to derive the averaged equations of motion and the one- and two-point moment or correlation equations. The terms in the various equations are interpreted. The closure problem of the averaged equations is discussed, and possible closure schemes are considered. Those schemes usually require an input of supplemental information unless the averaged equations are closed by calculating their terms by a numerical solution of the original unaveraged equations. The law of the wall for velocities and temperatures, the velocity- and temperature-defect laws, and the logarithmic laws for velocities and temperatures are derived. Various notions of randomness and their relation to turbulence are considered in light of ergodic theory.

Application of discontinuous Galerkin method for solving a compressible five-equation two-phase flow model

NASA Astrophysics Data System (ADS)

Saleem, M. Rehan; Ali, Ishtiaq; Qamar, Shamsul

2018-03-01

In this article, a reduced five-equation two-phase flow model is numerically investigated. The formulation of the model is based on the conservation and energy exchange laws. The model is non-conservative and the governing equations contain two equations for the mass conservation, one for the over all momentum and one for the total energy. The fifth equation is the energy equation for one of the two phases that includes a source term on the right hand side for incorporating energy exchange between the two fluids in the form of mechanical and thermodynamical works. A Runge-Kutta discontinuous Galerkin finite element method is applied to solve the model equations. The main attractive features of the proposed method include its formal higher order accuracy, its nonlinear stability, its ability to handle complicated geometries, and its ability to capture sharp discontinuities or strong gradients in the solutions without producing spurious oscillations. The proposed method is robust and well suited for large-scale time-dependent computational problems. Several case studies of two-phase flows are presented. For validation and comparison of the results, the same model equations are also solved by using a staggered central scheme. It was found that discontinuous Galerkin scheme produces better results as compared to the staggered central scheme.

Generalized continuity equations from two-field Schrödinger Lagrangians

NASA Astrophysics Data System (ADS)

Spourdalakis, A. G. B.; Pappas, G.; Morfonios, C. Â. V.; Kalozoumis, P. A.; Diakonos, F. K.; Schmelcher, P.

2016-11-01

A variational scheme for the derivation of generalized, symmetry-induced continuity equations for Hermitian and non-Hermitian quantum mechanical systems is developed. We introduce a Lagrangian which involves two complex wave fields and whose global invariance under dilation and phase variations leads to a mixed continuity equation for the two fields. In combination with discrete spatial symmetries of the underlying Hamiltonian, the mixed continuity equation is shown to produce bilocal conservation laws for a single field. This leads to generalized conserved charges for vanishing boundary currents and to divergenceless bilocal currents for stationary states. The formalism reproduces the bilocal continuity equation obtained in the special case of P T -symmetric quantum mechanics and paraxial optics.

Students' Understanding of Conservation of Matter, Stoichiometry and Balancing Equations in Indonesia

ERIC Educational Resources Information Center

Agung, Salamah; Schwartz, Marc S.

2007-01-01

This study examines Indonesian students' understanding of conservation of matter, balancing of equations and stoichiometry. Eight hundred and sixty-seven Grade 12 students from 22 schools across four different cities in two developed provinces in Indonesia participated in the study. Nineteen teachers also participated in order to validate the…

Spin-Orbit Coupling and the Conservation of Angular Momentum

ERIC Educational Resources Information Center

Hnizdo, V.

2012-01-01

In nonrelativistic quantum mechanics, the total (i.e. orbital plus spin) angular momentum of a charged particle with spin that moves in a Coulomb plus spin-orbit-coupling potential is conserved. In a classical nonrelativistic treatment of this problem, in which the Lagrange equations determine the orbital motion and the Thomas equation yields the…

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Lafitte, Pauline; Melis, Ward; Samaey, Giovanni

2017-07-01

We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff BGK source term. Then, this kinetic equation is integrated in time using a projective integration method. After taking a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time step restriction on the outer time step is similar to the CFL condition for the hyperbolic conservation law. Moreover, the number of inner time steps is also independent of the stiffness of the BGK source term. We discuss stability and consistency, and illustrate with numerical results (linear advection, Burgers' equation and the shallow water and Euler equations) in one and two spatial dimensions.

An energy and potential enstrophy conserving scheme for the shallow water equations. [orography effects on atmospheric circulation

NASA Technical Reports Server (NTRS)

Arakawa, A.; Lamb, V. R.

1979-01-01

A three-dimensional finite difference scheme for the solution of the shallow water momentum equations which accounts for the conservation of potential enstrophy in the flow of a homogeneous incompressible shallow atmosphere over steep topography as well as for total energy conservation is presented. The scheme is derived to be consistent with a reasonable scheme for potential vorticity advection in a long-term integration for a general flow with divergent mass flux. Numerical comparisons of the characteristics of the present potential enstrophy-conserving scheme with those of a scheme that conserves potential enstrophy only for purely horizontal nondivergent flow are presented which demonstrate the reduction of computational noise in the wind field with the enstrophy-conserving scheme and its convergence even in relatively coarse grids.

Agriculture’s Soil Conservation Programs Miss Full Potential in the Fight against Soil Erosion.

DTIC Science & Technology

1983-11-28

Soil Loss Equation ( USLE ) and Wind Erosion Equation can be used with a reasonable degree of accuracy. It is the intention of ASCS to expand VC/SL to...HD-R37 495 AGRICULTURE’S SOIL CONSERVATION PROGRAMS MISS FULL i/i POTENTIAL IN THE FIGHT.(U) GENERAL ACCOUNTING OFFICE WASHINGTON DC RESOURCES...GENERAL Report To The Congress OF THE UNITED STATES Agriculture’s Soil Conservation Programs Miss Full Potential In The Fight Against Soil Erosion

Second- and third-order upwind difference schemes for hyperbolic conservation laws

NASA Technical Reports Server (NTRS)

Yang, J. Y.

1984-01-01

Second- and third-order two time-level five-point explicit upwind-difference schemes are described for the numerical solution of hyperbolic systems of conservation laws and applied to the Euler equations of inviscid gas dynamics. Nonliner smoothing techniques are used to make the schemes total variation diminishing. In the method both hyperbolicity and conservation properties of the hyperbolic conservation laws are combined in a very natural way by introducing a normalized Jacobian matrix of the hyperbolic system. Entropy satisfying shock transition operators which are consistent with the upwind differencing are locally introduced when transonic shock transition is detected. Schemes thus constructed are suitable for shockcapturing calculations. The stability and the global order of accuracy of the proposed schemes are examined. Numerical experiments for the inviscid Burgers equation and the compressible Euler equations in one and two space dimensions involving various situations of aerodynamic interest are included and compared.

A new flux conserving Newton's method scheme for the two-dimensional, steady Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.; Chang, Sin-Chung

1993-01-01

A new numerical method is developed for the solution of the two-dimensional, steady Navier-Stokes equations. The method that is presented differs in significant ways from the established numerical methods for solving the Navier-Stokes equations. The major differences are described. First, the focus of the present method is on satisfying flux conservation in an integral formulation, rather than on simulating conservation laws in their differential form. Second, the present approach provides a unified treatment of the dependent variables and their unknown derivatives. All are treated as unknowns together to be solved for through simulating local and global flux conservation. Third, fluxes are balanced at cell interfaces without the use of interpolation or flux limiters. Fourth, flux conservation is achieved through the use of discrete regions known as conservation elements and solution elements. These elements are not the same as the standard control volumes used in the finite volume method. Fifth, the discrete approximation obtained on each solution element is a functional solution of both the integral and differential form of the Navier-Stokes equations. Finally, the method that is presented is a highly localized approach in which the coupling to nearby cells is only in one direction for each spatial coordinate, and involves only the immediately adjacent cells. A general third-order formulation for the steady, compressible Navier-Stokes equations is presented, and then a Newton's method scheme is developed for the solution of incompressible, low Reynolds number channel flow. It is shown that the Jacobian matrix is nearly block diagonal if the nonlinear system of discrete equations is arranged approximately and a proper pivoting strategy is used. Numerical results are presented for Reynolds numbers of 100, 1000, and 2000. Finally, it is shown that the present scheme can resolve the developing channel flow boundary layer using as few as six to ten cells per channel width, depending on the Reynolds number.

From Feynman rules to conserved quantum numbers, I

NASA Astrophysics Data System (ADS)

Nogueira, P.

2017-05-01

In the context of Quantum Field Theory (QFT) there is often the need to find sets of graph-like diagrams (the so-called Feynman diagrams) for a given physical model. If negative, the answer to the related problem 'Are there any diagrams with this set of external fields?' may settle certain physical questions at once. Here the latter problem is formulated in terms of a system of linear diophantine equations derived from the Lagrangian density, from which necessary conditions for the existence of the required diagrams may be obtained. Those conditions are equalities that look like either linear diophantine equations or linear modular (i.e. congruence) equations, and may be found by means of fairly simple algorithms that involve integer computations. The diophantine equations so obtained represent (particle) number conservation rules, and are related to the conserved (additive) quantum numbers that may be assigned to the fields of the model.

Ion-Conserving Modified Poisson-Boltzmann Theory Considering a Steric Effect in an Electrolyte

NASA Astrophysics Data System (ADS)

Sugioka, Hideyuki

2016-12-01

The modified Poisson-Nernst-Planck (MPNP) and modified Poisson-Boltzmann (MPB) equations are well known as fundamental equations that consider a steric effect, which prevents unphysical ion concentrations. However, it is unclear whether they are equivalent or not. To clarify this problem, we propose an improved free energy formulation that considers a steric limit with an ion-conserving condition and successfully derive the ion-conserving modified Poisson-Boltzmann (IC-MPB) equations that are equivalent to the MPNP equations. Furthermore, we numerically examine the equivalence by comparing between the IC-MPB solutions obtained by the Newton method and the steady MPNP solutions obtained by the finite-element finite-volume method. A surprising aspect of our finding is that the MPB solutions are much different from the MPNP (IC-MPB) solutions in a confined space. We consider that our findings will significantly contribute to understanding the surface science between solids and liquids.

Numerical Modeling of Flow Distribution in Micro-Fluidics Systems

NASA Technical Reports Server (NTRS)

Majumdar, Alok; Cole, Helen; Chen, C. P.

2005-01-01

This paper describes an application of a general purpose computer program, GFSSP (Generalized Fluid System Simulation Program) for calculating flow distribution in a network of micro-channels. GFSSP employs a finite volume formulation of mass and momentum conservation equations in a network consisting of nodes and branches. Mass conservation equation is solved for pressures at the nodes while the momentum conservation equation is solved at the branches to calculate flowrate. The system of equations describing the fluid network is solved by a numerical method that is a combination of the Newton-Raphson and successive substitution methods. The numerical results have been compared with test data and detailed CFD (computational Fluid Dynamics) calculations. The agreement between test data and predictions is satisfactory. The discrepancies between the predictions and test data can be attributed to the frictional correlation which does not include the effect of surface tension or electro-kinetic effect.

Governing equations for electro-conjugate fluid flow

NASA Astrophysics Data System (ADS)

Hosoda, K.; Takemura, K.; Fukagata, K.; Yokota, S.; Edamura, K.

2013-12-01

An electro-conjugation fluid (ECF) is a kind of dielectric liquid, which generates a powerful flow when high DC voltage is applied with tiny electrodes. This study deals with the derivation of the governing equations for electro-conjugate fluid flow based on the Korteweg-Helmholtz (KH) equation which represents the force in dielectric liquid subjected to high DC voltage. The governing equations consist of the Gauss's law, charge conservation with charge recombination, the KH equation, the continuity equation and the incompressible Navier-Stokes equations. The KH equation consists of coulomb force, dielectric constant gradient force and electrostriction force. The governing equation gives the distribution of electric field, charge density and flow velocity. In this study, direct numerical simulation (DNS) is used in order to get these distribution at arbitrary time. Successive over-relaxation (SOR) method is used in analyzing Gauss's law and constrained interpolation pseudo-particle (CIP) method is used in analyzing charge conservation with charge recombination. The third order Runge-Kutta method and conservative second-order-accurate finite difference method is used in analyzing the Navier-Stokes equations with the KH equation. This study also deals with the measurement of ECF ow generated with a symmetrical pole electrodes pair which are made of 0.3 mm diameter piano wire. Working fluid is FF-1EHA2 which is an ECF family. The flow is observed from the both electrodes, i.e., the flow collides in between the electrodes. The governing equation successfully calculates mean flow velocity in between the collector pole electrode and the colliding region by the numerical simulation.

A conservative implicit finite difference algorithm for the unsteady transonic full potential equation

NASA Technical Reports Server (NTRS)

Steger, J. L.; Caradonna, F. X.

1980-01-01

An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form. Computational efficiency is maintained by use of approximate factorization techniques. The numerical algorithm is first order in time and second order in space. A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm.

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

NASA Astrophysics Data System (ADS)

Motsepa, Tanki; Aziz, Taha; Fatima, Aeeman; Khalique, Chaudry Masood

2018-03-01

The optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

Dissipation-preserving spectral element method for damped seismic wave equations

NASA Astrophysics Data System (ADS)

Cai, Wenjun; Zhang, Huai; Wang, Yushun

2017-12-01

This article describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method. The conformal symplectic method is a variation of conventional symplectic methods to treat non-conservative time evolution problems, which has superior behaviors in long-time stability and dissipation preservation. To reveal the intrinsic dissipative properties of the model equations, we first reformulate the original systems in their equivalent conformal multi-symplectic structures and derive the corresponding conformal symplectic conservation laws. We thereafter separate each system into a conservative Hamiltonian system and a purely dissipative ordinary differential equation system. Based on the splitting methodology, we solve the two subsystems respectively. The dissipative one is cheaply solved by its analytic solution. While for the conservative system, we combine a fourth-order symplectic Nyström method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free-surface topography. The Strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed conformal symplectic method. A relative larger Courant number than that of the traditional Newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength. A benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate. The classical Lamb problem is used to demonstrate the ability of modeling Rayleigh wave in elastic wave propagation. More comprehensive numerical experiments are presented to investigate the long-time simulation, low dispersion and energy conservation properties of the conformal symplectic methods in both the attenuating homogeneous and heterogeneous media.

Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures

NASA Technical Reports Server (NTRS)

Scott, R. K.

1972-01-01

A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.

Multidimensional, fully implicit, exactly conserving electromagnetic particle-in-cell simulations

NASA Astrophysics Data System (ADS)

Chacon, Luis

2015-09-01

We discuss a new, conservative, fully implicit 2D-3V particle-in-cell algorithm for non-radiative, electromagnetic kinetic plasma simulations, based on the Vlasov-Darwin model. Unlike earlier linearly implicit PIC schemes and standard explicit PIC schemes, fully implicit PIC algorithms are unconditionally stable and allow exact discrete energy and charge conservation. This has been demonstrated in 1D electrostatic and electromagnetic contexts. In this study, we build on these recent algorithms to develop an implicit, orbit-averaged, time-space-centered finite difference scheme for the Darwin field and particle orbit equations for multiple species in multiple dimensions. The Vlasov-Darwin model is very attractive for PIC simulations because it avoids radiative noise issues in non-radiative electromagnetic regimes. The algorithm conserves global energy, local charge, and particle canonical-momentum exactly, even with grid packing. The nonlinear iteration is effectively accelerated with a fluid preconditioner, which allows efficient use of large timesteps, O(√{mi/me}c/veT) larger than the explicit CFL. In this presentation, we will introduce the main algorithmic components of the approach, and demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 1D and 2D. Support from the LANL LDRD program and the DOE-SC ASCR office.

Ensemble Averaged Probability Density Function (APDF) for Compressible Turbulent Reacting Flows

NASA Technical Reports Server (NTRS)

Shih, Tsan-Hsing; Liu, Nan-Suey

2012-01-01

In this paper, we present a concept of the averaged probability density function (APDF) for studying compressible turbulent reacting flows. The APDF is defined as an ensemble average of the fine grained probability density function (FG-PDF) with a mass density weighting. It can be used to exactly deduce the mass density weighted, ensemble averaged turbulent mean variables. The transport equation for APDF can be derived in two ways. One is the traditional way that starts from the transport equation of FG-PDF, in which the compressible Navier- Stokes equations are embedded. The resulting transport equation of APDF is then in a traditional form that contains conditional means of all terms from the right hand side of the Navier-Stokes equations except for the chemical reaction term. These conditional means are new unknown quantities that need to be modeled. Another way of deriving the transport equation of APDF is to start directly from the ensemble averaged Navier-Stokes equations. The resulting transport equation of APDF derived from this approach appears in a closed form without any need for additional modeling. The methodology of ensemble averaging presented in this paper can be extended to other averaging procedures: for example, the Reynolds time averaging for statistically steady flow and the Reynolds spatial averaging for statistically homogeneous flow. It can also be extended to a time or spatial filtering procedure to construct the filtered density function (FDF) for the large eddy simulation (LES) of compressible turbulent reacting flows.

A (2+1)-dimensional Korteweg-de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws

NASA Astrophysics Data System (ADS)

Adem, Abdullahi Rashid

2016-05-01

We consider a (2+1)-dimensional Korteweg-de Vries type equation which models the shallow-water waves, surface and internal waves. In the analysis, we use the Lie symmetry method and the multiple exp-function method. Furthermore, conservation laws are computed using the multiplier method.

The P1-RKDG method for two-dimensional Euler equations of gas dynamics

NASA Technical Reports Server (NTRS)

Cockburn, Bernardo; Shu, Chi-Wang

1991-01-01

A class of nonlinearly stable Runge-Kutta local projection discontinuous Galerkin (RKDG) finite element methods for conservation laws is investigated. Two dimensional Euler equations for gas dynamics are solved using P1 elements. The generalization of the local projections, which for scalar nonlinear conservation laws was designed to satisfy a local maximum principle, to systems of conservation laws such as the Euler equations of gas dynamics using local characteristic decompositions is discussed. Numerical examples include the standard regular shock reflection problem, the forward facing step problem, and the double Mach reflection problem. These preliminary numerical examples are chosen to show the capacity of the approach to obtain nonlinearly stable results comparable with the modern nonoscillatory finite difference methods.

Flux-vector splitting algorithm for chain-rule conservation-law form

NASA Technical Reports Server (NTRS)

Shih, T. I.-P.; Nguyen, H. L.; Willis, E. A.; Steinthorsson, E.; Li, Z.

1991-01-01

A flux-vector splitting algorithm with Newton-Raphson iteration was developed for the 'full compressible' Navier-Stokes equations cast in chain-rule conservation-law form. The algorithm is intended for problems with deforming spatial domains and for problems whose governing equations cannot be cast in strong conservation-law form. The usefulness of the algorithm for such problems was demonstrated by applying it to analyze the unsteady, two- and three-dimensional flows inside one combustion chamber of a Wankel engine under nonfiring conditions. Solutions were obtained to examine the algorithm in terms of conservation error, robustness, and ability to handle complex flows on time-dependent grid systems.

Evading the non-continuity equation in the f( R, T) cosmology

NASA Astrophysics Data System (ADS)

Moraes, P. H. R. S.; Correa, R. A. C.; Ribeiro, G.

2018-03-01

We present a new approach for the f( R, T) gravity formalism, by thoroughly exploring the extra terms of its effective energy-momentum tensor T_{μ ν }^eff, which we name \\tilde{T}_{μ ν }, so that T_{μ ν }^eff=T_{μ ν }+\\tilde{T}_{μ ν }, with T_{μ ν } being the usual energy-momentum tensor of matter. Purely from the Bianchi identities, we obtain the conservation of both parts of the effective energy-momentum tensor, rather than the non-conservation of T_{μ ν }, originally occurring in the f( R, T) theories. In this way, the intriguing scenario of matter creation, which still lacks observational evidence, is evaded. One is left, then, with two sets of cosmological equations to be solved: the Friedmann-like equations along with the conservation of T_{μ ν } and along with the conservation of \\tilde{T}_{μ ν }. We present a physical interpretation for the conservation of \\tilde{T}_{μ ν }, which can be related to the presence of stiff matter in the universe. The cosmological consequences of this approach are presented and discussed as well as the benefits of evading the matter energy-momentum tensor non-conservation.

Numerical Investigation of a Model Scramjet Combustor Using DDES

NASA Astrophysics Data System (ADS)

Shin, Junsu; Sung, Hong-Gye

2017-04-01

Non-reactive flows moving through a model scramjet were investigated using a delayed detached eddy simulation (DDES), which is a hybrid scheme combining Reynolds averaged Navier-Stokes scheme and a large eddy simulation. The three dimensional Navier-Stokes equations were solved numerically on a structural grid using finite volume methods. An in-house was developed. This code used a monotonic upstream-centered scheme for conservation laws (MUSCL) with an advection upstream splitting method by pressure weight function (AUSMPW+) for space. In addition, a 4th order Runge-Kutta scheme was used with preconditioning for time integration. The geometries and boundary conditions of a scramjet combustor operated by DLR, a German aerospace center, were considered. The profiles of the lower wall pressure and axial velocity obtained from a time-averaged solution were compared with experimental results. Also, the mixing efficiency and total pressure recovery factor were provided in order to inspect the performance of the combustor.

Comparison of Fully-Compressible Equation Sets for Atmospheric Dynamics

NASA Technical Reports Server (NTRS)

Ahmad, Nashat N.

2016-01-01

Traditionally, the equation for the conservation of energy used in atmospheric models is based on potential temperature and is used in place of the total energy conservation. This paper compares the application of the two equations sets for both the Euler and the Navier-Stokes solutions using several benchmark test cases. A high-resolution wave-propagation method which accurately takes into account the source term due to gravity is used for computing the non-hydrostatic atmospheric flows. It is demonstrated that there is little to no difference between the results obtained using the two different equation sets for Euler as well as Navier-Stokes solutions.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field

NASA Astrophysics Data System (ADS)

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-01

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

On buffer layers as non-reflecting computational boundaries

NASA Technical Reports Server (NTRS)

Hayder, M. Ehtesham; Turkel, Eli L.

1996-01-01

We examine an absorbing buffer layer technique for use as a non-reflecting boundary condition in the numerical simulation of flows. One such formulation was by Ta'asan and Nark for the linearized Euler equations. They modified the flow inside the buffer zone to artificially make it supersonic in the layer. We examine how this approach can be extended to the nonlinear Euler equations. We consider both a conservative and a non-conservative form modifying the governing equations in the buffer layer. We compare this with the case that the governing equations in the layer are the same as in the interior domain. We test the effectiveness of these buffer layers by a simulation of an excited axisymmetric jet based on a nonlinear compressible Navier-Stokes equations.

Fokker-Planck equation for the non-Markovian Brownian motion in the presence of a magnetic field.

PubMed

Das, Joydip; Mondal, Shrabani; Bag, Bidhan Chandra

2017-10-28

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

Conservative DEC Discretization of Incompressible Navier-Stokes Equations on Arbitrary Surface Simplicial Meshes

NASA Astrophysics Data System (ADS)

Mohamed, Mamdouh; Hirani, Anil; Samtaney, Ravi

2017-11-01

A conservative discretization of incompressible Navier-Stokes equations over surfaces is developed using discrete exterior calculus (DEC). The mimetic character of many of the DEC operators provides exact conservation of both mass and vorticity, in addition to superior kinetic energy conservation. The employment of signed diagonal Hodge star operators, while using the circumcentric dual defined on arbitrary meshes, is shown to produce correct solutions even when many non-Delaunay triangles pairs exist. This allows the DEC discretization to admit arbitrary surface simplicial meshes, in contrast to the previously held notion that DEC was limited only to Delaunay meshes. The discretization scheme is presented along with several numerical test cases demonstrating its numerical convergence and conservation properties. Recent developments regarding the extension to conservative higher order methods are also presented. KAUST Baseline Research Funds of R. Samtaney.

Pressure and Chemical Potential: Effects Hydrophilic Soils Have on Adsorption and Transport

NASA Astrophysics Data System (ADS)

Bennethum, L. S.; Weinstein, T.

2003-12-01

Using the assumption that thermodynamic properties of fluid is affected by its proximity to the solid phase, a theoretical model has been developed based on upscaling and fundamental thermodynamic principles (termed Hybrid Mixture Theory). The theory indicates that Darcy's law and the Darcy-scale chemical potential (which determines the rate of adsorption and diffusion) need to be modified in order to apply to soils containing hydrophilic soils. In this talk we examine the Darcy-scale definition of pressure and chemical potential, especially as it applies to hydrophilic soils. To arrive at our model, we used hybrid mixture theory - first pioneered by Hassanizadeh and Gray in 1979. The technique involves averaging the field equations (i.e. conservation of mass, momentum balance, energy balance, etc.) to obtain macroscopic field equations, where each field variable is defined precisely in terms of its microscale counterpart. To close the system consistently with classical thermodynamics, the entropy inequality is exploited in the sense of Coleman and Noll. With the exceptions that the macroscale field variables are defined precisely in terms of their microscale counterparts and that microscopic interfacial equations can also be treated in a similar manner, the resulting system of equations is consistent with those derived using classical mixture theory. Hence the terminology, Hybrid Mixture Theory.

A three-dimensional meso-macroscopic model for Li-Ion intercalation batteries

DOE PAGES

Allu, S.; Kalnaus, S.; Simunovic, S.; ...

2016-06-09

Through this study, we present a three-dimensional computational formulation for electrode-electrolyte-electrode system of Li-Ion batteries. The physical consistency between electrical, thermal and chemical equations is enforced at each time increment by driving the residual of the resulting coupled system of nonlinear equations to zero. The formulation utilizes a rigorous volume averaging approach typical of multiphase formulations used in other fields and recently extended to modeling of supercapacitors [1]. Unlike existing battery modeling methods which use segregated solution of conservation equations and idealized geometries, our unified approach can model arbitrary battery and electrode configurations. The consistency of multi-physics solution also allowsmore » for consideration of a wide array of initial conditions and load cases. The formulation accounts for spatio-temporal variations of material and state properties such as electrode/void volume fractions and anisotropic conductivities. The governing differential equations are discretized using the finite element method and solved using a nonlinearly consistent approach that provides robust stability and convergence. The new formulation was validated for standard Li-ion cells and compared against experiments. Finally, its scope and ability to capture spatio-temporal variations of potential and lithium distribution is demonstrated on a prototypical three-dimensional electrode problem.« less

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

NASA Astrophysics Data System (ADS)

Gao, Peng

2018-06-01

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.

Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

NASA Astrophysics Data System (ADS)

Gao, Peng

2018-04-01

This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.

Two-level schemes for the advection equation

NASA Astrophysics Data System (ADS)

Vabishchevich, Petr N.

2018-06-01

The advection equation is the basis for mathematical models of continuum mechanics. In the approximate solution of nonstationary problems it is necessary to inherit main properties of the conservatism and monotonicity of the solution. In this paper, the advection equation is written in the symmetric form, where the advection operator is the half-sum of advection operators in conservative (divergent) and non-conservative (characteristic) forms. The advection operator is skew-symmetric. Standard finite element approximations in space are used. The standard explicit two-level scheme for the advection equation is absolutely unstable. New conditionally stable regularized schemes are constructed, on the basis of the general theory of stability (well-posedness) of operator-difference schemes, the stability conditions of the explicit Lax-Wendroff scheme are established. Unconditionally stable and conservative schemes are implicit schemes of the second (Crank-Nicolson scheme) and fourth order. The conditionally stable implicit Lax-Wendroff scheme is constructed. The accuracy of the investigated explicit and implicit two-level schemes for an approximate solution of the advection equation is illustrated by the numerical results of a model two-dimensional problem.

Two-Layer Viscous Shallow-Water Equations and Conservation Laws

NASA Astrophysics Data System (ADS)

Kanayama, Hiroshi; Dan, Hiroshi

In our previous papers, the two-layer viscous shallow-water equations were derived from the three-dimensional Navier-Stokes equations under the hydrostatic assumption. Also, it was noted that the combination of upper and lower equations in the two-layer model produces the classical one-layer equations if the density of each layer is the same. Then, the two-layer equations were approximated by a finite element method which followed our numerical scheme established for the one-layer model in 1978. Also, it was numerically demonstrated that the interfacial instability generated when the densities are the same can be eliminated by providing a sufficient density difference. In this paper, we newly show that conservation laws are still valid in the two-layer model. Also, we show results of a new physical experiment for the interfacial instability.

A discrete model on Sierpinski gasket substrate for a conserved current equation with a conservative noise

NASA Astrophysics Data System (ADS)

Kim, Dae Ho; Kim, Jin Min

2012-09-01

A conserved discrete model on the Sierpinski gasket substrate is studied. The interface width W in the model follows the Family-Vicsek dynamic scaling form with growth exponent β ≈ 0.0542, roughness exponent α ≈ 0.240 and dynamic exponent z ≈ 4.42. They satisfy a scaling relation α + z = 2zrw, where zrw is the random walk exponent of the fractal substrate. Also, they are in a good agreement with the predicted values of a fractional Langevin equation \\frac{\\partial h}{\\partial t}={\

L1-Based Approximations of PDEs and Applications

DTIC Science & Technology

2012-09-05

the analysis of the Navier-Stokes equations. The early versions of artificial vis- cosities being overly dissipative, the interest for these technique ...Guermond, and B. Popov. Stability analysis of explicit en- tropy viscosity methods for non-linear scalar conservation equations. Math. Comp., 2012... methods for solv- ing mathematical models of nonlinear phenomena such as nonlinear conservation laws, surface/image/data reconstruction problems

Transient Performance of a Vertical Axis Wind Turbine

NASA Astrophysics Data System (ADS)

Onol, Aykut; Yesilyurt, Serhat

2016-11-01

A coupled CFD/rotor dynamics modeling approach is presented for the analysis of realistic transient behavior of a height-normalized, three-straight-bladed VAWT subject to inertial effects of the rotor and generator load which is manipulated by a feedback control under standardized wind gusts. The model employs the k- ɛ turbulence model to approximate unsteady Reynolds-averaged Navier-Stokes equations and is validated with data from field measurements. As distinct from related studies, here, the angular velocity is calculated from the rotor's equation of motion; thus, the dynamic response of the rotor is taken into account. Results include the following: First, the rotor's inertia filters large amplitude oscillations in the wind torque owing to the first-order dynamics. Second, the generator and wind torques differ especially during wind transients subject to the conservation of angular momentum of the rotor. Third, oscillations of the power coefficient exceed the Betz limit temporarily due to the energy storage in the rotor, which acts as a temporary buffer that stores the kinetic energy like a flywheel in short durations. Last, average of transient power coefficients peaks at a smaller tip-speed ratio for wind gusts than steady winds. This work was supported by the Sabanci University Internal Research Grant Program (SU-IRG-985).

Non-dispersive conservative regularisation of nonlinear shallow water (and isentropic Euler equations)

NASA Astrophysics Data System (ADS)

Clamond, Didier; Dutykh, Denys

2018-02-01

A new regularisation of the shallow water (and isentropic Euler) equations is proposed. The regularised equations are non-dissipative, non-dispersive and posses a variational structure; thus, the mass, the momentum and the energy are conserved. Hence, for instance, regularised hydraulic jumps are smooth and non-oscillatory. Another particularly interesting feature of this regularisation is that smoothed 'shocks' propagates at exactly the same speed as the original discontinuous ones. The performance of the new model is illustrated numerically on some dam-break test cases, which are classical in the hyperbolic realm.

Averaging problem in general relativity, macroscopic gravity and using Einstein's equations in cosmology.

NASA Astrophysics Data System (ADS)

Zalaletdinov, R. M.

1998-04-01

The averaging problem in general relativity is briefly discussed. A new setting of the problem as that of macroscopic description of gravitation is proposed. A covariant space-time averaging procedure is described. The structure of the geometry of macroscopic space-time, which follows from averaging Cartan's structure equations, is described and the correlation tensors present in the theory are discussed. The macroscopic field equations (averaged Einstein's equations) derived in the framework of the approach are presented and their structure is analysed. The correspondence principle for macroscopic gravity is formulated and a definition of the stress-energy tensor for the macroscopic gravitational field is proposed. It is shown that the physical meaning of using Einstein's equations with a hydrodynamic stress-energy tensor in looking for cosmological models means neglecting all gravitational field correlations. The system of macroscopic gravity equations to be solved when the correlations are taken into consideration is given and described.

Multi-Hamiltonian structure of the Born-Infeld equation

NASA Astrophysics Data System (ADS)

Arik, Metin; Neyzi, Fahrünisa; Nutku, Yavuz; Olver, Peter J.; Verosky, John M.

1989-06-01

The multi-Hamiltonian structure, conservation laws, and higher order symmetries for the Born-Infeld equation are exhibited. A new transformation of the Born-Infeld equation to the equations of a Chaplygin gas is presented and explored. The Born-Infeld equation is distinguished among two-dimensional hyperbolic systems by its wealth of such multi-Hamiltonian structures.

Frequency modulation at a moving material interface and a conservation law for wave number. [acoustic wave reflection and transmission

NASA Technical Reports Server (NTRS)

Kleinstein, G. G.; Gunzburger, M. D.

1976-01-01

An integral conservation law for wave numbers is considered. In order to test the validity of the proposed conservation law, a complete solution for the reflection and transmission of an acoustic wave impinging normally on a material interface moving at a constant speed is derived. The agreement between the frequency condition thus deduced from the dynamic equations of motion and the frequency condition derived from the jump condition associated with the integral equation supports the proposed law as a true conservation law. Additional comparisons such as amplitude discontinuities and Snells' law in a moving media further confirm the stated proposition. Results are stated concerning frequency and wave number relations across a shock front as predicted by the proposed conservation law.

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

NASA Astrophysics Data System (ADS)

Zhang, L.; Tang, G.; Xun, Z.; Han, K.; Chen, H.; Hu, B.

2008-05-01

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.

An Asymptotic and Stochastic Theory for the Effects of Surface Gravity Waves on Currents and Infragravity Waves

NASA Astrophysics Data System (ADS)

McWilliams, J. C.; Lane, E.; Melville, K.; Restrepo, J.; Sullivan, P.

