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NASA NTRS1975

Viscous flow solutions with a cubic spline approximation

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

Rubin, S. G.↗

NASA NTRS1977

Over-relaxation applied to the MacCormack finite-difference scheme

An over-relaxation procedure is applied to the MacCormack finite-difference scheme in order to reduce the computation time required to obtain a steady-state solution. The implementation of this acceleration procedure to an existing computer program using the regular MacCormack method is extremely simple and does not require additional storage. The over-relaxation procedure does not alter the steady-state solution, which is second-order accurate. The method is first applied to Burgers' equation. A stability condition and an expression for the increase in the rate of convergence are derived. The method is then applied to the calculation of the hypersonic viscous flow over a flat plate, using the complete Navier-Stokes equations, and the inviscid flow over a wedge. Reductions in computing time by factors of 3 and 1.5, respectively, are obtained by over-relaxation.

Desideri, J.-A.↗

NASA NTRS1979

Optimal nodal point distribution for improved accuracy in computational fluid dynamics

In applying finite-difference techniques to flow field problems, the accuracy attained for a fixed number of node points can be improved using unequally-spaced node points. The distribution of these node points is chosen here by minimizing a measure of local truncation error with respect to the parameters which define a transformation between the computational space of equally-spaced node points and the physical space of unequally-spaced node points. The problem then becomes a nonlinear programming problem. Numerical results are presented for two one-dimensional test problems: the Blasius boundary layer problem and the inviscid Burgers' equation.

Pierson, B. L.↗

NASA NTRS1977

Evolution of weak disturbances in inert binary mixtures

The evolution of weak disturbances in inert binary mixtures is determined for the one-dimensional piston problem. The interaction of the dissipative and nonlinear mechanisms is described by Burgers' equation. The binary mixture diffusion mechanisms enter as an additive term in an effective diffusivity. Results for the impulsive motion of a piston moving into an ambient medium and the sinusoidally oscillating piston are used to illustrate the results and elucidate the incorrect behavior pertaining to the associated linear theory.

Rasmussen, M. L.↗

NASA NTRS1981

Grid evolution in time asymptotic problems

The selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration. A new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh. The technique has been used to solve the simple viscous Burgers' equation in one and two dimensions. Substantial error reductions are demonstrated. The treatment of boundary points and the effect of using different error criteria in generating grids are discussed.

Rai, M. M.↗

NASA NTRS1984

Construction of finite difference schemes having special properties for ordinary and partial differential equations

Work on the construction of finite difference models of differential equations having zero truncation errors is summarized. Both linear and nonlinear unidirectional wave equations are discussed. Results regarding the construction of zero truncation error schemes for the full wave equation and Burger's equation are also briefly reported.

Mickens, R. E.↗

NASA NTRS1984

Development of a dynamically adaptive grid method for multidimensional problems

An approach to solution adaptive grid generation for use with finite difference techniques, previously demonstrated on model problems in one space dimension, has been extended to multidimensional problems. The method is based on the popular elliptic steady grid generators, but is 'dynamically' adaptive in the sense that a grid is maintained at all times satisfying the steady grid law driven by a solution-dependent source term. Testing has been carried out on Burgers' equation in one and two space dimensions. Results appear encouraging both for inviscid wave propagation cases and viscous boundary layer cases, suggesting that application to practical flow problems is now possible. In the course of the work, obstacles relating to grid correction, smoothing of the solution, and elliptic equation solvers have been largely overcome. Concern remains, however, about grid skewness, boundary layer resolution and the need for implicit integration methods. Also, the method in 3-D is expected to be very demanding of computer resources.

Holcomb, J. E.↗

NASA NTRS1981

On the relation between upwind-differencing schemes of Godunov, Engquist-Osher and Roe

The upwind differencing first order schemes of Godunov, Engquist-Osher and Roe are discussed on the basis of the inviscid Burgers equations. The differences between the schemes are interpreted as differences between the approximate Riemann solutions on which their numerical flux functions are based. Special attention is given to the proper formulation of these schemes when a source term is present. Second order two step schemes, based on the numerical flux functions of the first order schemes are also described. The schemes are compared in a numerical experiment, and recommendations on their use are included.

Vanleer, B.↗

NASA NTRS1985

Higher order methods for convection-diffusion problems

This paper applies C1 cubic Hermite polynomials embedded in an orthogonal collocation scheme to the spatial discretization of the unsteady nonlinear Burgers equation as a model of the equations of fluid mechanics. The temporal discretization is carried out by means of either a noniterative finite difference or an iterative finite difference procedure. Results of this method are compared with those of a second-order finite difference scheme and a splined-cubic Taylor's series scheme. Stability limits are derived and the matrix structure of the several schemes are compared.

