Turkel, Eli — Page 2 | Engineering Papers

NASA NTRS1990

Mappings and accuracy for Chebyshev pseudo-spectral approximations

The effect of mappings on the approximation, by Chebyshev collocation, of functions which exhibit localized regions of rapid variation is studied. A general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials which can be accurately approximated by Chebyshev polynomial expansions. A particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in approximations. It is shown that the mapped function can be approximated much more accurately by Chebyshev polynomial approximations than in physical space or where mappings constructed from other strategies are employed.

Bayliss, Alvin↗

NASA NTRS1990

Multigrid for hypersonic inviscid flows

The use of multigrid methods to solve the Euler equations for hypersonic flow is discussed. The steady state equations are considered with a Runge-Kutta smoother based on the time accurate equations together with local time stepping and residual smoothing. The effect of the Runge-Kutta coefficients on the convergence rate was examined considering both damping characteristics and convection properties. The importance of boundary conditions on the convergence rate for hypersonic flow is discussed. Also of importance are the switch between the second and fourth difference viscosity. Solutions are given for flow around the bump in a channel and flow around a biconic section.

Decker, Naomi H.↗

NASA NTRS1990

On central-difference and upwind schemes

A class of numerical dissipation models for central-difference schemes constructed with second- and fourth-difference terms is considered. The notion of matrix dissipation associated with upwind schemes is used to establish improved shock capturing capability for these models. In addition, conditions are given that guarantee that such dissipation models produce a Total Variation Diminishing (TVD) scheme. Appropriate switches for this type of model to ensure satisfaction of the TVD property are presented. Significant improvements in the accuracy of a central-difference scheme are demonstrated by computing both inviscid and viscous transonic airfoil flows.

Swanson, R. C.↗

NASA NTRS1990

Effect of artificial viscosity on three-dimensional flow solutions

Artificial viscosity is added either implicitly or explicitly in practically every numerical scheme for suppressing spurious oscillations in the solution of fluid dynamics equations. In the present central-difference scheme, artificial viscosity is added explicitly for suppressing high frequency oscillations and achieving good convergence properties. The amount of artificial viscosity added is controlled through the use of pre-selected coefficients. In the standard scheme, scalar coefficients based on the spectral radii of the Jacobian of convective fluxes are used. However, this can add too much viscosity to the slower waves. Hence, use the matrix-valued coefficients, which give appropriate viscosity for each wave component is suggested. With the matrix-valued coefficients, the central-difference scheme produces more accurate solutions on a given grid, particularly in the vicinity of shocks, while still maintaining good convergence properties.

Turkel, Eli↗

NASA NTRS1990

Asynchronous numerical solution of PDEs on parallel computers

Asynchronous and partially-synchronous finite difference schemes are investigated for the one-dimensional steady-state and nonsteady-state heat equation, respectively. It is shown experimentally that the asynchronous scheme runs twice as fast as the parallel synchronous scheme. For relatively small values of time levels, however, the asynchronous scheme produces the least accurate results, while the partially-asynchronous scheme is almost as accurate as its synchronous counterpart.

Amitai, Dganit↗

NASA NTRS1989

The 3-D Euler and Navier-Stokes calculations for aircraft components

An explicit multistage Runge-Kutta type of time-stepping scheme is used for solving transonic flow past a transport type wing/fuselage configuration. Solutions for both Euler and Navier-Stokes equations are obtained for quantitative assessment of boundary layer interaction effects. The viscous solutions are obtained on both a medium resolution grid of approximately 270,000 points and a find grid of 460,000 points to assess the effects of grid density on the solution. Computed pressure distributions are compared with the experimental data.

Vatsa, Veer N.↗

NASA NTRS1988

Improving the accuracy of central difference schemes

General difference approximations to the fluid dynamic equations require an artificial viscosity in order to converge to a steady state. This artificial viscosity serves two purposes. One is to suppress high frequency noise which is not damped by the central differences. The second purpose is to introduce an entropy-like condition so that shocks can be captured. These viscosities need a coefficient to measure the amount of viscosity to be added. In the standard scheme, a scalar coefficient is used based on the spectral radius of the Jacobian of the convective flux. However, this can add too much viscosity to the slower waves. Hence, it is suggested that a matrix viscosity be used. This gives an appropriate viscosity for each wave component. With this matrix valued coefficient, the central difference scheme becomes closer to upwind biased methods.

Turkel, Eli↗

NASA NTRS1988

Accuracy versus convergence rates for a three dimensional multistage Euler code

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.