2004-12-01

Oceanic surface gravity waves are approximately irrotational, weakly nonlinear, and conservative, and they have a much shorter time scale than oceanic currents and longer waves (e.g., infragravity waves) --- except where the primary surface waves break. This provides a framework for an asymptotic theory, based on separation of time (and space) scales, of wave-averaged effects associated with the conservative primary wave dynamics combined with a stochastic representation of the momentum transfer and induced mixing associated with non-conservative wave breaking. Such a theory requires only modest information about the primary wave field from measurements or operational model forecasts and thus avoids the enormous burden of calculating the waves on their intrinsically small space and time scales. For the conservative effects, the result is a vortex force associated with the primary wave's Stokes drift; a wave-averaged Bernoulli head and sea-level set-up; and an incremental material advection by the Stokes drift. This can be compared to the "radiation stress" formalism of Longuet-Higgins, Stewart, and Hasselmann; it is shown to be a preferable representation since the radiation stress is trivial at its apparent leading order. For the non-conservative breaking effects, a population of stochastic impulses is added to the current and infragravity momentum equations with distribution functions taken from measurements. In offshore wind-wave equilibria, these impulses replace the conventional surface wind stress and cause significant differences in the surface boundary layer currents and entrainment rate, particularly when acting in combination with the conservative vortex force. In the surf zone, where breaking associated with shoaling removes nearly all of the primary wave momentum and energy, the stochastic forcing plays an analogous role as the widely used nearshore radiation stress parameterizations. This talk describes the theoretical framework and presents some preliminary solutions using it. McWilliams, J.C., J.M. Restrepo, & E.M. Lane, 2004: An asymptotic theory for the interaction of waves and currents in coastal waters. J. Fluid Mech. 511, 135-178. Sullivan, P.P., J.C. McWilliams, & W.K. Melville, 2004: The oceanic boundary layer driven by wave breaking with stochastic variability. J. Fluid Mech. 507, 143-174.

Determination of gas & liquid two-phase flow regime transitions in wellbore annulus by virtual mass force coefficient when gas cut

NASA Astrophysics Data System (ADS)

Qu, Junbo; Yan, Tie; Sun, Xiaofeng; Chen, Ye; Pan, Yi

2017-10-01

With the development of drilling technology to deeper stratum, overflowing especially gas cut occurs frequently, and then flow regime in wellbore annulus is from the original drilling fluid single-phase flow into gas & liquid two-phase flow. By using averaged two-fluid model equations and the basic principle of fluid mechanics to establish the continuity equations and momentum conservation equations of gas phase & liquid phase respectively. Relationship between pressure and density of gas & liquid was introduced to obtain hyperbolic equation, and get the expression of the dimensionless eigenvalue of the equation by using the characteristic line method, and analyze wellbore flow regime to get the critical gas content under different virtual mass force coefficients. Results show that the range of equation eigenvalues is getting smaller and smaller with the increase of gas content. When gas content reaches the critical point, the dimensionless eigenvalue of equation has no real solution, and the wellbore flow regime changed from bubble flow to bomb flow. When virtual mass force coefficients are 0.50, 0.60, 0.70 and 0.80 respectively, the critical gas contents are 0.32, 0.34, 0.37 and 0.39 respectively. The higher the coefficient of virtual mass force, the higher gas content in wellbore corresponding to the critical point of transition flow regime, which is in good agreement with previous experimental results. Therefore, it is possible to determine whether there is a real solution of the dimensionless eigenvalue of equation by virtual mass force coefficient and wellbore gas content, from which we can obtain the critical condition of wellbore flow regime transformation. It can provide theoretical support for the accurate judgment of the annular flow regime.

Numerical methods for the weakly compressible Generalized Langevin Model in Eulerian reference frame

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

Azarnykh, Dmitrii, E-mail: d.azarnykh@tum.de; Litvinov, Sergey; Adams, Nikolaus A.

2016-06-01

A well established approach for the computation of turbulent flow without resolving all turbulent flow scales is to solve a filtered or averaged set of equations, and to model non-resolved scales by closures derived from transported probability density functions (PDF) for velocity fluctuations. Effective numerical methods for PDF transport employ the equivalence between the Fokker–Planck equation for the PDF and a Generalized Langevin Model (GLM), and compute the PDF by transporting a set of sampling particles by GLM (Pope (1985) [1]). The natural representation of GLM is a system of stochastic differential equations in a Lagrangian reference frame, typically solvedmore » by particle methods. A representation in a Eulerian reference frame, however, has the potential to significantly reduce computational effort and to allow for the seamless integration into a Eulerian-frame numerical flow solver. GLM in a Eulerian frame (GLMEF) formally corresponds to the nonlinear fluctuating hydrodynamic equations derived by Nakamura and Yoshimori (2009) [12]. Unlike the more common Landau–Lifshitz Navier–Stokes (LLNS) equations these equations are derived from the underdamped Langevin equation and are not based on a local equilibrium assumption. Similarly to LLNS equations the numerical solution of GLMEF requires special considerations. In this paper we investigate different numerical approaches to solving GLMEF with respect to the correct representation of stochastic properties of the solution. We find that a discretely conservative staggered finite-difference scheme, adapted from a scheme originally proposed for turbulent incompressible flow, in conjunction with a strongly stable (for non-stochastic PDE) Runge–Kutta method performs better for GLMEF than schemes adopted from those proposed previously for the LLNS. We show that equilibrium stochastic fluctuations are correctly reproduced.« less

Discrete Variational Approach for Modeling Laser-Plasma Interactions

NASA Astrophysics Data System (ADS)

Reyes, J. Paxon; Shadwick, B. A.

2014-10-01

The traditional approach for fluid models of laser-plasma interactions begins by approximating fields and derivatives on a grid in space and time, leading to difference equations that are manipulated to create a time-advance algorithm. In contrast, by introducing the spatial discretization at the level of the action, the resulting Euler-Lagrange equations have particular differencing approximations that will exactly satisfy discrete versions of the relevant conservation laws. For example, applying a spatial discretization in the Lagrangian density leads to continuous-time, discrete-space equations and exact energy conservation regardless of the spatial grid resolution. We compare the results of two discrete variational methods using the variational principles from Chen and Sudan and Brizard. Since the fluid system conserves energy and momentum, the relative errors in these conserved quantities are well-motivated physically as figures of merit for a particular method. This work was supported by the U. S. Department of Energy under Contract No. DE-SC0008382 and by the National Science Foundation under Contract No. PHY-1104683.

Periodic motions of generalized conservative mechanical systems whose equations of motion contain a large parameter

NASA Astrophysics Data System (ADS)

Sazonov, V. V.

An analysis is made of a generalized conservative mechanical system whose equations of motion contain a large parameter characterizing local forces acting along certain generalized coordinates. It is shown that the equations have periodic solutions which are close to periodic solutions to the corresponding degenerate equations. As an example, the periodic motions of a satellite with respect to its center of mass due to gravitational and restoring aerodynamic moments are examined for the case where the aerodynamic moment is much larger than the gravitational moment. Such motions can be treated as nominal unperturbed motions of a satellite under conditions of single-axis aerodynamic attitude control.

Modeling interfacial area transport in multi-fluid systems

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

Yarbro, Stephen Lee

1996-11-01

Many typical chemical engineering operations are multi-fluid systems. They are carried out in distillation columns (vapor/liquid), liquid-liquid contactors (liquid/liquid) and other similar devices. An important parameter is interfacial area concentration, which determines the rate of interfluid heat, mass and momentum transfer and ultimately, the overall performance of the equipment. In many cases, the models for determining interfacial area concentration are empirical and can only describe the cases for which there is experimental data. In an effort to understand multiphase reactors and the mixing process better, a multi-fluid model has been developed as part of a research effort to calculate interfacialmore » area transport in several different types of in-line static mixers. For this work, the ensemble-averaged property conservation equations have been derived for each fluid and for the mixture. These equations were then combined to derive a transport equation for the interfacial area concentration. The final, one-dimensional model was compared to interfacial area concentration data from two sizes of Kenics in-line mixer, two sizes of concurrent jet and a Tee mixer. In all cases, the calculated and experimental data compared well with the highest scatter being with the Tee mixer comparison.« less

Progress in the development of PDF turbulence models for combustion

NASA Technical Reports Server (NTRS)

Hsu, Andrew T.

1991-01-01

A combined Monte Carlo-computational fluid dynamic (CFD) algorithm was developed recently at Lewis Research Center (LeRC) for turbulent reacting flows. In this algorithm, conventional CFD schemes are employed to obtain the velocity field and other velocity related turbulent quantities, and a Monte Carlo scheme is used to solve the evolution equation for the probability density function (pdf) of species mass fraction and temperature. In combustion computations, the predictions of chemical reaction rates (the source terms in the species conservation equation) are poor if conventional turbulence modles are used. The main difficulty lies in the fact that the reaction rate is highly nonlinear, and the use of averaged temperature produces excessively large errors. Moment closure models for the source terms have attained only limited success. The probability density function (pdf) method seems to be the only alternative at the present time that uses local instantaneous values of the temperature, density, etc., in predicting chemical reaction rates, and thus may be the only viable approach for more accurate turbulent combustion calculations. Assumed pdf's are useful in simple problems; however, for more general combustion problems, the solution of an evolution equation for the pdf is necessary.

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

NASA Technical Reports Server (NTRS)

Towne, Charles E.; Schwab, John R.; Benson, Thomas J.; Suresh, Ambady

1990-01-01

A new computer code was developed to solve the two-dimensional or axisymmetric, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 2 is the User's Guide, and describes the program's general 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.

Tar Production from Biomass Pyrolysis in a Fluidized Bed Reactor: A Novel Turbulent Multiphase Flow Formulation

NASA Technical Reports Server (NTRS)

Bellan, J.; Lathouwers, D.

2000-01-01

A novel multiphase flow model is presented for describing the pyrolysis of biomass in a 'bubbling' fluidized bed reactor. The mixture of biomass and sand in a gaseous flow is conceptualized as a particulate phase composed of two classes interacting with the carrier gaseous flow. The solid biomass is composed of three initial species: cellulose, hemicellulose and lignin. From each of these initial species, two new solid species originate during pyrolysis: an 'active' species and a char, thus totaling seven solid-biomass species. The gas phase is composed of the original carrier gas (steam), tar and gas; the last two species originate from the volumetric pyrolysis reaction. The conservation equations are derived from the Boltzmann equations through ensemble averaging. Stresses in the gaseous phase are the sum of the Newtonian and Reynolds (turbulent) contributions. The particulate phase stresses are the sum of collisional and Reynolds contributions. Heat transfer between phases, and heat transfer between classes in the particulate phase is modeled, the last resulting from collisions between sand and biomass. Closure of the equations must be performed by modeling the Reynolds stresses for both phases. The results of a simplified version (first step) of the model are presented.

A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

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

Gamba, Irene M.; ICES, The University of Texas at Austin, 201 E. 24th St., Stop C0200, Austin, TX 78712; Haack, Jeffrey R.

2014-08-01

We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit tomore » the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.« less

Lie Symmetry Analysis, Conservation Laws and Exact Power Series Solutions for Time-Fractional Fordy-Gibbons Equation

NASA Astrophysics Data System (ADS)

Feng, Lian-Li; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian

2016-09-01

In this paper, the time fractional Fordy-Gibbons equation is investigated with Riemann-Liouville derivative. The equation can be reduced to the Caudrey-Dodd-Gibbon equation, Savada-Kotera equation and the Kaup-Kupershmidt equation, etc. By means of the Lie group analysis method, the invariance properties and symmetry reductions of the equation are derived. Furthermore, by means of the power series theory, its exact power series solutions of the equation are also constructed. Finally, two kinds of conservation laws of the equation are well obtained with aid of the self-adjoint method. Supported by the Fundamental Research Funds for Key Discipline Construction under Grant No. XZD201602, the Fundamental Research Funds for the Central Universities under Grant Nos. 2015QNA53 and 2015XKQY14, the Fundamental Research Funds for Postdoctoral at the Key Laboratory of Gas and Fire Control for Coal Mines, the General Financial Grant from the China Postdoctoral Science Foundation under Grant No. 2015M570498, and Natural Sciences Foundation of China under Grant No. 11301527

Use of transport models for wildfire behavior simulations

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

Linn, R.R.; Harlow, F.H.

1998-01-01

Investigators have attempted to describe the behavior of wildfires for over fifty years. Current models for numerical description are mainly algebraic and based on statistical or empirical ideas. The authors have developed a transport model called FIRETEC. The use of transport formulations connects the propagation rates to the full conservation equations for energy, momentum, species concentrations, mass, and turbulence. In this paper, highlights of the model formulation and results are described. The goal of the FIRETEC model is to describe most probable average behavior of wildfires in a wide variety of conditions. FIRETEC represents the essence of the combination ofmore » many small-scale processes without resolving each process in complete detail.« less

On recent advances and future research directions for computational fluid dynamics

NASA Technical Reports Server (NTRS)

Baker, A. J.; Soliman, M. O.; Manhardt, P. D.

1986-01-01

This paper highlights some recent accomplishments regarding CFD numerical algorithm constructions for generation of discrete approximate solutions to classes of Reynolds-averaged Navier-Stokes equations. Following an overview of turbulent closure modeling, and development of appropriate conservation law systems, a Taylor weak-statement semi-discrete approximate solution algorithm is developed. Various forms for completion to the final linear algebra statement are cited, as are a range of candidate numerical linear algebra solution procedures. This development sequence emphasizes the key building blocks of a CFD RNS algorithm, including solution trial and test spaces, integration procedure and added numerical stability mechanisms. A range of numerical results are discussed focusing on key topics guiding future research directions.

SToRM: A Model for Unsteady Surface Hydraulics Over Complex Terrain

USGS Publications Warehouse

Simoes, Francisco J.

2014-01-01

A two-dimensional (depth-averaged) finite volume Godunov-type shallow water model developed for flow over complex topography is presented. The model is based on an unstructured cellcentered finite volume formulation and a nonlinear strong stability preserving Runge-Kutta time stepping scheme. The numerical discretization is founded on the classical and well established shallow water equations in hyperbolic conservative form, but the convective fluxes are calculated using auto-switching Riemann and diffusive numerical fluxes. The model’s implementation within a graphical user interface is discussed. Field application of the model is illustrated by utilizing it to estimate peak flow discharges in a flooding event of historic significance in Colorado, U.S.A., in 2013.

A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

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

Manzini, Gianmarco; Delzanno, Gian Luca; Vencels, Juris

In this study, we present the design and implementation of an L 2-stable spectral method for the discretization of the Vlasov–Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank–Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations ismore » iteratively solved at any time cycle by a Jacobian-Free Newton–Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre–Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L 2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.« less

A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

DOE PAGES

Manzini, Gianmarco; Delzanno, Gian Luca; Vencels, Juris; ...

2016-04-22

In this study, we present the design and implementation of an L 2-stable spectral method for the discretization of the Vlasov–Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank–Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations ismore » iteratively solved at any time cycle by a Jacobian-Free Newton–Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre–Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L 2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.« less

Effect of the target power density on high-power impulse magnetron sputtering of copper

NASA Astrophysics Data System (ADS)

Kozák, Tomáš

2012-04-01

We present a model analysis of high-power impulse magnetron sputtering of copper. We use a non-stationary global model based on the particle and energy conservation equations in two zones (the high density plasma ring above the target racetrack and the bulk plasma region), which makes it possible to calculate time evolutions of the averaged process gas and target material neutral and ion densities, as well as the fluxes of these particles to the target and substrate during a pulse period. We study the effect of the increasing target power density under conditions corresponding to a real experimental system. The calculated target current waveforms show a long steady state and are in good agreement with the experimental results. For an increasing target power density, an analysis of the particle densities shows a gradual transition to a metal dominated discharge plasma with an increasing degree of ionization of the depositing flux. The average fraction of target material ions in the total ion flux onto the substrate is more than 90% for average target power densities higher than 500 W cm-2 in a pulse. The average ionized fraction of target material atoms in the flux onto the substrate reaches 80% for a maximum average target power density of 3 kW cm-2 in a pulse.

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods - Part 1: Derivation and properties

NASA Astrophysics Data System (ADS)

Eldred, Christopher; Randall, David

2017-02-01

The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.

A simple mass-conserved level set method for simulation of multiphase flows

NASA Astrophysics Data System (ADS)

Yuan, H.-Z.; Shu, C.; Wang, Y.; Shu, S.

2018-04-01

In this paper, a modified level set method is proposed for simulation of multiphase flows with large density ratio and high Reynolds number. The present method simply introduces a source or sink term into the level set equation to compensate the mass loss or offset the mass increase. The source or sink term is derived analytically by applying the mass conservation principle with the level set equation and the continuity equation of flow field. Since only a source term is introduced, the application of the present method is as simple as the original level set method, but it can guarantee the overall mass conservation. To validate the present method, the vortex flow problem is first considered. The simulation results are compared with those from the original level set method, which demonstrates that the modified level set method has the capability of accurately capturing the interface and keeping the mass conservation. Then, the proposed method is further validated by simulating the Laplace law, the merging of two bubbles, a bubble rising with high density ratio, and Rayleigh-Taylor instability with high Reynolds number. Numerical results show that the mass is a well-conserved by the present method.

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.

1994-01-01

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Couplings of gravitational currents with Chern-Simons gravities

NASA Astrophysics Data System (ADS)

Ertem, Ümit; Açık, Özgür

2013-02-01

The coupling of conserved p-brane currents with non-Abelian gauge theories is done consistently by using Chern-Simons forms. Conserved currents localized on p-branes that have a gravitational origin can be constructed from Killing-Yano forms of the underlying spacetime. We propose a generalization of the coupling procedure with Chern-Simons gravities to the case of gravitational conserved currents. In odd dimensions, the field equations of coupled Chern-Simons gravities that describe the local curvature on p-branes are obtained. In special cases of three and five dimensions, the field equations are investigated in detail.

Divergent conservation laws in hyperbolic thermoelasticity

NASA Astrophysics Data System (ADS)

Murashkin, E. V.; Radayev, Y. N.

2018-05-01

The present study is devoted to the problem of formulation of conservation laws in divergent form for hyperbolic thermoelastic continua. The field formalism is applied to study the problem. A natural density of thermoelastic action and the corresponding variational least action principle are formulated. A special form of the first variation of the action is employed to obtain 4-covariant divergent conservation laws. Differential field equations and constitutive laws are derived from a special form of the first variation of the action integral. The objectivity of constitutive equations is provided by the rotationally invariant forms of the Lagrangian employed.

A New Formulation of Time Domain Boundary Integral Equation for Acoustic Wave Scattering in the Presence of a Uniform Mean Flow

NASA Technical Reports Server (NTRS)

Hu, Fang; Pizzo, Michelle E.; Nark, Douglas M.

2017-01-01

It has been well-known that under the assumption of a constant uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation, in both the time domain and the frequency domain. Compared with solving partial differential equations, numerical methods based on the boundary integral equation have the advantage of a reduced spatial dimension and, hence, requiring only a surface mesh. However, the constant uniform mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the uniform mean flow. In this paper, we argue that the proper boundary condition for the acoustic wave should not have its normal velocity be zero everywhere on the solid surfaces, as has been applied in the literature. A careful study of the acoustic energy conservation equation is presented that shows such a boundary condition in fact leads to erroneous source or sink points on solid surfaces not aligned with the mean flow. A new solid wall boundary condition is proposed that conserves the acoustic energy and a new time domain boundary integral equation is derived. In addition to conserving the acoustic energy, another significant advantage of the new equation is that it is considerably simpler than previous formulations. In particular, tangential derivatives of the solution on the solid surfaces are no longer needed in the new formulation, which greatly simplifies numerical implementation. Furthermore, stabilization of the new integral equation by Burton-Miller type reformulation is presented. The stability of the new formulation is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the new formulation.

Direct numerical simulations of fluid flow, heat transfer and phase changes

NASA Technical Reports Server (NTRS)

Juric, D.; Tryggvason, G.; Han, J.

1997-01-01

Direct numerical simulations of fluid flow, heat transfer, and phase changes are presented. The simulations are made possible by a recently developed finite difference/front tracking method based on the one-field formulation of the governing equations where a single set of conservation equations is written for all the phases involved. The conservation equations are solved on a fixed rectangular grid, but the phase boundaries are kept sharp by tracking them explicitly by a moving grid of lower dimension. The method is discussed and applications to boiling heat transfer and the solidification of drops colliding with a wall are shown.

Applications of a Property of the Schrödinger Equation to the Modeling of Conservative Discrete Systems

NASA Astrophysics Data System (ADS)

Popa, Alexandru

1998-08-01

Recently we have demonstrated in a mathematical paper the following property: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. We present now an accurate analysis model of the conservative discrete systems, that is based on this property. The theory is checked for a lot of atomic systems. The experimental data, which are ionization energies, are taken from well known books.

Energy conservation and H theorem for the Enskog-Vlasov equation

NASA Astrophysics Data System (ADS)

Benilov, E. S.; Benilov, M. S.

2018-06-01

The Enskog-Vlasov (EV) equation is a widely used semiphenomenological model of gas-liquid phase transitions. We show that it does not generally conserve energy, although there exists a restriction on its coefficients for which it does. Furthermore, if an energy-preserving version of the EV equation satisfies an H theorem as well, it can be used to rigorously derive the so-called Maxwell construction which determines the parameters of liquid-vapor equilibria. Finally, we show that the EV model provides an accurate description of the thermodynamics of noble fluids, and there exists a version simple enough for use in applications.

A low-dispersion, exactly energy-charge-conserving semi-implicit relativistic particle-in-cell algorithm

NASA Astrophysics Data System (ADS)

Chen, Guangye; Luis, Chacon; Bird, Robert; Stark, David; Yin, Lin; Albright, Brian

2017-10-01

Leap-frog based explicit algorithms, either ``energy-conserving'' or ``momentum-conserving'', do not conserve energy discretely. Time-centered fully implicit algorithms can conserve discrete energy exactly, but introduce large dispersion errors in the light-wave modes, regardless of timestep sizes. This can lead to intolerable simulation errors where highly accurate light propagation is needed (e.g. laser-plasma interactions, LPI). In this study, we selectively combine the leap-frog and Crank-Nicolson methods to produce a low-dispersion, exactly energy-and-charge-conserving PIC algorithm. Specifically, we employ the leap-frog method for Maxwell equations, and the Crank-Nicolson method for particle equations. Such an algorithm admits exact global energy conservation, exact local charge conservation, and preserves the dispersion properties of the leap-frog method for the light wave. The algorithm has been implemented in a code named iVPIC, based on the VPIC code developed at LANL. We will present numerical results that demonstrate the properties of the scheme with sample test problems (e.g. Weibel instability run for 107 timesteps, and LPI applications.

Spatial averaging of a dissipative particle dynamics model for active suspensions

NASA Astrophysics Data System (ADS)

Panchenko, Alexander; Hinz, Denis F.; Fried, Eliot

2018-03-01

Starting from a fine-scale dissipative particle dynamics (DPD) model of self-motile point particles, we derive meso-scale continuum equations by applying a spatial averaging version of the Irving-Kirkwood-Noll procedure. Since the method does not rely on kinetic theory, the derivation is valid for highly concentrated particle systems. Spatial averaging yields stochastic continuum equations similar to those of Toner and Tu. However, our theory also involves a constitutive equation for the average fluctuation force. According to this equation, both the strength and the probability distribution vary with time and position through the effective mass density. The statistics of the fluctuation force also depend on the fine scale dissipative force equation, the physical temperature, and two additional parameters which characterize fluctuation strengths. Although the self-propulsion force entering our DPD model contains no explicit mechanism for aligning the velocities of neighboring particles, our averaged coarse-scale equations include the commonly encountered cubically nonlinear (internal) body force density.

Two-dimensional solitons in conservative and parity-time-symmetric triple-core waveguides with cubic-quintic nonlinearity

NASA Astrophysics Data System (ADS)

Feijoo, David; Zezyulin, Dmitry A.; Konotop, Vladimir V.

2015-12-01

We analyze a system of three two-dimensional nonlinear Schrödinger equations coupled by linear terms and with the cubic-quintic (focusing-defocusing) nonlinearity. We consider two versions of the model: conservative and parity-time (PT ) symmetric. These models describe triple-core nonlinear optical waveguides, with balanced gain and losses in the PT -symmetric case. We obtain families of soliton solutions and discuss their stability. The latter study is performed using a linear stability analysis and checked with direct numerical simulations of the evolutional system of equations. Stable solitons are found in the conservative and PT -symmetric cases. Interactions and collisions between the conservative and PT -symmetric solitons are briefly investigated, as well.

Lagrangian averages, averaged Lagrangians, and the mean effects of fluctuations in fluid dynamics.

PubMed

Holm, Darryl D.

2002-06-01

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincare (EP) variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler-Poincare (LAEP) theorem. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics. (c) 2002 American Institute of Physics.

A critical analysis of the accuracy of several numerical techniques for combustion kinetic rate equations

NASA Technical Reports Server (NTRS)

Radhadrishnan, Krishnan

1993-01-01

A detailed analysis of the accuracy of several techniques recently developed for integrating stiff ordinary differential equations is presented. The techniques include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP4 developed specifically to solve chemical kinetic rate equations. The accuracy study is made by application of these codes to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. To illustrate the error variation in the different combustion regimes the species are divided into three types (reactants, intermediates, and products), and error versus time plots are presented for each species type and the temperature. These plots show that CHEMEQ is the most accurate code during induction and early heat release. During late heat release and equilibration, however, the other codes are more accurate. A single global quantity, a mean integrated root-mean-square error, that measures the average error incurred in solving the complete problem is used to compare the accuracy of the codes. Among the codes examined, LSODE is the most accurate for solving chemical kinetics problems. It is also the most efficient code, in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that use of the algebraic enthalpy conservation equation to compute the temperature can be more accurate and efficient than integrating the temperature differential equation.

Development of a locally mass flux conservative computer code for calculating 3-D viscous flow in turbomachines

NASA Technical Reports Server (NTRS)

Walitt, L.

1982-01-01

The VANS successive approximation numerical method was extended to the computation of three dimensional, viscous, transonic flows in turbomachines. A cross-sectional computer code, which conserves mass flux at each point of the cross-sectional surface of computation was developed. In the VANS numerical method, the cross-sectional computation follows a blade-to-blade calculation. Numerical calculations were made for an axial annular turbine cascade and a transonic, centrifugal impeller with splitter vanes. The subsonic turbine cascade computation was generated in blade-to-blade surface to evaluate the accuracy of the blade-to-blade mode of marching. Calculated blade pressures at the hub, mid, and tip radii of the cascade agreed with corresponding measurements. The transonic impeller computation was conducted to test the newly developed locally mass flux conservative cross-sectional computer code. Both blade-to-blade and cross sectional modes of calculation were implemented for this problem. A triplet point shock structure was computed in the inducer region of the impeller. In addition, time-averaged shroud static pressures generally agreed with measured shroud pressures. It is concluded that the blade-to-blade computation produces a useful engineering flow field in regions of subsonic relative flow; and cross-sectional computation, with a locally mass flux conservative continuity equation, is required to compute the shock waves in regions of supersonic relative flow.

Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian

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

Cruz, Hans, E-mail: hans@ciencias.unam.mx; Schuch, Dieter; Castaños, Octavio, E-mail: ocasta@nucleares.unam.mx

2015-09-15

The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.

FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations

NASA Astrophysics Data System (ADS)

Ibragimov, N. H.; Torrisi, M.; Tracinà, R.

2010-11-01

In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

Lie symmetry analysis, conservation laws, solitary and periodic waves for a coupled Burger equation

NASA Astrophysics Data System (ADS)

Xu, Mei-Juan; Tian, Shou-Fu; Tu, Jian-Min; Zhang, Tian-Tian

2017-01-01

Under investigation in this paper is a generalized (2 + 1)-dimensional coupled Burger equation with variable coefficients, which describes lots of nonlinear physical phenomena in geophysical fluid dynamics, condense matter physics and lattice dynamics. By employing the Lie group method, the symmetry reductions and exact explicit solutions are obtained, respectively. Based on a direct method, the conservations laws of the equation are also derived. Furthermore, by virtue of the Painlevé analysis, we successfully obtain the integrable condition on the variable coefficients, which plays an important role in further studying the integrability of the equation. Finally, its auto-Bäcklund transformation as well as some new analytic solutions including solitary and periodic waves are also presented via algebraic and differential manipulation.

RAZORBACK - A Research Reactor Transient Analysis Code Version 1.0 - Volume 3: Verification and Validation Report.

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

Talley, Darren G.

2017-04-01

This report describes the work and results of the verification and validation (V&V) of the version 1.0 release of the Razorback code. Razorback is a computer code designed to simulate the operation of a research reactor (such as the Annular Core Research Reactor (ACRR)) by a coupled numerical solution of the point reactor kinetics equations, the energy conservation equation for fuel element heat transfer, the equation of motion for fuel element thermal expansion, and the mass, momentum, and energy conservation equations for the water cooling of the fuel elements. This V&V effort was intended to confirm that the code showsmore » good agreement between simulation and actual ACRR operations.« less

The charge conserving Poisson-Boltzmann equations: Existence, uniqueness, and maximum principle

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

Lee, Chiun-Chang, E-mail: chlee@mail.nhcue.edu.tw

2014-05-15

The present article is concerned with the charge conserving Poisson-Boltzmann (CCPB) equation in high-dimensional bounded smooth domains. The CCPB equation is a Poisson-Boltzmann type of equation with nonlocal coefficients. First, under the Robin boundary condition, we get the existence of weak solutions to this equation. The main approach is variational, based on minimization of a logarithm-type energy functional. To deal with the regularity of weak solutions, we establish a maximum modulus estimate for the standard Poisson-Boltzmann (PB) equation to show that weak solutions of the CCPB equation are essentially bounded. Then the classical solutions follow from the elliptic regularity theorem.more » Second, a maximum principle for the CCPB equation is established. In particular, we show that in the case of global electroneutrality, the solution achieves both its maximum and minimum values at the boundary. However, in the case of global non-electroneutrality, the solution may attain its maximum value at an interior point. In addition, under certain conditions on the boundary, we show that the global non-electroneutrality implies pointwise non-electroneutrality.« less

A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces

NASA Astrophysics Data System (ADS)

Henry de Frahan, Marc T.; Varadan, Sreenivas; Johnsen, Eric

2015-01-01

Although the Discontinuous Galerkin (DG) method has seen widespread use for compressible flow problems in a single fluid with constant material properties, it has yet to be implemented in a consistent fashion for compressible multiphase flows with shocks and interfaces. Specifically, it is challenging to design a scheme that meets the following requirements: conservation, high-order accuracy in smooth regions and non-oscillatory behavior at discontinuities (in particular, material interfaces). Following the interface-capturing approach of Abgrall [1], we model flows of multiple fluid components or phases using a single equation of state with variable material properties; discontinuities in these properties correspond to interfaces. To represent compressible phenomena in solids, liquids, and gases, we present our analysis for equations of state belonging to the Mie-Grüneisen family. Within the DG framework, we propose a conservative, high-order accurate, and non-oscillatory limiting procedure, verified with simple multifluid and multiphase problems. We show analytically that two key elements are required to prevent spurious pressure oscillations at interfaces and maintain conservation: (i) the transport equation(s) describing the material properties must be solved in a non-conservative weak form, and (ii) the suitable variables must be limited (density, momentum, pressure, and appropriate properties entering the equation of state), coupled with a consistent reconstruction of the energy. Further, we introduce a physics-based discontinuity sensor to apply limiting in a solution-adaptive fashion. We verify this approach with one- and two-dimensional problems with shocks and interfaces, including high pressure and density ratios, for fluids obeying different equations of state to illustrate the robustness and versatility of the method. The algorithm is implemented on parallel graphics processing units (GPU) to achieve high speedup.

Numerical Limitations of 1D Hydraulic Models Using MIKE11 or HEC-RAS software - Case study of Baraolt River, Romania

NASA Astrophysics Data System (ADS)

Andrei, Armas; Robert, Beilicci; Erika, Beilicci

2017-10-01

MIKE 11 is an advanced hydroinformatic tool, a professional engineering software package for simulation of one-dimensional flows in estuaries, rivers, irrigation systems, channels and other water bodies. MIKE 11 is a 1-dimensional river model. It was developed by DHI Water · Environment · Health, Denmark. The basic computational procedure of HEC-RAS for steady flow is based on the solution of the one-dimensional energy equation. Energy losses are evaluated by friction and contraction / expansion. The momentum equation may be used in situations where the water surface profile is rapidly varied. These situations include hydraulic jumps, hydraulics of bridges, and evaluating profiles at river confluences. For unsteady flow, HEC-RAS solves the full, dynamic, 1-D Saint Venant Equation using an implicit, finite difference method. The unsteady flow equation solver was adapted from Dr. Robert L. Barkau’s UNET package. Fluid motion is controlled by the basic principles of conservation of mass, energy and momentum, which form the basis of fluid mechanics and hydraulic engineering. Complex flow situations must be solved using empirical approximations and numerical models, which are based on derivations of the basic principles (backwater equation, Navier-Stokes equation etc.). All numerical models are required to make some form of approximation to solve these principles, and consequently all have their limitations. The study of hydraulics and fluid mechanics is founded on the three basic principles of conservation of mass, energy and momentum. Real-life situations are frequently too complex to solve without the aid of numerical models. There is a tendency among some engineers to discard the basic principles taught at university and blindly assume that the results produced by the model are correct. Regardless of the complexity of models and despite the claims of their developers, all numerical models are required to make approximations. These may be related to geometric limitations, numerical simplification, or the use of empirical correlations. Some are obvious: one-dimensional models must average properties over the two remaining directions. It is the less obvious and poorly advertised approximations that pose the greatest threat to the novice user. Some of these, such as the inability of one-dimensional unsteady models to simulate supercritical flow can cause significant inaccuracy in the model predictions.

On the Global Regularity of a Helical-Decimated Version of the 3D Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Biferale, Luca; Titi, Edriss S.

2013-06-01

We study the global regularity, for all time and all initial data in H 1/2, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the L 2-scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the H 1/2-Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the H 1/2 and the time average of the square of the H 3/2 norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato (Arch. Ration. Mech. Anal. 16:269-315, 1964; Rend. Semin. Mat. Univ. Padova 32:243-260, 1962), to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences.

A model for closing the inviscid form of the average-passage equation system

NASA Technical Reports Server (NTRS)

Adamczyk, J. J.; Mulac, R. A.; Celestina, M. L.

1985-01-01

A mathematical model is proposed for closing or mathematically completing the system of equations which describes the time average flow field through the blade passages of multistage turbomachinery. These equations referred to as the average passage equation system govern a conceptual model which has proven useful in turbomachinery aerodynamic design and analysis. The closure model is developed so as to insure a consistency between these equations and the axisymmetric through flow equations. The closure model was incorporated into a computer code for use in simulating the flow field about a high speed counter rotating propeller and a high speed fan stage. Results from these simulations are presented.

A GENERAL MASS-CONSERVATIVE NUMERICAL SOLUTION FOR THE UNSATURATED FLOW EQUATION

EPA Science Inventory

Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h-based form of Richards equation generally yields poor results, ch...

