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NASA NTRS1986

On symmetric and upwind TVD schemes

The performance of the upwind and symmetric total variation diminishing (TVD) schemes in viscous and inviscid airfoil steady-state calculations is considered, and the extension of the implicit second-order-accurate TVD scheme for hyperbolic systems of conservative laws in curvilinear coordinates is discussed. For two-dimensional steady-state applications, schemes are implemented in a conservative noniterative alternating direction implicit form, and results illustrate that the algorithm produces a fairly good solution for an RAE2822 airfoil calculation. The study demonstrates that the symmetric TVD scheme is as accurate as the upwind TVD scheme, while requiring less computational effort than it.

Yee, H. C.↗

NASA NTRS1987

Navier-Stokes simulations of blade-vortex interaction using high-order accurate upwind schemes

Conventional, spatially second-order-accurate, finite-difference schemes are much too dissipative for calculations involving vortices that travel large distances (relative to some measure of the size of the vortex). This study presents a fifth-order-accurate upwind-biased scheme that preserves vortex structure for much longer times than existing second-order-accurate central and upwind difference schemes. Vortex calculations demonstrating this aspect of the fifth-order scheme are also presented. The method is then applied to the blade-vortex interaction problem. Results for strong interactions wherein the vortex impinges directly on the airfoil or a shock associated with the airfoil are presented. None of these calculations required any modeling of the shape, size, and trajectory of the interacting vortex.

Rai, Man Mohan↗

NASA NTRS1987

High resolution upwind schemes for the three-dimensional incompressible Navier-Stokes equations

Based on flux-difference splitting, implicit high resolution schemes are constructed for efficient computations of steady-state solutions to the three-dimensional, incompressible Navier-Stokes equations in curvilinear coordinates. These schemes use first-order accurate Euler backward-time differencing and second-order central differencing for the viscous shear fluxes. Up to third-order accurate upwind differencing is achieved through a reconstruction of the solution from its cell averages. The reconstruction is accomplished by linear interpolation, where the node stencils are selected such that in regions of smooth solution the flow is highly resolved while spurious oscillations in regions of rapid changes in gradient are still suppressed. Fairly rapid convergence to steady-state solutions is attained with a completely vectorizable hybrid time-marching method. Flows around a sharp-edged delta wing are computed with the maximum accuracy of the upwind-differencing restricted to first-, second-, and third-order, to illustrate the effect of accuracy on the global and on the local vortical flow fields. The results are validated with experimental data.

Hartwich, PETER-M.↗

NASA NTRS1988

A streamwise upwind algorithm for the Euler and Navier-Stokes equations applied to transonic flows

A new algorithm was developed for the Euler and Navier-Stokes equations that uses upwind differencing based on the streawise direction. This algorithm is time accurate and can be used in codes for calculating unsteady transonic flows over wings. Such codes can be used for the flutter analysis of wings. In this algorithm, the coordinate system is locally rotated to align with the streamwise direction. For differencing the convective terms in the streamwise direction, a new form of flux splitting is employed, in which the biasing depends on the local Mach number. In the plane perpendicular to the stream direction, the new flux splitting uses the condition of no flow in that local plane. By using a locally rotated coordinate system, the convective flux vector biasing depends on the total Mach number. Hence, the switching of the flux vector biasing occurs across shock waves and the proper domain of dependence is used in supersonic regions. For comparison, many other upwind methods switch differencing based on Mach number of shock waves in multidimensional flows. The formulas for the convective flux vector differencing do not contain any user specified parameters. So, the amount of numerical dissipation is automatically determined.

Goorjian, Peter M.↗

NASA NTRS1988

An upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations are solved in a time-accurate manner in using the method of pseudocompressibility. Using this method, subiterations in pseudotime are required to satisfy the continuity equation at each time step. An upwind differencing scheme based on flux-difference splitting is used to compute the convective terms. The upwind differencing is biased based on the sign of the local eigenvalue of the Jacobian matrix. Third-order or fifth-order spatial accuracy is maintained throughout the interior grid points. The equations are solved using an implicit line-relaxation scheme. This solution scheme is stable and is capable of running at large time steps in pseudotime, leading to fast convergence for each physical time step. A variety of computed results are presented to validate the present scheme. Results for the flow over an oscillating plate are compared with the exact analytic solution, good agreement is seen. Excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure. Flow solutions over a circular cylinder with vortex shedding are also presented. Finally, the flow past an airfoil at -90 deg angle-of-attack is also computed.