Turkel, Eli↗

NASA NTRS1988

Effects of numerical dissipation on finite-volume solutions of compressible flow problems

The effects of numerical dissipation upon solutions to the Euler equations are considered, and results for transonic flows past airfoils are presented to demonstrate the effects of the dissipative terms. The equations are approximated using a finite-volume spatial approximation with added dissipation provided by an adaptive mixture of second and fourth differences. The resulting difference equations are solved using either an explicit multistage Runge-Kutta method or a diagonalized implicit method. It is found that errors in surface values can be introduced by the averaging required to calculate derived quantities of interest.

Caughey, David A.↗

NASA NTRS1988

Accuracy versus convergence rates for a three dimensional multistage Euler code

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.

Turkel, Eli↗

NASA NTRS1987

Preconditioned methods for solving the incompressible and low speed compressible equations

The introduction of artificial time derivatives which permit faster convergence to the steady state is used to precondition steady state incompressible equation-solving acceleration methods. An optimal value is obtained for beta, given constant alpha, where alpha and beta are arbitrary functions in the general preconditioning stage. A symmetric hyperbolic system is formed by the resulting incompressible equations. Compressible equations in conservation form with slow flow are also considered.

Turkel, Eli↗

NASA NTRS1987

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

Swanson, R. C.↗

NASA NTRS1987

Comparison of three explicit multigrid methods for the Euler and Navier-Stokes equations

Three explicit multigrid methods, Ni's method, Jameson's finite-volume method, and a finite-difference method based on Brandt's work, are described and compared for two model problems. All three methods use an explicit multistage Runge-Kutta scheme on the fine grid, and this scheme is also described. Convergence histories for inviscid flow over a bump in a channel for the fine-grid scheme alone show that convergence rate is proportional to Courant number and that implicit residual smoothing can significantly accelerate the scheme. Ni's method was slightly slower than the implicitly-smoothed scheme alone. Brandt's and Jameson's methods are shown to be equivalent in form but differ in their node versus cell-centered implementations. They are about 8.5 times faster than Ni's method in terms of CPU time. Results for an oblique shock/boundary layer interaction problem verify the accuracy of the finite-difference code. All methods slowed considerably on the stretched viscous grid but Brandt's method was still 2.1 times faster than Ni's method.

Chima, Rodrick V.↗

NASA NTRS1987

Comparison of three explicit multigrid methods for the Euler and Navier-Stokes equations

Three explicit multigrid methods, Ni's method, Jameson's finite-volume method, and a finite-difference method based on Brandt's work, are described and compared for two model problems. All three methods use an explicit multistage Runge-Kutta scheme on the fine grid, and this scheme is also described. Convergence histories for inviscid flow over a bump in a channel for the fine-grid scheme alone show that convergence rate is proportional to Courant number and that implicit residual smoothing can significantly accelerate the scheme. Ni's method was slightly slower than the implicitly-smoothed scheme alone. Brandt's and Jameson's methods are shown to be equivalent in form but differ in their node versus cell-centered implementations. They are about 8.5 times faster than Ni's method in terms of CPU time. Results for an oblique shock/boundary layer interaction problem verify the accuracy of the finite-difference code. All methods slowed considerably on the stretched viscous grid but Brandt's method was still 2.1 times faster than Ni's method.

Chima, Rodrick V.↗

NASA NTRS1987

Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations

An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined.Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flows is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.

Swanson, R. C.↗

NASA NTRS1986

Accuracy of schemes with nonuniform meshes for compressible fluid flows

The accuracy of the space discretization for time-dependent problems when a nonuniform mesh is used is considered. Many schemes reduce to first-order accuracy while a popular finite volume scheme is even inconsistent for general grids. This accuracy is based on physical variables. However, when accuracy is measured in computational variables then second-order accuracy can be obtained. This is meaningful only if the mesh accurately reflects the properties of the solution. In addition, the stability properties of some improved accurate schemes are analyzed and it can be shown that they also allow for larger time steps when Runge-Kutta type methods are used to advance in time.

Turkel, Eli↗

NASA NTRS1986

A multistage time-stepping scheme for the thin-layer Navier-Stokes equations

A finite-volume scheme for numerical integration of the Euler equations was extended to allow solution of the thin-layer Navier-Stokes equations in two and three dimensions. The extended algorithm, which is based on a class of four-stage Runge-Kutta time-stepping schemes, was made numerically efficient through the following convergence acceleration technique: (1) local time stepping, (2) enthalpy damping, and (3) residual smoothing. Also, the high degree of vectorization possible with the algorithm has yielded an efficient program for vector processors. The scheme was evaluated by solving laminar and turbulent flows. Numerical results have compared well with either theoretical or other numerical solutions and/or experimental data.

Swanson, R. C., Jr.↗

NASA NTRS1981

Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes

A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.

Jameson, A.↗