Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations.

PubMed

Zhang, Ling

2017-01-01

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Lagrangians and Euler morphisms from connections on the frame bundle

NASA Astrophysics Data System (ADS)

Kurek, J.; Mikulski, W. M.

2011-07-01

We classify all natural operators transforming torsion free classical linear connections ∇ on m-dimensional manifolds M into r-th order Lagrangians λ(∇) and Euler morphisms E(∇) on the linear frame bundle P1M. We also briefly write how this classification result can be generalized on higher order frame bundles PkM instead of P1M.

A Non-Linear Simulation for an Autonomous Unmanned Air Vehicle

DTIC Science & Technology

1993-09-01

4D cos T cos 4D cos T r These equations can now be integrated to find the time history of the Euler angles . 2. Quaternions Another choice for the...is associated with the Euler angles . Quaternions haxe been in 15 use for quite some time. having been discovered by Euler in a search for complex... quaternions has the following advantages over Euler angles in repre- senting spatial orientation of a rigid body: "* Four states required to express the

On the Use of Linearized Euler Equations in the Prediction of Jet Noise

NASA Technical Reports Server (NTRS)

Mankbadi, Reda R.; Hixon, R.; Shih, S.-H.; Povinelli, L. A.

1995-01-01

Linearized Euler equations are used to simulate supersonic jet noise generation and propagation. Special attention is given to boundary treatment. The resulting solution is stable and nearly free from boundary reflections without the need for artificial dissipation, filtering, or a sponge layer. The computed solution is in good agreement with theory and observation and is much less CPU-intensive as compared to large-eddy simulations.

Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method

NASA Technical Reports Server (NTRS)

Whitaker, David L.

1993-01-01

A method has been developed to solve the Euler equations on a three-dimensional unstructured grid composed of tetrahedra. The method uses an upwind flow solver with a linearized, backward-Euler time integration scheme. Each time step results in a sparse linear system of equations which is solved by an iterative, sparse matrix solver. Local-time stepping, switched evolution relaxation (SER), preconditioning and reuse of the Jacobian are employed to accelerate the convergence rate. Implicit boundary conditions were found to be extremely important for fast convergence. Numerical experiments have shown that convergence rates comparable to that of a multigrid, central-difference scheme are achievable on the same mesh. Results are presented for several grids about an ONERA M6 wing.

Numerical computation of linear instability of detonations

NASA Astrophysics Data System (ADS)

Kabanov, Dmitry; Kasimov, Aslan

2017-11-01

We propose a method to study linear stability of detonations by direct numerical computation. The linearized governing equations together with the shock-evolution equation are solved in the shock-attached frame using a high-resolution numerical algorithm. The computed results are processed by the Dynamic Mode Decomposition technique to generate dispersion relations. The method is applied to the reactive Euler equations with simple-depletion chemistry as well as more complex multistep chemistry. The results are compared with those known from normal-mode analysis. We acknowledge financial support from King Abdullah University of Science and Technology.

Alternate Solution to Generalized Bernoulli Equations via an Integrating Factor: An Exact Differential Equation Approach

ERIC Educational Resources Information Center

Tisdell, C. C.

2017-01-01

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…

Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

NASA Astrophysics Data System (ADS)

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

The most precise computations using Euler's method in standard floating-point arithmetic applied to modelling of biological systems.

PubMed

Kalinina, Elizabeth A

2013-08-01

The explicit Euler's method is known to be very easy and effective in implementation for many applications. This article extends results previously obtained for the systems of linear differential equations with constant coefficients to arbitrary systems of ordinary differential equations. Optimal (providing minimum total error) step size is calculated at each step of Euler's method. Several examples of solving stiff systems are included. Copyright © 2013 Elsevier Ireland Ltd. All rights reserved.

Numerical solution of Euler's equation by perturbed functionals

NASA Technical Reports Server (NTRS)

Dey, S. K.

1985-01-01

A perturbed functional iteration has been developed to solve nonlinear systems. It adds at each iteration level, unique perturbation parameters to nonlinear Gauss-Seidel iterates which enhances its convergence properties. As convergence is approached these parameters are damped out. Local linearization along the diagonal has been used to compute these parameters. The method requires no computation of Jacobian or factorization of matrices. Analysis of convergence depends on properties of certain contraction-type mappings, known as D-mappings. In this article, application of this method to solve an implicit finite difference approximation of Euler's equation is studied. Some representative results for the well known shock tube problem and compressible flows in a nozzle are given.

Rapid Aeroelastic Analysis of Blade Flutter in Turbomachines

NASA Technical Reports Server (NTRS)

Trudell, J. J.; Mehmed, O.; Stefko, G. L.; Bakhle, M. A.; Reddy, T. S. R.; Montgomery, M.; Verdon, J.

2006-01-01

The LINFLUX-AE computer code predicts flutter and forced responses of blades and vanes in turbomachines under subsonic, transonic, and supersonic flow conditions. The code solves the Euler equations of unsteady flow in a blade passage under the assumption that the blades vibrate harmonically at small amplitudes. The steady-state nonlinear Euler equations are solved by a separate program, then equations for unsteady flow components are obtained through linearization around the steady-state solution. A structural-dynamics analysis (see figure) is performed to determine the frequencies and mode shapes of blade vibrations, a preprocessor interpolates mode shapes from the structural-dynamics mesh onto the LINFLUX computational-fluid-dynamics mesh, and an interface code is used to convert the steady-state flow solution to a form required by LINFLUX. Then LINFLUX solves the linearized equations in the frequency domain to calculate the unsteady aerodynamic pressure distribution for a given vibration mode, frequency, and interblade phase angle. A post-processor uses the unsteady pressures to calculate generalized aerodynamic forces, response amplitudes, and eigenvalues (which determine the flutter frequency and damping). In comparison with the TURBO-AE aeroelastic-analysis code, which solves the equations in the time domain, LINFLUX-AE is 6 to 7 times faster.

A macroscopic plasma Lagrangian and its application to wave interactions and resonances

NASA Technical Reports Server (NTRS)

Peng, Y. K. M.

1974-01-01

The derivation of a macroscopic plasma Lagrangian is considered, along with its application to the description of nonlinear three-wave interaction in a homogeneous plasma and linear resonance oscillations in a inhomogeneous plasma. One approach to obtain the Lagrangian is via the inverse problem of the calculus of variations for arbitrary first and second order quasilinear partial differential systems. Necessary and sufficient conditions for the given equations to be Euler-Lagrange equations of a Lagrangian are obtained. These conditions are then used to determine the transformations that convert some classes of non-Euler-Lagrange equations to Euler-Lagrange equation form. The Lagrangians for a linear resistive transmission line and a linear warm collisional plasma are derived as examples. Using energy considerations, the correct macroscopic plasma Lagrangian is shown to differ from the velocity-integrated low Lagrangian by a macroscopic potential energy that equals twice the particle thermal kinetic energy plus the energy lost by heat conduction.

Euler technology assessment for preliminary aircraft design employing OVERFLOW code with multiblock structured-grid method

NASA Technical Reports Server (NTRS)

Treiber, David A.; Muilenburg, Dennis A.

1995-01-01

The viability of applying a state-of-the-art Euler code to calculate the aerodynamic forces and moments through maximum lift coefficient for a generic sharp-edge configuration is assessed. The OVERFLOW code, a method employing overset (Chimera) grids, was used to conduct mesh refinement studies, a wind-tunnel wall sensitivity study, and a 22-run computational matrix of flow conditions, including sideslip runs and geometry variations. The subject configuration was a generic wing-body-tail geometry with chined forebody, swept wing leading-edge, and deflected part-span leading-edge flap. The analysis showed that the Euler method is adequate for capturing some of the non-linear aerodynamic effects resulting from leading-edge and forebody vortices produced at high angle-of-attack through C(sub Lmax). Computed forces and moments, as well as surface pressures, match well enough useful preliminary design information to be extracted. Vortex burst effects and vortex interactions with the configuration are also investigated.

Linear Equations with the Euler Totient Function

DTIC Science & Technology

2007-02-13

unclassified c . THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 2 FLORIAN LUCA, PANTELIMON STĂNICĂ...of positive integers n such that φ(n) = φ(n+ 1), and that the set of Phibonacci numbers is A(1,1,−1) + 2. Theorem 2.1. Let C (t, a) = t3 logH(a). Then...the estimate #Aa(x) C (t, a) x log log log x√ log log x LINEAR EQUATIONS WITH THE EULER TOTIENT FUNCTION 3 holds uniformly in a and 1 ≤ t < y. Note

A Kinetic Approach to Propagation and Stability of Detonation Waves

NASA Astrophysics Data System (ADS)

Monaco, R.; Bianchi, M. Pandolfi; Soares, A. J.

2008-12-01

The problem of the steady propagation and linear stability of a detonation wave is formulated in the kinetic frame for a quaternary gas mixture in which a reversible bimolecular reaction takes place. The reactive Euler equations and related Rankine-Hugoniot conditions are deduced from the mesoscopic description of the process. The steady propagation problem is solved for a Zeldovich, von Neuman and Doering (ZND) wave, providing the detonation profiles and the wave thickness for different overdrive degrees. The one-dimensional stability of such detonation wave is then studied in terms of an initial value problem coupled with an acoustic radiation condition at the equilibrium final state. The stability equations and their initial data are deduced from the linearized reactive Euler equations and related Rankine-Hugoniot conditions through a normal mode analysis referred to the complex disturbances of the steady state variables. Some numerical simulations for an elementary reaction of the hydrogen-oxygen chain are proposed in order to describe the time and space evolution of the instabilities induced by the shock front perturbation.

Reprint of Solution of Ambrosio-Tortorelli model for image segmentation by generalized relaxation method

NASA Astrophysics Data System (ADS)

D'Ambra, Pasqua; Tartaglione, Gaetano

2015-04-01

Image segmentation addresses the problem to partition a given image into its constituent objects and then to identify the boundaries of the objects. This problem can be formulated in terms of a variational model aimed to find optimal approximations of a bounded function by piecewise-smooth functions, minimizing a given functional. The corresponding Euler-Lagrange equations are a set of two coupled elliptic partial differential equations with varying coefficients. Numerical solution of the above system often relies on alternating minimization techniques involving descent methods coupled with explicit or semi-implicit finite-difference discretization schemes, which are slowly convergent and poorly scalable with respect to image size. In this work we focus on generalized relaxation methods also coupled with multigrid linear solvers, when a finite-difference discretization is applied to the Euler-Lagrange equations of Ambrosio-Tortorelli model. We show that non-linear Gauss-Seidel, accelerated by inner linear iterations, is an effective method for large-scale image analysis as those arising from high-throughput screening platforms for stem cells targeted differentiation, where one of the main goal is segmentation of thousand of images to analyze cell colonies morphology.

Solution of Ambrosio-Tortorelli model for image segmentation by generalized relaxation method

NASA Astrophysics Data System (ADS)

D'Ambra, Pasqua; Tartaglione, Gaetano

2015-03-01

Image segmentation addresses the problem to partition a given image into its constituent objects and then to identify the boundaries of the objects. This problem can be formulated in terms of a variational model aimed to find optimal approximations of a bounded function by piecewise-smooth functions, minimizing a given functional. The corresponding Euler-Lagrange equations are a set of two coupled elliptic partial differential equations with varying coefficients. Numerical solution of the above system often relies on alternating minimization techniques involving descent methods coupled with explicit or semi-implicit finite-difference discretization schemes, which are slowly convergent and poorly scalable with respect to image size. In this work we focus on generalized relaxation methods also coupled with multigrid linear solvers, when a finite-difference discretization is applied to the Euler-Lagrange equations of Ambrosio-Tortorelli model. We show that non-linear Gauss-Seidel, accelerated by inner linear iterations, is an effective method for large-scale image analysis as those arising from high-throughput screening platforms for stem cells targeted differentiation, where one of the main goal is segmentation of thousand of images to analyze cell colonies morphology.

Boundary states at reflective moving boundaries

NASA Astrophysics Data System (ADS)

Acosta Minoli, Cesar A.; Kopriva, David A.

2012-06-01

We derive and evaluate boundary states for Maxwell's equations, the linear, and the nonlinear Euler gas-dynamics equations to compute wave reflection from moving boundaries. In this study we use a Discontinuous Galerkin Spectral Element method (DGSEM) with Arbitrary Lagrangian-Eulerian (ALE) mapping for the spatial approximation, but the boundary states can be used with other methods, like finite volume schemes. We present four studies using Maxwell's equations, one for the linear Euler equations, and one more for the nonlinear Euler equations. These are: reflection of light from a plane mirror moving at constant velocity, reflection of light from a moving cylinder, reflection of light from a vibrating mirror, reflection of sound from a plane wall and dipole sound generation by an oscillating cylinder in an inviscid flow. The studies show that the boundary states preserve spectral convergence in the solution and in derived quantities like divergence and vorticity.

Sound Radiated by a Wave-Like Structure in a Compressible Jet

NASA Technical Reports Server (NTRS)

Golubev, V. V.; Prieto, A. F.; Mankbadi, R. R.; Dahl, M. D.; Hixon, R.

2003-01-01

This paper extends the analysis of acoustic radiation from the source model representing spatially-growing instability waves in a round jet at high speeds. Compared to previous work, a modified approach to the sound source modeling is examined that employs a set of solutions to linearized Euler equations. The sound radiation is then calculated using an integral surface method.

Numerical solutions of the linearized Euler equations for unsteady vortical flows around lifting airfoils

NASA Technical Reports Server (NTRS)

Scott, James R.; Atassi, Hafiz M.

1990-01-01

A linearized unsteady aerodynamic analysis is presented for unsteady, subsonic vortical flows around lifting airfoils. The analysis fully accounts for the distortion effects of the nonuniform mean flow on the imposed vortical disturbances. A frequency domain numerical scheme which implements this linearized approach is described, and numerical results are presented for a large variety of flow configurations. The results demonstrate the effects of airfoil thickness, angle of attack, camber, and Mach number on the unsteady lift and moment of airfoils subjected to periodic vortical gusts. The results show that mean flow distortion can have a very strong effect on the airfoil unsteady response, and that the effect depends strongly upon the reduced frequency, Mach number, and gust wave numbers.

Computation of Large-Scale Structure Jet Noise Sources With Weak Nonlinear Effects Using Linear Euler

NASA Technical Reports Server (NTRS)

Dahl, Milo D.; Hixon, Ray; Mankbadi, Reda R.

2003-01-01

An approximate technique is presented for the prediction of the large-scale turbulent structure sound source in a supersonic jet. A linearized Euler equations code is used to solve for the flow disturbances within and near a jet with a given mean flow. Assuming a normal mode composition for the wave-like disturbances, the linear radial profiles are used in an integration of the Navier-Stokes equations. This results in a set of ordinary differential equations representing the weakly nonlinear self-interactions of the modes along with their interaction with the mean flow. Solutions are then used to correct the amplitude of the disturbances that represent the source of large-scale turbulent structure sound in the jet.

Euler/Navier-Stokes calculations of transonic flow past fixed- and rotary-wing aircraft configurations

NASA Technical Reports Server (NTRS)

Deese, J. E.; Agarwal, R. K.

1989-01-01

Computational fluid dynamics has an increasingly important role in the design and analysis of aircraft as computer hardware becomes faster and algorithms become more efficient. Progress is being made in two directions: more complex and realistic configurations are being treated and algorithms based on higher approximations to the complete Navier-Stokes equations are being developed. The literature indicates that linear panel methods can model detailed, realistic aircraft geometries in flow regimes where this approximation is valid. As algorithms including higher approximations to the Navier-Stokes equations are developed, computer resource requirements increase rapidly. Generation of suitable grids become more difficult and the number of grid points required to resolve flow features of interest increases. Recently, the development of large vector computers has enabled researchers to attempt more complex geometries with Euler and Navier-Stokes algorithms. The results of calculations for transonic flow about a typical transport and fighter wing-body configuration using thin layer Navier-Stokes equations are described along with flow about helicopter rotor blades using both Euler/Navier-Stokes equations.

A fast efficient implicit scheme for the gasdynamic equations using a matrix reduction technique

NASA Technical Reports Server (NTRS)

Barth, T. J.; Steger, J. L.

1985-01-01

An efficient implicit finite-difference algorithm for the gasdynamic equations utilizing matrix reduction techniques is presented. A significant reduction in arithmetic operations is achieved without loss of the stability characteristics generality found in the Beam and Warming approximate factorization algorithm. Steady-state solutions to the conservative Euler equations in generalized coordinates are obtained for transonic flows and used to show that the method offers computational advantages over the conventional Beam and Warming scheme. Existing Beam and Warming codes can be retrofit with minimal effort. The theoretical extension of the matrix reduction technique to the full Navier-Stokes equations in Cartesian coordinates is presented in detail. Linear stability, using a Fourier stability analysis, is demonstrated and discussed for the one-dimensional Euler equations.

An efficient method for solving the steady Euler equations

NASA Technical Reports Server (NTRS)

Liou, M.-S.

1986-01-01

An efficient numerical procedure for solving a set of nonlinear partial differential equations, the steady Euler equations, using Newton's linearization procedure is presented. A theorem indicating quadratic convergence for the case of differential equations is demonstrated. A condition for the domain of quadratic convergence Omega(2) is obtained which indicates that whether an approximation lies in Omega(2) depends on the rate of change and the smoothness of the flow vectors, and hence is problem-dependent. The choice of spatial differencing, of particular importance for the present method, is discussed. The treatment of boundary conditions is addressed, and the system of equations resulting from the foregoing analysis is summarized and solution strategies are discussed. The convergence of calculated solutions is demonstrated by comparing them with exact solutions to one and two-dimensional problems.

Flutter and Forced Response Analyses of Cascades using a Two-Dimensional Linearized Euler Solver

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Srivastava, R.; Mehmed, O.

1999-01-01

Flutter and forced response analyses for a cascade of blades in subsonic and transonic flow is presented. The structural model for each blade is a typical section with bending and torsion degrees of freedom. The unsteady aerodynamic forces due to bending and torsion motions. and due to a vortical gust disturbance are obtained by solving unsteady linearized Euler equations. The unsteady linearized equations are obtained by linearizing the unsteady nonlinear equations about the steady flow. The predicted unsteady aerodynamic forces include the effect of steady aerodynamic loading due to airfoil shape, thickness and angle of attack. The aeroelastic equations are solved in the frequency domain by coupling the un- steady aerodynamic forces to the aeroelastic solver MISER. The present unsteady aerodynamic solver showed good correlation with published results for both flutter and forced response predictions. Further improvements are required to use the unsteady aerodynamic solver in a design cycle.

Attitude control with realization of linear error dynamics

NASA Technical Reports Server (NTRS)

Paielli, Russell A.; Bach, Ralph E.

1993-01-01

An attitude control law is derived to realize linear unforced error dynamics with the attitude error defined in terms of rotation group algebra (rather than vector algebra). Euler parameters are used in the rotational dynamics model because they are globally nonsingular, but only the minimal three Euler parameters are used in the error dynamics model because they have no nonlinear mathematical constraints to prevent the realization of linear error dynamics. The control law is singular only when the attitude error angle is exactly pi rad about any eigenaxis, and a simple intuitive modification at the singularity allows the control law to be used globally. The forced error dynamics are nonlinear but stable. Numerical simulation tests show that the control law performs robustly for both initial attitude acquisition and attitude control.

Multiplicative Quaternion Extended Kalman Filtering for Nonspinning Guided Projectiles

DTIC Science & Technology

2013-07-01

tactical applications are inertial. The advantages of using quaternions rather than Euler angles to represent projectile attitude are discussed, and...projectiles generally don’t experience a wide range of heading angles , this has not a primary concern. The other major advantage of quaternions (or...DCMs) over Euler angles is their propagation equations are linear with respect to the quaternion and only depend on the IMU’s angular velocity. This

A novel approach to enhance the accuracy of vibration control of Frames

NASA Astrophysics Data System (ADS)

Toloue, Iraj; Shahir Liew, Mohd; Harahap, I. S. H.; Lee, H. E.

2018-03-01

All structures built within known seismically active regions are typically designed to endure earthquake forces. Despite advances in earthquake resistant structures, it can be inferred from hindsight that no structure is entirely immune to damage from earthquakes. Active vibration control systems, unlike the traditional methods which enlarge beams and columns, are highly effective countermeasures to reduce the effects of earthquake loading on a structure. It requires fast computation of nonlinear structural analysis in near time and has historically demanded advanced programming hosted on powerful computers. This research aims to develop a new approach for active vibration control of frames, which is applicable over both elastic and plastic material behavior. In this study, the Force Analogy Method (FAM), which is based on Hook's Law is further extended using the Timoshenko element which considers shear deformations to increase the reliability and accuracy of the controller. The proposed algorithm is applied to a 2D portal frame equipped with linear actuator, which is designed based on full state Linear Quadratic Regulator (LQR). For comparison purposes, the portal frame is analysed by both the Euler Bernoulli and Timoshenko element respectively. The results clearly demonstrate the superiority of the Timoshenko element over Euler Bernoulli for application in nonlinear analysis.

The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil

NASA Technical Reports Server (NTRS)

Meade, Andrew J., Jr.

1992-01-01

A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool.

Methodology for Sensitivity Analysis, Approximate Analysis, and Design Optimization in CFD for Multidisciplinary Applications

NASA Technical Reports Server (NTRS)

Taylor, Arthur C., III; Hou, Gene W.

1996-01-01

An incremental iterative formulation together with the well-known spatially split approximate-factorization algorithm, is presented for solving the large, sparse systems of linear equations that are associated with aerodynamic sensitivity analysis. This formulation is also known as the 'delta' or 'correction' form. For the smaller two dimensional problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. However, iterative methods are needed for larger two-dimensional and three dimensional applications because direct methods require more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioned coefficient matrix; this problem is overcome when these equations are cast in the incremental form. The methodology is successfully implemented and tested using an upwind cell-centered finite-volume formulation applied in two dimensions to the thin-layer Navier-Stokes equations for external flow over an airfoil. In three dimensions this methodology is demonstrated with a marching-solution algorithm for the Euler equations to calculate supersonic flow over the High-Speed Civil Transport configuration (HSCT 24E). The sensitivity derivatives obtained with the incremental iterative method from a marching Euler code are used in a design-improvement study of the HSCT configuration that involves thickness. camber, and planform design variables.

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.

A conformal approach for the analysis of the non-linear stability of radiation cosmologies

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

Luebbe, Christian, E-mail: c.luebbe@ucl.ac.uk; Department of Mathematics, University of Leicester, University Road, LE1 8RH; Valiente Kroon, Juan Antonio, E-mail: j.a.valiente-kroon@qmul.ac.uk

2013-01-15

The conformal Einstein equations for a trace-free (radiation) perfect fluid are derived in terms of the Levi-Civita connection of a conformally rescaled metric. These equations are used to provide a non-linear stability result for de Sitter-like trace-free (radiation) perfect fluid Friedman-Lemaitre-Robertson-Walker cosmological models. The solutions thus obtained exist globally towards the future and are future geodesically complete. - Highlights: Black-Right-Pointing-Pointer We study the Einstein-Euler system in General Relativity using conformal methods. Black-Right-Pointing-Pointer We analyze the structural properties of the associated evolution equations. Black-Right-Pointing-Pointer We establish the non-linear stability of pure radiation cosmological models.

Comparison of a Convected Helmholtz and Euler Model for Impedance Eduction in Flow

NASA Technical Reports Server (NTRS)

Watson, Willie R.; Jones, Michael G.

2006-01-01

Impedances educed from a well-tested convected Helmholtz model are compared to that of a recently developed linearized Euler model using two ceramic test liners under the assumed conditions or uniform flow and a plane wave source. The convected Helmholtz model is restricted to uniform mean flow whereas the linearized Euler model can account for the effect or the shear layer. Test data to educe the impedance is acquired from measurements obtained in the NASA Langley Research Center Grazing Incidence Tube for mean flow Mach numbers ranging from 0.0 to 0.5 and source frequencies ranging from 0.5 kHz to 3.0 kHz. The unknown impedance of the liner b educed by judiciously chooingth e impedance via an optimization method to match the measured acoustic pressure on the wall opposite the test liner. Results are presented on four spatial grids using three different optimization methods (contour deformation, Davidon-Fletcher Powell, and the Genetic Algorithm). All three optimization methods converge to the same impedance when used with the same model and to nearly identical impedances when used on different models. h anomaly was observed only at 0.5 kHz for high mean flow speeds. The anomaly is likely due to the use of measured data in a flow regime where shear layer effects are important but are neglected in the math models. Consistency between the impedances educed using the two models provides confidence that the linearized Euler model is ready For application to more realistic flows, such as those containing shear layers.

Modeling and Control of Intelligent Flexible Structures

DTIC Science & Technology

1994-03-26

can be approximated as a simply supported beam in transverse vibration. Assuming that the Euler- Bernoulli beam assumptions hold, linear equations of...The assumptions made during the derivation are that the element can be modeled as an Euler- Bernoulli beam, that the cross-section is symmetric, and...parametes A,. and ,%. andc input maces 3,,. The closed loop system. ecuation (7), is stable when the 3.. 8 and output gain mantices H1., H., H. for

Adaptively Refined Euler and Navier-Stokes Solutions with a Cartesian-Cell Based Scheme

NASA Technical Reports Server (NTRS)

Coirier, William J.; Powell, Kenneth G.

1995-01-01

A Cartesian-cell based scheme with adaptive mesh refinement for solving the Euler and Navier-Stokes equations in two dimensions has been developed and tested. Grids about geometrically complicated bodies were generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells were created using polygon-clipping algorithms. The grid was stored in a binary-tree data structure which provided a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations were solved on the resulting grids using an upwind, finite-volume formulation. The inviscid fluxes were found in an upwinded manner using a linear reconstruction of the cell primitives, providing the input states to an approximate Riemann solver. The viscous fluxes were formed using a Green-Gauss type of reconstruction upon a co-volume surrounding the cell interface. Data at the vertices of this co-volume were found in a linearly K-exact manner, which ensured linear K-exactness of the gradients. Adaptively-refined solutions for the inviscid flow about a four-element airfoil (test case 3) were compared to theory. Laminar, adaptively-refined solutions were compared to accepted computational, experimental and theoretical results.

Pulmonary Hypertension and Computed Tomography Measurement of Small Pulmonary Vessels in Severe Emphysema

PubMed Central

Matsuoka, Shin; Washko, George R.; Yamashiro, Tsuneo; Estepar, Raul San Jose; Diaz, Alejandro; Silverman, Edwin K.; Hoffman, Eric; Fessler, Henry E.; Criner, Gerard J.; Marchetti, Nathaniel; Scharf, Steven M.; Martinez, Fernando J.; Reilly, John J.; Hatabu, Hiroto

2010-01-01

Rationale: Vascular alteration of small pulmonary vessels is one of the characteristic features of pulmonary hypertension in chronic obstructive pulmonary disease. The in vivo relationship between pulmonary hypertension and morphological alteration of the small pulmonary vessels has not been assessed in patients with severe emphysema. Objectives: We evaluated the correlation of total cross-sectional area of small pulmonary vessels (CSA) assessed on computed tomography (CT) scans with the degree of pulmonary hypertension estimated by right heart catheterization. Methods: In 79 patients with severe emphysema enrolled in the National Emphysema Treatment Trial (NETT), we measured CSA less than 5 mm2 (CSA<5) and 5 to 10 mm2 (CSA5−10), and calculated the percentage of total CSA for the lung area (%CSA<5 and %CSA5–10, respectively). The correlations of %CSA<5 and %CSA5–10 with pulmonary arterial mean pressure (\\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}\\overline{Ppa}\\end{equation*}\\end{document}) obtained by right heart catheterization were evaluated. Multiple linear regression analysis using \\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}\\overline{Ppa}\\end{equation*}\\end{document} as the dependent outcome was also performed. Measurements and Main Results: The %CSA<5 had a significant negative correlation with \\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}\\overline{Ppa}\\end{equation*}\\end{document} (r = −0.512, P < 0.0001), whereas the correlation between %CSA5–10 and \\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}\\overline{Ppa}\\end{equation*}\\end{document} did not reach statistical significance (r = −0.196, P = 0.083). Multiple linear regression analysis showed that %CSA<5 and diffusing capacity of carbon monoxide (DlCO) % predicted were independent predictors of \\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}\\overline{Ppa}\\end{equation*}\\end{document} (r2 = 0.541): %CSA <5 (P < 0.0001), and DlCO % predicted (P = 0.022). Conclusions: The %CSA<5 measured on CT images is significantly correlated to \\documentclass[10pt]{article} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{pmc} \\usepackage[Euler]{upgreek} \\pagestyle{empty} \\oddsidemargin -1.0in \\begin{document} \\begin{equation*}\\overline{Ppa}\\end{equation*}\\end{document} in severe emphysema and can estimate the degree of pulmonary hypertension. PMID:19875683

Solution algorithms for the two-dimensional Euler equations on unstructured meshes

NASA Technical Reports Server (NTRS)

Whitaker, D. L.; Slack, David C.; Walters, Robert W.

1990-01-01

The objective of the study was to analyze implicit techniques employed in structured grid algorithms for solving two-dimensional Euler equations and extend them to unstructured solvers in order to accelerate convergence rates. A comparison is made between nine different algorithms for both first-order and second-order accurate solutions. Higher-order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The discussion is illustrated by results for flow over a transonic circular arc.

Application of Linear and Non-Linear Harmonic Methods for Unsteady Transonic Flow

NASA Astrophysics Data System (ADS)

Gundevia, Rayomand

This thesis explores linear and non-linear computational methods for solving unsteady flow. The eventual goal is to apply these methods to two-dimensional and three-dimensional flutter predictions. In this study the quasi-one-dimensional nozzle is used as a framework for understanding these methods and their limitations. Subsonic and transonic cases are explored as the back-pressure is forced to oscillate with known amplitude and frequency. A steady harmonic approach is used to solve this unsteady problem for which perturbations are said to be small in comparison to the mean flow. The use of a linearized Euler equations (LEE) scheme is good at capturing the flow characteristics but is limited by accuracy to relatively small amplitude perturbations. The introduction of time-averaged second-order terms in the Non-Linear Harmonic (NLH) method means that a better approximation of the mean-valued solution, upon which the linearization is based, can be made. The nonlinear time-accurate Euler solutions are used for comparison and to establish the regimes of unsteadiness for which these schemes fails. The usefulness of the LEE and NLH methods lie in the gains in computational efficiency over the full equations.

Boundary Layers for the Navier-Stokes Equations Linearized Around a Stationary Euler Flow

NASA Astrophysics Data System (ADS)

Gie, Gung-Min; Kelliher, James P.; Mazzucato, Anna L.

2018-03-01

We study the viscous boundary layer that forms at small viscosity near a rigid wall for the solution to the Navier-Stokes equations linearized around a smooth and stationary Euler flow (LNSE for short) in a smooth bounded domain Ω \\subset R^3 under no-slip boundary conditions. LNSE is supplemented with smooth initial data and smooth external forcing, assumed ill-prepared, that is, not compatible with the no-slip boundary condition. We construct an approximate solution to LNSE on the time interval [0, T], 0

Local existence of solutions to the Euler-Poisson system, including densities without compact support

NASA Astrophysics Data System (ADS)

Brauer, Uwe; Karp, Lavi

2018-01-01

Local existence and well posedness for a class of solutions for the Euler Poisson system is shown. These solutions have a density ρ which either falls off at infinity or has compact support. The solutions have finite mass, finite energy functional and include the static spherical solutions for γ = 6/5. The result is achieved by using weighted Sobolev spaces of fractional order and a new non-linear estimate which allows to estimate the physical density by the regularised non-linear matter variable. Gamblin also has studied this setting but using very different functional spaces. However we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the Newtonian limit.

Variation objective analyses for cyclone studies

NASA Technical Reports Server (NTRS)

Achtemeier, G. L.; Kidder, S. Q.; Ochs, H. T.

1985-01-01

The objectives were to: (1) develop an objective analysis technique that will maximize the information content of data available from diverse sources, with particular emphasis on the incorporation of observations from satellites with those from more traditional immersion techniques; and (2) to develop a diagnosis of the state of the synoptic scale atmosphere on a much finer scale over a much broader region than is presently possible to permit studies of the interactions and energy transfers between global, synoptic and regional scale atmospheric processes. The variational objective analysis model consists of the two horizontal momentum equations, the hydrostatic equation, and the integrated continuity equation for a dry hydrostatic atmosphere. Preliminary tests of the model with the SESMAE I data set are underway for 12 GMT 10 April 1979. At this stage of purpose of the analysis is not the diagnosis of atmospheric structures but rather the validation of the model. Model runs for rawinsonde data and with the precision modulus weights set to force most of the adjustment of the wind field to the mass field have produced 90 to 95 percent reductions in the imbalance of the initial data after only 4-cycles through the Euler-Lagrange equations. Sensitivity tests for linear stability of the 11 Euler-Lagrange equations that make up the VASP Model 1 indicate that there will be a lower limit to the scales of motion that can be resolved by this method. Linear stability criteria are violated where there is large horizontal wind shear near the upper tropospheric jet.

Asymptotic of the Solutions of Hyperbolic Equations with a Skew-Symmetric Perturbation

NASA Astrophysics Data System (ADS)

Gallagher, Isabelle

1998-12-01

Using methods introduced by S. Schochet inJ. Differential Equations114(1994), 476-512, we compute the first term of an asymptotic expansion of the solutions of hyperbolic equations perturbated by a skew-symmetric linear operator. That result is first applied to two systems describing the motion of geophysic fluids: the rotating Euler equations and the primitive system of the quasigeostrophic equations. Finally in the last part, we study the slightly compressible Euler equations by application of that same result.

Exploration of POD-Galerkin Techniques for Developing Reduced Order Models of the Euler Equations

DTIC Science & Technology

2015-07-01

modes [1]. Barone et al [15, 16] proposed to stabilize the reduced system by symmetrizing the higher-order PDE with a preconditioning matrix. Rowley et...advection scalar equation. The ROM is obtained by employing Galerkin’s method to reduce the high-order PDEs to a lower- order ODE system by means of POD...high-order PDEs to a lower-order ODE system by means of POD eigen-bases. For purposes of this study, a linearized version of the Euler equations is

Three dimensional steady subsonic Euler flows in bounded nozzles

NASA Astrophysics Data System (ADS)

Chen, Chao; Xie, Chunjing

The existence and uniqueness of three dimensional steady subsonic Euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the entrance are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic-sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli function are both small, the existence and uniqueness of subsonic Euler flows with non-zero vorticity are established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div-curl system, and delicate estimate for the transport equations.

Survey and analysis of research on supersonic drag-due-to-lift minimization with recommendations for wing design

NASA Technical Reports Server (NTRS)

Carlson, Harry W.; Mann, Michael J.

1992-01-01

A survey of research on drag-due-to-lift minimization at supersonic speeds, including a study of the effectiveness of current design and analysis methods was conducted. The results show that a linearized theory analysis with estimated attainable thrust and vortex force effects can predict with reasonable accuracy the lifting efficiency of flat wings. Significantly better wing performance can be achieved through the use of twist and camber. Although linearized theory methods tend to overestimate the amount of twist and camber required for a given application and provide an overly optimistic performance prediction, these deficiencies can be overcome by implementation of recently developed empirical corrections. Numerous examples of the correlation of experiment and theory are presented to demonstrate the applicability and limitations of linearized theory methods with and without empirical corrections. The use of an Euler code for the estimation of aerodynamic characteristics of a twisted and cambered wing and its application to design by iteration are discussed.

Eigenmode Analysis of Boundary Conditions for One-Dimensional Preconditioned Euler Equations

NASA Technical Reports Server (NTRS)

Darmofal, David L.

1998-01-01

An analysis of the effect of local preconditioning on boundary conditions for the subsonic, one-dimensional Euler equations is presented. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective with preconditioning, and, at low Mach numbers, disturbances do not decay. Other boundary conditions are investigated which are non-reflective with preconditioning and numerical results are presented confirming the analysis.

Chaotic dynamics of flexible Euler-Bernoulli beams

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

Awrejcewicz, J., E-mail: awrejcew@p.lodz.pl; Krysko, A. V., E-mail: anton.krysko@gmail.com; Kutepov, I. E., E-mail: iekutepov@gmail.com

2013-12-15

Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c{sup 2}) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions ismore » carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q{sub 0} and frequency ω{sub p} of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.« less

Global Regularity for the Fractional Euler Alignment System

NASA Astrophysics Data System (ADS)

Do, Tam; Kiselev, Alexander; Ryzhik, Lenya; Tan, Changhui

2018-04-01

We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian {(-partial _{xx})^{α/2}, α \\in (0, 1)}. The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all {α \\in (0, 1)}. To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.

A globally well-posed finite element algorithm for aerodynamics applications

NASA Technical Reports Server (NTRS)

Iannelli, G. S.; Baker, A. J.

1991-01-01

A finite element CFD algorithm is developed for Euler and Navier-Stokes aerodynamic applications. For the linear basis, the resultant approximation is at least second-order-accurate in time and space for synergistic use of three procedures: (1) a Taylor weak statement, which provides for derivation of companion conservation law systems with embedded dispersion-error control mechanisms; (2) a stiffly stable second-order-accurate implicit Rosenbrock-Runge-Kutta temporal algorithm; and (3) a matrix tensor product factorization that permits efficient numerical linear algebra handling of the terminal large-matrix statement. Thorough analyses are presented regarding well-posed boundary conditions for inviscid and viscous flow specifications. Numerical solutions are generated and compared for critical evaluation of quasi-one- and two-dimensional Euler and Navier-Stokes benchmark test problems.

Modelling and Closed-Loop System Identification of a Quadrotor-Based Aerial Manipulator

NASA Astrophysics Data System (ADS)

Dube, Chioniso; Pedro, Jimoh O.

2018-05-01

This paper presents the modelling and system identification of a quadrotor-based aerial manipulator. The aerial manipulator model is first derived analytically using the Newton-Euler formulation for the quadrotor and Recursive Newton-Euler formulation for the manipulator. The aerial manipulator is then simulated with the quadrotor under Proportional Derivative (PD) control, with the manipulator in motion. The simulation data is then used for system identification of the aerial manipulator. Auto Regressive with eXogenous inputs (ARX) models are obtained from the system identification for linear accelerations \\ddot{X} and \\ddot{Y} and yaw angular acceleration \\ddot{\\psi }. For linear acceleration \\ddot{Z}, and pitch and roll angular accelerations \\ddot{θ } and \\ddot{φ }, Auto Regressive Moving Average with eXogenous inputs (ARMAX) models are identified.

Non Lyapunov stability of a constant spatially developing 2-D gas flow

NASA Astrophysics Data System (ADS)

Balint, Agneta M.; Balint, Stefan; Tanasie, Loredana

2017-01-01

Different types of stabilities (global, local) and instabilities (global absolute, local convective) of the constant spatially developing 2-D gas flow are analyzed in a particular phase space of continuously differentiable functions, endowed with the usual algebraic operations and the topology generated by the uniform convergence on the plane. For this purpose the Euler equations linearized at the constant flow are used. The Lyapunov stability analysis was presented in [1] and this paper is a continuation of [1].

An approximation method for configuration optimization of trusses

NASA Technical Reports Server (NTRS)

Hansen, Scott R.; Vanderplaats, Garret N.

1988-01-01

Two- and three-dimensional elastic trusses are designed for minimum weight by varying the areas of the members and the location of the joints. Constraints on member stresses and Euler buckling are imposed and multiple static loading conditions are considered. The method presented here utilizes an approximate structural analysis based on first order Taylor series expansions of the member forces. A numerical optimizer minimizes the weight of the truss using information from the approximate structural analysis. Comparisons with results from other methods are made. It is shown that the method of forming an approximate structural analysis based on linearized member forces leads to a highly efficient method of truss configuration optimization.

A Direct and Non-Singular UKF Approach Using Euler Angle Kinematics for Integrated Navigation Systems

PubMed Central

Ran, Changyan; Cheng, Xianghong

2016-01-01

This paper presents a direct and non-singular approach based on an unscented Kalman filter (UKF) for the integration of strapdown inertial navigation systems (SINSs) with the aid of velocity. The state vector includes velocity and Euler angles, and the system model contains Euler angle kinematics equations. The measured velocity in the body frame is used as the filter measurement. The quaternion nonlinear equality constraint is eliminated, and the cross-noise problem is overcome. The filter model is simple and easy to apply without linearization. Data fusion is performed by an UKF, which directly estimates and outputs the navigation information. There is no need to process navigation computation and error correction separately because the navigation computation is completed synchronously during the filter time updating. In addition, the singularities are avoided with the help of the dual-Euler method. The performance of the proposed approach is verified by road test data from a land vehicle equipped with an odometer aided SINS, and a singularity turntable test is conducted using three-axis turntable test data. The results show that the proposed approach can achieve higher navigation accuracy than the commonly-used indirect approach, and the singularities can be efficiently removed as the result of dual-Euler method. PMID:27598169

Sensitivity analysis for aeroacoustic and aeroelastic design of turbomachinery blades

NASA Technical Reports Server (NTRS)

Lorence, Christopher B.; Hall, Kenneth C.

1995-01-01

A new method for computing the effect that small changes in the airfoil shape and cascade geometry have on the aeroacoustic and aeroelastic behavior of turbomachinery cascades is presented. The nonlinear unsteady flow is assumed to be composed of a nonlinear steady flow plus a small perturbation unsteady flow that is harmonic in time. First, the full potential equation is used to describe the behavior of the nonlinear mean (steady) flow through a two-dimensional cascade. The small disturbance unsteady flow through the cascade is described by the linearized Euler equations. Using rapid distortion theory, the unsteady velocity is split into a rotational part that contains the vorticity and an irrotational part described by a scalar potential. The unsteady vorticity transport is described analytically in terms of the drift and stream functions computed from the steady flow. Hence, the solution of the linearized Euler equations may be reduced to a single inhomogeneous equation for the unsteady potential. The steady flow and small disturbance unsteady flow equations are discretized using bilinear quadrilateral isoparametric finite elements. The nonlinear mean flow solution and streamline computational grid are computed simultaneously using Newton iteration. At each step of the Newton iteration, LU decomposition is used to solve the resulting set of linear equations. The unsteady flow problem is linear, and is also solved using LU decomposition. Next, a sensitivity analysis is performed to determine the effect small changes in cascade and airfoil geometry have on the mean and unsteady flow fields. The sensitivity analysis makes use of the nominal steady and unsteady flow LU decompositions so that no additional matrices need to be factored. Hence, the present method is computationally very efficient. To demonstrate how the sensitivity analysis may be used to redesign cascades, a compressor is redesigned for improved aeroelastic stability and two different fan exit guide vanes are redesigned for reduced downstream radiated noise. In addition, a framework detailing how the two-dimensional version of the method may be used to redesign three-dimensional geometries is presented.

Baseline Computational Fluid Dynamics Methodology for Longitudinal-Mode Liquid-Propellant Rocket Combustion Instability

NASA Technical Reports Server (NTRS)

Litchford, R. J.

2005-01-01

A computational method for the analysis of longitudinal-mode liquid rocket combustion instability has been developed based on the unsteady, quasi-one-dimensional Euler equations where the combustion process source terms were introduced through the incorporation of a two-zone, linearized representation: (1) A two-parameter collapsed combustion zone at the injector face, and (2) a two-parameter distributed combustion zone based on a Lagrangian treatment of the propellant spray. The unsteady Euler equations in inhomogeneous form retain full hyperbolicity and are integrated implicitly in time using second-order, high-resolution, characteristic-based, flux-differencing spatial discretization with Roe-averaging of the Jacobian matrix. This method was initially validated against an analytical solution for nonreacting, isentropic duct acoustics with specified admittances at the inflow and outflow boundaries. For small amplitude perturbations, numerical predictions for the amplification coefficient and oscillation period were found to compare favorably with predictions from linearized small-disturbance theory as long as the grid exceeded a critical density (100 nodes/wavelength). The numerical methodology was then exercised on a generic combustor configuration using both collapsed and distributed combustion zone models with a short nozzle admittance approximation for the outflow boundary. In these cases, the response parameters were varied to determine stability limits defining resonant coupling onset.

Residual Distribution Schemes for Conservation Laws Via Adaptive Quadrature

NASA Technical Reports Server (NTRS)

Barth, Timothy; Abgrall, Remi; Biegel, Bryan (Technical Monitor)

2000-01-01

This paper considers a family of nonconservative numerical discretizations for conservation laws which retains the correct weak solution behavior in the limit of mesh refinement whenever sufficient order numerical quadrature is used. Our analysis of 2-D discretizations in nonconservative form follows the 1-D analysis of Hou and Le Floch. For a specific family of nonconservative discretizations, it is shown under mild assumptions that the error arising from non-conservation is strictly smaller than the discretization error in the scheme. In the limit of mesh refinement under the same assumptions, solutions are shown to satisfy an entropy inequality. Using results from this analysis, a variant of the "N" (Narrow) residual distribution scheme of van der Weide and Deconinck is developed for first-order systems of conservation laws. The modified form of the N-scheme supplants the usual exact single-state mean-value linearization of flux divergence, typically used for the Euler equations of gasdynamics, by an equivalent integral form on simplex interiors. This integral form is then numerically approximated using an adaptive quadrature procedure. This renders the scheme nonconservative in the sense described earlier so that correct weak solutions are still obtained in the limit of mesh refinement. Consequently, we then show that the modified form of the N-scheme can be easily applied to general (non-simplicial) element shapes and general systems of first-order conservation laws equipped with an entropy inequality where exact mean-value linearization of the flux divergence is not readily obtained, e.g. magnetohydrodynamics, the Euler equations with certain forms of chemistry, etc. Numerical examples of subsonic, transonic and supersonic flows containing discontinuities together with multi-level mesh refinement are provided to verify the analysis.

Aerodynamics of a linear oscillating cascade

NASA Technical Reports Server (NTRS)

Buffum, Daniel H.; Fleeter, Sanford

1990-01-01

The steady and unsteady aerodynamics of a linear oscillating cascade are investigated using experimental and computational methods. Experiments are performed to quantify the torsion mode oscillating cascade aerodynamics of the NASA Lewis Transonic Oscillating Cascade for subsonic inlet flowfields using two methods: simultaneous oscillation of all the cascaded airfoils at various values of interblade phase angle, and the unsteady aerodynamic influence coefficient technique. Analysis of these data and correlation with classical linearized unsteady aerodynamic analysis predictions indicate that the wind tunnel walls enclosing the cascade have, in some cases, a detrimental effect on the cascade unsteady aerodynamics. An Euler code for oscillating cascade aerodynamics is modified to incorporate improved upstream and downstream boundary conditions and also the unsteady aerodynamic influence coefficient technique. The new boundary conditions are shown to improve the unsteady aerodynamic influence coefficient technique. The new boundary conditions are shown to improve the unsteady aerodynamic predictions of the code, and the computational unsteady aerodynamic influence coefficient technique is shown to be a viable alternative for calculation of oscillating cascade aerodynamics.

Non Lyapunov stability of the constant spatially developing 1-D gas flow in presence of solutions having strictly positive exponential growth rate

NASA Astrophysics Data System (ADS)

Balint, Stefan; Balint, Agneta M.

2017-01-01

Different types of stabilities (global, local) and instabilities (global absolute, local convective) of the constant spatially developing 1-D gas flow are analyzed in the phase space of continuously differentiable functions, endowed with the usual algebraic operations and the topology generated by the uniform convergence on the real axis. For this purpose the Euler equations linearized at the constant flow are used. The Lyapunov stability analysis was presented in [1] and this paper is a continuation of [1].

Linear stiff string vibrations in musical acoustics: Assessment and comparison of models.

PubMed

Ducceschi, Michele; Bilbao, Stefan

2016-10-01

Strings are amongst the most common elements found in musical instruments and an appropriate physical description of string dynamics is essential to modelling, analysis, and simulation. For linear vibration in a single polarisation, the most common model is based on the Euler-Bernoulli beam equation under tension. In spite of its simple form, such a model gives unbounded phase and group velocities at large wavenumbers, and such behaviour may be interpreted as unphysical. The Timoshenko model has, therefore, been employed in more recent works to overcome such shortcoming. This paper presents a third model based on the shear beam equations. The three models are here assessed and compared with regard to the perceptual considerations in musical acoustics.

On reinitializing level set functions

NASA Astrophysics Data System (ADS)

Min, Chohong

2010-04-01

In this paper, we consider reinitializing level functions through equation ϕt+sgn(ϕ0)(‖∇ϕ‖-1)=0[16]. The method of Russo and Smereka [11] is taken in the spatial discretization of the equation. The spatial discretization is, simply speaking, the second order ENO finite difference with subcell resolution near the interface. Our main interest is on the temporal discretization of the equation. We compare the three temporal discretizations: the second order Runge-Kutta method, the forward Euler method, and a Gauss-Seidel iteration of the forward Euler method. The fact that the time in the equation is fictitious makes a hypothesis that all the temporal discretizations result in the same result in their stationary states. The fact that the absolute stability region of the forward Euler method is not wide enough to include all the eigenvalues of the linearized semi-discrete system of the second order ENO spatial discretization makes another hypothesis that the forward Euler temporal discretization should invoke numerical instability. Our results in this paper contradict both the hypotheses. The Runge-Kutta and Gauss-Seidel methods obtain the second order accuracy, and the forward Euler method converges with order between one and two. Examining all their properties, we conclude that the Gauss-Seidel method is the best among the three. Compared to the Runge-Kutta, it is twice faster and requires memory two times less with the same accuracy.

A novel body frame based approach to aerospacecraft attitude tracking.

PubMed

Ma, Carlos; Chen, Michael Z Q; Lam, James; Cheung, Kie Chung

2017-09-01

In the common practice of designing an attitude tracker for an aerospacecraft, one transforms the Newton-Euler rotation equations to obtain the dynamic equations of some chosen inertial frame based attitude metrics, such as Euler angles and unit quaternions. A Lyapunov approach is then used to design a controller which ensures asymptotic convergence of the attitude to the desired orientation. Although this design methodology is pretty standard, it usually involves singularity-prone coordinate transformations which complicates the analysis process and controller design. A new, singularity free error feedback method is proposed in the paper to provide simple and intuitive stability analysis and controller synthesis. This new body frame based method utilizes the concept of Euleraxis and angles to generate the smallest error angles from a body frame perspective, without coordinate transformations. Global tracking convergence is illustrated with the use of a feedback linearizing PD tracker, a sliding mode controller, and a model reference adaptive controller. Experimental results are also obtained on a quadrotor platform with unknown system parameters and disturbances, using a boundary layer approximated sliding mode controller, a PIDD controller, and a unit sliding mode controller. Significant tracking quality is attained. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

An efficient iteration strategy for the solution of the Euler equations

NASA Technical Reports Server (NTRS)

Walters, R. W.; Dwoyer, D. L.

1985-01-01

A line Gauss-Seidel (LGS) relaxation algorithm in conjunction with a one-parameter family of upwind discretizations of the Euler equations in two-dimensions is described. The basic algorithm has the property that convergence to the steady-state is quadratic for fully supersonic flows and linear otherwise. This is in contrast to the block ADI methods (either central or upwind differenced) and the upwind biased relaxation schemes, all of which converge linearly, independent of the flow regime. Moreover, the algorithm presented here is easily enhanced to detect regions of subsonic flow embedded in supersonic flow. This allows marching by lines in the supersonic regions, converging each line quadratically, and iterating in the subsonic regions, thus yielding a very efficient iteration strategy. Numerical results are presented for two-dimensional supersonic and transonic flows containing both oblique and normal shock waves which confirm the efficiency of the iteration strategy.

Efficient solutions to the Euler equations for supersonic flow with embedded subsonic regions

NASA Technical Reports Server (NTRS)

Walters, Robert W.; Dwoyer, Douglas L.

1987-01-01

A line Gauss-Seidel (LGS) relaxation algorithm in conjunction with a one-parameter family of upwind discretizations of the Euler equations in two dimensions is described. Convergence of the basic algorithm to the steady state is quadratic for fully supersonic flows and is linear for other flows. This is in contrast to the block alternating direction implicit methods (either central or upwind differenced) and the upwind biased relaxation schemes, all of which converge linearly, independent of the flow regime. Moreover, the algorithm presented herein is easily coupled with methods to detect regions of subsonic flow embedded in supersonic flow. This allows marching by lines in the supersonic regions, converging each line quadratically, and iterating in the subsonic regions, and yields a very efficient iteration strategy. Numerical results are presented for two-dimensional supersonic and transonic flows containing oblique and normal shock waves which confirm the efficiency of the iteration strategy.

Pseudo-Linear Attitude Determination of Spinning Spacecraft

NASA Technical Reports Server (NTRS)

Bar-Itzhack, Itzhack Y.; Harman, Richard R.