An improved two-dimensional depth-integrated flow equation for rough-walled fractures

NASA Astrophysics Data System (ADS)

Mallikamas, Wasin; Rajaram, Harihar

2010-08-01

We present the development of an improved 2-D flow equation for rough-walled fractures. Our improved equation accounts for the influence of midsurface tortuosity and the fact that the aperture normal to the midsurface is in general smaller than the vertical aperture. It thus improves upon the well-known Reynolds equation that is widely used for modeling flow in fractures. Unlike the Reynolds equation, our approach begins from the lubrication approximation applied in an inclined local coordinate system tangential to the fracture midsurface. The local flow equation thus obtained is rigorously transformed to an arbitrary global Cartesian coordinate system, invoking the concepts of covariant and contravariant transformations for vectors defined on surfaces. Unlike previously proposed improvements to the Reynolds equation, our improved flow equation accounts for tortuosity both along and perpendicular to a flow path. Our approach also leads to a well-defined anisotropic local transmissivity tensor relating the representations of the flux and head gradient vectors in a global Cartesian coordinate system. We show that the principal components of the transmissivity tensor and the orientation of its principal axes depend on the directional local midsurface slopes. In rough-walled fractures, the orientations of the principal axes of the local transmissivity tensor will vary from point to point. The local transmissivity tensor also incorporates the influence of the local normal aperture, which is uniquely defined at each point in the fracture. Our improved flow equation is a rigorous statement of mass conservation in any global Cartesian coordinate system. We present three examples of simple geometries to compare our flow equation to analytical solutions obtained using the exact Stokes equations: an inclined parallel plate, and circumferential and axial flows in an incomplete annulus. The effective transmissivities predicted by our flow equation agree very well with values obtained using the exact Stokes equations in all these cases. We discuss potential limitations of our depth-integrated equation, which include the neglect of convergence/divergence and the inaccuracies implicit in any depth-averaging process near sharp corners where the wall and midsurface curvatures are large.

Using computational modeling of river flow with remotely sensed data to infer channel bathymetry

USGS Publications Warehouse

Nelson, Jonathan M.; McDonald, Richard R.; Kinzel, Paul J.; Shimizu, Y.

2012-01-01

As part of an ongoing investigation into the use of computational river flow and morphodynamic models for the purpose of correcting and extending remotely sensed river datasets, a simple method for inferring channel bathymetry is developed and discussed. The method is based on an inversion of the equations expressing conservation of mass and momentum to develop equations that can be solved for depth given known values of vertically-averaged velocity and water-surface elevation. The ultimate goal of this work is to combine imperfect remotely sensed data on river planform, water-surface elevation and water-surface velocity in order to estimate depth and other physical parameters of river channels. In this paper, the technique is examined using synthetic data sets that are developed directly from the application of forward two-and three-dimensional flow models. These data sets are constrained to satisfy conservation of mass and momentum, unlike typical remotely sensed field data sets. This provides a better understanding of the process and also allows assessment of how simple inaccuracies in remotely sensed estimates might propagate into depth estimates. The technique is applied to three simple cases: First, depth is extracted from a synthetic dataset of vertically averaged velocity and water-surface elevation; second, depth is extracted from the same data set but with a normally-distributed random error added to the water-surface elevation; third, depth is extracted from a synthetic data set for the same river reach using computed water-surface velocities (in place of depth-integrated values) and water-surface elevations. In each case, the extracted depths are compared to the actual measured depths used to construct the synthetic data sets (with two- and three-dimensional flow models). Errors in water-surface elevation and velocity that are very small degrade depth estimates and cannot be recovered. Errors in depth estimates associated with assuming water-surface velocities equal to depth-integrated velocities are substantial, but can be reduced with simple corrections.

A fast Laplace solver approach to pore scale permeability

NASA Astrophysics Data System (ADS)

Arns, Christoph; Adler, Pierre

2017-04-01

The permeability of a porous medium can be derived by solving the Stokes equations in the pore space with no slip at the walls. The resulting velocity averaged over the pore volume yields the permeability KS by application of the Darcy law. The Stokes equations can be solved by a number of different techniques such as finite differences, finite volume, Lattice Boltzmann, but whatever the technique it remains a heavy task since there are four unknowns at each node (the three velocity components and the pressure) which necessitate the solution of four equations (the projection of Newton's law on each axis and mass conservation). By comparison, the Laplace equation is scalar with a single unknown at each node. The objective of this work is to replace the Stokes equations by an elliptical equation with a space dependent permeability. More precisely, the local permeability k is supposed to be proportional to (r-alpha)**2 where r is the distance of the voxel to the closest wall, and alpha a constant; k is zero in the solid phase. The elliptical equation is div(k gradp)=0. A macroscopic pressure gradient is assumed to be exerted on the medium and again the resulting velocity averaged over space yields a permeability K_L. In order to validate this method, systematic calculations have been performed. First, elementary shapes (plane channel, circular pipe, rectangular channels) were studied for which flow occurs along parallel lines in which case KL is the arithmetic average of the k's. KL was calculated for various discretizations of the pore space and various values of alpha. For alpha=0.5, the agreement with the exact analytical value of KS is excellent for the plane and rectangular channels while it is only approximate for circular pipes. Second, the permeability KL of channels with sinusoidal walls was calculated and compared with analytical results and numerical ones provided by a Lattice Boltzmann algorithm. Generally speaking, the discrepancy does not exceed 25% when alpha=0.5. Third, the most important test was performed on two types of real media that were used for previous studies. A fracture network measured by FIB/SEM in a low permeability sandstone was used for that purpose; the two dimensionless permeabilities KS and KL are equal to 9.3d-3 and 8.5d-3. Similar calculations were performed on 256 samples of Fontainebleau sandstones and the agreement was in general excellent, except may be for very low permeabilities. To conclude, the Laplace solver is significantly more stable than the lattice Boltzmann approach, uses less memory, and is significantly faster. Permeabilities are in excellent agreement over a wide range of porosities.

Generalized energy and potential enstrophy conserving finite difference schemes for the shallow water equations

NASA Technical Reports Server (NTRS)

Abramopoulos, Frank

1988-01-01

The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.

Discrete conservation laws and the convergence of long time simulations of the mkdv equation

NASA Astrophysics Data System (ADS)

Gorria, C.; Alejo, M. A.; Vega, L.

2013-02-01

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.

Numerical artifacts in the Generalized Porous Medium Equation: Why harmonic averaging itself is not to blame

NASA Astrophysics Data System (ADS)

Maddix, Danielle C.; Sampaio, Luiz; Gerritsen, Margot

2018-05-01

The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to self-sharpening and its sharp corner solutions. For these problems, we show results for two subclasses of the GPME with differentiable k (p) with respect to p, namely the Porous Medium Equation (PME) and the superslow diffusion equation. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature. These issues have been attributed to harmonic averaging of the coefficient k (p) for small p, and arithmetic averaging has been suggested as an alternative. We show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues. Here, we investigate the causes of these numerical artifacts using modified equation analysis. The modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. The observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically through a Modified Harmonic Method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts.

Entropy Splitting for High Order Numerical Simulation of Vortex Sound at Low Mach Numbers

NASA Technical Reports Server (NTRS)

Mueller, B.; Yee, H. C.; Mansour, Nagi (Technical Monitor)

2001-01-01

A method of minimizing numerical errors, and improving nonlinear stability and accuracy associated with low Mach number computational aeroacoustics (CAA) is proposed. The method consists of two levels. From the governing equation level, we condition the Euler equations in two steps. The first step is to split the inviscid flux derivatives into a conservative and a non-conservative portion that satisfies a so called generalized energy estimate. This involves the symmetrization of the Euler equations via a transformation of variables that are functions of the physical entropy. Owing to the large disparity of acoustic and stagnation quantities in low Mach number aeroacoustics, the second step is to reformulate the split Euler equations in perturbation form with the new unknowns as the small changes of the conservative variables with respect to their large stagnation values. From the numerical scheme level, a stable sixth-order central interior scheme with a third-order boundary schemes that satisfies the discrete analogue of the integration-by-parts procedure used in the continuous energy estimate (summation-by-parts property) is employed.

Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma

NASA Astrophysics Data System (ADS)

EL-Kalaawy, O. H.; Moawad, S. M.; Wael, Shrouk

The propagation of nonlinear waves in unmagnetized strongly coupled dusty plasma with Boltzmann distributed electrons, iso-nonthermal distributed ions and negatively charged dust grains is considered. The basic set of fluid equations is reduced to the Schamel Kadomtsev-Petviashvili (S-KP) equation by using the reductive perturbation method. The variational principle and conservation laws of S-KP equation are obtained. It is shown that the S-KP equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by auto-Bäcklund transformations. The stability analysis is discussed for the existence of dust acoustic solitary waves (DASWs) and it is found that the physical parameters have strong effects on the stability criterion. In additional to, the electric field and the true Mach number of this solution are investigated. Finally, we will study the physical meanings of solutions.

f(R)-gravity from Killing tensors

NASA Astrophysics Data System (ADS)

Paliathanasis, Andronikos

2016-04-01

We consider f(R)-gravity in a Friedmann-Lemaître-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of f(R) and for the field equations to be invariant under Lie-Bäcklund transformations, which are linear in momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether’s theorem. We find three new integrable f(R)-models, for which, with the application of the conservation laws, we reduce the field equations to a system of two first-order ordinary differential equations. For each model we study the evolution of the cosmological fluid. We find that for each integrable model the cosmological fluid has an equation of state parameter, in which there is linear behavior in terms of the scale factor which describes the Chevallier, Polarski and Linder parametric dark energy model.

Kinetic theory of oxygen isotopic exchange between minerals and water

USGS Publications Warehouse

Criss, R.E.; Gregory, R.T.; Taylor, H.P.

1987-01-01

Kinetic and mass conservation equations are used to describe oxygen isotopic exchange between minerals and water in "closed" and open hydrothermal systems. In cases where n coexisting mineral phases having different reaction rates are present, the exchange process is described by a system of n + 1 simultaneous differential equations consisting of n pseudo first-order rate equations and a conservation of mass equation. The simultaneous solutions to these equations generate curved exchange trajectories on ??-?? plots. Families of such trajectories generated under conditions allowing for different fluid mole fractions, different fluid isotopic compositions, or different fluid flow rates are connected by positive-sloped isochronous lines. These isochrons reproduce the effects observed in hydrothermally exchanged mineral pairs including 1) steep positive slopes, 2) common reversals in the measured fractionation factors (??), and 3) measured fractionations that are highly variable over short distances where no thermal gradient can be geologically demonstrated. ?? 1987.

Three-Dimensional High-Order Spectral Volume Method for Solving Maxwell's Equations on Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel; Wang, Z. J.

2004-01-01

A three-dimensional, high-order, conservative, and efficient discontinuous spectral volume (SV) method for the solutions of Maxwell's equations on unstructured grids is presented. The concept of discontinuous 2nd high-order loca1 representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) method, but instead of using a Galerkin finite-element formulation, the SV method is based on a finite-volume approach to attain a simpler formulation. Conventional unstructured finite-volume methods require data reconstruction based on the least-squares formulation using neighboring cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In the SV method, one starts with a relatively coarse grid of triangles or tetrahedra, called spectral volumes (SVs), and partition each SV into a number of structured subcells, called control volumes (CVs), that support a polynomial expansion of a desired degree of precision. The unknowns are cell averages over CVs. If all the SVs are partitioned in a geometrically similar manner, the reconstruction becomes universal as a weighted sum of unknowns, and only a few universal coefficients need to be stored for the surface integrals over CV faces. Since the solution is discontinuous across the SV boundaries, a Riemann solver is thus necessary to maintain conservation. In the paper, multi-parameter and symmetric SV partitions, up to quartic for triangle and cubic for tetrahedron, are first presented. The corresponding weight coefficients for CV face integrals in terms of CV cell averages for each partition are analytically determined. These discretization formulas are then applied to the integral form of the Maxwell equations. All numerical procedures for outer boundary, material interface, zonal interface, and interior SV face are unified with a single characteristic formulation. The load balancing in a massive parallel computing environment is therefore easier to achieve. A parameter is introduced in the Riemann solver to control the strength of the smoothing term. Important aspects of the data structure and its effects to communication and the optimum use of cache memory are discussed. Results will be presented for plane TE and TM waves incident on a perfectly conducting cylinder for up to fifth order of accuracy, and a plane wave incident on a perfectly conducting sphere for up to fourth order of accuracy. Comparisons are made with exact solutions for these cases.

Structure-preserving operators for thermal-nonequilibrium hydrodynamics

NASA Astrophysics Data System (ADS)

Shiroto, Takashi; Kawai, Soshi; Ohnishi, Naofumi

2018-07-01

Radiation hydrodynamics simulations based on a single fluid two-temperature model may violate the law of energy conservation, because the governing equations are expressed in a nonconservative formulation. In this study, we maintain the important physical requirements by employing a strategy based on the key concept that mathematical structures associated with conservative and nonconservative equations are preserved, even at the discrete level. To this end, we discretize the conservation laws and transform them using exact algebraic operations. The proposed scheme maintains global conservation errors within the round-off level. In addition, a numerical experiment concerning the shock tube problem suggests that the proposed scheme agrees well with the jump conditions at the discontinuities regulated by the Rankine-Hugoniot relationship. The generalized derivation allows us to employ arbitrary central difference, artificial dissipation, and Runge-Kutta methods.

A Numerical Model for Trickle Bed Reactors

NASA Astrophysics Data System (ADS)

Propp, Richard M.; Colella, Phillip; Crutchfield, William Y.; Day, Marcus S.

2000-12-01

Trickle bed reactors are governed by equations of flow in porous media such as Darcy's law and the conservation of mass. Our numerical method for solving these equations is based on a total-velocity splitting, sequential formulation which leads to an implicit pressure equation and a semi-implicit mass conservation equation. We use high-resolution finite-difference methods to discretize these equations. Our solution scheme extends previous work in modeling porous media flows in two ways. First, we incorporate physical effects due to capillary pressure, a nonlinear inlet boundary condition, spatial porosity variations, and inertial effects on phase mobilities. In particular, capillary forces introduce a parabolic component into the recast evolution equation, and the inertial effects give rise to hyperbolic nonconvexity. Second, we introduce a modification of the slope-limiting algorithm to prevent our numerical method from producing spurious shocks. We present a numerical algorithm for accommodating these difficulties, show the algorithm is second-order accurate, and demonstrate its performance on a number of simplified problems relevant to trickle bed reactor modeling.

A new multi-symplectic scheme for the generalized Kadomtsev-Petviashvili equation

NASA Astrophysics Data System (ADS)

Li, Haochen; Sun, Jianqiang

2012-09-01

We propose a new scheme for the generalized Kadomtsev-Petviashvili (KP) equation. The multi-symplectic conservation property of the new scheme is proved. Back error analysis shows that the new multi-symplectic scheme has second order accuracy in space and time. Numerical application on studying the KPI equation and the KPII equation are presented in detail.

Steady inviscid transonic flows over planar airfoils: A search for a simplified procedure

NASA Technical Reports Server (NTRS)

Magnus, R.; Yoshihara, H.

1973-01-01

A finite difference procedure based upon a system of unsteady equations in proper conservation form with either exact or small disturbance steady terms is used to calculate the steady flows over several classes of airfoils. The airfoil condition is fulfilled on a slab whose upstream extremity is a semi-circle overlaying the airfoil leading edge circle. The limitations of the small disturbance equations are demonstrated in an extreme example of a blunt-nosed, aft-cambered airfoil. The necessity of using the equations in proper conservation form to capture the shock properly is stressed. Ability of the steady relaxation procedures to capture the shock is briefly examined.

Multigrid Relaxation of a Factorizable, Conservative Discretization of the Compressible Flow Equations

NASA Technical Reports Server (NTRS)

Roberts, Thomas W.; Sidilkover, David; Thomas, J. L.

2000-01-01

The second-order factorizable discretization of the compressible Euler equations developed by Sidilkover is extended to conservation form on general curvilinear body-fitted grids. The discrete equations are solved by symmetric collective Gauss-Seidel relaxation and FAS multigrid. Solutions for flow in a channel with Mach numbers ranging from 0.0001 to a supercritical Mach number are shown, demonstrating uniform convergence rates and no loss of accuracy in the incompressible limit. A solution for the flow around the leading edge of a semi-infinite parabolic body demonstrates that the scheme maintains rapid convergence for a flow containing a stagnation point.

Relativistic fluid dynamics with spin

NASA Astrophysics Data System (ADS)

Florkowski, Wojciech; Friman, Bengt; Jaiswal, Amaresh; Speranza, Enrico

2018-04-01

Using the conservation laws for charge, energy, momentum, and angular momentum, we derive hydrodynamic equations for the charge density, local temperature, and fluid velocity, as well as for the polarization tensor, starting from local equilibrium distribution functions for particles and antiparticles with spin 1/2. The resulting set of differential equations extends the standard picture of perfect-fluid hydrodynamics with a conserved entropy current in a minimal way. This framework can be used in space-time analyses of the evolution of spin and polarization in various physical systems including high-energy nuclear collisions. We demonstrate that a stationary vortex, which exhibits vorticity-spin alignment, corresponds to a special solution of the spin-hydrodynamical equations.

Transonic flow solutions using a composite velocity procedure for potential, Euler and RNS equations

NASA Technical Reports Server (NTRS)

Gordnier, R. E.; Rubin, S. G.

1986-01-01

Solutions for transonic viscous and inviscid flows using a composite velocity procedure are presented. The velocity components of the compressible flow equations are written in terms of a multiplicative composite consisting of a viscous or rotational velocity and an inviscid, irrotational, potential-like function. This provides for an efficient solution procedure that is locally representative of both asymptotic inviscid and boundary layer theories. A modified conservative form of the axial momentum equation that is required to obtain rotational solutions in the inviscid region is presented and a combined conservation/nonconservation form is applied for evaluation of the reduced Navier-Stokes (RNS), Euler and potential equations. A variety of results is presented and the effects of the approximations on entropy production, shock capturing, and viscous interaction are discussed.

Variational principles for relativistic smoothed particle hydrodynamics

NASA Astrophysics Data System (ADS)

Monaghan, J. J.; Price, D. J.

2001-12-01

In this paper we show how the equations of motion for the smoothed particle hydrodynamics (SPH) method may be derived from a variational principle for both non-relativistic and relativistic motion when there is no dissipation. Because the SPH density is a function of the coordinates the derivation of the equations of motion through variational principles is simpler than in the continuum case where the density is defined through the continuity equation. In particular, the derivation of the general relativistic equations is more direct and simpler than that of Fock. The symmetry properties of the Lagrangian lead immediately to the familiar additive conservation laws of linear and angular momentum and energy. In addition, we show that there is an approximately conserved quantity which, in the continuum limit, is the circulation.

Total energy and potential enstrophy conserving schemes for the shallow water equations using Hamiltonian methods $-$ Part 1: Derivation and properties

DOE PAGES

Eldred, Christopher; Randall, David

2017-02-17

The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restrictedmore » to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Lastly, detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.« less

Meridional overturning and large-scale circulation of the Indian Ocean

NASA Astrophysics Data System (ADS)

Ganachaud, Alexandre; Wunsch, Carl; Marotzke, Jochem; Toole, John

2000-11-01

The large scale Indian Ocean circulation is estimated from a global hydrographic inverse geostrophic box model with a focus on the meridional overturning circulation (MOC). The global model is based on selected recent World Ocean Circulation Experiment (WOCE) sections which in the Indian Basin consist of zonal sections at 32°S, 20°S and 8°S, and a section between Bali and Australia from the Java-Australia Dynamic Experiment (JADE). The circulation is required to conserve mass, salinity, heat, silica and "PO" (170PO4+O2). Near-conservation is imposed within layers bounded by neutral surfaces, while permitting advective and diffusive exchanges between the layers. Conceptually, the derived circulation is an estimate of the average circulation for the period 1987-1995. A deep inflow into the Indian Basin of 11±4 Sv is found, which is in the lower range of previous estimates, but consistent with conservation requirements and the global data set. The Indonesian Throughflow (ITF) is estimated at 15±5 Sv. The flow in the Mozambique Channel is of the same magnitude, implying a weak net flow between Madagascar and Australia. A net evaporation of -0.6±0.4 Sv is found between 32°S and 8°S, consistent with independent estimates. No net heat gain is found over the Indian Basin (0.1 ± 0.2PW north of 32°S) as a consequence of the large warm water influx from the ITF. Through the use of anomaly equations, the average dianeutral upwelling and diffusion between the sections are required and resolved, with values in the range 1-3×10-5 cm s-1 for the upwelling and 2-10 cm2 s-1 for the diffusivity.

Group-theoretical model of developed turbulence and renormalization of the Navier-Stokes equation.

PubMed

Saveliev, V L; Gorokhovski, M A

2005-07-01

On the basis of the Euler equation and its symmetry properties, this paper proposes a model of stationary homogeneous developed turbulence. A regularized averaging formula for the product of two fields is obtained. An equation for the averaged turbulent velocity field is derived from the Navier-Stokes equation by renormalization-group transformation.

Methods for estimating flood frequency in Montana based on data through water year 1998

USGS Publications Warehouse

Parrett, Charles; Johnson, Dave R.

2004-01-01

Annual peak discharges having recurrence intervals of 2, 5, 10, 25, 50, 100, 200, and 500 years (T-year floods) were determined for 660 gaged sites in Montana and in adjacent areas of Idaho, Wyoming, and Canada, based on data through water year 1998. The updated flood-frequency information was subsequently used in regression analyses, either ordinary or generalized least squares, to develop equations relating T-year floods to various basin and climatic characteristics, equations relating T-year floods to active-channel width, and equations relating T-year floods to bankfull width. The equations can be used to estimate flood frequency at ungaged sites. Montana was divided into eight regions, within which flood characteristics were considered to be reasonably homogeneous, and the three sets of regression equations were developed for each region. A measure of the overall reliability of the regression equations is the average standard error of prediction. The average standard errors of prediction for the equations based on basin and climatic characteristics ranged from 37.4 percent to 134.1 percent. Average standard errors of prediction for the equations based on active-channel width ranged from 57.2 percent to 141.3 percent. Average standard errors of prediction for the equations based on bankfull width ranged from 63.1 percent to 155.5 percent. In most regions, the equations based on basin and climatic characteristics generally had smaller average standard errors of prediction than equations based on active-channel or bankfull width. An exception was the Southeast Plains Region, where all equations based on active-channel width had smaller average standard errors of prediction than equations based on basin and climatic characteristics or bankfull width. Methods for weighting estimates derived from the basin- and climatic-characteristic equations and the channel-width equations also were developed. The weights were based on the cross correlation of residuals from the different methods and the average standard errors of prediction. When all three methods were combined, the average standard errors of prediction ranged from 37.4 percent to 120.2 percent. Weighting of estimates reduced the standard errors of prediction for all T-year flood estimates in four regions, reduced the standard errors of prediction for some T-year flood estimates in two regions, and provided no reduction in average standard error of prediction in two regions. A computer program for solving the regression equations, weighting estimates, and determining reliability of individual estimates was developed and placed on the USGS Montana District World Wide Web page. A new regression method, termed Region of Influence regression, also was tested. Test results indicated that the Region of Influence method was not as reliable as the regional equations based on generalized least squares regression. Two additional methods for estimating flood frequency at ungaged sites located on the same streams as gaged sites also are described. The first method, based on a drainage-area-ratio adjustment, is intended for use on streams where the ungaged site of interest is located near a gaged site. The second method, based on interpolation between gaged sites, is intended for use on streams that have two or more streamflow-gaging stations.

Comparison of Numerical Approaches to a Steady-State Landscape Equation

NASA Astrophysics Data System (ADS)

Bachman, S.; Peckham, S.

2008-12-01

A mathematical model of an idealized fluvial landscape has been developed, in which a land surface will evolve to preserve dendritic channel networks as the surface is lowered. The physical basis for this model stems from the equations for conservation of mass for water and sediment. These equations relate the divergence of the 2D vector fields showing the unit-width discharge of water and sediment to the excess rainrate and tectonic uplift on the land surface. The 2D flow direction is taken to be opposite to the water- surface gradient vector. These notions are combined with a generalized Manning-type flow resistance formula and a generalized sediment transport law to give a closed mathematical system that can, in principle, be solved for all variables of interest: discharge of water and sediment, land surface height, vertically- averaged flow velocity, water depth, and shear stress. The hydraulic geometry equations (Leopold et. al, 1964, 1995) are used to incorporate width, depth, velocity, and slope of river channels as powers of the mean-annual river discharge. Combined, they give the unit- width discharge of the stream as a power, γ, of the water surface slope. The simplified steady-state model takes into account three components among those listed above: conservation of mass for water, flow opposite the gradient, and a slope-discharge exponent γ = -1 to reflect mature drainage networks. The mathematical representation of this model appears as a second-order hyperbolic partial differential equation (PDE) where the diffusivity is inversely proportional to the square of the local surface slope. The highly nonlinear nature of this PDE has made it very difficult to solve both analytically and numerically. We present simplistic analytic solutions to this equation which are used to test the validity of the numerical algorithms. We also present three such numerical approaches which have been used in solving the differential equation. The first is based on a nonlinear diffusion filtering technique (Welk et. al, 2007) that has been applied successfully in the context of image processing. The second uses a Ritz finite element approach to the Euler-Lagrange formulation of the PDE in which an eighth degree polynomial is solved whose coefficients are locally dependent on slope and elevation. Lastly, we show a variant to the diffusion filtering approach in which a single-stage Runge-Kutta method is used to iterate a time-derivative to steady- state. The relative merits and drawbacks of these approaches are discussed, as well as stability and consistency requirements.

Variational objective analysis for cyclone studies

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.

1989-01-01

Significant accomplishments during 1987 to 1988 are summarized with regard to each of the major project components. Model 1 requires satisfaction of two nonlinear horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation. Model 2 requires satisfaction of model 1 plus the thermodynamic equation for a dry atmosphere. Model 3 requires satisfaction of model 2 plus the radiative transfer equation. Model 4 requires satisfaction of model 3 plus a moisture conservation equation and a parameterization for moist processes.

Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations

NASA Astrophysics Data System (ADS)

He, Wei

2015-02-01

The Gauge/Bethe correspondence relates Omega-deformed N = 2 supersymmetric gauge theories to some quantum integrable models, in simple cases the integrable models can be treated as solvable quantum mechanics models. For SU(2) gauge theory with an adjoint matter, or with 4 fundamental matters, the potential of corresponding quantum model is the elliptic function. If the mass of matter takes special value then the potential is an elliptic solution of KdV hierarchy. We show that the deformed prepotential of gauge theory can be obtained from the average densities of conserved charges of the classical KdV solution, the UV gauge coupling dependence is assembled into the Eisenstein series. The gauge theory with adjoint mass is taken as the example.

Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model.

PubMed

Schuss, Z; Nadler, B; Eisenberg, R S

2001-09-01

Permeation of ions from one electrolytic solution to another, through a protein channel, is a biological process of considerable importance. Permeation occurs on a time scale of micro- to milliseconds, far longer than the femtosecond time scales of atomic motion. Direct simulations of atomic dynamics are not yet possible for such long-time scales; thus, averaging is unavoidable. The question is what and how to average. In this paper, we average a Langevin model of ionic motion in a bulk solution and protein channel. The main result is a coupled system of averaged Poisson and Nernst-Planck equations (CPNP) involving conditional and unconditional charge densities and conditional potentials. The resulting NP equations contain the averaged force on a single ion, which is the sum of two components. The first component is the gradient of a conditional electric potential that is the solution of Poisson's equation with conditional and permanent charge densities and boundary conditions of the applied voltage. The second component is the self-induced force on an ion due to surface charges induced only by that ion at dielectric interfaces. The ion induces surface polarization charge that exerts a significant force on the ion itself, not present in earlier PNP equations. The proposed CPNP system is not complete, however, because the electric potential satisfies Poisson's equation with conditional charge densities, conditioned on the location of an ion, while the NP equations contain unconditional densities. The conditional densities are closely related to the well-studied pair-correlation functions of equilibrium statistical mechanics. We examine a specific closure relation, which on the one hand replaces the conditional charge densities by the unconditional ones in the Poisson equation, and on the other hand replaces the self-induced force in the NP equation by an effective self-induced force. This effective self-induced force is nearly zero in the baths but is approximately equal to the self-induced force in and near the channel. The charge densities in the NP equations are interpreted as time averages over long times of the motion of a quasiparticle that diffuses with the same diffusion coefficient as that of a real ion, but is driven by the averaged force. In this way, continuum equations with averaged charge densities and mean-fields can be used to describe permeation through a protein channel.

Integrability and Poisson Structures of Three Dimensional Dynamical Systems and Equations of Hydrodynamic Type

NASA Astrophysics Data System (ADS)

Gumral, Hasan

Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We take advantage of this fact and use the theory of foliations in discussing the geometrical structure underlying complete and partial integrability. We show that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a non-trivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of 3-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sl_2 structure is a quadratic unfolding of an integrable 1-form in 3 + 1 dimensions. We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation and a continuum limit of Toda lattice. We present further infinite sequences of conserved quantities for shallow water equations and show that their generalizations by Kodama admit bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.

The method of averages applied to the KS differential equations

NASA Technical Reports Server (NTRS)

Graf, O. F., Jr.; Mueller, A. C.; Starke, S. E.

1977-01-01

A new approach for the solution of artificial satellite trajectory problems is proposed. The basic idea is to apply an analytical solution method (the method of averages) to an appropriate formulation of the orbital mechanics equations of motion (the KS-element differential equations). The result is a set of transformed equations of motion that are more amenable to numerical solution.

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov-Poisson Equation during Violent Relaxation

NASA Astrophysics Data System (ADS)

Beraldo e Silva, Leandro; de Siqueira Pedra, Walter; Sodré, Laerte; Perico, Eder L. D.; Lima, Marcos

2017-09-01

The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov-Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes, NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening), and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor, and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov-Poisson equation, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called “fundamental paradox of stellar dynamics.” The long-term evolution is well-described by the orbit-averaged Fokker-Planck model, with Coulomb logarithm values in the expected range 10{--}12. By means of NBODY-2, we also study the dependence of the two-body relaxation timescale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of N-body systems.

Slip and barodiffusion phenomena in slow flows of a gas mixture

NASA Astrophysics Data System (ADS)

Zhdanov, V. M.

2017-03-01

The slip and barodiffusion problems for the slow flows of a gas mixture are investigated on the basis of the linearized moment equations following from the Boltzmann equation. We restrict ourselves to the set of the third-order moment equations and state two general relations (resembling conservation equations) for the moments of the distribution function similar to the conditions used by Loyalka [S. K. Loyalka, Phys. Fluids 14, 2291 (1971), 10.1063/1.1693331] in his approximation method (the modified Maxwell method). The expressions for the macroscopic velocities of the gas mixture species, the partial viscous stress tensors, and the reduced heat fluxes for the stationary slow flow of a gas mixture in the semi-infinite space over a plane wall are obtained as a result of the exact solution of the linearized moment equations in the 10- and 13-moment approximations. The general expression for the slip velocity and the simple and accurate expressions for the viscous, thermal, diffusion slip, and baroslip coefficients, which are given in terms of the basic transport coefficients, are derived by using the modified Maxwell method. The solutions of moment equations are also used for investigation of the flow and diffusion of a gas mixture in a channel formed by two infinite parallel plates. A fundamental result is that the barodiffusion factor in the cross-section-averaged expression for the diffusion flux contains contributions associated with the viscous transfer of momentum in the gas mixture and the effect of the Knudsen layer. Our study revealed that the barodiffusion factor is equal to the diffusion slip coefficient (correct to the opposite sign). This result is consistent with the Onsager's reciprocity relations for kinetic coefficients following from nonequilibrium thermodynamics of the discontinuous systems.

18 CFR 301.7 - Average System Cost methodology functionalization.

Code of Federal Regulations, 2014 CFR

2014-04-01

... 18 Conservation of Power and Water Resources 1 2014-04-01 2014-04-01 false Average System Cost methodology functionalization. 301.7 Section 301.7 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.7 Average System Cost methodology functionalization. (a) Functionalization of each Account...

18 CFR 301.7 - Average System Cost methodology functionalization.

Code of Federal Regulations, 2012 CFR

2012-04-01

... 18 Conservation of Power and Water Resources 1 2012-04-01 2012-04-01 false Average System Cost methodology functionalization. 301.7 Section 301.7 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.7 Average System Cost methodology functionalization. (a) Functionalization of each Account...

18 CFR 301.7 - Average System Cost methodology functionalization.

Code of Federal Regulations, 2013 CFR

2013-04-01

... 18 Conservation of Power and Water Resources 1 2013-04-01 2013-04-01 false Average System Cost methodology functionalization. 301.7 Section 301.7 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.7 Average System Cost methodology functionalization. (a) Functionalization of each Account...

18 CFR 301.7 - Average System Cost methodology functionalization.

Code of Federal Regulations, 2011 CFR

2011-04-01

... 18 Conservation of Power and Water Resources 1 2011-04-01 2011-04-01 false Average System Cost methodology functionalization. 301.7 Section 301.7 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.7 Average System Cost methodology functionalization. (a) Functionalization of each Account...

Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates

NASA Astrophysics Data System (ADS)

Constantin, A.; Johnson, R. S.

2017-04-01

Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.

Noether symmetries and stability of ideal gas solutions in Galileon cosmology

NASA Astrophysics Data System (ADS)

Dimakis, N.; Giacomini, Alex; Jamal, Sameerah; Leon, Genly; Paliathanasis, Andronikos

2017-03-01

A class of generalized Galileon cosmological models, which can be described by a pointlike Lagrangian, is considered in order to utilize Noether's theorem to determine conservation laws for the field equations. In the Friedmann-Lemaître-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Because of the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

Research in computational fluid dynamics

NASA Technical Reports Server (NTRS)

Murman, Earll M.

1987-01-01

The numerical integration of quasi-one-dimensional unsteady flow problems which involve finite rate chemistry are discussed, and are expressed in terms of conservative form Euler and species conservation equations. Hypersonic viscous calculations for delta wing geometries is also examined. The conical Navier-Stokes equations model was selected in order to investigate the effects of viscous-inviscid interations. The more complete three-dimensional model is beyond the available computing resources. The flux vector splitting method with van Leer's MUSCL differencing is being used. Preliminary results were computed for several conditions.

Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrodinger equations

DOE PAGES

Liang, Xiao; Khaliq, Abdul Q. M.; Xing, Yulong

2015-01-23

In this paper, we study a local discontinuous Galerkin method combined with fourth order exponential time differencing Runge-Kutta time discretization and a fourth order conservative method for solving the nonlinear Schrödinger equations. Based on different choices of numerical fluxes, we propose both energy-conserving and energy-dissipative local discontinuous Galerkin methods, and have proven the error estimates for the semi-discrete methods applied to linear Schrödinger equation. The numerical methods are proven to be highly efficient and stable for long-range soliton computations. Finally, extensive numerical examples are provided to illustrate the accuracy, efficiency and reliability of the proposed methods.

