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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 253 records · Page 14
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NASA NTRS1981

A modified Dodge algorithm for the parabolized Navier-Stokes equation and compressible duct flows

A revised version of Dodge's split-velocity method for numerical calculation of compressible duct flow was developed. The revision incorporates balancing of mass flow rates on each marching step in order to maintain front-to-back continuity during the calculation. The (checkerboard) zebra algorithm is applied to solution of the three dimensional continuity equation in conservative form. A second-order A-stable linear multistep method is employed in effecting a marching solution of the parabolized momentum equations. A checkerboard iteration is used to solve the resulting implicit nonlinear systems of finite-difference equations which govern stepwise transition. Qualitive agreement with analytical predictions and experimental results was obtained for some flows with well-known solutions.

Cooke, C. H.↗
NASA NTRS1983

A modified Dodge algorithm for the parabolized Navier-Stokes equations and compressible duct flows

A revised version of Dodge's split-velocity method for numerical calculation of compressible duct flow was developed. The revision incorporates balancing of mass flow rates on each marching step in order to maintain front-to-back continuity during the calculation. The (checkerboard) zebra algorithm is applied to solution of the three dimensional continuity equation in conservative form. A second-order A-stable linear multistep method is employed in effecting a marching solution of the parabolized momentum equations. A checkerboard iteration is used to solve the resulting implicit nonlinear systems of finite-difference equations which govern stepwise transition. Qualitative agreement with analytical predictions and experimental results was obtained for some flows with well-known solutions. Previously announced in STAR as N82-16363

Cooke, C. H.↗
NASA NTRS1981

Applications of estimation theory to numerical weather prediction

Numerical weather prediction (NWP) is an initial value problem for a system of nonlinear partial differential equations in which the initial values are known only incompletely and inaccurately. Data at initial time can be supplemented, however, by observations of the system distributed over a time interval preceding it. Estimation theory was successful in approaching such problems for models governed by systems of ordinary differential equations and of linear PDEs. Estimation-theoretic methods for NWP are developed. A model exhibiting many features of large scale atmospheric flow important in NWP is the one governed by the shallow fluid equations. The estimation problem for a linearized formulation of these equations is studied. A finite difference version of the equations is used as a forecast model to simulate the numerical models used in NWP.

Cohn, S.↗
NASA NTRS1977

A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations

An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of the nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The continuous cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

Cooke, C. H.↗
NASA NTRS1981

Analysis of capacitive heat exchangers, part 1

Governing equations are developed for a simple capacitive heat exchanger. This type of heat exchanger consists of hot spherical particles falling through an ascending cold gas stream. The assumptions made in deriving the continuity, momentum and energy equations are clearly stated. The analysis yields a system of first order, ordinary, nonlinear equations which form a complex boundary value problem. The method of solution is presented together with a comparison between the performance of capacitive heat exchangers and conventional counter flow ones.

Schonfeld, D.↗
NASA NTRS1981

Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations

Implicit, noniterative, finite difference schemes were recently developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture was that unconditional stability requires implicit numerical boundary conditions. An apparent counter example was the space time extrapolation considered by Gustafsson, Kreiss, and Sunstrom. Spatial (implicit) and space time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals are examined. The results indicate that for unconditional stability with a finite number of spatial mesh intervals, the numerical boundary conditions must be implicit.

Beam, R. M.↗
NASA NTRS1987

Parallel computations and their impact on mechanics; Proceedings of the Symposium, ASME Winter Annual Meeting, Boston, MA, Dec. 13-18, 1987

The conference presents papers on parallel architectures and the programming environment, parallel numerical algorithms, structural mechanics applications, and fluid dynamics applications. Topics include concurrent computer architecture, the state-of-the-art in highly parallel computer systems, supercomputer programming environments, uniquely parallel algorithms, the parallel solution of nonlinear elliptic equations, highly parallel banded systems solvers, and optimal mapping of irregular finite element domains to parallel processors. Consideration is also given to parallel processing in finite element structural analysis, heirarchical parallelism in a finite element CFD algorithm, and a parallelized elliptic solver for reacting flows.

Noor, Ahmed K.↗
NASA NTRS1987

Solitary waves in the resonant phenomenon between a surface gravity wave packet and an internal gravity wave

A two-layer inviscid incompressible fluid system of intermediate depth is considered. A multiple-scales perturbation technique is applied to the basic equations and boundary conditions for a two-layer fluid system to derive a system of weakly nonlinear partial integrodifferential equations governing the resonant interaction between a surface gravity wave packet and an internal gravity wave at an intermediate depth, providing a bridge between the existing shallow and deep fluid theories. The convolution integral term in these equations accounts for the dispersion in the lower-layer fluid. An iterative fast Fourier transform scheme is developed to find solitary wave solutions to this system of equations. The overtaking collision of two pairs of solitary waves, simulated using a spectral method, is found to be inelastic. It is found that the amplitude of the solitary waves changes slightly after the collision. The phase shifts these solitary waves undergo was calculated numerically.