2004-01-01

This paper presents the overall mathematical model and results from pseudo linear recursive estimators of attitude and rate for a spinning spacecraft. The measurements considered are vector measurements obtained by sun-sensors, fixed head star trackers, horizon sensors, and three axis magnetometers. Two filters are proposed for estimating the attitude as well as the angular rate vector. One filter, called the q-Filter, yields the attitude estimate as a quaternion estimate, and the other filter, called the D-Filter, yields the estimated direction cosine matrix. Because the spacecraft is gyro-less, Euler s equation of angular motion of rigid bodies is used to enable the estimation of the angular velocity. A simpler Markov model is suggested as a replacement for Euler's equation in the case where the vector measurements are obtained at high rates relative to the spacecraft angular rate. The performance of the two filters is examined using simulated data.

A hybrid approach for nonlinear computational aeroacoustics predictions

NASA Astrophysics Data System (ADS)

Sassanis, Vasileios; Sescu, Adrian; Collins, Eric M.; Harris, Robert E.; Luke, Edward A.

2017-01-01

In many aeroacoustics applications involving nonlinear waves and obstructions in the far-field, approaches based on the classical acoustic analogy theory or the linearised Euler equations are unable to fully characterise the acoustic field. Therefore, computational aeroacoustics hybrid methods that incorporate nonlinear wave propagation have to be constructed. In this study, a hybrid approach coupling Navier-Stokes equations in the acoustic source region with nonlinear Euler equations in the acoustic propagation region is introduced and tested. The full Navier-Stokes equations are solved in the source region to identify the acoustic sources. The flow variables of interest are then transferred from the source region to the acoustic propagation region, where the full nonlinear Euler equations with source terms are solved. The transition between the two regions is made through a buffer zone where the flow variables are penalised via a source term added to the Euler equations. Tests were conducted on simple acoustic and vorticity disturbances, two-dimensional jets (Mach 0.9 and 2), and a three-dimensional jet (Mach 1.5), impinging on a wall. The method is proven to be effective and accurate in predicting sound pressure levels associated with the propagation of linear and nonlinear waves in the near- and far-field regions.

Solving Nonlinear Euler Equations with Arbitrary Accuracy

NASA Technical Reports Server (NTRS)

Dyson, Rodger W.

2005-01-01

A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.

Euler Strut: A Mechanical Analogy for Dynamics in the Vicinity of a Critical Point

ERIC Educational Resources Information Center

Bobnar, Jaka; Susman, Katarina; Parsegian, V. Adrian; Rand, Peter R.; Cepic, Mojca; Podgornik, Rudolf

2011-01-01

An anchored elastic filament (Euler strut) under an external point load applied to its free end is a simple model for a second-order phase transition. In the static case, a load greater than the critical load causes a Euler buckling instability, leading to a change in the filament's shape. The analysis of filament dynamics with an external point…

Alternate solution to generalized Bernoulli equations via an integrating factor: an exact differential equation approach

NASA Astrophysics Data System (ADS)

Tisdell, C. C.

2017-08-01

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using 'exact methods', illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.

Designing pinhole vacancies in graphene towards functionalization: Effects on critical buckling load

NASA Astrophysics Data System (ADS)

Georgantzinos, S. K.; Markolefas, S.; Giannopoulos, G. I.; Katsareas, D. E.; Anifantis, N. K.

2017-03-01

The effect of size and placement of pinhole-type atom vacancies on Euler's critical load on free-standing, monolayer graphene, is investigated. The graphene is modeled by a structural spring-based finite element approach, in which every interatomic interaction is approached as a linear spring. The geometry of graphene and the pinhole size lead to the assembly of the stiffness matrix of the nanostructure. Definition of the boundary conditions of the problem leads to the solution of the eigenvalue problem and consequently to the critical buckling load. Comparison to results found in the literature illustrates the validity and accuracy of the proposed method. Parametric analysis regarding the placement and size of the pinhole-type vacancy, as well as the graphene geometry, depicts the effects on critical buckling load. Non-linear regression analysis leads to empirical-analytical equations for predicting the buckling behavior of graphene, with engineered pinhole-type atom vacancies.

The Scaling Group of the 1-D Invisicid Euler Equations

NASA Astrophysics Data System (ADS)

Schmidt, Emma; Ramsey, Scott; Boyd, Zachary; Baty, Roy

2017-11-01

The one dimensional (1-D) compressible Euler equations in non-ideal media support scale invariant solutions under a variety of initial conditions. Famous scale invariant solutions include the Noh, Sedov, Guderley, and collapsing cavity hydrodynamic test problems. We unify many classical scale invariant solutions under a single scaling group analysis. The scaling symmetry group generator provides a framework for determining all scale invariant solutions emitted by the 1-D Euler equations for arbitrary geometry, initial conditions, and equation of state. We approach the Euler equations from a geometric standpoint, and conduct scaling analyses for a broad class of materials.

A Novel Nonlinear Piezoelectric Energy Harvesting System Based on Linear-Element Coupling: Design, Modeling and Dynamic Analysis.

PubMed

Zhou, Shengxi; Yan, Bo; Inman, Daniel J

2018-05-09

This paper presents a novel nonlinear piezoelectric energy harvesting system which consists of linear piezoelectric energy harvesters connected by linear springs. In principle, the presented nonlinear system can improve broadband energy harvesting efficiency where magnets are forbidden. The linear spring inevitably produces the nonlinear spring force on the connected harvesters, because of the geometrical relationship and the time-varying relative displacement between two adjacent harvesters. Therefore, the presented nonlinear system has strong nonlinear characteristics. A theoretical model of the presented nonlinear system is deduced, based on Euler-Bernoulli beam theory, Kirchhoff’s law, piezoelectric theory and the relevant geometrical relationship. The energy harvesting enhancement of the presented nonlinear system (when n = 2, 3) is numerically verified by comparing with its linear counterparts. In the case study, the output power area of the presented nonlinear system with two and three energy harvesters is 268.8% and 339.8% of their linear counterparts, respectively. In addition, the nonlinear dynamic response characteristics are analyzed via bifurcation diagrams, Poincare maps of the phase trajectory, and the spectrum of the output voltage.

On Bi-Grid Local Mode Analysis of Solution Techniques for 3-D Euler and Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Ibraheem, S. O.; Demuren, A. O.

1994-01-01

A procedure is presented for utilizing a bi-grid stability analysis as a practical tool for predicting multigrid performance in a range of numerical methods for solving Euler and Navier-Stokes equations. Model problems based on the convection, diffusion and Burger's equation are used to illustrate the superiority of the bi-grid analysis as a predictive tool for multigrid performance in comparison to the smoothing factor derived from conventional von Neumann analysis. For the Euler equations, bi-grid analysis is presented for three upwind difference based factorizations, namely Spatial, Eigenvalue and Combination splits, and two central difference based factorizations, namely LU and ADI methods. In the former, both the Steger-Warming and van Leer flux-vector splitting methods are considered. For the Navier-Stokes equations, only the Beam-Warming (ADI) central difference scheme is considered. In each case, estimates of multigrid convergence rates from the bi-grid analysis are compared to smoothing factors obtained from single-grid stability analysis. Effects of grid aspect ratio and flow skewness are examined. Both predictions are compared with practical multigrid convergence rates for 2-D Euler and Navier-Stokes solutions based on the Beam-Warming central scheme.

Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems

NASA Astrophysics Data System (ADS)

Konopelchenko, B. G.; Ortenzi, G.

2013-12-01

The structure and properties of families of critical points for classes of functions W(z,{\\overline{z}}) obeying the elliptic Euler-Poisson-Darboux equation E(1/2, 1/2) are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented. There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the ‘inverse’ hierarchy and equations associated with the real-analytic Eisenstein series E(\\beta ,{\\overline{\\beta }};1/2) among them. The specific bi-Hamiltonian structure of these equations is also discussed.

Compressible cavitation with stochastic field method

NASA Astrophysics Data System (ADS)

Class, Andreas; Dumond, Julien

2012-11-01

Non-linear phenomena can often be well described using probability density functions (pdf) and pdf transport models. Traditionally the simulation of pdf transport requires Monte-Carlo codes based on Lagrange particles or prescribed pdf assumptions including binning techniques. Recently, in the field of combustion, a novel formulation called the stochastic field method solving pdf transport based on Euler fields has been proposed which eliminates the necessity to mix Euler and Lagrange techniques or prescribed pdf assumptions. In the present work, part of the PhD Design and analysis of a Passive Outflow Reducer relying on cavitation, a first application of the stochastic field method to multi-phase flow and in particular to cavitating flow is presented. The application considered is a nozzle subjected to high velocity flow so that sheet cavitation is observed near the nozzle surface in the divergent section. It is demonstrated that the stochastic field formulation captures the wide range of pdf shapes present at different locations. The method is compatible with finite-volume codes where all existing physical models available for Lagrange techniques, presumed pdf or binning methods can be easily extended to the stochastic field formulation.

On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence

NASA Astrophysics Data System (ADS)

Moura, R. C.; Mengaldo, G.; Peiró, J.; Sherwin, S. J.

2017-02-01

We present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The '1% rule' based on linear dispersion-diffusion analysis introduced by Moura et al. (2015) [10] is here adapted for 3D energy spectra and validated through the inviscid Taylor-Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.

Testing for a Signal with Unknown Location and Scale in a Stationary Gaussian Random Field

DTIC Science & Technology

1994-01-07

Secondary 60D05, 52A22. Key words and phrases. Euler characteristic, integral geometry, image analysis , Gaussian fields, volume of tubes. SUMMARY We...words and phrases. Euler characteristic, integral geometry. image analysis . Gaussian fields. volume of tubes. 20. AMST RACT (Coith..o an revmreo ef* It

Towards a Comprehensive Model of Jet Noise Using an Acoustic Analogy and Steady RANS Solutions

NASA Technical Reports Server (NTRS)

Miller, Steven A. E.

2013-01-01

An acoustic analogy is developed to predict the noise from jet flows. It contains two source models that independently predict the noise from turbulence and shock wave shear layer interactions. The acoustic analogy is based on the Euler equations and separates the sources from propagation. Propagation effects are taken into account by calculating the vector Green's function of the linearized Euler equations. The sources are modeled following the work of Tam and Auriault, Morris and Boluriaan, and Morris and Miller. A statistical model of the two-point cross-correlation of the velocity fluctuations is used to describe the turbulence. The acoustic analogy attempts to take into account the correct scaling of the sources for a wide range of nozzle pressure and temperature ratios. It does not make assumptions regarding fine- or large-scale turbulent noise sources, self- or shear-noise, or convective amplification. The acoustic analogy is partially informed by three-dimensional steady Reynolds-Averaged Navier-Stokes solutions that include the nozzle geometry. The predictions are compared with experiments of jets operating subsonically through supersonically and at unheated and heated temperatures. Predictions generally capture the scaling of both mixing noise and BBSAN for the conditions examined, but some discrepancies remain that are due to the accuracy of the steady RANS turbulence model closure, the equivalent sources, and the use of a simplified vector Green's function solver of the linearized Euler equations.

Influence of airfoil thickness on convected gust interaction noise

NASA Technical Reports Server (NTRS)

Kerschen, E. J.; Tsai, C. T.

1989-01-01

The case of a symmetric airfoil at zero angle of attack is considered in order to determine the influence of airfoil thickness on sound generated by interaction with convected gusts. The analysis is based on a linearization of the Euler equations about the subsonic mean flow past the airfoil. Primary sound generation is found to occur in a local region surrounding the leading edge, with the size of the local region scaling on the gust wavelength. For a parabolic leading edge, moderate leading edge thickness is shown to decrease the noise level in the low Mach number limit.

A new fractional nonlocal model and its application in free vibration of Timoshenko and Euler-Bernoulli beams

NASA Astrophysics Data System (ADS)

Rahimi, Zaher; Sumelka, Wojciech; Yang, Xiao-Jun

2017-11-01

The application of fractional calculus in fractional models (FMs) makes them more flexible than integer models inasmuch they can conclude all of integer and non-integer operators. In other words FMs let us use more potential of mathematics to modeling physical phenomena due to the use of both integer and fractional operators to present a better modeling of problems, which makes them more flexible and powerful. In the present work, a new fractional nonlocal model has been proposed, which has a simple form and can be used in different problems due to the simple form of numerical solutions. Then the model has been used to govern equations of the motion of the Timoshenko beam theory (TBT) and Euler-Bernoulli beam theory (EBT). Next, free vibration of the Timoshenko and Euler-Bernoulli simply-supported (S-S) beam has been investigated. The Galerkin weighted residual method has been used to solve the non-linear governing equations.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

NASA Astrophysics Data System (ADS)

Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

2009-09-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

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.

Experimental and analytical investigations of longitudinal combustion instability in a continuously variable resonance combustor (CVRC)

NASA Astrophysics Data System (ADS)

Yu, Yen Ching

An analytical model based on linearized Euler equations (LEE) is developed and used in conjunction with a validating experiment to study combustion instability. The LEE model features mean flow effects, entropy waves, adaptability for more physically-realistic boundary conditions, and is generalized for multiple-domain conditions. The model calculates spatial modes, resonant frequencies and linear growth rates of the overall system. The predicted resonant frequencies and spatially-resolved mode shapes agree with the experimental data from a longitudinally-unstable model rocket combustor to within 7%. Different gaseous fuels (methane, ethylene, and hydrogen) were tested under fixed geometry. Tests with hydrogen were stable, whereas ethylene, methane, and JP-8 were increasingly unstable. A novel method for obtaining large amounts of stability data under variable resonance conditions in a single test was demonstrated. The continuously variable resonance combustor (CVRC) incorporates a traversing choked axial oxidizer inlet to vary the overall combustion system resonance. The CVRC experiment successfully demonstrates different level of instability, transitions between stability levels, and identifies the most stable and unstable geometric combination. Pressure oscillation amplitudes ranged from less than 10% of mean pressure to greater than 60%. At low amplitudes, measured resonant frequency changed with inlet location but at high amplitude the measured resonance frequency matched the frequency of the combustion chamber. As the system transitions from linear to non-linear instability, the higher harmonics of the fundamental resonant mode appear nearly simultaneously. Transient, high-amplitude, broadband noise, at lower frequencies (on the order of 200 Hz) are also observed. Conversely, as the system transitions back to a more linear stability regime, the higher harmonics disappear sequentially, led by the highest order. Good agreements between analytical and experimental results are attained by treating the experiment as quasi-stationary. The stability characteristics from the high frequency measurements are further analyzed using filtered pressure traces, spectrograms, power spectral density plots, and oscillation decrements. Future works recommended include: direct measurements, such as chemiluminescence or high-speed imaging to examine the unsteady combustion processes; three-way comparisons between the acoustic-based, linear Euler-based, and non-linear Euler/RANS model; use the high fidelity computation to investigate the forcing terms modeled in the acoustic-based model.

Accurate upwind methods for the Euler equations

NASA Technical Reports Server (NTRS)

Huynh, Hung T.

1993-01-01

A new class of piecewise linear methods for the numerical solution of the one-dimensional Euler equations of gas dynamics is presented. These methods are uniformly second-order accurate, and can be considered as extensions of Godunov's scheme. With an appropriate definition of monotonicity preservation for the case of linear convection, it can be shown that they preserve monotonicity. Similar to Van Leer's MUSCL scheme, they consist of two key steps: a reconstruction step followed by an upwind step. For the reconstruction step, a monotonicity constraint that preserves uniform second-order accuracy is introduced. Computational efficiency is enhanced by devising a criterion that detects the 'smooth' part of the data where the constraint is redundant. The concept and coding of the constraint are simplified by the use of the median function. A slope steepening technique, which has no effect at smooth regions and can resolve a contact discontinuity in four cells, is described. As for the upwind step, existing and new methods are applied in a manner slightly different from those in the literature. These methods are derived by approximating the Euler equations via linearization and diagonalization. At a 'smooth' interface, Harten, Lax, and Van Leer's one intermediate state model is employed. A modification for this model that can resolve contact discontinuities is presented. Near a discontinuity, either this modified model or a more accurate one, namely, Roe's flux-difference splitting. is used. The current presentation of Roe's method, via the conceptually simple flux-vector splitting, not only establishes a connection between the two splittings, but also leads to an admissibility correction with no conditional statement, and an efficient approximation to Osher's approximate Riemann solver. These reconstruction and upwind steps result in schemes that are uniformly second-order accurate and economical at smooth regions, and yield high resolution at discontinuities.

Nonlinear truncation error analysis of finite difference schemes for the Euler equations

NASA Technical Reports Server (NTRS)

Klopfer, G. H.; Mcrae, D. S.

1983-01-01

It is pointed out that, in general, dissipative finite difference integration schemes have been found to be quite robust when applied to the Euler equations of gas dynamics. The present investigation considers a modified equation analysis of both implicit and explicit finite difference techniques as applied to the Euler equations. The analysis is used to identify those error terms which contribute most to the observed solution errors. A technique for analytically removing the dominant error terms is demonstrated, resulting in a greatly improved solution for the explicit Lax-Wendroff schemes. It is shown that the nonlinear truncation errors are quite large and distributed quite differently for each of the three conservation equations as applied to a one-dimensional shock tube problem.

Dark energy and modified gravity in the Effective Field Theory of Large-Scale Structure

NASA Astrophysics Data System (ADS)

Cusin, Giulia; Lewandowski, Matthew; Vernizzi, Filippo

2018-04-01

We develop an approach to compute observables beyond the linear regime of dark matter perturbations for general dark energy and modified gravity models. We do so by combining the Effective Field Theory of Dark Energy and Effective Field Theory of Large-Scale Structure approaches. In particular, we parametrize the linear and nonlinear effects of dark energy on dark matter clustering in terms of the Lagrangian terms introduced in a companion paper [1], focusing on Horndeski theories and assuming the quasi-static approximation. The Euler equation for dark matter is sourced, via the Newtonian potential, by new nonlinear vertices due to modified gravity and, as in the pure dark matter case, by the effects of short-scale physics in the form of the divergence of an effective stress tensor. The effective fluid introduces a counterterm in the solution to the matter continuity and Euler equations, which allows a controlled expansion of clustering statistics on mildly nonlinear scales. We use this setup to compute the one-loop dark-matter power spectrum.

IMEX HDG-DG: A coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for Euler systems on cubed sphere.

NASA Astrophysics Data System (ADS)

Kang, S.; Muralikrishnan, S.; Bui-Thanh, T.

2017-12-01

We propose IMEX HDG-DG schemes for Euler systems on cubed sphere. Of interest is subsonic flow, where the speed of the acoustic wave is faster than that of the nonlinear advection. In order to simulate these flows efficiently, we split the governing system into stiff part describing the fast waves and non-stiff part associated with nonlinear advection. The former is discretized implicitly with HDG method while explicit Runge-Kutta DG discretization is employed for the latter. The proposed IMEX HDG-DG framework: 1) facilitates high-order solution both in time and space; 2) avoids overly small time stepsizes; 3) requires only one linear system solve per time step; and 4) relatively to DG generates smaller and sparser linear system while promoting further parallelism owing to HDG discretization. Numerical results for various test cases demonstrate that our methods are comparable to explicit Runge-Kutta DG schemes in terms of accuracy, while allowing for much larger time stepsizes.

Three dimensional unstructured multigrid for the Euler equations

NASA Technical Reports Server (NTRS)

Mavriplis, D. J.

1991-01-01

The three dimensional Euler equations are solved on unstructured tetrahedral meshes using a multigrid strategy. The driving algorithm consists of an explicit vertex-based finite element scheme, which employs an edge-based data structure to assemble the residuals. The multigrid approach employs a sequence of independently generated coarse and fine meshes to accelerate the convergence to steady-state of the fine grid solution. Variables, residuals and corrections are passed back and forth between the various grids of the sequence using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using linear interpolation. The addresses and weights for interpolation are determined in a preprocessing stage using an efficient graph traversal algorithm. The preprocessing operation is shown to require a negligible fraction of the CPU time required by the overall solution procedure, while gains in overall solution efficiencies greater than an order of magnitude are demonstrated on meshes containing up to 350,000 vertices. Solutions using globally regenerated fine meshes as well as adaptively refined meshes are given.

Perfectly Matched Layer for Linearized Euler Equations in Open and Ducted Domains

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Auriault, Laurent; Cambuli, Francesco

1998-01-01

Recently, perfectly matched layer (PML) as an absorbing boundary condition has widespread applications. The idea was first introduced by Berenger for electromagnetic waves computations. In this paper, it is shown that the PML equations for the linearized Euler equations support unstable solutions when the mean flow has a component normal to the layer. To suppress such unstable solutions so as to render the PML concept useful for this class of problems, it is proposed that artificial selective damping terms be added to the discretized PML equations. It is demonstrated that with a proper choice of artificial mesh Reynolds number, the PML equations can be made stable. Numerical examples are provided to illustrate that the stabilized PML performs well as an absorbing boundary condition. In a ducted environment, the wave mode are dispersive. It will be shown that the group velocity and phase velocity of these modes can have opposite signs. This results in a confined environment, PML may not be suitable as an absorbing boundary condition.

A Computational/Experimental Study of Two Optimized Supersonic Transport Designs and the Reference H Baseline

NASA Technical Reports Server (NTRS)

Cliff, Susan E.; Baker, Timothy J.; Hicks, Raymond M.; Reuther, James J.

1999-01-01

Two supersonic transport configurations designed by use of non-linear aerodynamic optimization methods are compared with a linearly designed baseline configuration. One optimized configuration, designated Ames 7-04, was designed at NASA Ames Research Center using an Euler flow solver, and the other, designated Boeing W27, was designed at Boeing using a full-potential method. The two optimized configurations and the baseline were tested in the NASA Langley Unitary Plan Supersonic Wind Tunnel to evaluate the non-linear design optimization methodologies. In addition, the experimental results are compared with computational predictions for each of the three configurations from the Enter flow solver, AIRPLANE. The computational and experimental results both indicate moderate to substantial performance gains for the optimized configurations over the baseline configuration. The computed performance changes with and without diverters and nacelles were in excellent agreement with experiment for all three models. Comparisons of the computational and experimental cruise drag increments for the optimized configurations relative to the baseline show excellent agreement for the model designed by the Euler method, but poorer comparisons were found for the configuration designed by the full-potential code.

Recursive Newton-Euler formulation of manipulator dynamics

NASA Technical Reports Server (NTRS)

Nasser, M. G.

1989-01-01

A recursive Newton-Euler procedure is presented for the formulation and solution of manipulator dynamical equations. The procedure includes rotational and translational joints and a topological tree. This model was verified analytically using a planar two-link manipulator. Also, the model was tested numerically against the Walker-Orin model using the Shuttle Remote Manipulator System data. The hinge accelerations obtained from both models were identical. The computational requirements of the model vary linearly with the number of joints. The computational efficiency of this method exceeds that of Walker-Orin methods. This procedure may be viewed as a considerable generalization of Armstrong's method. A six-by-six formulation is adopted which enhances both the computational efficiency and simplicity of the model.

Linearized Unsteady Aerodynamic Analysis of the Acoustic Response to Wake/Blade-Row Interaction

NASA Technical Reports Server (NTRS)

Verdon, Joseph M.; Huff, Dennis L. (Technical Monitor)

2001-01-01

The three-dimensional, linearized Euler analysis, LINFLUX, is being developed to provide a comprehensive and efficient unsteady aerodynamic scheme for predicting the aeroacoustic and aeroelastic responses of axial-flow turbomachinery blading. LINFLUX couples a near-field, implicit, wave-split, finite-volume solution to far-field acoustic eigensolutions, to predict the aerodynamic responses of a blade row to prescribed structural and aerodynamic excitations. It is applied herein to predict the acoustic responses of a fan exit guide vane (FEGV) to rotor wake excitations. The intent is to demonstrate and assess the LINFLUX analysis via application to realistic wake/blade-row interactions. Numerical results are given for the unsteady pressure responses of the FEGV, including the modal pressure responses at inlet and exit. In addition, predictions for the modal and total acoustic power levels at the FEGV exit are compared with measurements. The present results indicate that the LINFLUX analysis should be useful in the aeroacoustic design process, and for understanding the three-dimensional flow physics relevant to blade-row noise generation and propagation.

Transonic flow analysis for rotors. Part 3: Three-dimensional, quasi-steady, Euler calculation

NASA Technical Reports Server (NTRS)

Chang, I-Chung

1990-01-01

A new method is presented for calculating the quasi-steady transonic flow over a lifting or non-lifting rotor blade in both hover and forward flight by using Euler equations. The approach is to solve Euler equations in a rotor-fixed frame of reference using a finite volume method. A computer program was developed and was then verified by comparison with wind-tunnel data. In all cases considered, good agreement was found with published experimental data.

An outflow boundary condition for aeroacoustic computations

NASA Technical Reports Server (NTRS)

Hayder, M. Ehtesham; Hagstrom, Thomas

1995-01-01

A formulation of boundary condition for flows with small disturbances is presented. The authors test their methodology in an axisymmetric jet flow calculation, using both the Navier-Stokes and Euler equations. Solutions in the far field are assumed to be oscillatory. If the oscillatory disturbances are small, the growth of the solution variables can be predicted by linear theory. Eigenfunctions of the linear theory are used explicitly in the formulation of the boundary conditions. This guarantees correct solutions at the boundary in the limit where the predictions of linear theory are valid.

APPLE - An aeroelastic analysis system for turbomachines and propfans

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, Milind A.; Srivastava, R.; Mehmed, Oral

1992-01-01

This paper reviews aeroelastic analysis methods for propulsion elements (advanced propellers, compressors and turbines) being developed and used at NASA Lewis Research Center. These aeroelastic models include both structural and aerodynamic components. The structural models include the typical section model, the beam model with and without disk flexibility, and the finite element blade model with plate bending elements. The aerodynamic models are based on the solution of equations ranging from the two-dimensional linear potential equation for a cascade to the three-dimensional Euler equations for multi-blade configurations. Typical results are presented for each aeroelastic model. Suggestions for further research are indicated. All the available aeroelastic models and analysis methods are being incorporated into a unified computer program named APPLE (Aeroelasticity Program for Propulsion at LEwis).

Experimental Comparison Between Mahoney and Complementary Sensor Fusion Algorithm for Attitude Determination by Raw Sensor Data of Xsens Imu on Buoy

NASA Astrophysics Data System (ADS)

Jouybari, A.; Ardalan, A. A.; Rezvani, M.-H.

2017-09-01

The accurate measurement of platform orientation plays a critical role in a range of applications including marine, aerospace, robotics, navigation, human motion analysis, and machine interaction. We used Mahoney filter, Complementary filter and Xsens Kalman filter for achieving Euler angle of a dynamic platform by integration of gyroscope, accelerometer, and magnetometer measurements. The field test has been performed in Kish Island using an IMU sensor (Xsens MTi-G-700) that installed onboard a buoy so as to provide raw data of gyroscopes, accelerometers, magnetometer measurements about 25 minutes. These raw data were used to calculate the Euler angles by Mahoney filter and Complementary filter, while the Euler angles collected by XSense IMU sensor become the reference of the Euler angle estimations. We then compared Euler angles which calculated by Mahoney Filter and Complementary Filter with reference to the Euler angles recorded by the XSense IMU sensor. The standard deviations of the differences between the Mahoney Filter, Complementary Filter Euler angles and XSense IMU sensor Euler angles were about 0.5644, 0.3872, 0.4990 degrees and 0.6349, 0.2621, 2.3778 degrees for roll, pitch, and heading, respectively, so the numerical result assert that Mahoney filter is precise for roll and heading angles determination and Complementary filter is precise only for pitch determination, it should be noted that heading angle determination by Complementary filter has more error than Mahoney filter.

A Review of Recent Aeroelastic Analysis Methods for Propulsion at NASA Lewis Research Center

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, Milind A.; Srivastava, R.; Mehmed, Oral; Stefko, George L.

1993-01-01

This report reviews aeroelastic analyses for propulsion components (propfans, compressors and turbines) being developed and used at NASA LeRC. These aeroelastic analyses include both structural and aerodynamic models. The structural models include a typical section, a beam (with and without disk flexibility), and a finite-element blade model (with plate bending elements). The aerodynamic models are based on the solution of equations ranging from the two-dimensional linear potential equation to the three-dimensional Euler equations for multibladed configurations. Typical calculated results are presented for each aeroelastic model. Suggestions for further research are made. Many of the currently available aeroelastic models and analysis methods are being incorporated in a unified computer program, APPLE (Aeroelasticity Program for Propulsion at LEwis).

An h-p Taylor-Galerkin finite element method for compressible Euler equations

NASA Technical Reports Server (NTRS)

Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O.

1991-01-01

An extension of the familiar Taylor-Galerkin method to arbitrary h-p spatial approximations is proposed. Boundary conditions are analyzed, and a linear stability result for arbitrary meshes is given, showing the unconditional stability for the parameter of implicitness alpha not less than 0.5. The wedge and blunt body problems are solved with both linear, quadratic, and cubic elements and h-adaptivity, showing the feasibility of higher orders of approximation for problems with shocks.

A Generalized Acoustic Analogy

NASA Technical Reports Server (NTRS)

Goldstein, M. E.

2003-01-01

The purpose of this article is to show that the Navier-Stokes equations can be rewritten as a set of linearized inhomogeneous Euler equations (in convective form) with source terms that are exactly the same as those that would result from externally imposed shear stress and energy flux perturbations. These results are used to develop a mathematical basis for some existing and potential new jet noise models by appropriately choosing the base flow about which the linearization is carried out.

Time-Domain Impedance Boundary Conditions for Computational Aeroacoustics

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Auriault, Laurent

1996-01-01

It is an accepted practice in aeroacoustics to characterize the properties of an acoustically treated surface by a quantity known as impedance. Impedance is a complex quantity. As such, it is designed primarily for frequency-domain analysis. Time-domain boundary conditions that are the equivalent of the frequency-domain impedance boundary condition are proposed. Both single frequency and model broadband time-domain impedance boundary conditions are provided. It is shown that the proposed boundary conditions, together with the linearized Euler equations, form well-posed initial boundary value problems. Unlike ill-posed problems, they are free from spurious instabilities that would render time-marching computational solutions impossible.

On the Wind Generation of Water Waves

NASA Astrophysics Data System (ADS)

Bühler, Oliver; Shatah, Jalal; Walsh, Samuel; Zeng, Chongchun

2016-11-01

In this work, we consider the mathematical theory of wind generated water waves. This entails determining the stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We present a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, we give a complete proof of the instability criterion of M iles [16]. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air-sea interface). We are thus able to give a unified equation connecting the Kelvin-Helmholtz and quasi-laminar models of wave generation.

Faddeev-Jackiw quantization of topological invariants: Euler and Pontryagin classes

NASA Astrophysics Data System (ADS)

Escalante, Alberto; Medel-Portugal, C.

2018-04-01

The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev-Jackiw context. The Faddeev-Jackiw constraints and the generalized Faddeev-Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev-Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions.

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.

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

NASA Astrophysics Data System (ADS)

Ansari, R.; Faraji Oskouie, M.; Gholami, R.

2016-01-01

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler-Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler-Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor-corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.

User's Guide for ECAP2D: an Euler Unsteady Aerodynamic and Aeroelastic Analysis Program for Two Dimensional Oscillating Cascades, Version 1.0

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.

1995-01-01

This guide describes the input data required for using ECAP2D (Euler Cascade Aeroelastic Program-Two Dimensional). ECAP2D can be used for steady or unsteady aerodynamic and aeroelastic analysis of two dimensional cascades. Euler equations are used to obtain aerodynamic forces. The structural dynamic equations are written for a rigid typical section undergoing pitching (torsion) and plunging (bending) motion. The solution methods include harmonic oscillation method, influence coefficient method, pulse response method, and time integration method. For harmonic oscillation method, example inputs and outputs are provided for pitching motion and plunging motion. For the rest of the methods, input and output for pitching motion only are given.

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).

CFD analysis of multiphase blood flow within aorta and its thoracic branches of patient with coarctation of aorta using multiphase Euler - Euler approach

NASA Astrophysics Data System (ADS)

Ostrowski, Z.; Melka, B.; Adamczyk, W.; Rojczyk, M.; Golda, A.; Nowak, A. J.

2016-09-01

In the research a numerical Computational Fluid Dynamics (CFD) model of the pulsatile blood flow was created and analyzed. A real geometry of aorta and its thoracic branches of 8-year old patient diagnosed with a congenital heart defect - coarctation of aorta was used. The inlet boundary condition were implemented as the User Define Function according to measured values of volumetric blood flow. The blood flow was treated as multiphase: plasma, set as the primary fluid phase, was dominant with volume fraction of 0.585 and morphological elements of blood were treated in Euler-Euler approach as dispersed phases (with 90% Red Blood Cells and White Blood Cells as remaining solid volume fraction).

Construction of reduced order models for the non-linear Navier-Stokes equations using the proper orthogonal fecomposition (POD)/Galerkin method.

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

Fike, Jeffrey A.

2013-08-01

The construction of stable reduced order models using Galerkin projection for the Euler or Navier-Stokes equations requires a suitable choice for the inner product. The standard L2 inner product is expected to produce unstable ROMs. For the non-linear Navier-Stokes equations this means the use of an energy inner product. In this report, Galerkin projection for the non-linear Navier-Stokes equations using the L2 inner product is implemented as a first step toward constructing stable ROMs for this set of physics.

Symmetry investigations on the incompressible stationary axisymmetric Euler equations with swirl

NASA Astrophysics Data System (ADS)

Frewer, M.; Oberlack, M.; Guenther, S.

2007-08-01

We discuss the incompressible stationary axisymmetric Euler equations with swirl, for which we derive via a scalar stream function an equivalent representation, the Bragg-Hawthorne equation [Bragg, S.L., Hawthorne, W.R., 1950. Some exact solutions of the flow through annular cascade actuator discs. J. Aero. Sci. 17, 243]. Despite this obvious equivalence, we will show that under a local Lie point symmetry analysis the Bragg-Hawthorne equation exposes itself as not being fully equivalent to the original Euler equations. This is reflected in the way that it possesses additional symmetries not being admitted by its counterpart. In other words, a symmetry of the Bragg-Hawthorne equation is in general not a symmetry of the Euler equations. Not the differential Euler equations but rather a set of integro-differential equations attains full equivalence to the Bragg-Hawthorne equation. For these intermediate Euler equations, it is interesting to note that local symmetries of the Bragg-Hawthorne equation transform to local as well as to nonlocal symmetries. This behaviour, on the one hand, is in accordance with Zawistowski's result [Zawistowski, Z.J., 2001. Symmetries of integro-differential equations. Rep. Math. Phys. 48, 269; Zawistowski, Z.J., 2004. General criterion of invariance for integro-differential equations. Rep. Math. Phys. 54, 341] that it is possible for integro-differential equations to admit local Lie point symmetries. On the other hand, with this transformation process we collect symmetries which cannot be obtained when carrying out a usual local Lie point symmetry analysis. Finally, the symmetry classification of the Bragg-Hawthorne equation is used to find analytical solutions for the phenomenon of vortex breakdown.

Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and its Application to Discontinuous Galerkin Discretizations of the Euler Equations

DTIC Science & Technology

2015-06-01

cient parallel code for applying the operator. Our method constructs a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show...apply the underlying operator. Such a preconditioner can be very attractive in scenarios where one has a highly efficient parallel code for applying...repeatedly solve a large system of linear equations where one has an extremely fast parallel code for applying an underlying fixed linear operator

Numerical calculations of two dimensional, unsteady transonic flows with circulation

NASA Technical Reports Server (NTRS)

Beam, R. M.; Warming, R. F.

1974-01-01

The feasibility of obtaining two-dimensional, unsteady transonic aerodynamic data by numerically integrating the Euler equations is investigated. An explicit, third-order-accurate, noncentered, finite-difference scheme is used to compute unsteady flows about airfoils. Solutions for lifting and nonlifting airfoils are presented and compared with subsonic linear theory. The applicability and efficiency of the numerical indicial function method are outlined. Numerically computed subsonic and transonic oscillatory aerodynamic coefficients are presented and compared with those obtained from subsonic linear theory and transonic wind-tunnel data.

An approach to the development of numerical algorithms for first order linear hyperbolic systems in multiple space dimensions: The constant coefficient case

NASA Technical Reports Server (NTRS)

Goodrich, John W.

1995-01-01

Two methods for developing high order single step explicit algorithms on symmetric stencils with data on only one time level are presented. Examples are given for the convection and linearized Euler equations with up to the eighth order accuracy in both space and time in one space dimension, and up to the sixth in two space dimensions. The method of characteristics is generalized to nondiagonalizable hyperbolic systems by using exact local polynominal solutions of the system, and the resulting exact propagator methods automatically incorporate the correct multidimensional wave propagation dynamics. Multivariate Taylor or Cauchy-Kowaleskaya expansions are also used to develop algorithms. Both of these methods can be applied to obtain algorithms of arbitrarily high order for hyperbolic systems in multiple space dimensions. Cross derivatives are included in the local approximations used to develop the algorithms in this paper in order to obtain high order accuracy, and improved isotropy and stability. Efficiency in meeting global error bounds is an important criterion for evaluating algorithms, and the higher order algorithms are shown to be up to several orders of magnitude more efficient even though they are more complex. Stable high order boundary conditions for the linearized Euler equations are developed in one space dimension, and demonstrated in two space dimensions.

Stability Results for Idealized Shear Flows on a Rectangular Periodic Domain

NASA Astrophysics Data System (ADS)

Dullin, Holger R.; Worthington, Joachim

2018-06-01

We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain [0,2π )× [0,2π / κ ) for κ \\in R^+, the Euler equations admit a family of stationary solutions given by the vorticity profiles Ω ^*(x)= Γ cos (p_1x_1+ κ p_2x_2). We show linear stability for such flows when p_2=0 and κ ≥ |p_1| (equivalently p_1=0 and κ {|p_2|}≤ {1}). The classical result due to Arnold is that for p_1 = 1, p_2 = 0 and κ ≥ 1 the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for κ ≥ |p_1| ≥ 2, p_2 = 0 the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying p_1^2+κ ^2{p_2^2}>{3(κ ^2+1)}/4(7-4√{3)}. The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case κ ≠ 1. This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results.

Low-Dispersion Scheme for Nonlinear Acoustic Waves in Nonuniform Flow

NASA Technical Reports Server (NTRS)

Baysal, Oktay; Kaushik, Dinesh K.; Idres, Moumen

1997-01-01

The linear dispersion-relation-preserving scheme and its boundary conditions have been extended to the nonlinear Euler equations. This allowed computing, a nonuniform flowfield and a nonlinear acoustic wave propagation in such a medium, by the same scheme. By casting all the equations, boundary conditions, and the solution scheme in generalized curvilinear coordinates, the solutions were made possible for non-Cartesian domains and, for the better deployment of the grid points, nonuniform grid step sizes could be used. It has been tested for a number of simple initial-value and periodic-source problems. A simple demonstration of the difference between a linear and nonlinear propagation was conducted. The wall boundary condition, derived from the momentum equations and implemented through a pressure at a ghost point, and the radiation boundary condition, derived from the asymptotic solution to the Euler equations, have proven to be effective for the nonlinear equations and nonuniform flows. The nonreflective characteristic boundary conditions also have shown success but limited to the nonlinear waves in no mean flow, and failed for nonlinear waves in nonuniform flow.

Evaluating a linearized Euler equations model for strong turbulence effects on sound propagation.

PubMed

Ehrhardt, Loïc; Cheinet, Sylvain; Juvé, Daniel; Blanc-Benon, Philippe

2013-04-01

Sound propagation outdoors is strongly affected by atmospheric turbulence. Under strongly perturbed conditions or long propagation paths, the sound fluctuations reach their asymptotic behavior, e.g., the intensity variance progressively saturates. The present study evaluates the ability of a numerical propagation model based on the finite-difference time-domain solving of the linearized Euler equations in quantitatively reproducing the wave statistics under strong and saturated intensity fluctuations. It is the continuation of a previous study where weak intensity fluctuations were considered. The numerical propagation model is presented and tested with two-dimensional harmonic sound propagation over long paths and strong atmospheric perturbations. The results are compared to quantitative theoretical or numerical predictions available on the wave statistics, including the log-amplitude variance and the probability density functions of the complex acoustic pressure. The match is excellent for the evaluated source frequencies and all sound fluctuations strengths. Hence, this model captures these many aspects of strong atmospheric turbulence effects on sound propagation. Finally, the model results for the intensity probability density function are compared with a standard fit by a generalized gamma function.

The Prediction of Scattered Broadband Shock-Associated Noise

NASA Technical Reports Server (NTRS)

Miller, Steven A. E.

2015-01-01

A mathematical model is developed for the prediction of scattered broadband shock-associated noise. Model arguments are dependent on the vector Green's function of the linearized Euler equations, steady Reynolds-averaged Navier-Stokes solutions, and the two-point cross-correlation of the equivalent source. The equivalent source is dependent on steady Reynolds-averaged Navier-Stokes solutions of the jet flow, that capture the nozzle geometry and airframe surface. Contours of the time-averaged streamwise velocity component and turbulent kinetic energy are examined with varying airframe position relative to the nozzle exit. Propagation effects are incorporated by approximating the vector Green's function of the linearized Euler equations. This approximation involves the use of ray theory and an assumption that broadband shock-associated noise is relatively unaffected by the refraction of the jet shear layer. A non-dimensional parameter is proposed that quantifies the changes of the broadband shock-associated noise source with varying jet operating condition and airframe position. Scattered broadband shock-associated noise possesses a second set of broadband lobes that are due to the effect of scattering. Presented predictions demonstrate relatively good agreement compared to a wide variety of measurements.

Computational fluid dynamic modelling of cavitation

NASA Technical Reports Server (NTRS)

Deshpande, Manish; Feng, Jinzhang; Merkle, Charles L.

1993-01-01

Models in sheet cavitation in cryogenic fluids are developed for use in Euler and Navier-Stokes codes. The models are based upon earlier potential-flow models but enable the cavity inception point, length, and shape to be determined as part of the computation. In the present paper, numerical solutions are compared with experimental measurements for both pressure distribution and cavity length. Comparisons between models are also presented. The CFD model provides a relatively simple modification to an existing code to enable cavitation performance predictions to be included. The analysis also has the added ability of incorporating thermodynamic effects of cryogenic fluids into the analysis. Extensions of the current two-dimensional steady state analysis to three-dimensions and/or time-dependent flows are, in principle, straightforward although geometrical issues become more complicated. Linearized models, however offer promise of providing effective cavitation modeling in three-dimensions. This analysis presents good potential for improved understanding of many phenomena associated with cavity flows.

A Discrete Analysis of Non-reflecting Boundary Conditions for Discontinuous Galerkin Method

NASA Technical Reports Server (NTRS)

Hu, Fang Q.; Atkins, Harold L.

2003-01-01

We present a discrete analysis of non-reflecting boundary conditions for the discontinuous Galerkin method. The boundary conditions considered in this paper include the recently proposed Perfectly Matched Layer absorbing boundary condition for the linearized Euler equation and two non-reflecting boundary conditions based on the characteristic decomposition of the flux on the boundary. The analyses for the three boundary conditions are carried out in a unifled way. In each case, eigensolutions of the discrete system are obtained and applied to compute the numerical reflection coefficients of a specified out-going wave. The dependencies of the reflections at the boundary on the out-going wave angle and frequency as well as the mesh sizes arc? studied. Comparisons with direct numerical simulation results are also presented.

Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model

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

Yang, Xiaofeng, E-mail: xfyang@math.sc.edu; Han, Daozhi, E-mail: djhan@iu.edu

2017-02-01

In this paper, we develop a series of linear, unconditionally energy stable numerical schemes for solving the classical phase field crystal model. The temporal discretizations are based on the first order Euler method, the second order backward differentiation formulas (BDF2) and the second order Crank–Nicolson method, respectively. The schemes lead to linear elliptic equations to be solved at each time step, and the induced linear systems are symmetric positive definite. We prove that all three schemes are unconditionally energy stable rigorously. Various classical numerical experiments in 2D and 3D are performed to validate the accuracy and efficiency of the proposedmore » schemes.« less

Non-linear interaction of a detonation/vorticity wave

NASA Technical Reports Server (NTRS)

Lasseigne, D. G.; Jackson, T. L.; Hussaini, M. Y.

1991-01-01

The interaction of an oblique, overdriven detonation wave with a vorticity disturbance is investigated by a direct two-dimensional numerical simulation using a multi-domain, finite-difference solution of the compressible Euler equations. The results are compared to those of linear theory, which predict that the effect of exothermicity on the interaction is relatively small except possibly near a critical angle where linear theory no longer holds. It is found that the steady-state computational results agree with the results of linear theory. However, for cases with incident angle near the critical angle, moderate disturbance amplitudes, and/or sudden transient encounter with a disturbance, the effects of exothermicity are more pronounced than predicted by linear theory. Finally, it is found that linear theory correctly determines the critical angle.

Generation and Radiation of Acoustic Waves from a 2-D Shear Layer using the CE/SE Method

NASA Technical Reports Server (NTRS)

Loh, Ching Y.; Wang, Xiao Y.; Chang, Sin-Chung; Jorgenson, Philip C. E.

2000-01-01

In the present work, the generation and radiation of acoustic waves from a 2-D shear layer problem is considered. An acoustic source inside of a 2-D jet excites an instability wave in the shear layer, resulting in sound Mach radiation. The numerical solution is obtained by solving the Euler equations using the space time conservation element and solution element (CE/SE) method. Linearization is achieved through choosing a small acoustic source amplitude. The Euler equations are nondimensionalized as instructed in the problem statement. All other conditions are the same except that the Crocco's relation has a slightly different form. In the following, after a brief sketch of the CE/SE method, the numerical results for this problem are presented.

An interactive adaptive remeshing algorithm for the two-dimensional Euler equations

NASA Technical Reports Server (NTRS)

Slack, David C.; Walters, Robert W.; Lohner, R.

1990-01-01

An interactive adaptive remeshing algorithm utilizing a frontal grid generator and a variety of time integration schemes for the two-dimensional Euler equations on unstructured meshes is presented. Several device dependent interactive graphics interfaces have been developed along with a device independent DI-3000 interface which can be employed on any computer that has the supporting software including the Cray-2 supercomputers Voyager and Navier. The time integration methods available include: an explicit four stage Runge-Kutta and a fully implicit LU decomposition. A cell-centered finite volume upwind scheme utilizing Roe's approximate Riemann solver is developed. To obtain higher order accurate results a monotone linear reconstruction procedure proposed by Barth is utilized. Results for flow over a transonic circular arc and flow through a supersonic nozzle are examined.

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Chen, Gui-Qiang; Wang, Ya-Guang

2008-03-01

Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number M > sqrt{2} and instability when M < sqrt{2} ; and Artola and Majda’s analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when M > sqrt{2} . For the Euler equations in three dimensions, every compressible vortex sheet is violently unstable and this instability is the analogue of the Kelvin Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash Moser Hörmander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in three-dimensional MHD.

Multidisciplinary aeroelastic analysis of a generic hypersonic vehicle

NASA Technical Reports Server (NTRS)

Gupta, K. K.; Petersen, K. L.

1993-01-01

This paper presents details of a flutter and stability analysis of aerospace structures such as hypersonic vehicles. Both structural and aerodynamic domains are discretized by the common finite element technique. A vibration analysis is first performed by the STARS code employing a block Lanczos solution scheme. This is followed by the generation of a linear aerodynamic grid for subsequent linear flutter analysis within subsonic and supersonic regimes of the flight envelope; the doublet lattice and constant pressure techniques are employed to generate the unsteady aerodynamic forces. Flutter analysis is then performed for several representative flight points. The nonlinear flutter solution is effected by first implementing a CFD solution of the entire vehicle. Thus, a 3-D unstructured grid for the entire flow domain is generated by a moving front technique. A finite element Euler solution is then implemented employing a quasi-implicit as well as an explicit solution scheme. A novel multidisciplinary analysis is next effected that employs modal and aerodynamic data to yield aerodynamic damping characteristics. Such analyses are performed for a number of flight points to yield a large set of pertinent data that define flight flutter characteristics of the vehicle. This paper outlines the finite-element-based integrated analysis procedures in detail, which is followed by the results of numerical analyses of flight flutter simulation.

Waves in magnetized quark matter

NASA Astrophysics Data System (ADS)

Fogaça, D. A.; Sanches, S. M.; Navarra, F. S.

2018-05-01

We study wave propagation in a non-relativistic cold quark-gluon plasma immersed in a constant magnetic field. Starting from the Euler equation we derive linear wave equations and investigate their stability and causality. We use a generic form for the equation of state, the EOS derived from the MIT bag model and also a variant of the this model which includes gluon degrees of freedom. The results of this analysis may be relevant for perturbations propagating through the quark matter phase in the core of compact stars and also for perturbations propagating in the low temperature quark-gluon plasma formed in low energy heavy ion collisions, to be carried out at FAIR and NICA.

Canonical forms of multidimensional steady inviscid flows

NASA Technical Reports Server (NTRS)

Taasan, Shlomo

1993-01-01

Canonical forms and canonical variables for inviscid flow problems are derived. In these forms the components of the system governed by different types of operators (elliptic and hyperbolic) are separated. Both the incompressible and compressible cases are analyzed, and their similarities and differences are discussed. The canonical forms obtained are block upper triangular operator form in which the elliptic and non-elliptic parts reside in different blocks. The full nonlinear equations are treated without using any linearization process. This form enables a better analysis of the equations as well as better numerical treatment. These forms are the analog of the decomposition of the one dimensional Euler equations into characteristic directions and Riemann invariants.

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Coirier, William J.; Powell, Kenneth G.

1994-01-01

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: a gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Coirier, William J.; Powell, Kenneth G.

1995-01-01

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: A gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment and other accepted computational results for a series of low and moderate Reynolds number flows.

A general multiblock Euler code for propulsion integration. Volume 3: User guide for the Euler code

NASA Technical Reports Server (NTRS)

Chen, H. C.; Su, T. Y.; Kao, T. J.

1991-01-01

This manual explains the procedures for using the general multiblock Euler (GMBE) code developed under NASA contract NAS1-18703. The code was developed for the aerodynamic analysis of geometrically complex configurations in either free air or wind tunnel environments (vol. 1). The complete flow field is divided into a number of topologically simple blocks within each of which surface fitted grids and efficient flow solution algorithms can easily be constructed. The multiblock field grid is generated with the BCON procedure described in volume 2. The GMBE utilizes a finite volume formulation with an explicit time stepping scheme to solve the Euler equations. A multiblock version of the multigrid method was developed to accelerate the convergence of the calculations. This user guide provides information on the GMBE code, including input data preparations with sample input files and a sample Unix script for program execution in the UNICOS environment.

On the stability analysis of approximate factorization methods for 3D Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Demuren, A. O.; Ibraheem, S. O.

1993-01-01

The convergence characteristics of various approximate factorizations for the 3D Euler and Navier-Stokes equations are examined using the von-Neumann stability analysis method. Three upwind-difference based factorizations and several central-difference based factorizations are considered for the Euler equations. In the upwind factorizations both the flux-vector splitting methods of Steger and Warming and van Leer are considered. Analysis of the Navier-Stokes equations is performed only on the Beam and Warming central-difference scheme. The range of CFL numbers over which each factorization is stable is presented for one-, two-, and three-dimensional flow. Also presented for each factorization is the CFL number at which the maximum eigenvalue is minimized, for all Fourier components, as well as for the high frequency range only. The latter is useful for predicting the effectiveness of multigrid procedures with these schemes as smoothers. Further, local mode analysis is performed to test the suitability of using a uniform flow field in the stability analysis. Some inconsistencies in the results from previous analyses are resolved.

Development of a grid-independent approximate Riemannsolver. Ph.D. Thesis - Michigan Univ.

NASA Technical Reports Server (NTRS)

Rumsey, Christopher Lockwood

1991-01-01

A grid-independent approximate Riemann solver for use with the Euler and Navier-Stokes equations was introduced and explored. The two-dimensional Euler and Navier-Stokes equations are described in Cartesian and generalized coordinates, as well as the traveling wave form of the Euler equations. The spatial and temporal discretization are described for both explicit and implicit time-marching schemes. The grid-aligned flux function of Roe is outlined, while the 5-wave grid-independent flux function is derived. The stability and monotonicity analysis of the 5-wave model are presented. Two-dimensional results are provided and extended to three dimensions. The corresponding results are presented.

PROP3D: A Program for 3D Euler Unsteady Aerodynamic and Aeroelastic (Flutter and Forced Response) Analysis of Propellers. Version 1.0

NASA Technical Reports Server (NTRS)

Srivastava, R.; Reddy, T. S. R.

1996-01-01

This guide describes the input data required, for steady or unsteady aerodynamic and aeroelastic analysis of propellers and the output files generated, in using PROP3D. The aerodynamic forces are obtained by solving three dimensional unsteady, compressible Euler equations. A normal mode structural analysis is used to obtain the aeroelastic equations, which are solved using either time domain or frequency domain solution method. Sample input and output files are included in this guide for steady aerodynamic analysis of single and counter-rotation propellers, and aeroelastic analysis of single-rotation propeller.