A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation

NASA Technical Reports Server (NTRS)

Chuang, Shun-Lien

1987-01-01

Two sets of coupled-mode equations for multiwaveguide systems are derived using a generalized reciprocity relation; one set for a lossless system, and the other for a general lossy or lossless system. The second set of equations also reduces to those of the first set in the lossless case under the condition that the transverse field components are chosen to be real. Analytical relations between the coupling coefficients are shown and applied to the coupling of mode equations. It is shown analytically that these results satisfy exactly both the reciprocity theorem and power conservation. New orthogonal relations between the supermodes are derived in matrix form, with the overlap integrals taken into account.

Geometry of Lax pairs: Particle motion and Killing-Yano tensors

NASA Astrophysics Data System (ADS)

Cariglia, Marco; Frolov, Valeri P.; Krtouš, Pavel; Kubizňák, David

2013-01-01

A geometric formulation of the Lax pair equation on a curved manifold is studied using the phase-space formalism. The corresponding (covariantly conserved) Lax tensor is defined and the method of generation of constants of motion from it is discussed. It is shown that when the Hamilton equations of motion are used, the conservation of the Lax tensor translates directly to the well-known Lax pair equation, with one matrix identified with components of the Lax tensor and the other matrix constructed from the (metric) connection. A generalization to Clifford objects is also discussed. Nontrivial examples of Lax tensors for geodesic and charged particle motion are found in spacetimes admitting a hidden symmetry of Killing-Yano tensors.

Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle.

PubMed

Shalymov, Dmitry S; Fradkov, Alexander L

2016-01-01

We propose dynamics equations which describe the behaviour of non-stationary processes that follow the maximum Rényi entropy principle. The equations are derived on the basis of the speed-gradient principle originated in the control theory. The maximum of the Rényi entropy principle is analysed for discrete and continuous cases, and both a discrete random variable and probability density function (PDF) are used. We consider mass conservation and energy conservation constraints and demonstrate the uniqueness of the limit distribution and asymptotic convergence of the PDF for both cases. The coincidence of the limit distribution of the proposed equations with the Rényi distribution is examined.

Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle

PubMed Central

2016-01-01

We propose dynamics equations which describe the behaviour of non-stationary processes that follow the maximum Rényi entropy principle. The equations are derived on the basis of the speed-gradient principle originated in the control theory. The maximum of the Rényi entropy principle is analysed for discrete and continuous cases, and both a discrete random variable and probability density function (PDF) are used. We consider mass conservation and energy conservation constraints and demonstrate the uniqueness of the limit distribution and asymptotic convergence of the PDF for both cases. The coincidence of the limit distribution of the proposed equations with the Rényi distribution is examined. PMID:26997886

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

NASA Technical Reports Server (NTRS)

Rosenfeld, Moshe; Kwak, Dochan; Vinokur, Marcel

1992-01-01

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

Numerical viscosity and the entropy condition for conservative difference schemes

NASA Technical Reports Server (NTRS)

Tadmor, E.

1983-01-01

Consider a scalar, nonlinear conservative difference scheme satisfying the entropy condition. It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservation equation. In particular, entropy satisfying convergence follows for E schemes - those containing more numerical viscosity than Godunov's scheme.

Equations for estimating bankfull channel geometry and discharge for streams in Massachusetts

USGS Publications Warehouse

Bent, Gardner C.; Waite, Andrew M.

2013-01-01

Regression equations were developed for estimating bankfull geometry—width, mean depth, cross-sectional area—and discharge for streams in Massachusetts. The equations provide water-resource and conservation managers with methods for estimating bankfull characteristics at specific stream sites in Massachusetts. This information can be used for the adminstration of the Commonwealth of Massachusetts Rivers Protection Act of 1996, which establishes a protected riverfront area extending from the mean annual high-water line corresponding to the elevation of bankfull discharge along each side of a perennial stream. Additionally, information on bankfull channel geometry and discharge are important to Federal, State, and local government agencies and private organizations involved in stream assessment and restoration projects. Regression equations are based on data from stream surveys at 33 sites (32 streamgages and 1 crest-stage gage operated by the U.S. Geological Survey) in and near Massachusetts. Drainage areas of the 33 sites ranged from 0.60 to 329 square miles (mi2). At 27 of the 33 sites, field data were collected and analyses were done to determine bankfull channel geometry and discharge as part of the present study. For 6 of the 33 sites, data on bankfull channel geometry and discharge were compiled from other studies done by the U.S. Geological Survey, Natural Resources Conservation Service of the U.S. Department of Agriculture, and the Vermont Department of Environmental Conservation. Similar techniques were used for field data collection and analysis for bankfull channel geometry and discharge at all 33 sites. Recurrence intervals of the bankfull discharge, which represent the frequency with which a stream fills its channel, averaged 1.53 years (median value 1.34 years) at the 33 sites. Simple regression equations were developed for bankfull width, mean depth, cross-sectional area, and discharge using drainage area, which is the most significant explanatory variable in estimating these bankfull characteristics. The use of drainage area as an explanatory variable is also the most commonly published method for estimating these bankfull characteristics. Regional curves (graphic plots) of bankfull channel geometry and discharge by drainage area are presented. The regional curves are based on the simple regression equations and can be used to estimate bankfull characteristics from drainage area. Multiple regression analysis, which includes basin characteristics in addition to drainage area, also was used to develop equations. Variability in bankfull width, mean depth, cross-sectional area, and discharge was more fully explained by the multiple regression equations that include mean-basin slope and drainage area than was explained by equations based on drainage area alone. The Massachusetts regional curves and equations developed in this study are similar, in terms of values of slopes and intercepts, to those developed for other parts of the northeastern United States. Limitations associated with site selection and development of the equations resulted in some constraints for the application of equations and regional curves presented in this report. The curves and equations are applicable to stream sites that have (1) less than about 25 percent of their drainage basin area occupied by urban land use (commercial, industrial, transportation, and high-density residential), (2) little to no streamflow regulation, especially from flood-control structures, (3) drainage basin areas greater than 0.60 mi2 and less than 329 mi2, and (4) a mean basin slope greater than 2.2 percent and less than 23.9 percent. The equations may not be applicable where streams flow through extensive wetlands. The equations also may not apply in areas of Cape Cod and the Islands and the area of southeastern Massachusetts close to Cape Cod with extensive areas of coarse-grained glacial deposits where none of the study sites are located. Regardless of the setting, the regression equations are not intended for use as the sole method of estimating bankfull characteristics; however, they may supplement field identification of the bankfull channel when used in conjunction with field verified bankfull indicators, flood-frequency analysis, or other supporting evidence.

Quantum asymmetry between time and space

PubMed Central

2016-01-01

An asymmetry exists between time and space in the sense that physical systems inevitably evolve over time, whereas there is no corresponding ubiquitous translation over space. The asymmetry, which is presumed to be elemental, is represented by equations of motion and conservation laws that operate differently over time and space. If, however, the asymmetry was found to be due to deeper causes, this conventional view of time evolution would need reworking. Here we show, using a sum-over-paths formalism, that a violation of time reversal (T) symmetry might be such a cause. If T symmetry is obeyed, then the formalism treats time and space symmetrically such that states of matter are localized both in space and in time. In this case, equations of motion and conservation laws are undefined or inapplicable. However, if T symmetry is violated, then the same sum over paths formalism yields states that are localized in space and distributed without bound over time, creating an asymmetry between time and space. Moreover, the states satisfy an equation of motion (the Schrödinger equation) and conservation laws apply. This suggests that the time–space asymmetry is not elemental as currently presumed, and that T violation may have a deep connection with time evolution. PMID:26997899

Effects of Nonequilibrium Chemistry and Darcy-Forchheimer Pyrolysis Flow for Charring Ablator

NASA Technical Reports Server (NTRS)

Chen, Yih-Kanq; Milos, Frank S.

2013-01-01

The fully implicit ablation and thermal response code simulates pyrolysis and ablation of thermal protection materials and systems. The governing equations, which include energy conservation, a three-component decomposition model, and a surface energy balance, are solved with a moving grid.This work describes new modeling capabilities that are added to a special version of code. These capabilities include a time-dependent pyrolysis gas flow momentum equation with Darcy-Forchheimer terms and pyrolysis gas species conservation equations with finite rate homogeneous chemical reactions. The total energy conservation equation is also enhanced for consistency with these new additions. Two groups of parametric studies of the phenolic impregnated carbon ablator are performed. In the first group, an Orion flight environment for a proposed lunar-return trajectory is considered. In the second group, various test conditions for arcjet models are examined. The central focus of these parametric studies is to understand the effect of pyrolysis gas momentum transfer on material in-depth thermal responses with finite-rate, equilibrium, or frozen homogeneous gas chemistry. Results indicate that the presence of chemical nonequilibrium pyrolysis gas flow does not significantly alter the in-depth thermal response performance predicted using the chemical equilibrium gas model.

Thermal conductivity and phonon transport properties of silicon using perturbation theory and the environment-dependent interatomic potential

NASA Astrophysics Data System (ADS)

Pascual-Gutiérrez, José A.; Murthy, Jayathi Y.; Viskanta, Raymond

2009-09-01

Silicon thermal conductivities are obtained from the solution of the linearized phonon Boltzmann transport equation without the use of any parameter-fitting. Perturbation theory is used to compute the strength of three-phonon and isotope scattering mechanisms. Matrix elements based on Fermi's golden rule are computed exactly without assuming either average or mode-dependent Grüeisen parameters, and with no underlying assumptions of crystal isotropy. The environment-dependent interatomic potential is employed to describe the interatomic force constants and the perturbing Hamiltonians. A detailed methodology to accurately find three-phonon processes satisfying energy- and momentum-conservation rules is also described. Bulk silicon thermal conductivity values are computed across a range of temperatures and shown to match experimental data very well. It is found that about two-thirds of the heat transport in bulk silicon may be attributed to transverse acoustic modes. Effective relaxation times and mean free paths are computed in order to provide a more complete picture of the detailed transport mechanisms and for use with carrier transport models based on the Boltzmann transport equation.

This Is Rocket Science!

NASA Astrophysics Data System (ADS)

Keith, Wayne; Martin, Cynthia; Veltkamp, Pamela

2013-09-01

Using model rockets to teach physics can be an effective way to engage students in learning. In this paper, we present a curriculum developed in response to an expressed need for helping high school students review physics equations in preparation for a state-mandated exam. This required a mode of teaching that was more advanced and analytical than that offered by Estes Industries, but more basic than the analysis of Nelson et al. In particular, drag is neglected until the very end of the exercise, which allows the concept of conservation of energy to be shown when predicting the rocket's flight. Also, the variable mass of the rocket motor is assumed to decrease linearly during the flight (while the propulsion charge and recovery delay charge are burning) and handled simplistically by using an average mass value. These changes greatly simplify the equations needed to predict the times and heights at various stages of flight, making it more useful as a review of basic physics. Details about model rocket motors, range safety, and other supplemental information may be found online at Apogee Components4 and the National Association of Rocketry.5

Gravitational self-force on generic bound geodesics in Kerr spacetime

NASA Astrophysics Data System (ADS)

van de Meent, Maarten

2018-05-01

In this work we present the first calculation of the gravitational self-force on generic bound geodesics in Kerr spacetime to first order in the mass ratio. That is, the local correction to equations of motion for a compact object orbiting a larger rotating black hole due to its own impact on the gravitational field. This includes both dissipative and conservative effects. Our method builds on and extends earlier methods for calculating the gravitational self-force on equatorial orbits. In particular we reconstruct the local metric perturbation in the outgoing radiation gauge from the Weyl scalar ψ4 , which in turn is obtained by solving the Teukolsky equation using semianalytical frequency domain methods. The gravitational self-force is subsequently obtained using (spherical) l -mode regularization. We test our implementation by comparing the large l -behavior against the analytically known regularization parameters. In addition we validate our results by comparing the long-term average changes to the energy, angular momentum, and Carter constant to changes to these constants of motion inferred from the gravitational wave flux to infinity and down the horizon.

18 CFR 301.5 - Changes in Average System Cost methodology.

Code of Federal Regulations, 2012 CFR

2012-04-01

... 18 Conservation of Power and Water Resources 1 2012-04-01 2012-04-01 false Changes in Average System Cost methodology. 301.5 Section 301.5 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.5 Changes in Average System Cost methodology. (a) The Administrator, at his or her...

18 CFR 301.5 - Changes in Average System Cost methodology.

Code of Federal Regulations, 2014 CFR

2014-04-01

... 18 Conservation of Power and Water Resources 1 2014-04-01 2014-04-01 false Changes in Average System Cost methodology. 301.5 Section 301.5 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.5 Changes in Average System Cost methodology. (a) The Administrator, at his or her...

18 CFR 301.5 - Changes in Average System Cost methodology.

Code of Federal Regulations, 2013 CFR

2013-04-01

... 18 Conservation of Power and Water Resources 1 2013-04-01 2013-04-01 false Changes in Average System Cost methodology. 301.5 Section 301.5 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.5 Changes in Average System Cost methodology. (a) The Administrator, at his or her...

18 CFR 301.5 - Changes in Average System Cost methodology.

Code of Federal Regulations, 2011 CFR

2011-04-01

... 18 Conservation of Power and Water Resources 1 2011-04-01 2011-04-01 false Changes in Average System Cost methodology. 301.5 Section 301.5 Conservation of Power and Water Resources FEDERAL ENERGY... ACT § 301.5 Changes in Average System Cost methodology. (a) The Administrator, at his or her...

Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes

NASA Astrophysics Data System (ADS)

Mohamed, Mamdouh S.; Hirani, Anil N.; Samtaney, Ravi

2016-05-01

A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.

Numerical Simulation of a Solar Domestic Hot Water System

NASA Astrophysics Data System (ADS)

Mongibello, L.; Bianco, N.; Di Somma, M.; Graditi, G.; Naso, V.

2014-11-01

An innovative transient numerical model is presented for the simulation of a solar Domestic Hot Water (DHW) system. The solar collectors have been simulated by using a zerodimensional analytical model. The temperature distributions in the heat transfer fluid and in the water inside the tank have been evaluated by one-dimensional models. The reversion elimination algorithm has been used to include the effects of natural convection among the water layers at different heights in the tank on the thermal stratification. A finite difference implicit scheme has been implemented to solve the energy conservation equation in the coil heat exchanger, and the energy conservation equation in the tank has been solved by using the finite difference Euler implicit scheme. Energy conservation equations for the solar DHW components models have been coupled by means of a home-made implicit algorithm. Results of the simulation performed using as input data the experimental values of the ambient temperature and the solar irradiance in a summer day are presented and discussed.

Simulations of incompressible Navier Stokes equations on curved surfaces using discrete exterior calculus

NASA Astrophysics Data System (ADS)

Samtaney, Ravi; Mohamed, Mamdouh; Hirani, Anil

2015-11-01

We present examples of numerical solutions of incompressible flow on 2D curved domains. The Navier-Stokes equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. A conservative discretization of Navier-Stokes equations on simplicial meshes is developed based on discrete exterior calculus (DEC). The discretization is then carried out by substituting the corresponding discrete operators based on the DEC framework. By construction, the method is conservative in that both the discrete divergence and circulation are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step. Numerical examples include Taylor vortices on a sphere, Stuart vortices on a sphere, and flow past a cylinder on domains with varying curvature. Supported by the KAUST Office of Competitive Research Funds under Award No. URF/1/1401-01.

Development of monitoring system of helium leakage from canister

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

Toriu, D.; Ushijima, S.; Takeda, H.

2013-07-01

This paper presents a computational method for the helium leakage from a canister. The governing equations for compressible fluids consist of mass conservation equation in Eulerian description, momentum equations and energy equation. The numerical procedures are divided into three phases, advection, diffusion and acoustic phases, and the equations of compressible fluids are discretized with a finite volume method. Thus, the mass conservation law is sufficiently satisfied in the calculation region. In particular, our computational method enables us to predict the change of the temperature distributions around the canister boundaries by calculating the governing equations for the compressible gas flows, whichmore » are leaked out from a slight crack on the canister boundary. In order to confirm the validity of our method, it was applied to the basic problem, 2-dimensional natural convection flows in a rectangular cavity. As a result, it was shown that the naturally convected flows can be reasonably simulated by our method. Furthermore, numerical experiments were conducted for the helium leakage from canister and we derived a close relationship between the inner pressure and the boundary temperature distributions.« less

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.

Nonlinear (time domain) and linearized (time and frequency domain) solutions to the compressible Euler equations in conservation law form

NASA Technical Reports Server (NTRS)

Sreenivas, Kidambi; Whitfield, David L.

1995-01-01

Two linearized solvers (time and frequency domain) based on a high resolution numerical scheme are presented. The basic approach is to linearize the flux vector by expressing it as a sum of a mean and a perturbation. This allows the governing equations to be maintained in conservation law form. A key difference between the time and frequency domain computations is that the frequency domain computations require only one grid block irrespective of the interblade phase angle for which the flow is being computed. As a result of this and due to the fact that the governing equations for this case are steady, frequency domain computations are substantially faster than the corresponding time domain computations. The linearized equations are used to compute flows in turbomachinery blade rows (cascades) arising due to blade vibrations. Numerical solutions are compared to linear theory (where available) and to numerical solutions of the nonlinear Euler equations.

Central Upwind Scheme for a Compressible Two-Phase Flow Model

PubMed Central

Ahmed, Munshoor; Saleem, M. Rehan; Zia, Saqib; Qamar, Shamsul

2015-01-01

In this article, a compressible two-phase reduced five-equation flow model is numerically investigated. The model is non-conservative and the governing equations consist of two equations describing the conservation of mass, one for overall momentum and one for total energy. The fifth equation is the energy equation for one of the two phases and it includes source term on the right-hand side which represents the energy exchange between two fluids in the form of mechanical and thermodynamical work. For the numerical approximation of the model a high resolution central upwind scheme is implemented. This is a non-oscillatory upwind biased finite volume scheme which does not require a Riemann solver at each time step. Few numerical case studies of two-phase flows are presented. For validation and comparison, the same model is also solved by using kinetic flux-vector splitting (KFVS) and staggered central schemes. It was found that central upwind scheme produces comparable results to the KFVS scheme. PMID:26039242

Central upwind scheme for a compressible two-phase flow model.

PubMed

Ahmed, Munshoor; Saleem, M Rehan; Zia, Saqib; Qamar, Shamsul

2015-01-01

In this article, a compressible two-phase reduced five-equation flow model is numerically investigated. The model is non-conservative and the governing equations consist of two equations describing the conservation of mass, one for overall momentum and one for total energy. The fifth equation is the energy equation for one of the two phases and it includes source term on the right-hand side which represents the energy exchange between two fluids in the form of mechanical and thermodynamical work. For the numerical approximation of the model a high resolution central upwind scheme is implemented. This is a non-oscillatory upwind biased finite volume scheme which does not require a Riemann solver at each time step. Few numerical case studies of two-phase flows are presented. For validation and comparison, the same model is also solved by using kinetic flux-vector splitting (KFVS) and staggered central schemes. It was found that central upwind scheme produces comparable results to the KFVS scheme.

A mass-conserving mixed Fourier-Galerkin B-Spline-collocation method for Direct Numerical Simulation of the variable-density Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Reuter, Bryan; Oliver, Todd; Lee, M. K.; Moser, Robert

2017-11-01

We present an algorithm for a Direct Numerical Simulation of the variable-density Navier-Stokes equations based on the velocity-vorticity approach introduced by Kim, Moin, and Moser (1987). In the current work, a Helmholtz decomposition of the momentum is performed. Evolution equations for the curl and the Laplacian of the divergence-free portion are formulated by manipulation of the momentum equations and the curl-free portion is reconstructed by enforcing continuity. The solution is expanded in Fourier bases in the homogeneous directions and B-Spline bases in the inhomogeneous directions. Discrete equations are obtained through a mixed Fourier-Galerkin and collocation weighted residual method. The scheme is designed such that the numerical solution conserves mass locally and globally by ensuring the discrete divergence projection is exact through the use of higher order splines in the inhomogeneous directions. The formulation is tested on multiple variable-density flow problems.

Structure-preserving spectral element method in attenuating seismic wave modeling

NASA Astrophysics Data System (ADS)

Cai, Wenjun; Zhang, Huai

2016-04-01

This work describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method. The conformal symplectic method is a variation of conventional symplectic methods to treat non-conservative time evolution problems which has superior behaviors in long-time stability and dissipation preservation. To construct the conformal symplectic method, we first reformulate the damped acoustic wave equation and the elastic wave equations in their equivalent conformal multi-symplectic structures, which naturally reveal the intrinsic properties of the original systems, especially, the dissipation laws. We thereafter separate each structures into a conservative Hamiltonian system and a purely dissipative ordinary differential equation system. Based on the splitting methodology, we solve the two subsystems respectively. The dissipative one is cheaply solved by its analytic solution. While for the conservative system, we combine a fourth-order symplectic Nyström method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free-surface topography. The Strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed numerical scheme, which is conformal symplectic and can therefore guarantee the numerical stability and dissipation preservation after a large time modeling. Additionally, a relative larger Courant number than that of the traditional Newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength. A benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate. The classical Lamb problem is used to demonstrate the ability of modeling Rayleigh-wave propagation. More comprehensive numerical experiments are presented to investigate the long-time simulation, low dispersion and energy conservation properties of the conformal symplectic method in both the attenuating homogeneous and heterogeneous mediums.

Mechanical balance laws for fully nonlinear and weakly dispersive water waves

NASA Astrophysics Data System (ADS)

Kalisch, Henrik; Khorsand, Zahra; Mitsotakis, Dimitrios

2016-10-01

The Serre-Green-Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and two-dimensional. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre-Green-Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre-Green-Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre-Green-Naghdi equations using a finite-element discretization coupled with an adaptive Runge-Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre-Green-Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.

Laminar flamelets, conserved scalars, and non-unity Lewis numbers: What does this have to do with chemistry?

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

Miller, H.J.; Marro, M.A.T.; Smooke, M.

1994-12-31

In general, computation of laminar flame structure involves the simultaneous solution of the conservation equations for mass, energy, momentum, and chemical species. It has been proposed and confirmed in numerous experiments that flame species concentrations can be considered as functions of a conserved scalar (a quantity such as elemental mass fraction, that has no chemical source term). One such conserved scalar is the mixture fraction which is normalized to be zero in the air stream and one in the fuel stream. This allows the species conservation equations to be rewritten as a function of the mixture fraction (itself a conservedmore » scalar) which significantly simplifies the calculation of flame structure. Despite the widespread acceptance that the conserved scalar description of diffusion flame structure has found in the combustion community, there has been surprisingly little effort expended in the development of a detailed evaluation of how well it actually works. In this presentation we compare the results of a {open_quotes}full{close_quotes} transport and chemical calculation performed by Smooke with the predictions of the conserved scalar approach. Our results show that the conserved scalar approach works because some species` concentrations are not dependent only on mixture fraction.« less

A Model to Couple Flow, Thermal and Reactive Chemical Transport, and Geo-mechanics in Variably Saturated Media

NASA Astrophysics Data System (ADS)

Yeh, G. T.; Tsai, C. H.

2015-12-01

This paper presents the development of a THMC (thermal-hydrology-mechanics-chemistry) process model in variably saturated media. The governing equations for variably saturated flow and reactive chemical transport are obtained based on the mass conservation principle of species transport supplemented with Darcy's law, constraint of species concentration, equation of states, and constitutive law of K-S-P (Conductivity-Degree of Saturation-Capillary Pressure). The thermal transport equation is obtained based on the conservation of energy. The geo-mechanic displacement is obtained based on the assumption of equilibrium. Conventionally, these equations have been implicitly coupled via the calculations of secondary variables based on primary variables. The mechanisms of coupling have not been obvious. In this paper, governing equations are explicitly coupled for all primary variables. The coupling is accomplished via the storage coefficients, transporting velocities, and conduction-dispersion-diffusion coefficient tensor; one set each for every primary variable. With this new system of equations, the coupling mechanisms become clear. Physical interpretations of every term in the coupled equations will be discussed. Examples will be employed to demonstrate the intuition and superiority of these explicit coupling approaches. Keywords: Variably Saturated Flow, Thermal Transport, Geo-mechanics, Reactive Transport.

Time-dependent spectral renormalization method

NASA Astrophysics Data System (ADS)

Cole, Justin T.; Musslimani, Ziad H.

2017-11-01

The spectral renormalization method was introduced by Ablowitz and Musslimani (2005) as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel's principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable PT symmetric nonlocal NLS and the viscous Burgers' equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

Conservation laws with coinciding smooth solutions but different conserved variables

NASA Astrophysics Data System (ADS)

Colombo, Rinaldo M.; Guerra, Graziano

2018-04-01

Consider two hyperbolic systems of conservation laws in one space dimension with the same eigenvalues and (right) eigenvectors. We prove that solutions to Cauchy problems with the same initial data differ at third order in the total variation of the initial datum. As a first application, relying on the classical Glimm-Lax result (Glimm and Lax in Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs of the American Mathematical Society, No. 101. American Mathematical Society, Providence, 1970), we obtain estimates improving those in Saint-Raymond (Arch Ration Mech Anal 155(3):171-199, 2000) on the distance between solutions to the isentropic and non-isentropic inviscid compressible Euler equations, under general equations of state. Further applications are to the general scalar case, where rather precise estimates are obtained, to an approximation by Di Perna of the p-system and to a traffic model.

Estimating soil erosion risk and evaluating erosion control measures for soil conservation planning at Koga watershed in the highlands of Ethiopia

NASA Astrophysics Data System (ADS)

Molla, Tegegne; Sisheber, Biniam

2017-01-01

Soil erosion is one of the major factors affecting sustainability of agricultural production in Ethiopia. The objective of this paper is to estimate soil erosion using the universal soil loss equation (RUSLE) model and to evaluate soil conservation practices in a data-scarce watershed region. For this purpose, soil data, rainfall, erosion control practices, satellite images and topographic maps were collected to determine the RUSLE factors. In addition, measurements of randomly selected soil and water conservation structures were done at three sub-watersheds (Asanat, Debreyakob and Rim). This study was conducted in Koga watershed at upper part of the Blue Nile basin which is affected by high soil erosion rates. The area is characterized by undulating topography caused by intensive agricultural practices with poor soil conservation practices. The soil loss rates were determined and conservation strategies have been evaluated under different slope classes and land uses. The results showed that the watershed is affected by high soil erosion rates (on average 42 t ha-1 yr-1), greater than the maximum tolerable soil loss (18 t ha-1 yr-1). The highest soil loss (456 t ha-1 yr-1) estimated from the upper watershed occurred on cultivated lands of steep slopes. As a result, soil erosion is mainly aggravated by land-use conflicts and topographic factors and the rugged topographic land forms of the area. The study also demonstrated that the contribution of existing soil conservation structures to erosion control is very small due to incorrect design and poor management. About 35 % out of the existing structures can reduce soil loss significantly since they were constructed correctly. Most of the existing structures were demolished due to the sediment overload, vulnerability to livestock damage and intense rainfall. Therefore, appropriate and standardized soil and water conservation measures for different erosion-prone land uses and land forms need to be implemented in Koga watershed.

Optical solitons, nonlinear self-adjointness and conservation laws for the cubic nonlinear Shrödinger's equation with repulsive delta potential

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi

2017-11-01

In this paper, the complex envelope function ansatz method is used to acquire the optical solitons to the cubic nonlinear Shrödinger's equation with repulsive delta potential (δ-NLSE). The method reveals dark and bright optical solitons. The necessary constraint conditions which guarantee the existence of the solitons are also presented. We studied the δ-NLSE by analyzing a system of partial differential equations (PDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system and prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conserved vectors for the system using the general Cls theorem presented by Ibragimov. Some interesting figures for the acquired solutions are also presented.

Lie Symmetry Analysis and Conservation Laws of a Generalized Time Fractional Foam Drainage Equation

NASA Astrophysics Data System (ADS)

Wang, Li; Tian, Shou-Fu; Zhao, Zhen-Tao; Song, Xiao-Qiu

2016-07-01

In this paper, a generalized time fractional nonlinear foam drainage equation is investigated by means of the Lie group analysis method. Based on the Riemann—Liouville derivative, the Lie point symmetries and symmetry reductions of the equation are derived, respectively. Furthermore, conservation laws with two kinds of independent variables of the equation are performed by making use of the nonlinear self-adjointness method. Supported by the National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant No. 201410290039, the Fundamental Research Funds for the Central Universities under Grant Nos. 2015QNA53 and 2015XKQY14, the Fundamental Research Funds for Postdoctoral at the Key Laboratory of Gas and Fire Control for Coal Mines, the General Financial Grant from the China Postdoctoral Science Foundation under Grant No. 2015M570498, and Natural Sciences Foundation of China under Grant No. 11301527

The development of flux-split algorithms for flows with non-equilibrium thermodynamics and chemical reactions

NASA Technical Reports Server (NTRS)

Grossman, B.; Cinella, P.

1988-01-01

A finite-volume method for the numerical computation of flows with nonequilibrium thermodynamics and chemistry is presented. A thermodynamic model is described which simplifies the coupling between the chemistry and thermodynamics and also results in the retention of the homogeneity property of the Euler equations (including all the species continuity and vibrational energy conservation equations). Flux-splitting procedures are developed for the fully coupled equations involving fluid dynamics, chemical production and thermodynamic relaxation processes. New forms of flux-vector split and flux-difference split algorithms are embodied in a fully coupled, implicit, large-block structure, including all the species conservation and energy production equations. Several numerical examples are presented, including high-temperature shock tube and nozzle flows. The methodology is compared to other existing techniques, including spectral and central-differenced procedures, and favorable comparisons are shown regarding accuracy, shock-capturing and convergence rates.

Material point method modeling in oil and gas reservoirs

DOEpatents

Vanderheyden, William Brian; Zhang, Duan

2016-06-28

A computer system and method of simulating the behavior of an oil and gas reservoir including changes in the margins of frangible solids. A system of equations including state equations such as momentum, and conservation laws such as mass conservation and volume fraction continuity, are defined and discretized for at least two phases in a modeled volume, one of which corresponds to frangible material. A material point model technique for numerically solving the system of discretized equations, to derive fluid flow at each of a plurality of mesh nodes in the modeled volume, and the velocity of at each of a plurality of particles representing the frangible material in the modeled volume. A time-splitting technique improves the computational efficiency of the simulation while maintaining accuracy on the deformation scale. The method can be applied to derive accurate upscaled model equations for larger volume scale simulations.

Pair correlation function and nonlinear kinetic equation for a spatially uniform polarizable nonideal plasma

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

Belyi, V.V.; Kukharenko, Y.A.; Wallenborn, J.

Taking into account the first non-Markovian correction to the Balescu-Lenard equation, we have derived an expression for the pair correlation function and a nonlinear kinetic equation valid for a nonideal polarized classical plasma. This last equation allows for the description of the correlational energy evolution and shows the global conservation of energy with dynamical polarization. {copyright} {ital 1996 The American Physical Society.}

Dark optical, singular solitons and conservation laws to the nonlinear Schrödinger’s equation with spatio-temporal dispersion

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi

2017-05-01

This paper studies the dynamics of solitons to the nonlinear Schrödinger’s equation (NLSE) with spatio-temporal dispersion (STD). The integration algorithm that is employed in this paper is the Riccati-Bernoulli sub-ODE method. This leads to dark and singular soliton solutions that are important in the field of optoelectronics and fiber optics. The soliton solutions appear with all necessary constraint conditions that are necessary for them to exist. There are four types of nonlinear media studied in this paper. They are Kerr law, power law, parabolic law and dual law. The conservation laws (Cls) for the Kerr law and parabolic law nonlinear media are constructed using the conservation theorem presented by Ibragimov.

Code Development of Three-Dimensional General Relativistic Hydrodynamics with AMR (Adaptive-Mesh Refinement) and Results from Special and General Relativistic Hydrodynamics

NASA Astrophysics Data System (ADS)

Dönmez, Orhan

2004-09-01

In this paper, the general procedure to solve the general relativistic hydrodynamical (GRH) equations with adaptive-mesh refinement (AMR) is presented. In order to achieve, the GRH equations are written in the conservation form to exploit their hyperbolic character. The numerical solutions of GRH equations are obtained by high resolution shock Capturing schemes (HRSC), specifically designed to solve nonlinear hyperbolic systems of conservation laws. These schemes depend on the characteristic information of the system. The Marquina fluxes with MUSCL left and right states are used to solve GRH equations. First, different test problems with uniform and AMR grids on the special relativistic hydrodynamics equations are carried out to verify the second-order convergence of the code in one, two and three dimensions. Results from uniform and AMR grid are compared. It is found that adaptive grid does a better job when the number of resolution is increased. Second, the GRH equations are tested using two different test problems which are Geodesic flow and Circular motion of particle In order to do this, the flux part of GRH equations is coupled with source part using Strang splitting. The coupling of the GRH equations is carried out in a treatment which gives second order accurate solutions in space and time.

Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach

NASA Astrophysics Data System (ADS)

Camassa, R.; Falqui, G.; Ortenzi, G.

2017-02-01

The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite two-dimensional channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids’ inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, albeit in a non-trivial way. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, a family of approximate constants of the motion are explicitly constructed and used to find local solutions of the evolution equations by means of hodograph-like formulae.

HYDRA-II: A hydrothermal analysis computer code: Volume 2, User's manual

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

McCann, R.A.; Lowery, P.S.; Lessor, D.L.

1987-09-01

HYDRA-II is a hydrothermal computer code capable of three-dimensional analysis of coupled conduction, convection, and thermal radiation problems. This code is especially appropriate for simulating the steady-state performance of spent fuel storage systems. The code has been evaluated for this application for the US Department of Energy's Commercial Spent Fuel Management Program. HYDRA-II provides a finite-difference solution in cartesian coordinates to the equations governing the conservation of mass, momentum, and energy. A cylindrical coordinate system may also be used to enclose the cartesian coordinate system. This exterior coordinate system is useful for modeling cylindrical cask bodies. The difference equations formore » conservation of momentum incorporate directional porosities and permeabilities that are available to model solid structures whose dimensions may be smaller than the computational mesh. The equation for conservation of energy permits modeling of orthotropic physical properties and film resistances. Several automated methods are available to model radiation transfer within enclosures and from fuel rod to fuel rod. The documentation of HYDRA-II is presented in three separate volumes. Volume 1 - Equations and Numerics describes the basic differential equations, illustrates how the difference equations are formulated, and gives the solution procedures employed. This volume, Volume 2 - User's Manual, contains code flow charts, discusses the code structure, provides detailed instructions for preparing an input file, and illustrates the operation of the code by means of a sample problem. The final volume, Volume 3 - Verification/Validation Assessments, provides a comparison between the analytical solution and the numerical simulation for problems with a known solution. 6 refs.« less

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

Băloi, Mihaela-Andreea, E-mail: mihaela.baloi88@e-uvt.ro; Crucean, Cosmin

The production of fermions in dipolar electric fields on de Sitter universe is studied. The amplitude and probability of pair production are computed using the exact solution of the Dirac equation in de Sitter spacetime. The form of the dipolar fields is established using the conformal invariance of the Maxwell equations. We obtain that the momentum conservation law is broken in the process of pair production in dipolar electric fields. Also we establish that there are nonvanishing probabilities for processes in which the helicity is conserved/nonconserved. The Minkowski limit is recovered when the expansion factor becomes zero.