Sepulveda, Nicasio↗
NASA NTRS1973

A sequence solution to the Fokker-Planck equation.

In this paper a new approach to nonlinear system analysis based on the Fokker-Planck equation is developed. The development begins as a modification of the parametrix method of partial differential equation theory and provides a sequence solution to the multidimensional time-varying Fokker-Planck equation. This sequence is simplified for the case where only a steady-state solution or a solution for the variance of a particular combination of states is desired. Equations are presented in a form suitable for obtaining numerical results using a digital computer.

Mayfield, W. W.↗
NASA NTRS1996

Multigrid Acceleration of Time-Accurate DNS of Compressible Turbulent Flow

An efficient scheme for the direct numerical simulation of 3D transitional and developed turbulent flow is presented. Explicit and implicit time integration schemes for the compressible Navier-Stokes equations are compared. The nonlinear system resulting from the implicit time discretization is solved with an iterative method and accelerated by the application of a multigrid technique. Since we use central spatial discretizations and no artificial dissipation is added to the equations, the smoothing method is less effective than in the more traditional use of multigrid in steady-state calculations. Therefore, a special prolongation method is needed in order to obtain an effective multigrid method. This simulation scheme was studied in detail for compressible flow over a flat plate. In the laminar regime and in the first stages of turbulent flow the implicit method provides a speed-up of a factor 2 relative to the explicit method on a relatively coarse grid. At increased resolution this speed-up is enhanced correspondingly.

Broeze, Jan↗
NASA NTRS1981

A modified dodge algorithm for the parabolized Navier-Stokes equations and compressible duct flows

A revised version of a split-velocity method for numerical calculation of compressible duct flow was developed. The revision incorporates balancing of mass flow rates on each marching step in order to maintain front-to-back continuity during the calculation. The (checkerboard) zebra algorithm is applied to solution of the three-dimensional continuity equation in conservative form. A second-order A-stable linear multistep method is employed in effecting a marching solution of the parabolized momentum equations. A checkerboard successive overrelaxation iteration is used to solve the resulting implicit nonlinear systems of finite-difference equations which govern stepwise transition.

Cooke, C. H.↗
NASA NTRS1980

Iterative methods based upon residual averaging

Iterative methods for solving boundary value problems for systems of nonlinear partial differential equations are discussed. The methods involve subtracting an average of residuals from one approximation in order to arrive at a subsequent approximation. Two abstract methods in Hilbert space are given and application of these methods to quasilinear systems to give numerical schemes for such problems is demonstrated. Potential theoretic matters related to the iteration schemes are discussed.

Neuberger, J. W.↗
NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - 2

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.

Yee, H. C.↗
NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - Part 2

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Yee, H. C.↗
NASA NTRS1988

Stability analysis and control of tethered satellites

Issues of stability and stabilization of the Shuttle/Tethered Satellite System (TSS) are considered. The basic, nonlinear equations of motion of the TSS are derived using the system Lagrangian. A tension control strategy in feedback form is introduced to stabilize the TSS during station-keeping. In addition, the stability properties of a constant angle control method for deployment and retrieval are investigated using Liapunov stability theory.

Liaw, D.-C.↗
NASA NTRS1981

Accuracy and convergence of a finite element algorithm for turbulent boundary layer flow

The Galerkin-Weighted Residuals formulation is employed to derive an implicit finite element solution algorithm for the nonlinear parabolic partial differential equation system governing turbulent boundary layer flow. Solution accuracy and convergence with discretization refinement are quantized in several error norms using linear and quadratic basis functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the nonlinear equations characteristic of turbulent boundary layer flow.

Soliman, M. O.↗
NASA NTRS1991

Maneuver simulations of flexible spacecraft by solving TPBVP

The optimal control of large angle rapid maneuvers and vibrations of a Shuttle mast reflector system is considered. The nonlinear equations of motion are formulated by using Lagrange's formula, with the mast modeled as a continuous beam. The nonlinear terms in the equations come from the coupling between the angular velocities, the modal coordinates, and the modal rates. Pontryagin's Maximum Principle is applied to the slewing problem, to derive the necessary conditions for the optimal controls, which are bounded by given saturation levels. The resulting two point boundary value problem (TPBVP) is then solved by using the quasilinearization algorithm and the method of particular solutions. In the numerical simulations, the structural parameters and the control limits from the Spacecraft Control Lab Experiment (SCOLE) are used. In the 2-D case, only the motion in the plane of an Earth orbit or the single axis slewing motion is discussed. In the 3-D slewing, the mast is modeled as a continuous beam subjected to 3-D deformations. The numerical results for both the linearized system and the nonlinear system are presented to compare the differences in their time response.

Bainum, Peter M.↗