Rogers, Stuart E.↗

NASA NTRS1988

Use of high-resolution upwind scheme for vortical flow simulations

For vortical flow simulations at high Reynolds numbers, it is important to keep the artificial dissipation as small as possible since it induces unphysical decay of the vortex strength. One way to accomplish this is to decrease the grid spacing. Another way is to use computational schemes having little dissipation. Here, one of the high-resolution upwind schemes called MUSCL with Roe's average is applied to vortical flow fields. Two examples are considered. One is the leading-edge separation-vortex flow over a strake-delta wing. The other is a high-angle of attack supersonic flow over a spaceplane-like geometry. Comparison with the central difference solutions indicates that the present upwind scheme is less dissipative and thus has better resolution for the vortical flows.

Fujii, Kozo↗

NASA NTRS1989

Use of high-resolution upwind scheme for vortical flow simulations

For vortical flow simulations at high Reynolds number, it is important to keep the artificial dissipation as small as possible since it induces unphysical decay of the vortex strength. One way to accomplish this is to decrease the grid spacing. Another way is to use computational schemes having little dissipation. In the present paper, one of the high-resolution upwind schemes called 'MUSCL with Roe's average'is applied to vortical flow simulations. Two examples are considered. One is the leading-edge separation-vortex flow over a strake-delta wing. The other is a high-angle of attack supersonic flow over a spaceplane-like configuration. The comparison with the central difference solutions indicates that the present upwind scheme is less dissipative and thus has better resolution for the vortical flows.

Fujii, Kozo↗

NASA NTRS1989

Numerical simulation of flow over a hypersonic aircraft using an explicit upwind PNS solver

A hypersonic flow field over a generic airplane configuration is simulated by solving the Parabolized Navier-Stokes (PNS) equations. The finite difference solution of the PNS equations is calculated using a noniterative space marching, explicit, upwind scheme recently developed by the authors. Special gridding techniques are used which allowed the sharp changes in surface geometry of the airplane configuration to be modelled without smoothing of corners. Comparisons of the PNS results to a solution of the Navier-Stokes equations demonstrates a good agreement of the numerical results in approximately 1/6 of the cpu time. This paper demonstrates that the explicit upwind algorithm for solving the PNS equations is an efficient method for simulating hypersonic flow fields about complete airplane configurations and should be considered as an alternative to solving the Navier-Stokes equations for flow fields where the PNS equations are valid.

Korte, John J.↗

NASA NTRS1988

A streamwise upwind algorithm for the Euler and Navier-Stokes equations applied to transonic flows

A new algorithm was developed for the Euler and Navier-Stokes equations that uses upwind differencing based on the streamwise direction. This algorithm is time accurate and can be used in codes for calculating unsteady transonic flows over wings. Such codes can be used for the flutter analysis of wings. In this algorithm, the coordinate system is locally rotated to align with the streamwise direction. For differencing the convective terms in the streamwise direction, a new form of flux splitting is employed, in which the biasing depends on the local Mach number. In the plane perpendicular to the stream direction, the new flux splitting uses the condition of no flow in that local plane. By using a locally rotated coordinate system, the convective flux vector biasing depends on the total Mach number. Hence, the switching of the flux vector biasing occurs across shock waves and the proper domain of dependence is used in supersonic regions. For comparison, many other upwind methods switch differencing based on Mach number of shock wave in multidimensional flows. The formulas for the convective flux vector differencing do not contain any user specified parameters. So, the amount of numerical dissipation is automatically determined.

Goorjian, P. M.↗

NASA NTRS1989

A streamwise upwind algorithm applied to vortical flow over a delta wing

Improvements were made to a streamwise upwind algorithm so that it can be used for calculating flows with vortices. A calculation is shown of flow over a delta wing at an angle of attack. The laminar, thin layer, Navier-Stokes equations are used for the calculation. The results are compared with another upwind method, a central differencing method, and experimental data. The present method shows improvements in accuracy and convergence properties.

Goorjian, Peter M.↗

NASA NTRS1990

Towards a genuinely multi-dimensional upwind scheme

Methods of incorporating multi-dimensional ideas into algorithms for the solution of Euler equations are presented. Three schemes are developed and tested: a scheme based on a downwind distribution, a scheme based on a rotated Riemann solver and a scheme based on a generalized Riemann solver. The schemes show an improvement over first-order, grid-aligned upwind schemes, but the higher-order performance is less impressive. An outlook for the future of multi-dimensional upwind schemes is given.