Entropy Analysis of Kinetic Flux Vector Splitting Schemes for the Compressible Euler Equations

NASA Technical Reports Server (NTRS)

Shiuhong, Lui; Xu, Jun

1999-01-01

Flux Vector Splitting (FVS) scheme is one group of approximate Riemann solvers for the compressible Euler equations. In this paper, the discretized entropy condition of the Kinetic Flux Vector Splitting (KFVS) scheme based on the gas-kinetic theory is proved. The proof of the entropy condition involves the entropy definition difference between the distinguishable and indistinguishable particles.

A general multiblock Euler code for propulsion integration. Volume 1: Theory document

NASA Technical Reports Server (NTRS)

Chen, H. C.; Su, T. Y.; Kao, T. J.

1991-01-01

A general multiblock Euler solver was developed for the analysis of flow fields over geometrically complex configurations either in free air or in a wind tunnel. In this approach, the external space around a complex configuration was divided into a number of topologically simple blocks, so that surface-fitted grids and an efficient flow solution algorithm could be easily applied in each block. The computational grid in each block is generated using a combination of algebraic and elliptic methods. A grid generation/flow solver interface program was developed to facilitate the establishment of block-to-block relations and the boundary conditions for each block. The flow solver utilizes a finite volume formulation and an explicit time stepping scheme to solve the Euler equations. A multiblock version of the multigrid method was developed to accelerate the convergence of the calculations. The generality of the method was demonstrated through the analysis of two complex configurations at various flow conditions. Results were compared to available test data. Two accompanying volumes, user manuals for the preparation of multi-block grids (vol. 2) and for the Euler flow solver (vol. 3), provide information on input data format and program execution.

Analysis of real-time numerical integration methods applied to dynamic clamp experiments.

PubMed

Butera, Robert J; McCarthy, Maeve L

2004-12-01

Real-time systems are frequently used as an experimental tool, whereby simulated models interact in real time with neurophysiological experiments. The most demanding of these techniques is known as the dynamic clamp, where simulated ion channel conductances are artificially injected into a neuron via intracellular electrodes for measurement and stimulation. Methodologies for implementing the numerical integration of the gating variables in real time typically employ first-order numerical methods, either Euler or exponential Euler (EE). EE is often used for rapidly integrating ion channel gating variables. We find via simulation studies that for small time steps, both methods are comparable, but at larger time steps, EE performs worse than Euler. We derive error bounds for both methods, and find that the error can be characterized in terms of two ratios: time step over time constant, and voltage measurement error over the slope factor of the steady-state activation curve of the voltage-dependent gating variable. These ratios reliably bound the simulation error and yield results consistent with the simulation analysis. Our bounds quantitatively illustrate how measurement error restricts the accuracy that can be obtained by using smaller step sizes. Finally, we demonstrate that Euler can be computed with identical computational efficiency as EE.

Implicit flux-split Euler schemes for unsteady aerodynamic analysis involving unstructured dynamic meshes

NASA Technical Reports Server (NTRS)

Batina, John T.

1990-01-01

Improved algorithms for the solution of the time-dependent Euler equations are presented for unsteady aerodynamic analysis involving unstructured dynamic meshes. The improvements have been developed recently to the spatial and temporal discretizations used by unstructured grid flow solvers. The spatial discretization involves a flux-split approach which is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves an implicit time-integration shceme using a Gauss-Seidel relaxation procedure which is computationally efficient for either steady or unsteady flow problems. For example, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady and unsteady flow results are presented for the NACA 0012 airfoil to demonstrate applications of the new Euler solvers. The unsteady results were obtained for the airfoil pitching harmonically about the quarter chord. The resulting instantaneous pressure distributions and lift and moment coefficients during a cycle of motion compare well with experimental data. The paper presents a description of the Euler solvers along with results and comparisons which assess the capability.

Implicit flux-split Euler schemes for unsteady aerodynamic analysis involving unstructured dynamic meshes

NASA Technical Reports Server (NTRS)

Batina, John T.

1990-01-01

Improved algorithm for the solution of the time-dependent Euler equations are presented for unsteady aerodynamic analysis involving unstructured dynamic meshes. The improvements were developed recently to the spatial and temporal discretizations used by unstructured grid flow solvers. The spatial discretization involves a flux-split approach which is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves an implicit time-integration scheme using a Gauss-Seidel relaxation procedure which is computationally efficient for either steady or unsteady flow problems. For example, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady and unsteady flow results are presented for the NACA 0012 airfoil to demonstrate applications of the new Euler solvers. The unsteady results were obtained for the airfoil pitching harmonically about the quarter chord. The resulting instantaneous pressure distributions and lift and moment coefficients during a cycle of motion compare well with experimental data. A description of the Euler solvers is presented along with results and comparisons which assess the capability.

Couple stress theory of curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-01-01

New models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby, all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

User's Manual for DuctE3D: A Program for 3D Euler Unsteady Aerodynamic and Aeroelastic Analysis of Ducted Fans

NASA Technical Reports Server (NTRS)

Srivastava, R.; Reddy, T. S. R.

1997-01-01

The program DuctE3D is used for steady or unsteady aerodynamic and aeroelastic analysis of ducted fans. This guide describes the input data required and the output files generated, in using DuctE3D. The analysis solves three dimensional unsteady, compressible Euler equations to obtain the aerodynamic forces. A normal mode structural analysis is used to obtain the aeroelastic equations, which are solved using either the time domain or the frequency domain solution method. Sample input and output files are included in this guide for steady aerodynamic analysis and aeroelastic analysis of an isolated fan row.

Adjoint Method and Predictive Control for 1-D Flow in NASA Ames 11-Foot Transonic Wind Tunnel

NASA Technical Reports Server (NTRS)

Nguyen, Nhan; Ardema, Mark

2006-01-01

This paper describes a modeling method and a new optimal control approach to investigate a Mach number control problem for the NASA Ames 11-Foot Transonic Wind Tunnel. The flow in the wind tunnel is modeled by the 1-D unsteady Euler equations whose boundary conditions prescribe a controlling action by a compressor. The boundary control inputs to the compressor are in turn controlled by a drive motor system and an inlet guide vane system whose dynamics are modeled by ordinary differential equations. The resulting Euler equations are thus coupled to the ordinary differential equations via the boundary conditions. Optimality conditions are established by an adjoint method and are used to develop a model predictive linear-quadratic optimal control for regulating the Mach number due to a test model disturbance during a continuous pitch

Poly-Frobenius-Euler polynomials

NASA Astrophysics Data System (ADS)

Kurt, Burak

2017-07-01

Hamahata [3] defined poly-Euler polynomials and the generalized poly-Euler polynomials. He proved some relations and closed formulas for the poly-Euler polynomials. By this motivation, we define poly-Frobenius-Euler polynomials. We give some relations for this polynomials. Also, we prove the relationships between poly-Frobenius-Euler polynomials and Stirling numbers of the second kind.

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.

The extended Fourier pseudospectral time-domain method for atmospheric sound propagation.

PubMed

Hornikx, Maarten; Waxler, Roger; Forssén, Jens

2010-10-01

An extended Fourier pseudospectral time-domain (PSTD) method is presented to model atmospheric sound propagation by solving the linearized Euler equations. In this method, evaluation of spatial derivatives is based on an eigenfunction expansion. Evaluation on a spatial grid requires only two spatial points per wavelength. Time iteration is done using a low-storage optimized six-stage Runge-Kutta method. This method is applied to two-dimensional non-moving media models, one with screens and one for an urban canyon, with generally high accuracy in both amplitude and phase. For a moving atmosphere, accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen. The method has also been validated for three-dimensional sound propagation over a screen. For that application, the developed method is in the order of 100 times faster than the second-order-accurate FDTD solution to the linearized Euler equations. The method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated.

Numerical Boundary Conditions for Computational Aeroacoustics Benchmark Problems

NASA Technical Reports Server (NTRS)

Tam, Chritsopher K. W.; Kurbatskii, Konstantin A.; Fang, Jun

1997-01-01

Category 1, Problems 1 and 2, Category 2, Problem 2, and Category 3, Problem 2 are solved computationally using the Dispersion-Relation-Preserving (DRP) scheme. All these problems are governed by the linearized Euler equations. The resolution requirements of the DRP scheme for maintaining low numerical dispersion and dissipation as well as accurate wave speeds in solving the linearized Euler equations are now well understood. As long as 8 or more mesh points per wavelength is employed in the numerical computation, high quality results are assured. For the first three categories of benchmark problems, therefore, the real challenge is to develop high quality numerical boundary conditions. For Category 1, Problems 1 and 2, it is the curved wall boundary conditions. For Category 2, Problem 2, it is the internal radiation boundary conditions inside the duct. For Category 3, Problem 2, they are the inflow and outflow boundary conditions upstream and downstream of the blade row. These are the foci of the present investigation. Special nonhomogeneous radiation boundary conditions that generate the incoming disturbances and at the same time allow the outgoing reflected or scattered acoustic disturbances to leave the computation domain without significant reflection are developed. Numerical results based on these boundary conditions are provided.

A Zonal Approach for Prediction of Jet Noise

NASA Technical Reports Server (NTRS)

Shih, S. H.; Hixon, D. R.; Mankbadi, Reda R.

1995-01-01

A zonal approach for direct computation of sound generation and propagation from a supersonic jet is investigated. The present work splits the computational domain into a nonlinear, acoustic-source regime and a linear acoustic wave propagation regime. In the nonlinear regime, the unsteady flow is governed by the large-scale equations, which are the filtered compressible Navier-Stokes equations. In the linear acoustic regime, the sound wave propagation is described by the linearized Euler equations. Computational results are presented for a supersonic jet at M = 2. 1. It is demonstrated that no spurious modes are generated in the matching region and the computational expense is reduced substantially as opposed to fully large-scale simulation.

Three Dimensional Aerodynamic Analysis of a High-Lift Transport Configuration

NASA Technical Reports Server (NTRS)

Dodbele, Simha S.

1993-01-01

Two computational methods, a surface panel method and an Euler method employing unstructured grid methodology, were used to analyze a subsonic transport aircraft in cruise and high-lift conditions. The computational results were compared with two separate sets of flight data obtained for the cruise and high-lift configurations. For the cruise configuration, the surface pressures obtained by the panel method and the Euler method agreed fairly well with results from flight test. However, for the high-lift configuration considerable differences were observed when the computational surface pressures were compared with the results from high-lift flight test. On the lower surface of all the elements with the exception of the slat, both the panel and Euler methods predicted pressures which were in good agreement with flight data. On the upper surface of all the elements the panel method predicted slightly higher suction compared to the Euler method. On the upper surface of the slat, pressure coefficients obtained by both the Euler and panel methods did not agree with the results of the flight tests. A sensitivity study of the upward deflection of the slat from the 40 deg. flap setting suggested that the differences in the slat deflection between the computational model and the flight configuration could be one of the sources of this discrepancy. The computation time for the implicit version of the Euler code was about 1/3 the time taken by the explicit version though the implicit code required 3 times the memory taken by the explicit version.

Wave Propagation Analysis of Edge Cracked Circular Beams under Impact Force

PubMed Central

Akbaş, Şeref Doğuşcan

2014-01-01

This paper presents responses of an edge circular cantilever beam under the effect of an impact force. The beam is excited by a transverse triangular force impulse modulated by a harmonic motion. The Kelvin–Voigt model for the material of the beam is used. The cracked beam is modelled as an assembly of two sub-beams connected through a massless elastic rotational spring. The considered problem is investigated within the Bernoulli-Euler beam theory by using energy based finite element method. The system of equations of motion is derived by using Lagrange's equations. The obtained system of linear differential equations is reduced to a linear algebraic equation system and solved in the time domain by using Newmark average acceleration method. In the study, the effects of the location of crack, the depth of the crack, on the characteristics of the reflected waves are investigated in detail. Also, the positions of the cracks are calculated by using reflected waves. PMID:24972050

Interaction of a shock with a longitudinal vortex

NASA Technical Reports Server (NTRS)

Erlebacher, Gordon; Hussaini, M. Y.; Shu, Chi-Wang

1996-01-01

In this paper we study the shock/longitudinal vortex interaction problem in axisymmetric geometry. Linearized analysis for small vortex strength is performed, and compared with results from a high order axisymmetric shock-fitted Euler solution obtained for this purpose. It is confirmed that for weak vortices, predictions from linear theory agree well with results from nonlinear numerical simulations at the shock location. To handle very strong longitudinal vortices, which may ultimately break the shock, we use an axisymmetric high order essentially non-oscillatory (ENO) shock capturing scheme. Comparison of shock-captured and shock-fitted results are performed in their regions of common validity. We also study the vortex breakdown as a function of Mach number ranging from 1.3 to 10, thus extending the range of existing results. For vortex strengths above a critical value. a triple point forms on the shock and a secondary shock forms to provide the necessary deceleration so that the fluid velocity can adjust to downstream conditions at the shock.

Incompressible limit of the degenerate quantum compressible Navier-Stokes equations with general initial data

NASA Astrophysics Data System (ADS)

Kwon, Young-Sam; Li, Fucai

2018-03-01

In this paper we study the incompressible limit of the degenerate quantum compressible Navier-Stokes equations in a periodic domain T3 and the whole space R3 with general initial data. In the periodic case, by applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of velocity, we prove rigorously that the gradient part of the weak solutions (velocity) of the degenerate quantum compressible Navier-Stokes equations converge to the strong solution of the incompressible Navier-Stokes equations. Our results improve considerably the ones obtained by Yang, Ju and Yang [25] where only the well-prepared initial data case is considered. While for the whole space case, thanks to the Strichartz's estimates of linear wave equations, we can obtain the convergence of the weak solutions of the degenerate quantum compressible Navier-Stokes equations to the strong solution of the incompressible Navier-Stokes/Euler equations with a linear damping term. Moreover, the convergence rates are also given.

Bending, longitudinal and torsional wave transmission on Euler-Bernoulli and Timoshenko beams with high propagation losses.

PubMed

Wang, X; Hopkins, C

2016-10-01

Advanced Statistical Energy Analysis (ASEA) is used to predict vibration transmission across coupled beams which support multiple wave types up to high frequencies where Timoshenko theory is valid. Bending-longitudinal and bending-torsional models are considered for an L-junction and rectangular beam frame. Comparisons are made with measurements, Finite Element Methods (FEM) and Statistical Energy Analysis (SEA). When beams support at least two local modes for each wave type in a frequency band and the modal overlap factor is at least 0.1, measurements and FEM have relatively smooth curves. Agreement between measurements, FEM, and ASEA demonstrates that ASEA is able to predict high propagation losses which are not accounted for with SEA. These propagation losses tend to become more important at high frequencies with relatively high internal loss factors and can occur when there is more than one wave type. At such high frequencies, Timoshenko theory, rather than Euler-Bernoulli theory, is often required. Timoshenko theory is incorporated in ASEA and SEA using wave theory transmission coefficients derived assuming Euler-Bernoulli theory, but using Timoshenko group velocity when calculating coupling loss factors. The changeover between theories is appropriate above the frequency where there is a 26% difference between Euler-Bernoulli and Timoshenko group velocities.

New fundamental parameters for attitude representation

NASA Astrophysics Data System (ADS)

Patera, Russell P.

2017-08-01

A new attitude parameter set is developed to clarify the geometry of combining finite rotations in a rotational sequence and in combining infinitesimal angular increments generated by angular rate. The resulting parameter set of six Pivot Parameters represents a rotation as a great circle arc on a unit sphere that can be located at any clocking location in the rotation plane. Two rotations are combined by linking their arcs at either of the two intersection points of the respective rotation planes. In a similar fashion, linking rotational increments produced by angular rate is used to derive the associated kinematical equations, which are linear and have no singularities. Included in this paper is the derivation of twelve Pivot Parameter elements that represent all twelve Euler Angle sequences, which enables efficient conversions between Pivot Parameters and any Euler Angle sequence. Applications of this new parameter set include the derivation of quaternions and the quaternion composition rule, as well as, the derivation of the analytical solution to time dependent coning motion. The relationships between Pivot Parameters and traditional parameter sets are included in this work. Pivot Parameters are well suited for a variety of aerospace applications due to their effective composition rule, singularity free kinematic equations, efficient conversion to and from Euler Angle sequences and clarity of their geometrical foundation.

Conical Euler analysis and active roll suppression for unsteady vortical flows about rolling delta wings

NASA Technical Reports Server (NTRS)

Lee-Rausch, Elizabeth M.; Batina, John T.

1993-01-01

A conical Euler code was developed to study unsteady vortex-dominated flows about rolling, highly swept delta wings undergoing either forced motions or free-to-roll motions that include active roll suppression. The flow solver of the code involves a multistage, Runge-Kutta time-stepping scheme that uses a cell-centered, finite-volume, spatial discretization of the Euler equations on an unstructured grid of triangles. The code allows for the additional analysis of the free to-roll case by simultaneously integrating in time the rigid-body equation of motion with the governing flow equations. Results are presented for a delta wing with a 75 deg swept, sharp leading edge at a free-stream Mach number of 1.2 and at 10 deg, 20 deg, and 30 deg angle of attack alpha. At the lower angles of attack (10 and 20 deg), forced-harmonic analyses indicate that the rolling-moment coefficients provide a positive damping, which is verified by free-to-roll calculations. In contrast, at the higher angle of attack (30 deg), a forced-harmonic analysis indicates that the rolling-moment coefficient provides negative damping at the small roll amplitudes. A free-to-roll calculation for this case produces an initially divergent response, but as the amplitude of motion grows with time, the response transitions to a wing-rock type of limit cycle oscillation, which is characteristic of highly swept delta wings. This limit cycle oscillation may be actively suppressed through the use of a rate-feedback control law and antisymmetrically deflected leading-edge flaps. Descriptions of the conical Euler flow solver and the free-to roll analysis are included in this report. Results are presented that demonstrate how the systematic analysis of the forced response of the delta wing can be used to predict the stable, neutrally stable, and unstable free response of the delta wing. These results also give insight into the flow physics associated with unsteady vortical flows about delta wings undergoing forced motions and free-to-roll motions, including the active suppression of the wing-rock type phenomenon. The conical Euler methodology developed is directly extend able to three-dimensional calculations.

Congruence Approximations for Entrophy Endowed Hyperbolic Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy J.; Saini, Subhash (Technical Monitor)

1998-01-01

Building upon the standard symmetrization theory for hyperbolic systems of conservation laws, congruence properties of the symmetrized system are explored. These congruence properties suggest variants of several stabilized numerical discretization procedures for hyperbolic equations (upwind finite-volume, Galerkin least-squares, discontinuous Galerkin) that benefit computationally from congruence approximation. Specifically, it becomes straightforward to construct the spatial discretization and Jacobian linearization for these schemes (given a small amount of derivative information) for possible use in Newton's method, discrete optimization, homotopy algorithms, etc. Some examples will be given for the compressible Euler equations and the nonrelativistic MHD equations using linear and quadratic spatial approximation.

A method for exponential propagation of large systems of stiff nonlinear differential equations

NASA Technical Reports Server (NTRS)

Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright C.; Russo, Thomas V.

1989-01-01

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

Dynamic modelling and control of a rotating Euler-Bernoulli beam

NASA Astrophysics Data System (ADS)

Yang, J. B.; Jiang, L. J.; Chen, D. CH.

2004-07-01

Flexible motion of a uniform Euler-Bernoulli beam attached to a rotating rigid hub is investigated. Fully coupled non-linear integro-differential equations, describing axial, transverse and rotational motions of the beam, are derived by using the extended Hamilton's principle. The centrifugal stiffening effect is included in the derivation. A finite-dimensional model, including couplings of axial and transverse vibrations, and of elastic deformations and rigid motions, is obtained by the finite element method. By neglecting the axial motion, a simplified modelling, suitable for studying the transverse vibration and control of a beam with large angle and high-speed rotation, is presented. And suppressions of transverse vibrations of a rotating beam are simulated with the model by combining positive position feedback and momentum exchange feedback control laws. It is indicated that an improved performance for vibration control can be achieved with the method.

Least-squares finite element methods for compressible Euler equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Carey, G. F.

1990-01-01

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

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.

Solution of the surface Euler equations for accurate three-dimensional boundary-layer analysis of aerodynamic configurations

NASA Technical Reports Server (NTRS)

Iyer, V.; Harris, J. E.

1987-01-01

The three-dimensional boundary-layer equations in the limit as the normal coordinate tends to infinity are called the surface Euler equations. The present paper describes an accurate method for generating edge conditions for three-dimensional boundary-layer codes using these equations. The inviscid pressure distribution is first interpolated to the boundary-layer grid. The surface Euler equations are then solved with this pressure field and a prescribed set of initial and boundary conditions to yield the velocities along the two surface coordinate directions. Results for typical wing and fuselage geometries are presented. The smoothness and accuracy of the edge conditions obtained are found to be superior to the conventional interpolation procedures.

The Magellan seamount trail: implications for Cretaceous hotspot volcanism and absolute Pacific plate motion

NASA Astrophysics Data System (ADS)

Koppers, Anthony A. P.; Staudigel, Hubert; Wijbrans, Jan R.; Pringle, Malcolm S.

1998-11-01

The Magellan Seamount Trail (MST) delineates a northwest trending chain of four Cretaceous guyots in the West Pacific Seamount Province (WPSP). Seamount morphology, 40Ar/ 39Ar geochronology and Sr-Nd-Pb geochemistry of the MST provides evidence for a hotspot origin between the Samoa, Rarotonga and Society hotspots of the South Pacific Isotopic and Thermal Anomaly (SOPITA). The MST yields an excellent linear age progression of 47.6±1.6 mm/yr ( r2=1.000; MSWD = 0.23; 1 σ SE) including Vlinder guyot (95.1±0.5 Ma, n=5; 2 σ SD), Pako guyot (91.3±0.3 Ma, n=3) and Ioah guyot (87.1±0.3 Ma, n=2). The MST also exhibits a small range in Sr-Nd-Pb isotopic compositions indicating enriched mantle sources with an affinity of EMI. Nevertheless, three volcanic events are found out of sequence with linear MST hotspot volcanism: (1) an independent volcanic pedestal was formed 4-7 Myr before shield-volcanism started at Vlinder guyot, (2) a post-erosional volcanic cone was formed at least 20-30 Myr after drowning of Vlinder guyot, and (3) Ita Mai Tai guyot (118.1±0.5 Ma, n=3) was formed 34-36 Myr before the MST hotspot arrived at the predicted location of this guyot. By identifying and ruling out discordant volcanic events, we can use the age progression in MST to test the fixity of its hotspot. When presuming the fixed hotspot hypothesis, the local age progressions of the MST (47.6±1.6 mm/yr) and the copolar Musicians seamount trail (55.8±6.4 mm/yr) are not compatible with their 100-80 Ma Euler pole. We investigate two options: (1) acceptance of a `forced' Euler pole obeying the hotspot hypothesis by using both the age progressions and the azimuths of the studied seamount trails, or (2) acceptance of a `best-fit' Euler pole by using the azimuths of the studied seamount trail exclusively. In the first option, the angular speed of the Pacific plate during the 100-80 Ma stage pole is calculated at 0.502±0.017°/Myr. In the second option, the `best-fit' Euler pole is found approximately 35° different from the `forced' Euler pole. We argue that the observed age progressions can only be reconciled with the `best-fit' pole when allowing for the relative movement of the MST and Musicians mantle plumes with respect to one another. The calculated maximum velocity component parallel to the line of age progression could then be as much as 23 mm/yr for the mantle plumes — when assuming one fixed hotspot in this alternate model.

Nonlocal theory of curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-09-01

New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.

Instability waves and low-frequency noise radiation in the subsonic chevron jet

NASA Astrophysics Data System (ADS)

Ran, Lingke; Ye, Chuangchao; Wan, Zhenhua; Yang, Haihua; Sun, Dejun

2017-11-01

Spatial instability waves associated with low-frequency noise radiation at shallow polar angles in the chevron jet are investigated and are compared to the round counterpart. The Reynolds-averaged Navier-Stokes equations are solved to obtain the mean flow fields, which serve as the baseflow for linear stability analysis. The chevron jet has more complicated instability waves than the round jet, where three types of instability modes are identified in the vicinity of the nozzle, corresponding to radial shear, azimuthal shear, and their integrated effect of the baseflow, respectively. The most unstable frequency of all chevron modes and round modes in both jets decrease as the axial location moves downstream. Besides, the azimuthal shear effect related modes are more unstable than radial shear effect related modes at low frequencies. Compared to a round jet, a chevron jet reduces the growth rate of the most unstable modes at downstream locations. Moreover, linearized Euler equations are employed to obtain the beam pattern of pressure generated by spatially evolving instability waves at a dominant low frequency St=0.3 , and the acoustic efficiencies of these linear wavepackets are evaluated for both jets. It is found that the acoustic efficiency of linear wavepacket is able to be reduced greatly in the chevron jet, compared to the round jet.

Instability waves and low-frequency noise radiation in the subsonic chevron jet

NASA Astrophysics Data System (ADS)

Ran, Lingke; Ye, Chuangchao; Wan, Zhenhua; Yang, Haihua; Sun, Dejun

2018-06-01

Spatial instability waves associated with low-frequency noise radiation at shallow polar angles in the chevron jet are investigated and are compared to the round counterpart. The Reynolds-averaged Navier-Stokes equations are solved to obtain the mean flow fields, which serve as the baseflow for linear stability analysis. The chevron jet has more complicated instability waves than the round jet, where three types of instability modes are identified in the vicinity of the nozzle, corresponding to radial shear, azimuthal shear, and their integrated effect of the baseflow, respectively. The most unstable frequency of all chevron modes and round modes in both jets decrease as the axial location moves downstream. Besides, the azimuthal shear effect related modes are more unstable than radial shear effect related modes at low frequencies. Compared to a round jet, a chevron jet reduces the growth rate of the most unstable modes at downstream locations. Moreover, linearized Euler equations are employed to obtain the beam pattern of pressure generated by spatially evolving instability waves at a dominant low frequency St=0.3, and the acoustic efficiencies of these linear wavepackets are evaluated for both jets. It is found that the acoustic efficiency of linear wavepacket is able to be reduced greatly in the chevron jet, compared to the round jet.

Eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method

NASA Astrophysics Data System (ADS)

Bekhoucha, F.; Rechak, S.; Cadou, J. M.

2016-12-01

In this paper, free vibrations of a rotating clamped Euler-Bernoulli beams with uniform cross section are studied using continuation method, namely asymptotic numerical method. The governing equations of motion are derived using Lagrange's method. The kinetic and strain energy expression are derived from Rayleigh-Ritz method using a set of hybrid variables and based on a linear deflection assumption. The derived equations are transformed in two eigenvalue problems, where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms. While the second is standard eigenvalue problem and corresponds to the flapping motion. Those two eigenvalue problems are transformed into two functionals treated by continuation method, the Asymptotic Numerical Method. New method proposed for the solution of the linear gyroscopic system based on an augmented system, which transforms the original problem to a standard form with real symmetric matrices. By using some techniques to resolve these singular problems by the continuation method, evolution curves of the natural frequencies against dimensionless angular velocity are determined. At high angular velocity, some singular points, due to the linear elastic assumption, are computed. Numerical tests of convergence are conducted and the obtained results are compared to the exact values. Results obtained by continuation are compared to those computed with discrete eigenvalue problem.

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially.

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.

Semi-discrete Galerkin solution of the compressible boundary-layer equations with viscous-inviscid interaction

NASA Technical Reports Server (NTRS)

Day, Brad A.; Meade, Andrew J., Jr.

1993-01-01

A semi-discrete Galerkin (SDG) method is under development to model attached, turbulent, and compressible boundary layers for transonic airfoil analysis problems. For the boundary-layer formulation the method models the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby providing high resolution near the wall and permitting the use of a uniform finite element grid which automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past RAE 2822 and NACA 0012 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack.

Regularized estimation of Euler pole parameters

NASA Astrophysics Data System (ADS)

Aktuğ, Bahadir; Yildirim, Ömer

2013-07-01

Euler vectors provide a unified framework to quantify the relative or absolute motions of tectonic plates through various geodetic and geophysical observations. With the advent of space geodesy, Euler parameters of several relatively small plates have been determined through the velocities derived from the space geodesy observations. However, the available data are usually insufficient in number and quality to estimate both the Euler vector components and the Euler pole parameters reliably. Since Euler vectors are defined globally in an Earth-centered Cartesian frame, estimation with the limited geographic coverage of the local/regional geodetic networks usually results in highly correlated vector components. In the case of estimating the Euler pole parameters directly, the situation is even worse, and the position of the Euler pole is nearly collinear with the magnitude of the rotation rate. In this study, a new method, which consists of an analytical derivation of the covariance matrix of the Euler vector in an ideal network configuration, is introduced and a regularized estimation method specifically tailored for estimating the Euler vector is presented. The results show that the proposed method outperforms the least squares estimation in terms of the mean squared error.

An Application of the A* Search to Trajectory Optimization

DTIC Science & Technology

1990-05-11

linearized model of orbital motion called the Clohessy - Wiltshire Equations and a node search technique called A*. The planner discussed in this thesis starts...states while transfer time is left unspecified. 13 Chapter 2. Background HILL’S ( CLOHESSY - WILTSHIRE ) EQUATIONS The Euler-Hill equations describe... Clohessy - Wiltshire equations. The coordinate system used in this thesis is commonly referred to as Local Vertical, Local Horizontal or LVLH reference frame

Micropolar curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-01-01

New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and special hypothesis based on assumptions that take into account the fact that the rod is thin.High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First stress and strain tensors,vectors of displacements and rotation and body force shave been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking in to account micropolar couple stress and rotation effects.

The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations

NASA Astrophysics Data System (ADS)

Wei, Changhua

2017-10-01

This paper investigates the lower bound of the lifespan of three-dimensional spherically symmetric solutions to the non-isentropic relativistic Euler equations, when the initial data are prescribed as a small perturbation with compact support to a constant state. Based on the structure of the hyperbolic system, we show the almost global existence of the smooth solutions to Eulerian flows (polytropic gases and generalized Chaplygin gases) with genuinely nonlinear characteristics. While for the Eulerian flows (Chaplygin gas and stiff matter) with mild linearly degenerate characteristics, we show the global existence of the radial solutions, moreover, for the non-strictly hyperbolic system (pressureless perfect fluid) satisfying the mild linearly degenerate condition, we prove the blowup phenomenon of the radial solutions and show that the lifespan of the solutions is of order O(ɛ ^{-1}), where ɛ denotes the width of the perturbation. This work can be seen as a complement of our work (Lei and Wei in Math Ann 367:1363-1401, 2017) for relativistic Chaplygin gas and can also be seen as a generalization of the classical Eulerian fluids (Godin in Arch Ration Mech Anal 177:497-511, 2005, J Math Pures Appl 87:91-117, 2007) to the relativistic Eulerian fluids.

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

Mendenhall, M.R.

The present volume discusses tactical missile aerodynamic drag, drag-prediction methods for axisymmetric missile bodies, an aerodynamic heating analysis for supersonic missiles, a component buildup method for engineering analysis of missiles at low-to-high angles of attack, experimental and analytical methods for missiles with noncircular fuselages, and a vortex-cloud model for body vortex shedding and tracking. Also discussed are panel methods with vorticity effects and corrections for nonlinear compressibility, supersonic full-potential methods for missile body analysis, space-marching Euler solvers, the time-asymptotic Euler/Navier-Stokes methods for subsonic and transonic flows, 3D boundary layers on missiles, Navier-Stokes analyses of flows over slender airframes, and themore » interaction of exhaust plumes with missile airframes.« less

Developing and utilizing an Euler computational method for predicting the airframe/propulsion effects for an aft-mounted turboprop transport. Volume 2: User guide

NASA Technical Reports Server (NTRS)

Chen, H. C.; Neback, H. E.; Kao, T. J.; Yu, N. Y.; Kusunose, K.

1991-01-01

This manual explains how to use an Euler based computational method for predicting the airframe/propulsion integration effects for an aft-mounted turboprop transport. The propeller power effects are simulated by the actuator disk concept. This method consists of global flow field analysis and the embedded flow solution for predicting the detailed flow characteristics in the local vicinity of an aft-mounted propfan engine. The computational procedure includes the use of several computer programs performing four main functions: grid generation, Euler solution, grid embedding, and streamline tracing. This user's guide provides information for these programs, including input data preparations with sample input decks, output descriptions, and sample Unix scripts for program execution in the UNICOS environment.

Euler Technology Assessment - SPLITFLOW Code Applications for Stability and Control Analysis on an Advanced Fighter Model Employing Innovative Control Concepts

NASA Technical Reports Server (NTRS)

Jordan, Keith J.

1998-01-01

This report documents results from the NASA-Langley sponsored Euler Technology Assessment Study conducted by Lockheed-Martin Tactical Aircraft Systems (LMTAS). The purpose of the study was to evaluate the ability of the SPLITFLOW code using viscous and inviscid flow models to predict aerodynamic stability and control of an advanced fighter model. The inviscid flow model was found to perform well at incidence angles below approximately 15 deg, but not as well at higher angles of attack. The results using a turbulent, viscous flow model matched the trends of the wind tunnel data, but did not show significant improvement over the Euler solutions. Overall, the predictions were found to be useful for stability and control design purposes.

A free boundary approach to the Rosensweig instability of ferrofluids

NASA Astrophysics Data System (ADS)

Parini, Enea; Stylianou, Athanasios

2018-04-01

We establish the existence of saddle points for a free boundary problem describing the two-dimensional free surface of a ferrofluid undergoing normal field instability. The starting point is the ferrohydrostatic equations for the magnetic potentials in the ferrofluid and air, and the function describing their interface. These constitute the strong form for the Euler-Lagrange equations of a convex-concave functional, which we extend to include interfaces that are not necessarily graphs of functions. Saddle points are then found by iterating the direct method of the calculus of variations and applying classical results of convex analysis. For the existence part, we assume a general nonlinear magnetization law; for a linear law, we also show, via convex duality, that the saddle point is a constrained minimizer of the relevant energy functional.

An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations. Ph.D. Thesis - Michigan Univ.

NASA Technical Reports Server (NTRS)

Coirier, William John

1994-01-01

A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal 'cut' cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a different formulation of the viscous terms are shown to be necessary. A hybrid Cartesian/body-fitted grid generation approach is demonstrated. In addition, a grid-generation procedure based on body-aligned cell cutting coupled with a viscous stensil-construction procedure based on quadratic programming is presented.

Investigation of instabilities affecting detonations: Improving the resolution using block-structured adaptive mesh refinement

NASA Astrophysics Data System (ADS)

Ravindran, Prashaanth

The unstable nature of detonation waves is a result of the critical relationship between the hydrodynamic shock and the chemical reactions sustaining the shock. A perturbative analysis of the critical point is quite challenging due to the multiple spatio-temporal scales involved along with the non-linear nature of the shock-reaction mechanism. The author's research attempts to provide detailed resolution of the instabilities at the shock front. Another key aspect of the present research is to develop an understanding of the causality between the non-linear dynamics of the front and the eventual breakdown of the sub-structures. An accurate numerical simulation of detonation waves requires a very efficient solution of the Euler equations in conservation form with detailed, non-equilibrium chemistry. The difference in the flow and reaction length scales results in very stiff source terms, requiring the problem to be solved with adaptive mesh refinement. For this purpose, Berger-Colella's block-structured adaptive mesh refinement (AMR) strategy has been developed and applied to time-explicit finite volume methods. The block-structured technique uses a hierarchy of parent-child sub-grids, integrated recursively over time. One novel approach to partition the problem within a large supercomputer was the use of modified Peano-Hilbert space filling curves. The AMR framework was merged with CLAWPACK, a package providing finite volume numerical methods tailored for wave-propagation problems. The stiffness problem is bypassed by using a 1st order Godunov or a 2nd order Strang splitting technique, where the flow variables and source terms are integrated independently. A linearly explicit fourth-order Runge-Kutta integrator is used for the flow, and an ODE solver was used to overcome the numerical stiffness. Second-order spatial resolution is obtained by using a second-order Roe-HLL scheme with the inclusion of numerical viscosity to stabilize the solution near the discontinuity. The scheme is made monotonic by coupling the van Albada limiter with the higher order MUSCL-Hancock extrapolation to the primitive variables of the Euler equations. Simulations using simplified single-step and detailed chemical kinetics have been provided. In detonations with simplified chemistry, the one-dimensional longitudinal instabilities have been simulated, and a mechanism forcing the collapse of the period-doubling modes was identified. The transverse instabilities were simulated for a 2D detonation, and the corresponding transverse wave was shown to be unstable with a periodic normal mode. Also, a Floquet analysis was carried out with the three-dimensional inviscid Euler equations for a longitudinally stable case. Using domain decomposition to identify the global eigenfunctions corresponding to the two least stable eigenvalues, it was found that the bifurcation of limit cycles in three dimensions follows a period doubling process similar to that proven to occur in one dimension and it is because of transverse instabilities. For detonations with detailed chemistry, the one dimensional simulations for two cases were presented and validated with experimental results. The 2D simulation shows the re-initiation of the triple point leading to the formation of cellular structure of the detonation wave. Some of the important features in the front were identified and explained.

Stress analysis of rotating propellers subject to forced excitations

NASA Astrophysics Data System (ADS)

Akgun, Ulas

Turbine blades experience vibrations due to the flow disturbances. These vibrations are the leading cause for fatigue failure in turbine blades. This thesis presents the finite element analysis methods to estimate the maximum vibrational stresses of rotating structures under forced excitation. The presentation included starts with the derived equations of motion for vibration of rotating beams using energy methods under the Euler Bernoulli beam assumptions. The nonlinear large displacement formulation captures the centrifugal stiffening and gyroscopic effects. The weak form of the equations and their finite element discretization are shown. The methods implemented were used for normal modes analyses and forced vibration analyses of rotating beam structures. The prediction of peak stresses under simultaneous multi-mode excitation show that the maximum vibrational stresses estimated using the linear superposition of the stresses can greatly overestimate the stresses if the phase information due to damping (physical and gyroscopic effects) are neglected. The last section of this thesis also presents the results of a practical study that involves finite element analysis and redesign of a composite propeller.

Investigation of advanced counterrotation blade configuration concepts for high speed turboprop systems, task 1: Ducted propfan analysis

NASA Technical Reports Server (NTRS)

Hall, Edward J.; Delaney, Robert A.; Bettner, James L.

1990-01-01

The time-dependent three-dimensional Euler equations of gas dynamics were solved numerically to study the steady compressible transonic flow about ducted propfan propulsion systems. Aerodynamic calculations were based on a four-stage Runge-Kutta time-marching finite volume solution technique with added numerical dissipation. An implicit residual smoothing operator was used to aid convergence. Two calculation grids were employed in this study. The first grid utilized an H-type mesh network with a branch cut opening to represent the axisymmetric cowl. The second grid utilized a multiple-block mesh system with a C-type grid about the cowl. The individual blocks were numerically coupled in the Euler solver. Grid systems were generated by a combined algebraic/elliptic algortihm developed specifically for ducted propfans. Numerical calculations were initially performed for unducted propfans to verify the accuracy of the three-dimensional Euler formulation. The Euler analyses were then applied for the calculation of ducted propfan flows, and predicted results were compared with experimental data for two cases. The three-dimensional Euler analyses displayed exceptional accuracy, although certain parameters were observed to be very sensitive to geometric deflections. Both solution schemes were found to be very robust and demonstrated nearly equal efficiency and accuracy, although it was observed that the multi-block C-grid formulation provided somewhat better resolution of the cowl leading edge region.

Numerical simulation of the fluid-structure interaction between air blast waves and soil structure

NASA Astrophysics Data System (ADS)

Umar, S.; Risby, M. S.; Albert, A. Luthfi; Norazman, M.; Ariffin, I.; Alias, Y. Muhamad

2014-03-01

Normally, an explosion threat on free field especially from high explosives is very dangerous due to the ground shocks generated that have high impulsive load. Nowadays, explosion threats do not only occur in the battlefield, but also in industries and urban areas. In industries such as oil and gas, explosion threats may occur on logistic transportation, maintenance, production, and distribution pipeline that are located underground to supply crude oil. Therefore, the appropriate blast resistances are a priority requirement that can be obtained through an assessment on the structural response, material strength and impact pattern of material due to ground shock. A highly impulsive load from ground shocks is a dynamic load due to its loading time which is faster than ground response time. Of late, almost all blast studies consider and analyze the ground shock in the fluid-structure interaction (FSI) because of its influence on the propagation and interaction of ground shock. Furthermore, analysis in the FSI integrates action of ground shock and reaction of ground on calculations of velocity, pressure and force. Therefore, this integration of the FSI has the capability to deliver the ground shock analysis on simulation to be closer to experimental investigation results. In this study, the FSI was implemented on AUTODYN computer code by using Euler-Godunov and the arbitrary Lagrangian-Eulerian (ALE). Euler-Godunov has the capability to deliver a structural computation on a 3D analysis, while ALE delivers an arbitrary calculation that is appropriate for a FSI analysis. In addition, ALE scheme delivers fine approach on little deformation analysis with an arbitrary motion, while the Euler-Godunov scheme delivers fine approach on a large deformation analysis. An integrated scheme based on Euler-Godunov and the arbitrary Lagrangian-Eulerian allows us to analyze the blast propagation waves and structural interaction simultaneously.

A highly parallel multigrid-like method for the solution of the Euler equations

NASA Technical Reports Server (NTRS)

Tuminaro, Ray S.

1989-01-01

We consider a highly parallel multigrid-like method for the solution of the two dimensional steady Euler equations. The new method, introduced as filtering multigrid, is similar to a standard multigrid scheme in that convergence on the finest grid is accelerated by iterations on coarser grids. In the filtering method, however, additional fine grid subproblems are processed concurrently with coarse grid computations to further accelerate convergence. These additional problems are obtained by splitting the residual into a smooth and an oscillatory component. The smooth component is then used to form a coarse grid problem (similar to standard multigrid) while the oscillatory component is used for a fine grid subproblem. The primary advantage in the filtering approach is that fewer iterations are required and that most of the additional work per iteration can be performed in parallel with the standard coarse grid computations. We generalize the filtering algorithm to a version suitable for nonlinear problems. We emphasize that this generalization is conceptually straight-forward and relatively easy to implement. In particular, no explicit linearization (e.g., formation of Jacobians) needs to be performed (similar to the FAS multigrid approach). We illustrate the nonlinear version by applying it to the Euler equations, and presenting numerical results. Finally, a performance evaluation is made based on execution time models and convergence information obtained from numerical experiments.

Development of Computational Aeroacoustics Code for Jet Noise and Flow Prediction

NASA Astrophysics Data System (ADS)

Keith, Theo G., Jr.; Hixon, Duane R.

2002-07-01

Accurate prediction of jet fan and exhaust plume flow and noise generation and propagation is very important in developing advanced aircraft engines that will pass current and future noise regulations. In jet fan flows as well as exhaust plumes, two major sources of noise are present: large-scale, coherent instabilities and small-scale turbulent eddies. In previous work for the NASA Glenn Research Center, three strategies have been explored in an effort to computationally predict the noise radiation from supersonic jet exhaust plumes. In order from the least expensive computationally to the most expensive computationally, these are: 1) Linearized Euler equations (LEE). 2) Very Large Eddy Simulations (VLES). 3) Large Eddy Simulations (LES). The first method solves the linearized Euler equations (LEE). These equations are obtained by linearizing about a given mean flow and the neglecting viscous effects. In this way, the noise from large-scale instabilities can be found for a given mean flow. The linearized Euler equations are computationally inexpensive, and have produced good noise results for supersonic jets where the large-scale instability noise dominates, as well as for the tone noise from a jet engine blade row. However, these linear equations do not predict the absolute magnitude of the noise; instead, only the relative magnitude is predicted. Also, the predicted disturbances do not modify the mean flow, removing a physical mechanism by which the amplitude of the disturbance may be controlled. Recent research for isolated airfoils' indicates that this may not affect the solution greatly at low frequencies. The second method addresses some of the concerns raised by the LEE method. In this approach, called Very Large Eddy Simulation (VLES), the unsteady Reynolds averaged Navier-Stokes equations are solved directly using a high-accuracy computational aeroacoustics numerical scheme. With the addition of a two-equation turbulence model and the use of a relatively coarse grid, the numerical solution is effectively filtered into a directly calculated mean flow with the small-scale turbulence being modeled, and an unsteady large-scale component that is also being directly calculated. In this way, the unsteady disturbances are calculated in a nonlinear way, with a direct effect on the mean flow. This method is not as fast as the LEE approach, but does have many advantages to recommend it; however, like the LEE approach, only the effect of the largest unsteady structures will be captured. An initial calculation was performed on a supersonic jet exhaust plume, with promising results, but the calculation was hampered by the explicit time marching scheme that was employed. This explicit scheme required a very small time step to resolve the nozzle boundary layer, which caused a long run time. Current work is focused on testing a lower-order implicit time marching method to combat this problem.

CFD research and systems in Kawasaki Heavy Industries and its future prospects

NASA Astrophysics Data System (ADS)

Hiraoka, Koichi

1990-09-01

KHI Computational Fluid Dynamics (CFD) system is composed of VP100 computer and 2-D and 3-D Euler and/or Navier-Stokes (NS) analysis softwares. For KHI, this system has become a very powerful aerodynamic tool together with the Kawasaki 1 m Transonic Wind Tunnel. The 2-D Euler/NS software, developed in-house, is fully automated, requires no special skill, and was successfully applied to the design of YXX high lift devices and SST supersonic inlet, etc. The 3-D Euler/NS software, developed under joint research with NAL, has an interactively operated Multi-Block type grid generator and can effectively generate grids around complex airplane shapes. Due to the main memory size limitation, 3-D analysis of relatively simple shape, such as SST wing-body, was computed in-house on VP100, otherwise, such as detailed 3-D analyses of ASUKA and HOPE, were computed on NAL VP400, which is 10 times more powerful than VP100, under KHI-NAL joint research. These analysis results have very good correlation with experimental results. However, the present CFD system is less productive than wind tunnel and has applicability limitations.

Estimating locations and total magnetization vectors of compact magnetic sources from scalar, vector, or tensor magnetic measurements through combined Helbig and Euler analysis

USGS Publications Warehouse

Phillips, J.D.; Nabighian, M.N.; Smith, D.V.; Li, Y.

2007-01-01

The Helbig method for estimating total magnetization directions of compact sources from magnetic vector components is extended so that tensor magnetic gradient components can be used instead. Depths of the compact sources can be estimated using the Euler equation, and their dipole moment magnitudes can be estimated using a least squares fit to the vector component or tensor gradient component data. ?? 2007 Society of Exploration Geophysicists.

An Astronomical Test of CCD Photometric Precision

NASA Technical Reports Server (NTRS)

Koch, David; Dunham, Edward; Borucki, William; Jenkins, Jon; DeVingenzi, D. (Technical Monitor)

1998-01-01

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques. we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

An algorithm for solving the perturbed gas dynamic equations

NASA Technical Reports Server (NTRS)

Davis, Sanford

1993-01-01

The present application of a compact, higher-order central-difference approximation to the linearized Euler equations illustrates the multimodal character of these equations by means of computations for acoustic, vortical, and entropy waves. Such dissipationless central-difference methods are shown to propagate waves exhibiting excellent phase and amplitude resolution on the basis of relatively large time-steps; they can be applied to wave problems governed by systems of first-order partial differential equations.

Development of a numerical model for vehicle-bridge interaction analysis of railway bridges

NASA Astrophysics Data System (ADS)

Kim, Hee Ju; Cho, Eun Sang; Ham, Jun Su; Park, Ki Tae; Kim, Tae Heon

2016-04-01

In the field of civil engineering, analyzing dynamic response was main concern for a long time. These analysis methods can be divided into moving load analysis method and moving mass analysis method, and formulating each an equation of motion has recently been studied after dividing vehicles and bridges. In this study, the numerical method is presented, which can consider the various train types and can solve the equations of motion for a vehicle-bridge interaction analysis by non-iteration procedure through formulating the coupled equations for motion. Also, 3 dimensional accurate numerical models was developed by KTX-vehicle in order to analyze dynamic response characteristics. The equations of motion for the conventional trains are derived, and the numerical models of the conventional trains are idealized by a set of linear springs and dashpots with 18 degrees of freedom. The bridge models are simplified by the 3 dimensional space frame element which is based on the Euler-Bernoulli theory. The rail irregularities of vertical and lateral directions are generated by PSD functions of the Federal Railroad Administration (FRA).

Development of Implicit Methods in CFD NASA Ames Research Center 1970's - 1980's

NASA Technical Reports Server (NTRS)

Pulliam, Thomas H.

2010-01-01

The focus here is on the early development (mid 1970's-1980's) at NASA Ames Research Center of implicit methods in Computational Fluid Dynamics (CFD). A class of implicit finite difference schemes of the Beam and Warming approximate factorization type will be addressed. The emphasis will be on the Euler equations. A review of material pertinent to the solution of the Euler equations within the framework of implicit methods will be presented. The eigensystem of the equations will be used extensively in developing a framework for various methods applied to the Euler equations. The development and analysis of various aspects of this class of schemes will be given along with the motivations behind many of the choices. Various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes will be presented.

Development of an Aeroelastic Code Based on an Euler/Navier-Stokes Aerodynamic Solver

NASA Technical Reports Server (NTRS)

Bakhle, Milind A.; Srivastava, Rakesh; Keith, Theo G., Jr.; Stefko, George L.; Janus, Mark J.

1996-01-01

This paper describes the development of an aeroelastic code (TURBO-AE) based on an Euler/Navier-Stokes unsteady aerodynamic analysis. A brief review of the relevant research in the area of propulsion aeroelasticity is presented. The paper briefly describes the original Euler/Navier-Stokes code (TURBO) and then details the development of the aeroelastic extensions. The aeroelastic formulation is described. The modeling of the dynamics of the blade using a modal approach is detailed, along with the grid deformation approach used to model the elastic deformation of the blade. The work-per-cycle approach used to evaluate aeroelastic stability is described. Representative results used to verify the code are presented. The paper concludes with an evaluation of the development thus far, and some plans for further development and validation of the TURBO-AE code.

CFD Approaches for Simulation of Wing-Body Stage Separation

NASA Technical Reports Server (NTRS)

Buning, Pieter G.; Gomez, Reynaldo J.; Scallion, William I.

2004-01-01

A collection of computational fluid dynamics tools and techniques are being developed and tested for application to stage separation and abort simulation for next-generation launch vehicles. In this work, an overset grid Navier-Stokes flow solver has been enhanced and demonstrated on a matrix of proximity cases and on a dynamic separation simulation of a belly-to-belly wing-body configuration. Steady cases show excellent agreement between Navier-Stokes results, Cartesian grid Euler solutions, and wind tunnel data at Mach 3. Good agreement has been obtained between Navier-Stokes, Euler, and wind tunnel results at Mach 6. An analysis of a dynamic separation at Mach 3 demonstrates that unsteady aerodynamic effects are not important for this scenario. Results provide an illustration of the relative applicability of Euler and Navier-Stokes methods to these types of problems.

Physical lumping methods for developing linear reduced models for high speed propulsion systems

NASA Technical Reports Server (NTRS)

Immel, S. M.; Hartley, Tom T.; Deabreu-Garcia, J. Alex

1991-01-01

In gasdynamic systems, information travels in one direction for supersonic flow and in both directions for subsonic flow. A shock occurs at the transition from supersonic to subsonic flow. Thus, to simulate these systems, any simulation method implemented for the quasi-one-dimensional Euler equations must have the ability to capture the shock. In this paper, a technique combining both backward and central differencing is presented. The equations are subsequently linearized about an operating point and formulated into a linear state space model. After proper implementation of the boundary conditions, the model order is reduced from 123 to less than 10 using the Schur method of balancing. Simulations comparing frequency and step response of the reduced order model and the original system models are presented.

Development of Euler's ideas at the Moscow State Regional University

NASA Astrophysics Data System (ADS)

Vysikaylo, P. I.; Belyaev, V. V.

2018-03-01

In honor of the 250th anniversary of Euler's discovery of three libration points in Russia in 1767 in the area of two rotating gravitational attractors in 2017 an International Interdisciplinary Conference “Euler Readings MRSU 2017” was held in Moscow Region State University (MRSU). The Conference demonstrated that the Euler's ideas continue to remain relevant at the present time. This paper summarizes the main achievements on the basis of Leonard Euler's ideas presented at the Conference.

Gain scheduled linear quadratic control for quadcopter

NASA Astrophysics Data System (ADS)

Okasha, M.; Shah, J.; Fauzi, W.; Hanouf, Z.