Murphy, J. D.↗

NASA NTRS1985

Recent developments in FEM-CFD

The current status of CFD with regard to unstructured grids employing finite element methods and Eulerian frames is reviewed. Algorithms suitable for the computation of large three-dimensional problems involving flow past arbitrary geometries are developed. Adaptive mesh refinement strategy is reviewed, and domain splitting or local time-stepping are briefly addressed. The development of search algorithms of optimal order, variable time-stepping Jacobi smoothers for elliptic problems, and transport concepts for hyperbolics to help achieve good performance for unstructured multigrid processes is discussed. As examples, transient supersonic flow in a channel, regular shock reflection of a wall, viscous flow past a protruberance, potential flow past a cylinder, and Burgers equation are considered.

Loehner, R.↗

NASA NTRS1988

Analysis of the spectral vanishing viscosity method for periodic conservation laws

The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers' equation is analyzed. It is proven that this kind of vanishing viscosity is responsible for a spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small independent of whether the underlying solution is smooth or not. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.

Maday, Yvon↗

NASA NTRS1988

Karhunen-Loeve expansion of Burgers' model of turbulence

The properties of the Karhunen-Loeve expansion of a strongly inhomogeneous random process are examined with emphasis on applications to turbulent flow fields. The ability of the KL expansion to represent functions that have both slow and rapid variations in a relatively small number of expansion terms is tested on a one-dimensional model based on the forced Burgers' equation. The rate of the convergence of the expansion is evaluated, and its dependence on the Reynolds number is determined. It is shown that the KL eigenfunctions possess wall boundary layers attached to outer structures that are independent of the Reynolds number (at high Reynolds numbers). It is also shown that the spectrum of eigenvalues is broad at large Reynolds numbers, requiring many terms to represent higher-order derivatives of the function.

Chambers, D. H.↗

NASA NTRS1990

Triangle based TVD schemes for hyperbolic conservation laws

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

Durlofsky, Louis J.↗

NASA NTRS1990

Comparison of several finite-difference methods

The present numerical study establishes that, with the MacCormack and 'two-four' methods, the solution accuracy obtainable in problems involving wave propagation, shock-wave and contact discontinuities, and viscous effects, will be strongly dependent on Courant number. The application of flux correction to the MacCormack and two-four methods is noted to significantly attenuate dispersion errors; the ensuing solutions capture the discontinuities in the shock-tube problem with improved accuracy and resolution, and are free of dispersion errors for the viscous Burgers' equation.

Biringen, S.↗

NASA NTRS1990

Calculation of shocked one-dimensional flows on abruptly changing grids by mathematical programming

Cell-centered finite differences on cells of an arbitrarily spaced grid are presently used to discretize the steady-state inviscid and nearly inviscid Burgers' equations for steady-state shocked flow in a quasi-one-dimensional nozzle. The everdetermined system of nonlinear algebraic equations thus obtained is solved by a procedure which minimizes a weighted sum of the residuals' absolute values. Numerical solutions for both viscous and inviscid problems are accurate and nonoscillatory, on grids whose abrupt mesh lengths refinements are as great as a factor of 10,000.

Lavery, John E.↗

NASA NTRS1991

Alfven shock trains

The Cohen-Kulsrud-Burgers equation (CKB) is used to consider the nonlinear evolution of resistive, quasi-parallel Alfven waves subject to a long-wavelength, plane-polarized, monochromatic instability. The instability saturates by nonlinear steepening, which proceeds until the periodic waveform develops an interior scale length comparable to the dissipation length; a fast or an intermediate shock then forms. The result is a periodic train of Alfven shocks of one or the other type. For propagation strictly parallel to the magnetic field, there will be two shocks per instability wavelength. Numerical integration of the time-dependent CKB equation shows that an initial, small-amplitude growing wave asymptotes to a stable, periodic stationary wave whose analytic solution specifies how the type of shock embedded in the shock train, and the amplitude and speed of the shock train, depend on the strength and phase of the instability. Waveforms observed upstream of the earth's bowshock and cometary shocks resemble those calculated here.

Malkov, M. A.↗

NASA NTRS1992

Structure and evolution of time-dependent intermediate shocks

A quantitative description of time-dependent intermediate shocks is formulated using the Cohen-Kulsrud-Burgers equations. In noncoplanar Riemann problems, time-dependent two-three transition intermediate shocks evolve in time as a localized self-similar structure whose strength decreases as 1/the square root of t, and whose width expands as the square root of t. Time-dependent intermediate shocks offer a way of solving the noncoplanar MHD Riemann problem.

Wu, C. C.↗

NASA NTRS1992

Triangle based adaptive stencils for the solution of hyperbolic conservation laws

Durlofsky, Louis J.↗