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NASA NTRS1986

Upwind relaxation algorithms for Euler/Navier-Stokes equations

A description of and results from a solution algorithm for the compressible Navier-Stokes equations are presented. The main features of the algorithm are second or third order accurate upwind discretization of the convection and pressure derivatives and a relaxation scheme for the unfactored implicit backward Euler time method, implemented in a finite-volume formulation. Upwind methods were successfully used to obtain solutions to the Euler equations for flows with strong shock waves. The particular upwind method being used is based on the flux vector splitting technique developed by Van Leer and both second and third order accurate discretizations were developed. Currently, the most widely used implicit solution technique for the Navier-Stokes equations use approximate factorization (AF) methods to treat multidimensional problems. The time integration scheme being used in the present algorithm corresponds to a line Gauss-Seidel relaxation method. This method produces good convergence rates for steady-state flows, and most of the algorithm was vectorized on the NASA Langley VPS 32 computer. The Navier-Stokes algorithm was tested for several two-dimensional flow problems. Solutions for the problems gave excellent results. The presented effort is directed toward the extension of the scheme to the full three-dimensional Navier-Stokes equations.

Thomas, J. L.↗

NASA NTRS1988

Hypersonic blunt body computations including real gas effects

The recently developed second-order explicit and implicit total variation diminishing (TVD) shock-capturing methods of the Harten and Yee, Yee, and van Leer types in conjunction with a generalized Roe's approximate Riemann solver of Vinokur and the generalized flux-vector splittings of Vinokur and Montagne for two-dimensional hypersonic real gas flows are studied. A previous study on one-dimensional unsteady problems indicated that these schemes produce good shock-capturing capability and that the state equation does not have a large effect on the general behavior of these methods for a wide range of flow conditions for equilibrium air. The objective of this paper is to investigate the applicability and shock resolution of these schemes for two-dimensional steady-state hypersonic blunt body flows. The main contribution of this paper is to identify some of the elements and parameters which can affect the convergence rate for high Mach numbers or real gases but have negligible effect for low Mach number cases for steady-state inviscid blunt body flows.

Montagne, J.-L.↗

NASA NTRS1988

Improved numerical methods for turbulent viscous recirculating flows

The performance of discrete methods for the prediction of fluid flows can be enhanced by improving the convergence rate of solvers and by increasing the accuracy of the discrete representation of the equations of motion. This report evaluates the gains in solver performance that are available when various acceleration methods are applied. Various discretizations are also examined and two are recommended because of their accuracy and robustness. Insertion of the improved discretization and solver accelerator into a TEACH mode, that has been widely applied to combustor flows, illustrates the substantial gains to be achieved.

Turan, A.↗

NASA NTRS1988

Accelerating the computation of steady state solutions for non-equilibrium, chemically reacting flow fields

The objective of the study is to develop a technique for accelerating the computation of steady state solutions for nonequilibrium chemically reacting flow fields using the time marching method. To achieve this, a better initial condition is introduced for the time marching scheme to achieve a faster convergence to the steady state solution. This is done by replacing the source terms of the original system of equations, which are responsible for the sitffness, by approximate source terms. It is shown that the saving in computation time increases in direct proportion to the stiffness of the flow model, with a factor of 10 increase in convergence rate observed for the models tested in this study.

Krispin, Jacob↗

NASA NTRS1988

Efficient solutions of two-dimensional incompressible steady viscous flows

A simple, efficient, and robust numerical technique is provided for solving two dimensional incompressible steady viscous flows at moderate to high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established.

Morrison, J. H.↗

NASA NTRS1988

A mixed finite difference/Galerkin method for three-dimensional Rayleigh-Benard convection

A fast and accurate numerical method, for nonlinear conservation equation systems whose solutions are periodic in two of the three spatial dimensions, is presently implemented for the case of Rayleigh-Benard convection between two rigid parallel plates in the parameter region where steady, three-dimensional convection is known to be stable. High-order streamfunctions secure the reduction of the system of five partial differential equations to a system of only three. Numerical experiments are presented which verify both the expected convergence rates and the absolute accuracy of the method.

Buell, Jeffrey C.↗

NASA NTRS1988

Shape optimization of 3D continuum structures via force approximation techniques

The existing need to develop methods whereby the shape design efficiency can be improved through the use of high quality approximation methods is addressed. An efficient approximation method for stress constraints in 3D shape design problems is proposed based on expanding the nodal forces in Taylor series with respect to shape variations. The significance of this new method is shown through elementary beam theory calculations and via numerical computations using 3D solid finite elements. Numerical examples including the classical cantilever beam structure and realistic automotive parts like the engine connecting rod are designed for optimum shape using the proposed method. The numerical results obtained from these methods are compared with other published results, to assess the efficiency and the convergence rate of the proposed method.

Vanderplaats, Garret N.↗

NASA NTRS1988

Spectral multigrid methods for the solution of homogeneous turbulence problems

New three-dimensional spectral multigrid algorithms are analyzed and implemented to solve the variable coefficient Helmholtz equation. Periodicity is assumed in all three directions which leads to a Fourier collocation representation. Convergence rates are theoretically predicted and confirmed through numerical tests. Residual averaging results in a spectral radius of 0.2 for the variable coefficient Poisson equation. In general, non-stationary Richardson must be used for the Helmholtz equation. The algorithms developed are applied to the large-eddy simulation of incompressible isotropic turbulence.