2017-12-01

This study exploits the dynamics and control of quadcopters using Linear Quadratic Regulator (LQR) control approach. The quadcopter’s mathematical model is derived using the Newton-Euler method. It is a highly manoeuvrable, nonlinear, coupled with six degrees of freedom (DOF) model, which includes aerodynamics and detailed gyroscopic moments that are often ignored in many literatures. The linearized model is obtained and characterized by the heading angle (i.e. yaw angle) of the quadcopter. The adopted control approach utilizes LQR method to track several reference trajectories including circle and helix curves with significant variation in the yaw angle. The controller is modified to overcome difficulties related to the continuous changes in the operating points and eliminate chattering and discontinuity that is observed in the control input signal. Numerical non-linear simulations are performed using MATLAB and Simulink to illustrate to accuracy and effectiveness of the proposed controller.

A Reconstructed Discontinuous Galerkin Method for the Euler Equations on Arbitrary Grids

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

Hong Luo; Luqing Luo; Robert Nourgaliev

2012-11-01

A reconstruction-based discontinuous Galerkin (RDG(P1P2)) method, a variant of P1P2 method, is presented for the solution of the compressible Euler equations on arbitrary grids. In this method, an in-cell reconstruction, designed to enhance the accuracy of the discontinuous Galerkin method, is used to obtain a quadratic polynomial solution (P2) from the underlying linear polynomial (P1) discontinuous Galerkin solution using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The developed RDG method is used to compute a variety of flow problems onmore » arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG(P1P2) method is third-order accurate, and outperforms the third-order DG method (DG(P2)) in terms of both computing costs and storage requirements.« less

Optimized growth and reorientation of anisotropic material based on evolution equations

NASA Astrophysics Data System (ADS)

Jantos, Dustin R.; Junker, Philipp; Hackl, Klaus

2018-07-01

Modern high-performance materials have inherent anisotropic elastic properties. The local material orientation can thus be considered to be an additional design variable for the topology optimization of structures containing such materials. In our previous work, we introduced a variational growth approach to topology optimization for isotropic, linear-elastic materials. We solved the optimization problem purely by application of Hamilton's principle. In this way, we were able to determine an evolution equation for the spatial distribution of density mass, which can be evaluated in an iterative process within a solitary finite element environment. We now add the local material orientation described by a set of three Euler angles as additional design variables into the three-dimensional model. This leads to three additional evolution equations that can be separately evaluated for each (material) point. Thus, no additional field unknown within the finite element approach is needed, and the evolution of the spatial distribution of density mass and the evolution of the Euler angles can be evaluated simultaneously.

Progressive wave expansions and open boundary problems

NASA Technical Reports Server (NTRS)

Hagstrom, T.; Hariharan, S. I.

1995-01-01

In this paper we construct progressive wave expansions and asymptotic boundary conditions for wave-like equations in exterior domains, including applications to electromagnetics, compressible flows and aero-acoustics. The development of the conditions will be discussed in two parts. The first part will include derivations of asymptotic conditions based on the well-known progressive wave expansions for the two-dimensional wave equations. A key feature in the derivations is that the resulting family of boundary conditions involves a single derivative in the direction normal to the open boundary. These conditions are easy to implement and an application in electromagnetics will be presented. The second part of the paper will discuss the theory for hyperbolic systems in two dimensions. Here, the focus will be to obtain the expansions in a general way and to use them to derive a class of boundary conditions that involve only time derivatives or time and tangential derivatives. Maxwell's equations and the compressible Euler equations are used as examples. Simulations with the linearized Euler equations are presented to validate the theory.

Distributed adaptive asymptotically consensus tracking control of uncertain Euler-Lagrange systems under directed graph condition.

PubMed

Wang, Wei; Wen, Changyun; Huang, Jiangshuai; Fan, Huijin

2017-11-01

In this paper, a backstepping based distributed adaptive control scheme is proposed for multiple uncertain Euler-Lagrange systems under directed graph condition. The common desired trajectory is allowed totally unknown by part of the subsystems and the linearly parameterized trajectory model assumed in currently available results is no longer needed. To compensate the effects due to unknown trajectory information, a smooth function of consensus errors and certain positive integrable functions are introduced in designing virtual control inputs. Besides, to overcome the difficulty of completely counteracting the coupling terms of distributed consensus errors and parameter estimation errors in the presence of asymmetric Laplacian matrix, extra information transmission of local parameter estimates are introduced among linked subsystem and adaptive gain technique is adopted to generate distributed torque inputs. It is shown that with the proposed distributed adaptive control scheme, global uniform boundedness of all the closed-loop signals and asymptotically output consensus tracking can be achieved. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations

NASA Astrophysics Data System (ADS)

Dobrev, V.; Kolev, Tz.; Kuzmin, D.; Rieben, R.; Tomov, V.

2018-03-01

We present a new predictor-corrector approach to enforcing local maximum principles in piecewise-linear finite element schemes for the compressible Euler equations. The new element-based limiting strategy is suitable for continuous and discontinuous Galerkin methods alike. In contrast to synchronized limiting techniques for systems of conservation laws, we constrain the density, momentum, and total energy in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum gradients are adjusted to incorporate the irreversible effect of density changes. Antidiffusive corrections to bounds-compatible low-order approximations are limited to satisfy inequality constraints for the specific total and kinetic energy. An accuracy-preserving smoothness indicator is introduced to gradually adjust lower bounds for the element-based correction factors. The employed smoothness criterion is based on a Hessian determinant test for the density. A numerical study is performed for test problems with smooth and discontinuous solutions.

Newton-Euler Dynamic Equations of Motion for a Multi-body Spacecraft

NASA Technical Reports Server (NTRS)

Stoneking, Eric

2007-01-01

The Magnetospheric MultiScale (MMS) mission employs a formation of spinning spacecraft with several flexible appendages and thruster-based control. To understand the complex dynamic interaction of thruster actuation, appendage motion, and spin dynamics, each spacecraft is modeled as a tree of rigid bodies connected by spherical or gimballed joints. The method presented facilitates assembling by inspection the exact, nonlinear dynamic equations of motion for a multibody spacecraft suitable for solution by numerical integration. The building block equations are derived by applying Newton's and Euler's equations of motion to an "element" consisting of two bodies and one joint (spherical and gimballed joints are considered separately). Patterns in the "mass" and L'force" matrices guide assembly by inspection of a general N-body tree-topology system. Straightforward linear algebra operations are employed to eliminate extraneous constraint equations, resulting in a minimum-dimension system of equations to solve. This method thus combines a straightforward, easily-extendable, easily-mechanized formulation with an efficient computer implementation.

Vorticity Transport and Wave Emission in the Protoplanetary Nebula

NASA Technical Reports Server (NTRS)

Davis, S. S.; DeVincenzi, Donald (Technical Monitor)

2001-01-01

Higher order numerical algorithms (4th order in time, 3rd order in space) are applied to the Euler/Energy equations and are used to examine vorticity transport and wave motion in a non-self gravitating, initially isentropic Keplerian disk. In this talk we will examine the response of the nebula to an isolated vortex with a circulation about equal to the rotation rate of Jupiter. The vortex is located on the 4 AU circle and the nebula is simulated from 1 to 24 AU. We show that the vortex emits pressure-supported density and Rossby-type wave packets before it decays within a few orbits. The acoustic density waves evolve into weak (non entropy preserving) shock waves that propagate over the entire disk. The Rossby waves remain in the vicinity of the initial vortex disturbance, but are rapidly damped. Temporal frequencies and spatial wavenumbers are derived using the simulation data and compared with analytical dispersion relations from the linearized Euler/Energy equations.

Vorticity Transport and Wave Emission In A Protoplanetary Disk

NASA Technical Reports Server (NTRS)

Davis, S. S.; Davis, Sanford (Technical Monitor)

2002-01-01

Higher order numerical algorithms (4th order in time, 3rd order in space) are applied to the Euler equations and are used to examine vorticity transport and wave motion in a non-self gravitating, initially isentropic Keplerian disk. In this talk we will examine the response of the disk to an isolated vortex with a circulation about equal to the rotation rate of Jupiter. The vortex is located on the 4 AU circle and the nebula is simulated from 1 to 24 AU. We show that the vortex emits pressure-supported density and Rossby-type wave packets before it decays within a few orbits. The acoustic density waves evolve into weak (non entropy preserving) shock waves that propagate over the entire disk. The Rossby waves remain in the vicinity of the initial vortex disturbance, but are rapidly damped. Temporal frequencies and spatial wavenumbers are derived from the nonlinear simulation data and correlated with analytical dispersion relations from the linearized Euler and energy equations.

The Euler-Poisson-Darboux equation for relativists

NASA Astrophysics Data System (ADS)

Stewart, John M.

2009-09-01

The Euler-Poisson-Darboux (EPD) equation is the simplest linear hyperbolic equation in two independent variables whose coefficients exhibit singularities, and as such must be of interest as a paradigm to relativists. Sadly it receives scant treatment in the textbooks. The first half of this review is didactic in nature. It discusses in the simplest terms possible the nature of solutions of the EPD equation for the timelike and spacelike singularity cases. Also covered is the Riemann representation of solutions of the characteristic initial value problem, which is hard to find in the literature. The second half examines a few of the possible applications, ranging from explicit computation of the leading terms in the far-field backscatter from predominantly outgoing radiation in a Schwarzschild space-time, to computing explicitly the leading terms in the matter-induced singularities in plane symmetric space-times. There are of course many other applications and the aim of this article is to encourage relativists to investigate this underrated paradigm.

Fast preconditioned multigrid solution of the Euler and Navier-Stokes equations for steady, compressible flows

NASA Astrophysics Data System (ADS)

Caughey, David A.; Jameson, Antony

2003-10-01

New versions of implicit algorithms are developed for the efficient solution of the Euler and Navier-Stokes equations of compressible flow. The methods are based on a preconditioned, lower-upper (LU) implementation of a non-linear, symmetric Gauss-Seidel (SGS) algorithm for use as a smoothing algorithm in a multigrid method. Previously, this method had been implemented for flows in quasi-one-dimensional ducts and for two-dimensional flows past airfoils on boundary-conforming O-type grids for a variety of symmetric limited positive (SLIP) spatial approximations, including the scalar dissipation and convective upwind split pressure (CUSP) schemes. Here results are presented for both inviscid and viscous (laminar) flows past airfoils on boundary-conforming C-type grids. The method is significantly faster than earlier explicit or implicit methods for inviscid problems, allowing solution of these problems to the level of truncation error in three to five multigrid cycles. Viscous solutions still require as many as twenty multigrid cycles.

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

DOE PAGES

Larios, Adam; Petersen, Mark R.; Titi, Edriss S.; ...

2017-04-29

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formationmore » in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.« less

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

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

Larios, Adam; Petersen, Mark R.; Titi, Edriss S.

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for xed values of the regularization parameter α > 0. Therefore, the new blow-up criteria allow one to gain information about possible singularity formationmore » in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be suficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.« less

A Universal Formula for Extracting the Euler Angles

NASA Technical Reports Server (NTRS)

Shuster, Malcolm D.; Markley, F. Landis

2004-01-01

Recently, the authors completed a study of the Davenport angles, which are a generalization of the Euler angles for which the initial and final Euler axes need not be either mutually parallel or mutually perpendicular or even along the coordinate axes. During the conduct of that study, those authors discovered a relationship which can be used to compute straightforwardly the Euler angles characterizing a proper-orthogonal direction-cosine matrix for an arbitrary Euler-axis set satisfying n(sub 1) x n(sub 2) = 0 and n(sub 3) x n(sub 1) = 0, which is also satisfied by the more usual Euler angles we encounter commonly in the practice of Astronautics. Rather than leave that relationship hidden in an article with very different focus from the present Engineering note, we present it and the universal algorithm derived from it for extracting the Euler angles from the direction-cosine matrix here. We also offer literal "code" for performing the operations, numerical examples, and general considerations about the extraction of Euler angles which are not universally known, particularly, the treatment of statistical error.

A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams

NASA Astrophysics Data System (ADS)

Andreaus, Ugo; Spagnuolo, Mario; Lekszycki, Tomasz; Eugster, Simon R.

2018-04-01

We present a finite element discrete model for pantographic lattices, based on a continuous Euler-Bernoulli beam for modeling the fibers composing the pantographic sheet. This model takes into account large displacements, rotations and deformations; the Euler-Bernoulli beam is described by using nonlinear interpolation functions, a Green-Lagrange strain for elongation and a curvature depending on elongation. On the basis of the introduced discrete model of a pantographic lattice, we perform some numerical simulations. We then compare the obtained results to an experimental BIAS extension test on a pantograph printed with polyamide PA2200. The pantographic structures involved in the numerical as well as in the experimental investigations are not proper fabrics: They are composed by just a few fibers for theoretically allowing the use of the Euler-Bernoulli beam theory in the description of the fibers. We compare the experiments to numerical simulations in which we allow the fibers to elastically slide one with respect to the other in correspondence of the interconnecting pivot. We present as result a very good agreement between the numerical simulation, based on the introduced model, and the experimental measures.

Accuracy of an unstructured-grid upwind-Euler algorithm for the ONERA M6 wing

NASA Technical Reports Server (NTRS)

Batina, John T.

1991-01-01

Improved algorithms for the solution of the three-dimensional, time-dependent Euler equations are presented for aerodynamic analysis involving unstructured dynamic meshes. The improvements have been developed recently to the spatial and temporal discretizations used by unstructured-grid flow solvers. The spatial discretization involves a flux-split approach that is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves either an explicit time-integration scheme using a multistage Runge-Kutta procedure or an implicit time-integration scheme using a Gauss-Seidel relaxation procedure, which is computationally efficient for either steady or unsteady flow problems. With the implicit Gauss-Seidel procedure, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady flow results are presented for both the NACA 0012 airfoil and the Office National d'Etudes et de Recherches Aerospatiales M6 wing to demonstrate applications of the new Euler solvers. The paper presents a description of the Euler solvers along with results and comparisons that assess the capability.

Numerical simulation of vortical ideal fluid flow through curved channel

NASA Astrophysics Data System (ADS)

Moshkin, N. P.; Mounnamprang, P.

2003-04-01

A numerical algorithm to study the boundary-value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co-ordinate system. The convergence of the finite-difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka-Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two-dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented.

eulerAPE: Drawing Area-Proportional 3-Venn Diagrams Using Ellipses

PubMed Central

Micallef, Luana; Rodgers, Peter

2014-01-01

Venn diagrams with three curves are used extensively in various medical and scientific disciplines to visualize relationships between data sets and facilitate data analysis. The area of the regions formed by the overlapping curves is often directly proportional to the cardinality of the depicted set relation or any other related quantitative data. Drawing these diagrams manually is difficult and current automatic drawing methods do not always produce appropriate diagrams. Most methods depict the data sets as circles, as they perceptually pop out as complete distinct objects due to their smoothness and regularity. However, circles cannot draw accurate diagrams for most 3-set data and so the generated diagrams often have misleading region areas. Other methods use polygons to draw accurate diagrams. However, polygons are non-smooth and non-symmetric, so the curves are not easily distinguishable and the diagrams are difficult to comprehend. Ellipses are more flexible than circles and are similarly smooth, but none of the current automatic drawing methods use ellipses. We present eulerAPE as the first method and software that uses ellipses for automatically drawing accurate area-proportional Venn diagrams for 3-set data. We describe the drawing method adopted by eulerAPE and we discuss our evaluation of the effectiveness of eulerAPE and ellipses for drawing random 3-set data. We compare eulerAPE and various other methods that are currently available and we discuss differences between their generated diagrams in terms of accuracy and ease of understanding for real world data. PMID:25032825

eulerAPE: drawing area-proportional 3-Venn diagrams using ellipses.

PubMed

Micallef, Luana; Rodgers, Peter

2014-01-01

Venn diagrams with three curves are used extensively in various medical and scientific disciplines to visualize relationships between data sets and facilitate data analysis. The area of the regions formed by the overlapping curves is often directly proportional to the cardinality of the depicted set relation or any other related quantitative data. Drawing these diagrams manually is difficult and current automatic drawing methods do not always produce appropriate diagrams. Most methods depict the data sets as circles, as they perceptually pop out as complete distinct objects due to their smoothness and regularity. However, circles cannot draw accurate diagrams for most 3-set data and so the generated diagrams often have misleading region areas. Other methods use polygons to draw accurate diagrams. However, polygons are non-smooth and non-symmetric, so the curves are not easily distinguishable and the diagrams are difficult to comprehend. Ellipses are more flexible than circles and are similarly smooth, but none of the current automatic drawing methods use ellipses. We present eulerAPE as the first method and software that uses ellipses for automatically drawing accurate area-proportional Venn diagrams for 3-set data. We describe the drawing method adopted by eulerAPE and we discuss our evaluation of the effectiveness of eulerAPE and ellipses for drawing random 3-set data. We compare eulerAPE and various other methods that are currently available and we discuss differences between their generated diagrams in terms of accuracy and ease of understanding for real world data.

Conical Euler simulation and active suppression of delta wing rocking motion

NASA Technical Reports Server (NTRS)

Lee, Elizabeth M.; Batina, John T.

1990-01-01

A conical Euler code was developed to study unsteady vortex-dominated flows about rolling highly-swept delta wings, undergoing either forced or free-to-roll motions including active roll suppression. The flow solver of the code involves a multistage Runge-Kutta time-stepping scheme which uses a finite volume spatial discretization of the Euler equations on an unstructured grid of triangles. The code allows for the additional analysis of the free-to-roll case, by including the rigid-body equation of motion for its simultaneous time integration with the governing flow equations. Results are presented for a 75 deg swept sharp leading edge delta wing at a freestream Mach number of 1.2 and at alpha equal to 10 and 30 deg angle of attack. A forced harmonic analysis indicates that the rolling moment coefficient provides: (1) a positive damping at the lower angle of attack equal to 10 deg, which is verified in a free-to-roll calculation; (2) a negative damping at the higher angle of attack equal to 30 deg at the small roll amplitudes. A free-to-roll calculation for the latter case produces an initially divergent response, but as the amplitude of motion grows with time, the response transitions to a wing-rock type of limit cycle oscillation. The wing rocking motion may be actively suppressed, however, through the use of a rate-feedback control law and antisymmetrically deflected leading edge flaps. The descriptions of the conical Euler flow solver and the free-to-roll analysis are presented. Results are also presented which give insight into the flow physics associated with unsteady vortical flows about forced and free-to-roll delta wings, including the active roll suppression of this wing-rock phenomenon.

Application of Linearized Kalman Filter-Smoother to Aircraft Trajectory Estimation.

DTIC Science & Technology

1988-06-01

the report). The kinematic relationships between wind-axis Euler angles and angular rates are given below (Etkin, 1972: 150): q w OS r w s i n* * (4...I values, and those for RP-2 were chosen in order to explore less accurate range measurements combined with more accurate angular measurements. This...was of interest because of the uncertainty in position introduced by large angular measurement uncertainties at long ranges. Finally, radar models RR

A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

NASA Technical Reports Server (NTRS)

Larson, Mats G.; Barth, Timothy J.

1999-01-01

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

High-Order Non-Reflecting Boundary Conditions for the Linearized Euler Equations

DTIC Science & Technology

2008-09-01

rotational effect. Now this rotational effect can be simplified. The atmosphere is thin compared to the radius of the Earth . Furthermore, atmospheric flows...error norm of the discrete solution. Blayo and Debreu [13] considered a characteristic variable ap- proach to NRBCs in first-order systems for ocean and...Third Edition, John Wiley and Sons, New York, 1995. [77] Jensen, T., “Open Boundary Conditions in Stratified Ocean Models,” Journal of Marine Systems

Comparison with Analytical Solution: Generation and Radiation of Acoustic Waves from a 2-D Shear Layer

NASA Technical Reports Server (NTRS)

Dahl, Milo D.

2000-01-01

An acoustic source inside of a 2-D jet excites an instability wave in the shear layer resulting in sound radiating away from the shear layer. Solve the linearized Euler equations to predict the sound radiation outside of the jet. The jet static pressure is assumed to be constant. The jet flow is parallel and symmetric about the x-axis. Use a symmetry boundary condition along the x-axis.

Finite element analysis of large transient elastic-plastic deformations of simple structures, with application to the engine rotor fragment containment/deflection problem

NASA Technical Reports Server (NTRS)

Wu, R. W.; Witmer, E. A.

1972-01-01

Assumed-displacement versions of the finite-element method are developed to predict large-deformation elastic-plastic transient deformations of structures. Both the conventional and a new improved finite-element variational formulation are derived. These formulations are then developed in detail for straight-beam and curved-beam elements undergoing (1) Bernoulli-Euler-Kirchhoff or (2) Timoshenko deformation behavior, in one plane. For each of these categories, several types of assumed-displacement finite elements are developed, and transient response predictions are compared with available exact solutions for small-deflection, linear-elastic transient responses. The present finite-element predictions for large-deflection elastic-plastic transient responses are evaluated via several beam and ring examples for which experimental measurements of transient strains and large transient deformations and independent finite-difference predictions are available.

Aerodynamic and heat transfer analysis of the low aspect ratio turbine

NASA Astrophysics Data System (ADS)

Sharma, O. P.; Nguyen, P.; Ni, R. H.; Rhie, C. M.; White, J. A.

1987-06-01

The available two- and three-dimensional codes are used to estimate external heat loads and aerodynamic characteristics of a highly loaded turbine stage in order to demonstrate state-of-the-art methodologies in turbine design. By using data for a low aspect ratio turbine, it is found that a three-dimensional multistage Euler code gives good averall predictions for the turbine stage, yielding good estimates of the stage pressure ratio, mass flow, and exit gas angles. The nozzle vane loading distribution is well predicted by both the three-dimensional multistage Euler and three-dimensional Navier-Stokes codes. The vane airfoil surface Stanton number distributions, however, are underpredicted by both two- and three-dimensional boundary value analysis.

PID position regulation in one-degree-of-freedom Euler-Lagrange systems actuated by a PMSM

NASA Astrophysics Data System (ADS)

Verastegui-Galván, J.; Hernández-Guzmán, V. M.; Orrante-Sakanassi, J.

2018-02-01

This paper is concerned with position regulation in one-degree-of-freedom Euler-Lagrange Systems. We consider that the mechanical subsystem is actuated by a permanent magnet synchronous motor (PMSM). Our proposal consists of a Proportional-Integral-Derivative (PID) controller for the mechanical subsystem and a slight variation of field oriented control for the PMSM. We take into account the motor electric dynamics during the stability analysis. We present, for the first time, a global asymptotic stability proof for such a control scheme without requiring the mechanical subsystem to naturally possess viscous friction. Finally, as a corollary of our main result we prove global asymptotic stability for output feedback PID regulation of one-degree-of-freedom Euler-Lagrange systems when generated torque is considered as the system input, i.e. when the electric dynamics of PMSM's is not taken into account.

Conical Euler solution for a highly-swept delta wing undergoing wing-rock motion

NASA Technical Reports Server (NTRS)

Lee, Elizabeth M.; Batina, John T.

1990-01-01

Modifications to an unsteady conical Euler code for the free-to-roll analysis of highly-swept delta wings are described. The modifications involve the addition of the rolling rigid-body equation of motion for its simultaneous time-integration with the governing flow equations. The flow solver utilized in the Euler code includes a multistage Runge-Kutta time-stepping scheme which uses a finite-volume spatial discretization on an unstructured mesh made up of triangles. Steady and unsteady results are presented for a 75 deg swept delta wing at a freestream Mach number of 1.2 and an angle of attack of 30 deg. The unsteady results consist of forced harmonic and free-to-roll calculations. The free-to-roll case exhibits a wing rock response produced by unsteady aerodynamics consistent with the aerodynamics of the forced harmonic results. Similarities are shown with a wing-rock time history from a low-speed wind tunnel test.

On the dynamics of viscous masonry beams

NASA Astrophysics Data System (ADS)

Lucchesi, M.; Pintucchi, B.; Šilhavý, M.; Zani, N.

2015-05-01

In this paper, we consider the longitudinal and transversal vibrations of the masonry beams and arches. The basic motivation is the seismic vulnerability analysis of masonry structures that can be modeled as monodimensional elements. The Euler-Bernoulli hypothesis is employed for the system of forces in the beam. The axial force and the bending moment are assumed to consist of the elastic and viscous parts. The elastic part is described by the no-tension material, i.e., the material with no resistance to tension and which accounts for the cases of limitless, as well as bounded compressive strength. The adaptation of this material to beams has been developed in Orlandi (Analisi non lineare di strutture ad arco in muratura. Thesis, 1999) and Zani (Eur J Mech A/Solids 23:467-484, 2004). The viscous part amounts to the Kelvin-Voigt damping depending linearly on the time derivatives of the linearized strain and curvature. The dynamical equations are formulated, and a mathematical analysis of them is presented. Specifically, following Gajewski et al. (Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974), the theorems of existence, uniqueness and regularity of the solution of the dynamical equations are recapitulated and specialized for our purposes, to support the numerical analysis applied previously in Lucchesi and Pintucchi (Eur J Mech A/Solids 26:88-105, 2007 ). As usual, for that the Galerkin method has been used. As an illustration, two numerical examples (slender masonry tower and masonry arch) are presented in this paper with the applied forces corresponding to the acceleration in the earthquake in Emilia Romagna in May 29, 2012.

The Natural Neighbour Radial Point Interpolation Meshless Method Applied to the Non-Linear Analysis

NASA Astrophysics Data System (ADS)

Dinis, L. M. J. S.; Jorge, R. M. Natal; Belinha, J.

2011-05-01

In this work the Natural Neighbour Radial Point Interpolation Method (NNRPIM), is extended to large deformation analysis of elastic and elasto-plastic structures. The NNPRIM uses the Natural Neighbour concept in order to enforce the nodal connectivity and to create a node-depending background mesh, used in the numerical integration of the NNRPIM interpolation functions. Unlike the FEM, where geometrical restrictions on elements are imposed for the convergence of the method, in the NNRPIM there are no such restrictions, which permits a random node distribution for the discretized problem. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed using the Radial Point Interpolators, with some differences that modify the method performance. In the construction of the NNRPIM interpolation functions no polynomial base is required and the used Radial Basis Function (RBF) is the Multiquadric RBF. The NNRPIM interpolation functions posses the delta Kronecker property, which simplify the imposition of the natural and essential boundary conditions. One of the scopes of this work is to present the validation the NNRPIM in the large-deformation elasto-plastic analysis, thus the used non-linear solution algorithm is the Newton-Rapson initial stiffness method and the efficient "forward-Euler" procedure is used in order to return the stress state to the yield surface. Several non-linear examples, exhibiting elastic and elasto-plastic material properties, are studied to demonstrate the effectiveness of the method. The numerical results indicated that NNRPIM handles large material distortion effectively and provides an accurate solution under large deformation.

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.

p-Euler equations and p-Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Li, Lei; Liu, Jian-Guo

2018-04-01

We propose in this work new systems of equations which we call p-Euler equations and p-Navier-Stokes equations. p-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the 'momentum' is the signed (p - 1)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier-Stokes equations. If γ = p, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier-Stokes equations in Rd for γ = p and p ≥ d ≥ 2 through a compactness criterion.

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.; Ochs, Harry T., III

1988-01-01

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Application of Artificial Intelligence For Euler Solutions Clustering

NASA Astrophysics Data System (ADS)

Mikhailov, V.; Galdeano, A.; Diament, M.; Gvishiani, A.; Agayan, S.; Bogoutdinov, Sh.; Graeva, E.; Sailhac, P.

Results of Euler deconvolution strongly depend on the selection of viable solutions. Synthetic calculations using multiple causative sources show that Euler solutions clus- ter in the vicinity of causative bodies even when they do not group densely about perimeter of the bodies. We have developed a clustering technique to serve as a tool for selecting appropriate solutions. The method RODIN, employed in this study, is based on artificial intelligence and was originally designed for problems of classification of large data sets. It is based on a geometrical approach to study object concentration in a finite metric space of any dimension. The method uses a formal definition of cluster and includes free parameters that facilitate the search for clusters of given proper- ties. Test on synthetic and real data showed that the clustering technique successfully outlines causative bodies more accurate than other methods of discriminating Euler solutions. In complicated field cases such as the magnetic field in the Gulf of Saint Malo region (Brittany, France), the method provides geologically insightful solutions. Other advantages of the clustering method application are: - Clusters provide solutions associated with particular bodies or parts of bodies permitting the analysis of different clusters of Euler solutions separately. This may allow computation of average param- eters for individual causative bodies. - Those measurements of the anomalous field that yield clusters also form dense clusters themselves. The application of cluster- ing technique thus outlines areas where the influence of different causative sources is more prominent. This allows one to focus on areas for reinterpretation, using different window sizes, structural indices and so on.

Linearized potential solution for an airfoil in nonuniform parallel streams

NASA Technical Reports Server (NTRS)

Prabhu, R. K.; Tiwari, S. N.

1983-01-01

A small perturbation potential flow theory is applied to the problem of determining the chordwise pressure distribution, lift and pitching moment of a thin airfoil in the middle of five parallel streams. This theory is then extended to the case of an undisturbed stream having a given smooth velocity profile. Two typical examples are considered and the results obtained are compared with available solutions of Euler's equations. The agreement between these two results is not quite satisfactory. Possible reasons for the differences are indicated.

Numerical solution of fluid-structure interaction represented by human vocal folds in airflow

NASA Astrophysics Data System (ADS)

Valášek, J.; Sváček, P.; Horáček, J.

2016-03-01

The paper deals with the human vocal folds vibration excited by the fluid flow. The vocal fold is modelled as an elastic body assuming small displacements and therefore linear elasticity theory is used. The viscous incompressible fluid flow is considered. For purpose of numerical solution the arbitrary Lagrangian-Euler method (ALE) is used. The whole problem is solved by the finite element method (FEM) based solver. Results of numerical experiments with different boundary conditions are presented.

Multiscale techniques for parabolic equations.

PubMed

Målqvist, Axel; Persson, Anna

2018-01-01

We use the local orthogonal decomposition technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583-2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the [Formula: see text]-norm. We present numerical examples, which confirm our theoretical findings.

Hydrodynamics of the Dirac spectrum

DOE PAGES

Liu, Yizhuang; Warchoł, Piotr; Zahed, Ismail

2015-12-15

We discuss a hydrodynamical description of the eigenvalues of the Dirac spectrum in even dimensions in the vacuum and in the large N (volume) limit. The linearized hydrodynamics supports sound waves. The hydrodynamical relaxation of the eigenvalues is captured by a hydrodynamical (tunneling) minimum configuration which follows from a pertinent form of Euler equation. As a result, the relaxation from a phase of unbroken chiral symmetry to a phase of broken chiral symmetry occurs over a time set by the speed of sound.

Supersonic Coaxial Jets: Noise Predictions and Measurements

NASA Technical Reports Server (NTRS)

Dahl, Milo D.; Papamoschou, Dimitri; Hixon, Ray

1998-01-01

The noise from perfectly expanded coaxial jets was measured in an anechoic chamber for different operating conditions with the same total thrust, mass flow, and exit area. The shape of the measured noise spectrum at different angles to the jet axis was found to agree with spectral shapes for single, axisymmetric jets. Based on these spectra, the sound was characterized as being generated by large turbulent structures or fine-scale turbulence. Modeling the large scale structures as instability waves, a stability analysis was conducted for the coaxial jets to identify the growing and decaying instability waves in each shear layer and predict their noise radiation pattern outside the jet. When compared to measured directivity, the analysis identified the region downstream of the outer potential core, where the two shear layers were merging, as the source of the peak radiated noise where instability waves, with their origin in the inner shear layer, reach their maximum amplitude. Numerical computations were also performed using a linearized Euler equation solver. Those results were compared to both the results from the instability wave analysis and to measured data.

Application of computational aeroacoustic methodologies to advanced propeller configurations - A review

NASA Technical Reports Server (NTRS)

Korkan, Kenneth D.; Eagleson, Lisa A.; Griffiths, Robert C.

1991-01-01

Current research in the area of advanced propeller configurations for performance and acoustics are briefly reviewed. Particular attention is given to the techniques of Lock and Theodorsen modified for use in the design of counterrotating propeller configurations; a numerical method known as SSTAGE, which is a Euler solver for the unducted fan concept; the NASPROP-E numerical analysis also based on a Euler solver and used to study the near acoustic fields for the SR series propfan configurations; and a counterrotating propeller test rig designed to obtain an experimental performance/acoustic data base for various propeller configurations.

Domain decomposition methods in aerodynamics

NASA Technical Reports Server (NTRS)

Venkatakrishnan, V.; Saltz, Joel

1990-01-01

Compressible Euler equations are solved for two-dimensional problems by a preconditioned conjugate gradient-like technique. An approximate Riemann solver is used to compute the numerical fluxes to second order accuracy in space. Two ways to achieve parallelism are tested, one which makes use of parallelism inherent in triangular solves and the other which employs domain decomposition techniques. The vectorization/parallelism in triangular solves is realized by the use of a recording technique called wavefront ordering. This process involves the interpretation of the triangular matrix as a directed graph and the analysis of the data dependencies. It is noted that the factorization can also be done in parallel with the wave front ordering. The performances of two ways of partitioning the domain, strips and slabs, are compared. Results on Cray YMP are reported for an inviscid transonic test case. The performances of linear algebra kernels are also reported.

Parallelization of implicit finite difference schemes in computational fluid dynamics

NASA Technical Reports Server (NTRS)

Decker, Naomi H.; Naik, Vijay K.; Nicoules, Michel

1990-01-01

Implicit finite difference schemes are often the preferred numerical schemes in computational fluid dynamics, requiring less stringent stability bounds than the explicit schemes. Each iteration in an implicit scheme involves global data dependencies in the form of second and higher order recurrences. Efficient parallel implementations of such iterative methods are considerably more difficult and non-intuitive. The parallelization of the implicit schemes that are used for solving the Euler and the thin layer Navier-Stokes equations and that require inversions of large linear systems in the form of block tri-diagonal and/or block penta-diagonal matrices is discussed. Three-dimensional cases are emphasized and schemes that minimize the total execution time are presented. Partitioning and scheduling schemes for alleviating the effects of the global data dependencies are described. An analysis of the communication and the computation aspects of these methods is presented. The effect of the boundary conditions on the parallel schemes is also discussed.

On homogeneous second order linear general quantum difference equations.

PubMed

Faried, Nashat; Shehata, Enas M; El Zafarani, Rasha M

2017-01-01

In this paper, we prove the existence and uniqueness of solutions of the β -Cauchy problem of second order β -difference equations [Formula: see text] [Formula: see text], in a neighborhood of the unique fixed point [Formula: see text] of the strictly increasing continuous function β , defined on an interval [Formula: see text]. These equations are based on the general quantum difference operator [Formula: see text], which is defined by [Formula: see text], [Formula: see text]. We also construct a fundamental set of solutions for the second order linear homogeneous β -difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β -difference equation.

Solution of steady and unsteady transonic-vortex flows using Euler and full-potential equations

NASA Technical Reports Server (NTRS)

Kandil, Osama A.; Chuang, Andrew H.; Hu, Hong

1989-01-01

Two methods are presented for inviscid transonic flows: unsteady Euler equations in a rotating frame of reference for transonic-vortex flows and integral solution of full-potential equation with and without embedded Euler domains for transonic airfoil flows. The computational results covered: steady and unsteady conical vortex flows; 3-D steady transonic vortex flow; and transonic airfoil flows. The results are in good agreement with other computational results and experimental data. The rotating frame of reference solution is potentially efficient as compared with the space fixed reference formulation with dynamic gridding. The integral equation solution with embedded Euler domain is computationally efficient and as accurate as the Euler equations.

Paleomagnetic Euler Poles and the Apparent Polar Wander and Absolute Motion of North America Since the Carboniferous

NASA Astrophysics Data System (ADS)

Gordon, Richard G.; Cox, Allan; O'Hare, Scott

1984-10-01

The apparent polar wander path for a plate is determined from paleomagnetic data by plotting a time sequence of paleomagnetic poles, each representing the location of the earth's spin axis as seen from the plate. Apparent polar wander paths consist of long, gently curved segments termed tracks linked by short segments with sharp curvature termed cusps. The tracks correspond to time intervals when the direction of plate motion was constant, and the cusps correspond to time intervals when the direction of plate motion was changing. Apparent polar wander tracks, like hot spot tracks, tend to lie along small circles. The center of a circle is called a hot spot Euler pole in the case of hot spot tracks and a paleomagnetic Euler pole in the case of paleomagnetic apparent polar wander paths. Both types of tracks mark the motion of a plate with respect to a point, a rising mantle plume in the case of hot spot tracks and the earth's paleomagnetic axis in the case of apparent polar wander paths. Unlike approaches uced in previous studies, paleomagnetic Euler pole analysis yields all three components of motion—including the east-west motion—of a plate with respect to the paleomagnetic axis. A new method for analyzing paleomagnetic poles along a track by using a maximum likelihood criterion gives the best fit paleomagnetic Euler pole and an ellipsoid of 95% confidence about the paleomagnetic Euler pole. In analyzing synthetic and real data, we found that the ellipsoids are elongate, the long axes being aligned with a great circle drawn from the paleomagnetic Euler pole to the center of the apparent polar wander track. This elongation is caused by the azimuths of circular tracks being better defined than their radii of curvature. A Jurassic-Cretaceous paleomagnetic Euler pole for North America was determined from 13 paleomagnetic poles. This track begins with the Wingate and Kayenta formations (about 200 Ma) and ends with the Niobrara Formation (about 87 Ma). Morgan's hot spot Euler pole for 200-90 Ma lies only 15° outside the 95% confidence ellipsoid of the paleomagnetic Euler pole. The good but not perfect agreement reflects displacement between the hot spot and paleomagnetic reference frames at an average rate that is smaller by an order of magnitude than the rate at which the faster plates are moving. The angular velocity of North America about the Jurassic-Cretaceous paleomagnetic Euler pole was determined by plotting the angular positions of paleomagnetic poles along the track as a function of age. For the Cretaceous the angular velocity was too small to measure. During the Jurassic the angular velocity was high, corresponding to a root-mean-square velocity of 70 km/m.y. for the North American plate. A short time interval of even more rapid movement during the Middle and Late Jurassic, possibly corresponding to the beginning of rapid displacement between North America and Africa, is suggested by the data. The direction of absolute motion of North America during the Jurassic was toward the northwest. A Carboniferous-Permian-Triassic paleomagnetic Euler pole was determined from 26 paleomagnetic poles. The progression of poles along this track is consistent with known ages and stratigraphy, except for some systematic differences between poles from Triassic rocks on the Colorado Plateau and poles from Triassic rocks off the Colorado Plateau. These differences could be due to a small clockwise rotation of the Colorado Plateau with respect to cratonal North America, or to miscorrelations between Triassic rocks on the Colorado Plateau and off the Colorado Plateau, or to large lag times between the deposition and magnetization of some rock units, or to some combination of these possibilities. Despite these ambiguities in interpreting paleomagnetic data from Triassic rocks, the general pattern of apparent polar wander and plate motion during the Carboniferous through Triassic is clear: The root-mean-square velocity of North America was slow (about 20 km/m.y.) during the Carboniferous, probably slow (about 20 km/m.y.) during the Permian, but rapid (60-100 km/m.y.) during the Triassic. Paleomagnetic Euler pole analysis establishes that the present slow (less than 30 km/m.y.) velocity of large continental plates like North America is not an intrinsic property of the plates. Occasionally these plates have, for intervals of 50 ± 20 m.y., moved as rapidly as the oceanic plates are moving today. In our interpretation, during times of rapid motion the continents were attached along a passive margin to oceanic lithosphere that was being subducted at some distance from the continent. Rapid motion stopped when the oceanic lithosphere had been consumed by subduction. If North America, Greenland, and Eurasia were joined as a single land mass during the Jurassic, then a likely location for the subducting oceanic plate attached to this landmass is along the southern margin of the cratonal core of Asia with the oceanic plate extending into Tethys. At the cusp between the Carboniferous-Permian-Triassic track and the Jurassic-Cretaceous track, the trend of the path changes by 160°. The western point of the cusp, which is delineated by paleomagnetic poles from the Chinle, Wingate, and Kayenta formations, is 13° farther west in our analysis than it is in commonly accepted apparent polar wander paths for North America. An implication for terrane analysis is that northward displacements found by using our Late Triassic and Early Jurassic poles are up to 2000 km smaller than are those found by using previously published Late Triassic and Early Jurassic cratonal poles.

An exponential time-integrator scheme for steady and unsteady inviscid flows

NASA Astrophysics Data System (ADS)

Li, Shu-Jie; Luo, Li-Shi; Wang, Z. J.; Ju, Lili

2018-07-01

An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge-Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

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.

A grid generation and flow solution method for the Euler equations on unstructured grids

NASA Astrophysics Data System (ADS)

Anderson, W. Kyle

1994-01-01

A grid generation and flow solution algorithm for the Euler equations on unstructured grids is presented. The grid generation scheme utilizes Delaunay triangulation and self-generates the field points for the mesh based on cell aspect ratios and allows for clustering near solid surfaces. The flow solution method is an implicit algorithm in which the linear set of equations arising at each time step is solved using a Gauss Seidel procedure which is completely vectorizable. In addition, a study is conducted to examine the number of subiterations required for good convergence of the overall algorithm. Grid generation results are shown in two dimensions for a National Advisory Committee for Aeronautics (NACA) 0012 airfoil as well as a two-element configuration. Flow solution results are shown for two-dimensional flow over the NACA 0012 airfoil and for a two-element configuration in which the solution has been obtained through an adaptation procedure and compared to an exact solution. Preliminary three-dimensional results are also shown in which subsonic flow over a business jet is computed.

Generation of circular polarization in CMB radiation via nonlinear photon-photon interaction

NASA Astrophysics Data System (ADS)

Sadegh, Mahdi; Mohammadi, Rohoollah; Motie, Iman

2018-01-01

Standard cosmological models do predict a measurable amount of anisotropies in the intensity and linear polarization of the cosmic microwave background radiation (CMB) via Thomson scattering, even though these theoretical models do not predict circular polarization for CMB radiation. In other hand, the circular polarization of CMB has not been excluded in observational evidences. Here we estimate the circular polarization power spectrum ClV (S ) in CMB radiation due to Compton scattering and nonlinear photon-photon forward scattering via Euler-Heisenberg effective Lagrangian. We have estimated the average value of circular power spectrum is l (l +1 )ClV (S )/(2 π )˜10-4 (μ K) 2 for l ˜300 at present time which is smaller than recently reported data for upper limit of circular polarization (SPIDER collaboration). As a result to test our results, the ability to detect nano-Kelvin level signals of CMB circular polarization requires. We also show that the generation of B-mode polarization for CMB photons in the presence of the primordial scalar perturbation via Euler-Heisenberg interaction is possible however this contribution for B-mode polarization is not remarkable.

Simple Test Functions in Meshless Local Petrov-Galerkin Methods

NASA Technical Reports Server (NTRS)

Raju, Ivatury S.

2016-01-01

Two meshless local Petrov-Galerkin (MLPG) methods based on two different trial functions but that use a simple linear test function were developed for beam and column problems. These methods used generalized moving least squares (GMLS) and radial basis (RB) interpolation functions as trial functions. These two methods were tested on various patch test problems. Both methods passed the patch tests successfully. Then the methods were applied to various beam vibration problems and problems involving Euler and Beck's columns. Both methods yielded accurate solutions for all problems studied. The simple linear test function offers considerable savings in computing efforts as the domain integrals involved in the weak form are avoided. The two methods based on this simple linear test function method produced accurate results for frequencies and buckling loads. Of the two methods studied, the method with radial basis trial functions is very attractive as the method is simple, accurate, and robust.

Testing the equivalence principle on cosmological scales

NASA Astrophysics Data System (ADS)

Bonvin, Camille; Fleury, Pierre

2018-05-01

The equivalence principle, that is one of the main pillars of general relativity, is very well tested in the Solar system; however, its validity is more uncertain on cosmological scales, or when dark matter is concerned. This article shows that relativistic effects in the large-scale structure can be used to directly test whether dark matter satisfies Euler's equation, i.e. whether its free fall is characterised by geodesic motion, just like baryons and light. After having proposed a general parametrisation for deviations from Euler's equation, we perform Fisher-matrix forecasts for future surveys like DESI and the SKA, and show that such deviations can be constrained with a precision of order 10%. Deviations from Euler's equation cannot be tested directly with standard methods like redshift-space distortions and gravitational lensing, since these observables are not sensitive to the time component of the metric. Our analysis shows therefore that relativistic effects bring new and complementary constraints to alternative theories of gravity.

A High Order Finite Difference Scheme with Sharp Shock Resolution for the Euler Equations

NASA Technical Reports Server (NTRS)

Gerritsen, Margot; Olsson, Pelle

1996-01-01

We derive a high-order finite difference scheme for the Euler equations that satisfies a semi-discrete energy estimate, and present an efficient strategy for the treatment of discontinuities that leads to sharp shock resolution. The formulation of the semi-discrete energy estimate is based on a symmetrization of the Euler equations that preserves the homogeneity of the flux vector, a canonical splitting of the flux derivative vector, and the use of difference operators that satisfy a discrete analogue to the integration by parts procedure used in the continuous energy estimate. Around discontinuities or sharp gradients, refined grids are created on which the discrete equations are solved after adding a newly constructed artificial viscosity. The positioning of the sub-grids and computation of the viscosity are aided by a detection algorithm which is based on a multi-scale wavelet analysis of the pressure grid function. The wavelet theory provides easy to implement mathematical criteria to detect discontinuities, sharp gradients and spurious oscillations quickly and efficiently.

Embedding methods for the steady Euler equations

NASA Technical Reports Server (NTRS)

Chang, S. H.; Johnson, G. M.

1983-01-01

An approach to the numerical solution of the steady Euler equations is to embed the first-order Euler system in a second-order system and then to recapture the original solution by imposing additional boundary conditions. Initial development of this approach and computational experimentation with it were previously based on heuristic physical reasoning. This has led to the construction of a relaxation procedure for the solution of two-dimensional steady flow problems. The theoretical justification for the embedding approach is addressed. It is proven that, with the appropriate choice of embedding operator and additional boundary conditions, the solution to the embedded system is exactly the one to the original Euler equations. Hence, solving the embedded version of the Euler equations will not produce extraneous solutions.

On Euler's Theorem for Homogeneous Functions and Proofs Thereof.

ERIC Educational Resources Information Center

Tykodi, R. J.

1982-01-01

Euler's theorem for homogenous functions is useful when developing thermodynamic distinction between extensive and intensive variables of state and when deriving the Gibbs-Duhem relation. Discusses Euler's theorem and thermodynamic applications. Includes six-step instructional strategy for introducing the material to students. (Author/JN)

Nonlinear earthquake analysis of reinforced concrete frames with fiber and Bernoulli-Euler beam-column element.

PubMed

Karaton, Muhammet

2014-01-01

A beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamic analysis of reinforced concrete (RC) structural element. Stiffness matrix of this element is obtained by using rigidity method. A solution technique that included nonlinear dynamic substructure procedure is developed for dynamic analyses of RC frames. A predicted-corrected form of the Bossak-α method is applied for dynamic integration scheme. A comparison of experimental data of a RC column element with numerical results, obtained from proposed solution technique, is studied for verification the numerical solutions. Furthermore, nonlinear cyclic analysis results of a portal reinforced concrete frame are achieved for comparing the proposed solution technique with Fibre element, based on flexibility method. However, seismic damage analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumped/distributed mass and load. Damage region, propagation, and intensities according to both approaches are researched.

Analysis of the Hessian for Aerodynamic Optimization: Inviscid Flow

NASA Technical Reports Server (NTRS)

Arian, Eyal; Ta'asan, Shlomo

1996-01-01

In this paper we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However the optimization problems in two and three dimensions are inherently different. While the two dimensional optimization problems are well-posed the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. A natural choice of such a regularization is derived. The analysis also gives an estimate of the Hessian's condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient based methods are derived from these approximate Hessians.

Stability analysis of internally damped rotating composite shafts using a finite element formulation

NASA Astrophysics Data System (ADS)

Ben Arab, Safa; Rodrigues, José Dias; Bouaziz, Slim; Haddar, Mohamed

2018-04-01

This paper deals with the stability analysis of internally damped rotating composite shafts. An Euler-Bernoulli shaft finite element formulation based on Equivalent Single Layer Theory (ESLT), including the hysteretic internal damping of composite material and transverse shear effects, is introduced and then used to evaluate the influence of various parameters: stacking sequences, fiber orientations and bearing properties on natural frequencies, critical speeds, and instability thresholds. The obtained results are compared with those available in the literature using different theories. The agreement in the obtained results show that the developed Euler-Bernoulli finite element based on ESLT including hysteretic internal damping and shear transverse effects can be effectively used for the stability analysis of internally damped rotating composite shafts. Furthermore, the results revealed that rotor stability is sensitive to the laminate parameters and to the properties of the bearings.

A higher-order conservation element solution element method for solving hyperbolic differential equations on unstructured meshes

NASA Astrophysics Data System (ADS)

Bilyeu, David

This dissertation presents an extension of the Conservation Element Solution Element (CESE) method from second- to higher-order accuracy. The new method retains the favorable characteristics of the original second-order CESE scheme, including (i) the use of the space-time integral equation for conservation laws, (ii) a compact mesh stencil, (iii) the scheme will remain stable up to a CFL number of unity, (iv) a fully explicit, time-marching integration scheme, (v) true multidimensionality without using directional splitting, and (vi) the ability to handle two- and three-dimensional geometries by using unstructured meshes. This algorithm has been thoroughly tested in one, two and three spatial dimensions and has been shown to obtain the desired order of accuracy for solving both linear and non-linear hyperbolic partial differential equations. The scheme has also shown its ability to accurately resolve discontinuities in the solutions. Higher order unstructured methods such as the Discontinuous Galerkin (DG) method and the Spectral Volume (SV) methods have been developed for one-, two- and three-dimensional application. Although these schemes have seen extensive development and use, certain drawbacks of these methods have been well documented. For example, the explicit versions of these two methods have very stringent stability criteria. This stability criteria requires that the time step be reduced as the order of the solver increases, for a given simulation on a given mesh. The research presented in this dissertation builds upon the work of Chang, who developed a fourth-order CESE scheme to solve a scalar one-dimensional hyperbolic partial differential equation. The completed research has resulted in two key deliverables. The first is a detailed derivation of a high-order CESE methods on unstructured meshes for solving the conservation laws in two- and three-dimensional spaces. The second is the code implementation of these numerical methods in a computer code. For code development, a one-dimensional solver for the Euler equations was developed. This work is an extension of Chang's work on the fourth-order CESE method for solving a one-dimensional scalar convection equation. A generic formulation for the nth-order CESE method, where n ≥ 4, was derived. Indeed, numerical implementation of the scheme confirmed that the order of convergence was consistent with the order of the scheme. For the two- and three-dimensional solvers, SOLVCON was used as the basic framework for code implementation. A new solver kernel for the fourth-order CESE method has been developed and integrated into the framework provided by SOLVCON. The main part of SOLVCON, which deals with unstructured meshes and parallel computing, remains intact. The SOLVCON code for data transmission between computer nodes for High Performance Computing (HPC). To validate and verify the newly developed high-order CESE algorithms, several one-, two- and three-dimensional simulations where conducted. For the arbitrary order, one-dimensional, CESE solver, three sets of governing equations were selected for simulation: (i) the linear convection equation, (ii) the linear acoustic equations, (iii) the nonlinear Euler equations. All three systems of equations were used to verify the order of convergence through mesh refinement. In addition the Euler equations were used to solve the Shu-Osher and Blastwave problems. These two simulations demonstrated that the new high-order CESE methods can accurately resolve discontinuities in the flow field.For the two-dimensional, fourth-order CESE solver, the Euler equation was employed in four different test cases. The first case was used to verify the order of convergence through mesh refinement. The next three cases demonstrated the ability of the new solver to accurately resolve discontinuities in the flows. This was demonstrated through: (i) the interaction between acoustic waves and an entropy pulse, (ii) supersonic flow over a circular blunt body, (iii) supersonic flow over a guttered wedge. To validate and verify the three-dimensional, fourth-order CESE solver, two different simulations where selected. The first used the linear convection equations to demonstrate fourth-order convergence. The second used the Euler equations to simulate supersonic flow over a spherical body to demonstrate the scheme's ability to accurately resolve shocks. All test cases used are well known benchmark problems and as such, there are multiple sources available to validate the numerical results. Furthermore, the simulations showed that the high-order CESE solver was stable at a CFL number near unity.

Leonhard Euler and his contributions to fluid mechanics

NASA Technical Reports Server (NTRS)

Salas, M. D.