Four-level conservative finite-difference schemes for Boussinesq paradigm equation

NASA Astrophysics Data System (ADS)

Kolkovska, N.

2013-10-01

In this paper a two-parametric family of four level conservative finite difference schemes is constructed for the multidimensional Boussinesq paradigm equation. The schemes are explicit in the sense that no inner iterations are needed for evaluation of the numerical solution. The preservation of the discrete energy with this method is proved. The schemes have been numerically tested on one soliton propagation model and two solitons interaction model. The numerical experiments demonstrate that the proposed family of schemes has second order of convergence in space and time steps in the discrete maximal norm.

Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique

NASA Technical Reports Server (NTRS)

Nordmann, R.; Weiser, P.

1989-01-01

The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.

Spherization of the remnants of asymmetrical SN explosions in a uniform medium

NASA Astrophysics Data System (ADS)

Bisnovatyi-Kogan, G. S.; Blinnikov, S. I.

A 'snow-plow' approximation is used to project a spherical shape for a supernova remnant (SNR) after a shock wave has traveled through a uniform medium following an asymmetrical SN explosion. The asymmetry arises as magnetorotation causes the explosion. It is assumed that the main part of the mass remains in a thin layer after the explosion and that the layer can be described by 1,5-dimensional hydrodynamics. The cavity pressure inside the shock is assumed much greater than the pressure of the outside medium. The snow-plow model accounts for asymmetrical particle velocities in the expanding layer and the tangential velocity averaged across the shock. The equations are configured to conserve mass and momentum and have specific initial conditions. The calculations are in agreement with observations of Cas A.

Simulation of mixing in the quick quench region of a rich burn-quick quench mix-lean burn combustor

NASA Technical Reports Server (NTRS)

Shih, Tom I.-P.; Nguyen, H. Lee; Howe, Gregory W.; Li, Z.

1991-01-01

A computer program was developed to study the mixing process in the quick quench region of a rich burn-quick quench mix-lean burn combustor. The computer program developed was based on the density-weighted, ensemble-averaged conservation equations of mass, momentum (full compressible Navier-Stokes), total energy, and species, closed by a k-epsilon turbulence model with wall functions. The combustion process was modeled by a two-step global reaction mechanism, and NO(x) formation was modeled by the Zeldovich mechanism. The formulation employed in the computer program and the essence of the numerical method of solution are described. Some results obtained for nonreacting and reacting flows with different main-flow to dilution-jet momentum flux ratios are also presented.

Detachment and diffusive-convective transport in an evolving heterogeneous two-dimensional biofilm hybrid model.

PubMed

Luna, E; Domínguez-Zacarias, G; Ferreira, C Pio; Velasco-Hernandez, J X

2004-12-01

Under the hypothesis of correlation between biofilm survival and nutrient availability, by considering fluid drag forces and mortality due to nutrient depletion, a biofilm detachment/breaking condition is derived. The mechanisms leading to biofilm detachment/breaking are discussed. We construct and describe a hybrid model for a heterogeneous biofilm attached to walls in a channel where liquid is flowing. The model is called hybrid because it couples conservation equations with a cellular automaton. The biofilm layer is viewed as a porous medium with variable porosity, tortuosity, and permeability. The model is solved using asymptotic and finite differences methods. Results for porosity, nutrient distribution, and average surface location are presented. The model is capable of reproducing biofilm heterogeneity as well as the typical surface fingering (mushroomlike structure).

HAM2D: 2D Shearing Box Model

NASA Astrophysics Data System (ADS)

Gammie, Charles F.; Guan, Xiaoyue

2012-10-01

HAM solves non-relativistic hyperbolic partial differential equations in conservative form using high-resolution shock-capturing techniques. This version of HAM has been configured to solve the magnetohydrodynamic equations of motion in axisymmetry to evolve a shearing box model.

Thermal evolution of plutons: a parameterized approach.

PubMed

Spera, F

1980-01-18

A conservation-of-energy equation has been derived for the spatially averaged magma temperature in a spherical pluton undergoing simultaneous crystallization and both internal (magma) and external (hydrothermal fluid) thermal convection. The model accounts for the dependence of magma viscosity on crystallinity, temperature, and bulk composition; it includes latent heat effects and the effects of different initial water concentrations in the melt and quantitatively considers the role that large volumes of circulatory hydrothermal fluids play in dissipating heat. The nonlinear ordinary differential equation describing these processes has been solved for a variety of magma compositions, initial termperatures, initial crystallinities, volume ratios of hydrothermal fluid to magma, and pluton sizes. These calculations are graphically summarized in plots of the average magma temperature versus time after emplacement. Solidification times, defined as the time necessary for magma to cool from the initial emplacement temperature to the solidus temperature vary as R(1,3), where R is the pluton radius. The solidification time of a pluton with a radius of 1 kilometer is 5 x 10(4) years; for an otherwise identical pluton with a radius of 10 kilometers, the solidification time is approximately 10(6) years. The water content has a marked effect on the solidification time. A granodiorite pluton with a radius of 5 kilometers and either 0.5 or 4 percent (by weight) water cools in 3.3 x 10(5) or 5 x 10(4) years, respectively. Convection solidification times are usually but not always less than conduction cooling times.

A finite-volume Eulerian-Lagrangian Localized Adjoint Method for solution of the advection-dispersion equation

USGS Publications Warehouse

Healy, R.W.; Russell, T.F.

1993-01-01

A new mass-conservative method for solution of the one-dimensional advection-dispersion equation is derived and discussed. Test results demonstrate that the finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) outperforms standard finite-difference methods, in terms of accuracy and efficiency, for solute transport problems that are dominated by advection. For dispersion-dominated problems, the performance of the method is similar to that of standard methods. Like previous ELLAM formulations, FVELLAM systematically conserves mass globally with all types of boundary conditions. FVELLAM differs from other ELLAM approaches in that integrated finite differences, instead of finite elements, are used to approximate the governing equation. This approach, in conjunction with a forward tracking scheme, greatly facilitates mass conservation. The mass storage integral is numerically evaluated at the current time level, and quadrature points are then tracked forward in time to the next level. Forward tracking permits straightforward treatment of inflow boundaries, thus avoiding the inherent problem in backtracking, as used by most characteristic methods, of characteristic lines intersecting inflow boundaries. FVELLAM extends previous ELLAM results by obtaining mass conservation locally on Lagrangian space-time elements. Details of the integration, tracking, and boundary algorithms are presented. Test results are given for problems in Cartesian and radial coordinates.

Turbulence modeling for hypersonic flows

NASA Technical Reports Server (NTRS)

Marvin, J. G.; Coakley, T. J.

1989-01-01

Turbulence modeling for high speed compressible flows is described and discussed. Starting with the compressible Navier-Stokes equations, methods of statistical averaging are described by means of which the Reynolds-averaged Navier-Stokes equations are developed. Unknown averages in these equations are approximated using various closure concepts. Zero-, one-, and two-equation eddy viscosity models, algebraic stress models and Reynolds stress transport models are discussed. Computations of supersonic and hypersonic flows obtained using several of the models are discussed and compared with experimental results. Specific examples include attached boundary layer flows, shock wave boundary layer interactions and compressible shear layers. From these examples, conclusions regarding the status of modeling and recommendations for future studies are discussed.

Effects of Non-Equilibrium Chemistry and Darcy-Forchheimer Flow of Pyrolysis Gas for a Charring Ablator

NASA Technical Reports Server (NTRS)

Chen, Yih-Kanq; Milos, Frank S.

2011-01-01

The Fully Implicit Ablation and Thermal Response code, FIAT, simulates pyrolysis and ablation of thermal protection materials and systems. The governing equations, which include energy conservation, a three-component decomposition model, and a surface energy balance, are solved with a moving grid. This work describes new modeling capabilities that are added to a special version of FIAT. These capabilities include a time-dependent pyrolysis gas flow momentum equation with Darcy-Forchheimer terms and pyrolysis gas species conservation equations with finite-rate homogeneous chemical reactions. The total energy conservation equation is also enhanced for consistency with these new additions. Parametric studies are performed using this enhanced version of FIAT. Two groups of analyses of Phenolic Impregnated Carbon Ablator (PICA) are presented. In the first group, an Orion flight environment for a proposed Lunar-return trajectory is considered. In the second group, various test conditions for arcjet models are examined. The central focus of these parametric studies is to understand the effect of pyrolysis gas momentum transfer on PICA material in-depth thermal responses with finite-rate, equilibrium, or frozen homogeneous gas chemistry. Results are presented, discussed, and compared with those predicted by the baseline PICA/FIAT ablation and thermal response model developed by the Orion Thermal Protection System Advanced Development Project.

A direct Primitive Variable Recovery Scheme for hyperbolic conservative equations: The case of relativistic hydrodynamics.

PubMed

Aguayo-Ortiz, A; Mendoza, S; Olvera, D

2018-01-01

In this article we develop a Primitive Variable Recovery Scheme (PVRS) to solve any system of coupled differential conservative equations. This method obtains directly the primitive variables applying the chain rule to the time term of the conservative equations. With this, a traditional finite volume method for the flux is applied in order avoid violation of both, the entropy and "Rankine-Hugoniot" jump conditions. The time evolution is then computed using a forward finite difference scheme. This numerical technique evades the recovery of the primitive vector by solving an algebraic system of equations as it is often used and so, it generalises standard techniques to solve these kind of coupled systems. The article is presented bearing in mind special relativistic hydrodynamic numerical schemes with an added pedagogical view in the appendix section in order to easily comprehend the PVRS. We present the convergence of the method for standard shock-tube problems of special relativistic hydrodynamics and a graphical visualisation of the errors using the fluctuations of the numerical values with respect to exact analytic solutions. The PVRS circumvents the sometimes arduous computation that arises from standard numerical methods techniques, which obtain the desired primitive vector solution through an algebraic polynomial of the charges.

Applying Boundary Conditions Using a Time-Dependent Lagrangian for Modeling Laser-Plasma Interactions

NASA Astrophysics Data System (ADS)

Reyes, Jonathan; Shadwick, B. A.

2016-10-01

Modeling the evolution of a short, intense laser pulse propagating through an underdense plasma is of particular interest in the physics of laser-plasma interactions. Numerical models are typically created by first discretizing the equations of motion and then imposing boundary conditions. Using the variational principle of Chen and Sudan, we spatially discretize the Lagrangian density to obtain discrete equations of motion and a discrete energy conservation law which is exactly satisfied regardless of the spatial grid resolution. Modifying the derived equations of motion (e.g., enforcing boundary conditions) generally ruins energy conservation. However, time-dependent terms can be added to the Lagrangian which force the equations of motion to have the desired boundary conditions. Although some foresight is needed to choose these time-dependent terms, this approach provides a mechanism for energy to exit the closed system while allowing the conservation law to account for the loss. An appropriate time discretization scheme is selected based on stability analysis and resolution requirements. We present results using this variational approach in a co-moving coordinate system and compare such results to those using traditional second-order methods. This work was supported by the U. S. Department of Energy under Contract No. DE-SC0008382 and by the National Science Foundation under Contract No. PHY- 1104683.

A direct Primitive Variable Recovery Scheme for hyperbolic conservative equations: The case of relativistic hydrodynamics

PubMed Central

Mendoza, S.; Olvera, D.

2018-01-01

In this article we develop a Primitive Variable Recovery Scheme (PVRS) to solve any system of coupled differential conservative equations. This method obtains directly the primitive variables applying the chain rule to the time term of the conservative equations. With this, a traditional finite volume method for the flux is applied in order avoid violation of both, the entropy and “Rankine-Hugoniot” jump conditions. The time evolution is then computed using a forward finite difference scheme. This numerical technique evades the recovery of the primitive vector by solving an algebraic system of equations as it is often used and so, it generalises standard techniques to solve these kind of coupled systems. The article is presented bearing in mind special relativistic hydrodynamic numerical schemes with an added pedagogical view in the appendix section in order to easily comprehend the PVRS. We present the convergence of the method for standard shock-tube problems of special relativistic hydrodynamics and a graphical visualisation of the errors using the fluctuations of the numerical values with respect to exact analytic solutions. The PVRS circumvents the sometimes arduous computation that arises from standard numerical methods techniques, which obtain the desired primitive vector solution through an algebraic polynomial of the charges. PMID:29659602

The stationary flow in a heterogeneous compliant vessel network

NASA Astrophysics Data System (ADS)

Filoche, Marcel; Florens, Magali

2011-09-01

We introduce a mathematical model of the hydrodynamic transport into systems consisting in a network of connected flexible pipes. In each pipe of the network, the flow is assumed to be steady and one-dimensional. The fluid-structure interaction is described through tube laws which relate the pipe diameter to the pressure difference across the pipe wall. We show that the resulting one-dimensional differential equation describing the flow in the pipe can be exactly integrated if one is able to estimate averages of the Reynolds number along the pipe. The differential equation is then transformed into a non linear scalar equation relating pressures at both ends of the pipe and the flow rate in the pipe. These equations are coupled throughout the network with mass conservation equations for the flow and zero pressure losses at the branching points of the network. This allows us to derive a general model for the computation of the flow into very large inhomogeneous networks consisting of several thousands of flexible pipes. This model is then applied to perform numerical simulations of the human lung airway system at exhalation. The topology of the system and the tube laws are taken from morphometric and physiological data in the literature. We find good qualitative and quantitative agreement between the simulation results and flow-volume loops measured in real patients. In particular, expiratory flow limitation which is an essential characteristic of forced expiration is found to be well reproduced by our simulations. Finally, a mathematical model of a pathology (Chronic Obstructive Pulmonary Disease) is introduced which allows us to quantitatively assess the influence of a moderate or severe alteration of the airway compliances.

Taguchi method for partial differential equations with application in tumor growth.

PubMed

Ilea, M; Turnea, M; Rotariu, M; Arotăriţei, D; Popescu, Marilena

2014-01-01

The growth of tumors is a highly complex process. To describe this process, mathematical models are needed. A variety of partial differential mathematical models for tumor growth have been developed and studied. Most of those models are based on the reaction-diffusion equations and mass conservation law. A variety of modeling strategies have been developed, each focusing on tumor growth. Systems of time-dependent partial differential equations occur in many branches of applied mathematics. The vast majority of mathematical models in tumor growth are formulated in terms of partial differential equations. We propose a mathematical model for the interactions between these three cancer cell populations. The Taguchi methods are widely used by quality engineering scientists to compare the effects of multiple variables, together with their interactions, with a simple and manageable experimental design. In Taguchi's design of experiments, variation is more interesting to study than the average. First, Taguchi methods are utilized to search for the significant factors and the optimal level combination of parameters. Except the three parameters levels, other factors levels other factors levels would not be considered. Second, cutting parameters namely, cutting speed, depth of cut, and feed rate are designed using the Taguchi method. Finally, the adequacy of the developed mathematical model is proved by ANOVA. According to the results of ANOVA, since the percentage contribution of the combined error is as small. Many mathematical models can be quantitatively characterized by partial differential equations. The use of MATLAB and Taguchi method in this article illustrates the important role of informatics in research in mathematical modeling. The study of tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

A conservation law for virus infection kinetics in vitro.

PubMed

Kakizoe, Yusuke; Morita, Satoru; Nakaoka, Shinji; Takeuchi, Yasuhiro; Sato, Kei; Miura, Tomoyuki; Beauchemin, Catherine A A; Iwami, Shingo

2015-07-07

Conservation laws are among the most important properties of a physical system, but are not commonplace in biology. We derived a conservation law from the basic model for viral infections which consists in a small set of ordinary differential equations. We challenged the conservation law experimentally for the case of a virus infection in a cell culture. We found that the derived, conserved quantity remained almost constant throughout the infection period, implying that the derived conservation law holds in this biological system. We also suggest a potential use for the conservation law in evaluating the accuracy of experimental measurements. Copyright © 2015 Elsevier Ltd. All rights reserved.

Distributed Relaxation for Conservative Discretizations

NASA Technical Reports Server (NTRS)

Diskin, Boris; Thomas, James L.

2001-01-01

A multigrid method is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. The way to achieve this efficiency is the distributed relaxation approach. TME solvers employing distributed relaxation have already been demonstrated for nonconservative formulations of high-Reynolds-number viscous incompressible and subsonic compressible flow regimes. The purpose of this paper is to provide foundations for applications of distributed relaxation to conservative discretizations. A direct correspondence between the primitive variable interpolations for calculating fluxes in conservative finite-volume discretizations and stencils of the discretized derivatives in the nonconservative formulation has been established. Based on this correspondence, one can arrive at a conservative discretization which is very efficiently solved with a nonconservative relaxation scheme and this is demonstrated for conservative discretization of the quasi one-dimensional Euler equations. Formulations for both staggered and collocated grid arrangements are considered and extensions of the general procedure to multiple dimensions are discussed.

Modified Maturity Offset Prediction Equations: Validation in Independent Longitudinal Samples of Boys and Girls.

PubMed

Kozieł, Sławomir M; Malina, Robert M

2018-01-01

Predicted maturity offset and age at peak height velocity are increasingly used with youth athletes, although validation studies of the equations indicated major limitations. The equations have since been modified and simplified. The objective of this study was to validate the new maturity offset prediction equations in independent longitudinal samples of boys and girls. Two new equations for boys with chronological age and sitting height and chronological age and stature as predictors, and one equation for girls with chronological age and stature as predictors were evaluated in serial data from the Wrocław Growth Study, 193 boys (aged 8-18 years) and 198 girls (aged 8-16 years). Observed age at peak height velocity for each youth was estimated with the Preece-Baines Model 1. The original prediction equations were included for comparison. Predicted age at peak height velocity was the difference between chronological age at prediction and maturity offset. Predicted ages at peak height velocity with the new equations approximated observed ages at peak height velocity in average maturing boys near the time of peak height velocity; a corresponding window for average maturing girls was not apparent. Compared with observed age at peak height velocity, predicted ages at peak height velocity with the new and original equations were consistently later in early maturing youth and earlier in late maturing youth of both sexes. Predicted ages at peak height velocity with the new equations had reduced variation compared with the original equations and especially observed ages at peak height velocity. Intra-individual variation in predicted ages at peak height velocity with all equations was considerable. The new equations are useful for average maturing boys close to the time of peak height velocity; there does not appear to be a clear window for average maturing girls. The new and original equations have major limitations with early and late maturing boys and girls.

Lie-Mei symmetry and conserved quantities of the Rosenberg problem

NASA Astrophysics Data System (ADS)

Liu, Xiao-Wei; Li, Yuan-Cheng

2011-07-01

The Rosenberg problem is a typical but not too complicated problem of nonholonomic mechanical systems. The Lie—Mei symmetry and the conserved quantities of the Rosenberg problem are studied. For the Rosenberg problem, the Lie and the Mei symmetries for the equation are obtained, the conserved quantities are deduced from them and then the definition and the criterion for the Lie—Mei symmetry of the Rosenberg problem are derived. Finally, the Hojman conserved quantity and the Mei conserved quantity are deduced from the Lie—Mei symmetry.

Numerical Modeling of Conjugate Heat Transfer in Fluid Network

NASA Technical Reports Server (NTRS)

Majumdar, Alok

2004-01-01

Fluid network modeling with conjugate heat transfer has many applications in Aerospace engineering. In modeling unsteady flow with heat transfer, it is important to know the variation of wall temperature in time and space to calculate heat transfer between solid to fluid. Since wall temperature is a function of flow, a coupled analysis of temperature of solid and fluid is necessary. In cryogenic applications, modeling of conjugate heat transfer is of great importance to correctly predict boil-off rate in propellant tanks and chill down of transfer lines. In TFAWS 2003, the present author delivered a paper to describe a general-purpose computer program, GFSSP (Generalized Fluid System Simulation Program). GFSSP calculates flow distribution in complex flow circuit for compressible/incompressible, with or without heat transfer or phase change in all real fluids or mixtures. The flow circuit constitutes of fluid nodes and branches. The mass, energy and specie conservation equations are solved at the nodes where as momentum conservation equations are solved at the branches. The proposed paper describes the extension of GFSSP to model conjugate heat transfer. The network also includes solid nodes and conductors in addition to fluid nodes and branches. The energy conservation equations for solid nodes solves to determine the temperatures of the solid nodes simultaneously with all conservation equations governing fluid flow. The numerical scheme accounts for conduction, convection and radiation heat transfer. The paper will also describe the applications of the code to predict chill down of cryogenic transfer line and boil-off rate of cryogenic propellant storage tank.

Wave equations on anti self dual (ASD) manifolds

NASA Astrophysics Data System (ADS)

Bashingwa, Jean-Juste; Kara, A. H.

2017-11-01

In this paper, we study and perform analyses of the wave equation on some manifolds with non diagonal metric g_{ij} which are of neutral signatures. These include the invariance properties, variational symmetries and conservation laws. In the recent past, wave equations on the standard (space time) Lorentzian manifolds have been performed but not on the manifolds from metrics of neutral signatures.

Einstein's osmotic equilibrium of colloidal suspensions in conservative force fields

NASA Astrophysics Data System (ADS)

Fu, Jinxin; Ou-Yang, H. Daniel

2014-09-01

Predicted by Einstein in his 1905 paper on Brownian motion, colloidal particles in suspension reach osmotic equilibrium under gravity. The idea was demonstrated by J.B. Perrin to win Nobel Prize in Physics in 1926. We show Einstein's equation for osmotic equilibrium can be applied to colloids in a conservative force field generated by optical gradient forces. We measure the osmotic equation of state of 100nm Polystyrene latex particles in the presence of KCl salt and PEG polymer. We also obtain the osmotic compressibility, which is important for determining colloidal stability and the internal chemical potential, which is useful for predicting the phase transition of colloidal systems. This generalization allows for the use of any conservative force fields for systems ranging from colloidal systems to macromolecular solutions.

Lectures series in computational fluid dynamics

NASA Technical Reports Server (NTRS)

Thompson, Kevin W.

1987-01-01

The lecture notes cover the basic principles of computational fluid dynamics (CFD). They are oriented more toward practical applications than theory, and are intended to serve as a unified source for basic material in the CFD field as well as an introduction to more specialized topics in artificial viscosity and boundary conditions. Each chapter in the test is associated with a videotaped lecture. The basic properties of conservation laws, wave equations, and shock waves are described. The duality of the conservation law and wave representations is investigated, and shock waves are examined in some detail. Finite difference techniques are introduced for the solution of wave equations and conservation laws. Stability analysis for finite difference approximations are presented. A consistent description of artificial viscosity methods are provided. Finally, the problem of nonreflecting boundary conditions are treated.

Properties of bright solitons in averaged and unaveraged models for SDG fibres

NASA Astrophysics Data System (ADS)

Kumar, Ajit; Kumar, Atul

1996-04-01

Using the slowly varying envelope approximation and averaging over the fibre cross-section the evolution equation for optical pulses in semiconductor-doped glass (SDG) fibres is derived from the nonlinear wave equation. Bright soliton solutions of this equation are obtained numerically and their properties are studied and compared with those of the bright solitons in the unaveraged model.

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.

Textbook Multigrid Efficiency for Computational Fluid Dynamics Simulations

NASA Technical Reports Server (NTRS)

Brandt, Achi; Thomas, James L.; Diskin, Boris

2001-01-01

Considerable progress over the past thirty years has been made in the development of large-scale computational fluid dynamics (CFD) solvers for the Euler and Navier-Stokes equations. Computations are used routinely to design the cruise shapes of transport aircraft through complex-geometry simulations involving the solution of 25-100 million equations; in this arena the number of wind-tunnel tests for a new design has been substantially reduced. However, simulations of the entire flight envelope of the vehicle, including maximum lift, buffet onset, flutter, and control effectiveness have not been as successful in eliminating the reliance on wind-tunnel testing. These simulations involve unsteady flows with more separation and stronger shock waves than at cruise. The main reasons limiting further inroads of CFD into the design process are: (1) the reliability of turbulence models; and (2) the time and expense of the numerical simulation. Because of the prohibitive resolution requirements of direct simulations at high Reynolds numbers, transition and turbulence modeling is expected to remain an issue for the near term. The focus of this paper addresses the latter problem by attempting to attain optimal efficiencies in solving the governing equations. Typically current CFD codes based on the use of multigrid acceleration techniques and multistage Runge-Kutta time-stepping schemes are able to converge lift and drag values for cruise configurations within approximately 1000 residual evaluations. An optimally convergent method is defined as having textbook multigrid efficiency (TME), meaning the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in the discretized system of equations (residual equations). In this paper, a distributed relaxation approach to achieving TME for Reynolds-averaged Navier-Stokes (RNAS) equations are discussed along with the foundations that form the basis of this approach. Because the governing equations are a set of coupled nonlinear conservation equations with discontinuities (shocks, slip lines, etc.) and singularities (flow- or grid-induced), the difficulties are many. This paper summarizes recent progress towards the attainment of TME in basic CFD simulations.

Method and Apparatus for Predicting Unsteady Pressure and Flow Rate Distribution in a Fluid Network

NASA Technical Reports Server (NTRS)

Majumdar, Alok K. (Inventor)

2009-01-01

A method and apparatus for analyzing steady state and transient flow in a complex fluid network, modeling phase changes, compressibility, mixture thermodynamics, external body forces such as gravity and centrifugal force and conjugate heat transfer. In some embodiments, a graphical user interface provides for the interactive development of a fluid network simulation having nodes and branches. In some embodiments, mass, energy, and specific conservation equations are solved at the nodes, and momentum conservation equations are solved in the branches. In some embodiments, contained herein are data objects for computing thermodynamic and thermophysical properties for fluids. In some embodiments, the systems of equations describing the fluid network are solved by a hybrid numerical method that is a combination of the Newton-Raphson and successive substitution methods.

Numerical investigation of shock induced bubble collapse in water

NASA Astrophysics Data System (ADS)

Apazidis, N.

2016-04-01

A semi-conservative, stable, interphase-capturing numerical scheme for shock propagation in heterogeneous systems is applied to the problem of shock propagation in liquid-gas systems. The scheme is based on the volume-fraction formulation of the equations of motion for liquid and gas phases with separate equations of state. The semi-conservative formulation of the governing equations ensures the absence of spurious pressure oscillations at the material interphases between liquid and gas. Interaction of a planar shock in water with a single spherical bubble as well as twin adjacent bubbles is investigated. Several stages of the interaction process are considered, including focusing of the transmitted shock within the deformed bubble, creation of a water-hammer shock as well as generation of high-speed liquid jet in the later stages of the process.

Initial verification and validation of RAZORBACK - A research reactor transient analysis code

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

Talley, Darren G.

2015-09-01

This report describes the work and results of the initial verification and validation (V&V) of the beta release of the Razorback code. Razorback is a computer code designed to simulate the operation of a research reactor (such as the Annular Core Research Reactor (ACRR)) by a coupled numerical solution of the point reactor kinetics equations, the energy conservation equation for fuel element heat transfer, and the mass, momentum, and energy conservation equations for the water cooling of the fuel elements. This initial V&V effort was intended to confirm that the code work to-date shows good agreement between simulation and actualmore » ACRR operations, indicating that the subsequent V&V effort for the official release of the code will be successful.« less

An Entropy-Based Approach to Nonlinear Stability

NASA Technical Reports Server (NTRS)

Merriam, Marshal L.

1989-01-01

Many numerical methods used in computational fluid dynamics (CFD) incorporate an artificial dissipation term to suppress spurious oscillations and control nonlinear instabilities. The same effect can be accomplished by using upwind techniques, sometimes augmented with limiters to form Total Variation Diminishing (TVD) schemes. An analysis based on numerical satisfaction of the second law of thermodynamics allows many such methods to be compared and improved upon. A nonlinear stability proof is given for discrete scalar equations arising from a conservation law. Solutions to such equations are bounded in the L sub 2 norm if the second law of thermodynamics is satisfied in a global sense over a periodic domain. It is conjectured that an analogous statement is true for discrete equations arising from systems of conservation laws. Analysis and numerical experiments suggest that a more restrictive condition, a positive entropy production rate in each cell, is sufficient to exclude unphysical phenomena such as oscillations and expansion shocks. Construction of schemes which satisfy this condition is demonstrated for linear and nonlinear wave equations and for the one-dimensional Euler equations.

Brownian microhydrodynamics of active filaments.

PubMed

Laskar, Abhrajit; Adhikari, R

2015-12-21

Slender bodies capable of spontaneous motion in the absence of external actuation in an otherwise quiescent fluid are common in biological, physical and technological contexts. The interplay between the spontaneous fluid flow, Brownian motion, and the elasticity of the body presents a challenging fluid-structure interaction problem. Here, we model this problem by approximating the slender body as an elastic filament that can impose non-equilibrium velocities or stresses at the fluid-structure interface. We derive equations of motion for such an active filament by enforcing momentum conservation in the fluid-structure interaction and assuming slow viscous flow in the fluid. The fluid-structure interaction is obtained, to any desired degree of accuracy, through the solution of an integral equation. A simplified form of the equations of motion, which allows for efficient numerical solutions, is obtained by applying the Kirkwood-Riseman superposition approximation to the integral equation. We use this form of equation of motion to study dynamical steady states in free and hinged minimally active filaments. Our model provides the foundation to study collective phenomena in momentum-conserving, Brownian, active filament suspensions.

On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation

NASA Astrophysics Data System (ADS)

Amadori, Debora; Ha, Seung-Yeal; Park, Jinyeong

2017-01-01

The Kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit-cycle oscillators. When the number of oscillators is sufficiently large, the dynamics of Kuramoto ensemble can be effectively approximated by the corresponding mean-field equation, namely "the Kuramoto-Sakaguchi (KS) equation". This KS equation is a kind of scalar conservation law with a nonlocal flux function due to the mean-field interactions among oscillators. In this paper, we provide a unique global solvability of bounded variation (BV) weak solutions to the kinetic KS equation for identical oscillators using the method of front-tracking in hyperbolic conservation laws. Moreover, we also show that our BV weak solutions satisfy local-in-time L1-stability with respect to BV-initial data. For the ensemble of identical Kuramoto oscillators, we explicitly construct an exponentially growing BV weak solution generated from BV perturbation of incoherent state for any positive coupling strength. This implies the nonlinear instability of incoherent state in a positive coupling strength regime. We provide several numerical examples and compare them with our analytical results.

A conservative, thermodynamically consistent numerical approach for low Mach number combustion. Part I: Single-level integration

NASA Astrophysics Data System (ADS)

Nonaka, Andrew; Day, Marcus S.; Bell, John B.

2018-01-01

We present a numerical approach for low Mach number combustion that conserves both mass and energy while remaining on the equation of state to a desired tolerance. We present both unconfined and confined cases, where in the latter the ambient pressure changes over time. Our overall scheme is a projection method for the velocity coupled to a multi-implicit spectral deferred corrections (SDC) approach to integrate the mass and energy equations. The iterative nature of SDC methods allows us to incorporate a series of pressure discrepancy corrections naturally that lead to additional mass and energy influx/outflux in each finite volume cell in order to satisfy the equation of state. The method is second order, and satisfies the equation of state to a desired tolerance with increasing iterations. Motivated by experimental results, we test our algorithm on hydrogen flames with detailed kinetics. We examine the morphology of thermodiffusively unstable cylindrical premixed flames in high-pressure environments for confined and unconfined cases. We also demonstrate that our algorithm maintains the equation of state for premixed methane flames and non-premixed dimethyl ether jet flames.

Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Blinkov, Yuri A.; Mozzhilkin, Vladimir V.

2006-05-01

In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.

Volume-Of-Fluid Simulation for Predicting Two-Phase Cooling in a Microchannel

NASA Astrophysics Data System (ADS)

Gorle, Catherine; Parida, Pritish; Houshmand, Farzad; Asheghi, Mehdi; Goodson, Kenneth

2014-11-01

Two-phase flow in microfluidic geometries has applications of increasing interest for next generation electronic and optoelectronic systems, telecommunications devices, and vehicle electronics. While there has been progress on comprehensive simulation of two-phase flows in compact geometries, validation of the results in different flow regimes should be considered to determine the predictive capabilities. In the present study we use the volume-of-fluid method to model the flow through a single micro channel with cross section 100 × 100 μm and length 10 mm. The channel inlet mass flux and the heat flux at the lower wall result in a subcooled boiling regime in the first 2.5 mm of the channel and a saturated flow regime further downstream. A conservation equation for the vapor volume fraction, and a single set of momentum and energy equations with volume-averaged fluid properties are solved. A reduced-physics phase change model represents the evaporation of the liquid and the corresponding heat loss, and the surface tension is accounted for by a source term in the momentum equation. The phase change model used requires the definition of a time relaxation parameter, which can significantly affect the solution since it determines the rate of evaporation. The results are compared to experimental data available from literature, focusing on the capability of the reduced-physics phase change model to predict the correct flow pattern, temperature profile and pressure drop.

Implicit solution of three-dimensional internal turbulent flows

NASA Technical Reports Server (NTRS)

Michelassi, V.; Liou, M.-S.; Povinelli, Louis A.; Martelli, F.

1991-01-01

The scalar form of the approximate factorization method was used to develop a new code for the solution of three dimensional internal laminar and turbulent compressible flows. The Navier-Stokes equations in their Reynolds-averaged form were iterated in time until a steady solution was reached. Evidence was given to the implicit and explicit artificial damping schemes that proved to be particularly efficient in speeding up convergence and enhancing the algorithm robustness. A conservative treatment of these terms at the domain boundaries was proposed in order to avoid undesired mass and/or momentum artificial fluxes. Turbulence effects were accounted for by the zero-equation Baldwin-Lomax turbulence model and the q-omega two-equation model. The flow in a developing S-duct was then solved in the laminar regime in a Reynolds number (Re) of 790 and in the turbulent regime at Re equals 40,000 by using the Baldwin-Lomax model. The Stanitz elbow was then solved by using an invicid version of the same code at M sub inlet equals 0.4. Grid dependence and convergence rate were investigated, showing that for this solver the implicit damping scheme may play a critical role for convergence characteristics. The same flow at Re equals 2.5 times 10(exp 6) was solved with the Baldwin-Lomax and the q-omega models. Both approaches show satisfactory agreement with experiments, although the q-omega model was slightly more accurate.

A fully-neoclassical finite-orbit-width version of the CQL3D Fokker–Planck code

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

Petrov, Yu V.; Harvey, R. W.