Erlebacher, G.↗

NASA NTRS1988

An adaptive remeshing method for finite element thermal analysis

A finite element remeshing approach that makes use of quadrilateral and triangular elements is described. The approach uses the solution on a previous mesh to create a new mesh. Meshes are completely unstructured with highly refined elements in regions of steep gradients and larger elements where gradients are smaller. Studies of convergence rates for heat conduction problems with exact solutions show that for problems with highly localized solution variations, the remeshing approach gives smaller solution errors with fewer unknowns than refinement of uniform, structured meshes.

Thornton, Earl A.↗

NASA NTRS1988

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

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

Grossman, B.↗

NASA NTRS1988

A survey of the state-of-the-art and focused research in range systems

In this one-year renewal of NASA Contract No. 2-304, basic research, development, and implementation in the areas of modern estimation algorithms and digital communication systems have been performed. In the first area, basic study on the conversion of general classes of practical signal processing algorithms into systolic array algorithms is considered, producing four publications. Also studied were the finite word length effects and convergence rates of lattice algorithms, producing two publications. In the second area of study, the use of efficient importance sampling simulation technique for the evaluation of digital communication system performances were studied, producing two publications.

Kung, Yao↗

NASA NTRS1989

A cell-vertex multigrid method for the Navier-Stokes equations

A cell-vertex scheme for the Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing, a multigrid method, and carefully controlled artificial dissipative terms, very good convergence rates are obtained for a wide range of two- and three-dimensional flows over airfoils and wings. The accuracy of the code is examined by grid refinement studies and comparison with experimental data. For an accurate prediction of turbulent flows with strong separations, a modified version of the nonequilibrium turbulence model of Johnson and King is introduced, which is well suited for an implementation into three-dimensional Navier-Stokes codes. It is shown that the solutions for three-dimensional flows with strong separations can be dramatically improved, when a nonequilibrium model of turbulence is used.

Radespiel, R.↗

NASA NTRS1989

Zonal multigrid solution of compressible flow problems on unstructured and adaptive meshes

The simultaneous use of adaptive meshing techniques with a multigrid strategy for solving the 2-D Euler equations in the context of unstructured meshes is studied. To obtain optimal efficiency, methods capable of computing locally improved solutions without recourse to global recalculations are pursued. A method for locally refining an existing unstructured mesh, without regenerating a new global mesh is employed, and the domain is automatically partitioned into refined and unrefined regions. Two multigrid strategies are developed. In the first, time-stepping is performed on a global fine mesh covering the entire domain, and convergence acceleration is achieved through the use of zonal coarse grid accelerator meshes, which lie under the adaptively refined regions of the global fine mesh. Both schemes are shown to produce similar convergence rates to each other, and also with respect to a previously developed global multigrid algorithm, which performs time-stepping throughout the entire domain, on each mesh level. However, the present schemes exhibit higher computational efficiency due to the smaller number of operations on each level.

Mavriplis, Dimitri J.↗

NASA NTRS1989

Control of the errors of discretization and idealization in finite element analysis

Understanding of the basic principles which control errors of discretization in finite element analysis has increased very substantially since 1980. The main milestones were: (1) development of the theoretical basis of p-extensions (1981); (2) understanding of the proper interplay between mesh design and assignment of polynomial degree to elements. Practical realization of exponential convergence rates, independently of the smoothness of the exact solution (1984); and (3) industrial experience with the new finite element technology known as the p- or hp-version of the finite element method: General Dynamics reported thirty- to forty-fold savings in terms of human time and large savings in computer time (1986). Lockheed reported favorably on their evaluation of error estimation and quality control capabilities of the p-version in industrial settings (1987). The gains in our understanding of how to control the errors of discretization represent only half of the control necessary to ensure that a numerical model is in fact an accurate representation of the corresponding physical system. Control of the errors of idealization is equally important. A brief overview of the main ideas of how to ensure the quality and reliability of mathematical models of structural systems is presented.

Szabo, Barna A.↗

NASA NTRS1988

General methods for determining the linear stability of coronal magnetic fields

A time integration of a linearized plasma equation of motion has been performed to calculate the ideal linear stability of arbitrary three-dimensional magnetic fields. The convergence rates of the explicit and implicit power methods employed are speeded up by using sequences of cyclic shifts. Growth rates are obtained for Gold-Hoyle force-free equilibria, and the corkscrew-kink instability is found to be very weak.

Craig, I. J. D.↗

NASA NTRS1988

Backward differentiation approximations of nonlinear differential/algebraic systems

Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

Brenan, Kathryn E.↗

NASA NTRS1989

An investigation of cell centered and cell vertex multigrid schemes for the Navier-Stokes equations

Two efficient and robust finite-volume multigrid schemes for solving the Navier-Stokes equations are investigated. These schemes employ either a cell centered or a cell vertex discretization technique. An explicit Runge-Kutta algorithm is used to advance the solution in time. Acceleration techniques are applied to obtain faster steady-state convergence. Accuracy and convergence of the schemes are examined. Computational results for transonic airfoil flows are essentially the same, even for a coarse mesh. Both schemes exhibit good convergence rates for a broad range of artificial dissipation coefficients.

Radespiel, R.↗

NASA NTRS1988

High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows

The development of robust, accurate, and efficient implicit shock-capturing schemes for multidimensional compressible Navier-Stokes equations in the hypersonic and real gas flow regimes is presently undertaken by extending a class of implicit total variation-diminishing (TVD) schemes suitable for transonic and supersonic, multidimensional Euler and Navier-Stokes equations to hypersonic computations. Numerical aspects of TVD schemes are identified which affect the convergence rate for hypersonic Mach numbers and real gas flows, but which have a negligible effect on low Mach number or perfect gas flows.

Yee, H. C.↗