1988-01-01

The career of Leonhard Euler, one of the world's most gifted scientists, is reviewed. The paper focuses on Euler's contributions to fluid mechanics and gives a perspective of how this science was born. A bibliography is included to provide the history enthusiast with a starting point for further study.

Beyond Euler's Method: Implicit Finite Differences in an Introductory ODE Course

ERIC Educational Resources Information Center

Kull, Trent C.

2011-01-01

A typical introductory course in ordinary differential equations (ODEs) exposes students to exact solution methods. However, many differential equations must be approximated with numerical methods. Textbooks commonly include explicit methods such as Euler's and Improved Euler's. Implicit methods are typically introduced in more advanced courses…

A new Euler scheme based on harmonic-polygon approach for solving first order ordinary differential equation

NASA Astrophysics Data System (ADS)

Yusop, Nurhafizah Moziyana Mohd; Hasan, Mohammad Khatim; Wook, Muslihah; Amran, Mohd Fahmi Mohamad; Ahmad, Siti Rohaidah

2017-10-01

There are many benefits to improve Euler scheme for solving the Ordinary Differential Equation Problems. Among the benefits are simple implementation and low-cost computational. However, the problem of accuracy in Euler scheme persuade scholar to use complex method. Therefore, the main purpose of this research are show the construction a new modified Euler scheme that improve accuracy of Polygon scheme in various step size. The implementing of new scheme are used Polygon scheme and Harmonic mean concept that called as Harmonic-Polygon scheme. This Harmonic-Polygon can provide new advantages that Euler scheme could offer by solving Ordinary Differential Equation problem. Four set of problems are solved via Harmonic-Polygon. Findings show that new scheme or Harmonic-Polygon scheme can produce much better accuracy result.

A Comparison of Three PML Treatments for CAA (and CFD)

NASA Technical Reports Server (NTRS)

Goodrich, John W.

2008-01-01

In this paper we compare three Perfectly Matched Layer (PML) treatments by means of a series of numerical experiments, using common numerical algorithms, computational grids, and code implementations. These comparisons are with the Linearized Euler Equations, for base uniform base flow. We see that there are two very good PML candidates, and that can both control the introduced error. Furthermore, we also show that corners can be handled with essentially no increase in the introduced error, and that with a good PML, the outer boundary is the most significant source of err

Accurate Monotonicity - Preserving Schemes With Runge-Kutta Time Stepping

NASA Technical Reports Server (NTRS)

Suresh, A.; Huynh, H. T.

1997-01-01

A new class of high-order monotonicity-preserving schemes for the numerical solution of conservation laws is presented. The interface value in these schemes is obtained by limiting a higher-order polynominal reconstruction. The limiting is designed to preserve accuracy near extrema and to work well with Runge-Kutta time stepping. Computational efficiency is enhanced by a simple test that determines whether the limiting procedure is needed. For linear advection in one dimension, these schemes are shown as well as the Euler equations also confirm their high accuracy, good shock resolution, and computational efficiency.

On multigrid solution of the implicit equations of hydrodynamics. Experiments for the compressible Euler equations in general coordinates

NASA Astrophysics Data System (ADS)

Kifonidis, K.; Müller, E.

2012-08-01

Aims: We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics. Schemes of this class are free of the Courant-Friedrichs-Lewy condition. They are intended for simulations in which widely differing wave propagation timescales are present. A preferred solver in this class is identified. Applications to some simple stiff test problems that are governed by the compressible Euler equations, are presented to evaluate the convergence behavior, and the stability properties of this solver. Algorithmic areas are determined where further work is required to make the method sufficiently efficient and robust for future application to difficult astrophysical flow problems. Methods: The basic equations are formulated and discretized on non-orthogonal, structured curvilinear meshes. Roe's approximate Riemann solver and a second-order accurate reconstruction scheme are used for spatial discretization. Implicit Runge-Kutta (ESDIRK) schemes are employed for temporal discretization. The resulting discrete equations are solved with a full-coarsening, non-linear multigrid method. Smoothing is performed with multistage-implicit smoothers. These are applied here to the time-dependent equations by means of dual time stepping. Results: For steady-state problems, our results show that the efficiency of the present approach is comparable to the best implicit solvers for conservative discretizations of the compressible Euler equations that can be found in the literature. The use of red-black as opposed to symmetric Gauss-Seidel iteration in the multistage-smoother is found to have only a minor impact on multigrid convergence. This should enable scalable parallelization without having to seriously compromise the method's algorithmic efficiency. For time-dependent test problems, our results reveal that the multigrid convergence rate degrades with increasing Courant numbers (i.e. time step sizes). Beyond a Courant number of nine thousand, even complete multigrid breakdown is observed. Local Fourier analysis indicates that the degradation of the convergence rate is associated with the coarse-grid correction algorithm. An implicit scheme for the Euler equations that makes use of the present method was, nevertheless, able to outperform a standard explicit scheme on a time-dependent problem with a Courant number of order 1000. Conclusions: For steady-state problems, the described approach enables the construction of parallelizable, efficient, and robust implicit hydrodynamics solvers. The applicability of the method to time-dependent problems is presently restricted to cases with moderately high Courant numbers. This is due to an insufficient coarse-grid correction of the employed multigrid algorithm for large time steps. Further research will be required to help us to understand and overcome the observed multigrid convergence difficulties for time-dependent problems.

Computational fluid dynamics applications at McDonnel Douglas

NASA Technical Reports Server (NTRS)

Hakkinen, R. J.

1987-01-01

Representative examples are presented of applications and development of advanced Computational Fluid Dynamics (CFD) codes for aerodynamic design at the McDonnell Douglas Corporation (MDC). Transonic potential and Euler codes, interactively coupled with boundary layer computation, and solutions of slender-layer Navier-Stokes approximation are applied to aircraft wing/body calculations. An optimization procedure using evolution theory is described in the context of transonic wing design. Euler methods are presented for analysis of hypersonic configurations, and helicopter rotors in hover and forward flight. Several of these projects were accepted for access to the Numerical Aerodynamic Simulation (NAS) facility at the NASA-Ames Research Center.

Statistical Extremes of Turbulence and a Cascade Generalisation of Euler's Gyroscope Equation

NASA Astrophysics Data System (ADS)

Tchiguirinskaia, Ioulia; Scherzer, Daniel

2016-04-01

Turbulence refers to a rather well defined hydrodynamical phenomenon uncovered by Reynolds. Nowadays, the word turbulence is used to designate the loss of order in many different geophysical fields and the related fundamental extreme variability of environmental data over a wide range of scales. Classical statistical techniques for estimating the extremes, being largely limited to statistical distributions, do not take into account the mechanisms generating such extreme variability. An alternative approaches to nonlinear variability are based on a fundamental property of the non-linear equations: scale invariance, which means that these equations are formally invariant under given scale transforms. Its specific framework is that of multifractals. In this framework extreme variability builds up scale by scale leading to non-classical statistics. Although multifractals are increasingly understood as a basic framework for handling such variability, there is still a gap between their potential and their actual use. In this presentation we discuss how to dealt with highly theoretical problems of mathematical physics together with a wide range of geophysical applications. We use Euler's gyroscope equation as a basic element in constructing a complex deterministic system that preserves not only the scale symmetry of the Navier-Stokes equations, but some more of their symmetries. Euler's equation has been not only the object of many theoretical investigations of the gyroscope device, but also generalised enough to become the basic equation of fluid mechanics. Therefore, there is no surprise that a cascade generalisation of this equation can be used to characterise the intermittency of turbulence, to better understand the links between the multifractal exponents and the structure of a simplified, but not simplistic, version of the Navier-Stokes equations. In a given way, this approach is similar to that of Lorenz, who studied how the flap of a butterfly wing could generate a cyclone with the help of a 3D ordinary differential system. Being well supported by the extensive numerical results, the cascade generalisation of Euler's gyroscope equation opens new horizons for predictability and predictions of processes having long-range dependences.

Effect of Prestresses on the Dispersion of Quasi-Lamb Waves in the System Consisting of an Ideal Liquid Layer and a Compressible Elastic Layer

NASA Astrophysics Data System (ADS)

Bagno, A. M.

2017-03-01

The propagation of quasi-Lamb waves in a prestrained compressible elastic layer interacting with a layer of an ideal compressible fluid is studied. The three-dimensional equations of linearized elasticity and the assumption of finite strains for the elastic layer and the three-dimensional linearized Euler equations for the fluid are used. The dispersion curves for the quasi-Lamb modes are plotted over a wide frequency range. The effect of prestresses and the thickness of the elastic and liquid layers on the frequency spectrum of normal quasi-Lamb waves is analyzed. The localization properties of the lower quasi-Lamb modes in the elastic-fluid waveguides are studied. The numerical results are presented in the form of graphs and analyzed

Weak Galerkin method for the Biot’s consolidation model

DOE PAGES

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

2017-08-23

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure withoutmore » special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.« less

Weak Galerkin method for the Biot’s consolidation model

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

Hu, Xiaozhe; Mu, Lin; Ye, Xiu

In this study, we develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure withoutmore » special treatment. Lastlyl, numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.« less

A variational approach to dynamics of flexible multibody systems

NASA Technical Reports Server (NTRS)

Wu, Shih-Chin; Haug, Edward J.; Kim, Sung-Soo

1989-01-01

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body references frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed.

Euler force actuation mechanism for siphon valving in compact disk-like microfluidic chips.

PubMed

Deng, Yongbo; Fan, Jianhua; Zhou, Song; Zhou, Teng; Wu, Junfeng; Li, Yin; Liu, Zhenyu; Xuan, Ming; Wu, Yihui

2014-03-01

Based on the Euler force induced by the acceleration of compact disk (CD)-like microfluidic chip, this paper presents a novel actuation mechanism for siphon valving. At the preliminary stage of acceleration, the Euler force in the tangential direction of CD-like chip takes the primary place compared with the centrifugal force to function as the actuation of the flow, which fills the siphon and actuates the siphon valving. The Euler force actuation mechanism is demonstrated by the numerical solution of the phase-field based mathematical model for the flow in siphon valve. In addition, experimental validation is implemented in the polymethylmethacrylate-based CD-like microfluidic chip manufactured using CO2 laser engraving technique. To prove the application of the proposed Euler force actuation mechanism, whole blood separation and plasma extraction has been conducted using the Euler force actuated siphon valving. The newly introduced actuation mechanism overcomes the dependence on hydrophilic capillary filling of siphon by avoiding external manipulation or surface treatments of polymeric material. The sacrifice for highly integrated processing in pneumatic pumping technique is also prevented by excluding the volume-occupied compressed air chamber.

Blade design and analysis using a modified Euler solver

NASA Technical Reports Server (NTRS)

Leonard, O.; Vandenbraembussche, R. A.

1991-01-01

An iterative method for blade design based on Euler solver and described in an earlier paper is used to design compressor and turbine blades providing shock free transonic flows. The method shows a rapid convergence, and indicates how much the flow is sensitive to small modifications of the blade geometry, that the classical iterative use of analysis methods might not be able to define. The relationship between the required Mach number distribution and the resulting geometry is discussed. Examples show how geometrical constraints imposed upon the blade shape can be respected by using free geometrical parameters or by relaxing the required Mach number distribution. The same code is used both for the design of the required geometry and for the off-design calculations. Examples illustrate the difficulty of designing blade shapes with optimal performance also outside of the design point.

Axion as a cold dark matter candidate: analysis to third order perturbation for classical axion

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

Noh, Hyerim; Hwang, Jai-chan; Park, Chan-Gyung, E-mail: hr@kasi.re.kr, E-mail: jchan@knu.ac.kr, E-mail: park.chan.gyung@gmail.com

2015-12-01

We investigate aspects of axion as a coherently oscillating massive classical scalar field by analyzing third order perturbations in Einstein's gravity in the axion-comoving gauge. The axion fluid has its characteristic pressure term leading to an axion Jeans scale which is cosmologically negligible for a canonical axion mass. Our classically derived axion pressure term in Einstein's gravity is identical to the one derived in the non-relativistic quantum mechanical context in the literature. We present the general relativistic continuity and Euler equations for an axion fluid valid up to third order perturbation. Equations for axion are exactly the same as thatmore » of a zero-pressure fluid in Einstein's gravity except for an axion pressure term in the Euler equation. Our analysis includes the cosmological constant.« 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.

The Green's matrix and the boundary integral equations for analysis of time-harmonic dynamics of elastic helical springs.

PubMed

Sorokin, Sergey V

2011-03-01

Helical springs serve as vibration isolators in virtually any suspension system. Various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions. The method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time-harmonic dynamics of elastic springs in the framework of Bernoulli-Euler beam theory. In this paper, the derivations of the Green's matrix, of the Somigliana's identities, and of the boundary integral equations are presented. The vibrational power transmission in an infinitely long spring is analyzed by means of the Green's matrix. The eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations. In the course of analysis, the essential features and advantages of the method of boundary integral equations are highlighted. The reported analytical results may be used to study the time-harmonic motion in any wave guide governed by a system of linear differential equations in a single spatial coordinate along its axis. © 2011 Acoustical Society of America

An experiment for determining the Euler load by direct computation

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.; Stein, Peter A.

1986-01-01

A direct algorithm is presented for computing the Euler load of a column from experimental data. The method is based on exact inextensional theory for imperfect columns, which predicts two distinct deflected shapes at loads near the Euler load. The bending stiffness of the column appears in the expression for the Euler load along with the column length, therefore the experimental data allows a direct computation of bending stiffness. Experiments on graphite-epoxy columns of rectangular cross-section are reported in the paper. The bending stiffness of each composite column computed from experiment is compared with predictions from laminated plate theory.

Mathematical Investigation of Fluid Flow, Mass Transfer, and Slag-steel Interfacial Behavior in Gas-stirred Ladles

NASA Astrophysics Data System (ADS)

Cao, Qing; Nastac, Laurentiu

2018-06-01

In this study, the Euler-Euler and Euler-Lagrange modeling approaches were applied to simulate the multiphase flow in the water model and gas-stirred ladle systems. Detailed comparisons of the computational and experimental results were performed to establish which approach is more accurate for predicting the gas-liquid multiphase flow phenomena. It was demonstrated that the Euler-Lagrange approach is more accurate than the Euler-Euler approach. The Euler-Lagrange approach was applied to study the effects of the free surface setup, injected bubble size, gas flow rate, and slag layer thickness on the slag-steel interaction and mass transfer behavior. Detailed discussions on the flat/non-flat free surface assumption were provided. Significant inaccuracies in the prediction of the surface fluid flow characteristics were found when the flat free surface was assumed. The variations in the main controlling parameters (bubble size, gas flow rate, and slag layer thickness) and their potential impact on the multiphase fluid flow and mass transfer characteristics (turbulent intensity, mass transfer rate, slag-steel interfacial area, flow patterns, etc.,) in gas-stirred ladles were quantitatively determined to ensure the proper increase in the ladle refining efficiency. It was revealed that by injecting finer bubbles as well as by properly increasing the gas flow rate and the slag layer thickness, the ladle refining efficiency can be enhanced significantly.

Spherical Coordinate Systems for Streamlining Suited Mobility Analysis

NASA Technical Reports Server (NTRS)

Benson, Elizabeth; Cowley, Matthew S.; Harvill. Lauren; Rajulu, Sudhakar

2014-01-01

When describing human motion, biomechanists generally report joint angles in terms of Euler angle rotation sequences. However, there are known limitations in using this method to describe complex motions such as the shoulder joint during a baseball pitch. Euler angle notation uses a series of three rotations about an axis where each rotation is dependent upon the preceding rotation. As such, the Euler angles need to be regarded as a set to get accurate angle information. Unfortunately, it is often difficult to visualize and understand these complex motion representations. One of our key functions is to help design engineers understand how a human will perform with new designs and all too often traditional use of Euler rotations becomes as much of a hindrance as a help. It is believed that using a spherical coordinate system will allow ABF personnel to more quickly and easily transmit important mobility data to engineers, in a format that is readily understandable and directly translatable to their design efforts. Objectives: The goal of this project is to establish new analysis and visualization techniques to aid in the examination and comprehension of complex motions. Methods: This project consisted of a series of small sub-projects, meant to validate and verify the method before it was implemented in the ABF's data analysis practices. The first stage was a proof of concept, where a mechanical test rig was built and instrumented with an inclinometer, so that its angle from horizontal was known. The test rig was tracked in 3D using an optical motion capture system, and its position and orientation were reported in both Euler and spherical reference systems. The rig was meant to simulate flexion/extension, transverse rotation and abduction/adduction of the human shoulder, but without the variability inherent in human motion. In the second phase of the project, the ABF estimated the error inherent in a spherical coordinate system, and evaluated how this error would vary within the reference frame. This stage also involved expanding a kinematic model of the shoulder, to include the torso, knees, ankle, elbows, wrists and neck. Part of this update included adding a representation of 'roll' about an axis, for upper arm and lower leg rotations. The third stage of the project involved creating visualization methods to assist in interpreting motion in a spherical frame. This visualization method will be incorporated in a tool to evaluate a database of suited mobility data, which is currently in development.

Flow prediction for propfan engine installation effects on transport aircraft at transonic speeds

NASA Technical Reports Server (NTRS)

Samant, S. S.; Yu, N. J.

1986-01-01

An Euler-based method for aerodynamic analysis of turboprop transport aircraft at transonic speeds has been developed. In this method, inviscid Euler equations are solved over surface-fitted grids constructed about aircraft configurations. Propeller effects are simulated by specifying sources of momentum and energy on an actuator disc located in place of the propeller. A stripwise boundary layer procedure is included to account for the viscous effects. A preliminary version of an approach to embed the exhaust plume within the global Euler solution has also been developed for more accurate treatment of the exhaust flow. The resulting system of programs is capable of handling wing-body-nacelle-propeller configurations. The propeller disks may be tractors or pushers and may represent single or counterrotation propellers. Results from analyses of three test cases of interest (a wing alone, a wing-body-nacelle model, and a wing-nacelle-endplate model) are presented. A user's manual for executing the system of computer programs with formats of various input files, sample job decks, and sample input files is provided in appendices.

Euler and His Contribution Number Theory

ERIC Educational Resources Information Center

Len, Amy; Scott, Paul

2004-01-01

Born in 1707, Leonhard Euler was the son of a Protestant minister from the vicinity of Basel, Switzerland. With the aim of pursuing a career in theology, Euler entered the University of Basel at the age of thirteen, where he was tutored in mathematics by Johann Bernoulli (of the famous Bernoulli family of mathematicians). He developed an interest…

CFD validation needs for advanced concepts at Northrop Corporation

NASA Technical Reports Server (NTRS)

George, Michael W.

1987-01-01

Information is given in viewgraph form on the Computational Fluid Dynamics (CFD) Workshop held July 14 - 16, 1987. Topics covered include the philosophy of CFD validation, current validation efforts, the wing-body-tail Euler code, F-20 Euler simulated oil flow, and Euler Navier-Stokes code validation for 2D and 3D nozzle afterbody applications.

Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES

NASA Astrophysics Data System (ADS)

Marras, Simone; Nazarov, Murtazo; Giraldo, Francis X.

2015-11-01

The high order spectral element approximation of the Euler equations is stabilized via a dynamic sub-grid scale model (Dyn-SGS). This model was originally designed for linear finite elements to solve compressible flows at large Mach numbers. We extend its application to high-order spectral elements to solve the Euler equations of low Mach number stratified flows. The major justification of this work is twofold: stabilization and large eddy simulation are achieved via one scheme only. Because the diffusion coefficients of the regularization stresses obtained via Dyn-SGS are residual-based, the effect of the artificial diffusion is minimal in the regions where the solution is smooth. The direct consequence is that the nominal convergence rate of the high-order solution of smooth problems is not degraded. To our knowledge, this is the first application in atmospheric modeling of a spectral element model stabilized by an eddy viscosity scheme that, by construction, may fulfill stabilization requirements, can model turbulence via LES, and is completely free of a user-tunable parameter. From its derivation, it will be immediately clear that Dyn-SGS is independent of the numerical method; it could be implemented in a discontinuous Galerkin, finite volume, or other environments alike. Preliminary discontinuous Galerkin results are reported as well. The straightforward extension to non-linear scalar problems is also described. A suite of 1D, 2D, and 3D test cases is used to assess the method, with some comparison against the results obtained with the most known Lilly-Smagorinsky SGS model.

Determination of regional Euler pole parameters for Eastern Austria

NASA Astrophysics Data System (ADS)

Umnig, Elke; Weber, Robert; Schartner, Matthias; Brueckl, Ewald

2017-04-01

The horizontal motion of lithospheric plates can be described as rotations around a rotation axes through the Earth's center. The two possible points where this axes intersects the surface of the Earth are called Euler poles. The rotation is expressed by the Euler parameters in terms of angular velocities together with the latitude and longitude of the Euler pole. Euler parameters were calculated from GPS data for a study area in Eastern Austria. The observation network is located along the Mur-Mürz Valley and the Vienna Basin. This zone is part of the Vienna Transfer Fault, which is the major fault system between the Eastern Alps and the Carpathians. The project ALPAACT (seismological and geodetic monitoring of ALpine-PAnnonian ACtive Tectonics) investigated intra plate tectonic movements within the Austrian part in order to estimate the seismic hazard. Precise site coordinate time series established from processing 5 years of GPS observations are available for the regional network spanning the years from 2010.0 to 2015.0. Station velocities with respect to the global reference frame ITRF2008 have been computed for 23 sites. The common Euler vector was estimated on base of a subset of reliable site velocities, for stations directly located within the area of interest. In a further step a geokinematic interpretation shall be carried out. Therefore site motions with respect to the Eurasian Plate are requested. To obtain this motion field different variants are conceivable. In a simple approach the mean ITRF2008 velocity of IGS site GRAZ can be adopted as Eurasian rotational velocity. An improved alternative is to calculate site-specific velocity differences between the Euler rotation and the individual site velocities. In this poster presentation the Euler parameters, the residual motion field as well as first geokinematic interpretation results are presented.

Leonhard Euler and the mechanics of rigid bodies

NASA Astrophysics Data System (ADS)

Marquina, J. E.; Marquina, M. L.; Marquina, V.; Hernández-Gómez, J. J.

2017-01-01

In this work we present the original ideas and the construction of the rigid bodies theory realised by Leonhard Euler between 1738 and 1775. The number of treatises written by Euler on this subject is enormous, including the most notorious Scientia Navalis (1749), Decouverte d’un noveau principe de mecanique (1752), Du mouvement de rotation des corps solides autour d’un axe variable (1765), Theoria motus corporum solidorum seu rigidorum (1765) and Nova methodus motu corporum rigidorum determinandi (1776), in which he developed the ideas of the instantaneous rotation axis, the so-called Euler equations and angles, the components of what is now known as the inertia tensor, the principal axes of inertia, and, finally, the generalisation of the translation and rotation movement equations for any system. Euler, the man who ‘put most of mechanics into its modern form’ (Truesdell 1968 Essays in the History of Mechanics (Berlin: Springer) p 106).

An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

NASA Astrophysics Data System (ADS)

Drivas, Theodore D.; Eyink, Gregory L.

2017-12-01

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our L 3-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.

Rotations in the actively colliding Finisterre Arc Terrane: paleomagnetic constraints on Plio-Pleistocene evolution of the South Bismarck microplate, northeastern Papua New Guinea

NASA Astrophysics Data System (ADS)

Weiler, P. D.; Coe, R. S.

2000-01-01

We report paleomagnetic results from 12 Plio-Pleistocene localities in the actively colliding Finisterre Arc Terrane of northeastern Papua New Guinea (PNG). Calcareous, hemipelagic cover rocks possess a stable, syn-collisional remagnetization indicating a clockwise rotation of the colliding terrane through about 40° in post-Miocene time. A decrease in paleomagnetic declination anomalies as a function of along-strike distance in the Finisterre Arc Terrane, analyzed by our preferred model of a linear remagnetization and a migrating Euler pole, suggests an average rotation rate of 8° Ma -1, in good agreement with the instantaneous rate from global positioning system geodesy. Thus, we propose that this rotation results from a coherent, rigid-body rotation of the Finisterre Terrane rather than from sequential docking of independently colliding blocks of the terrane. Moreover, we conclude that these paleomagnetic declinations result mainly from South Bismarck Plate motion, and not decoupled rotation of the crustal terrane independent of the underlying lithosphere. We examine models of a syn-collisional remagnetization with both fixed and migrating Euler poles of South Bismarck/Australia plate relative motion, and suggest that the Euler pole describing South Bismarck Plate motion has migrated southwestward to its present location on the collision suture in response to the propagating collision. This plate kinematic model agrees with the variability in depth of the seismogenic slab beneath the collision zone. Our best-fit model of pole migration describes South Bismarck/Australia relative motion producing a highly oblique collision in its early stages, with the Finisterre Arc Terrane converging along a left-lateral Ramu-Markham suture, gradually changing to the nearly orthogonal convergence observed today.

A Simplified Mesh Deformation Method Using Commercial Structural Analysis Software

NASA Technical Reports Server (NTRS)

Hsu, Su-Yuen; Chang, Chau-Lyan; Samareh, Jamshid

2004-01-01

Mesh deformation in response to redefined or moving aerodynamic surface geometries is a frequently encountered task in many applications. Most existing methods are either mathematically too complex or computationally too expensive for usage in practical design and optimization. We propose a simplified mesh deformation method based on linear elastic finite element analyses that can be easily implemented by using commercially available structural analysis software. Using a prescribed displacement at the mesh boundaries, a simple structural analysis is constructed based on a spatially varying Young s modulus to move the entire mesh in accordance with the surface geometry redefinitions. A variety of surface movements, such as translation, rotation, or incremental surface reshaping that often takes place in an optimization procedure, may be handled by the present method. We describe the numerical formulation and implementation using the NASTRAN software in this paper. The use of commercial software bypasses tedious reimplementation and takes advantage of the computational efficiency offered by the vendor. A two-dimensional airfoil mesh and a three-dimensional aircraft mesh were used as test cases to demonstrate the effectiveness of the proposed method. Euler and Navier-Stokes calculations were performed for the deformed two-dimensional meshes.

Analysis

NASA Astrophysics Data System (ADS)

Abdelazeem, Maha; El-Sawy, El-Sawy K.; Gobashy, Mohamed M.

2013-06-01

Ar Rika fault zone constitutes one of the two major parts of the NW-SE Najd fault system (NFS), which is one of the most prominent structural features located in the east of the center of the Arabian Shield, Saudi Arabia. By using Enhancement Thematic Mapper data (ETM+) and Principle Component Analysis (PCA), surface geological characteristics, distribution of rock types, and the different trends of linear features and faults are determined in the study area. First and second order magnetic gradients of the geomagnetic field at the North East of Wadi Ar Rika have been calculated in the frequency domain to map both surface and subsurface lineaments and faults. Lineaments as deduced from previous studies, suggest an extension of the NFS beneath the cover rocks in the study area. In the present study, integration of magnetic gradients and remote sensing analysis that resulted in different valuable derivative maps confirm the subsurface extension of some of the surface features. The 3D Euler deconvolution, the total gradient, and the tilt angle maps have been utilized to determine accurately the distribution of shear zones, the tectonic implications, and the internal structures of the terranes in the Ar Rika quadrangle in three dimensions.

Aeroelastic Uncertainty Quantification Studies Using the S4T Wind Tunnel Model

NASA Technical Reports Server (NTRS)

Nikbay, Melike; Heeg, Jennifer

2017-01-01

This paper originates from the joint efforts of an aeroelastic study team in the Applied Vehicle Technology Panel from NATO Science and Technology Organization, with the Task Group number AVT-191, titled "Application of Sensitivity Analysis and Uncertainty Quantification to Military Vehicle Design." We present aeroelastic uncertainty quantification studies using the SemiSpan Supersonic Transport wind tunnel model at the NASA Langley Research Center. The aeroelastic study team decided treat both structural and aerodynamic input parameters as uncertain and represent them as samples drawn from statistical distributions, propagating them through aeroelastic analysis frameworks. Uncertainty quantification processes require many function evaluations to asses the impact of variations in numerous parameters on the vehicle characteristics, rapidly increasing the computational time requirement relative to that required to assess a system deterministically. The increased computational time is particularly prohibitive if high-fidelity analyses are employed. As a remedy, the Istanbul Technical University team employed an Euler solver in an aeroelastic analysis framework, and implemented reduced order modeling with Polynomial Chaos Expansion and Proper Orthogonal Decomposition to perform the uncertainty propagation. The NASA team chose to reduce the prohibitive computational time by employing linear solution processes. The NASA team also focused on determining input sample distributions.

Analytical Kinematics and Coupled Vibrations Analysis of Mechanical System Operated by Solar Array Drive Assembly

NASA Astrophysics Data System (ADS)

Sattar, M.; Wei, C.; Jalali, A.; Sattar, R.

2017-07-01

To address the impact of solar array (SA) anomalies and vibrations on performance of precision space-based operations, it is important to complete its accurate jitter analysis. This work provides mathematical modelling scheme to approximate kinematics and coupled micro disturbance dynamics of rigid load supported and operated by solar array drive assembly (SADA). SADA employed in analysis provides a step wave excitation torque to activate the system. Analytical investigations into kinematics is accomplished by using generalized linear and Euler angle coordinates, applying multi-body dynamics concepts and transformations principles. Theoretical model is extended, to develop equations of motion (EoM), through energy method (Lagrange equation). The main emphasis is to research coupled frequency response by determining energies dissipated and observing dynamic behaviour of internal vibratory systems of SADA. The disturbance model captures discrete active harmonics of SADA, natural modes and vibration amplifications caused by interactions between active harmonics and structural modes of mechanical assembly. The proposed methodology can help to predict true micro disturbance nature of SADA operating rigid load. Moreover, performance outputs may be compared against actual mission requirements to assess precise spacecraft controller design to meet next space generation stringent accuracy goals.

Analyzing Aeroelasticity in Turbomachines

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Srivastava, R.

2003-01-01

ASTROP2-LE is a computer program that predicts flutter and forced responses of blades, vanes, and other components of such turbomachines as fans, compressors, and turbines. ASTROP2-LE is based on the ASTROP2 program, developed previously for analysis of stability of turbomachinery components. In developing ASTROP2- LE, ASTROP2 was modified to include a capability for modeling forced responses. The program was also modified to add a capability for analysis of aeroelasticity with mistuning and unsteady aerodynamic solutions from another program, LINFLX2D, that solves the linearized Euler equations of unsteady two-dimensional flow. Using LINFLX2D to calculate unsteady aerodynamic loads, it is possible to analyze effects of transonic flow on flutter and forced response. ASTROP2-LE can be used to analyze subsonic, transonic, and supersonic aerodynamics and structural mistuning for rotors with blades of differing structural properties. It calculates the aerodynamic damping of a blade system operating in airflow so that stability can be assessed. The code also predicts the magnitudes and frequencies of the unsteady aerodynamic forces on the airfoils of a blade row from incoming wakes. This information can be used in high-cycle fatigue analysis to predict the fatigue lives of the blades.

An explicit plate kinematic model for the orogeny in the southern Uralides

NASA Astrophysics Data System (ADS)

Görz, Ines; Hielscher, Peggy

2010-10-01

The Palaeozoic Uralides formed in a three plate constellation between Europe, Siberia and Kazakhstan-Tarim. Starting from the first plate tectonic concepts, it was controversially discussed, whether the Uralide orogeny was the result of a relative plate motion between Europe and Siberia or between Europe and Kazakhstan. In this study, we use a new approach to address this problem. We perform a structural analysis on the sphere, reconstruct the positions of the Euler poles of the relative plate rotation Siberia-Europe and Tarim-Europe and describe Uralide structures by their relation to small circles about the two Euler poles. Using this method, changes in the strike of tectonic elements that are caused by the spherical geometry of the Earth's surface are eliminated and structures that are compatible with one of the relative plate motions can be identified. We show that only two Euler poles controlled the Palaeozoic tectonic evolution in the whole West Siberian region, but that they acted diachronously in different regions. We provide an explicit model describing the tectonism in West Siberia by an Euler pole, a sense of rotation and an approximate rotation angle. In the southern Uralides, Devonian structures resulted from a plate rotation of Siberia with respect to Europe, while the Permian structures were caused by a relative plate motion of Kazakhstan-Tarim with respect to Europe. The tectonic pause in the Carboniferous period correlates with a reorganization of the plate kinematics.

Droplets size evolution of dispersion in a stirred tank

NASA Astrophysics Data System (ADS)

Kysela, Bohus; Konfrst, Jiri; Chara, Zdenek; Sulc, Radek; Jasikova, Darina

2018-06-01

Dispersion of two immiscible liquids is commonly used in chemical industry as wall as in metallurgical industry e. g. extraction process. The governing property is droplet size distribution. The droplet sizes are given by the physical properties of both liquids and flow properties inside a stirred tank. The first investigation stage is focused on in-situ droplet size measurement using image analysis and optimizing of the evaluation method to achieve maximal result reproducibility. The obtained experimental results are compared with multiphase flow simulation based on Euler-Euler approach combined with PBM (Population Balance Modelling). The population balance model was, in that specific case, simplified with assumption of pure breakage of droplets.

Langley's Computational Efforts in Sonic-Boom Softening of the Boeing HSCT

NASA Technical Reports Server (NTRS)

Fouladi, Kamran

1999-01-01

NASA Langley's computational efforts in the sonic-boom softening of the Boeing high-speed civil transport are discussed in this paper. In these efforts, an optimization process using a higher order Euler method for analysis was employed to reduce the sonic boom of a baseline configuration through fuselage camber and wing dihedral modifications. Fuselage modifications did not provide any improvements, but the dihedral modifications were shown to be an important tool for the softening process. The study also included aerodynamic and sonic-boom analyses of the baseline and some of the proposed "softened" configurations. Comparisons of two Euler methodologies and two propagation programs for sonic-boom predictions are also discussed in the present paper.

Mode Identification of High-Amplitude Pressure Waves in Liquid Rocket Engines

NASA Astrophysics Data System (ADS)

EBRAHIMI, R.; MAZAHERI, K.; GHAFOURIAN, A.

2000-01-01

Identification of existing instability modes from experimental pressure measurements of rocket engines is difficult, specially when steep waves are present. Actual pressure waves are often non-linear and include steep shocks followed by gradual expansions. It is generally believed that interaction of these non-linear waves is difficult to analyze. A method of mode identification is introduced. After presumption of constituent modes, they are superposed by using a standard finite difference scheme for solution of the classical wave equation. Waves are numerically produced at each end of the combustion tube with different wavelengths, amplitudes, and phases with respect to each other. Pressure amplitude histories and phase diagrams along the tube are computed. To determine the validity of the presented method for steep non-linear waves, the Euler equations are numerically solved for non-linear waves, and negligible interactions between these waves are observed. To show the applicability of this method, other's experimental results in which modes were identified are used. Results indicate that this simple method can be used in analyzing complicated pressure signal measurements.

Optimized Hypernetted-Chain Solutions for Helium -4 Surfaces and Metal Surfaces

NASA Astrophysics Data System (ADS)

Qian, Guo-Xin

This thesis is a study of inhomogeneous Bose systems such as liquid ('4)He slabs and inhomogeneous Fermi systems such as the electron gas in metal films, at zero temperature. Using a Jastrow-type many-body wavefunction, the ground state energy is expressed by means of Bogoliubov-Born-Green-Kirkwood -Yvon and Hypernetted-Chain techniques. For Bose systems, Euler-Lagrange equations are derived for the one- and two -body functions and systematic approximation methods are physically motivated. It is shown that the optimized variational method includes a self-consistent summation of ladder- and ring-diagrams of conventional many-body theory. For Fermi systems, a linear potential model is adopted to generate the optimized Hartree-Fock basis. Euler-Lagrange equations are derived for the two-body correlations which serve to screen the strong bare Coulomb interaction. The optimization of the pair correlation leads to an expression of correlation energy in which the state averaged RPA part is separated. Numerical applications are presented for the density profile and pair distribution function for both ('4)He surfaces and metal surfaces. Both the bulk and surface energies are calculated in good agreement with experiments.

Solution-Adaptive Cartesian Cell Approach for Viscous and Inviscid Flows

NASA Technical Reports Server (NTRS)

Coirier, William J.; Powell, Kenneth G.

1996-01-01

A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

The Riemann problem for the relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation

NASA Astrophysics Data System (ADS)

Shao, Zhiqiang

2018-04-01

The relativistic full Euler system with generalized Chaplygin proper energy density-pressure relation is studied. The Riemann problem is solved constructively. The delta shock wave arises in the Riemann solutions, provided that the initial data satisfy some certain conditions, although the system is strictly hyperbolic and the first and third characteristic fields are genuinely nonlinear, while the second one is linearly degenerate. There are five kinds of Riemann solutions, in which four only consist of a shock wave and a centered rarefaction wave or two shock waves or two centered rarefaction waves, and a contact discontinuity between the constant states (precisely speaking, the solutions consist in general of three waves), and the other involves delta shocks on which both the rest mass density and the proper energy density simultaneously contain the Dirac delta function. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. The formation mechanism, generalized Rankine-Hugoniot relation and entropy condition are clarified for this type of delta shock wave. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of solutions involving delta shocks for the Riemann problem.

Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by preconditioned nonlinear conjugate gradient methods

NASA Astrophysics Data System (ADS)

Antoine, Xavier; Levitt, Antoine; Tang, Qinglin

2017-08-01

We propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial discretization scheme for computing the ground states (GS) of rotating Bose-Einstein condensates (BEC), modeled by the Gross-Pitaevskii Equation (GPE). We first start by reviewing the classical gradient flow (also known as imaginary time (IMT)) method which considers the problem from the PDE standpoint, leading to numerically solve a dissipative equation. Based on this IMT equation, we analyze the forward Euler (FE), Crank-Nicolson (CN) and the classical backward Euler (BE) schemes for linear problems and recognize classical power iterations, allowing us to derive convergence rates. By considering the alternative point of view of minimization problems, we propose the preconditioned steepest descent (PSD) and conjugate gradient (PCG) methods for the GS computation of the GPE. We investigate the choice of the preconditioner, which plays a key role in the acceleration of the convergence process. The performance of the new algorithms is tested in 1D, 2D and 3D. We conclude that the PCG method outperforms all the previous methods, most particularly for 2D and 3D fast rotating BECs, while being simple to implement.

The History of the Planar Elastica: Insights into Mechanics and Scientific Method

ERIC Educational Resources Information Center

Goss, Victor Geoffrey Alan

2009-01-01

Euler's formula for the buckling of an elastic column is widely used in engineering design. However, only a handful of engineers will be familiar with Euler's classic paper "De Curvis Elasticis" in which the formula is derived. In addition to the Euler Buckling Formula, "De Curvis Elasticis" classifies all the bent configurations of elastic rod--a…

The importance of Leonhard Euler's discoveries in the field of shipbuilding for the scientific evolution of academician A. N. Krylov

NASA Astrophysics Data System (ADS)

Sharkov, N. A.; Sharkova, O. A.

2018-05-01

The paper identifies the importance of the Leonhard Euler's discoveries in the field of shipbuilding for the scientific evolution of academician A. N. Krylov and for the modern knowledge in survivability and safety of ships. The works by Leonard Euler "Marine Science" and "The Moon Motion New Theory" are discussed.

On the statistical mechanics of the 2D stochastic Euler equation

NASA Astrophysics Data System (ADS)

Bouchet, Freddy; Laurie, Jason; Zaboronski, Oleg

2011-12-01

The dynamics of vortices and large scale structures is qualitatively very different in two dimensional flows compared to its three dimensional counterparts, due to the presence of multiple integrals of motion. These are believed to be responsible for a variety of phenomena observed in Euler flow such as the formation of large scale coherent structures, the existence of meta-stable states and random abrupt changes in the topology of the flow. In this paper we study stochastic dynamics of the finite dimensional approximation of the 2D Euler flow based on Lie algebra su(N) which preserves all integrals of motion. In particular, we exploit rich algebraic structure responsible for the existence of Euler's conservation laws to calculate the invariant measures and explore their properties and also study the approach to equilibrium. Unexpectedly, we find deep connections between equilibrium measures of finite dimensional su(N) truncations of the stochastic Euler equations and random matrix models. Our work can be regarded as a preparation for addressing the questions of large scale structures, meta-stability and the dynamics of random transitions between different flow topologies in stochastic 2D Euler flows.

3D GIS spatial operation based on extended Euler operators

NASA Astrophysics Data System (ADS)

Xu, Hongbo; Lu, Guonian; Sheng, Yehua; Zhou, Liangchen; Guo, Fei; Shang, Zuoyan; Wang, Jing

2008-10-01

The implementation of 3 dimensions spatial operations, based on certain data structure, has a lack of universality and is not able to treat with non-manifold cases, at present. ISO/DIS 19107 standard just presents the definition of Boolean operators and set operators for topological relationship query, and OGC GeoXACML gives formal definitions for several set functions without implementation detail. Aiming at these problems, based mathematical foundation on cell complex theory, supported by non-manifold data structure and using relevant research in the field of non-manifold geometry modeling for reference, firstly, this paper according to non-manifold Euler-Poincaré formula constructs 6 extended Euler operators and inverse operators to carry out creating, updating and deleting 3D spatial elements, as well as several pairs of supplementary Euler operators to convenient for implementing advanced functions. Secondly, we change topological element operation sequence of Boolean operation and set operation as well as set functions defined in GeoXACML into combination of extended Euler operators, which separates the upper functions and lower data structure. Lastly, we develop underground 3D GIS prototype system, in which practicability and credibility of extended Euler operators faced to 3D GIS presented by this paper are validated.

Euler force actuation mechanism for siphon valving in compact disk-like microfluidic chips

PubMed Central

Deng, Yongbo; Fan, Jianhua; Zhou, Song; Zhou, Teng; Wu, Junfeng; Li, Yin; Liu, Zhenyu; Xuan, Ming; Wu, Yihui

2014-01-01

Based on the Euler force induced by the acceleration of compact disk (CD)-like microfluidic chip, this paper presents a novel actuation mechanism for siphon valving. At the preliminary stage of acceleration, the Euler force in the tangential direction of CD-like chip takes the primary place compared with the centrifugal force to function as the actuation of the flow, which fills the siphon and actuates the siphon valving. The Euler force actuation mechanism is demonstrated by the numerical solution of the phase-field based mathematical model for the flow in siphon valve. In addition, experimental validation is implemented in the polymethylmethacrylate-based CD-like microfluidic chip manufactured using CO2 laser engraving technique. To prove the application of the proposed Euler force actuation mechanism, whole blood separation and plasma extraction has been conducted using the Euler force actuated siphon valving. The newly introduced actuation mechanism overcomes the dependence on hydrophilic capillary filling of siphon by avoiding external manipulation or surface treatments of polymeric material. The sacrifice for highly integrated processing in pneumatic pumping technique is also prevented by excluding the volume-occupied compressed air chamber. PMID:24753736

A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov

NASA Astrophysics Data System (ADS)

Greenough, J. A.; Rider, W. J.

2004-05-01

A numerical study is undertaken comparing a fifth-order version of the weighted essentially non-oscillatory numerical (WENO5) method to a modern piecewise-linear, second-order, version of Godunov's (PLMDE) method for the compressible Euler equations. A series of one-dimensional test problems are examined beginning with classical linear problems and ending with complex shock interactions. The problems considered are: (1) linear advection of a Gaussian pulse in density, (2) Sod's shock tube problem, (3) the "peak" shock tube problem, (4) a version of the Shu and Osher shock entropy wave interaction and (5) the Woodward and Colella interacting shock wave problem. For each problem and method, run times, density error norms and convergence rates are reported for each method as produced from a common code test-bed. The linear problem exhibits the advertised convergence rate for both methods as well as the expected large disparity in overall error levels; WENO5 has the smaller errors and an enormous advantage in overall efficiency (in accuracy per unit CPU time). For the nonlinear problems with discontinuities, however, we generally see both first-order self-convergence of error as compared to an exact solution, or when an analytic solution is not available, a converged solution generated on an extremely fine grid. The overall comparison of error levels shows some variation from problem to problem. For Sod's shock tube, PLMDE has nearly half the error, while on the peak problem the errors are nearly the same. For the interacting blast wave problem the two methods again produce a similar level of error with a slight edge for the PLMDE. On the other hand, for the Shu-Osher problem, the errors are similar on the coarser grids, but favors WENO by a factor of nearly 1.5 on the finer grids used. In all cases holding mesh resolution constant though, PLMDE is less costly in terms of CPU time by approximately a factor of 6. If the CPU cost is taken as fixed, that is run times are equal for both numerical methods, then PLMDE uniformly produces lower errors than WENO for the fixed computation cost on the test problems considered here.

"Astronomica" in the Correspondence between Leonhard Euler and Daniel Bernoull (German Title: "Astronomica" im Briefwechsel zwischen Leonhard Euler und Daniel Bernoulli)

NASA Astrophysics Data System (ADS)

Verdun, Andreas

2010-12-01

The Euler Commission of the Swiss Academy of Sciences intends to terminate the edition of Leonhard Euler's works in the next year 2011 after nearly one hundred years since the beginning of the editorial works. These works include, e.g., Volume 3 of the Series quarta A which will contain the correspondence between Leonhard Euler (1707-1783) and Daniel Bernoulli (1700-1783) and which is currently being edited by Dr. Emil A. Fellmann (Basel) and Prof. Dr. Gleb K. Mikhailov (Moscow). This correspondence contains more than hundred letters, principally from Daniel Bernoulli to Euler. Parts of this correspondence were published uncommented already in 1843. It is astonishing that, apart from mathematics and physics (mainly mechanics and hydrodynamics), many topics addressed concern astronomy. The major part of the preserved correspondence between Euler and Daniel Bernoulli, in which astronomical themes are discussed, concerns celestial mechanics as the dominant discipline of theoretical astronomy of the eighteenth century. It was triggered and coined mainly by the prize questions of the Paris Academy of Science. In more than two thirds of the letters current problems and questions concerning celestial mechanics of that time are treated, focusing on the lunar theory and the great inequality in the motions of Jupiter and Saturn as special applications of the three body problem. In the remaining letters, problems concerning spherical astronomy are solved and attempts are made to explain certain phenomena in the field of "cosmic physics" concerning astronomical observations.

Nonlinear Earthquake Analysis of Reinforced Concrete Frames with Fiber and Bernoulli-Euler Beam-Column Element

PubMed Central

Karaton, Muhammet

2014-01-01

A beam-column element based on the Euler-Bernoulli beam theory is researched for nonlinear dynamic analysis of reinforced concrete (RC) structural element. Stiffness matrix of this element is obtained by using rigidity method. A solution technique that included nonlinear dynamic substructure procedure is developed for dynamic analyses of RC frames. A predicted-corrected form of the Bossak-α method is applied for dynamic integration scheme. A comparison of experimental data of a RC column element with numerical results, obtained from proposed solution technique, is studied for verification the numerical solutions. Furthermore, nonlinear cyclic analysis results of a portal reinforced concrete frame are achieved for comparing the proposed solution technique with Fibre element, based on flexibility method. However, seismic damage analyses of an 8-story RC frame structure with soft-story are investigated for cases of lumped/distributed mass and load. Damage region, propagation, and intensities according to both approaches are researched. PMID:24578667

Paleomagnetic study of an active arc-continent collision, Finisterre Arc Terrane, Papua New Guinea

NASA Astrophysics Data System (ADS)

Weiler, Peter Donald

1999-12-01

This dissertation includes 3 studies from the active collision zone between the Finisterre volcanic arc and Papua New Guinea. Chapter 1 is a paleomagnetic study of thrust sheets of the fold and thrust belt north of the Ramu-Markham suture indicating very rapid vertical-axis rotations related to tectonic transport of thrust units. Our data indicate that rotations as great as 90° since 1 Ma have occurred locally in the Erap Valley area. Such rapid rotations during thrust sheet emplacement may be more common in fold and thrust belts than is presently recognized. Anisotropy of magnetic susceptibility (AMS) lineations are rendered parallel by the same rotations used to restore the paleomagnetic remanence to N-S thus independently confirming the rapid rotations. In Chapter 2, we compare the AMS fabrics from the Erap Valley with microscopic shape fabrics obtained through digital image analysis. We find that the orientations of principal axes found by the two techniques agree very well, but that the maximum and intermediate axes of the magnetic fabric are inverted relative to the grain shape. We interpret the shape fabric as a primary depositional fabric, and the magnetic fabric as the result of a weak tectonic strain overprinting a depositional fabric. Thus, comparison of these fabrics detects the earliest transition from depositional to tectonic strain fabric. Finally, in Chapter 3, we turn to larger scale paleomagnetic results from the colliding Finisterre Arc. Hemipelagic rocks possess a syn-collisional remagnetization indicating a clockwise rotation of the colliding terrane through about 40° in post-Miocene time. Decreasing paleomagnetic declination anomalies as a function of along-strike distance in the Finisterre Terrane, analyzed by our preferred model of a linear remagnetization and a migrating Euler pole, suggests an average rotation rate of 8°/Ma. Thus, we propose that the rotation results from a rigid-body rotation of the Finisterre Terrane rather than from sequential docking of independently colliding blocks. We examine models of a syn-collisional remagnetization with both fixed and migrating Euler poles, and suggest that the Euler pole describing Bismarck/Australia plate motion may have migrated 675 km through post-Miocene time to its present location at the collision suture.

Pressure wave propagation studies for oscillating cascades

NASA Technical Reports Server (NTRS)

Huff, Dennis L.

1992-01-01

The unsteady flow field around an oscillating cascade of flat plates is studied using a time marching Euler code. Exact solutions based on linear theory serve as model problems to study pressure wave propagation in the numerical solution. The importance of using proper unsteady boundary conditions, grid resolution, and time step is demonstrated. Results show that an approximate non-reflecting boundary condition based on linear theory does a good job of minimizing reflections from the inflow and outflow boundaries and allows the placement of the boundaries to be closer than cases using reflective boundary conditions. Stretching the boundary to dampen the unsteady waves is another way to minimize reflections. Grid clustering near the plates does a better job of capturing the unsteady flow field than cases using uniform grids as long as the CFL number is less than one for a sufficient portion of the grid. Results for various stagger angles and oscillation frequencies show good agreement with linear theory as long as the grid is properly resolved.

Research in Parallel Algorithms and Software for Computational Aerosciences

DOT National Transportation Integrated Search

1996-04-01

Phase I is complete for the development of a Computational Fluid Dynamics : with automatic grid generation and adaptation for the Euler : analysis of flow over complex geometries. SPLITFLOW, an unstructured Cartesian : grid code developed at Lockheed...

Analysis of gravity data beneath Endut geothermal prospect using horizontal gradient and Euler deconvolution

NASA Astrophysics Data System (ADS)

Supriyanto, Noor, T.; Suhanto, E.

2017-07-01

The Endut geothermal prospect is located in Banten Province, Indonesia. The geological setting of the area is dominated by quaternary volcanic, tertiary sediments and tertiary rock intrusion. This area has been in the preliminary study phase of geology, geochemistry, and geophysics. As one of the geophysical study, the gravity data measurement has been carried out and analyzed in order to understand geological condition especially subsurface fault structure that control the geothermal system in Endut area. After precondition applied to gravity data, the complete Bouguer anomaly have been analyzed using advanced derivatives method such as Horizontal Gradient (HG) and Euler Deconvolution (ED) to clarify the existance of fault structures. These techniques detected boundaries of body anomalies and faults structure that were compared with the lithologies in the geology map. The analysis result will be useful in making a further realistic conceptual model of the Endut geothermal area.

Accuracy of AFM force distance curves via direct solution of the Euler-Bernoulli equation

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

Eppell, Steven J., E-mail: steven.eppell@case.edu; Liu, Yehe; Zypman, Fredy R.