The time-dependent bounce-averaged CQL3D flux-conservative finite-difference Fokker–Planck equation (FPE) solver has been upgraded to include finite-orbit-width (FOW) capabilities which are necessary for an accurate description of neoclassical transport, losses to the walls, and transfer of particles, momentum, and heat to the scrape-off layer. The FOW modifications are implemented in the formulation of the neutral beam source, collision operator, RF quasilinear diffusion operator, and in synthetic particle diagnostics. The collisional neoclassical radial transport appears naturally in the FOW version due to the orbit-averaging of local collision coefficients coupled with transformation coefficients from local (R, Z) coordinates along each guiding-center orbit tomore » the corresponding midplane computational coordinates, where the FPE is solved. In a similar way, the local quasilinear RF diffusion terms give rise to additional radial transport of orbits. We note that the neoclassical results are obtained for ‘full’ orbits, not dependent on a common small orbit-width approximation. Results of validation tests for the FOW version are also presented.« less

A fully-neoclassical finite-orbit-width version of the CQL3D Fokker–Planck code

DOE PAGES

Petrov, Yu V.; Harvey, R. W.

2016-09-08

The time-dependent bounce-averaged CQL3D flux-conservative finite-difference Fokker–Planck equation (FPE) solver has been upgraded to include finite-orbit-width (FOW) capabilities which are necessary for an accurate description of neoclassical transport, losses to the walls, and transfer of particles, momentum, and heat to the scrape-off layer. The FOW modifications are implemented in the formulation of the neutral beam source, collision operator, RF quasilinear diffusion operator, and in synthetic particle diagnostics. The collisional neoclassical radial transport appears naturally in the FOW version due to the orbit-averaging of local collision coefficients coupled with transformation coefficients from local (R, Z) coordinates along each guiding-center orbit tomore » the corresponding midplane computational coordinates, where the FPE is solved. In a similar way, the local quasilinear RF diffusion terms give rise to additional radial transport of orbits. We note that the neoclassical results are obtained for ‘full’ orbits, not dependent on a common small orbit-width approximation. Results of validation tests for the FOW version are also presented.« less

The impact of the form of the Euler equations for radial flow in cylindrical and spherical coordinates on numerical conservation and accuracy

NASA Astrophysics Data System (ADS)

Crittenden, P. E.; Balachandar, S.

2018-07-01

The radial one-dimensional Euler equations are often rewritten in what is known as the geometric source form. The differential operator is identical to the Cartesian case, but source terms result. Since the theory and numerical methods for the Cartesian case are well-developed, they are often applied without modification to cylindrical and spherical geometries. However, numerical conservation is lost. In this article, AUSM^+-up is applied to a numerically conservative (discrete) form of the Euler equations labeled the geometric form, a nearly conservative variation termed the geometric flux form, and the geometric source form. The resulting numerical methods are compared analytically and numerically through three types of test problems: subsonic, smooth, steady-state solutions, Sedov's similarity solution for point or line-source explosions, and shock tube problems. Numerical conservation is analyzed for all three forms in both spherical and cylindrical coordinates. All three forms result in constant enthalpy for steady flows. The spatial truncation errors have essentially the same order of convergence, but the rate constants are superior for the geometric and geometric flux forms for the steady-state solutions. Only the geometric form produces the correct shock location for Sedov's solution, and a direct connection between the errors in the shock locations and energy conservation is found. The shock tube problems are evaluated with respect to feature location using an approximation with a very fine discretization as the benchmark. Extensions to second order appropriate for cylindrical and spherical coordinates are also presented and analyzed numerically. Conclusions are drawn, and recommendations are made. A derivation of the steady-state solution is given in the Appendix.

The impact of the form of the Euler equations for radial flow in cylindrical and spherical coordinates on numerical conservation and accuracy

NASA Astrophysics Data System (ADS)

Crittenden, P. E.; Balachandar, S.

2018-03-01

The radial one-dimensional Euler equations are often rewritten in what is known as the geometric source form. The differential operator is identical to the Cartesian case, but source terms result. Since the theory and numerical methods for the Cartesian case are well-developed, they are often applied without modification to cylindrical and spherical geometries. However, numerical conservation is lost. In this article, AUSM^+ -up is applied to a numerically conservative (discrete) form of the Euler equations labeled the geometric form, a nearly conservative variation termed the geometric flux form, and the geometric source form. The resulting numerical methods are compared analytically and numerically through three types of test problems: subsonic, smooth, steady-state solutions, Sedov's similarity solution for point or line-source explosions, and shock tube problems. Numerical conservation is analyzed for all three forms in both spherical and cylindrical coordinates. All three forms result in constant enthalpy for steady flows. The spatial truncation errors have essentially the same order of convergence, but the rate constants are superior for the geometric and geometric flux forms for the steady-state solutions. Only the geometric form produces the correct shock location for Sedov's solution, and a direct connection between the errors in the shock locations and energy conservation is found. The shock tube problems are evaluated with respect to feature location using an approximation with a very fine discretization as the benchmark. Extensions to second order appropriate for cylindrical and spherical coordinates are also presented and analyzed numerically. Conclusions are drawn, and recommendations are made. A derivation of the steady-state solution is given in the Appendix.

Mathematical modeling of impact of two metal plates using two-fluid approach

NASA Astrophysics Data System (ADS)

Utkin, P. S.; Fortova, S. V.

2018-01-01

The paper is devoted to the development of the two-fluid mathematical model and the computational algorithm for the modeling of two metal plates impact. In one-dimensional case the governing system of equations comprises seven equations: three conservation laws for each fluid and transfer equation for the volume fraction of one of the fluids. Both fluids are considered to be compressible and equilibrium on velocities. Pressures equilibrium is used as fluids interface condition. The system has hyperbolic type but could not be written in the conservative form because of nozzling terms in the right-hand side of the equations. The algorithm is based on the Harten-Lax-van Leer numerical flux function. The robust computation in the presence of the interface boundary is carried out due to the special pressure relaxation procedure. The problem is solved using stiffened gas equations of state for each fluid. The parameters in the equations of state are calibrated using the results of computations using wide-range equations of state for the metals. In simulations of metal plates impact we get two shocks after the initial impact that propagate to the free surfaces of the samples. The characteristics of shock waves are close (maximum relative error in characteristics of shocks is not greater than 7%) to the data from the wide-range equations of states computations.

Computation techniques and computer programs to analyze Stirling cycle engines using characteristic dynamic energy equations

NASA Technical Reports Server (NTRS)

Larson, V. H.

1982-01-01

The basic equations that are used to describe the physical phenomena in a Stirling cycle engine are the general energy equations and equations for the conservation of mass and conversion of momentum. These equations, together with the equation of state, an analytical expression for the gas velocity, and an equation for mesh temperature are used in this computer study of Stirling cycle characteristics. The partial differential equations describing the physical phenomena that occurs in a Stirling cycle engine are of the hyperbolic type. The hyperbolic equations have real characteristic lines. By utilizing appropriate points along these curved lines the partial differential equations can be reduced to ordinary differential equations. These equations are solved numerically using a fourth-fifth order Runge-Kutta integration technique.

Alfvén wave interactions in the solar wind

NASA Astrophysics Data System (ADS)

Webb, G. M.; McKenzie, J. F.; Hu, Q.; le Roux, J. A.; Zank, G. P.

2012-11-01

Alfvén wave mixing (interaction) equations used in locally incompressible turbulence transport equations in the solar wind are analyzed from the perspective of linear wave theory. The connection between the wave mixing equations and non-WKB Alfven wave driven wind theories are delineated. We discuss the physical wave energy equation and the canonical wave energy equation for non-WKB Alfven waves and the WKB limit. Variational principles and conservation laws for the linear wave mixing equations for the Heinemann and Olbert non-WKB wind model are obtained. The connection with wave mixing equations used in locally incompressible turbulence transport in the solar wind are discussed.

Composite scheme using localized relaxation with non-standard finite difference method for hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Kumar, Vivek; Raghurama Rao, S. V.

2008-04-01

Non-standard finite difference methods (NSFDM) introduced by Mickens [ Non-standard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994] are interesting alternatives to the traditional finite difference and finite volume methods. When applied to linear hyperbolic conservation laws, these methods reproduce exact solutions. In this paper, the NSFDM is first extended to hyperbolic systems of conservation laws, by a novel utilization of the decoupled equations using characteristic variables. In the second part of this paper, the NSFDM is studied for its efficacy in application to nonlinear scalar hyperbolic conservation laws. The original NSFDMs introduced by Mickens (1994) were not in conservation form, which is an important feature in capturing discontinuities at the right locations. Mickens [Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equations, Journal of Sound and Vibration 257 (4) (2002) 791-797] recently introduced a NSFDM in conservative form. This method captures the shock waves exactly, without any numerical dissipation. In this paper, this algorithm is tested for the case of expansion waves with sonic points and is found to generate unphysical expansion shocks. As a remedy to this defect, we use the strategy of composite schemes [R. Liska, B. Wendroff, Composite schemes for conservation laws, SIAM Journal of Numerical Analysis 35 (6) (1998) 2250-2271] in which the accurate NSFDM is used as the basic scheme and localized relaxation NSFDM is used as the supporting scheme which acts like a filter. Relaxation schemes introduced by Jin and Xin [The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications in Pure and Applied Mathematics 48 (1995) 235-276] are based on relaxation systems which replace the nonlinear hyperbolic conservation laws by a semi-linear system with a stiff relaxation term. The relaxation parameter ( λ) is chosen locally on the three point stencil of grid which makes the proposed method more efficient. This composite scheme overcomes the problem of unphysical expansion shocks and captures the shock waves with an accuracy better than the upwind relaxation scheme, as demonstrated by the test cases, together with comparisons with popular numerical methods like Roe scheme and ENO schemes.

Prolongation structures of nonlinear evolution equations

NASA Technical Reports Server (NTRS)

Wahlquist, H. D.; Estabrook, F. B.

1975-01-01

A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

Discontinuous Spectral Difference Method for Conservation Laws on Unstructured Grids

NASA Technical Reports Server (NTRS)

Liu, Yen; Vinokur, Marcel

2004-01-01

A new, high-order, conservative, and efficient discontinuous spectral finite difference (SD) method for conservation laws on unstructured grids is developed. The concept of discontinuous and high-order local representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) and the Spectral Volume (SV) methods, but while these methods are based on the integrated forms of the equations, the new method is based on the differential form to attain a simpler formulation and higher efficiency. Conventional unstructured finite-difference and finite-volume methods require data reconstruction based on the least-squares formulation using neighboring point or cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every point or cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In addition, the finite-difference method does not satisfy the integral conservation in general. By contrast, the DG and SV methods employ a local, universal reconstruction of a given order of accuracy in each cell in terms of internally defined conservative unknowns. Since the solution is discontinuous across cell boundaries, a Riemann solver is necessary to evaluate boundary flux terms and maintain conservation. In the DG method, a Galerkin finite-element method is employed to update the nodal unknowns within each cell. This requires the inversion of a mass matrix, and the use of quadratures of twice the order of accuracy of the reconstruction to evaluate the surface integrals and additional volume integrals for nonlinear flux functions. In the SV method, the integral conservation law is used to update volume averages over subcells defined by a geometrically similar partition of each grid cell. As the order of accuracy increases, the partitioning for 3D requires the introduction of a large number of parameters, whose optimization to achieve convergence becomes increasingly more difficult. Also, the number of interior facets required to subdivide non-planar faces, and the additional increase in the number of quadrature points for each facet, increases the computational cost greatly.

Final Technical Report

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

Brizard, Alain J

Final Technical Report for U.S. Department of Energy Grant No. DE-FG02-09ER55005 Nonlinear FLR Effects in Reduced Fluid Models Alain J. Brizard, Saint Michael's College The above-mentioned DoE grant was used to support research activities by the PI during a sabbatical leave from Saint Michael's College in 2009. The major focus of the work was the role played by guiding-center and gyrocenter (linear and nonlinear) polarization and magnetization effects in understanding transport processes in turbulent magnetized plasmas. The theoretical tools used for this work include Lie-transform perturbation methods and Lagrangian (variational) methods developed by the PI in previous work. The presentmore » final technical report lists (I) the peer-reviewed publications that were written based on work funded by the Grant; (II) invited and contributed conference presentations during the period funded by the Grant; and (III) seminars presented during the period funded by the Grant. I. Peer-reviewed Publications A.J. Brizard and N. Tronko, 2011, Exact momentum conservation for the gyrokinetic Vlasov- Poisson equations, Physics of Plasmas 18 , 082307:1-14 [http://dx.doi.org/10.1063/1.3625554 ]. J. Decker, Y. Peysson, A.J. Brizard, and F.-X. Duthoit, 2010, Orbit-averaged guiding-center Fokker-Planck operator for numerical applications, Physics of Plasmas 17, 112513:1-12 [http://dx.doi.org/10.1063/1.3519514]. A.J. Brizard, 2010, Noether derivation of exact conservation laws for dissipationless reduced fluid models, Physics of Plasmas 17, 112503:1-8 [http://dx.doi.org/10.1063/1.3515303]. F.-X. Duthoit, A.J. Brizard, Y. Peysson, and J. Decker, 2010, Perturbation analysis of trapped particle dynamics in axisymmetric dipole geometry, Physics of Plasmas 17, 102903:1-9 [http://dx.doi.org/10.1063/1.3486554]. A.J. Brizard, 2010, Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations, Physics of Plasmas 17, 042303:1-11 [http://dx.doi.org/10.1063/1.3374428]. A.J. Brizard, J. Decker, Y. Peysson, and F.-X. Duthoit, 2009, Orbit-averaged guiding-center Fokker-Planck operator, Physics of Plasmas 16, 102304:1-9[http://dx.doi.org/10.1063/1.3249627]. A.J. Brizard, 2009, Variational Principles for Reduced Plasma Physics, Journal of Physics: Conference Series 169, 012003 [http://dx.doi.org/10.1088/1742-6596/169/1/012003]. II. Invited and Contributed Conference Presentations A.J. Brizard and N. Tronko, Momentum conservation law for the gyrokinetic Vlasov-Poisson equations, 53rd Annual Meeting of the APS Division of Plasma Physics, Salt Lake City (Utah), November 14-18, 2011. A.J. Brizard, P.J. Morrison, C. Chandre, and E. Tassi, On the road to the Hamiltonian formulation of gyrokinetic theory, 52nd Annual Meeting of the APS Division of Plasma Physics, Chicago (Illinois), November 8-12, 2010. F.-X. Duthoit, A.J. Brizard, Y. Peysson, and J. Decker, Lie-transform perturbation analysis of trapped-particle dynamics in axisymmetric dipole geometry, 2010 International Sherwood Fusion Theory Conference, Seattle (Washington), April 19-21, 2010. N. Tronko and A.J. Brizard, Gyrokinetic momentum conservation law, 2010 International Sherwood Fusion Theory Conference, Seattle (Washington), April 19-21, 2010. C. Chandre and A.J. Brizard, Hamiltonian formulation of reduced Vlasov-Maxwell equations, 50th Annual Meeting of the APS Division of Plasma Physics, Dallas (Texas), November 17-21, 2008. A.J. Brizard, Nonlinear FLR effects in reduced fluid models, Invited Presentation at 11th Easter Plasma Meeting, Torino (Italy), April 15-17, 2009. III. Seminars Reduced Fokker-Planck operators for advanced plasma simulations, seminar given at CEA Cadarache (France), May 25, 2009. Ray phase-space methods in linear mode conversion, seminar given at CPT Luminy (France), April 1, 2009. Old and new methods in gyrokinetic theory, seminar given at CEA Cadarache (France), March 20, 2009. Hamiltonian theory of adiabatic motion of relativistic charged particles, seminar given at CPT Luminy (France), March 11, 2009. Noether method for fluids and plasmas, seminar given at CEA Cadarache (France), February 5, 2009. Nonlinear FLR effects in reduced fluid models, invited speaker at the Journee de la Dynamique Non Lineaire, Centre de Physique Theorique, CNRS Luminy (Marseille, France), June 3, 2008.« less

A direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume scheme on unstructured tetrahedral meshes for conservative and non-conservative hyperbolic systems in 3D

NASA Astrophysics Data System (ADS)

Boscheri, Walter; Dumbser, Michael

2014-10-01

In this paper we present a new family of high order accurate Arbitrary-Lagrangian-Eulerian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations with stiff source terms on moving tetrahedral meshes in three space dimensions. A WENO reconstruction technique is used to achieve high order of accuracy in space, while an element-local space-time Discontinuous Galerkin finite element predictor on moving curved meshes is used to obtain a high order accurate one-step time discretization. Within the space-time predictor the physical element is mapped onto a reference element using a high order isoparametric approach, where the space-time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space-time nodes. Since our algorithm is cell-centered, the final mesh motion is computed by using a suitable node solver algorithm. A rezoning step as well as a flattener strategy are used in some of the test problems to avoid mesh tangling or excessive element deformations that may occur when the computation involves strong shocks or shear waves. The ALE algorithm presented in this article belongs to the so-called direct ALE methods because the final Lagrangian finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, with the rezoned geometry taken already into account during the computation of the fluxes. We apply our new high order unstructured ALE schemes to the 3D Euler equations of compressible gas dynamics, for which a set of classical numerical test problems has been solved and for which convergence rates up to sixth order of accuracy in space and time have been obtained. We furthermore consider the equations of classical ideal magnetohydrodynamics (MHD) as well as the non-conservative seven-equation Baer-Nunziato model of compressible multi-phase flows with stiff relaxation source terms.

Evaluating the first Brazilian program of payments for environmental services: An approach for optimizing soil conservation using GIS

NASA Astrophysics Data System (ADS)

Zolin, C. A.; Folegatti, M. V.; Mingoti, R.; Paulino, J.; Sánchez-Román, R. M.; González, A. M.

2013-12-01

Brazil possesses one of the most important water assets in the world, however, the country experiences vast differences among its hydrographic regions. Although Brazil has the largest water reserves in the world, those reserves are not distributed according to the concentration of the population. In addition, the largest portions of these water reserves are not always located where the highest urban concentrations and demands occur, which causes serious problems in maintaining water supply within the country's most populous regions (Zolin et al. 2011). It has become evident that policies aimed at mitigating the growing water resources and water use conflicts in Brazil are crucial. The municipality of Extrema in Minas Gerais state in Brazil pioneered the first Brazilian municipal PES initiative (Conservador das Águas program), based on the relationship between forests and the benefits they provide. This study aimed to assess soil loss in the Posses sub-basin, where the Conservador das Águas program began. Additionally, we aimed to determine the potential that this PES initiative has for soil conservation, as well as to optimize the environmental services provided as a function of forest area size and location. In this sense, considering the prescribed conservation practices, land use situation, and soil cover in the Posses sub-basin, we analyzed the effectiveness of the Conservador das Águas program before and after implementation in relation to reduced soil loss under different land use and soil cover scenarios. We used a geographic information system (GIS) for spatializing and producing different information plans and the Revised Universal Soil Loss Equation (RUSLE) for estimating soil loss. As a result, we found that optimized soil conservation may be obtained by adopting pasture conservation practices. Additionally the expected average soil loss in the Posses sub-basin under conditions of land use and soil cover, before and after implementing the water conservation program, was 30.63 and 7.06 Mg ha-1 year-1, respectively.

Flame-conditioned turbulence modeling for reacting flows

NASA Astrophysics Data System (ADS)

Macart, Jonathan F.; Mueller, Michael E.

2017-11-01

Conventional approaches to turbulence modeling in reacting flows rely on unconditional averaging or filtering, that is, consideration of the momentum equations only in physical space, implicitly assuming that the flame only weakly affects the turbulence, aside from a variation in density. Conversely, for scalars, which are strongly coupled to the flame structure, their evolution equations are often projected onto a reduced-order manifold, that is, conditionally averaged or filtered, on a flame variable such as a mixture fraction or progress variable. Such approaches include Conditional Moment Closure (CMC) and related variants. However, recent observations from Direct Numerical Simulation (DNS) have indicated that the flame can strongly affect turbulence in premixed combustion at low Karlovitz number. In this work, a new approach to turbulence modeling for reacting flows is investigated in which conditionally averaged or filtered equations are evolved for the momentum. The conditionally-averaged equations for the velocity and its covariances are derived, and budgets are evaluated from DNS databases of turbulent premixed planar jet flames. The most important terms in these equations are identified, and preliminary closure models are proposed.

Initialization and assimilation of cloud and rainwater in a regional model

NASA Technical Reports Server (NTRS)

Raymond, William H.; Olson, William S.

1990-01-01

The initialization and assimilation of cloud and rainwater quantities in a mesoscale regional model was examined. Forecasts of explicit cloud and rainwater are made using conservation equations. The physical processes include condensation, evaporation, autoconversion, accretion, and the removal of rainwater by fallout. These physical processes, some of which are parameterized, represent source and sink in terms in the conservation equations. The question of how to initialize the explicit liquid water calculations in numerical models and how to retain information about precipitation processes during the 4-D assimilation cycle are important issues that are addressed.

Symmetrization of conservation laws with entropy for high-temperature hypersonic computations

NASA Technical Reports Server (NTRS)

Chalot, F.; Hughes, T. J. R.; Shakib, F.

1990-01-01

Results of Hughes, France, and Mallet are generalized to conservation law systems taking into account high-temperature effects. Symmetric forms of different equation sets are derived in terms of entropy variables. First, the case of a general divariant gas is studied; it can be specialized to the usual Navier-Stokes equations, as well as to situations where the gas is vibrationally excited, and undergoes equilibrium chemical reactions. The case of gas in thermochemical nonequilibrium is considered next. Transport phenomena, and in particular mass diffusion, are examined in the framework of symmetric advective-diffusive systems.

An approximation of herd effect due to vaccinating children against seasonal influenza - a potential solution to the incorporation of indirect effects into static models.

PubMed

Van Vlaenderen, Ilse; Van Bellinghen, Laure-Anne; Meier, Genevieve; Nautrup, Barbara Poulsen

2013-01-22

Indirect herd effect from vaccination of children offers potential for improving the effectiveness of influenza prevention in the remaining unvaccinated population. Static models used in cost-effectiveness analyses cannot dynamically capture herd effects. The objective of this study was to develop a methodology to allow herd effect associated with vaccinating children against seasonal influenza to be incorporated into static models evaluating the cost-effectiveness of influenza vaccination. Two previously published linear equations for approximation of herd effects in general were compared with the results of a structured literature review undertaken using PubMed searches to identify data on herd effects specific to influenza vaccination. A linear function was fitted to point estimates from the literature using the sum of squared residuals. The literature review identified 21 publications on 20 studies for inclusion. Six studies provided data on a mathematical relationship between effective vaccine coverage in subgroups and reduction of influenza infection in a larger unvaccinated population. These supported a linear relationship when effective vaccine coverage in a subgroup population was between 20% and 80%. Three studies evaluating herd effect at a community level, specifically induced by vaccinating children, provided point estimates for fitting linear equations. The fitted linear equation for herd protection in the target population for vaccination (children) was slightly less conservative than a previously published equation for herd effects in general. The fitted linear equation for herd protection in the non-target population was considerably less conservative than the previously published equation. This method of approximating herd effect requires simple adjustments to the annual baseline risk of influenza in static models: (1) for the age group targeted by the childhood vaccination strategy (i.e. children); and (2) for other age groups not targeted (e.g. adults and/or elderly). Two approximations provide a linear relationship between effective coverage and reduction in the risk of infection. The first is a conservative approximation, recommended as a base-case for cost-effectiveness evaluations. The second, fitted to data extracted from a structured literature review, provides a less conservative estimate of herd effect, recommended for sensitivity analyses.

Some new solutions for the Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Ramírez, J.; Romero, J. L.; Tracinà, R.

2013-09-01

The well-known Derrida-Lebowitz-Speer-Spohn equation is investigated. By using specific ansätze and the classical symmetries of the equation, several families of new exact solutions have been found. In particular, there appear traveling waves that include compactons and soliton-compactons. Some other solutions conserve the mass and exhibit diffusion and convection processes from an instantaneous source and localized peakons.

Numerical Solution for Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Warsi, Z. U. A.; Weed, R. A.; Thompson, J. F.

1982-01-01

Carefully selected blend of computational techniques solves complete set of equations for viscous, unsteady, hypersonic flow in general curvilinear coordinates. New algorithm has tested computation of axially directed flow about blunt body having shape similar to that of such practical bodies as wide-body aircraft or artillery shells. Method offers significant computational advantages because of conservation-law form of equations and because it reduces amount of metric data required.

Volatilization of organic compounds from streams

USGS Publications Warehouse

Rathburn, R.E.; Tai, D.Y.

1982-01-01

Mass-transfer coefficients for the volatilization of ethylene and propane were correlated with the hydraulic and geometric properties of seven streams, and predictive equations were developed. The equations were evaluated using a normalized root-mean-square error as the criterion of comparison. The two best equations were a two-variable equation containing the energy dissipated per unit mass per unit time and the average depth of flow and a three-variable equation containing the average velocity, the average depth of flow, and the slope of the stream. Procedures for adjusting the ethylene and propane coefficients for other organic compounds were evaluated. These procedures are based on molecular diffusivity, molecular diameter, or molecular weight. Because of limited data, none of these procedures have been extensively verified. Therefore, until additional data become available, it is suggested that the mass-transfer coefficient be assumed to be inversely proportional to the square root of the molecular weight.

The ion-acoustic soliton: A gas-dynamic viewpoint

NASA Astrophysics Data System (ADS)

McKenzie, J. F.

2002-03-01

The properties of fully nonlinear ion-acoustic solitons are investigated by interpreting conservation of total momentum as the structure equation for the proton flow in the wave. In most studies momentum conservation is regarded as the first integral of the Poisson equation for the electric potential and is interpreted as being analogous to a particle moving in a pseudo-potential well. By adopting an essentially gas-dynamic viewpoint, which emphasizes momentum conservation and the properties of the Bernoulli-type energy equations, the crucial role played by the proton sonic point becomes apparent. The relationship (implied by energy conservation) between the electron and proton speeds in the transition yields a locus—the hodograph of the system-which shows that, in the first half of the soliton, the electrons initially lag behind the protons until the charge neutral point is reached, after which they run ahead of the protons. The system reaches an equilibrium point (the center of the soliton) before the proton flow goes sonic. It follows that the critical ion-acoustic Mach number, Mc, above which smooth, continuous solitons cannot be constructed, stems from the requirement that the two equilibrium points of the structure equation coalesce at the proton sonic point of the flow. In general the range of the ion-acoustic Mach numbers, Mep, in which solitons exist, is extended beyond the classical range 1

A geometric viewpoint on generalized hydrodynamics

NASA Astrophysics Data System (ADS)

Doyon, Benjamin; Spohn, Herbert; Yoshimura, Takato

2018-01-01

Generalized hydrodynamics (GHD) is a large-scale theory for the dynamics of many-body integrable systems. It consists of an infinite set of conservation laws for quasi-particles traveling with effective ("dressed") velocities that depend on the local state. We show that these equations can be recast into a geometric dynamical problem. They are conservation equations with state-independent quasi-particle velocities, in a space equipped with a family of metrics, parametrized by the quasi-particles' type and speed, that depend on the local state. In the classical hard rod or soliton gas picture, these metrics measure the free length of space as perceived by quasi-particles; in the quantum picture, they weigh space with the density of states available to them. Using this geometric construction, we find a general solution to the initial value problem of GHD, in terms of a set of integral equations where time appears explicitly. These integral equations are solvable by iteration and provide an extremely efficient solution algorithm for GHD.

On the non-stationary generalized Langevin equation

NASA Astrophysics Data System (ADS)

Meyer, Hugues; Voigtmann, Thomas; Schilling, Tanja

2017-12-01

In molecular dynamics simulations and single molecule experiments, observables are usually measured along dynamic trajectories and then averaged over an ensemble ("bundle") of trajectories. Under stationary conditions, the time-evolution of such averages is described by the generalized Langevin equation. By contrast, if the dynamics is not stationary, it is not a priori clear which form the equation of motion for an averaged observable has. We employ the formalism of time-dependent projection operator techniques to derive the equation of motion for a non-equilibrium trajectory-averaged observable as well as for its non-stationary auto-correlation function. The equation is similar in structure to the generalized Langevin equation but exhibits a time-dependent memory kernel as well as a fluctuating force that implicitly depends on the initial conditions of the process. We also derive a relation between this memory kernel and the autocorrelation function of the fluctuating force that has a structure similar to a fluctuation-dissipation relation. In addition, we show how the choice of the projection operator allows us to relate the Taylor expansion of the memory kernel to data that are accessible in MD simulations and experiments, thus allowing us to construct the equation of motion. As a numerical example, the procedure is applied to Brownian motion initialized in non-equilibrium conditions and is shown to be consistent with direct measurements from simulations.

Calculation of unsteady airfoil loads with and without flap deflection at -90 degrees incidence

NASA Technical Reports Server (NTRS)

Stremel, Paul M.

1991-01-01

A method has been developed for calculating the viscous flow about airfoils with and without deflected flaps at -90 deg incidence. This unique method provides for the direct solution of the incompressible Navier-Stokes equations by means of a fully coupled implicit technique. The solution is calculated on a body-fitted computational mesh incorporating a staggered grid method. The vorticity is determined at the node points, and the velocity components are defined at the mesh-cell sides. The staggered-grid orientation provides for accurate representation of vorticity at the node points and for the conservation of mass at the mesh-cell centers. The method provides for the direct solution of the flow field and satisfies the conservation of mass to machine zero at each time-step. The results of the present analysis and experimental results obtained for a XV-15 airfoil are compared. The comparisons indicate that the calculated drag reduction caused by flap deflection and the calculated average surface pressure are in excellent agreement with the measured results. Comparisons of the numerical results of the present method for several airfoils demonstrate the significant influence of airfoil curvature and flap deflection on the predicted download.

Coarse-grained forms for equations describing the microscopic motion of particles in a fluid.

PubMed

Das, Shankar P; Yoshimori, Akira

2013-10-01

Exact equations of motion for the microscopically defined collective density ρ(x,t) and the momentum density ĝ(x,t) of a fluid have been obtained in the past starting from the corresponding Langevin equations representing the dynamics of the fluid particles. In the present work we average these exact equations of microscopic dynamics over the local equilibrium distribution to obtain stochastic partial differential equations for the coarse-grained densities with smooth spatial and temporal dependence. In particular, we consider Dean's exact balance equation for the microscopic density of a system of interacting Brownian particles to obtain the basic equation of the dynamic density functional theory with noise. Our analysis demonstrates that on thermal averaging the dependence of the exact equations on the bare interaction potential is converted to dependence on the corresponding thermodynamic direct correlation functions in the coarse-grained equations.

A two-phase micromorphic model for compressible granular materials

NASA Astrophysics Data System (ADS)

Paolucci, Samuel; Li, Weiming; Powers, Joseph

2009-11-01

We introduce a new two-phase continuum model for compressible granular material based on micromorphic theory and treat it as a two-phase mixture with inner structure. By taking an appropriate number of moments of the local micro scale balance equations, the average phase balance equations result from a systematic averaging procedure. In addition to equations for mass, momentum and energy, the balance equations also include evolution equations for microinertia and microspin tensors. The latter equations combine to yield a general form of a compaction equation when the material is assumed to be isotropic. When non-linear and inertial effects are neglected, the generalized compaction equation reduces to that originally proposed by Bear and Nunziato. We use the generalized compaction equation to numerically model a mixture of granular high explosive and interstitial gas. One-dimensional shock tube and piston-driven solutions are presented and compared with experimental results and other known solutions.

7 CFR 610.14 - Use of USLE, RUSLE, and WEQ.

Code of Federal Regulations, 2011 CFR

2011-01-01

..., DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations..., this includes the soil loss tolerance values used in those formulas for determining HEL. The soil loss tolerance value is used as one of the criteria for planning soil conservation systems. These values are...

7 CFR 610.11 - Purpose and scope.

Code of Federal Regulations, 2012 CFR

2012-01-01

... AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.11 Purpose... by the Natural Resources Conservation Service (NRCS) to predict soil erosion due to water and wind... Act, as amended, 16 U.S.C. 3801-3813 specified that the Secretary would publish the universal soil...

7 CFR 610.14 - Use of USLE, RUSLE, and WEQ.

Code of Federal Regulations, 2013 CFR

2013-01-01

..., DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations..., this includes the soil loss tolerance values used in those formulas for determining HEL. The soil loss tolerance value is used as one of the criteria for planning soil conservation systems. These values are...

7 CFR 610.11 - Purpose and scope.

Code of Federal Regulations, 2013 CFR

2013-01-01

... AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.11 Purpose... by the Natural Resources Conservation Service (NRCS) to predict soil erosion due to water and wind... Act, as amended, 16 U.S.C. 3801-3813 specified that the Secretary would publish the universal soil...

7 CFR 610.11 - Purpose and scope.

Code of Federal Regulations, 2010 CFR

2010-01-01

... AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.11 Purpose... by the Natural Resources Conservation Service (NRCS) to predict soil erosion due to water and wind... Act, as amended, 16 U.S.C. 3801-3813 specified that the Secretary would publish the universal soil...

7 CFR 610.14 - Use of USLE, RUSLE, and WEQ.

Code of Federal Regulations, 2014 CFR

2014-01-01

..., DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations..., this includes the soil loss tolerance values used in those formulas for determining HEL. The soil loss tolerance value is used as one of the criteria for planning soil conservation systems. These values are...

7 CFR 610.11 - Purpose and scope.

Code of Federal Regulations, 2011 CFR

2011-01-01

... AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.11 Purpose... by the Natural Resources Conservation Service (NRCS) to predict soil erosion due to water and wind... Act, as amended, 16 U.S.C. 3801-3813 specified that the Secretary would publish the universal soil...

7 CFR 610.14 - Use of USLE, RUSLE, and WEQ.

Code of Federal Regulations, 2012 CFR

2012-01-01

..., DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations..., this includes the soil loss tolerance values used in those formulas for determining HEL. The soil loss tolerance value is used as one of the criteria for planning soil conservation systems. These values are...

7 CFR 610.14 - Use of USLE, RUSLE, and WEQ.

Code of Federal Regulations, 2010 CFR

2010-01-01

..., DEPARTMENT OF AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations..., this includes the soil loss tolerance values used in those formulas for determining HEL. The soil loss tolerance value is used as one of the criteria for planning soil conservation systems. These values are...

7 CFR 610.11 - Purpose and scope.

Code of Federal Regulations, 2014 CFR

2014-01-01

... AGRICULTURE CONSERVATION OPERATIONS TECHNICAL ASSISTANCE Soil Erosion Prediction Equations § 610.11 Purpose... by the Natural Resources Conservation Service (NRCS) to predict soil erosion due to water and wind... Act, as amended, 16 U.S.C. 3801-3813 specified that the Secretary would publish the universal soil...

