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NASA NTRS1991

Derivative interface conditions for multiblock grids

Methods are developed for computing numerical solutions along block boundaries, even when there is a discontinuity in the grid lines or slopes. The technique is based on matching derivatives and does not require overlapping and interpolation of solution values at block boundaries. The comparison of block boundary values is implicit and has proven to be stable for both implicit and explicit numerical algorithms. Examples are included for the numerical solution of the Euler equations for compressible flow on grids with both grid line discontinuities and discontinuous slopes at block boundaries.

Mastin, C. W.↗
NASA NTRS1992

Evaluation of an optimal guidance algorithm for aero-assisted orbit transfer

A detailed evaluation is performed of a guidance algorithm for aeroassisted orbit transfer that was developed earlier based on the method of matched asymptotic expansions. It is shown that, by exploiting the structure of the matched asymptotic expansion solution procedure, the original problem which requires the solution of a set of 20 implicit algebraic equations can be reduced to a problem of six implicit equations in six unknowns. The main contribution here is that it was possible to obtain a solution that is near optimal, requires a minimum of computation, and thus can be implemented in real time and on-board the vehicle. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the matched asymptotic expansion problem to obtain the feedback controls.

Melamed, N.↗
NASA NTRS1993

Numerical simulation of free shear flows: Towards a predictive computational aeroacoustics capability

Implicit and explicit spatial differencing techniques with fourth order accuracy have been developed. The implicit technique is based on the Pade compact scheme. A Dispersion Relation Preserving concept has been incorporated into both of the numerical schemes. Two dimensional Euler computation of a spatially-developing free shear flow, with and without external excitation, has been performed to demonstrate the capability of numerical schemes developed. Results are in good agreement with theory and experimental observation regarding the growth rate of fluctuating velocity, the convective velocity, and the vortex-pairing process.

Soh, W. Y.↗
NASA NTRS1992

Calculation of fully three-dimensional separated flow with an unsteady viscous-inviscid interaction method

Three previous papers have shown that the viscous inviscid numerical methods were capable of calculating separated turbulent flows. The 'Semi-inverse method' and the models of the first author allow the computation of massive separations, stalled flows, and shock wave-boundary layer interactions, in two-dimensional or quasi-three-dimensional conditions, from low speeds to supersonic speeds. The 'Semi-implicit' method for time-consistent coupling allows for the computation of time-accurate transonic separated flow and buffer computations in two-dimensions. The present paper shows that the viscous-inviscid interaction approach is also able to compute the fully three-dimensional flow separation phenomena. The method is based on a thin-layer approximation of the theory of 'Defect-Formulation' that provides the viscous-inviscid splitting of the Navier-Stokes equations. A parametric analytical modelling of the 3D-turbulent velocity profiles is involved. Numerically, the 3D-velocity profiles are discretized in the normal z-direction and driven by parametric integral 3D-equations in direct or inverse modes in the x-direction. The viscous-inviscid coupling is fully 3D and solved the time-consistency problem with an extension of the 'Semi-implicit' method previously suggested in two-dimensions. A 3D inviscid subroutine with TSP approximation is used. Results are obtained for transonic steady flows over wings with shock-induced transonic separation. The method provides results for highly three-dimensional flow separations, such as induced by a three-dimensional through at the wall of a flat plate. The 3D viscous-inviscid coupling and the 3D model of the velocity field provide three-dimensional instantaneous skin-friction lines whose patterns exhibit the same complex topology as Navier-Stokes solvers, with foci, nodes, and saddle-points.

Leballeur, J. C.↗
NASA NTRS1993

Application of fast Fourier transforms to the direct solution of a class of two-dimensional separable elliptic equations on the sphere

An efficient, direct, second-order solver for the discrete solution of a class of two-dimensional separable elliptic equations on the sphere (which generally arise in implicit and semi-implicit atmospheric models) is presented. The method involves a Fourier transformation in longitude and a direct solution of the resulting coupled second-order finite-difference equations in latitude. The solver is made efficient by vectorizing over longitudinal wave-number and by using a vectorized fast Fourier transform routine. It is evaluated using a prescribed solution method and compared with a multigrid solver and the standard direct solver from FISHPAK.

Moorthi, Shrinivas↗
NASA NTRS1991

Finite element computation of compressible flows with the SUPG formulation

Finite element computation of compressible Euler equations is presented in the context of the streamline-upwind/Petrov-Galerkin (SUPG) formulation. The SUPG formulation, which is based on adding stabilizing terms to the Galerkin formulation, is further supplemented with a shock capturing operator which addresses the difficulty in maintaining a satisfactory solution near discontinuities in the solution field. The shock capturing operator, which has been derived from work done in entropy variables for a similar operator, is shown to lead to an appropriate level of additional stabilization near shocks, without resulting in excessive numerical diffusion. An implicit treatment of the impermeable wall boundary condition is also presented. This treatment of the no-penetration condition offers increased stability for large Courant numbers, and accelerated convergence of the computations for both implicit and explicit applications. Several examples are presented to demonstrate the ability of this method to solve the equations governing compressible fluid flow.