2016-03-15

In an effort to improve the accuracy of force-separation curves obtained from atomic force microscope data, we compare force-separation curves computed using two methods to solve the Euler-Bernoulli equation. A recently introduced method using a direct sequential forward solution, Causal Time-Domain Analysis, is compared against a previously introduced Tikhonov Regularization method. Using the direct solution as a benchmark, it is found that the regularization technique is unable to reproduce accurate curve shapes. Using L-curve analysis and adjusting the regularization parameter, λ, to match either the depth or the full width at half maximum of the force curves, the two techniquesmore » are contrasted. Matched depths result in full width at half maxima that are off by an average of 27% and matched full width at half maxima produce depths that are off by an average of 109%.« less

Spherical Coordinate Systems for Streamlining Suited Mobility Analysis

NASA Technical Reports Server (NTRS)

Benson, Elizabeth; Cowley, Matthew; Harvill, Lauren; Rajulu. Sudhakar

2015-01-01

Introduction: When describing human motion, biomechanists generally report joint angles in terms of Euler angle rotation sequences. However, there are known limitations in using this method to describe complex motions such as the shoulder joint during a baseball pitch. Euler angle notation uses a series of three rotations about an axis where each rotation is dependent upon the preceding rotation. As such, the Euler angles need to be regarded as a set to get accurate angle information. Unfortunately, it is often difficult to visualize and understand these complex motion representations. It has been shown that using a spherical coordinate system allows Anthropometry and Biomechanics Facility (ABF) personnel to increase their ability to transmit important human mobility data to engineers, in a format that is readily understandable and directly translatable to their design efforts. Objectives: The goal of this project was to use innovative analysis and visualization techniques to aid in the examination and comprehension of complex motions. Methods: This project consisted of a series of small sub-projects, meant to validate and verify a new method before it was implemented in the ABF's data analysis practices. A mechanical test rig was built and tracked in 3D using an optical motion capture system. Its position and orientation were reported in both Euler and spherical reference systems. In the second phase of the project, the ABF estimated the error inherent in a spherical coordinate system, and evaluated how this error would vary within the reference frame. This stage also involved expanding a kinematic model of the shoulder to include the rest of the joints of the body. The third stage of the project involved creating visualization methods to assist in interpreting motion in a spherical frame. These visualization methods will be incorporated in a tool to evaluate a database of suited mobility data, which is currently in development. Results: Initial results demonstrated that a spherical coordinate system is helpful in describing and visualizing the motion of a space suit. The system is particularly useful in describing the motion of the shoulder, where multiple degrees of freedom can lead to very complex motion paths.

Splitting methods for low Mach number Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Abarbanel, Saul; Dutt, Pravir; Gottlieb, David

1987-01-01

Examined are some splitting techniques for low Mach number Euler flows. Shortcomings of some of the proposed methods are pointed out and an explanation for their inadequacy suggested. A symmetric splitting for both the Euler and Navier-Stokes equations is then presented which removes the stiffness of these equations when the Mach number is small. The splitting is shown to be stable.

A hierarchy for modeling high speed propulsion systems

NASA Technical Reports Server (NTRS)

Hartley, Tom T.; Deabreu, Alex

1991-01-01

General research efforts on reduced order propulsion models for control systems design are overviewed. Methods for modeling high speed propulsion systems are discussed including internal flow propulsion systems that do not contain rotating machinery such as inlets, ramjets, and scramjets. The discussion is separated into four sections: (1) computational fluid dynamics model for the entire nonlinear system or high order nonlinear models; (2) high order linearized model derived from fundamental physics; (3) low order linear models obtained from other high order models; and (4) low order nonlinear models. Included are special considerations on any relevant control system designs. The methods discussed are for the quasi-one dimensional Euler equations of gasdynamic flow. The essential nonlinear features represented are large amplitude nonlinear waves, moving normal shocks, hammershocks, subsonic combustion via heat addition, temperature dependent gases, detonation, and thermal choking.

Three-Dimensional Viscous Alternating Direction Implicit Algorithm and Strategies for Shape Optimization

NASA Technical Reports Server (NTRS)

Pandya, Mohagna J.; Baysal, Oktay

1997-01-01

A gradient-based shape optimization based on quasi-analytical sensitivities has been extended for practical three-dimensional aerodynamic applications. The flow analysis has been rendered by a fully implicit, finite-volume formulation of the Euler and Thin-Layer Navier-Stokes (TLNS) equations. Initially, the viscous laminar flow analysis for a wing has been compared with an independent computational fluid dynamics (CFD) code which has been extensively validated. The new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4 with coarse- and fine-grid based computations performed with Euler and TLNS equations. The influence of the initial constraints on the geometry and aerodynamics of the optimized shape has been explored. Various final shapes generated for an identical initial problem formulation but with different optimization path options (coarse or fine grid, Euler or TLNS), have been aerodynamically evaluated via a common fine-grid TLNS-based analysis. The initial constraint conditions show significant bearing on the optimization results. Also, the results demonstrate that to produce an aerodynamically efficient design, it is imperative to include the viscous physics in the optimization procedure with the proper resolution. Based upon the present results, to better utilize the scarce computational resources, it is recommended that, a number of viscous coarse grid cases using either a preconditioned bi-conjugate gradient (PbCG) or an alternating-direction-implicit (ADI) method, should initially be employed to improve the optimization problem definition, the design space and initial shape. Optimized shapes should subsequently be analyzed using a high fidelity (viscous with fine-grid resolution) flow analysis to evaluate their true performance potential. Finally, a viscous fine-grid-based shape optimization should be conducted, using an ADI method, to accurately obtain the final optimized shape.

Navier-Stokes analysis of an oxidizer turbine blade with tip clearance

NASA Technical Reports Server (NTRS)

Gibeling, Howard J.; Sabnis, Jayant S.

1992-01-01

The Gas Generator Oxidizer Turbine (GGOT) Blade is being analyzed by various investigators under the NASA MSFC sponsored Turbine Stage Technology Team design effort. The present work concentrates on the tip clearance region flow and associated losses; however, flow details for the passage region are also obtained in the simulations. The present calculations simulate the rotor blade row in a rotating reference frame with the appropriate coriolis and centrifugal acceleration terms included in the momentum equation. The upstream computational boundary is located about one axial chord from the blade leading edge. The boundary conditions at this location were determined by using a Euler analysis without the vanes to obtain approximately the same flow profiles at the rotor as were obtained with the Euler stage analysis including the vanes. Inflow boundary layer profiles are then constructed assuming the skin friction coefficient at both the hub and the casing. The downstream computational boundary is located about one axial chord from the blade trailing edge, and the circumferentially averaged static pressure at this location was also obtained from the Euler analysis. Results were obtained for the 3-D baseline GGOT geometry at the full scale design Reynolds number. Details of the clearance region flow behavior and blade pressure distributions were computed. The spanwise variation in blade loading distributions are shown, and circumferentially averaged spanwise distributions of total pressure, total temperature, Mach number, and flow angle are shown at several axial stations. The spanwise variation of relative total pressure loss shows a region of high loss in the region near the casing. Particle traces in the near tip region show vortical behavior of the fluid which passes through the clearance region and exits at the downstream edge of the gap.

Quantum calculus of classical vortex images, integrable models and quantum states

NASA Astrophysics Data System (ADS)

Pashaev, Oktay K.

2016-10-01

From two circle theorem described in terms of q-periodic functions, in the limit q→1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrodinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.

Radiation of sound from unflanged cylindrical ducts

NASA Technical Reports Server (NTRS)

Hartharan, S. L.; Bayliss, A.

1983-01-01

Calculations of sound radiated from unflanged cylindrical ducts are presented. The numerical simulation models the problem of an aero-engine inlet. The time dependent linearized Euler equations are solved from a state of rest until a harmonic solution is attained. A fourth order accurate finite difference scheme is used and solutions are obtained from a fully vectorized Cyber-203 computer program. Cases of both plane waves and spin modes are treated. Spin modes model the sound generated by a turbofan engine. Boundary conditions for both plane waves and spin modes are treated. Solutions obtained are compared with experiments conducted at NASA Langley Research Center.

Multigrid methods in structural mechanics

NASA Technical Reports Server (NTRS)

Raju, I. S.; Bigelow, C. A.; Taasan, S.; Hussaini, M. Y.

1986-01-01

Although the application of multigrid methods to the equations of elasticity has been suggested, few such applications have been reported in the literature. In the present work, multigrid techniques are applied to the finite element analysis of a simply supported Bernoulli-Euler beam, and various aspects of the multigrid algorithm are studied and explained in detail. In this study, six grid levels were used to model half the beam. With linear prolongation and sequential ordering, the multigrid algorithm yielded results which were of machine accuracy with work equivalent to 200 standard Gauss-Seidel iterations on the fine grid. Also with linear prolongation and sequential ordering, the V(1,n) cycle with n greater than 2 yielded better convergence rates than the V(n,1) cycle. The restriction and prolongation operators were derived based on energy principles. Conserving energy during the inter-grid transfers required that the prolongation operator be the transpose of the restriction operator, and led to improved convergence rates. With energy-conserving prolongation and sequential ordering, the multigrid algorithm yielded results of machine accuracy with a work equivalent to 45 Gauss-Seidel iterations on the fine grid. The red-black ordering of relaxations yielded solutions of machine accuracy in a single V(1,1) cycle, which required work equivalent to about 4 iterations on the finest grid level.

Green's function and Bloch theory for the analysis of the dynamic response of a periodically supported beam to a moving load

NASA Astrophysics Data System (ADS)

Lassoued, R.; Lecheheb, M.; Bonnet, G.

2012-08-01

This paper describes an analytical method for the wave field induced by a moving load on a periodically supported beam. The Green's function for an Euler beam without support is evaluated by using the direct integration. Afterwards, it introduces the supports into the model established by using the superposition principle which states that the response from all the sleeper points and from the external point force add up linearly to give a total response. The periodicity of the supports is described by Bloch's theorem. The homogeneous system thus obtained represents a linear differential equation which governs rail response. It is initially solved in the homogeneous case, and it admits a no null solution if its determinant is null, this permits the establishment the dispersion equation to Bloch waves and wave bands. The Bloch waves and dispersion curves contain all the physics of the dynamic problem and the wave field induced by a dynamic load applied to the system is finally obtained by decomposition into Bloch waves, similarly to the usual decomposition into dynamic modes on a finite structure. The method is applied to obtain the field induced by a load moving at constant velocity on a thin beam supported by periodic elastic supports.

Impact of Glacial Isostatic Adjustment on North America Plate Specific Terrestrial Reference Frame

NASA Astrophysics Data System (ADS)

Herring, Thomas; Melbourne, Tim; Murray, Mark; Floyd, Mike; Szeliga, Walter; King, Robert; Phillips, David; Puskas, Christine

2017-04-01

We examine the impact of incorporating glacial isostatic adjustment (GIA) models in determining the Euler poles for plate specific terrestrial reference frames. We will specifically examine the impact of GIA models on the realization of a North America Reference frame. We use a combination of the velocity fields determined by the Geodesy Advancing Geosciences and EarthScope (GAGE) Facility which analyzes GPS data from the Plate Boundary Observatory (PBO) and other geodetic quality GPS sites in North America, and from the ITRF2014 re-analysis. Initial analysis of the GAGE velocity field shows reduced root-mean-square (RMS) scatter of velocity estimate residuals when the North America Euler pole is estimated including the ICE-6G GIA mode. The reduction in the north-south direction is from 0.69 mm/yr to 0.52 mm/yr, in the east-west direction from 0.34 mm/yr to 0.30 mm/yr and in height from 0.93 mm/yr to 0.72 mm/yr. The reduction in the height RMS is not surprising since the contemporary geodetic height velocity estimates are used in the developing the ICE-6G model. Contemporary horizontal motions are not used the GIA model development, and the reduction in horizontal RMS provides a partial validation of the model. There is no reduction in the horizontal velocity residual when the ICE-5G model is used. Although removing the ICE-6G model before fitting an Euler pole for the North American plate reduces the RMS of the residuals, the pattern of residuals is still systematic suggesting possibly that a spherically symmetric viscosity model might not be adequate for accurate modeling of the horizontal motions associated with GIA in North America. This presentation in focus on the prospects and impacts of incorporating GIA models in plate-specific Euler poles with emphasis on North America.

Closed-form solution for static pull-in voltage of electrostatically actuated clamped-clamped micro/nano beams under the effect of fringing field and van der Waals force

NASA Astrophysics Data System (ADS)

Bhojawala, V. M.; Vakharia, D. P.

2017-12-01

This investigation provides an accurate prediction of static pull-in voltage for clamped-clamped micro/nano beams based on distributed model. The Euler-Bernoulli beam theory is used adapting geometric non-linearity of beam, internal (residual) stress, van der Waals force, distributed electrostatic force and fringing field effects for deriving governing differential equation. The Galerkin discretisation method is used to make reduced-order model of the governing differential equation. A regime plot is presented in the current work for determining the number of modes required in reduced-order model to obtain completely converged pull-in voltage for micro/nano beams. A closed-form relation is developed based on the relationship obtained from curve fitting of pull-in instability plots and subsequent non-linear regression for the proposed relation. The output of regression analysis provides Chi-square (χ 2) tolerance value equals to 1  ×  10-9, adjusted R-square value equals to 0.999 29 and P-value equals to zero, these statistical parameters indicate the convergence of non-linear fit, accuracy of fitted data and significance of the proposed model respectively. The closed-form equation is validated using available data of experimental and numerical results. The relative maximum error of 4.08% in comparison to several available experimental and numerical data proves the reliability of the proposed closed-form equation.

Aerodynamic Shape Sensitivity Analysis and Design Optimization of Complex Configurations Using Unstructured Grids

NASA Technical Reports Server (NTRS)

Taylor, Arthur C., III; Newman, James C., III; Barnwell, Richard W.

1997-01-01

A three-dimensional unstructured grid approach to aerodynamic shape sensitivity analysis and design optimization has been developed and is extended to model geometrically complex configurations. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater efficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. In this work the nonlinear Euler equations are solved using an upwind, cell-centered, finite-volume scheme. The discrete, linearized systems which result from this scheme are solved iteratively by a preconditioned conjugate-gradient-like algorithm known as GMRES for the two-dimensional geometry and a Gauss-Seidel algorithm for the three-dimensional; similar procedures are used to solve the accompanying linear aerodynamic sensitivity equations in incremental iterative form. As shown, this particular form of the sensitivity equation makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory efficient methods to construct exact Jacobian matrix-vector products. Simple parameterization techniques are utilized for demonstrative purposes. Once the surface has been deformed, the unstructured grid is adapted by considering the mesh as a system of interconnected springs. Grid sensitivities are obtained by differentiating the surface parameterization and the grid adaptation algorithms with ADIFOR (which is an advanced automatic-differentiation software tool). To demonstrate the ability of this procedure to analyze and design complex configurations of practical interest, the sensitivity analysis and shape optimization has been performed for a two-dimensional high-lift multielement airfoil and for a three-dimensional Boeing 747-200 aircraft.

Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation

NASA Technical Reports Server (NTRS)

Park, Michael A.

2002-01-01

Engineering computational fluid dynamics (CFD) analysis and design applications focus on output functions (e.g., lift, drag). Errors in these output functions are generally unknown and conservatively accurate solutions may be computed. Computable error estimates can offer the possibility to minimize computational work for a prescribed error tolerance. Such an estimate can be computed by solving the flow equations and the linear adjoint problem for the functional of interest. The computational mesh can be modified to minimize the uncertainty of a computed error estimate. This robust mesh-adaptation procedure automatically terminates when the simulation is within a user specified error tolerance. This procedure for estimating and adapting to error in a functional is demonstrated for three-dimensional Euler problems. An adaptive mesh procedure that links to a Computer Aided Design (CAD) surface representation is demonstrated for wing, wing-body, and extruded high lift airfoil configurations. The error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly refined grids with ten times as many grid points.

Dynamic modelling and adaptive robust tracking control of a space robot with two-link flexible manipulators under unknown disturbances

NASA Astrophysics Data System (ADS)

Yang, Xinxin; Ge, Shuzhi Sam; He, Wei

2018-04-01

In this paper, both the closed-form dynamics and adaptive robust tracking control of a space robot with two-link flexible manipulators under unknown disturbances are developed. The dynamic model of the system is described with assumed modes approach and Lagrangian method. The flexible manipulators are represented as Euler-Bernoulli beams. Based on singular perturbation technique, the displacements/joint angles and flexible modes are modelled as slow and fast variables, respectively. A sliding mode control is designed for trajectories tracking of the slow subsystem under unknown but bounded disturbances, and an adaptive sliding mode control is derived for slow subsystem under unknown slowly time-varying disturbances. An optimal linear quadratic regulator method is proposed for the fast subsystem to damp out the vibrations of the flexible manipulators. Theoretical analysis validates the stability of the proposed composite controller. Numerical simulation results demonstrate the performance of the closed-loop flexible space robot system.

Sonic Boom Prediction and Minimization of the Douglas Reference OPT5 Configuration

NASA Technical Reports Server (NTRS)

Siclari, Michael J.

1999-01-01

Conventional CFD methods and grids do not yield adequate resolution of the complex shock flow pattern generated by a real aircraft geometry. As a result, a unique grid topology and supersonic flow solver was developed at Northrop Grumman based on the characteristic behavior of supersonic wave patterns emanating from the aircraft. Using this approach, it was possible to compute flow fields with adequate resolution several body lengths below the aircraft. In this region, three-dimensional effects are diminished and conventional two-dimensional modified linear theory (MLT) can be applied to estimate ground pressure signatures or sonic booms. To accommodate real aircraft geometries and alleviate the burdensome grid generation task, an implicit marching multi-block, multi-grid finite-volume Euler code was developed as the basis for the sonic boom prediction methodology. The Thomas two-dimensional extrapolation method is built into the Euler code so that ground signatures can be obtained quickly and efficiently with minimum computational effort suitable to the aircraft design environment. The loudness levels of these signatures can then be determined using a NASA generated noise code. Since the Euler code is a three-dimensional flow field solver, the complete circumferential region below the aircraft is computed. The extrapolation of all this field data from a cylinder of constant radius leads to the definition of the entire boom corridor occurring directly below and off to the side of the aircraft's flight path yielding an estimate for the entire noise "annoyance" corridor in miles as well as its magnitude. An automated multidisciplinary sonic boom design optimization software system was developed during the latter part of HSR Phase 1. Using this system, it was found that sonic boom signatures could be reduced through optimization of a variety of geometric aircraft parameters. This system uses a gradient based nonlinear optimizer as the driver in conjunction with a computationally efficient Euler CFD solver (NIIM3DSB) for computing the three-dimensional near-field characteristics of the aircraft. The intent of the design system is to identify and optimize geometric design variables that have a beneficial impact on the ground sonic boom. The system uses a simple wave drag data format to specify the aircraft geometry. The geometry is internally enhanced and analytic methods are used to generate marching grids suitable for the multi-block Euler solver. The Thomas extrapolation method is integrated into this system, and hence, the aircraft's centerline ground sonic boom signature is also automatically computed for a specified cruise altitude and yields the parameters necessary to evaluate the design function. The entire design system has been automated since the gradient based optimization software requires many flow analyses in order to obtain the required sensitivity derivatives for each design variable in order to converge on an optimal solution. Hence, once the problem is defined which includes defining the objective function and geometric and aerodynamic constraints, the system will automatically regenerate the perturbed geometry, the necessary grids, the Euler solution, and finally the ground sonic boom signature at the request of the optimizer.

Voidage correction algorithm for unresolved Euler-Lagrange simulations

NASA Astrophysics Data System (ADS)

Askarishahi, Maryam; Salehi, Mohammad-Sadegh; Radl, Stefan

2018-04-01

The effect of grid coarsening on the predicted total drag force and heat exchange rate in dense gas-particle flows is investigated using Euler-Lagrange (EL) approach. We demonstrate that grid coarsening may reduce the predicted total drag force and exchange rate. Surprisingly, exchange coefficients predicted by the EL approach deviate more significantly from the exact value compared to results of Euler-Euler (EE)-based calculations. The voidage gradient is identified as the root cause of this peculiar behavior. Consequently, we propose a correction algorithm based on a sigmoidal function to predict the voidage experienced by individual particles. Our correction algorithm can significantly improve the prediction of exchange coefficients in EL models, which is tested for simulations involving Euler grid cell sizes between 2d_p and 12d_p . It is most relevant in simulations of dense polydisperse particle suspensions featuring steep voidage profiles. For these suspensions, classical approaches may result in an error of the total exchange rate of up to 30%.

Recursive linearization of multibody dynamics equations of motion

NASA Technical Reports Server (NTRS)

Lin, Tsung-Chieh; Yae, K. Harold

1989-01-01

The equations of motion of a multibody system are nonlinear in nature, and thus pose a difficult problem in linear control design. One approach is to have a first-order approximation through the numerical perturbations at a given configuration, and to design a control law based on the linearized model. Here, a linearized model is generated analytically by following the footsteps of the recursive derivation of the equations of motion. The equations of motion are first written in a Newton-Euler form, which is systematic and easy to construct; then, they are transformed into a relative coordinate representation, which is more efficient in computation. A new computational method for linearization is obtained by applying a series of first-order analytical approximations to the recursive kinematic relationships. The method has proved to be computationally more efficient because of its recursive nature. It has also turned out to be more accurate because of the fact that analytical perturbation circumvents numerical differentiation and other associated numerical operations that may accumulate computational error, thus requiring only analytical operations of matrices and vectors. The power of the proposed linearization algorithm is demonstrated, in comparison to a numerical perturbation method, with a two-link manipulator and a seven degrees of freedom robotic manipulator. Its application to control design is also demonstrated.

Unsteady transonic viscous-inviscid interaction using Euler and boundary-layer equations

NASA Technical Reports Server (NTRS)

Pirzadeh, Shahyar; Whitfield, Dave

1989-01-01

The Euler code is used extensively for computation of transonic unsteady aerodynamics. The boundary layer code solves the 3-D, compressible, unsteady, mean flow kinetic energy integral boundary layer equations in the direct mode. Inviscid-viscous coupling is handled using porosity boundary conditions. Some of the advantages and disadvantages of using the Euler and boundary layer equations for investigating unsteady viscous-inviscid interaction is examined.

Discrete maximal regularity of time-stepping schemes for fractional evolution equations.

PubMed

Jin, Bangti; Li, Buyang; Zhou, Zhi

2018-01-01

In this work, we establish the maximal [Formula: see text]-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order [Formula: see text], [Formula: see text], in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank-Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735-758, 2001. doi:10.1007/PL00004457) and its discrete analogue due to Blunck (Stud Math 146:157-176, 2001. doi:10.4064/sm146-2-3). These results generalize the corresponding results for parabolic problems.

Efficient Numerical Methods for Nonlinear-Facilitated Transport and Exchange in a Blood-Tissue Exchange Unit

PubMed Central

Poulain, Christophe A.; Finlayson, Bruce A.; Bassingthwaighte, James B.

2010-01-01

The analysis of experimental data obtained by the multiple-indicator method requires complex mathematical models for which capillary blood-tissue exchange (BTEX) units are the building blocks. This study presents a new, nonlinear, two-region, axially distributed, single capillary, BTEX model. A facilitated transporter model is used to describe mass transfer between plasma and intracellular spaces. To provide fast and accurate solutions, numerical techniques suited to nonlinear convection-dominated problems are implemented. These techniques are the random choice method, an explicit Euler-Lagrange scheme, and the MacCormack method with and without flux correction. The accuracy of the numerical techniques is demonstrated, and their efficiencies are compared. The random choice, Euler-Lagrange and plain MacCormack method are the best numerical techniques for BTEX modeling. However, the random choice and Euler-Lagrange methods are preferred over the MacCormack method because they allow for the derivation of a heuristic criterion that makes the numerical methods stable without degrading their efficiency. Numerical solutions are also used to illustrate some nonlinear behaviors of the model and to show how the new BTEX model can be used to estimate parameters from experimental data. PMID:9146808

Euler solutions for an unbladed jet engine configuration

NASA Technical Reports Server (NTRS)

Stewart, Mark E. M.

1991-01-01

A Euler solution for an axisymmetric jet engine configuration without blade effects is presented. The Euler equations are solved on a multiblock grid which covers a domain including the inlet, bypass duct, core passage, nozzle, and the far field surrounding the engine. The simulation is verified by considering five theoretical properties of the solution. The solution demonstrates both multiblock grid generation techniques and a foundation for a full jet engine throughflow calculation.

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.

Nonlinear Stochastic PDEs: Analysis and Approximations

DTIC Science & Technology

2016-05-23

numerical performance. Main theoretical and experimental advances include: 1.Introduction of a number of effective approaches to numerical analysis of...Stokes and Euler SPDEs, quasi -geostrophic SPDE, Ginzburg-Landau SPDE and Duffing oscillator REPORT DOCUMENTATION PAGE 11. SPONSOR/MONITOR’S REPORT...compare their numerical performance. Main theoretical and experimental advances include: 1.Introduction of a number of effective approaches to

Thickness ratio effects on quasistatic actuation and sensing behavior of laminate magnetoelectric cantilevers

NASA Astrophysics Data System (ADS)

Wang, Yezuo; Atulasimha, Jayasimha; Clarke, Joshua; Sundaresan, Vishnu B.

2010-04-01

In this work, the magnetoelectric cantilever composed of a layer of Galfenol and a layer of PZT-5H is studied for novel applications such as surgical ablation tools and cutting tools for machining applications. For developing a suitable model for the magnetoelectric cantilever, an energy based approach for the non-linear constitutive behavior of the magnetostrictive material and linear piezoelectric constitutive equations will be coupled with Euler Bernoulli model for composite beams. The cantilever is held in a uniform magnetic field and the magnetic field is measured by a Gaussmeter. The tip-deflection of the cantilever is detected by a laser triangulation sensor. The piezoelectric response can be studied with low noise preamplifier. Four PZT-5H layers with different thickness are separately bonded on the top of the same Galfenol layer and characterized to study the thickness ratio effects on the quasistatic actuation and sensing behavior of the composite cantilever.

Dissipation models for central difference schemes

NASA Astrophysics Data System (ADS)

Eliasson, Peter

1992-12-01

In this paper different flux limiters are used to construct dissipation models. The flux limiters are usually of Total Variation Diminishing (TVD type and are applied to the characteristic variables for the hyperbolic Euler equations in one, two or three dimensions. A number of simplified dissipation models with a reduced number of limiters are considered to reduce the computational effort. The most simplified methods use only one limiter, the dissipation model by Jameson belongs to this class since the Jameson pressure switch is considered as a limiter, not TVD though. Other one-limiter models with TVD limiters are also investigated. Models in between the most simplified one-limiter models and the full model with limiters on all the different characteristics are considered where different dissipation models are applied to the linear and non-linear characteristcs. In this paper the theory by Yee is extended to a general explicit Runge-Kutta type of schemes.

The dynamics and optimal control of spinning spacecraft with movable telescoping appendages. Part C: Effect of flexibility during boom deployment

NASA Technical Reports Server (NTRS)

Bainum, P. M.; James, P. K.

1977-01-01

The dynamics of a spinning symmetrical spacecraft system during the deployment (or retraction) of flexible boom-type appendages were investigated. The effect of flexibility during boom deployment is treated by modelling the deployable members as compound spherical pendula of varying length (according to a control law). The orientation of the flexible booms with respect to the hub, is described by a sequence of two Euler angles. The boom members contain a flexural stiffness which can be related to an assumed effective restoring linear spring constant, and structural damping which effects the entire system. Linearized equations of motion for this system, when the boom length is constant, involve periodic coefficients with the frequency of the hub spin. A bounded transformation is found which converts this system into a kinematically equivalent one involving only constant coefficients.

Mathematical simulation of sound propagation in a flow channel with impedance walls

NASA Astrophysics Data System (ADS)

Osipov, A. A.; Reent, K. S.

2012-07-01

The paper considers the specifics of calculating tonal sound propagating in a flow channel with an installed sound-absorbing device. The calculation is performed on the basis of numerical integrating on linearized nonstationary Euler equations using a code developed by the authors based on the so-called discontinuous Galerkin method. Using the linear theory of small perturbations, the effect of the sound-absorbing lining of the channel walls is described with the modified value of acoustic impedance proposed by the authors, for which, under flow channel conditions, the traditional classification of the active and reactive types of lining in terms of the real and imaginary impedance values, respectively, remains valid. To stabilize the computation process, a generalized impedance boundary condition is proposed in which, in addition to the impedance value itself, some additional parameters are introduced characterizing certain fictitious properties of inertia and elasticity of the impedance surface.

A Computational Study of Shear Layer Receptivity

NASA Astrophysics Data System (ADS)

Barone, Matthew; Lele, Sanjiva

2002-11-01

The receptivity of two-dimensional, compressible shear layers to local and external excitation sources is examined using a computational approach. The family of base flows considered consists of a laminar supersonic stream separated from nearly quiescent fluid by a thin, rigid splitter plate with a rounded trailing edge. The linearized Euler and linearized Navier-Stokes equations are solved numerically in the frequency domain. The flow solver is based on a high order finite difference scheme, coupled with an overset mesh technique developed for computational aeroacoustics applications. Solutions are obtained for acoustic plane wave forcing near the most unstable shear layer frequency, and are compared to the existing low frequency theory. An adjoint formulation to the present problem is developed, and adjoint equation calculations are performed using the same numerical methods as for the regular equation sets. Solutions to the adjoint equations are used to shed light on the mechanisms which control the receptivity of finite-width compressible shear layers.

A Lagrangian meshfree method applied to linear and nonlinear elasticity.

PubMed

Walker, Wade A

2017-01-01

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code.

Coupling between plate vibration and acoustic radiation

NASA Technical Reports Server (NTRS)

Frendi, Abdelkader; Maestrello, Lucio; Bayliss, Alvin

1992-01-01

A detailed numerical investigation of the coupling between the vibration of a flexible plate and the acoustic radiation is performed. The nonlinear Euler equations are used to describe the acoustic fluid while the nonlinear plate equation is used to describe the plate vibration. Linear, nonlinear, and quasi-periodic or chaotic vibrations and the resultant acoustic radiation are analyzed. We find that for the linear plate response, acoustic coupling is negligible. However, for the nonlinear and chaotic responses, acoustic coupling has a significant effect on the vibration level as the loading increases. The radiated pressure from a plate undergoing nonlinear or chaotic vibrations is found to propagate nonlinearly into the far-field. However, the nonlinearity due to wave propagation is much weaker than that due to the plate vibrations. As the acoustic wave propagates into the far-field, the relative difference in level between the fundamental and its harmonics and subharmonics decreases with distance.

A dimensionally split Cartesian cut cell method for hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Gokhale, Nandan; Nikiforakis, Nikos; Klein, Rupert

2018-07-01

We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a full description of its three-dimensional implementation in the dimensionally split framework of Klein et al. [1]. The convergence and stability of the method are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. When compared to the cut cell flux of Klein et al., it was found that the new flux alleviates the problem of oscillatory boundary solutions produced by the former at higher Courant numbers, and also enables the computation of more accurate solutions near stagnation points. Being dimensionally split, the method is simple to implement and extends readily to multiple dimensions.

A Lagrangian meshfree method applied to linear and nonlinear elasticity

PubMed Central

2017-01-01

The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code. PMID:29045443

Researched applied to transonic compressors in numerical fluid mechanics of inviscid flow and viscous flow

NASA Technical Reports Server (NTRS)

Thompkins, W. T., Jr.

1985-01-01

A streamline Euler solver which combines high accuracy and good convergence rates with capabilities for inverse or direct mode solution modes and an analysis technique for finite difference models of hyperbolic partial difference equations were developed.

Euler polynomials and identities for non-commutative operators

NASA Astrophysics Data System (ADS)

De Angelis, Valerio; Vignat, Christophe

2015-12-01

Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.

Computational aeroelastic analysis of aircraft wings including geometry nonlinearity

NASA Astrophysics Data System (ADS)

Tian, Binyu

The objective of the present study is to show the ability of solving fluid structural interaction problems more realistically by including the geometric nonlinearity of the structure so that the aeroelastic analysis can be extended into the onset of flutter, or in the post flutter regime. A nonlinear Finite Element Analysis software is developed based on second Piola-Kirchhoff stress and Green-Lagrange strain. The second Piola-Kirchhoff stress and Green-Lagrange strain is a pair of energetically conjugated tensors that can accommodate arbitrary large structural deformations and deflection, to study the flutter phenomenon. Since both of these tensors are objective tensors, i.e., the rigid-body motion has no contribution to their components, the movement of the body, including maneuvers and deformation, can be included. The nonlinear Finite Element Analysis software developed in this study is verified with ANSYS, NASTRAN, ABAQUS, and IDEAS for the linear static, nonlinear static, linear dynamic and nonlinear dynamic structural solutions. To solve the flow problems by Euler/Navier equations, the current nonlinear structural software is then embedded into ENSAERO, which is an aeroelastic analysis software package developed at NASA Ames Research Center. The coupling of the two software, both nonlinear in their own field, is achieved by domain decomposition method first proposed by Guruswamy. A procedure has been set for the aeroelastic analysis process. The aeroelastic analysis results have been obtained for fight wing in the transonic regime for various cases. The influence dynamic pressure on flutter has been checked for a range of Mach number. Even though the current analysis matches the general aeroelastic characteristic, the numerical value not match very well with previous studies and needs farther investigations. The flutter aeroelastic analysis results have also been plotted at several time points. The influences of the deforming wing geometry can be well seen in those plots. The movement of shock changes the aerodynamic load distribution on the wing. The effect of viscous on aeroelastic analysis is also discussed. Also compared are the flutter solutions with, or without the structural nonlinearity. As can be seen, linear structural solution goes to infinite, which can not be true in reality. The nonlinear solution is more realistic and can be used to understand the fluid and structure interaction behavior, to control, or prevent disastrous events. (Abstract shortened by UMI.)

Vibrational analysis of vertical axis wind turbine blades

NASA Astrophysics Data System (ADS)

Kapucu, Onur

The goal of this research is to derive a vibration model for a vertical axis wind turbine blade. This model accommodates the affects of varying relative flow angle caused by rotating the blade in the flow field, uses a simple aerodynamic model that assumes constant wind speed and constant rotation rate, and neglects the disturbance of wind due to upstream blade or post. The blade is modeled as elastic Euler-Bernoulli beam under transverse bending and twist deflections. Kinetic and potential energy equations for a rotating blade under deflections are obtained, expressed in terms of assumed modal coordinates and then plugged into Lagrangian equations where the non-conservative forces are the lift and drag forces and moments. An aeroelastic model for lift and drag forces, approximated with third degree polynomials, on the blade are obtained assuming an airfoil under variable angle of attack and airflow magnitudes. A simplified quasi-static airfoil theory is used, in which the lift and drag coefficients are not dependent on the history of the changing angle of attack. Linear terms on the resulting equations of motion will be used to conduct a numerical analysis and simulation, where numeric specifications are modified from the Sandia-17m Darrieus wind turbine by Sandia Laboratories.

Analytical Solution for the Aeroelastic Response of a Two-Dimensional Elastic Plate in Axial Flow

NASA Astrophysics Data System (ADS)

Medina, Cory; Kang, Chang-Kwon

2017-11-01

The aeroelastic response of an elastic plate in an unsteady flow describes many engineering problems from bio-locomotion, deforming airfoils, to energy harvesting. However, the analysis is challenging because the shape of the plate is a priori unknown. This study presents an analytical model that can predict the two-way tightly coupled aeroelastic response of a two-dimensional elastic plate including the effects of plate curvature along the flow direction. The plate deforms due to the dynamic balance of wing inertia, elastic restoring force, and aerodynamic force. The coupled model utilizes the linearized Euler-Bernoulli beam theory for the structural model and thin airfoil theory as presented by Theodorsen, which assumes incompressible potential flow, for the aerodynamic model. The coupled equations of motion are solved via Galerkin's method, where closed form solutions for the plate deformation are obtained by deriving the unsteady aerodynamic pressure with respect to the plate normal functions, expressed in a Chebyshev polynomial expansion. Stability analysis is performed for a range of mass ratios obtaining the flutter velocities and corresponding frequencies and the results agree well with the results reported in the literature.

Assessment of the effects of azimuthal mode number perturbations upon the implosion processes of fluids in cylinders

NASA Astrophysics Data System (ADS)

Lindstrom, Michael

2017-06-01

Fluid instabilities arise in a variety of contexts and are often unwanted results of engineering imperfections. In one particular model for a magnetized target fusion reactor, a pressure wave is propagated in a cylindrical annulus comprised of a dense fluid before impinging upon a plasma and imploding it. Part of the success of the apparatus is a function of how axially-symmetric the final pressure pulse is upon impacting the plasma. We study a simple model for the implosion of the system to study how imperfections in the pressure imparted on the outer circumference grow due to geometric focusing. Our methodology entails linearizing the compressible Euler equations for mass and momentum conservation about a cylindrically symmetric problem and analysing the perturbed profiles at different mode numbers. The linearized system gives rise to singular shocks and through analysing the perturbation profiles at various times, we infer that high mode numbers are dampened through the propagation. We also study the Linear Klein-Gordon equation in the context of stability of linear cylindrical wave formation whereby highly oscillatory, bounded behaviour is observed in a far field solution.

Prediction of Broadband Shock-Associated Noise Including Propagation Effects Originating NASA

NASA Technical Reports Server (NTRS)

Miller, Steven; Morris, Philip J.

2012-01-01

An acoustic analogy is developed based on the Euler equations for broadband shock-associated noise (BBSAN) that directly incorporates the vector Green s function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) to describe the mean flow. The vector Green s function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation rate. An adjoint vector Green s function solver is implemented to determine the vector Green s function based on a locally parallel mean flow at different streamwise locations. The newly developed acoustic analogy can be simplified to one that uses the Green s function associated with the Helmholtz equation, which is consistent with a previous formulation by the authors. A large number of predictions are generated using three different nozzles over a wide range of fully-expanded jet Mach numbers and jet stagnation temperatures. These predictions are compared with experimental data from multiple jet noise experimental facilities. In addition, two models for the so-called fine-scale mixing noise are included in the comparisons. Improved BBSAN predictions are obtained relative to other models that do not include propagation effects.

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

Free time minimizers for the three-body problem

NASA Astrophysics Data System (ADS)

Moeckel, Richard; Montgomery, Richard; Sánchez Morgado, Héctor

2018-03-01

Free time minimizers of the action (called "semi-static" solutions by Mañe in International congress on dynamical systems in Montevideo (a tribute to Ricardo Mañé), vol 362, pp 120-131, 1996) play a central role in the theory of weak KAM solutions to the Hamilton-Jacobi equation (Fathi in Weak KAM Theorem in Lagrangian Dynamics Preliminary Version Number 10, 2017). We prove that any solution to Newton's three-body problem which is asymptotic to Lagrange's parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange's solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz and Maderna (Math Proc Camb Philos Soc 156:209-227, 1980) which showed that every free time minimizer for the N-body problem is parabolic and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler's central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.

Time integration algorithms for the two-dimensional Euler equations on unstructured meshes

NASA Technical Reports Server (NTRS)

Slack, David C.; Whitaker, D. L.; Walters, Robert W.

1994-01-01

Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.

Computation of transonic viscous-inviscid interacting flow

NASA Technical Reports Server (NTRS)

Whitfield, D. L.; Thomas, J. L.; Jameson, A.; Schmidt, W.

1983-01-01

Transonic viscous-inviscid interaction is considered using the Euler and inverse compressible turbulent boundary-layer equations. Certain improvements in the inverse boundary-layer method are mentioned, along with experiences in using various Runge-Kutta schemes to solve the Euler equations. Numerical conditions imposed on the Euler equations at a surface for viscous-inviscid interaction using the method of equivalent sources are developed, and numerical solutions are presented and compared with experimental data to illustrate essential points. Previously announced in STAR N83-17829

Equations with Arithmetic Functions of Pell Numbers

DTIC Science & Technology

2014-01-01

Bull. Math. Soc. Sci. Math. Roumanie Tome 57(105) No. 4, 2014, 409–413 Equations with arithmetic functions of Pell numbers by 1Florian Luca...2Pantelimon Stănică Abstract Here, we prove some diophantine results about the Euler function of Pell numbers and their Pell –Lucas companion sequence. For...example, if the Euler function of the nth Pell number Pn or Pell –Lucas companion number Qn is a power of 2, then n ≤ 8. Key Words: Euler function, Pell

Remarks on Heisenberg-Euler-type electrodynamics

NASA Astrophysics Data System (ADS)

Kruglov, S. I.

2017-05-01

We consider Heisenberg-Euler-type model of nonlinear electrodynamics with two parameters. Heisenberg-Euler electrodynamics is a particular case of this model. Corrections to Coulomb’s law at r →∞ are obtained and energy conditions are studied. The total electrostatic energy of charged particles is finite. The charged black hole solution in the framework of nonlinear electrodynamics is investigated. We find the asymptotic of the metric and mass functions at r →∞. Corrections to the Reissner-Nordström solution are obtained.

Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds

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

Guth, Larry, E-mail: lguth@math.mit.edu; Lubotzky, Alexander, E-mail: alex.lubotzky@mail.huji.ac.il

2014-08-15

Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance n{sup ε}. Their rate is evaluated via Euler characteristic arguments and their distance using Z{sub 2}-systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), Lecture Notes in Computer Science Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.

Control of sound radiation from a wavepacket over a curved surface

NASA Technical Reports Server (NTRS)

Maestrello, Lucio; El Hady, Nabil M.

1989-01-01

Active control of acoustic pressure in the far field resulting from the growth and decay of a wavepacket convecting in a boundary layer over a concave-convex surface is investigated numerically using direct computations of the Navier-Stokes equations. The resulting sound radiation is computed using linearized Euler equations with the pressure from the Navier-Stokes solution as a time-dependent boundary condition. The acoustic far field exhibits directivity type of behavior that points upstream to the flow direction. A fixed control algorithm is used where the attenuation signal is synthesized by a filter which actively adapt it to the amplitude-time response of the outgoing acoustic wave.

Strictly stable high order difference approximations for computational aeroacoustics

NASA Astrophysics Data System (ADS)

Müller, Bernhard; Johansson, Stefan

2005-09-01

High order finite difference approximations with improved accuracy and stability properties have been developed for computational aeroacoustics (CAA). One of our new difference operators corresponds to Tam and Webb's DRP scheme in the interior, but is modified near the boundaries to be strictly stable. A unified formulation of the nonlinear and linearized Euler equations is used, which can be extended to the Navier-Stokes equations. The approach has been verified for 1D, 2D and axisymmetric test problems. We have simulated the sound propagation from a rocket launch before lift-off. To cite this article: B. Müller, S. Johansson, C. R. Mecanique 333 (2005).

A fourth order Euler/Navier-Stokes prediction method for the aerodynamics and aeroelasticity of hovering rotor blades

NASA Astrophysics Data System (ADS)

Smith, Marilyn Jones

Some of the computational issues relating to the development of a three-dimensional fourth-order compact Euler/Navier-Stokes methodology for rotary wing flows and its coupling with an elastic rotor blade beam structural model have been explored. The compact Euler/NavierStokes method is used to predict the aerodynamic loads on an isolated rotor blade. Because the scheme is fourth-order, fewer grid nodes are necessary to predict loads with the same accuracy as traditional second order methodologies on finer grids. Grid and numerical parameter optimizations were performed to examine the changes in the predictive capabilities of the higher-order scheme. Comparisons were made with experimental data for a rotor using NACA 0012 airfoil sections and a rectangular planform with no twist. Simulations for both lifting and non-lifting configurations at various tip Mach numbers were performed. This Euler/Navier-Stokes methodology can be applied to rotor blades with either rigid-blade or elastic-beam-structural models to determine the steady-state response in hovering flight. The blade is represented by a geometrically nonlinear beam model which accounts for coupled flap bending, lead-lag bending and torsion. Moderately large displacements and rotations due to structural deformations can be simulated. The analysis has been performed for blade configurations having uniform mass and stiffness, no twist, and no chordwise offsets of the elastic and tension axes, as well as the center of mass. The results are compared with a panel method coupled with the same structural dynamics model. Computations have been made to predict the aerodynamic deflections for the rotor in hover. A starting solution using initial deflections predicted by aeroelastic analyses with a two-dimensional aerodynamic model was investigated. The present Euler/Navier-Stokes method using a momentum wake and a contracting vortex wake shows the impact on the aeroelastic deflections of a three-dimensional aerodynamic module which includes rotational and viscous effects, particularly at higher collective pitch angles. The differences in the aeroelastic predictions using fully coupled and loosely coupled aerodynamic analyses are examined. The induced wake plays a critical role in determining the final equilibrium tip deflections.

Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

PubMed Central

Cheung, Ka Luen; Wong, Sen

2016-01-01

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)|x|α−1 x + b(t)(x/|x|) for any value of α ≠ 1 or any positive integer N ≠ 1. Then, we show that blowup phenomenon occurs when α = N = 1 and c2(0)+c˙(0)<0. As a corollary, the blowup properties of solutions with velocity of the form (a˙t/at)x+b(t)(x/x) are obtained. Our analysis includes both the isentropic case (γ > 1) and the isothermal case (γ = 1). PMID:27066528

Dark photon decay beyond the Euler-Heisenberg limit

NASA Astrophysics Data System (ADS)

McDermott, Samuel D.; Patel, Hiren H.; Ramani, Harikrishnan

2018-04-01

We calculate the exact width for a dark photon decaying to three photons at one-loop order for dark photon masses m' below the e+e- production threshold of 2 me. We find substantial deviations from previous results derived from the lowest order Euler-Heisenberg effective Lagrangian in the range me≲m'≤2 me, where higher order terms in the derivative expansion are non-negligible. This mass range is precisely where the three photon decay takes place on cosmologically relevant timescales. Our improved analysis reveals a window for dark photons in the range 850 KeV ≲m'≤2 me , 10-5≲ɛ ≲10-4 that is only constrained by possibly model-dependent bounds on the number of light degrees of freedom in the early Universe.

Stability analysis of the Euler discretization for SIR epidemic model

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

Suryanto, Agus

2014-06-19

In this paper we consider a discrete SIR epidemic model obtained by the Euler method. For that discrete model, existence of disease free equilibrium and endemic equilibrium is established. Sufficient conditions on the local asymptotical stability of both disease free equilibrium and endemic equilibrium are also derived. It is found that the local asymptotical stability of the existing equilibrium is achieved only for a small time step size h. If h is further increased and passes the critical value, then both equilibriums will lose their stability. Our numerical simulations show that a complex dynamical behavior such as bifurcation or chaosmore » phenomenon will appear for relatively large h. Both analytical and numerical results show that the discrete SIR model has a richer dynamical behavior than its continuous counterpart.« less

Three-Dimensional Aeroelastic and Aerothermoelastic Behavior in Hypersonic Flow

NASA Technical Reports Server (NTRS)

McNamara, Jack J.; Friedmann, Peretz P.; Powell, Kenneth G.; Thuruthimattam, Biju J.; Bartels, Robert E.

2005-01-01

The aeroelastic and aerothermoelastic behavior of three-dimensional configurations in hypersonic flow regime are studied. The aeroelastic behavior of a low aspect ratio wing, representative of a fin or control surface on a generic hypersonic vehicle, is examined using third order piston theory, Euler and Navier-Stokes aerodynamics. The sensitivity of the aeroelastic behavior generated using Euler and Navier-Stokes aerodynamics to parameters governing temporal accuracy is also examined. Also, a refined aerothermoelastic model, which incorporates the heat transfer between the fluid and structure using CFD generated aerodynamic heating, is used to examine the aerothermoelastic behavior of the low aspect ratio wing in the hypersonic regime. Finally, the hypersonic aeroelastic behavior of a generic hypersonic vehicle with a lifting-body type fuselage and canted fins is studied using piston theory and Euler aerodynamics for the range of 2.5 less than or equal to M less than or equal to 28, at altitudes ranging from 10,000 feet to 80,000 feet. This analysis includes a study on optimal mesh selection for use with Euler aerodynamics. In addition to the aeroelastic and aerothermoelastic results presented, three time domain flutter identification techniques are compared, namely the moving block approach, the least squares curve fitting method, and a system identification technique using an Auto-Regressive model of the aeroelastic system. In general, the three methods agree well. The system identification technique, however, provided quick damping and frequency estimations with minimal response record length, and therefore o ers significant reductions in computational cost. In the present case, the computational cost was reduced by 75%. The aeroelastic and aerothermoelastic results presented illustrate the applicability of the CFL3D code for the hypersonic flight regime.

A multiblock multigrid three-dimensional Euler equation solver

NASA Technical Reports Server (NTRS)

Cannizzaro, Frank E.; Elmiligui, Alaa; Melson, N. Duane; Vonlavante, E.

1990-01-01

Current aerodynamic designs are often quite complex (geometrically). Flexible computational tools are needed for the analysis of a wide range of configurations with both internal and external flows. In the past, geometrically dissimilar configurations required different analysis codes with different grid topologies in each. The duplicity of codes can be avoided with the use of a general multiblock formulation which can handle any grid topology. Rather than hard wiring the grid topology into the program, it is instead dictated by input to the program. In this work, the compressible Euler equations, written in a body-fitted finite-volume formulation, are solved using a pseudo-time-marching approach. Two upwind methods (van Leer's flux-vector-splitting and Roe's flux-differencing) were investigated. Two types of explicit solvers (a two-step predictor-corrector and a modified multistage Runge-Kutta) were used with multigrid acceleration to enhance convergence. A multiblock strategy is used to allow greater geometric flexibility. A report on simple explicit upwind schemes for solving compressible flows is included.

[Improved euler algorithm for trend forecast model and its application to oil spectrum analysis].

PubMed

Zheng, Chang-song; Ma, Biao

2009-04-01

The oil atomic spectrometric analysis technology is one of the most important methods for fault diagnosis and state monitoring of large machine equipment. The gray method is preponderant in the trend forecast at the same time. With the use of oil atomic spectrometric analysis result and combining the gray forecast theory, the present paper established a gray forecast model of the Fe/Cu concentration trend in the power-shift steering transmission. Aiming at the shortage of the gray method used in the trend forecast, the improved Euler algorithm was put forward for the first time to resolve the problem of the gray model and avoid the non-precision that the old gray model's forecast value depends on the first test value. This new method can make the forecast value more precision as shown in the example. Combined with the threshold value of the oil atomic spectrometric analysis, the new method was applied on the Fe/Cu concentration forecast and the premonition of fault information was obtained. So we can take steps to prevent the fault and this algorithm can be popularized to the state monitoring in the industry.

Novel Euler-LaCoste linkage as a very low frequency vertical vibration isolator.

PubMed

Hosain, M A; Sirr, A; Ju, L; Blair, D G

2012-08-01

LaCoste linkage vibration isolators have shown excellent performance for ultra-low frequency vertical vibration isolation. However, such isolators depend on the use of conventional pre-stressed coil springs, which suffer from creep. Here, we show that compressional Euler springs can be configured to create a stable tension unit for use in a LaCoste structure. In a proof of concept experiment, we demonstrate a vertical resonance frequency of 0.15 Hz in an Euler-LaCoste configuration with 200 mm height. The system enables the use of very low creep maraging steel as spring elements to eliminate the creep while minimising spring mass and reducing the effect of parasitic resonances. Larger scale systems with optimized Euler spring boundary conditions should achieve performance suitable for applications on third generation gravitational wave detectors such as the proposed Einstein telescope.

Euler equation computations for the flow over a hovering helicopter rotor

NASA Technical Reports Server (NTRS)

Roberts, Thomas Wesley

1988-01-01

A numerical solution technique is developed for computing the flow field around an isolated helicopter rotor in hover. The flow is governed by the compressible Euler equations which are integrated using a finite volume approach. The Euler equations are coupled to a free wake model of the rotary wing vortical wake. This wake model is incorporated into the finite volume solver using a prescribed flow, or perturbation, technique which eliminates the numerical diffusion of vorticity due to the artificial viscosity of the scheme. The work is divided into three major parts: (1) comparisons of Euler solutions to experimental data for the flow around isolated wings show good agreement with the surface pressures, but poor agreement with the vortical wake structure; (2) the perturbation method is developed and used to compute the interaction of a streamwise vortex with a semispan wing. The rapid diffusion of the vortex when only the basic Euler solver is used is illustrated, and excellent agreement with experimental section lift coefficients is demonstrated when using the perturbation approach; and (3) the free wake solution technique is described and the coupling of the wake to the Euler solver for an isolated rotor is presented. Comparisons with experimental blade load data for several cases show good agreement, with discrepancies largely attributable to the neglect of viscous effects. The computed wake geometries agree less well with experiment, the primary difference being that too rapid a wake contraction is predicted for all the cases.

Hybrid finite element method for describing the electrical response of biological cells to applied fields.

PubMed

Ying, Wenjun; Henriquez, Craig S

2007-04-01

A novel hybrid finite element method (FEM) for modeling the response of passive and active biological membranes to external stimuli is presented. The method is based on the differential equations that describe the conservation of electric flux and membrane currents. By introducing the electric flux through the cell membrane as an additional variable, the algorithm decouples the linear partial differential equation part from the nonlinear ordinary differential equation part that defines the membrane dynamics of interest. This conveniently results in two subproblems: a linear interface problem and a nonlinear initial value problem. The linear interface problem is solved with a hybrid FEM. The initial value problem is integrated by a standard ordinary differential equation solver such as the Euler and Runge-Kutta methods. During time integration, these two subproblems are solved alternatively. The algorithm can be used to model the interaction of stimuli with multiple cells of almost arbitrary geometries and complex ion-channel gating at the plasma membrane. Numerical experiments are presented demonstrating the uses of the method for modeling field stimulation and action potential propagation.