Assessment of carbon in woody plants and soil across a vineyard-woodland landscape

PubMed Central

2011-01-01

Background Quantification of ecosystem services, such as carbon (C) storage, can demonstrate the benefits of managing for both production and habitat conservation in agricultural landscapes. In this study, we evaluated C stocks and woody plant diversity across vineyard blocks and adjoining woodland ecosystems (wildlands) for an organic vineyard in northern California. Carbon was measured in soil from 44 one m deep pits, and in aboveground woody biomass from 93 vegetation plots. These data were combined with physical landscape variables to model C stocks using a geographic information system and multivariate linear regression. Results Field data showed wildlands to be heterogeneous in both C stocks and woody tree diversity, reflecting the mosaic of several different vegetation types, and storing on average 36.8 Mg C/ha in aboveground woody biomass and 89.3 Mg C/ha in soil. Not surprisingly, vineyard blocks showed less variation in above- and belowground C, with an average of 3.0 and 84.1 Mg C/ha, respectively. Conclusions This research demonstrates that vineyards managed with practices that conserve some fraction of adjoining wildlands yield benefits for increasing overall C stocks and species and habitat diversity in integrated agricultural landscapes. For such complex landscapes, high resolution spatial modeling is challenging and requires accurate characterization of the landscape by vegetation type, physical structure, sufficient sampling, and allometric equations that relate tree species to each landscape. Geographic information systems and remote sensing techniques are useful for integrating the above variables into an analysis platform to estimate C stocks in these working landscapes, thereby helping land managers qualify for greenhouse gas mitigation credits. Carbon policy in California, however, shows a lack of focus on C stocks compared to emissions, and on agriculture compared to other sectors. Correcting these policy shortcomings could create incentives for ecosystem service provision, including C storage, as well as encourage better farm stewardship and habitat conservation. PMID:22070870

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.

Exploring the propagation of relativistic quantum wavepackets in the trajectory-based formulation

NASA Astrophysics Data System (ADS)

Tsai, Hung-Ming; Poirier, Bill

2016-03-01

In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [Phys. Rev. X, 4, 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a “many interacting worlds” interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.

Theoretical analysis of multiphase flow during oil-well drilling by a conservative model

NASA Astrophysics Data System (ADS)

Nicolas-Lopez, Ruben

2005-11-01

In order to decrease cost and improve drilling operations is necessary a better understood of the flow mechanisms. Therefore, it was carried out a multiphase conservative model that includes three mass equations and a momentum equation. Also, the measured geothermal gradient is utilized by state equations for estimating physical properties of the phases flowing. The mathematical model is solved by numerical conservative schemes. It is used to analyze the interaction among solid-liquid-gas phases. The circulating system consists as follow, the circulating fluid is pumped downward into the drilling pipe until the bottom of the open hole then it flows through the drill bit, and at this point formation cuttings are incorporated to the circulating fluid and carried upward to the surface. The mixture returns up to the surface by an annular flow area. The real operational conditions are fed to conservative model and the results are matched up to field measurements in several oil wells. Mainly, flow rates, drilling rate, well and tool geometries are data to estimate the profiles of pressure, mixture density, equivalent circulating density, gas fraction and solid carrying capacity. Even though the problem is very complex, the model describes, properly, the hydrodynamics of drilling techniques applied at oil fields. *Authors want to thank to Instituto Mexicano del Petroleo and Petroleos Mexicanos for supporting this research.

Extra compressibility terms for Favre-averaged two-equation models of inhomogeneous turbulent flows

NASA Technical Reports Server (NTRS)

Rubesin, Morris W.

1990-01-01

Forms of extra-compressibility terms that result from use of Favre averaging of the turbulence transport equations for kinetic energy and dissipation are derived. These forms introduce three new modeling constants, a polytropic coefficient that defines the interrelationships of the pressure, density, and enthalpy fluctuations and two constants in the dissipation equation that account for the non-zero pressure-dilitation and mean pressure gradients.

Unsteady Aerodynamic Modeling of A Maneuvering Aircraft Using Indicial Functions

DTIC Science & Technology

2016-03-30

indicial functions are directly calculated using the results of unsteady Reynolds-averaged Navier - Stokes simulation and a grid-movement tool. Results are...but meanwhile, the full-order model based on Unsteady Reynolds-averaged Navier - Stokes (URANS) equation is too computationally expensive to be used...The flow solver used in this study solves the unsteady, three-dimensional and compressible Navier - Stokes equations. The equations in terms of

FANS Simulation of Propeller Wash at Navy Harbors (ESTEP Project ER-201031)

DTIC Science & Technology

2016-08-01

this study, the Finite-Analytic Navier–Stokes code was employed to solve the Reynolds-Averaged Navier–Stokes equations in conjunction with advanced...site-specific harbor configurations, it is desirable to perform propeller wash study by solving the Navier–Stokes equations directly in conjunction ...Analytic Navier–Stokes code was employed to solve the Reynolds-Averaged Navier–Stokes equations in conjunction with advanced near-wall turbulence

The physical basis of glacier volume-area scaling

USGS Publications Warehouse

Bahr, D.B.; Meier, M.F.; Peckham, S.D.

1997-01-01

Ice volumes are known for only a few of the roughly 160,000 glaciers worldwide but are important components of many climate and sea level studies which require water flux estimates. A scaling analysis of the mass and momentum conservation equations shows that glacier volumes can be related by a power law to more easily observed glacier surface areas. The relationship requires four closure choices for the scaling behavior of glacier widths, slopes, side drag and mass balance. Reasonable closures predict a volume-area scaling exponent which is consistent with observations, giving a physical and practical basis for estimating ice volumes. Glacier volume is insensitive to perturbations in the mass balance scaling, but changes in average accumulation area ratios reflect significant changes in the scaling of both mass balance and ice volume. Copyright 1997 by the American Geophysical Union.

Deformed coset models from gauged WZW actions

NASA Astrophysics Data System (ADS)

Park, Q.-Han

1994-06-01

A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term Tr(gTg -1overlineT) , where algebra elements T, overlineT belong to the center of the algebra h associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equation, we give a systematic way to construct infinitely many conserved currents as well as soliton solutions. In the case of the parafermionic SU(2)/U(1)-coset model, we derive n-solitons and conserved currents explicitly.

Unified Framework for Deriving Simultaneous Equation Algorithms for Water Distribution Networks

EPA Science Inventory

The known formulations for steady state hydraulics within looped water distribution networks are re-derived in terms of linear and non-linear transformations of the original set of partly linear and partly non-linear equations that express conservation of mass and energy. All of ...

Martian tidal pressure and wind fields obtained from the Mariner 9 infrared spectroscopy experiment

NASA Technical Reports Server (NTRS)

Pirraglia, J. A.; Conrath, B. J.

1973-01-01

Using temperature fields derived from the Mariner 9 infrared spectroscopy experiment, the Martian atmospheric tidal pressure and wind fields are calculated. Temperature as a function of local time, latitude, and atmospheric pressure level is obtained by secular and longitudinal averaging of the data. The resulting temperature field is approximated by a spherical harmonic expansion, retaining one symmetric and one asymmetric term for wavenumber zero and wavenumber one. Vertical averaging of the linearized momentum and continuity equations results in an inhomogeneous tidal equation for surface pressure fluctuations with the driving function related to the temperature field through the geopotential function and the hydrostatic equation. Solutions of the tidal equation show a diurnal fractional pressure amplitude approximately equal to one half of the vertically averaged diurnal fractional temperature amplitude.

Martian tidal pressure and wind fields obtained from the Mariner 9 infrared spectroscopy experiment

NASA Technical Reports Server (NTRS)

Pirraglia, J. A.; Conrath, B. J.

1974-01-01

Using temperature fields derived from the Mariner 9 infrared spectroscopy experiment, the Martian atmospheric tidal pressure and wind fields are calculated. Temperature as a function of local time, latitude, and atmospheric pressure level is obtained by secular and longitudinal averaging of the data. The resulting temperature field is approximated by a spherical harmonic expansion, retaining one symmetric and one asymmetric term each for wavenumber zero and wavenumber one. Vertical averaging of the linearized momentum and continuity equations results in an inhomogeneous tidal equation for surface pressure fluctuations with the driving function related to the temperature field through the geopotential function and the hydrostatic equation. Solutions of the tidal equation show a diurnal fractional pressure amplitude approximately equal to one-half the vertically averaged diurnal fractional temperature amplitude.

Linearization of Conservative Nonlinear Oscillators

ERIC Educational Resources Information Center

Belendez, A.; Alvarez, M. L.; Fernandez, E.; Pascual, I.

2009-01-01

A linearization method of the nonlinear differential equation for conservative nonlinear oscillators is analysed and discussed. This scheme is based on the Chebyshev series expansion of the restoring force which allows us to obtain a frequency-amplitude relation which is valid not only for small but also for large amplitudes and, sometimes, for…

E = Mc(super 2) for the Chemist: When is Mass Conserved?

ERIC Educational Resources Information Center

Treptow. Richard S.

2005-01-01

An equation derived by Albert Einstein in 1905 that expresses a relationship between mass and energy, formulated as E = mc(super 2) is discussed with reference to the extent mass is conserved. This query can be used to challenge students and develop their language and critical thinking skills.

Wave-current interactions in three dimensions: why 3D radiation stresses are not practical

NASA Astrophysics Data System (ADS)

Ardhuin, Fabrice

2017-04-01

The coupling of ocean circulation and wave models is based on a wave-averaged mass and momentum conservation equations. Whereas several equivalent equations for the evolution of the current momentum have been proposed, implemented, and used, the possibility to formulate practical equations for the total momentum, which is the sum of the current and wave momenta, has been obscured by a series of publications. In a recent update on previous derivations, Mellor (J. Phys. Oceanogr. 2015) proposed a new set of wave-forced total momentum equations. Here we show that this derivation misses a term that integrates to zero over the vertical. This is because he went from his depth-integrated eq. (28) to the 3D equation (30) by simply removing the integral, but any extra zero-integrating term can be added. Corrected for this omission, the equations of motion are equivalent to the earlier equations by Mellor (2003) which are correct when expressed in terms of wave-induced pressure, horizontal velocity and vertical displacement. Namely the total momentum evolution is driven by the horizontal divergence of a horizontal momentum flux, ----- --- ∂^s- Sαβ = ^uα^uβ + δαβ ∂ς (^p- g^s) (1) and the vertical divergence of a vertical flux, Sαz = (p^-g^s)∂^s/∂xα, (2) where p is the wave-induced non-hydrostatic pressure, s is the wave-induced vertical displacement, and u^ α is the horizontal wave-induced velocity in direction α. So far, so good. Problems arise when p and s are evaluated. Indeend, Ardhuin et al. (J. Phys. Oceanogr. 2008) showed that, over a sloping bottom ∂Sαβ/∂xβ is of order of the slope, hence a consistent wave forcing requires an estimation of Sαz that must be estimated to first order in the bottom slope. For this, Airy wave theory, i.e. cosh(kz-+-kh) p ≃ ga cosh (kD ) cosψ, (3) is not enough. Ardhuin et al. (2008) has shown that using an exact solution of the Laplace equations the vertical flux can indeed be computed. The alternative of neglecting completely Sαz, as suggested by Mellor (2011) for small slopes, will always generate spurious currents because of the unbalanced forcing ∂Sαβ/∂xβ. Fortunately, there are many explicit versions of the wave-averaged equations without the wave momentum in them (Suzuki and Fox-Kemper 2016), with or without vortex force which are all consistent with the exact 3D equations of Andrews and McIntyre (1978). There is thus no need to stumble again and again on this fundamental problem of vertical momentum flux, which is a flux of wave momentum. The problem simply goes away by writing the equations for the current momentum only, without the problematic wave momentum. The current and wave momentum are coupled by forcing terms, and the wave momentum can be solved in 2D, the vertical distribution of momentum being maintained by the complex flux Sαz.

An efficient numerical solution of the transient storage equations for solute transport in small streams

USGS Publications Warehouse

Runkel, Robert L.; Chapra, Steven C.

1993-01-01

Several investigators have proposed solute transport models that incorporate the effects of transient storage. Transient storage occurs in small streams when portions of the transported solute become isolated in zones of water that are immobile relative to water in the main channel (e.g., pools, gravel beds). Transient storage is modeled by adding a storage term to the advection-dispersion equation describing conservation of mass for the main channel. In addition, a separate mass balance equation is written for the storage zone. Although numerous applications of the transient storage equations may be found in the literature, little attention has been paid to the numerical aspects of the approach. Of particular interest is the coupled nature of the equations describing mass conservation for the main channel and the storage zone. In the work described herein, an implicit finite difference technique is developed that allows for a decoupling of the governing differential equations. This decoupling method may be applied to other sets of coupled equations such as those describing sediment-water interactions for toxic contaminants. For the case at hand, decoupling leads to a 50% reduction in simulation run time. Computational costs may be further reduced through efficient application of the Thomas algorithm. These techniques may be easily incorporated into existing codes and new applications in which simulation run time is of concern.

Documentation of the Fourth Order Band Model

NASA Technical Reports Server (NTRS)

Kalnay-Rivas, E.; Hoitsma, D.

1979-01-01

A general circulation model is presented which uses quadratically conservative, fourth order horizontal space differences on an unstaggered grid and second order vertical space differences with a forward-backward or a smooth leap frog time scheme to solve the primitive equations of motion. The dynamic equations for motion, finite difference equations, a discussion of the structure and flow chart of the program code, a program listing, and three relevent papers are given.

Kinematics and depth-integrated terms in surf zone waves from laboratory measurement

NASA Astrophysics Data System (ADS)

Stansby, Peter K.; Feng, Tong

2005-04-01

Kinematics of nominally periodic surf zone waves have been measured in the laboratory using LDA (laser Doppler anemometry), above trough level as well as below, for weakly plunging breakers transforming into bores in shallower water. The aim was to determine, through phase- or ensemble-averaging, periodic flow structures in a two-dimensional vertical plane, from large-scale down to small-scale vortical structures. Coherent multiple vortical structures were evident at the initiation of breaking, becoming elongated along the surface during bore propagation. The initial region is likely to become more extensive as waves become more strongly plunging and could explain the difference in turbulence characteristics between plunging and spilling breakers observed elsewhere. Comparison of vorticity magnitudes with hydraulic-jump measurements showed some similarities during the initial stages of breaking, but these quickly grew less as breaking progressed into shallower water. Period-averaged kinematics and vorticity were also obtained showing shoreward mass transport above trough level and undertow below, with a thick layer of vorticity at trough level and a thin layer of vorticity of opposite rotation at the bed. There were also concentrated regions of mean vorticity near the end of the plunging region. Residual turbulence of relatively high frequency was presented as Reynolds stresses, showing marked anisotrophy. Dynamic pressure (pressure minus its hydrostatic component) was determined from the kinematics. The magnitudes of different effects were evaluated through the depth-integrated Reynolds-averaged Navier-Stokes (RANS) equations, which may be reduced to nine terms (the standard inviscid terms of the shallow-water equations conserving mass and momentum with hydrostatic pressure, and six additional terms), assuming that the complex, often aerated, free surface is treated as a simple interface. All terms were evaluated, assuming that a space/time transformation was justified with a slowly varying phase speed, and the net balance was always small in relation to the maxima of the larger terms. Terms due to dynamic pressure and vertical dispersion (due to the vertical variation of velocity) were as significant as the three terms in the inviscid shallow-water equations; terms involving residual turbulence were insignificant. The r.m.s. (root mean square) variation of each along the slope is highly irregular, with the inertia term due to (Eulerian) acceleration always greatest. This is consistent with complex, though repetitive, coherent structures. Modelling the flow with the shallow-water equations, using the surface elevation variation at the break point as input, nevertheless gave a good prediction of the wave height variation up the slope.

Cardiovascular risk prediction tools for populations in Asia.

PubMed

Barzi, F; Patel, A; Gu, D; Sritara, P; Lam, T H; Rodgers, A; Woodward, M

2007-02-01

Cardiovascular risk equations are traditionally derived from the Framingham Study. The accuracy of this approach in Asian populations, where resources for risk factor measurement may be limited, is unclear. To compare "low-information" equations (derived using only age, systolic blood pressure, total cholesterol and smoking status) derived from the Framingham Study with those derived from the Asian cohorts, on the accuracy of cardiovascular risk prediction. Separate equations to predict the 8-year risk of a cardiovascular event were derived from Asian and Framingham cohorts. The performance of these equations, and a subsequently "recalibrated" Framingham equation, were evaluated among participants from independent Chinese cohorts. Six cohort studies from Japan, Korea and Singapore (Asian cohorts); six cohort studies from China; the Framingham Study from the US. 172,077 participants from the Asian cohorts; 25,682 participants from Chinese cohorts and 6053 participants from the Framingham Study. In the Chinese cohorts, 542 cardiovascular events occurred during 8 years of follow-up. Both the Asian cohorts and the Framingham equations discriminated cardiovascular risk well in the Chinese cohorts; the area under the receiver-operator characteristic curve was at least 0.75 for men and women. However, the Framingham risk equation systematically overestimated risk in the Chinese cohorts by an average of 276% among men and 102% among women. The corresponding average overestimation using the Asian cohorts equation was 11% and 10%, respectively. Recalibrating the Framingham risk equation using cardiovascular disease incidence from the non-Chinese Asian cohorts led to an overestimation of risk by an average of 4% in women and underestimation of risk by an average of 2% in men. A low-information Framingham cardiovascular risk prediction tool, which, when recalibrated with contemporary data, is likely to estimate future cardiovascular risk with similar accuracy in Asian populations as tools developed from data on local cohorts.

STOCK Mechanics:. a General Theory and Method of Energy Conservation with Applications on Djia

NASA Astrophysics Data System (ADS)

Tuncay, Çağlar

A new method, based on the original theory of conservation of sum of kinetic and potential energy defined for prices is proposed and applied on the Dow Jones Industrials Average (DJIA). The general trends averaged over months or years gave a roughly conserved total energy, with three different potential energies, i.e., positive definite quadratic, negative definite quadratic and linear potential energy for exponential rises (and falls), sinusoidal oscillations and parabolic trajectories, respectively. Corresponding expressions for force (impact) are also given.

Electro-kinetically driven peristaltic transport of viscoelastic physiological fluids through a finite length capillary: Mathematical modeling.

PubMed

Tripathi, Dharmendra; Yadav, Ashu; Bég, O Anwar

2017-01-01

Analytical solutions are developed for the electro-kinetic flow of a viscoelastic biological liquid in a finite length cylindrical capillary geometry under peristaltic waves. The Jefferys' non-Newtonian constitutive model is employed to characterize rheological properties of the fluid. The unsteady conservation equations for mass and momentum with electro-kinetic and Darcian porous medium drag force terms are reduced to a system of steady linearized conservation equations in an axisymmetric coordinate system. The long wavelength, creeping (low Reynolds number) and Debye-Hückel linearization approximations are utilized. The resulting boundary value problem is shown to be controlled by a number of parameters including the electro-osmotic parameter, Helmholtz-Smoluchowski velocity (maximum electro-osmotic velocity), and Jefferys' first parameter (ratio of relaxation and retardation time), wave amplitude. The influence of these parameters and also time on axial velocity, pressure difference, maximum volumetric flow rate and streamline distributions (for elucidating trapping phenomena) is visualized graphically and interpreted in detail. Pressure difference magnitudes are enhanced consistently with both increasing electro-osmotic parameter and Helmholtz-Smoluchowski velocity, whereas they are only elevated with increasing Jefferys' first parameter for positive volumetric flow rates. Maximum time averaged flow rate is enhanced with increasing electro-osmotic parameter, Helmholtz-Smoluchowski velocity and Jefferys' first parameter. Axial flow is accelerated in the core (plug) region of the conduit with greater values of electro-osmotic parameter and Helmholtz-Smoluchowski velocity whereas it is significantly decelerated with increasing Jefferys' first parameter. The simulations find applications in electro-osmotic (EO) transport processes in capillary physiology and also bio-inspired EO pump devices in chemical and aerospace engineering. Copyright © 2016 Elsevier Inc. All rights reserved.

Titration Calculations with Computer Algebra Software

ERIC Educational Resources Information Center

Lachance, Russ; Biaglow, Andrew

2012-01-01

This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…

Equations of state and transport properties of mixtures in the warm dense regime

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

Hou, Yong; Dai, Jiayu; Kang, Dongdong

2015-02-15

We have performed average-atom molecular dynamics to simulate the CH and LiH mixtures in the warm dense regime, and obtained equations of state and the ionic transport properties. The electronic structures are calculated by using the modified average-atom model, which have included the broadening of energy levels, and the ion-ion pair potentials of mixtures are constructed based on the temperature-dependent density functional theory. The ionic transport properties, such as ionic diffusion and shear viscosity, are obtained through the ionic velocity correlation functions. The equations of state and transport properties for carbon, hydrogen and lithium, hydrogen mixtures in a wide regionmore » of density and temperature are calculated. Through our computing the average ionization degree, average ion-sphere diameter and transition properties in the mixture, it is shown that transport properties depend not only on the ionic mass but also on the average ionization degree.« less

Energy Stable Flux Formulas For The Discontinuous Galerkin Discretization Of First Order Nonlinear Conservation Laws

NASA Technical Reports Server (NTRS)

Barth, Timothy; Charrier, Pierre; Mansour, Nagi N. (Technical Monitor)

2001-01-01

We consider the discontinuous Galerkin (DG) finite element discretization of first order systems of conservation laws derivable as moments of the kinetic Boltzmann equation. This includes well known conservation law systems such as the Euler For the class of first order nonlinear conservation laws equipped with an entropy extension, an energy analysis of the DG method for the Cauchy initial value problem is developed. Using this DG energy analysis, several new variants of existing numerical flux functions are derived and shown to be energy stable.

Simulation of Synthetic Jets in Quiescent Air Using Unsteady Reynolds Averaged Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Vatsa, Veer N.; Turkel, Eli

2006-01-01

We apply an unsteady Reynolds-averaged Navier-Stokes (URANS) solver for the simulation of a synthetic jet created by a single diaphragm piezoelectric actuator in quiescent air. This configuration was designated as Case 1 for the CFDVAL2004 workshop held at Williamsburg, Virginia, in March 2004. Time-averaged and instantaneous data for this case were obtained at NASA Langley Research Center, using multiple measurement techniques. Computational results for this case using one-equation Spalart-Allmaras and two-equation Menter's turbulence models are presented along with the experimental data. The effect of grid refinement, preconditioning and time-step variation are also examined in this paper.

Evolution of statistical averages: An interdisciplinary proposal using the Chapman-Enskog method

NASA Astrophysics Data System (ADS)

Mariscal-Sanchez, A.; Sandoval-Villalbazo, A.

2017-08-01

This work examines the idea of applying the Chapman-Enskog (CE) method for approximating the solution of the Boltzmann equation beyond the realm of physics, using an information theory approach. Equations describing the evolution of averages and their fluctuations in a generalized phase space are established up to first-order in the Knudsen parameter which is defined as the ratio of the time between interactions (mean free time) and a characteristic macroscopic time. Although the general equations here obtained may be applied in a wide range of disciplines, in this paper, only a particular case related to the evolution of averages in speculative markets is examined.

A conservative finite difference algorithm for the unsteady transonic potential equation in generalized coordinates

NASA Technical Reports Server (NTRS)

Bridgeman, J. O.; Steger, J. L.; Caradonna, F. X.

1982-01-01

An implicit, approximate-factorization, finite-difference algorithm has been developed for the computation of unsteady, inviscid transonic flows in two and three dimensions. The computer program solves the full-potential equation in generalized coordinates in conservation-law form in order to properly capture shock-wave position and speed. A body-fitted coordinate system is employed for the simple and accurate treatment of boundary conditions on the body surface. The time-accurate algorithm is modified to a conventional ADI relaxation scheme for steady-state computations. Results from two- and three-dimensional steady and two-dimensional unsteady calculations are compared with existing methods.

One-dimensional thermohydraulic code THESEUS and its application to chilldown process simulation in two-phase hydrogen flows

NASA Astrophysics Data System (ADS)

Papadimitriou, P.; Skorek, T.

THESUS is a thermohydraulic code for the calculation of steady state and transient processes of two-phase cryogenic flows. The physical model is based on four conservation equations with separate liquid and gas phase mass conservation equations. The thermohydraulic non-equilibrium is calculated by means of evaporation and condensation models. The mechanical non-equilibrium is modeled by a full-range drift-flux model. Also heat conduction in solid structures and heat exchange for the full spectrum of heat transfer regimes can be simulated. Test analyses of two-channel chilldown experiments and comparisons with the measured data have been performed.

The method of space-time and conservation element and solution element: A new approach for solving the Navier-Stokes and Euler equations

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung

1995-01-01

A new numerical framework for solving conservation laws is being developed. This new framework differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is conceptually simple and designed to overcome several key limitations of the above traditional methods. A two-level scheme for solving the convection-diffusion equation is constructed and used to illuminate the major differences between the present method and those previously mentioned. This explicit scheme, referred to as the a-mu scheme, has two independent marching variables.

Lateral Load Capacity of Piles: A Comparative Study Between Indian Standards and Theoretical Approach

NASA Astrophysics Data System (ADS)

Jayasree, P. K.; Arun, K. V.; Oormila, R.; Sreelakshmi, H.

2018-05-01

As per Indian Standards, laterally loaded piles are usually analysed using the method adopted by IS 2911-2010 (Part 1/Section 2). But the practising engineers are of the opinion that the IS method is very conservative in design. This work aims at determining the extent to which the conventional IS design approach is conservative. This is done through a comparative study between IS approach and the theoretical model based on Vesic's equation. Bore log details for six different bridges were collected from the Kerala Public Works Department. Cast in situ fixed head piles embedded in three soil conditions both end bearing as well as friction piles were considered and analyzed separately. Piles were also modelled in STAAD.Pro software based on IS approach and the results were validated using Matlock and Reese (In Proceedings of fifth international conference on soil mechanics and foundation engineering, 1961) equation. The results were presented as the percentage variation in values of bending moment and deflection obtained by different methods. The results obtained from the mathematical model based on Vesic's equation and that obtained as per the IS approach were compared and the IS method was found to be uneconomical and conservative.

A point-centered arbitrary Lagrangian Eulerian hydrodynamic approach for tetrahedral meshes

DOE PAGES

Morgan, Nathaniel R.; Waltz, Jacob I.; Burton, Donald E.; ...

2015-02-24

We present a three dimensional (3D) arbitrary Lagrangian Eulerian (ALE) hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedral meshes. The new approach stores the conserved variables (mass, momentum, and total energy) at the nodes of the mesh and solves the conservation equations on a control volume surrounding the point. This type of an approach is termed a point-centered hydrodynamic (PCH) method. The conservation equations are discretized using an edge-based finite element (FE) approach with linear basis functions. All fluxes in the new approach are calculated at the center of each tetrahedron. A multidirectional Riemann-like problem is solved atmore » the center of the tetrahedron. The advective fluxes are calculated by solving a 1D Riemann problem on each face of the nodal control volume. A 2-stage Runge–Kutta method is used to evolve the solution forward in time, where the advective fluxes are part of the temporal integration. The mesh velocity is smoothed by solving a Laplacian equation. The details of the new ALE hydrodynamic scheme are discussed. Results from a range of numerical test problems are presented.« less

Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows.

PubMed

Abel, I G; Plunk, G G; Wang, E; Barnes, M; Cowley, S C; Dorland, W; Schekochihin, A A

2013-11-01

This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ε = ρi/α of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad-Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the 'high-flow' gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, is conserved and there is no net turbulent heating. Entropy is produced by the action of fluxes flattening gradients, Ohmic heating and the equilibration of interspecies temperature differences. This equilibration is found to include both turbulent and collisional contributions. Finally, this framework is condensed, in the low-Mach-number limit, to a more concise set of equations suitable for numerical implementation.

Energy and maximum norm estimates for nonlinear conservation laws

NASA Technical Reports Server (NTRS)

Olsson, Pelle; Oliger, Joseph

1994-01-01

We have devised a technique that makes it possible to obtain energy estimates for initial-boundary value problems for nonlinear conservation laws. The two major tools to achieve the energy estimates are a certain splitting of the flux vector derivative f(u)(sub x), and a structural hypothesis, referred to as a cone condition, on the flux vector f(u). These hypotheses are fulfilled for many equations that occur in practice, such as the Euler equations of gas dynamics. It should be noted that the energy estimates are obtained without any assumptions on the gradient of the solution u. The results extend to weak solutions that are obtained as point wise limits of vanishing viscosity solutions. As a byproduct we obtain explicit expressions for the entropy function and the entropy flux of symmetrizable systems of conservation laws. Under certain circumstances the proposed technique can be applied repeatedly so as to yield estimates in the maximum norm.

Localized states in the conserved Swift-Hohenberg equation with cubic nonlinearity

NASA Astrophysics Data System (ADS)

Thiele, Uwe; Archer, Andrew J.; Robbins, Mark J.; Gomez, Hector; Knobloch, Edgar

2013-04-01

The conserved Swift-Hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state. The resulting phase field crystal model describes a variety of spatially localized structures, in addition to different spatially extended periodic structures. The location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions. Localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases, and may lie outside of the binodal for these states. The results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential, and are expected to carry over to related models with a conserved order parameter.

Control volume based hydrocephalus research; analysis of human data

NASA Astrophysics Data System (ADS)

Cohen, Benjamin; Wei, Timothy; Voorhees, Abram; Madsen, Joseph; Anor, Tomer

2010-11-01

Hydrocephalus is a neuropathophysiological disorder primarily diagnosed by increased cerebrospinal fluid volume and pressure within the brain. To date, utilization of clinical measurements have been limited to understanding of the relative amplitude and timing of flow, volume and pressure waveforms; qualitative approaches without a clear framework for meaningful quantitative comparison. Pressure volume models and electric circuit analogs enforce volume conservation principles in terms of pressure. Control volume analysis, through the integral mass and momentum conservation equations, ensures that pressure and volume are accounted for using first principles fluid physics. This approach is able to directly incorporate the diverse measurements obtained by clinicians into a simple, direct and robust mechanics based framework. Clinical data obtained for analysis are discussed along with data processing techniques used to extract terms in the conservation equation. Control volume analysis provides a non-invasive, physics-based approach to extracting pressure information from magnetic resonance velocity data that cannot be measured directly by pressure instrumentation.

Three-pattern decomposition of global atmospheric circulation: part II—dynamical equations of horizontal, meridional and zonal circulations

NASA Astrophysics Data System (ADS)

Hu, Shujuan; Cheng, Jianbo; Xu, Ming; Chou, Jifan

2018-04-01

The three-pattern decomposition of global atmospheric circulation (TPDGAC) partitions three-dimensional (3D) atmospheric circulation into horizontal, meridional and zonal components to study the 3D structures of global atmospheric circulation. This paper incorporates the three-pattern decomposition model (TPDM) into primitive equations of atmospheric dynamics and establishes a new set of dynamical equations of the horizontal, meridional and zonal circulations in which the operator properties are studied and energy conservation laws are preserved, as in the primitive equations. The physical significance of the newly established equations is demonstrated. Our findings reveal that the new equations are essentially the 3D vorticity equations of atmosphere and that the time evolution rules of the horizontal, meridional and zonal circulations can be described from the perspective of 3D vorticity evolution. The new set of dynamical equations includes decomposed expressions that can be used to explore the source terms of large-scale atmospheric circulation variations. A simplified model is presented to demonstrate the potential applications of the new equations for studying the dynamics of the Rossby, Hadley and Walker circulations. The model shows that the horizontal air temperature anomaly gradient (ATAG) induces changes in meridional and zonal circulations and promotes the baroclinic evolution of the horizontal circulation. The simplified model also indicates that the absolute vorticity of the horizontal circulation is not conserved, and its changes can be described by changes in the vertical vorticities of the meridional and zonal circulations. Moreover, the thermodynamic equation shows that the induced meridional and zonal circulations and advection transport by the horizontal circulation in turn cause a redistribution of the air temperature. The simplified model reveals the fundamental rules between the evolution of the air temperature and the horizontal, meridional and zonal components of global atmospheric circulation.

Conservative 3 + 1 general relativistic variable Eddington tensor radiation transport equations

DOE PAGES

Cardall, Christian Y.; Endeve, Eirik; Mezzacappa, Anthony

2013-05-07

We present conservative 3+1 general relativistic variable Eddington tensor radiation transport equations, including greater elaboration of the momentum space divergence (that is, the energy derivative term) than in previous work. These equations are intended for use in simulations involving numerical relativity, particularly in the absence of spherical symmetry. The independent variables are the lab frame coordinate basis spacetime position coordinates and the particle energy measured in the comoving frame. With an eye towards astrophysical applications—such as core-collapse supernovae and compact object mergers—in which the fluid includes nuclei and/or nuclear matter at finite temperature, and in which the transported particles aremore » neutrinos, we pay special attention to the consistency of four-momentum and lepton number exchange between neutrinos and the fluid, showing the term-by-term cancellations that must occur for this consistency to be achieved.« less

Modulational instability: Conservation laws and bright soliton solution of ion-acoustic waves in electron-positron-ion-dust plasmas

NASA Astrophysics Data System (ADS)

EL-Kalaawy, O. H.

2018-02-01

We consider the nonlinear propagation of non-planar (cylindrical and spherical) ion-acoustic (IA) envelope solitary waves in an unmagnetized plasma consisting of electron-positron-ion-dust plasma with two-electron temperature distributions in the context of the non-extensive statistics. The basic set of fluid equations is reduced to the modified nonlinear Schrödinger (MNLS) equation in cylindrical and spherical geometry by using the reductive perturbation method (RPM). It is found that the nature of the modulational instabilities would be significantly modified due to the effects of the non-extensive and other plasma parameters as well as cylindrical and spherical geometry. Conservation laws of the MNLS equation are obtained by Lie symmetry and multiplier method. A new exact solution (envelope bright soliton) is obtained by the extended homogeneous balance method. Finally, we study the results of this article.

A spectral mimetic least-squares method

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

Bochev, Pavel; Gerritsma, Marc

We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are alsomore » satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.« less

A spectral mimetic least-squares method

DOE PAGES

Bochev, Pavel; Gerritsma, Marc

2014-09-01

We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are alsomore » satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.« less

Lax pair, conservation laws and solitons for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger equation governing an α-helical protein

NASA Astrophysics Data System (ADS)

Chai, Jun; Tian, Bo; Zhen, Hui-Ling; Sun, Wen-Rong

2015-11-01

Energy transfer through a (2+1)-dimensional α-helical protein can be described by a (2+1)-dimensional fourth-order nonlinear Schrödinger equation. For such an equation, a Lax pair and the infinitely-many conservation laws are derived. Using an auxiliary function and a bilinear formulation, we get the one-, two-, three- and N-soliton solutions via the Hirota method. The soliton velocity is linearly related to the lattice parameter γ, while the soliton' direction and amplitude do not depend on γ. Interactions between the two solitons are elastic, while those among the three solitons are pairwise elastic. Oblique, head-on and overtaking interactions between the two solitons are displayed. Oblique interaction among the three solitons and interactions among the two parallel solitons and a single one are presented as well.