Powell, Kenneth G.↗

NASA NTRS1990

A three-dimensional upwind parabolized Navier-Stokes code for chemically reacting flows

A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the three-dimensional flow of chemically reacting air around hypersonic vehicles. The code is a modification of the perfect gas, three-dimensional UPS code of Lawrence et al. (1986) which has been extended in the present study to permit the calculation of hypersonic, viscous flows in chemical nonequilibrium. The algorithm solves the PNS equations using a finite-volume, upwind TVD method based on Roe's approximate Riemann solver that has been modified to account for real gas effects. The present code solves the fluid dynamic and species continuity equations in a loosely-coupled manner. The fluid medium is assumed to be a chemically reacting mixture of thermally perfect (but calorically imperfect) gases in thermal equilibrium. Results are presented for the hypersonic laminar flow over a cone at 0- and 10-deg angles of attack and for a generic hypersonic vehicle. Calculations are performed assuming either perfect gas, equilibrium air, or finite-rate chemistry.

Buelow, Philip E.↗

NASA NTRS1989

Three-dimensional simulation of underexpanded plumes using upwind algorithms

Different numerical techniques have been examined for simulating the underexpanded plumes in supersonic flows. The results revealed that the monotone differentiable limiter is superior to the TVD (total variation diminishing) minmod limiter when the MUSCL (monotonic upstream schemes for conservation laws) approach is used for Roe's upwind scheme. Roe's upwind scheme is found to be superior to the central difference scheme not only for capturing shock waves, but also for capturing shear flows.

Obayashi, Shigeru↗

NASA NTRS1990

Computational results for flows over 2-D ramp and 3-D obstacle with an upwind Navier-Stokes solver

An implicit, finite-difference, upwind, full Navier-Stokes solver was applied to supersonic/hypersonic flows over two-dimensional ramps and three-dimensional obstacle. Some of the computed results are presented. The numerical scheme used in the study is an implicit, spacially second order accurate, upwind, LU-ADI scheme based on Roe's approximate Reimann solver with MUSCL differencing of Van Leer. An algebraic grid generation scheme based on generalized interpolation scheme was used in generating the grids for the various 2-D and 3-D problems.

Venkatapathy, Ethiraj↗

NASA NTRS1991

An upwind-biased space marching algorithm for supersonic viscous flow

The modifications are documented which were made to the Langley Aerothermodynamic Upwind Relaxation Algorithm which allow it to compute solutions in a space marching manner. The space marching flux is formulated to be either first- or second-order accurate, using Roe's upwind differencing or Symmetric Total Variation Diminishing differencing, respectively. The algorithm solves the thin layer Navier-Stokes equations, and is subject to the same flow restrictions as a parabolized Navier-Stokes solver. Each cross flow plane is locally iterated in time until converged, then marched in space to the next station. The algorithm is tested on a sphere-cone geometry and a geometry which models the windward side of the Space Shuttle Orbiter. Computational results for surface heating are compared to ground based experimental data. In addition, space marching predictions for surface pressure are compared against values from the original algorithm.

Greene, Francis A.↗

NASA NTRS1991

The prediction of viscous hypersonic flows about complex configurations using an upwind parabolized Navier-Stokes code

A three-dimensional parabolized Navier-Stokes computer code which employs an upwind algorithm is used to conduct a numerical study of an advanced maneuvering reentry vehicle configuration. Comparisons between numerical solutions and experimental data are presented for surface pressure, wall heat flux, and overall forces and moments. The effects of angle of attack, angle of yaw, and surface mass injection are investigated. Good agreement is observed between the calculated and measured data. The results of this investigation demonstrate the accuracy and efficiency of an upwind scheme in predicting the hypersonic flow field characteristics about a complex configuration.

Narain, J. P.↗

NASA NTRS1990

A streamwise upwind algorithm applied to vortical flow over a delta wing

Improvements have been made to a streamwise upwind algorithm so that it can be used for calculating flows with vortices. A calculation is shown of flow over a delta wing at an angle of attack. The laminar, thin-layer, Navier-Stokes equations are used for the calculation. The results are compared with another upwind method, a central-differencing method, and experimental data. The present method shows improvements in accuracy and convergence properties.

Goorjian, Peter M.↗

NASA NTRS1991

Efficient real gas upwinded Navier-Stokes computations of high speed flows

An efficient method to account for the chemically frozen thermodynamic and transport properties of air in three-dimensional Navier-Stokes calculations has been demonstrated. This approach uses an explicitly specified equation of state (EOS) so that the fluid pressure, temperature, and transport properties are directly related to the flow variables. The method is efficient since no subiterations are required to deduce the pressure and temperature from the flux variables and is modular by allowing different equations of state to be easily supplied to the code. The flexibility of the EOS approach is shown by its implementation into a high-order total variation diminishing upwinding scheme as well as a standard central-differencing scheme. The EOS approach is then demonstrated by computing the hypersonic flow through the corner region of two mutually perpendicular flat plates using both the upwind and central-differencing schemes.

Coirier, William J.↗