Compatible diagonal-norm staggered and upwind SBP operators

NASA Astrophysics Data System (ADS)

Mattsson, Ken; O'Reilly, Ossian

2018-01-01

The main motivation with the present study is to achieve a provably stable high-order accurate finite difference discretisation of linear first-order hyperbolic problems on a staggered grid. The use of a staggered grid makes it non-trivial to discretise advective terms. To overcome this difficulty we discretise the advective terms using upwind Summation-By-Parts (SBP) operators, while the remaining terms are discretised using staggered SBP operators. The upwind and staggered SBP operators (for each order of accuracy) are compatible, here meaning that they are based on the same diagonal norms, allowing for energy estimates to be formulated. The boundary conditions are imposed using a penalty (SAT) technique, to guarantee linear stability. The resulting SBP-SAT approximations lead to fully explicit ODE systems. The accuracy and stability properties are demonstrated for linear hyperbolic problems in 1D, and for the 2D linearised Euler equations with constant background flow. The newly derived upwind and staggered SBP operators lead to significantly more accurate numerical approximations, compared with the exclusive usage of (previously derived) central-difference first derivative SBP operators.

Unsteady-flow-field predictions for oscillating cascades

NASA Technical Reports Server (NTRS)

Huff, Dennis L.

1991-01-01

The unsteady flow field around an oscillating cascade of flat plates with zero stagger was studied by using a time marching Euler code. This case had an exact solution based on linear theory and served as a model problem for studying pressure wave propagation in the numerical solution. The importance of using proper unsteady boundary conditions, grid resolution, and time step size was shown for a moderate reduced frequency. Results show that an approximate nonreflecting boundary condition based on linear theory does a good job of minimizing reflections from the inflow and outflow boundaries and allows the placement of the boundaries to be closer to the airfoils than when reflective boundaries are used. Stretching the boundary to dampen the unsteady waves is another way to minimize reflections. Grid clustering near the plates captures the unsteady flow field better than when uniform grids are used as long as the 'Courant Friedrichs Levy' (CFL) number is less than 1 for a sufficient portion of the grid. Finally, a solution based on an optimization of grid, CFL number, and boundary conditions shows good agreement with linear theory.

Development of fault tolerant adaptive control laws for aerospace systems

NASA Astrophysics Data System (ADS)

Perez Rocha, Andres E.

The main topic of this dissertation is the design, development and implementation of intelligent adaptive control techniques designed to maintain healthy performance of aerospace systems subjected to malfunctions, external parameter changes and/or unmodeled dynamics. The dissertation is focused on the development of novel adaptive control configurations that rely on non-linear functions that appear in the immune system of living organisms as main source of adaptation. One of the main goals of this dissertation is to demonstrate that these novel adaptive control architectures are able to improve overall performance and protect the system while reducing control effort and maintaining adequate operation outside bounds of nominal design. This research effort explores several phases, ranging from theoretical stability analysis, simulation and hardware implementation on different types of aerospace systems including spacecraft, aircraft and quadrotor vehicles. The results presented in this dissertation are focused on two main adaptivity approaches, the first one is intended for aerospace systems that do not attain large angles and use exact feedback linearization of Euler angle kinematics. A proof of stability is presented by means of the circle Criterion and Lyapunov's direct method. The second approach is intended for aerospace systems that can attain large attitude angles (e.g. space systems in gravity-less environments), the adaptation is incorporated on a baseline architecture that uses partial feedback linearization of quaternions kinematics. In this case, the closed loop stability was analyzed using Lyapunov's direct method and Barbalat's Lemma. It is expected that some results presented in this dissertation can contribute towards the validation and certification of direct adaptive controllers.

3-D conditional hyperbolic method of moments for high-fidelity Euler-Euler simulations of particle-laden flows

NASA Astrophysics Data System (ADS)

Patel, Ravi; Kong, Bo; Capecelatro, Jesse; Fox, Rodney; Desjardins, Olivier

2017-11-01

Particle-laden turbulent flows are important features of many environmental and industrial processes. Euler-Euler (EE) simulations of these flows are more computationally efficient than Euler-Lagrange (EL) simulations. However, traditional EE methods, such as the two-fluid model, cannot faithfully capture dilute regions of flow with finite Stokes number particles. For this purpose, the multi-valued nature of the particle velocity field must be treated with a polykinetic description. Various quadrature-based moment methods (QBMM) can be used to approximate the full kinetic description by solving for a set of moments of the particle velocity distribution function (VDF) and providing closures for the higher-order moments. Early QBMM fail to maintain the strict hyperbolicity of the kinetic equations, producing unphysical delta shocks (i.e., mass accumulation at a point). In previous work, a 2-D conditional hyperbolic quadrature method of moments (CHyQMOM) was proposed as a fourth-order QBMM closure that maintains strict hyperbolicity. Here, we present the 3-D extension of CHyQMOM. We compare results from CHyQMOM to other QBMM and EL in the context of particle trajectory crossing, cluster-induced turbulence, and particle-laden channel flow. NSF CBET-1437903.

Oscillation Amplitude Growth for a Decelerating Object with Constant Pitch Damping

NASA Technical Reports Server (NTRS)

Schoenenberger, Mark; Queen, Eric M.; Litton, Daniel

2006-01-01

The equations governing the deceleration and oscillation of a blunt body moving along a planar trajectory are re-expressed in the form of the Euler-Cauchy equation. An analytic solution of this equation describes the oscillation amplitude growth and frequency dilation with time for a statically stable decelerating body with constant pitch damping. The oscillation histories for several constant pitch damping values, predicted by the solution of the Euler-Cauchy equation are compared to POST six degree-of-freedom (6-DoF) trajectory simulations. The simulations use simplified aerodynamic coefficients matching the Euler-Cauchy approximations. Agreement between the model predictions and simulation results are excellent. Euler-Cauchy curves are also fit through nonlinear 6-DoF simulations and ballistic range data to identify static stability and pitch damping coefficients. The model os shown to closely fit through the data points and capture the behavior of the blunt body observed in simulation and experiment. The extracted coefficients are in reasonable agreement with higher fidelity, nonlinear parameter identification results. Finally, a nondimensional version of the Euler-Cauchy equation is presented and shown to be a simple and effective tool for designing dynamically scaled experiments for decelerating blunt capsule flight.

One-dimensional high-order compact method for solving Euler's equations

NASA Astrophysics Data System (ADS)

Mohamad, M. A. H.; Basri, S.; Basuno, B.

2012-06-01

In the field of computational fluid dynamics, many numerical algorithms have been developed to simulate inviscid, compressible flows problems. Among those most famous and relevant are based on flux vector splitting and Godunov-type schemes. Previously, this system was developed through computational studies by Mawlood [1]. However the new test cases for compressible flows, the shock tube problems namely the receding flow and shock waves were not investigated before by Mawlood [1]. Thus, the objective of this study is to develop a high-order compact (HOC) finite difference solver for onedimensional Euler equation. Before developing the solver, a detailed investigation was conducted to assess the performance of the basic third-order compact central discretization schemes. Spatial discretization of the Euler equation is based on flux-vector splitting. From this observation, discretization of the convective flux terms of the Euler equation is based on a hybrid flux-vector splitting, known as the advection upstream splitting method (AUSM) scheme which combines the accuracy of flux-difference splitting and the robustness of flux-vector splitting. The AUSM scheme is based on the third-order compact scheme to the approximate finite difference equation was completely analyzed consequently. In one-dimensional problem for the first order schemes, an explicit method is adopted by using time integration method. In addition to that, development and modification of source code for the one-dimensional flow is validated with four test cases namely, unsteady shock tube, quasi-one-dimensional supersonic-subsonic nozzle flow, receding flow and shock waves in shock tubes. From these results, it was also carried out to ensure that the definition of Riemann problem can be identified. Further analysis had also been done in comparing the characteristic of AUSM scheme against experimental results, obtained from previous works and also comparative analysis with computational results generated by van Leer, KFVS and AUSMPW schemes. Furthermore, there is a remarkable improvement with the extension of the AUSM scheme from first-order to third-order accuracy in terms of shocks, contact discontinuities and rarefaction waves.

Perturbational blowup solutions to the compressible Euler equations with damping.

PubMed

Cheung, Ka Luen

2016-01-01

The N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions. In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases [Formula: see text] and [Formula: see text], where [Formula: see text] is the adiabatic constant. With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen. Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points. The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications.

Euler Technology Assessment program for preliminary aircraft design employing SPLITFLOW code with Cartesian unstructured grid method

NASA Technical Reports Server (NTRS)

Finley, Dennis B.

1995-01-01

This report documents results from the Euler Technology Assessment program. The objective was to evaluate the efficacy of Euler computational fluid dynamics (CFD) codes for use in preliminary aircraft design. Both the accuracy of the predictions and the rapidity of calculations were to be assessed. This portion of the study was conducted by Lockheed Fort Worth Company, using a recently developed in-house Cartesian-grid code called SPLITFLOW. The Cartesian grid technique offers several advantages for this study, including ease of volume grid generation and reduced number of cells compared to other grid schemes. SPLITFLOW also includes grid adaptation of the volume grid during the solution convergence to resolve high-gradient flow regions. This proved beneficial in resolving the large vortical structures in the flow for several configurations examined in the present study. The SPLITFLOW code predictions of the configuration forces and moments are shown to be adequate for preliminary design analysis, including predictions of sideslip effects and the effects of geometry variations at low and high angles of attack. The time required to generate the results from initial surface definition is on the order of several hours, including grid generation, which is compatible with the needs of the design environment.

Quality assessment of two- and three-dimensional unstructured meshes and validation of an upwind Euler flow solver

NASA Technical Reports Server (NTRS)

Woodard, Paul R.; Batina, John T.; Yang, Henry T. Y.

1992-01-01

Quality assessment procedures are described for two-dimensional unstructured meshes. The procedures include measurement of minimum angles, element aspect ratios, stretching, and element skewness. Meshes about the ONERA M6 wing and the Boeing 747 transport configuration are generated using an advancing front method grid generation package of programs. Solutions of Euler's equations for these meshes are obtained at low angle-of-attack, transonic conditions. Results for these cases, obtained as part of a validation study demonstrate accuracy of an implicit upwind Euler solution algorithm.

Refinement Of Hexahedral Cells In Euler Flow Computations

NASA Technical Reports Server (NTRS)

Melton, John E.; Cappuccio, Gelsomina; Thomas, Scott D.

1996-01-01

Topologically Independent Grid, Euler Refinement (TIGER) computer program solves Euler equations of three-dimensional, unsteady flow of inviscid, compressible fluid by numerical integration on unstructured hexahedral coordinate grid refined where necessary to resolve shocks and other details. Hexahedral cells subdivided, each into eight smaller cells, as needed to refine computational grid in regions of high flow gradients. Grid Interactive Refinement and Flow-Field Examination (GIRAFFE) computer program written in conjunction with TIGER program to display computed flow-field data and to assist researcher in verifying specified boundary conditions and refining grid.

On the Euler Function of the Catalan Numbers

DTIC Science & Technology

2012-02-26

ON THE EULER FUNCTION OF THE CATALAN NUMBERS FLORIAN LUCA AND PANTELIMON STĂNICĂ Abstract. We study the solutions of the equation φ(Cm)/φ(Cn) = r...where r is a fixed rational number , Ck is the kth Catalan number and φ is the Euler function. We note that the number r = 4 is special for this...observation concerning φ(Cn+1)/φ(Cn) For a positive integer n, let (1) Cn = 1 n+ 1 ( 2n n ) be the n-th Catalan number . For a positive integer m we put φ(m) for

Measuring (subglacial) bedform orientation, length, and longitudinal asymmetry - Method assessment.

PubMed

Jorge, Marco G; Brennand, Tracy A

2017-01-01

Geospatial analysis software provides a range of tools that can be used to measure landform morphometry. Often, a metric can be computed with different techniques that may give different results. This study is an assessment of 5 different methods for measuring longitudinal, or streamlined, subglacial bedform morphometry: orientation, length and longitudinal asymmetry, all of which require defining a longitudinal axis. The methods use the standard deviational ellipse (not previously applied in this context), the longest straight line fitting inside the bedform footprint (2 approaches), the minimum-size footprint-bounding rectangle, and Euler's approximation. We assess how well these methods replicate morphometric data derived from a manually mapped (visually interpreted) longitudinal axis, which, though subjective, is the most typically used reference. A dataset of 100 subglacial bedforms covering the size and shape range of those in the Puget Lowland, Washington, USA is used. For bedforms with elongation > 5, deviations from the reference values are negligible for all methods but Euler's approximation (length). For bedforms with elongation < 5, most methods had small mean absolute error (MAE) and median absolute deviation (MAD) for all morphometrics and thus can be confidently used to characterize the central tendencies of their distributions. However, some methods are better than others. The least precise methods are the ones based on the longest straight line and Euler's approximation; using these for statistical dispersion analysis is discouraged. Because the standard deviational ellipse method is relatively shape invariant and closely replicates the reference values, it is the recommended method. Speculatively, this study may also apply to negative-relief, and fluvial and aeolian bedforms.

Static aeroelastic analysis of wings using Euler/Navier-Stokes equations coupled with improved wing-box finite element structures

NASA Technical Reports Server (NTRS)

Guruswamy, Guru P.; MacMurdy, Dale E.; Kapania, Rakesh K.

1994-01-01

Strong interactions between flow about an aircraft wing and the wing structure can result in aeroelastic phenomena which significantly impact aircraft performance. Time-accurate methods for solving the unsteady Navier-Stokes equations have matured to the point where reliable results can be obtained with reasonable computational costs for complex non-linear flows with shock waves, vortices and separations. The ability to combine such a flow solver with a general finite element structural model is key to an aeroelastic analysis in these flows. Earlier work involved time-accurate integration of modal structural models based on plate elements. A finite element model was developed to handle three-dimensional wing boxes, and incorporated into the flow solver without the need for modal analysis. Static condensation is performed on the structural model to reduce the structural degrees of freedom for the aeroelastic analysis. Direct incorporation of the finite element wing-box structural model with the flow solver requires finding adequate methods for transferring aerodynamic pressures to the structural grid and returning deflections to the aerodynamic grid. Several schemes were explored for handling the grid-to-grid transfer of information. The complex, built-up nature of the wing-box complicated this transfer. Aeroelastic calculations for a sample wing in transonic flow comparing various simple transfer schemes are presented and discussed.

Fast-Running Aeroelastic Code Based on Unsteady Linearized Aerodynamic Solver Developed

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, Milind A.; Keith, T., Jr.

2003-01-01

The NASA Glenn Research Center has been developing aeroelastic analyses for turbomachines for use by NASA and industry. An aeroelastic analysis consists of a structural dynamic model, an unsteady aerodynamic model, and a procedure to couple the two models. The structural models are well developed. Hence, most of the development for the aeroelastic analysis of turbomachines has involved adapting and using unsteady aerodynamic models. Two methods are used in developing unsteady aerodynamic analysis procedures for the flutter and forced response of turbomachines: (1) the time domain method and (2) the frequency domain method. Codes based on time domain methods require considerable computational time and, hence, cannot be used during the design process. Frequency domain methods eliminate the time dependence by assuming harmonic motion and, hence, require less computational time. Early frequency domain analyses methods neglected the important physics of steady loading on the analyses for simplicity. A fast-running unsteady aerodynamic code, LINFLUX, which includes steady loading and is based on the frequency domain method, has been modified for flutter and response calculations. LINFLUX, solves unsteady linearized Euler equations for calculating the unsteady aerodynamic forces on the blades, starting from a steady nonlinear aerodynamic solution. First, we obtained a steady aerodynamic solution for a given flow condition using the nonlinear unsteady aerodynamic code TURBO. A blade vibration analysis was done to determine the frequencies and mode shapes of the vibrating blades, and an interface code was used to convert the steady aerodynamic solution to a form required by LINFLUX. A preprocessor was used to interpolate the mode shapes from the structural dynamic mesh onto the computational dynamics mesh. Then, we used LINFLUX to calculate the unsteady aerodynamic forces for a given mode, frequency, and phase angle. A postprocessor read these unsteady pressures and calculated the generalized aerodynamic forces, eigenvalues, and response amplitudes. The eigenvalues determine the flutter frequency and damping. As a test case, the flutter of a helical fan was calculated with LINFLUX and compared with calculations from TURBO-AE, a nonlinear time domain code, and from ASTROP2, a code based on linear unsteady aerodynamics.

Technical Evaluation Report on the Fluid Dynamics Panel Symposium on Aerodynamics and Acoustics of Propellers.

DTIC Science & Technology

1985-07-01

vortex filaments instead of the continuous sheet of vorticity used by Goldstein the propeller-nacelle interaction analysis also represents the wake by...the US Manufacturers in parallel with the development of the experimental propeller models , illustrated on Figre 0, these analysis methods range from...still poor, the difference between the two methods being mainly due to .,ifferent approaches used for obtaining lift. The Euler analysis of swirl angle

Analysis of Coaxial Injectors Using CFD++

DTIC Science & Technology

2011-09-12

turpentine droplets d = 1-90 um Uc = 2.4 m/s Low turbulence intensity of 1.4% Model Validation Case 9/12/2011 28Distribution A: Approved for Public...Approved for Public Release (Pending) Case Conditions: •Two species (Air, Turpentine ) •Base Equation Type: Compressible Real Gas Navier-Stokes/Euler

Three-dimensional multigrid algorithms for the flux-split Euler equations

NASA Technical Reports Server (NTRS)

Anderson, W. Kyle; Thomas, James L.; Whitfield, David L.

1988-01-01

The Full Approximation Scheme (FAS) multigrid method is applied to several implicit flux-split algorithms for solving the three-dimensional Euler equations in a body fitted coordinate system. Each of the splitting algorithms uses a variation of approximate factorization and is implemented in a finite volume formulation. The algorithms are all vectorizable with little or no scalar computation required. The flux vectors are split into upwind components using both the splittings of Steger-Warming and Van Leer. The stability and smoothing rate of each of the schemes are examined using a Fourier analysis of the complete system of equations. Results are presented for three-dimensional subsonic, transonic, and supersonic flows which demonstrate substantially improved convergence rates with the multigrid algorithm. The influence of using both a V-cycle and a W-cycle on the convergence is examined.

Free vibration analysis of microtubules based on the molecular mechanics and continuum beam theory.

PubMed

Zhang, Jin; Wang, Chengyuan

2016-10-01

A molecular structural mechanics (MSM) method has been implemented to investigate the free vibration of microtubules (MTs). The emphasis is placed on the effects of the configuration and the imperfect boundaries of MTs. It is shown that the influence of protofilament number on the fundamental frequency is strong, while the effect of helix-start number is almost negligible. The fundamental frequency is also found to decrease as the number of the blocked filaments at boundaries decreases. Subsequently, the Euler-Bernoulli beam theory is employed to reveal the physics behind the simulation results. Fitting the Euler-Bernoulli beam into the MSM data leads to an explicit formula for the fundamental frequency of MTs with various configurations and identifies a possible correlation between the imperfect boundary conditions and the length-dependent bending stiffness of MTs reported in experiments.

Multistage Schemes with Multigrid for Euler and Navier-Strokes Equations: Components and Analysis

NASA Technical Reports Server (NTRS)

Swanson, R. C.; Turkel, Eli

1997-01-01

A class of explicit multistage time-stepping schemes with centered spatial differencing and multigrids are considered for the compressible Euler and Navier-Stokes equations. These schemes are the basis for a family of computer programs (flow codes with multigrid (FLOMG) series) currently used to solve a wide range of fluid dynamics problems, including internal and external flows. In this paper, the components of these multistage time-stepping schemes are defined, discussed, and in many cases analyzed to provide additional insight into their behavior. Special emphasis is given to numerical dissipation, stability of Runge-Kutta schemes, and the convergence acceleration techniques of multigrid and implicit residual smoothing. Both the Baldwin and Lomax algebraic equilibrium model and the Johnson and King one-half equation nonequilibrium model are used to establish turbulence closure. Implementation of these models is described.

Dynamic Modeling and Simulation of a Rotational Inverted Pendulum

NASA Astrophysics Data System (ADS)

Duart, J. L.; Montero, B.; Ospina, P. A.; González, E.

2017-01-01

This paper presents an alternative way to the dynamic modeling of a rotational inverted pendulum using the classic mechanics known as Euler-Lagrange allows to find motion equations that describe our model. It also has a design of the basic model of the system in SolidWorks software, which based on the material and dimensions of the model provides some physical variables necessary for modeling. In order to verify the theoretical results, It was made a contrast between the solutions obtained by simulation SimMechanics-Matlab and the system of equations Euler-Lagrange, solved through ODE23tb method included in Matlab bookstores for solving equations systems of the type and order obtained. This article comprises a pendulum trajectory analysis by a phase space diagram that allows the identification of stable and unstable regions of the system.

Summing up the Euler [phi] Function

ERIC Educational Resources Information Center

Loomis, Paul; Plytage, Michael; Polhill, John

2008-01-01

The Euler [phi] function counts the number of positive integers less than and relatively prime to a positive integer n. Here we look at perfect totient numbers, number for which [phi](n) + [phi]([phi](n)) + [phi]([phi]([phi](n))) + ... + 1 = n.

Hydrodynamic Coherence and Vortex Solutions of the Euler-Helmholtz Equation

NASA Astrophysics Data System (ADS)

Fimin, N. N.; Chechetkin, V. M.

2018-03-01

The form of the general solution of the steady-state Euler-Helmholtz equation (reducible to the Joyce-Montgomery one) in arbitrary domains on the plane is considered. This equation describes the dynamics of vortex hydrodynamic structures.

Women and Mathematics in the Time of Euler

ERIC Educational Resources Information Center

Mayfield, Betty

2013-01-01

We explore mathematics written both by and for women in eighteenth-century Europe, and some of the interesting personalities involved: Maria Agnesi, Emilie du Chatelet, Laura Bassi, Princess Charlotte Ludovica Luisa, John Colson, Francesco Algarotti, and Leonhard Euler himself.

On Richardson extrapolation for low-dissipation low-dispersion diagonally implicit Runge-Kutta schemes

NASA Astrophysics Data System (ADS)

Havasi, Ágnes; Kazemi, Ehsan

2018-04-01

In the modeling of wave propagation phenomena it is necessary to use time integration methods which are not only sufficiently accurate, but also properly describe the amplitude and phase of the propagating waves. It is not clear if amending the developed schemes by extrapolation methods to obtain a high order of accuracy preserves the qualitative properties of these schemes in the perspective of dissipation, dispersion and stability analysis. It is illustrated that the combination of various optimized schemes with Richardson extrapolation is not optimal for minimal dissipation and dispersion errors. Optimized third-order and fourth-order methods are obtained, and it is shown that the proposed methods combined with Richardson extrapolation result in fourth and fifth orders of accuracy correspondingly, while preserving optimality and stability. The numerical applications include the linear wave equation, a stiff system of reaction-diffusion equations and the nonlinear Euler equations with oscillatory initial conditions. It is demonstrated that the extrapolated third-order scheme outperforms the recently developed fourth-order diagonally implicit Runge-Kutta scheme in terms of accuracy and stability.

On Flexible Tubes Conveying Fluid: Geometric Nonlinear Theory, Stability and Dynamics

NASA Astrophysics Data System (ADS)

Gay-Balmaz, François; Putkaradze, Vakhtang

2015-08-01

We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross section available to the fluid motion during the dynamics. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. We first derive the equations of motion directly, by using an Euler-Poincaré variational principle. We then justify this derivation with a more general theory elucidating the interesting mathematical concepts appearing in this problem, such as partial left (elastic) and right (fluid) invariance of the system, with the added holonomic constraint (volume). We analyze the fully nonlinear behavior of the model when the axis of the tube remains straight. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross section. Finally, we derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions.

Centrifuge Rotor Models: A Comparison of the Euler-Lagrange and the Bond Graph Modeling Approach

NASA Technical Reports Server (NTRS)

Granda, Jose J.; Ramakrishnan, Jayant; Nguyen, Louis H.

2006-01-01

A viewgraph presentation on centrifuge rotor models with a comparison using Euler-Lagrange and bond graph methods is shown. The topics include: 1) Objectives; 2) MOdeling Approach Comparisons; 3) Model Structures; and 4) Application.

On the commutator of C^{\\infty}} -symmetries and the reduction of Euler-Lagrange equations

NASA Astrophysics Data System (ADS)

Ruiz, A.; Muriel, C.; Olver, P. J.

2018-04-01

A novel procedure to reduce by four the order of Euler-Lagrange equations associated to nth order variational problems involving single variable integrals is presented. In preparation, a new formula for the commutator of two \

The 3D Euler solutions using automated Cartesian grid generation

NASA Technical Reports Server (NTRS)

Melton, John E.; Enomoto, Francis Y.; Berger, Marsha J.

1993-01-01

Viewgraphs on 3-dimensional Euler solutions using automated Cartesian grid generation are presented. Topics covered include: computational fluid dynamics (CFD) and the design cycle; Cartesian grid strategy; structured body fit; grid generation; prolate spheroid; and ONERA M6 wing.

Singularities of the Euler equation and hydrodynamic stability

NASA Technical Reports Server (NTRS)

Tanveer, S.; Speziale, Charles G.

1993-01-01

Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.

Singularities of the Euler equation and hydrodynamic stability

NASA Technical Reports Server (NTRS)

Tanveer, S.; Speziale, Charles G.

1992-01-01

Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin-Stuart Cat's Eye.

Quality assessment of two- and three-dimensional unstructured meshes and validation of an upwind Euler flow solver

NASA Technical Reports Server (NTRS)

Woodard, Paul R.; Yang, Henry T. Y.; Batina, John T.

1992-01-01

Quality assessment procedures are described for two-dimensional and three-dimensional unstructured meshes. The procedures include measurement of minimum angles, element aspect ratios, stretching, and element skewness. Meshes about the ONERA M6 wing and the Boeing 747 transport configuration are generated using an advancing front method grid generation package of programs. Solutions of Euler's equations for these meshes are obtained at low angle-of-attack, transonic conditions. Results for these cases, obtained as part of a validation study demonstrate the accuracy of an implicit upwind Euler solution algorithm.

On the Local Type I Conditions for the 3D Euler Equations

NASA Astrophysics Data System (ADS)

Chae, Dongho; Wolf, Jörg

2018-05-01

We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution {v\\in L^∞ (-1,0; L^2 ( B(x_0,r)))\\cap L^∞_{loc} (-1,0; W^{1, ∞} (B(x_0, r)))} of the 3D Euler equations, where {B(x_0,r)} is the ball with radius r and the center at x 0, if the limiting values of certain scale invariant quantities for a solution v(·, t) as {t\\to 0} are small enough, then { \

Unstructured Euler flow solutions using hexahedral cell refinement

NASA Technical Reports Server (NTRS)

Melton, John E.; Cappuccio, Gelsomina; Thomas, Scott D.

1991-01-01

An attempt is made to extend grid refinement into three dimensions by using unstructured hexahedral grids. The flow solver is developed using the TIGER (topologically Independent Grid, Euler Refinement) as the starting point. The program uses an unstructured hexahedral mesh and a modified version of the Jameson four-stage, finite-volume Runge-Kutta algorithm for integration of the Euler equations. The unstructured mesh allows for local refinement appropriate for each freestream condition, thereby concentrating mesh cells in the regions of greatest interest. This increases the computational efficiency because the refinement is not required to extend throughout the entire flow field.

Investigation of source location determination from Magsat magnetic anomalies: The Euler method approach

NASA Technical Reports Server (NTRS)

Ravat, Dhananjay

1996-01-01

The applicability of the Euler method of source location determination was investigated on several model situations pertinent to satellite-data scale situations as well as Magsat data of Europe. Our investigations enabled us to understand the end-member cases for which the Euler method will work with the present satellite magnetic data and also the cases for which the assumptions implicit in the Euler method will not be met by the present satellite magnetic data. These results have been presented in one invited lecture at the Indo-US workshop on Geomagnetism in Studies of the Earth's Interior in August 1994 in Pune, India, and at one presentation at the 21st General Assembly of the IUGG in July 1995 in Boulder, CO. A new method, called Anomaly Attenuation Rate (AAR) Method (based on the Euler method), was developed during this study. This method is scale-independent and is appropriate to locate centroids of semi-compact three dimensional sources of gravity and magnetic anomalies. The method was presented during 1996 Spring AGU meeting and a manuscript describing this method is being prepared for its submission to a high-ranking journal. The grant has resulted in 3 papers and presentations at national and international meetings and one manuscript of a paper (to be submitted shortly to a reputable journal).

Static and dynamic response of a sandwich structure under axial compression

NASA Astrophysics Data System (ADS)

Ji, Wooseok

This thesis is concerned with a combined experimental and theoretical investigation of the static and dynamic response of an axially compressed sandwich structure. For the static response problem of sandwich structures, a two-dimensional mechanical model is developed to predict the global and local buckling of a sandwich beam, using classical elasticity. The face sheet and the core are assumed as linear elastic orthotropic continua in a state of planar deformation. General buckling deformation modes (periodic and non-periodic) of the sandwich beam are considered. On the basis of the model developed here, validation and accuracy of several previous theories are discussed for different geometric and material properties of a sandwich beam. The appropriate incremental stress and conjugate incremental finite strain measure for the instability problem of the sandwich beam, and the corresponding constitutive model are addressed. The formulation used in the commercial finite element package is discussed in relation to the formulation adopted in the theoretical derivation. The Dynamic response problem of a sandwich structure subjected to axial impact by a falling mass is also investigated. The dynamic counterpart of the celebrated Euler buckling problem is formulated first and solved by considering the case of a slender column that is impacted by a falling mass. A new notion, that of the time to buckle, "t*" is introduced, which is the corresponding critical quantity analogous to the critical load in static Euler buckling. The dynamic bifurcation buckling analysis is extended to thick sandwich structures using an elastic foundation model. A comprehensive set of impact test results of sandwich columns with various configurations are presented. Failure mechanisms and the temporal history of how a sandwich column responds to axial impact are discussed through the experimental results. The experimental results are compared against analytical dynamic buckling studies and finite element based simulation of the impact event.

Error Control with Perfectly Matched Layer or Damping Layer Treatments for Computational Aeroacoustics with Jet Flows

NASA Technical Reports Server (NTRS)

Goodrich, John W.

2009-01-01

In this paper we show by means of numerical experiments that the error introduced in a numerical domain because of a Perfectly Matched Layer or Damping Layer boundary treatment can be controlled. These experimental demonstrations are for acoustic propagation with the Linearized Euler Equations with both uniform and steady jet flows. The propagating signal is driven by a time harmonic pressure source. Combinations of Perfectly Matched and Damping Layers are used with different damping profiles. These layer and profile combinations allow the relative error introduced by a layer to be kept as small as desired, in principle. Tradeoffs between error and cost are explored.

Progress on a Taylor weak statement finite element algorithm for high-speed aerodynamic flows

NASA Technical Reports Server (NTRS)

Baker, A. J.; Freels, J. D.

1989-01-01

A new finite element numerical Computational Fluid Dynamics (CFD) algorithm has matured to the point of efficiently solving two-dimensional high speed real-gas compressible flow problems in generalized coordinates on modern vector computer systems. The algorithm employs a Taylor Weak Statement classical Galerkin formulation, a variably implicit Newton iteration, and a tensor matrix product factorization of the linear algebra Jacobian under a generalized coordinate transformation. Allowing for a general two-dimensional conservation law system, the algorithm has been exercised on the Euler and laminar forms of the Navier-Stokes equations. Real-gas fluid properties are admitted, and numerical results verify solution accuracy, efficiency, and stability over a range of test problem parameters.

Poisson structure on a space with linear SU(2) fuzziness

NASA Astrophysics Data System (ADS)

Khorrami, Mohammad; Fatollahi, Amir H.; Shariati, Ahmad

2009-07-01

The Poisson structure is constructed for a model in which spatial coordinates of configuration space are noncommutative and satisfy the commutation relations of a Lie algebra. The case is specialized to that of the group SU(2), for which the counterpart of the angular momentum as well as the Euler parametrization of the phase space are introduced. SU(2)-invariant classical systems are discussed, and it is observed that the path of particle can be obtained by the solution of a first-order equation, as the case with such models on commutative spaces. The examples of free particle, rotationally invariant potentials, and specially the isotropic harmonic oscillator are investigated in more detail.

Finite area method for nonlinear supersonic conical flows

NASA Technical Reports Server (NTRS)

Sritharan, S. S.; Seebass, A. R.

1983-01-01

A fully conservative numerical method for the computation of steady inviscid supersonic flow about general conical bodies at incidence is described. The procedure utilizes the potential approximation and implements a body conforming mesh generator. The conical potential is assumed to have its best linear variation inside each mesh cell; a secondary interlocking cell system is used to establish the flux balance required to conserve mass. In the supersonic regions the scheme is symmetrized by adding artificial viscosity in conservation form. The algorithm is nearly an order of a magnitude faster than present Euler methods and predicts known results accurately and qualitative features such as nodal point lift off correctly. Results are compared with those of other investigators.

Finite area method for nonlinear conical flows

NASA Technical Reports Server (NTRS)

Sritharan, S. S.; Seebass, A. R.

1982-01-01

A fully conservative finite area method for the computation of steady inviscid flow about general conical bodies at incidence is described. The procedure utilizes the potential approximation and implements a body conforming mesh generator. The conical potential is assumed to have its best linear variation inside each mesh cell and a secondary interlocking cell system is used to establish the flux balance required to conserve mass. In the supersonic regions the scheme is desymmetrized by adding appropriate artificial viscosity in conservation form. The algorithm is nearly an order of a magnitude faster than present Euler methods and predicts known results accurately and qualitative features such as nodal point lift off correctly. Results are compared with those of other investigations.

Accuracy versus convergence rates for a three dimensional multistage Euler code

NASA Technical Reports Server (NTRS)

Turkel, Eli

1988-01-01

Using a central difference scheme, it is necessary to add an artificial viscosity in order to reach a steady state. This viscosity usually consists of a linear fourth difference to eliminate odd-even oscillations and a nonlinear second difference to suppress oscillations in the neighborhood of steep gradients. There are free constants in these differences. As one increases the artificial viscosity, the high modes are dissipated more and the scheme converges more rapidly. However, this higher level of viscosity smooths the shocks and eliminates other features of the flow. Thus, there is a conflict between the requirements of accuracy and efficiency. Examples are presented for a variety of three-dimensional inviscid solutions over isolated wings.

Study of water based nanofluid flows in annular tubes using numerical simulation and sensitivity analysis

NASA Astrophysics Data System (ADS)

Siadaty, Moein; Kazazi, Mohsen

2018-04-01

Convective heat transfer, entropy generation and pressure drop of two water based nanofluids (Cu-water and Al2O3-water) in horizontal annular tubes are scrutinized by means of computational fluids dynamics, response surface methodology and sensitivity analysis. First, central composite design is used to perform a series of experiments with diameter ratio, length to diameter ratio, Reynolds number and solid volume fraction. Then, CFD is used to calculate the Nusselt Number, Euler number and entropy generation. After that, RSM is applied to fit second order polynomials on responses. Finally, sensitivity analysis is conducted to manage the above mentioned parameters inside tube. Totally, 62 different cases are examined. CFD results show that Cu-water and Al2O3-water have the highest and lowest heat transfer rate, respectively. In addition, analysis of variances indicates that increase in solid volume fraction increases dimensionless pressure drop for Al2O3-water. Moreover, it has a significant negative and insignificant effects on Cu-water Nusselt and Euler numbers, respectively. Analysis of Bejan number indicates that frictional and thermal entropy generations are the dominant irreversibility in Al2O3-water and Cu-water flows, respectively. Sensitivity analysis indicates dimensionless pressure drop sensitivity to tube length for Cu-water is independent of its diameter ratio at different Reynolds numbers.

Bifurcation-enhanced ultrahigh sensitivity of a buckled cantilever

PubMed Central

An, Sangmin; Kim, Bongsu; Kwon, Soyoung; Moon, Geol; Lee, Manhee

2018-01-01

Buckling, first introduced by Euler in 1744 [Euler L (1744) Opera Omnia I 24:231], a sudden mechanical sideways deflection of a structural member under compressive stress, represents a bifurcation in the solution to the equations of static equilibrium. Although it has been investigated in diverse research areas, such a common nonlinear phenomenon may be useful to devise a unique mechanical sensor that addresses the still-challenging features, such as the enhanced sensitivity and polarization-dependent detection capability. We demonstrate the bifurcation-enhanced sensitive measurement of mechanical vibrations using the nonlinear buckled cantilever tip in ambient conditions. The cantilever, initially buckled with its tip pinned, flips its buckling near the bifurcation point (BP), where the buckled tip becomes softened. The enhanced mechanical sensitivity results from the increasing fluctuations, unlike the typical linear sensors, which facilitate the noise-induced buckling-to-flipping transition of the softened cantilever. This allows the in situ continuous or repeated single-shot detection of the surface acoustic waves of different polarizations without any noticeable wear of the tip. We obtained the sensitivity above 106 V(m/s)−1, a 1,000-fold enhancement over the conventional seismometers. Our results lead to development of mechanical sensors of high sensitivity, reproducibility, and durability, which may be applied to detect, e.g., the directional surface waves on the laboratory as well as the geological scale. PMID:29511105

Verification of a non-hydrostatic dynamical core using horizontally spectral element vertically finite difference method: 2-D aspects

NASA Astrophysics Data System (ADS)

Choi, S.-J.; Giraldo, F. X.; Kim, J.; Shin, S.

2014-06-01

The non-hydrostatic (NH) compressible Euler equations of dry atmosphere are solved in a simplified two dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss-Lobatto-Legendre (GLL) quadrature points. The FDM employs a third-order upwind biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative terms and quadrature. The Euler equations used here are in a flux form based on the hydrostatic pressure vertical coordinate, which are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma-pressure vertical coordinate is implemented in this model. We verified the model by conducting widely used standard benchmark tests: the inertia-gravity wave, rising thermal bubble, density current wave, and linear hydrostatic mountain wave. The results from those tests demonstrate that the horizontally spectral element vertically finite difference model is accurate and robust. By using the 2-D slice model, we effectively show that the combined spatial discretization method of the spectral element and finite difference method in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core.

Benchmark problems in computational aeroacoustics

NASA Technical Reports Server (NTRS)

Porter-Locklear, Freda

1994-01-01

A recent directive at NASA Langley is aimed at numerically predicting principal noise sources. During my summer stay, I worked with high-order ENO code, developed by Dr. Harold Atkins, for solving the unsteady compressible Navier-Stokes equations, as it applies to computational aeroacoustics (CAA). A CAA workshop, composed of six categories of benchmark problems, has been organized to test various numerical properties of code. My task was to determine the robustness of Atkins' code for these test problems. In one category, we tested the nonlinear wave propagation of the code for the one-dimensional Euler equations, with initial pressure, density, and velocity conditions. Using freestream boundary conditions, our results were plausible. In another category, we solved the linearized two-dimensional Euler equations to test the effectiveness of radiation boundary conditions. Here we utilized MAPLE to compute eigenvalues and eigenvectors of the Jacobian given variable and flux vectors. We experienced a minor problem with inflow and outflow boundary conditions. Next, we solved the quasi one dimensional unsteady flow equations with an incoming acoustic wave of amplitude 10(exp -6). The small amplitude sound wave was incident on a convergent-divergent nozzle. After finding a steady-state solution and then marching forward, our solution indicated that after 30 periods the acoustic wave had dissipated (a period is time required for sound wave to traverse one end of nozzle to other end).

An efficient method for solving the steady Euler equations

NASA Technical Reports Server (NTRS)

Liou, M. S.

1986-01-01

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

Monotonic Derivative Correction for Calculation of Supersonic Flows

ERIC Educational Resources Information Center

Bulat, Pavel V.; Volkov, Konstantin N.

2016-01-01

Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the…

Latin and Cross Latin Squares

ERIC Educational Resources Information Center

Emanouilidis, Emanuel

2008-01-01

Latin squares were first introduced and studied by the famous mathematician Leonhard Euler in the 1700s. Through the years, Latin squares have been used in areas such as statistics, graph theory, coding theory, the generation of random numbers as well as in the design and analysis of experiments. Recently, with the international popularity of…

The Legacy of Leonhard Euler--A Tricentennial Tribute

ERIC Educational Resources Information Center

Debnath, Lokenath

2009-01-01

This tricentennial tribute commemorates Euler's major contributions to mathematical and physical sciences. A brief biographical sketch is presented with his major contributions to certain selected areas of number theory, differential and integral calculus, differential equations, solid and fluid mechanics, topology and graph theory, infinite…

Euler's Identity, Leibniz Tables, and the Irrationality of Pi

ERIC Educational Resources Information Center

Jones, Timothy W.

2012-01-01

Using techniques that show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's identity. The proof involves evaluations of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.

Research on Distribution Characteristics of Lunar Faults

NASA Astrophysics Data System (ADS)

Lu, T.; Chen, S.; Lu, P.

2017-12-01

Circular and linear tectonics are two major types of tectonics on lunar surface. Tectonic characteristics are of significance for researching about lunar geological evolution. Linear tectonics refers to those structures extending linearly on a lunar surface. Their distribution are closely related to the internal geological actions of the moon. Linear tectonics can integrally or locally express the structural feature and the stress status as well as showing the geological information of the interior of the moon. Faults are of the largest number and are of a certain distribution regularity among the linear tectonics, and are always the focus of domestic and overseas lunar tectonic research. Based on remote sensing geology and theory of traditional tectonic geology, We use a variety of remote sensing data processing to establish lunar linear tectonic interpretation keys with lunar spectral, terrain and gravity data. On this basis, interpretation of faults of the whole moon was primarily conducted from Chang'e-2 CCD image data and reference to wide-angle camera data of LROC, laser altimeter data of LOLA and gravity data of GRAIL. Statistical analysis of the number and distribution characteristics of whole lunar faults are counted from three latitude ranges of low, middle and high latitudes, then analyze the azimuth characteristics of the faults at different latitudes. We concluded that S-N direction is a relatively developed orientation at low latitudes. Middle latitudes reveal six preferred orientations of N-E, N-W, NN-E, NN-W, N-EE and N-WW directions. There are sparse faults of E-W direction distribution at low and middle latitudes. Meanwhile, the largest number of faults of E-W direction on lunar surface are mainly distributed along high latitudes with continuity and regularity. Analyzing faults of Mare Imbrium by the method of Euler deconvolution. The result show that there are two different properties of faults in Mare Imbrium. In conclusion, we suggest that the dynamics mechanism of the formation of the lunar faults is mainly affected by despinning, followed by tidal force and global contraction.

Error Propagation in the four terrestrial reference frames of the 2022 Modernized National Spatial Reference System

NASA Astrophysics Data System (ADS)

Roman, D. R.; Smith, D. A.

2017-12-01

In 2022, the National Geodetic Survey will replace all three NAD 83 reference frames with four new terrestrial reference frames. Each frame will be named after a tectonic plate (North American, Pacific, Caribbean and Mariana) and each will be related to the IGS frame through three Euler Pole parameters (EPPs). This talk will focus on three main areas of error propagation when defining coordinates in these four frames. Those areas are (1) use of the small angle approximation to relate true rotation about an Euler Pole to small rotations about three Cartesian axes (2) The current state of the art in determining the Euler Poles of these four plates and (3) the combination of both IGS Cartesian coordinate uncertainties and EPP uncertainties into coordinate uncertainties in the four new frames. Discussion will also include recent efforts at improving the Euler Poles for these frames and expected dates when errors in the EPPs will cause an unacceptable level of uncertainty in the four new terrestrial reference frames.

A Revelation: Quantum-Statistics and Classical-Statistics are Analytic-Geometry Conic-Sections and Numbers/Functions: Euler, Riemann, Bernoulli Generating-Functions: Conics to Numbers/Functions Deep Subtle Connections

NASA Astrophysics Data System (ADS)

Descartes, R.; Rota, G.-C.; Euler, L.; Bernoulli, J. D.; Siegel, Edward Carl-Ludwig

2011-03-01

Quantum-statistics Dichotomy: Fermi-Dirac(FDQS) Versus Bose-Einstein(BEQS), respectively with contact-repulsion/non-condensation(FDCR) versus attraction/ condensationBEC are manifestly-demonstrated by Taylor-expansion ONLY of their denominator exponential, identified BOTH as Descartes analytic-geometry conic-sections, FDQS as Elllipse (homotopy to rectangle FDQS distribution-function), VIA Maxwell-Boltzmann classical-statistics(MBCS) to Parabola MORPHISM, VS. BEQS to Hyperbola, Archimedes' HYPERBOLICITY INEVITABILITY, and as well generating-functions[Abramowitz-Stegun, Handbook Math.-Functions--p. 804!!!], respectively of Euler-numbers/functions, (via Riemann zeta-function(domination of quantum-statistics: [Pathria, Statistical-Mechanics; Huang, Statistical-Mechanics]) VS. Bernoulli-numbers/ functions. Much can be learned about statistical-physics from Euler-numbers/functions via Riemann zeta-function(s) VS. Bernoulli-numbers/functions [Conway-Guy, Book of Numbers] and about Euler-numbers/functions, via Riemann zeta-function(s) MORPHISM, VS. Bernoulli-numbers/ functions, visa versa!!! Ex.: Riemann-hypothesis PHYSICS proof PARTLY as BEQS BEC/BEA!!!

Multigrid calculation of three-dimensional turbomachinery flows

NASA Technical Reports Server (NTRS)

Caughey, David A.

1989-01-01

Research was performed in the general area of computational aerodynamics, with particular emphasis on the development of efficient techniques for the solution of the Euler and Navier-Stokes equations for transonic flows through the complex blade passages associated with turbomachines. In particular, multigrid methods were developed, using both explicit and implicit time-stepping schemes as smoothing algorithms. The specific accomplishments of the research have included: (1) the development of an explicit multigrid method to solve the Euler equations for three-dimensional turbomachinery flows based upon the multigrid implementation of Jameson's explicit Runge-Kutta scheme (Jameson 1983); (2) the development of an implicit multigrid scheme for the three-dimensional Euler equations based upon lower-upper factorization; (3) the development of a multigrid scheme using a diagonalized alternating direction implicit (ADI) algorithm; (4) the extension of the diagonalized ADI multigrid method to solve the Euler equations of inviscid flow for three-dimensional turbomachinery flows; and also (5) the extension of the diagonalized ADI multigrid scheme to solve the Reynolds-averaged Navier-Stokes equations for two-dimensional turbomachinery flows.

Flow analysis for the nacelle of an advanced ducted propeller at high angle-of-attack and at cruise with boundary layer control

NASA Technical Reports Server (NTRS)

Hwang, D. P.; Boldman, D. R.; Hughes, C. E.

1994-01-01

An axisymmetric panel code and a three dimensional Navier-Stokes code (used as an inviscid Euler code) were verified for low speed, high angle of attack flow conditions. A three dimensional Navier-Stokes code (used as an inviscid code), and an axisymmetric Navier-Stokes code (used as both viscous and inviscid code) were also assessed for high Mach number cruise conditions. The boundary layer calculations were made by using the results from the panel code or Euler calculation. The panel method can predict the internal surface pressure distributions very well if no shock exists. However, only Euler and Navier-Stokes calculations can provide a good prediction of the surface static pressure distribution including the pressure rise across the shock. Because of the high CPU time required for a three dimensional Navier-Stokes calculation, only the axisymmetric Navier-Stokes calculation was considered at cruise conditions. The use of suction and tangential blowing boundary layer control to eliminate the flow separation on the internal surface was demonstrated for low free stream Mach number and high angle of attack cases. The calculation also shows that transition from laminar flow to turbulent flow on the external cowl surface can be delayed by using suction boundary layer control at cruise flow conditions. The results were compared with experimental data where possible.

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

NASA Astrophysics Data System (ADS)

Do, Tam; Kiselev, Alexander; Xu, Xiaoqian

2016-10-01

The question of the global regularity versus finite- time blowup in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow-up scenario for the 3D Euler equation proposed by Hou and Luo (Multiscale Model Simul 12:1722-1776, 2014) based on extensive numerical simulations. These models generalize the 1D Hou-Luo model suggested in Hou and Luo Luo and Hou (2014), for which finite-time blowup has been established in Choi et al. (arXiv preprint. arXiv:1407.4776, 2014). The main new aspects of this work are twofold. First, we establish finite-time blowup for a model that is a closer approximation of the three-dimensional case than the original Hou-Luo model, in the sense that it contains relevant lower-order terms in the Biot-Savart law that have been discarded in Hou and Luo Choi et al. (2014). Secondly, we show that the blow-up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou-Luo model. Such blow-up stability result may be useful in further work on understanding the 3D hyperbolic blow-up scenario.

High-Energy Vacuum Birefringence and Dichroism in an Ultrastrong Laser Field

NASA Astrophysics Data System (ADS)

Bragin, Sergey; Meuren, Sebastian; Keitel, Christoph H.; Di Piazza, Antonino

2017-12-01

A long-standing prediction of quantum electrodynamics, yet to be experimentally observed, is the interaction between real photons in vacuum. As a consequence of this interaction, the vacuum is expected to become birefringent and dichroic if a strong laser field polarizes its virtual particle-antiparticle dipoles. Here, we derive how a generally polarized probe photon beam is influenced by both vacuum birefringence and dichroism in a strong linearly polarized plane-wave laser field. Furthermore, we consider an experimental scheme to measure these effects in the nonperturbative high-energy regime, where the Euler-Heisenberg approximation breaks down. By employing circularly polarized high-energy probe photons, as opposed to the conventionally considered linearly polarized ones, the feasibility of quantitatively confirming the prediction of nonlinear QED for vacuum birefringence at the 5 σ confidence level on the time scale of a few days is demonstrated for upcoming 10 PW laser systems. Finally, dichroism and anomalous dispersion in vacuum are shown to be accessible at these facilities.

Stabilization Approaches for Linear and Nonlinear Reduced Order Models

NASA Astrophysics Data System (ADS)

Rezaian, Elnaz; Wei, Mingjun

2017-11-01

It has been a major concern to establish reduced order models (ROMs) as reliable representatives of the dynamics inherent in high fidelity simulations, while fast computation is achieved. In practice it comes to stability and accuracy of ROMs. Given the inviscid nature of Euler equations it becomes more challenging to achieve stability, especially where moving discontinuities exist. Originally unstable linear and nonlinear ROMs are stabilized here by two approaches. First, a hybrid method is developed by integrating two different stabilization algorithms. At the same time, symmetry inner product is introduced in the generation of ROMs for its known robust behavior for compressible flows. Results have shown a notable improvement in computational efficiency and robustness compared to similar approaches. Second, a new stabilization algorithm is developed specifically for nonlinear ROMs. This method adopts Particle Swarm Optimization to enforce a bounded ROM response for minimum discrepancy between the high fidelity simulation and the ROM outputs. Promising results are obtained in its application on the nonlinear ROM of an inviscid fluid flow with discontinuities. Supported by ARL.

A hierarchy for modeling high speed propulsion systems

NASA Technical Reports Server (NTRS)

Hartley, Tom T.; Deabreu, Alex

1991-01-01

General research efforts on reduced order propulsion models for control systems design are overviewed. Methods for modeling high speed propulsion systems are discussed including internal flow propulsion systems that do not contain rotating machinery, such as inlets, ramjets, and scramjets. The discussion is separated into four areas: (1) computational fluid dynamics models for the entire nonlinear system or high order nonlinear models; (2) high order linearized models derived from fundamental physics; (3) low order linear models obtained from the other high order models; and (4) low order nonlinear models (order here refers to the number of dynamic states). Included in the discussion are any special considerations based on the relevant control system designs. The methods discussed are for the quasi-one-dimensional Euler equations of gasdynamic flow. The essential nonlinear features represented are large amplitude nonlinear waves, including moving normal shocks, hammershocks, simple subsonic combustion via heat addition, temperature dependent gases, detonations, and thermal choking. The report also contains a comprehensive list of papers and theses generated by this grant.

An efficient model for coupling structural vibrations with acoustic radiation

NASA Technical Reports Server (NTRS)

Frendi, Abdelkader; Maestrello, Lucio; Ting, LU

1993-01-01

The scattering of an incident wave by a flexible panel is studied. The panel vibration is governed by the nonlinear plate equations while the loading on the panel, which is the pressure difference across the panel, depends on the reflected and transmitted waves. Two models are used to calculate this structural-acoustic interaction problem. One solves the three dimensional nonlinear Euler equations for the flow-field coupled with the plate equations (the fully coupled model). The second uses the linear wave equation for the acoustic field and expresses the load as a double integral involving the panel oscillation (the decoupled model). The panel oscillation governed by a system of integro-differential equations is solved numerically and the acoustic field is then defined by an explicit formula. Numerical results are obtained using the two models for linear and nonlinear panel vibrations. The predictions given by these two models are in good agreement but the computational time needed for the 'fully coupled model' is 60 times longer than that for 'the decoupled model'.

Use of the PARC code to estimate the off-design transonic performance of an over/under turboramjet nozzle

NASA Technical Reports Server (NTRS)

Lam, David W.