Pinching solutions of slender cylindrical jets

NASA Technical Reports Server (NTRS)

Papageorgiou, Demetrios T.; Orellana, Oscar

1993-01-01

Simplified equations for slender jets are derived for a circular jet of one fluid flowing into an ambient second fluid, the flow being confined in a circular tank. Inviscid flows are studied which include both surface tension effects and Kelvin-Helmholtz instability. For slender jets a coupled nonlinear system of equations is found for the jet shape and the axial velocity jump across it. The equations can break down after a finite time and similarity solutions are constructed, and studied analytically and numerically. The break-ups found pertain to the jet pinching after a finite time, without violation of the slender jet ansatz. The system is conservative and admissible singular solutions are those which conserve the total energy, mass, and momentum. Such solutions are constructed analytically and numerically, and in the case of vortex sheets with no surface tension certain solutions are given in closed form.

A Runge-Kutta discontinuous finite element method for high speed flows

NASA Technical Reports Server (NTRS)

Bey, Kim S.; Oden, J. T.

1991-01-01

A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.

Quantitative Compactness Estimates for Hamilton-Jacobi Equations

NASA Astrophysics Data System (ADS)

Ancona, Fabio; Cannarsa, Piermarco; Nguyen, Khai T.

2016-02-01

We study quantitative compactness estimates in {W^{1,1}_{loc}} for the map {S_t}, {t > 0} that is associated with the given initial data {u_0in Lip (R^N)} for the corresponding solution {S_t u_0} of a Hamilton-Jacobi equation u_t+Hbig(nabla_{x} ubig)=0, qquad t≥ 0,quad xinR^N, with a uniformly convex Hamiltonian {H=H(p)}. We provide upper and lower estimates of order {1/\\varepsilon^N} on the Kolmogorov {\\varepsilon}-entropy in {W^{1,1}} of the image through the map S t of sets of bounded, compactly supported initial data. Estimates of this type are inspired by a question posed by Lax (Course on Hyperbolic Systems of Conservation Laws. XXVII Scuola Estiva di Fisica Matematica, Ravello, 2002) within the context of conservation laws, and could provide a measure of the order of "resolution" of a numerical method implemented for this equation.

Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes

NASA Astrophysics Data System (ADS)

Chertock, Alina; Cui, Shumo; Kurganov, Alexander; Özcan, Şeyma Nur; Tadmor, Eitan

2018-04-01

We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global fluxes. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables combined with a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.

A three-dimensional, time-dependent model of Mobile Bay

NASA Technical Reports Server (NTRS)

Pitts, F. H.; Farmer, R. C.

1976-01-01

A three-dimensional, time-variant mathematical model for momentum and mass transport in estuaries was developed and its solution implemented on a digital computer. The mathematical model is based on state and conservation equations applied to turbulent flow of a two-component, incompressible fluid having a free surface. Thus, bouyancy effects caused by density differences between the fresh and salt water, inertia from thare river and tidal currents, and differences in hydrostatic head are taken into account. The conservation equations, which are partial differential equations, are solved numerically by an explicit, one-step finite difference scheme and the solutions displayed numerically and graphically. To test the validity of the model, a specific estuary for which scaled model and experimental field data are available, Mobile Bay, was simulated. Comparisons of velocity, salinity and water level data show that the model is valid and a viable means of simulating the hydrodynamics and mass transport in non-idealized estuaries.

Measuring the equations of state in a relaxed magnetohydrodynamic plasma.

PubMed

Kaur, M; Barbano, L J; Suen-Lewis, E M; Shrock, J E; Light, A D; Brown, M R; Schaffner, D A

2018-01-01

We report measurements of the equations of state of a fully relaxed magnetohydrodynamic (MHD) laboratory plasma. Parcels of magnetized plasma, called Taylor states, are formed in a coaxial magnetized plasma gun, and are allowed to relax and drift into a closed flux conserving volume. Density, ion temperature, and magnetic field are measured as a function of time as the Taylor states compress and heat. The theoretically predicted MHD and double adiabatic equations of state are compared to experimental measurements. We find that the MHD equation of state is inconsistent with our data.

On the various forms of the energy equation for a dilute, monatomic mixture of nonreacting gases

NASA Technical Reports Server (NTRS)

Kennedy, Christopher A.

1994-01-01

In the case of gas mixtures, the governing equations become rather formidable and a complete listing of the equations in their various forms and methods to evaluate the transport coefficients is difficult to find. This paper seeks to compile common, as well as less well known, results in a single document. Various relationships between equations describing conservation of energy for a dilute, monatomic, nonreacting gas in local equilibrium are provided. The gas is treated as nonrelativistic, not subject to magnetic or electric fields, or radiative effects.

Measuring the equations of state in a relaxed magnetohydrodynamic plasma

NASA Astrophysics Data System (ADS)

Kaur, M.; Barbano, L. J.; Suen-Lewis, E. M.; Shrock, J. E.; Light, A. D.; Brown, M. R.; Schaffner, D. A.

2018-01-01

We report measurements of the equations of state of a fully relaxed magnetohydrodynamic (MHD) laboratory plasma. Parcels of magnetized plasma, called Taylor states, are formed in a coaxial magnetized plasma gun, and are allowed to relax and drift into a closed flux conserving volume. Density, ion temperature, and magnetic field are measured as a function of time as the Taylor states compress and heat. The theoretically predicted MHD and double adiabatic equations of state are compared to experimental measurements. We find that the MHD equation of state is inconsistent with our data.

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

Eldred, Christopher; Randall, David

The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restrictedmore » to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Lastly, detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.« less

An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes

DOE PAGES

Taitano, William; Chacon, Luis; Simakov, Andrei Nikolaevich

2016-04-25

In this paper, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth–Fokker–Planck (RFP) equation, which is exactly mass-, momentum-, and energy-conserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local thermal velocity, v th (t), and explicitly considers the resulting inertial terms in the Fokker–Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker–Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic v th -ratio-based expansion ofmore » the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. Specifically, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.« less

An entropy correction method for unsteady full potential flows with strong shocks

NASA Technical Reports Server (NTRS)

Whitlow, W., Jr.; Hafez, M. M.; Osher, S. J.

1986-01-01

An entropy correction method for the unsteady full potential equation is presented. The unsteady potential equation is modified to account for entropy jumps across shock waves. The conservative form of the modified equation is solved in generalized coordinates using an implicit, approximate factorization method. A flux-biasing differencing method, which generates the proper amounts of artificial viscosity in supersonic regions, is used to discretize the flow equations in space. Comparisons between the present method and solutions of the Euler equations and between the present method and experimental data are presented. The comparisons show that the present method more accurately models solutions of the Euler equations and experiment than does the isentropic potential formulation.

Numerical solution of nonlinear partial differential equations of mixed type. [finite difference approximation

NASA Technical Reports Server (NTRS)

Jameson, A.

1976-01-01

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

Revisiting low-fidelity two-fluid models for gas-solids transport

NASA Astrophysics Data System (ADS)

Adeleke, Najeem; Adewumi, Michael; Ityokumbul, Thaddeus

2016-08-01

Two-phase gas-solids transport models are widely utilized for process design and automation in a broad range of industrial applications. Some of these applications include proppant transport in gaseous fracking fluids, air/gas drilling hydraulics, coal-gasification reactors and food processing units. Systems automation and real time process optimization stand to benefit a great deal from availability of efficient and accurate theoretical models for operations data processing. However, modeling two-phase pneumatic transport systems accurately requires a comprehensive understanding of gas-solids flow behavior. In this study we discuss the prevailing flow conditions and present a low-fidelity two-fluid model equation for particulate transport. The model equations are formulated in a manner that ensures the physical flux term remains conservative despite the inclusion of solids normal stress through the empirical formula for modulus of elasticity. A new set of Roe-Pike averages are presented for the resulting strictly hyperbolic flux term in the system of equations, which was used to develop a Roe-type approximate Riemann solver. The resulting scheme is stable regardless of the choice of flux-limiter. The model is evaluated by the prediction of experimental results from both pneumatic riser and air-drilling hydraulics systems. We demonstrate the effect and impact of numerical formulation and choice of numerical scheme on model predictions. We illustrate the capability of a low-fidelity one-dimensional two-fluid model in predicting relevant flow parameters in two-phase particulate systems accurately even under flow regimes involving counter-current flow.

Heuristic approach to capillary pressures averaging

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

Coca, B.P.

1980-10-01

Several methods are available to average capillary pressure curves. Among these are the J-curve and regression equations of the wetting-fluid saturation in porosity and permeability (capillary pressure held constant). While the regression equation seem completely empiric, the J-curve method seems to be theoretically sound due to its expression based on a relation between the average capillary radius and the permeability-porosity ratio. An analysis is given of each of these methods.

Kinetic description of ionospheric dynamics in the three-fluid approximation

NASA Technical Reports Server (NTRS)

Comfort, R. H.

1975-01-01

Conservation equations are developed in the three-fluid approximation for general application problems of ionospheric dynamics in the altitude region 90 km to 800 km for all geographic locations. These equations are applied to a detailed study of auroral E region neutral winds and their relationship to ionospheric plasma motions.

Program For Finite-Element Analyses Of Phase-Change Fluids

NASA Technical Reports Server (NTRS)

Viterna, L. A.

1995-01-01

PHASTRAN analyzes heat-transfer and flow behaviors of materials undergoing phase changes. Many phase changes operate over range of accelerations or effective gravitational fields. To analyze such thermal systems, it is necessary to obtain simultaneous solutions for equations of conservation of energy, momentum, and mass, and for equation of state. Written in APL2.

Steady-state heat conduction in quiescent fluids: Incompleteness of the Navier-Stokes-Fourier equations

NASA Astrophysics Data System (ADS)

Brenner, Howard

2011-10-01

Linear irreversible thermodynamic principles are used to demonstrate, by counterexample, the existence of a fundamental incompleteness in the basic pre-constitutive mass, momentum, and energy equations governing fluid mechanics and transport phenomena in continua. The demonstration is effected by addressing the elementary case of steady-state heat conduction (and transport processes in general) occurring in quiescent fluids. The counterexample questions the universal assumption of equality of the four physically different velocities entering into the basic pre-constitutive mass, momentum, and energy conservation equations. Explicitly, it is argued that such equality is an implicit constitutive assumption rather than an established empirical fact of unquestioned authority. Such equality, if indeed true, would require formal proof of its validity, currently absent from the literature. In fact, our counterexample shows the assumption of equality to be false. As the current set of pre-constitutive conservation equations appearing in textbooks are regarded as applicable both to continua and noncontinua (e.g., rarefied gases), our elementary counterexample negating belief in the equality of all four velocities impacts on all aspects of fluid mechanics and transport processes, continua and noncontinua alike.

Variational formulation for dissipative continua and an incremental J-integral

NASA Astrophysics Data System (ADS)

Rahaman, Md. Masiur; Dhas, Bensingh; Roy, D.; Reddy, J. N.

2018-01-01

Our aim is to rationally formulate a proper variational principle for dissipative (viscoplastic) solids in the presence of inertia forces. As a first step, a consistent linearization of the governing nonlinear partial differential equations (PDEs) is carried out. An additional set of complementary (adjoint) equations is then formed to recover an underlying variational structure for the augmented system of linearized balance laws. This makes it possible to introduce an incremental Lagrangian such that the linearized PDEs, including the complementary equations, become the Euler-Lagrange equations. Continuous groups of symmetries of the linearized PDEs are computed and an analysis is undertaken to identify the variational groups of symmetries of the linearized dissipative system. Application of Noether's theorem leads to the conservation laws (conserved currents) of motion corresponding to the variational symmetries. As a specific outcome, we exploit translational symmetries of the functional in the material space and recover, via Noether's theorem, an incremental J-integral for viscoplastic solids in the presence of inertia forces. Numerical demonstrations are provided through a two-dimensional plane strain numerical simulation of a compact tension specimen of annealed mild steel under dynamic loading.

Involution and Difference Schemes for the Navier-Stokes Equations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Blinkov, Yuri A.

In the present paper we consider the Navier-Stokes equations for the two-dimensional viscous incompressible fluid flows and apply to these equations our earlier designed general algorithmic approach to generation of finite-difference schemes. In doing so, we complete first the Navier-Stokes equations to involution by computing their Janet basis and discretize this basis by its conversion into the integral conservation law form. Then we again complete the obtained difference system to involution with eliminating the partial derivatives and extracting the minimal Gröbner basis from the Janet basis. The elements in the obtained difference Gröbner basis that do not contain partial derivatives of the dependent variables compose a conservative difference scheme. By exploiting arbitrariness in the numerical integration approximation we derive two finite-difference schemes that are similar to the classical scheme by Harlow and Welch. Each of the two schemes is characterized by a 5×5 stencil on an orthogonal and uniform grid. We also demonstrate how an inconsistent difference scheme with a 3×3 stencil is generated by an inappropriate numerical approximation of the underlying integrals.

Numerical Examination of Flux Correction for Solving the Navier-Stokes Equations on Unstructured Meshes

DTIC Science & Technology

2014-08-29

lends itself to parallelization due to it’s discontinuous nature [27]. It has well-established stability properties and is actively being researched...from considering Conservation of Mass , Conservation of Momentum, and Conservation of Energy. This well-known result, known as the Navier-Stokes...internal plus kinetic) per unit mass and h = e+ Pρ as the enthalpy. qj is the jth component of the heat flux vector. This can be related to temperature

Reduced Basis and Stochastic Modeling of Liquid Propellant Rocket Engine as a Complex System

DTIC Science & Technology

2015-07-02

additions, the approach will be extended to a real- gas system so that it can be used to investigate model multi-element liquid rocket combustors in a...Sirignano (2010). In the following discussion, we examine the various conservation principles for the gas and liquid phases. The hyperbolic nature of the...conservation equations for the gas and liquid phases. Mass conservation of individual chemical species or of individual classes of liquid droplets will

HYDRA-II: A hydrothermal analysis computer code: Volume 3, Verification/validation assessments

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

McCann, R.A.; Lowery, P.S.

1987-10-01

HYDRA-II is a hydrothermal computer code capable of three-dimensional analysis of coupled conduction, convection, and thermal radiation problems. This code is especially appropriate for simulating the steady-state performance of spent fuel storage systems. The code has been evaluated for this application for the US Department of Energy's Commercial Spent Fuel Management Program. HYDRA-II provides a finite difference solution in cartesian coordinates to the equations governing the conservation of mass, momentum, and energy. A cylindrical coordinate system may also be used to enclose the cartesian coordinate system. This exterior coordinate system is useful for modeling cylindrical cask bodies. The difference equationsmore » for conservation of momentum are enhanced by the incorporation of directional porosities and permeabilities that aid in modeling solid structures whose dimensions may be smaller than the computational mesh. The equation for conservation of energy permits modeling of orthotropic physical properties and film resistances. Several automated procedures are available to model radiation transfer within enclosures and from fuel rod to fuel rod. The documentation of HYDRA-II is presented in three separate volumes. Volume I - Equations and Numerics describes the basic differential equations, illustrates how the difference equations are formulated, and gives the solution procedures employed. Volume II - User's Manual contains code flow charts, discusses the code structure, provides detailed instructions for preparing an input file, and illustrates the operation of the code by means of a model problem. This volume, Volume III - Verification/Validation Assessments, provides a comparison between the analytical solution and the numerical simulation for problems with a known solution. This volume also documents comparisons between the results of simulations of single- and multiassembly storage systems and actual experimental data. 11 refs., 55 figs., 13 tabs.« less

High-Resolution Genuinely Multidimensional Solution of Conservation Laws by the Space-Time Conservation Element and Solution Element Method

NASA Technical Reports Server (NTRS)

Himansu, Ananda; Chang, Sin-Chung; Yu, Sheng-Tao; Wang, Xiao-Yen; Loh, Ching-Yuen; Jorgenson, Philip C. E.

1999-01-01

In this overview paper, we review the basic principles of the method of space-time conservation element and solution element for solving the conservation laws in one and two spatial dimensions. The present method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. In contrast to the modern upwind schemes, the approach here does not use the Riemann solver and the reconstruction procedure as the building blocks. The drawbacks of the upwind approach, such as the difficulty of rationally extending the 1D scalar approach to systems of equations and particularly to multiple dimensions is here contrasted with the uniformity and ease of generalization of the Conservation Element and Solution Element (CE/SE) 1D scalar schemes to systems of equations and to multiple spatial dimensions. The assured compatibility with the simplest type of unstructured meshes, and the uniquely simple nonreflecting boundary conditions of the present method are also discussed. The present approach has yielded high-resolution shocks, rarefaction waves, acoustic waves, vortices, ZND detonation waves, and shock/acoustic waves/vortices interactions. Moreover, since no directional splitting is employed, numerical resolution of two-dimensional calculations is comparable to that of the one-dimensional calculations. Some sample applications displaying the strengths and broad applicability of the CE/SE method are reviewed.

Estimating glomerular filtration rate (GFR) in children. The average between a cystatin C- and a creatinine-based equation improves estimation of GFR in both children and adults and enables diagnosing Shrunken Pore Syndrome.

PubMed

Leion, Felicia; Hegbrant, Josefine; den Bakker, Emil; Jonsson, Magnus; Abrahamson, Magnus; Nyman, Ulf; Björk, Jonas; Lindström, Veronica; Larsson, Anders; Bökenkamp, Arend; Grubb, Anders

2017-09-01

Estimating glomerular filtration rate (GFR) in adults by using the average of values obtained by a cystatin C- (eGFR cystatin C ) and a creatinine-based (eGFR creatinine ) equation shows at least the same diagnostic performance as GFR estimates obtained by equations using only one of these analytes or by complex equations using both analytes. Comparison of eGFR cystatin C and eGFR creatinine plays a pivotal role in the diagnosis of Shrunken Pore Syndrome, where low eGFR cystatin C compared to eGFR creatinine has been associated with higher mortality in adults. The present study was undertaken to elucidate if this concept can also be applied in children. Using iohexol and inulin clearance as gold standard in 702 children, we studied the diagnostic performance of 10 creatinine-based, 5 cystatin C-based and 3 combined cystatin C-creatinine eGFR equations and compared them to the result of the average of 9 pairs of a eGFR cystatin C and a eGFR creatinine estimate. While creatinine-based GFR estimations are unsuitable in children unless calibrated in a pediatric or mixed pediatric-adult population, cystatin C-based estimations in general performed well in children. The average of a suitable creatinine-based and a cystatin C-based equation generally displayed a better diagnostic performance than estimates obtained by equations using only one of these analytes or by complex equations using both analytes. Comparing eGFR cystatin and eGFR creatinine may help identify pediatric patients with Shrunken Pore Syndrome.

Nonlinear gyrokinetics: a powerful tool for the description of microturbulence in magnetized plasmas

NASA Astrophysics Data System (ADS)

Krommes, John A.

2010-12-01

Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained. The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a 'pull-back' (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and difficulties) of deriving nonlinear gyrofluid equations suitable for rapid numerical solution—although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.

MUSTA fluxes for systems of conservation laws

NASA Astrophysics Data System (ADS)

Toro, E. F.; Titarev, V. A.

2006-08-01

This paper is about numerical fluxes for hyperbolic systems and we first present a numerical flux, called GFORCE, that is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes. For the linear advection equation with constant coefficient, the new flux reduces identically to that of the Godunov first-order upwind method. Then we incorporate GFORCE in the framework of the MUSTA approach [E.F. Toro, Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations. Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003], resulting in a version that we call GMUSTA. For non-linear systems this gives results that are comparable to those of the Godunov method in conjunction with the exact Riemann solver or complete approximate Riemann solvers, noting however that in our approach, the solution of the Riemann problem in the conventional sense is avoided. Both the GFORCE and GMUSTA fluxes are extended to multi-dimensional non-linear systems in a straightforward unsplit manner, resulting in linearly stable schemes that have the same stability regions as the straightforward multi-dimensional extension of Godunov's method. The methods are applicable to general meshes. The schemes of this paper share with the family of centred methods the common properties of being simple and applicable to a large class of hyperbolic systems, but the schemes of this paper are distinctly more accurate. Finally, we proceed to the practical implementation of our numerical fluxes in the framework of high-order finite volume WENO methods for multi-dimensional non-linear hyperbolic systems. Numerical results are presented for the Euler equations and for the equations of magnetohydrodynamics.

Advanced Monte Carlo methods for thermal radiation transport

NASA Astrophysics Data System (ADS)

Wollaber, Allan B.

During the past 35 years, the Implicit Monte Carlo (IMC) method proposed by Fleck and Cummings has been the standard Monte Carlo approach to solving the thermal radiative transfer (TRT) equations. However, the IMC equations are known to have accuracy limitations that can produce unphysical solutions. In this thesis, we explicitly provide the IMC equations with a Monte Carlo interpretation by including particle weight as one of its arguments. We also develop and test a stability theory for the 1-D, gray IMC equations applied to a nonlinear problem. We demonstrate that the worst case occurs for 0-D problems, and we extend the results to a stability algorithm that may be used for general linearizations of the TRT equations. We derive gray, Quasidiffusion equations that may be deterministically solved in conjunction with IMC to obtain an inexpensive, accurate estimate of the temperature at the end of the time step. We then define an average temperature T* to evaluate the temperature-dependent problem data in IMC, and we demonstrate that using T* is more accurate than using the (traditional) beginning-of-time-step temperature. We also propose an accuracy enhancement to the IMC equations: the use of a time-dependent "Fleck factor". This Fleck factor can be considered an automatic tuning of the traditionally defined user parameter alpha, which generally provides more accurate solutions at an increased cost relative to traditional IMC. We also introduce a global weight window that is proportional to the forward scalar intensity calculated by the Quasidiffusion method. This weight window improves the efficiency of the IMC calculation while conserving energy. All of the proposed enhancements are tested in 1-D gray and frequency-dependent problems. These enhancements do not unconditionally eliminate the unphysical behavior that can be seen in the IMC calculations. However, for fixed spatial and temporal grids, they suppress them and clearly work to make the solution more accurate. Overall, the work presented represents first steps along several paths that can be taken to improve the Monte Carlo simulations of TRT problems.

Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

DOE PAGES

Su, Hongling; Li, Shengtai

2016-02-03

In this study, we propose two new energy/dissipation-preserving Birkhoffian multi-symplectic methods (Birkhoffian and Birkhoffian box) for Maxwell's equations with dissipation terms. After investigating the non-autonomous and autonomous Birkhoffian formalism for Maxwell's equations with dissipation terms, we first apply a novel generating functional theory to the non-autonomous Birkhoffian formalism to propose our Birkhoffian scheme, and then implement a central box method to the autonomous Birkhoffian formalism to derive the Birkhoffian box scheme. We have obtained four formal local conservation laws and three formal energy global conservation laws. We have also proved that both of our derived schemes preserve the discrete versionmore » of the global/local conservation laws. Furthermore, the stability, dissipation and dispersion relations are also investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, dissipation-preserving for Maxwell's equations in a perfectly matched layer (PML) medium and have second order accuracy in both time and space. Numerical experiments for problems with exact theoretical results are given to demonstrate that the Birkhoffian multi-symplectic schemes are much more accurate in preserving energy than both the exponential finite-difference time-domain (FDTD) method and traditional Hamiltonian scheme. Finally, we also solve the electromagnetic pulse (EMP) propagation problem and the numerical results show that the Birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation.« less

Energy/dissipation-preserving Birkhoffian multi-symplectic methods for Maxwell's equations with dissipation terms

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

Su, Hongling; Li, Shengtai

In this study, we propose two new energy/dissipation-preserving Birkhoffian multi-symplectic methods (Birkhoffian and Birkhoffian box) for Maxwell's equations with dissipation terms. After investigating the non-autonomous and autonomous Birkhoffian formalism for Maxwell's equations with dissipation terms, we first apply a novel generating functional theory to the non-autonomous Birkhoffian formalism to propose our Birkhoffian scheme, and then implement a central box method to the autonomous Birkhoffian formalism to derive the Birkhoffian box scheme. We have obtained four formal local conservation laws and three formal energy global conservation laws. We have also proved that both of our derived schemes preserve the discrete versionmore » of the global/local conservation laws. Furthermore, the stability, dissipation and dispersion relations are also investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, dissipation-preserving for Maxwell's equations in a perfectly matched layer (PML) medium and have second order accuracy in both time and space. Numerical experiments for problems with exact theoretical results are given to demonstrate that the Birkhoffian multi-symplectic schemes are much more accurate in preserving energy than both the exponential finite-difference time-domain (FDTD) method and traditional Hamiltonian scheme. Finally, we also solve the electromagnetic pulse (EMP) propagation problem and the numerical results show that the Birkhoffian scheme recovers the magnitude of the current source and reaction history very well even after long time propagation.« less

An approximation of herd effect due to vaccinating children against seasonal influenza – a potential solution to the incorporation of indirect effects into static models

PubMed Central

2013-01-01

Background Indirect herd effect from vaccination of children offers potential for improving the effectiveness of influenza prevention in the remaining unvaccinated population. Static models used in cost-effectiveness analyses cannot dynamically capture herd effects. The objective of this study was to develop a methodology to allow herd effect associated with vaccinating children against seasonal influenza to be incorporated into static models evaluating the cost-effectiveness of influenza vaccination. Methods Two previously published linear equations for approximation of herd effects in general were compared with the results of a structured literature review undertaken using PubMed searches to identify data on herd effects specific to influenza vaccination. A linear function was fitted to point estimates from the literature using the sum of squared residuals. Results The literature review identified 21 publications on 20 studies for inclusion. Six studies provided data on a mathematical relationship between effective vaccine coverage in subgroups and reduction of influenza infection in a larger unvaccinated population. These supported a linear relationship when effective vaccine coverage in a subgroup population was between 20% and 80%. Three studies evaluating herd effect at a community level, specifically induced by vaccinating children, provided point estimates for fitting linear equations. The fitted linear equation for herd protection in the target population for vaccination (children) was slightly less conservative than a previously published equation for herd effects in general. The fitted linear equation for herd protection in the non-target population was considerably less conservative than the previously published equation. Conclusions This method of approximating herd effect requires simple adjustments to the annual baseline risk of influenza in static models: (1) for the age group targeted by the childhood vaccination strategy (i.e. children); and (2) for other age groups not targeted (e.g. adults and/or elderly). Two approximations provide a linear relationship between effective coverage and reduction in the risk of infection. The first is a conservative approximation, recommended as a base-case for cost-effectiveness evaluations. The second, fitted to data extracted from a structured literature review, provides a less conservative estimate of herd effect, recommended for sensitivity analyses. PMID:23339290

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.

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations.

PubMed

Schüler, D; Alonso, S; Torcini, A; Bär, M

2014-12-01

Pattern formation often occurs in spatially extended physical, biological, and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue, we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular, revealing for the Swift-Hohenberg equations, a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of a weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

A theory of post-stall transients in axial compression systems. I - Development of equations

NASA Technical Reports Server (NTRS)

Moore, F. K.; Greitzer, E. M.

1985-01-01

An approximate theory is presented for post-stall transients in multistage axial compression systems. The theory leads to a set of three simultaneous nonlinear third-order partial differential equations for pressure rise, and average and disturbed values of flow coefficient, as functions of time and angle around the compressor. By a Galerkin procedure, angular dependence is averaged, and the equations become first order in time. These final equations are capable of describing the growth and possible decay of a rotating-stall cell during a compressor mass-flow transient. It is shown how rotating-stall-like and surgelike motions are coupled through these equations, and also how the instantaneous compressor pumping characteristic changes during the transient stall process.

Properties of the Residual Stress of the Temporally Filtered Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Pruett, C. D.; Gatski, T. B.; Grosch, C. E.; Thacker, W. D.

2002-01-01

The development of a unifying framework among direct numerical simulations, large-eddy simulations, and statistically averaged formulations of the Navier-Stokes equations, is of current interest. Toward that goal, the properties of the residual (subgrid-scale) stress of the temporally filtered Navier-Stokes equations are carefully examined. Causal time-domain filters, parameterized by a temporal filter width 0 less than Delta less than infinity, are considered. For several reasons, the differential forms of such filters are preferred to their corresponding integral forms; among these, storage requirements for differential forms are typically much less than for integral forms and, for some filters, are independent of Delta. The behavior of the residual stress in the limits of both vanishing and in infinite filter widths is examined. It is shown analytically that, in the limit Delta to 0, the residual stress vanishes, in which case the Navier-Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit Delta to infinity, the residual stress is equivalent to the long-time averaged stress, and the Reynolds-averaged Navier-Stokes equations are recovered from the temporally filtered equations. The predicted behavior at the asymptotic limits of filter width is further validated by numerical simulations of the temporally filtered forced, viscous Burger's equation. Finally, finite filter widths are also considered, and a priori analyses of temporal similarity and temporal approximate deconvolution models of the residual stress are conducted.

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

NASA Astrophysics Data System (ADS)

Joshi, Vaibhav; Jaiman, Rajeev K.

2018-05-01

We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio. The variational finite element technique relies on the Allen-Cahn phase-field equation for capturing the phase interface on a fixed Eulerian mesh with mass conservative and energy-stable discretization. The mass conservation is achieved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the underlying solution of the phase-field equation. To make the scheme energy-stable in a variational sense, we discretize the spatial part of the Lagrange multiplier in the phase-field equation by the mid-point approximation. The proposed variational technique is designed to reduce the spurious and unphysical oscillations in the solution while maintaining the second-order accuracy of both spatial and temporal discretizations. We integrate the Allen-Cahn phase-field equation with the incompressible Navier-Stokes equations for modeling a broad range of two-phase flow and fluid-fluid interface problems. The coupling of the implicit discretizations corresponding to the phase-field and the incompressible flow equations is achieved via nonlinear partitioned iterative procedure. Comparison of results between the standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen-Cahn two-phase solver. We examine the convergence and accuracy properties of the coupled phase-field solver through the standard benchmarks of the Laplace-Young law and a sloshing tank problem. Two- and three-dimensional dam break problems are simulated to assess the capability of the phase-field solver for complex air-water interfaces involving topological changes on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical offshore engineering application of wave-structure interaction.

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.

Real-Gas Effects on Binary Mixing Layers

NASA Technical Reports Server (NTRS)

Okong'o, Nora; Bellan, Josette

2003-01-01

This paper presents a computational study of real-gas effects on the mean flow and temporal stability of heptane/nitrogen and oxygen/hydrogen mixing layers at supercritical pressures. These layers consist of two counterflowing free streams of different composition, temperature, and density. As in related prior studies reported in NASA Tech Briefs, the governing conservation equations were the Navier-Stokes equations of compressible flow plus equations for the conservation of total energy and of chemical- species masses. In these equations, the expressions for heat fluxes and chemical-species mass fluxes were derived from fluctuation-dissipation theory and incorporate Soret and Dufour effects. Similarity equations for the streamwise velocity, temperature, and mass fractions were derived as approximations to the governing equations. Similarity profiles showed important real-gas, non-ideal-mixture effects, particularly for temperature, in departing from the error-function profile, which is the similarity solution for incompressible flow. The temperature behavior was attributed to real-gas thermodynamics and variations in Schmidt and Prandtl numbers. Temporal linear inviscid stability analyses were performed using the similarity and error-function profiles as the mean flow. For the similarity profiles, the growth rates were found to be larger and the wavelengths of highest instability shorter, relative to those of the errorfunction profiles and to those obtained from incompressible-flow stability analysis. The range of unstable wavelengths was found to be larger for the similarity profiles than for the error-function profiles

Well-posedness, linear perturbations, and mass conservation for the axisymmetric Einstein equations

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

Dain, Sergio; Ortiz, Omar E.; Facultad de Matematica, Astronomia y Fisica, FaMAF, Universidad Nacional de Cordoba, Instituto de Fisica Enrique Gaviola, IFEG, CONICET, Ciudad Universitaria

2010-02-15

For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution. In this gauge, Einstein equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. As a first step in analyzing this system of equations we study linear perturbations on a flat background. We prove that the linear equations reduce to a very simple system of equations which provide, thoughmore » the mass formula, useful insight into the structure of the full system. However, the singular behavior of the coefficients at the axis makes the study of this linear system difficult from the analytical point of view. In order to understand the behavior of the solutions, we study the numerical evolution of them. We provide strong numerical evidence that the system is well-posed and that its solutions have the expected behavior. Finally, this linear system allows us to formulate a model problem which is physically interesting in itself, since it is connected with the linear stability of black hole solutions in axial symmetry. This model can contribute significantly to solve the nonlinear problem and at the same time it appears to be tractable.« less

SToRM: A Model for 2D environmental hydraulics

USGS Publications Warehouse

Simões, Francisco J. M.

2017-01-01

A two-dimensional (depth-averaged) finite volume Godunov-type shallow water model developed for flow over complex topography is presented. The model, SToRM, is based on an unstructured cell-centered finite volume formulation and on nonlinear strong stability preserving Runge-Kutta time stepping schemes. The numerical discretization is founded on the classical and well established shallow water equations in hyperbolic conservative form, but the convective fluxes are calculated using auto-switching Riemann and diffusive numerical fluxes. Computational efficiency is achieved through a parallel implementation based on the OpenMP standard and the Fortran programming language. SToRM’s implementation within a graphical user interface is discussed. Field application of SToRM is illustrated by utilizing it to estimate peak flow discharges in a flooding event of the St. Vrain Creek in Colorado, U.S.A., in 2013, which reached 850 m3/s (~30,000 f3 /s) at the location of this study.

A numerical study of transient heat and mass transfer in crystal growth

NASA Technical Reports Server (NTRS)

Han, Samuel Bang-Moo

1987-01-01

A numerical analysis of transient heat and solute transport across a rectangular cavity is performed. Five nonlinear partial differential equations which govern the conservation of mass, momentum, energy and solute concentration related to crystal growth in solution, are simultaneously integrated by a numerical method based on the SIMPLE algorithm. Numerical results showed that the flow, temperature and solute fields are dependent on thermal and solutal Grashoff number, Prandtl number, Schmidt number and aspect ratio. The average Nusselt and Sherwood numbers evaluated at the center of the cavity decrease markedly when the solutal buoyancy force acts in the opposite direction to the thermal buoyancy force. When the solutal and thermal buoyancy forces act in the same direction, however, Sherwood number increases significantly and yet Nusselt number decreases. Overall effects of convection on the crystal growth are seen to be an enhancement of growth rate as expected but with highly nonuniform spatial growth variations.

Whole stand volume tables for quaking aspen in the Rocky Mountains

Treesearch

Wayne D. Shepperd; H. Todd Mowrer

1984-01-01

Linear regression equations were developed to predict stand volumes for aspen given average stand basal area and average stand height. Tables constructed from these equations allow easy field estimation of gross merchantable cubic and board foot Scribner Rules per acre, and cubic meters per hectare using simple prism cruise data.