1995-01-01

The transonic performance of a dual-throat, single-expansion-ramp nozzle (SERN) was investigated with a PARC computational fluid dynamics (CFD) code, an external flow Navier-Stokes solver. The nozzle configuration was from a conceptual Mach 5 cruise aircraft powered by four air-breathing turboramjets. Initial test cases used the two-dimensional version of PARC in Euler mode to investigate the effect of geometric variation on transonic performance. Additional cases used the two-dimensional version in viscous mode and the three-dimensional version in both Euler and viscous modes. Results of the analysis indicate low nozzle performance and a highly three-dimensional nozzle flow at transonic conditions. In another comparative study using the PARC code, a single-throat SERN configuration for which experimental data were available at transonic conditions was used to validate the results of the over/under turboramjet nozzle.

Vibrational algorithms for quantitative crystallographic analyses of hydroxyapatite-based biomaterials: I, theoretical foundations.

PubMed

Pezzotti, Giuseppe; Zhu, Wenliang; Boffelli, Marco; Adachi, Tetsuya; Ichioka, Hiroaki; Yamamoto, Toshiro; Marunaka, Yoshinori; Kanamura, Narisato

2015-05-01

The Raman spectroscopic method has quantitatively been applied to the analysis of local crystallographic orientation in both single-crystal hydroxyapatite and human teeth. Raman selection rules for all the vibrational modes of the hexagonal structure were expanded into explicit functions of Euler angles in space and six Raman tensor elements (RTE). A theoretical treatment has also been put forward according to the orientation distribution function (ODF) formalism, which allows one to resolve the statistical orientation patterns of the nm-sized hydroxyapatite crystallite comprised in the Raman microprobe. Close-form solutions could be obtained for the Euler angles and their statistical distributions resolved with respect to the direction of the average texture axis. Polarized Raman spectra from single-crystalline hydroxyapatite and textured polycrystalline (teeth enamel) samples were compared, and a validation of the proposed Raman method could be obtained through confirming the agreement between RTE values obtained from different samples.

Developing and utilizing an Euler computational method for predicting the airframe/propulsion effects for an aft-mounted turboprop transport. Volume 1: Theory document

NASA Technical Reports Server (NTRS)

Chen, H. C.; Yu, N. Y.

1991-01-01

An Euler flow solver was developed for predicting the airframe/propulsion integration effects for an aft-mounted turboprop transport. This solver employs a highly efficient multigrid scheme, with a successive mesh-refinement procedure to accelerate the convergence of the solution. A new dissipation model was also implemented to render solutions that are grid insensitive. The propeller power effects are simulated by the actuator disk concept. An embedded flow solution method was developed for predicting the detailed flow characteristics in the local vicinity of an aft-mounted propfan engine in the presence of a flow field induced by a complete aircraft. Results from test case analysis are presented. A user's guide for execution of computer programs, including format of various input files, sample job decks, and sample input files, is provided in an accompanying volume.

Sufficient condition for a finite-time singularity in a high-symmetry Euler flow: Analysis and statistics

NASA Astrophysics Data System (ADS)

Ng, C. S.; Bhattacharjee, A.

1996-08-01

A sufficient condition is obtained for the development of a finite-time singularity in a highly symmetric Euler flow, first proposed by Kida [J. Phys. Soc. Jpn. 54, 2132 (1995)] and recently simulated by Boratav and Pelz [Phys. Fluids 6, 2757 (1994)]. It is shown that if the second-order spatial derivative of the pressure (pxx) is positive following a Lagrangian element (on the x axis), then a finite-time singularity must occur. Under some assumptions, this Lagrangian sufficient condition can be reduced to an Eulerian sufficient condition which requires that the fourth-order spatial derivative of the pressure (pxxxx) at the origin be positive for all times leading up to the singularity. Analytical as well as direct numerical evaluation over a large ensemble of initial conditions demonstrate that for fixed total energy, pxxxx is predominantly positive with the average value growing with the numbers of modes.

Euler analysis comparison with LDV data for an advanced counter-rotation propfan at cruise

NASA Technical Reports Server (NTRS)

Miller, Christopher J.; Podboy, Gary G.

1990-01-01

A fine mesh Euler solution of the F4/A4 unducted fan (UDF) model flowfield is compared with laser Doppler velocimeter (LDV) data taken in the NASA Lewis 8- by 6-Foot Supersonic Wind Tunnel. The comparison is made primarily at one axial plane downstream of the front rotor where the LDV particle lag errors are reduced. The agreement between measured and predicted velocities in this axial plane is good. The results show that a dense mesh is needed in the centerbody stagnation region to minimize entropy generation that weakens the aft row passage shock. The predicted radial location of the tip vortex downstream of the front rotor agrees well with the experimental results but the strength is overpredicted. With 40 points per chord line, the integrated performance quantities are nearly converged, but more points are needed to resolve passage shocks and flow field details.

Investigation of dispersion-relation-preserving scheme and spectral analysis methods for acoustic waves

NASA Technical Reports Server (NTRS)

Vanel, Florence O.; Baysal, Oktay

1995-01-01

Important characteristics of the aeroacoustic wave propagation are mostly encoded in their dispersion relations. Hence, a computational aeroacoustic (CAA) algorithm, which reasonably preserves these relations, was investigated. It was derived using an optimization procedure to ensure, that the numerical derivatives preserved the wave number and angular frequency of the differential terms in the linearized, 2-D Euler equations. Then, simulations were performed to validate the scheme and a compatible set of discretized boundary conditions. The computational results were found to agree favorably with the exact solutions. The boundary conditions were transparent to the outgoing waves, except when the disturbance source was close to a boundary. The time-domain data generated by such CAA solutions were often intractable until their spectra was analyzed. Therefore, the relative merits of three different methods were included in the study. For simple, periodic waves, the periodogram method produced better estimates of the steep-sloped spectra than the Blackman-Tukey method. Also, for this problem, the Hanning window was more effective when used with the weighted-overlapped-segment-averaging and Blackman-Tukey methods gave better results than the periodogram method. Finally, it was demonstrated that the representation of time domain-data was significantly dependent on the particular spectral analysis method employed.

Determination of the free lunar libration modes from ephemeris DE430

NASA Astrophysics Data System (ADS)

Yang, Yong-Zhang; Li, Jin-Ling; Ping, Jin-Song; Hanada, Hideo

2017-12-01

The Moon’s physical librations have been extensively studied, and elaborate researches have been developed for the purpose of deriving accurate modes of free librations. Our motivation comes from the Planetary and Lunar Ephemeris DE430 by JPL/NASA, which was created in April 2013, and is reported to be the most accurate lunar ephemeris today using the data from Gravity Recovery and Interior Laboratory (GRAIL). Therefore, the residuals after fitting the model have reduced owing to improvement in the libration models, and the free librations embedded in the Euler angles have also improved. We use Fourier analysis to extract the approximate frequencies from DE430 and then a quadratic interpolation method is used to determine higher accuracy frequencies. With the frequencies, the linear least-squares fitting method is employed to fit the lunar physical librations to DE430. From this analysis we identified the three modes of free physical librations, and estimated the amplitudes as {1.471}\\prime\\prime in longitude, {0.025}\\prime\\prime in latitude and {8.19}\\prime\\prime× {3.31}\\prime\\prime for the wobble, with the respective periods of 1056.16, 8806.9 and 27262.99 d. Since the free librations damp with time, they require recent excitation or a continuous stimulating mechanism in order to sustain.

Symplectic analysis of vertical random vibration for coupled vehicle track systems

NASA Astrophysics Data System (ADS)

Lu, F.; Kennedy, D.; Williams, F. W.; Lin, J. H.

2008-10-01

A computational model for random vibration analysis of vehicle-track systems is proposed and solutions use the pseudo excitation method (PEM) and the symplectic method. The vehicle is modelled as a mass, spring and damping system with 10 degrees of freedom (dofs) which consist of vertical and pitching motion for the vehicle body and its two bogies and vertical motion for the four wheelsets. The track is treated as an infinite Bernoulli-Euler beam connected to sleepers and hence to ballast and is regarded as a periodic structure. Linear springs couple the vehicle and the track. Hence, the coupled vehicle-track system has only 26 dofs. A fixed excitation model is used, i.e. the vehicle does not move along the track but instead the track irregularity profile moves backwards at the vehicle velocity. This irregularity is assumed to be a stationary random process. Random vibration theory is used to obtain the response power spectral densities (PSDs), by using PEM to transform this random multiexcitation problem into a deterministic harmonic excitation one and then applying symplectic solution methodology. Numerical results for an example include verification of the proposed method by comparing with finite element method (FEM) results; comparison between the present model and the traditional rigid track model and; discussion of the influences of track damping and vehicle velocity.

Accurate electrostatic and van der Waals pull-in prediction for fully clamped nano/micro-beams using linear universal graphs of pull-in instability

NASA Astrophysics Data System (ADS)

Tahani, Masoud; Askari, Amir R.

2014-09-01

In spite of the fact that pull-in instability of electrically actuated nano/micro-beams has been investigated by many researchers to date, no explicit formula has been presented yet which can predict pull-in voltage based on a geometrically non-linear and distributed parameter model. The objective of present paper is to introduce a simple and accurate formula to predict this value for a fully clamped electrostatically actuated nano/micro-beam. To this end, a non-linear Euler-Bernoulli beam model is employed, which accounts for the axial residual stress, geometric non-linearity of mid-plane stretching, distributed electrostatic force and the van der Waals (vdW) attraction. The non-linear boundary value governing equation of equilibrium is non-dimensionalized and solved iteratively through single-term Galerkin based reduced order model (ROM). The solutions are validated thorough direct comparison with experimental and other existing results reported in previous studies. Pull-in instability under electrical and vdW loads are also investigated using universal graphs. Based on the results of these graphs, non-dimensional pull-in and vdW parameters, which are defined in the text, vary linearly versus the other dimensionless parameters of the problem. Using this fact, some linear equations are presented to predict pull-in voltage, the maximum allowable length, the so-called detachment length, and the minimum allowable gap for a nano/micro-system. These linear equations are also reduced to a couple of universal pull-in formulas for systems with small initial gap. The accuracy of the universal pull-in formulas are also validated by comparing its results with available experimental and some previous geometric linear and closed-form findings published in the literature.

On the estimation variance for the specific Euler-Poincaré characteristic of random networks.

PubMed

Tscheschel, A; Stoyan, D

2003-07-01

The specific Euler number is an important topological characteristic in many applications. It is considered here for the case of random networks, which may appear in microscopy either as primary objects of investigation or as secondary objects describing in an approximate way other structures such as, for example, porous media. For random networks there is a simple and natural estimator of the specific Euler number. For its estimation variance, a simple Poisson approximation is given. It is based on the general exact formula for the estimation variance. In two examples of quite different nature and topology application of the formulas is demonstrated.

On the Approximation of Generalized Lipschitz Function by Euler Means of Conjugate Series of Fourier Series

PubMed Central

Kushwaha, Jitendra Kumar

2013-01-01

Approximation theory is a very important field which has various applications in pure and applied mathematics. The present study deals with a new theorem on the approximation of functions of Lipschitz class by using Euler's mean of conjugate series of Fourier series. In this paper, the degree of approximation by using Euler's means of conjugate of functions belonging to Lip (ξ(t), p) class has been obtained. Lipα and Lip (α, p) classes are the particular cases of Lip (ξ(t), p) class. The main result of this paper generalizes some well-known results in this direction. PMID:24379744

Uniform high order spectral methods for one and two dimensional Euler equations

NASA Technical Reports Server (NTRS)

Cai, Wei; Shu, Chi-Wang

1991-01-01

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.

Remarks on High Reynolds Numbers Hydrodynamics and the Inviscid Limit

NASA Astrophysics Data System (ADS)

Constantin, Peter; Vicol, Vlad

2018-04-01

We prove that any weak space-time L^2 vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of R^2 satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that t-a.e. weak L^2 inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second-order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.

A Solution Adaptive Structured/Unstructured Overset Grid Flow Solver with Applications to Helicopter Rotor Flows

NASA Technical Reports Server (NTRS)

Duque, Earl P. N.; Biswas, Rupak; Strawn, Roger C.

1995-01-01

This paper summarizes a method that solves both the three dimensional thin-layer Navier-Stokes equations and the Euler equations using overset structured and solution adaptive unstructured grids with applications to helicopter rotor flowfields. The overset structured grids use an implicit finite-difference method to solve the thin-layer Navier-Stokes/Euler equations while the unstructured grid uses an explicit finite-volume method to solve the Euler equations. Solutions on a helicopter rotor in hover show the ability to accurately convect the rotor wake. However, isotropic subdivision of the tetrahedral mesh rapidly increases the overall problem size.

Functional equations for orbifold wreath products

NASA Astrophysics Data System (ADS)

Farsi, Carla; Seaton, Christopher

2017-10-01

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for the Euler and Euler-Satake characteristics, which we compute for a class of weighted projective spaces. This generalizes results known for global quotients by finite groups to all closed, effective orbifolds. We also describe a combinatorial approach to extensions of multiplicative invariants using decomposable functors that recovers the formula for the Euler-Satake characteristic of a wreath product of a global quotient orbifold.

Modelling non-linear effects of dark energy

NASA Astrophysics Data System (ADS)

Bose, Benjamin; Baldi, Marco; Pourtsidou, Alkistis

2018-04-01

We investigate the capabilities of perturbation theory in capturing non-linear effects of dark energy. We test constant and evolving w models, as well as models involving momentum exchange between dark energy and dark matter. Specifically, we compare perturbative predictions at 1-loop level against N-body results for four non-standard equations of state as well as varying degrees of momentum exchange between dark energy and dark matter. The interaction is modelled phenomenologically using a time dependent drag term in the Euler equation. We make comparisons at the level of the matter power spectrum and the redshift space monopole and quadrupole. The multipoles are modelled using the Taruya, Nishimichi and Saito (TNS) redshift space spectrum. We find perturbation theory does very well in capturing non-linear effects coming from dark sector interaction. We isolate and quantify the 1-loop contribution coming from the interaction and from the non-standard equation of state. We find the interaction parameter ξ amplifies scale dependent signatures in the range of scales considered. Non-standard equations of state also give scale dependent signatures within this same regime. In redshift space the match with N-body is improved at smaller scales by the addition of the TNS free parameter σv. To quantify the importance of modelling the interaction, we create mock data sets for varying values of ξ using perturbation theory. This data is given errors typical of Stage IV surveys. We then perform a likelihood analysis using the first two multipoles on these sets and a ξ=0 modelling, ignoring the interaction. We find the fiducial growth parameter f is generally recovered even for very large values of ξ both at z=0.5 and z=1. The ξ=0 modelling is most biased in its estimation of f for the phantom w=‑1.1 case.

Discretization vs. Rounding Error in Euler's Method

ERIC Educational Resources Information Center

Borges, Carlos F.

2011-01-01

Euler's method for solving initial value problems is an excellent vehicle for observing the relationship between discretization error and rounding error in numerical computation. Reductions in stepsize, in order to decrease discretization error, necessarily increase the number of steps and so introduce additional rounding error. The problem is…

Kalman filters for assimilating near-surface observations into the Richards equation - Part 1: Retrieving state profiles with linear and nonlinear numerical schemes

NASA Astrophysics Data System (ADS)

Chirico, G. B.; Medina, H.; Romano, N.

2014-07-01

This paper examines the potential of different algorithms, based on the Kalman filtering approach, for assimilating near-surface observations into a one-dimensional Richards equation governing soil water flow in soil. Our specific objectives are: (i) to compare the efficiency of different Kalman filter algorithms in retrieving matric pressure head profiles when they are implemented with different numerical schemes of the Richards equation; (ii) to evaluate the performance of these algorithms when nonlinearities arise from the nonlinearity of the observation equation, i.e. when surface soil water content observations are assimilated to retrieve matric pressure head values. The study is based on a synthetic simulation of an evaporation process from a homogeneous soil column. Our first objective is achieved by implementing a Standard Kalman Filter (SKF) algorithm with both an explicit finite difference scheme (EX) and a Crank-Nicolson (CN) linear finite difference scheme of the Richards equation. The Unscented (UKF) and Ensemble Kalman Filters (EnKF) are applied to handle the nonlinearity of a backward Euler finite difference scheme. To accomplish the second objective, an analogous framework is applied, with the exception of replacing SKF with the Extended Kalman Filter (EKF) in combination with a CN numerical scheme, so as to handle the nonlinearity of the observation equation. While the EX scheme is computationally too inefficient to be implemented in an operational assimilation scheme, the retrieval algorithm implemented with a CN scheme is found to be computationally more feasible and accurate than those implemented with the backward Euler scheme, at least for the examined one-dimensional problem. The UKF appears to be as feasible as the EnKF when one has to handle nonlinear numerical schemes or additional nonlinearities arising from the observation equation, at least for systems of small dimensionality as the one examined in this study.

An assessment of viscous effects in computational simulation of benign and burst vortex flows on generic fighter wind-tunnel models using TEAM code

NASA Technical Reports Server (NTRS)

Kinard, Tim A.; Harris, Brenda W.; Raj, Pradeep

1995-01-01

Vortex flows on a twin-tail and a single-tail modular transonic vortex interaction (MTVI) model, representative of a generic fighter configuration, are computationally simulated in this study using the Three-dimensional Euler/Navier-Stokes Aerodynamic Method (TEAM). The primary objective is to provide an assessment of viscous effects on benign (10 deg angle of attack) and burst (35 deg angle of attack) vortex flow solutions. This study was conducted in support of a NASA project aimed at assessing the viability of using Euler technology to predict aerodynamic characteristics of aircraft configurations at moderate-to-high angles of attack in a preliminary design environment. The TEAM code solves the Euler and Reynolds-average Navier-Stokes equations on patched multiblock structured grids. Its algorithm is based on a cell-centered finite-volume formulation with multistage time-stepping scheme. Viscous effects are assessed by comparing the computed inviscid and viscous solutions with each other and experimental data. Also, results of Euler solution sensitivity to grid density and numerical dissipation are presented for the twin-tail model. The results show that proper accounting of viscous effects is necessary for detailed design and optimization but Euler solutions can provide meaningful guidelines for preliminary design of flight vehicles which exhibit vortex flows in parts of their flight envelope.

Orthogonality of embedded wave functions for different states in frozen-density embedding theory

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

Zech, Alexander; Wesolowski, Tomasz A.; Aquilante, Francesco

2015-10-28

Other than lowest-energy stationary embedded wave functions obtained in Frozen-Density Embedding Theory (FDET) [T. A. Wesolowski, Phys. Rev. A 77, 012504 (2008)] can be associated with electronic excited states but they can be mutually non-orthogonal. Although this does not violate any physical principles — embedded wave functions are only auxiliary objects used to obtain stationary densities — working with orthogonal functions has many practical advantages. In the present work, we show numerically that excitation energies obtained using conventional FDET calculations (allowing for non-orthogonality) can be obtained using embedded wave functions which are strictly orthogonal. The used method preserves the mathematicalmore » structure of FDET and self-consistency between energy, embedded wave function, and the embedding potential (they are connected through the Euler-Lagrange equations). The orthogonality is built-in through the linearization in the embedded density of the relevant components of the total energy functional. Moreover, we show formally that the differences between the expectation values of the embedded Hamiltonian are equal to the excitation energies, which is the exact result within linearized FDET. Linearized FDET is shown to be a robust approximation for a large class of reference densities.« less

Baseline Experiments on Coulomb Damping due to Rotational Slip

DTIC Science & Technology

1992-12-01

by Griffe121 . As expected Equation (2-39) matches the result given by Griffel . 2.2.2. Euler-Bernoulli Beam versus Timeshenko Beam. Omitted from Euler...McGraw-Hill, Inc., 1983. 20. Clark, S. K., Dynamics of Continuous Elements, New Jersey, Prentice-Hall, Inc., 1972. 21. Griffel , W., Beam Formulas

Euler Teaches a Class in Structural Steel Design

ERIC Educational Resources Information Center

Boyajian, David M.

2009-01-01

Even before steel was a topic of formal study for structural engineers, the brilliant eighteenth century Swiss mathematician and physicist, Leonhard Euler (1707-1783), investigated the theory governing the elastic behaviour of columns, the results of which are incorporated into the American Institute of Steel Construction's (AISC's) Bible: the…

Transonic Unsteady Aerodynamics and Aeroelasticity 1987, part 1

NASA Technical Reports Server (NTRS)

Bland, Samuel R. (Compiler)

1989-01-01

Computational fluid dynamics methods have been widely accepted for transonic aeroelastic analysis. Previously, calculations with the TSD methods were used for 2-D airfoils, but now the TSD methods are applied to the aeroelastic analysis of the complete aircraft. The Symposium papers are grouped into five subject areas, two of which are covered in this part: (1) Transonic Small Disturbance (TSD) theory for complete aircraft configurations; and (2) Full potential and Euler equation methods.

Dynamic response of a viscoelastic Timoshenko beam

NASA Technical Reports Server (NTRS)

Kalyanasundaram, S.; Allen, D. H.; Schapery, R. A.

1987-01-01

The analysis presented in this study deals with the vibratory response of viscoelastic Timoshenko (1955) beams under the assumption of small material loss tangents. The appropriate method of analysis employed here may be applied to more complex structures. This study compares the damping ratios obtained from the Timoshenko and Euler-Bernoulli theories for a given viscoelastic material system. From this study the effect of shear deformation and rotary inertia on damping ratios can be identified.

Analysis of the wobbling effect in a lens-shaped body rotation

NASA Astrophysics Data System (ADS)

Kim, Minho

2017-03-01

We discuss the wobbling motion in a lens-shaped body rotation, focusing on the frequencies and the amplitude of nutation by filming the rotational motion and wobbling of the body. The friction coefficient of the surface is altered to examine its influence for two lenses with different curvature radii. MATLAB programs are developed to retrieve the Euler angles, which are graphed according to time. It is shown that the lens with a smaller curvature radius exhibits the wobbling effect in all cases, whereas the lens with a larger curvature radius shows such behaviour in limited circumstances. The study confirms that the friction coefficient has a negative linear correlation with the vertical axis declination amplitude with the R-squared value 0.878, showing that friction gives damping and causes smaller axis declination amplitudes. Negative linear correlation also exists with relation to the number of wobbles before the motion stops, where the R-squared value is 0.938, providing further evidence that friction and wobbling cause higher energy dissipation rates. The frequency of the wobbling motion only has a correlation with the curvature radius of the lens, showing no explicit correlation with the friction coefficient, with its R-squared value being 0.077. No losses of contact were observable in this motion. The overall process does not utilize particularly expensive apparatus and will be applicable for senior undergraduate students to experiment on and analyze the motion of a special situation regarding a rigid body that is both spinning and nutating.

Free Vibration Analysis of a Spinning Flexible DISK-SPINDLE System Supported by Ball Bearing and Flexible Shaft Using the Finite Element Method and Substructure Synthesis

NASA Astrophysics Data System (ADS)

JANG, G. H.; LEE, S. H.; JUNG, M. S.

2002-03-01

Free vibration of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft is analyzed by using Hamilton's principle, FEM and substructure synthesis. The spinning disk is described by using the Kirchhoff plate theory and von Karman non-linear strain. The rotating spindle and stationary shaft are modelled by Rayleigh beam and Euler beam respectively. Using Hamilton's principle and including the rigid body translation and tilting motion, partial differential equations of motion of the spinning flexible disk and spindle are derived consistently to satisfy the geometric compatibility in the internal boundary between substructures. FEM is used to discretize the derived governing equations, and substructure synthesis is introduced to assemble each component of the disk-spindle-bearing-shaft system. The developed method is applied to the spindle system of a computer hard disk drive with three disks, and modal testing is performed to verify the simulation results. The simulation result agrees very well with the experimental one. This research investigates critical design parameters in an HDD spindle system, i.e., the non-linearity of a spinning disk and the flexibility and boundary condition of a stationary shaft, to predict the free vibration characteristics accurately. The proposed method may be effectively applied to predict the vibration characteristics of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft in the various forms of computer storage device, i.e., FDD, CD, HDD and DVD.

Comparison of two- and three-dimensional flow computations with laser anemometer measurements in a transonic compressor rotor

NASA Technical Reports Server (NTRS)

Chima, R. V.; Strazisar, A. J.

1982-01-01

Two and three dimensional inviscid solutions for the flow in a transonic axial compressor rotor at design speed are compared with probe and laser anemometers measurements at near-stall and maximum-flow operating points. Experimental details of the laser anemometer system and computational details of the two dimensional axisymmetric code and three dimensional Euler code are described. Comparisons are made between relative Mach number and flow angle contours, shock location, and shock strength. A procedure for using an efficient axisymmetric code to generate downstream pressure input for computationally expensive Euler codes is discussed. A film supplement shows the calculations of the two operating points with the time-marching Euler code.

Navier-Stokes analysis of an oxidizer turbine blade with tip clearance with and without a mini-shroud

NASA Technical Reports Server (NTRS)

Chan, Tony; Dejong, Frederik J.

1993-01-01

The Gas Generator Oxidizer Turbine (GGOT) Blade is being analyzed by various investigators under the NASA MSFC-sponsored Turbine Stage Technology Team design effort. The present work concentrates on the tip clearance region flow and associated losses; however, flow details for the passage region are also obtained in the simulations. The present calculations simulate the rotor blade row in a rotating reference frame with the appropriate coriolis and centrifugal acceleration term included in the momentum equations. The upstream computational boundary is located about one axial chord from the blade leading edge. The boundary conditions at this location have been determined by Pratt & Whitney using an Euler analysis without the vanes to obtain approximately the same flow profiles at the rotor as were obtained with the Euler stage analysis including the vanes. Inflow boundary layer profiles are then constructed assuming the skin friction coefficient at both the hub and the casing. The downstream computational boundary is located about one axial chord from the blade trailing edge, and the circumferentially averaged static pressure at this location was also obtained from the P&W Euler analysis. Results obtained for the 3-D baseline GGOT geometry at the full scale design Reynolds number show a region of high loss in the region near the casing. Particle traces in the near tip region show vortical flow behavior of the fluid which passes through the clearance region and exits at the downstream edge of the gap. In an effort to reduce clearance flow losses, the mini-shroud concept was proposed by the Pratt & Whitney design team. Calculations were performed on the GGO geometry with the mini-shroud. Results of these calculations indicate that the mini-shroud does not significantly affect the flow in the passage region, and although the tip clearance flow is different, the mini-shroud does not seem to prevent the above-mentioned vortical flow behavior. Since both flow distortion and total pressure losses are similar for both geometries, the addition of the mini-shroud does not seem to reduce the tip clearance flow effects.

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Using High Resolution Design Spaces for Aerodynamic Shape Optimization Under Uncertainty

NASA Technical Reports Server (NTRS)

Li, Wu; Padula, Sharon

2004-01-01

This paper explains why high resolution design spaces encourage traditional airfoil optimization algorithms to generate noisy shape modifications, which lead to inaccurate linear predictions of aerodynamic coefficients and potential failure of descent methods. By using auxiliary drag constraints for a simultaneous drag reduction at all design points and the least shape distortion to achieve the targeted drag reduction, an improved algorithm generates relatively smooth optimal airfoils with no severe off-design performance degradation over a range of flight conditions, in high resolution design spaces parameterized by cubic B-spline functions. Simulation results using FUN2D in Euler flows are included to show the capability of the robust aerodynamic shape optimization method over a range of flight conditions.

Airburst-Generated Tsunamis

NASA Astrophysics Data System (ADS)

Berger, Marsha; Goodman, Jonathan

2018-04-01

This paper examines the questions of whether smaller asteroids that burst in the air over water can generate tsunamis that could pose a threat to distant locations. Such airburst-generated tsunamis are qualitatively different than the more frequently studied earthquake-generated tsunamis, and differ as well from tsunamis generated by asteroids that strike the ocean. Numerical simulations are presented using the shallow water equations in several settings, demonstrating very little tsunami threat from this scenario. A model problem with an explicit solution that demonstrates and explains the same phenomena found in the computations is analyzed. We discuss the question of whether compressibility and dispersion are important effects that should be included, and show results from a more sophisticated model problem using the linearized Euler equations that begins to addresses this.

Determination of in vivo mechanical properties of long bones from their impedance response curves

NASA Technical Reports Server (NTRS)

Borders, S. G.

1981-01-01

A mathematical model consisting of a uniform, linear, visco-elastic, Euler-Bernoulli beam to represent the ulna or tibia of the vibrating forearm or leg system is developed. The skin and tissue compressed between the probe and bone is represented by a spring in series with the beam. The remaining skin and tissue surrounding the bone is represented by a visco-elastic foundation with mass. An extensive parametric study is carried out to determine the effect of each parameter of the mathematical model on its impedance response. A system identification algorithm is developed and programmed on a digital computer to determine the parametric values of the model which best simulate the data obtained from an impedance test.

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

Kim, K.; Petersson, N. A.; Rodgers, A.

Acoustic waveform modeling is a computationally intensive task and full three-dimensional simulations are often impractical for some geophysical applications such as long-range wave propagation and high-frequency sound simulation. In this study, we develop a two-dimensional high-order accurate finite-difference code for acoustic wave modeling. We solve the linearized Euler equations by discretizing them with the sixth order accurate finite difference stencils away from the boundary and the third order summation-by-parts (SBP) closure near the boundary. Non-planar topographic boundary is resolved by formulating the governing equation in curvilinear coordinates following the interface. We verify the implementation of the algorithm by numerical examplesmore » and demonstrate the capability of the proposed method for practical acoustic wave propagation problems in the atmosphere.« less

Use of High Fidelity Methods in Multidisciplinary Optimization-A Preliminary Survey

NASA Technical Reports Server (NTRS)

Guruswamy, Guru P.; Kwak, Dochan (Technical Monitor)

2002-01-01

Multidisciplinary optimization is a key element of design process. To date multidiscipline optimization methods that use low fidelity methods are well advanced. Optimization methods based on simple linear aerodynamic equations and plate structural equations have been applied to complex aerospace configurations. However, use of high fidelity methods such as the Euler/ Navier-Stokes for fluids and 3-D (three dimensional) finite elements for structures has begun recently. As an activity of Multidiscipline Design Optimization Technical Committee (MDO TC) of AIAA (American Institute of Aeronautics and Astronautics), an effort was initiated to assess the status of the use of high fidelity methods in multidisciplinary optimization. Contributions were solicited through the members MDO TC committee. This paper provides a summary of that survey.

Instability and sound emission from a flow over a curved surface

NASA Technical Reports Server (NTRS)

Maestrello, L.; Parikh, P.; Bayliss, A.

1988-01-01

The growth and decay of a wavepacket convecting in a boundary layer over a concave-convex surface is studied numerically using direct computations of the Navier-Stokes equations. The resulting sound radiation is computed using the linearized Euler equations with the pressure from the Navier-Stokes solution as a time-dependent boundary condition. It is shown that on the concave portion the amplitude of the wavepacket increases and its bandwidth broadens while on the convex portion some of the components in the packet are stabilized. The pressure field decays exponentially away from the surface and then algebraically exhibits a decay characteristic of acoustic waves in two dimensions. The far-field acoustic pressure exhibits a peak at a frequency corresponding to the inflow instability frequency.

Inhomogeneous Radiation Boundary Conditions Simulating Incoming Acoustic Waves for Computational Aeroacoustics

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Fang, Jun; Kurbatskii, Konstantin A.

1996-01-01

A set of nonhomogeneous radiation and outflow conditions which automatically generate prescribed incoming acoustic or vorticity waves and, at the same time, are transparent to outgoing sound waves produced internally in a finite computation domain is proposed. This type of boundary condition is needed for the numerical solution of many exterior aeroacoustics problems. In computational aeroacoustics, the computation scheme must be as nondispersive ans nondissipative as possible. It must also support waves with wave speeds which are nearly the same as those of the original linearized Euler equations. To meet these requirements, a high-order/large-stencil scheme is necessary The proposed nonhomogeneous radiation and outflow boundary conditions are designed primarily for use in conjunction with such high-order/large-stencil finite difference schemes.

Generalization of the Euler-type solution to the wave equation

NASA Astrophysics Data System (ADS)

Borisov, Victor V.

2001-08-01

Generalization of the Euler-type solution to the wave equation is given. Peculiarities of the space-time structure of obtained waves are considered. For some particular cases interpretation of these waves as `subliminal' and `superluminal' is discussed. The possibility of description of electromagnetic waves by means of the scalar solutions is shown.

Discovering Euler Circuits and Paths through a Culturally Relevant Lesson

ERIC Educational Resources Information Center

Robichaux, Rebecca R.; Rodrigue, Paulette R.

2006-01-01

This article describes a middle school discrete mathematics lesson that uses the context of catching crawfish to provide students with a hands-on experience related to Euler circuits and paths. The lesson promotes mathematical communication through the use of cooperative learning as well as connections between mathematics and the real world…

Two Identities for the Bernoulli-Euler Numbers

ERIC Educational Resources Information Center

Gauthier, N.

2008-01-01

Two identities for the Bernoulli and for the Euler numbers are derived. These identities involve two special cases of central combinatorial numbers. The approach is based on a set of differential identities for the powers of the secant. Generalizations of the Mittag-Leffler series for the secant are introduced and used to obtain closed-form…

Newton's Laws, Euler's Laws and the Speed of Light

ERIC Educational Resources Information Center

Whitaker, Stephen

2009-01-01

Chemical engineering students begin their studies of mechanics in a department of physics where they are introduced to the mechanics of Newton. The approach presented by physicists differs in both perspective and substance from that encountered in chemical engineering courses where Euler's laws provide the foundation for studies of fluid and solid…

Generation of unstructured grids and Euler solutions for complex geometries

NASA Technical Reports Server (NTRS)

Loehner, Rainald; Parikh, Paresh; Salas, Manuel D.

1989-01-01

Algorithms are described for the generation and adaptation of unstructured grids in two and three dimensions, as well as Euler solvers for unstructured grids. The main purpose is to demonstrate how unstructured grids may be employed advantageously for the economic simulation of both geometrically as well as physically complex flow fields.

An installed nacelle design code using a multiblock Euler solver. Volume 1: Theory document

NASA Technical Reports Server (NTRS)

Chen, H. C.

1992-01-01

An efficient multiblock Euler design code was developed for designing a nacelle installed on geometrically complex airplane configurations. This approach employed a design driver based on a direct iterative surface curvature method developed at LaRC. A general multiblock Euler flow solver was used for computing flow around complex geometries. The flow solver used a finite-volume formulation with explicit time-stepping to solve the Euler Equations. It used a multiblock version of the multigrid method to accelerate the convergence of the calculations. The design driver successively updated the surface geometry to reduce the difference between the computed and target pressure distributions. In the flow solver, the change in surface geometry was simulated by applying surface transpiration boundary conditions to avoid repeated grid generation during design iterations. Smoothness of the designed surface was ensured by alternate application of streamwise and circumferential smoothings. The capability and efficiency of the code was demonstrated through the design of both an isolated nacelle and an installed nacelle at various flow conditions. Information on the execution of the computer program is provided in volume 2.

Tectonics and crustal structure of the Saurashtra peninsula: based on Gravity and Magnetic data

NASA Astrophysics Data System (ADS)

Mishra, A. K.; Singh, A.; Singh, U. K.

2016-12-01

The Saurashtra peninsula is located at the North Western margin of the Indian shield which occurs as a horst block between the rifts namely as Kachchh, Cambay and Narmada. It is important because of occurrence of moderate earthquake and presence of mesozoic sediments below the Deccan trap. The maps of bouguer gravity anomaly and the total intensity magnetic anomalies of Saurashtra have delineated six circular gravity highs of magnitudes 40-60 mGal and 800-1000 nT respectively. In order to understand the location, structure and depth of the source body, methods like continuous wavelet transform (CWT), Euler deconvolution and power spectrum analysis have been implemented in the potential field data. The CWT and Euler deconvolution give 16-18 km average depth of volcanic plug in Junagadh and Rajula region. From the power spectrum analysis, it is found that average Moho depth in the Saurashtra is about 36-38 km. Keeping the constraints obtained from geophysical studies like borehole, deep seismic survey, receiver function analysis and geological information, combined gravity and magnetic modeling have been performed. Detailed crustal structure of the Saurashtra region has been delineated along two profiles which pass from prominent geological features Junagadh and Rajula volcanic plugs respectively.

FORS2 observes a multi-epoch transmission spectrum of the hot Saturn-mass exoplanet WASP-49b

NASA Astrophysics Data System (ADS)

Lendl, M.; Delrez, L.; Gillon, M.; Madhusudhan, N.; Jehin, E.; Queloz, D.; Anderson, D. R.; Demory, B.-O.; Hellier, C.

2016-03-01

Context. Transmission spectroscopy has proven to be a useful tool for the study of exoplanet atmospheres, because the absorption and scattering signatures of the atmosphere manifest themselves as variations in the planetary transit depth. Several planets have been studied with this technique, leading to the detection of a small number of elements and molecules (Na, K, H2O), but also revealing that many planets show flat transmission spectra consistent with the presence of opaque high-altitude clouds. Aims: We apply this technique to the MP = 0.40MJ, Rp = 1.20RJ, P = 2.78 d planet WASP-49b, aiming to characterize its transmission spectrum between 0.73 and 1 ¯m and search for the features of K and H2O. Owing to its density and temperature, the planet is predicted to possess an extended atmosphere and is thus a good target for transmission spectroscopy. Methods: Three transits of WASP-49b have been observed with the FORS2 instrument installed at the VLT/UT1 telescope at the ESO Paranal site. We used FORS2 in MXU mode with grism GRIS_600z, producing simultaneous multiwavelength transit light curves throughout the I' and z' bands. We combined these data with independent broadband photometry from the Euler and TRAPPIST telescopes to obtain a good measurement of the transit shape. Strong correlated noise structures are present in the FORS2 light curves, which are due to rotating flat-field structures that are introduced by inhomogeneities of the linear atmospheric dispersion corrector's transparency. We accounted for these structures by constructing common noise models from the residuals of light curves bearing the same noise structures and used them together with simple parametric models to infer the transmission spectrum. Results: We present three independent transmission spectra of WASP-49b between 0.73 and 1.02 ¯m, as well as a transmission spectrum between 0.65 and 1.02 ¯m from the combined analysis of FORS2 and broadband data. The results obtained from the three individual epochs agree well. The transmission spectrum of WASP-49b is best fit by atmospheric models containing a cloud deck at pressure levels of 1 mbar or lower. Based on photometric observations made with FORS2 on the ESO VLT/UT1 (Prog. ID 090.C-0758), EulerCam on the Euler-Swiss telescope and the Belgian TRAPPIST telescope.The photometric time series data in this work are only available in electronic form at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/587/A67

Large displacement behavior of double parallelogram flexure mechanisms with underconstraint eliminators

DOE PAGES

Panas, Robert M.

2016-06-23

This paper presents a new analytical method for predicting the large displacement behavior of flexural double parallelogram (DP) bearings with underconstraint eliminator (UE) linkages. This closed-form perturbative Euler analysis method is able to – for the first time – directly incorporate the elastomechanics of a discrete UE linkage, which is a hybrid flexure element that is linked to ground as well as both stages on the bearing. The models are used to understand a nested linkage UE design, however the method is extensible to other UE linkages. Design rules and figures-of-merit are extracted from the analysis models, which provide powerfulmore » tools for accelerating the design process. The models, rules and figures-of-merit enable the rapid design of a UE for a desired large displacement behavior, as well as providing a means for determining the limits of UE and DP structures. This will aid in the adoption of UE linkages into DP bearings for precision mechanisms. Models are generated for a nested linkage UE design, and the performance of this DP with UE structure is compared to a DP-only bearing. As a result, the perturbative Euler analysis is shown to match existing theories for DP-only bearings with distributed compliance within ≈2%, and Finite Element Analysis for the DP with UE bearings within an average 10%.« less

Large displacement behavior of double parallelogram flexure mechanisms with underconstraint eliminators

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

Panas, Robert M.

This paper presents a new analytical method for predicting the large displacement behavior of flexural double parallelogram (DP) bearings with underconstraint eliminator (UE) linkages. This closed-form perturbative Euler analysis method is able to – for the first time – directly incorporate the elastomechanics of a discrete UE linkage, which is a hybrid flexure element that is linked to ground as well as both stages on the bearing. The models are used to understand a nested linkage UE design, however the method is extensible to other UE linkages. Design rules and figures-of-merit are extracted from the analysis models, which provide powerfulmore » tools for accelerating the design process. The models, rules and figures-of-merit enable the rapid design of a UE for a desired large displacement behavior, as well as providing a means for determining the limits of UE and DP structures. This will aid in the adoption of UE linkages into DP bearings for precision mechanisms. Models are generated for a nested linkage UE design, and the performance of this DP with UE structure is compared to a DP-only bearing. As a result, the perturbative Euler analysis is shown to match existing theories for DP-only bearings with distributed compliance within ≈2%, and Finite Element Analysis for the DP with UE bearings within an average 10%.« less

The Apollo Capsule Optimization for Improved Stability and Computational/Experimental Data Comparisons

NASA Technical Reports Server (NTRS)

Cliff, Susan E.; Thomas, Scott D.

2005-01-01

Numerical optimization was employed on the Apollo Command Module to modify its external shape. The Apollo Command Module (CM) that was used on all NASA human space flights during the Apollo Space Program is stable and trimmed in an apex forward (alpha of approximately 40 to 80 degrees) position. This poses a safety risk if the CM separates from the launch tower during abort. Optimization was employed on the Apollo CM to remedy the undesirable stability characteristics of the configuration. Geometric shape changes were limited to axisymmetric modifications that altered the radius of the apex (R(sub A)), base radius (R(sub O)), corner radius (R(sub C)), and the cone half angle (theta), while the maximum diameter of the CM was held constant. The results of multipoint optimization on the CM indicated that the cross-range performance can be improved while maintaining robust apex-aft stability with a single trim point. Navier-Stokes computations were performed on the baseline and optimized configurations and confirmed the Euler-based optimization results. Euler Analysis of ten alternative CM vehicles with different values of the above four parameters are compared with the published experimental results of numerous wind tunnel tests during the late 1960's. These comparisons cover a wide Mach number range and a full 180-degree pitch range and show that the Euler methods are capable of fairly accurate force and moment computations and can separate the vehicle characteristics of these ten alternative configurations.

The dynamics and control of large-flexible space structures, part 10

NASA Technical Reports Server (NTRS)

Bainum, Peter M.; Reddy, A. S. S. R.

1988-01-01

A mathematical model is developed to predict the dynamics of the proposed orbiting Spacecraft Control Laboratory Experiment (SCOLE) during the station keeping phase. The equations of motion are derived using a Newton-Euler formulation. The model includes the effects of gravity, flexibility, and orbital dynamics. The control is assumed to be provided to the system through the Shuttle's three torquers, and through six actuators located by pairs at two points on the mast and at the mass center of the reflector. The modal shape functions are derived using the fourth order beam equation. The generic mode equations are derived to account for the effects of the control forces on the modal shape and frequencies. The equations are linearized about a nominal equilibrium position. The linear regulator theory is used to derive control laws for both the linear model of the rigidized SCOLE as well as that of the actual SCOLE including the first four flexible modes. The control strategy previously derived for the linear model of the rigidized SCOLE is applied to the nonlinear model of the same configuration of the system and preliminary single axis slewing maneuvers conducted. The results obtained confirm the applicability of the intuitive and appealing two-stage control strategy which would slew the SCOLE system, as if rigid to its desired position and then concentrate on damping out the residual flexible motions.

The Prediction of Broadband Shock-Associated Noise Including Propagation Effects

NASA Technical Reports Server (NTRS)

Miller, Steven; Morris, Philip J.

2011-01-01

An acoustic analogy is developed based on the Euler equations for broadband shock- associated noise (BBSAN) that directly incorporates the vector Green's function of the linearized Euler equations and a steady Reynolds-Averaged Navier-Stokes solution (SRANS) as the mean flow. The vector Green's function allows the BBSAN propagation through the jet shear layer to be determined. The large-scale coherent turbulence is modeled by two-point second order velocity cross-correlations. Turbulent length and time scales are related to the turbulent kinetic energy and dissipation. An adjoint vector Green's function solver is implemented to determine the vector Green's function based on a locally parallel mean flow at streamwise locations of the SRANS solution. However, the developed acoustic analogy could easily be based on any adjoint vector Green's function solver, such as one that makes no assumptions about the mean flow. The newly developed acoustic analogy can be simplified to one that uses the Green's function associated with the Helmholtz equation, which is consistent with the formulation of Morris and Miller (AIAAJ 2010). A large number of predictions are generated using three different nozzles over a wide range of fully expanded Mach numbers and jet stagnation temperatures. These predictions are compared with experimental data from multiple jet noise labs. In addition, two models for the so-called 'fine-scale' mixing noise are included in the comparisons. Improved BBSAN predictions are obtained relative to other models that do not include the propagation effects, especially in the upstream direction of the jet.

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

NASA Astrophysics Data System (ADS)

Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri

2018-05-01

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank-Nicolson scheme and backward Euler scheme are utilized.

DCOMP Award Lecture (Metropolis): A 3D Spectral Anelastic Hydrodynamic Code for Shearing, Stratified Flows

NASA Astrophysics Data System (ADS)

Barranco, Joseph

2006-03-01

We have developed a three-dimensional (3D) spectral hydrodynamic code to study vortex dynamics in rotating, shearing, stratified systems (eg, the atmosphere of gas giant planets, protoplanetary disks around newly forming protostars). The time-independent background state is stably stratified in the vertical direction and has a unidirectional linear shear flow aligned with one horizontal axis. Superposed on this background state is an unsteady, subsonic flow that is evolved with the Euler equations subject to the anelastic approximation to filter acoustic phenomena. A Fourier-Fourier basis in a set of quasi-Lagrangian coordinates that advect with the background shear is used for spectral expansions in the two horizontal directions. For the vertical direction, two different sets of basis functions have been implemented: (1) Chebyshev polynomials on a truncated, finite domain, and (2) rational Chebyshev functions on an infinite domain. Use of this latter set is equivalent to transforming the infinite domain to a finite one with a cotangent mapping, and using cosine and sine expansions in the mapped coordinate. The nonlinear advection terms are time integrated explicitly, whereas the Coriolis force, buoyancy terms, and pressure/enthalpy gradient are integrated semi- implicitly. We show that internal gravity waves can be damped by adding new terms to the Euler equations. The code exhibits excellent parallel performance with the Message Passing Interface (MPI). As a demonstration of the code, we simulate vortex dynamics in protoplanetary disks and the Kelvin-Helmholtz instability in the dusty midplanes of protoplanetary disks.

A 3D spectral anelastic hydrodynamic code for shearing, stratified flows

NASA Astrophysics Data System (ADS)

Barranco, Joseph A.; Marcus, Philip S.

2006-11-01

We have developed a three-dimensional (3D) spectral hydrodynamic code to study vortex dynamics in rotating, shearing, stratified systems (e.g., the atmosphere of gas giant planets, protoplanetary disks around newly forming protostars). The time-independent background state is stably stratified in the vertical direction and has a unidirectional linear shear flow aligned with one horizontal axis. Superposed on this background state is an unsteady, subsonic flow that is evolved with the Euler equations subject to the anelastic approximation to filter acoustic phenomena. A Fourier Fourier basis in a set of quasi-Lagrangian coordinates that advect with the background shear is used for spectral expansions in the two horizontal directions. For the vertical direction, two different sets of basis functions have been implemented: (1) Chebyshev polynomials on a truncated, finite domain, and (2) rational Chebyshev functions on an infinite domain. Use of this latter set is equivalent to transforming the infinite domain to a finite one with a cotangent mapping, and using cosine and sine expansions in the mapped coordinate. The nonlinear advection terms are time-integrated explicitly, the pressure/enthalpy terms are integrated semi-implicitly, and the Coriolis force and buoyancy terms are treated semi-analytically. We show that internal gravity waves can be damped by adding new terms to the Euler equations. The code exhibits excellent parallel performance with the message passing interface (MPI). As a demonstration of the code, we simulate the merger of two 3D vortices in the midplane of a protoplanetary disk.

Fractional hereditariness of lipid membranes: Instabilities and linearized evolution.

PubMed

Deseri, L; Pollaci, P; Zingales, M; Dayal, K

2016-05-01

In this work lipid ordering phase changes arising in planar membrane bilayers is investigated both accounting for elasticity alone and for effective viscoelastic response of such assemblies. The mechanical response of such membranes is studied by minimizing the Gibbs free energy which penalizes perturbations of the changes of areal stretch and their gradients only (Deseri and Zurlo, 2013). As material instabilities arise whenever areal stretches characterizing homogeneous configurations lie inside the spinoidal zone of the free energy density, bifurcations from such configurations are shown to occur as oscillatory perturbations of the in-plane displacement. Experimental observations (Espinosa et al., 2011) show a power-law in-plane viscous behavior of lipid structures allowing for an effective viscoelastic behavior of lipid membranes, which falls in the framework of Fractional Hereditariness. A suitable generalization of the variational principle invoked for the elasticity is applied in this case, and the corresponding Euler-Lagrange equation is found together with a set of boundary and initial conditions. Separation of variables allows for showing how Fractional Hereditariness owes bifurcated modes with a larger number of spatial oscillations than the corresponding elastic analog. Indeed, the available range of areal stresses for material instabilities is found to increase with respect to the purely elastic case. Nevertheless, the time evolution of the perturbations solving the Euler-Lagrange equation above exhibits time-decay and the large number of spatial oscillation slowly relaxes, thereby keeping the features of a long-tail type time-response. Copyright © 2015 Elsevier Ltd. All rights reserved.

A Brief Historical Introduction to Euler's Formula for Polyhedra, Topology, Graph Theory and Networks

ERIC Educational Resources Information Center

Debnath, Lokenath

2010-01-01

This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Konigsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real…

Computer-Assisted Instruction in Engineering Dynamics. CAI-Systems Memo Number 18.

ERIC Educational Resources Information Center

Sheldon, John W.

A 90-minute computer-assisted instruction (CAI) unit course supplemented by a 1-hour lecture on the dynamic nature of three-dimensional rotations and Euler angles was given to 29 undergraduate engineering students. The area of Euler angles was selected because it is essential to problem-working in three-dimensional rotations of a rigid body, yet…

Implementation of a parallel unstructured Euler solver on shared and distributed memory architectures

NASA Technical Reports Server (NTRS)

Mavriplis, D. J.; Das, Raja; Saltz, Joel; Vermeland, R. E.

1992-01-01

An efficient three dimensional unstructured Euler solver is parallelized on a Cray Y-MP C90 shared memory computer and on an Intel Touchstone Delta distributed memory computer. This paper relates the experiences gained and describes the software tools and hardware used in this study. Performance comparisons between two differing architectures are made.

A new stream function formulation for the Euler equations

NASA Technical Reports Server (NTRS)

Atkins, H. L.; Hassan, H. A.

1983-01-01

A new stream function formulation is developed for the solution of Euler's equations in the transonic flow region. The stream function and the density are the dependent variables in this method, while the governing equations for adiabatic flow are the momentum equations which are solved in the strong conservation law form. The application of this method does not require a knowledge of the vorticity. The algorithm is combined with the automatic grid solver (GRAPE) of Steger and Sorenson (1979) in order to study arbitrary geometries. Results of the application of this method are presented for the NACA 0012 airfoil at various Mach numbers and angles of attack, and cylinders. In addition, detailed comparisons are made with other solutions of the Euler equations.

An installed nacelle design code using a multiblock Euler solver. Volume 2: User guide

NASA Technical Reports Server (NTRS)

Chen, H. C.

1992-01-01

This is a user manual for the general multiblock Euler design (GMBEDS) code. The code is for the design of a nacelle installed on a geometrically complex configuration such as a complete airplane with wing/body/nacelle/pylon. It consists of two major building blocks: a design module developed by LaRC using directive iterative surface curvature (DISC); and a general multiblock Euler (GMBE) flow solver. The flow field surrounding a complex configuration is divided into a number of topologically simple blocks to facilitate surface-fitted grid generation and improve flow solution efficiency. This user guide provides input data formats along with examples of input files and a Unix script for program execution in the UNICOS environment.

Use of a residual distribution Euler solver to study the occurrence of transonic flow in Wells turbine rotor blades

NASA Astrophysics Data System (ADS)

Henriques, J. C. C.; Gato, L. M. C.

The aim of the present study is to investigate the occurrence of transonic flow in several cascade geometries and blade sections that have been considered in the design of Wells turbine rotor blades. The calculations were performed using an implicit Euler solver for two-dimensional flow. The numerical method uses a multi-dimensional upwind matrix residual distribution scheme formulated on a new symmetrized form of the Euler equations, both in time and in space, that decouples the entropy and the enthalpy equations. Second-order accurate steady-state solutions where obtained using a compact three-point stencil. The results show that unwanted transonic flow may occur in the turbine rotor at relatively low mean-flow Mach numbers.