Le Beau, G. J.↗
NASA NTRS1993

High order finite difference and multigrid methods for spatially evolving instability in a planar channel

The fourth-order finite-difference scheme with fully implicit time-marching presently used to computationally study the spatial instability of planar Poiseuille flow incorporates a novel treatment for outflow boundary conditions that renders the buffer area as short as one wavelength. A semicoarsening multigrid method accelerates convergence for the implicit scheme at each time step; a line-distributive relaxation is developed as a robust fast solver that is efficient for anisotropic grids. Computational cost is no greater than that of explicit schemes, and excellent agreement with linear theory is obtained.

Liu, C.↗
NASA NTRS1991

Computational results for 2-D and 3-D ramp flows 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, spatially 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 NTRS1993

Line relaxation methods for the solution of 2D and 3D compressible flows

An implicit finite element based algorithm for the compressible Navier-Stokes equations is outlined, and the solution of the resulting equation by a line relaxation on general meshes of triangles or tetrahedra is described. The problem of generating and adapting unstructured meshes for viscous flows is reexamined, and an approach for both 2D and 3D simulations is proposed. An efficient approach appears to be the use of an implicit/explicit procedure, with the implicit treatment being restricted to those regions of the mesh where viscous effects are known to be dominant. Numerical examples demonstrating the computational performance of the proposed techniques are given.

Hassan, O.↗
NASA NTRS1993

A multigrid nonoscillatory method for computing high speed flows

A multigrid method using different smoothers has been developed to solve the Euler equations discretized by a nonoscillatory scheme up to fourth order accuracy. The best smoothing property is provided by a five-stage Runge-Kutta technique with optimized coefficients, yet the most efficient smoother is a backward Euler technique in factored and diagonalized form. The singlegrid solution for a hypersonic, viscous conic flow is in excellent agreement with the solution obtained by the third order MUSCL and Roe's method. Mach 8 inviscid flow computations for a complete entry probe have shown that the accuracy is at least as good as the symmetric TVD scheme of Yee and Harten. The implicit multigrid method is four times more efficient than the explicit multigrid technique and 3.5 times faster than the single-grid implicit technique. For a Mach 8.7 inviscid flow over a blunt delta wing at 30 deg incidence, the CPU reduction factor from the three-level multigrid computation is 2.2 on a grid of 37 x 41 x 73 nodes.

Li, C. P.↗
NASA NTRS1993

Guidance law development for aeroassisted transfer vehicles using matched asymptotic expansions

This report addresses and clarifies a number of issues related to the Matched Asymptotic Expansion (MAE) analysis of skip trajectories, or any class of problems that give rise to inner layers that are not associated directly with satisfying boundary conditions. The procedure for matching inner and outer solutions, and using the composite solution to satisfy boundary conditions is developed and rigorously followed to obtain a set of algebraic equations for the problem of inclination change with minimum energy loss. A detailed evaluation of the zeroth order guidance algorithm for aeroassisted orbit transfer is performed. It is shown that by exploiting the structure of the MAE solution procedure, the original problem, which requires the solution of a set of 20 implicit algebraic equations, can be reduced to a problem of 6 implicit equations in 6 unknowns. A solution that is near optimal, requires a minimum of computation, and thus can be implemented in real time and on-board the vehicle, has been obtained. Guidance law implementation entails treating the current state as a new initial state and repetitively solving the zeroth order MAE problem to obtain the feedback controls. Finally, a general procedure is developed for constructing a MAE solution up to first order, of the Hamilton-Jacobi-Bellman equation based on the method of characteristics. The development is valid for a class of perturbation problems whose solution exhibits two-time-scale behavior. A regular expansion for problems of this type is shown to be inappropriate since it is not valid over a narrow range of the independent variable. That is, it is not uniformly valid. Of particular interest here is the manner in which matching and boundary conditions are enforced when the expansion is carried out to first order. Two cases are distinguished-one where the left boundary condition coincides with, or lies to the right of, the singular region, and another one where the left boundary condition lies to the left of the singular region. A simple example is used to illustrate the procedure where the obtained solution is uniformly valid to O(Epsilon(exp 2)). The potential application of this procedure to aeroassisted plane change is also described and partially evaluated.

Calise, Anthony J.↗
NASA NTRS1993

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 2: User's guide

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the User's Guide, and describes the program's features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.

Towne, Charles E.↗
NASA NTRS1993

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 1: Analysis description

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the Analysis Description, and presents the equations and solution procedure. The governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models are described in detail.

Towne, Charles E.↗
NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 2: User's guide

A computer code called Proteus 3D was developed to solve the three-dimensional, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This User's Guide describes the program's features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.

Towne, Charles E.↗
NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 3: Programmer's reference

A computer code called Proteus 3D was developed to solve the three-dimensional, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.

Towne, Charles E.↗
NASA NTRS1993

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 3: Programmer's reference

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.

Towne, Charles E.↗
NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

A computer code called Proteus 3D has been developed to solve the three dimensional, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort has been to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation have been emphasized. The governing equations are solved in generalized non-orthogonal body-fitted coordinates by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the Analysis Description, and presents the equations and solution procedure. It describes in detail the governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models.

Towne, Charles E.↗
NASA NTRS1993

Computational techniques for high-speed flows with viscous and chemical effects

Algorithms for solving the Euler and the Navier-Stokes equations in conjunction with chemical kinetic equations are presented. The convective flux is estimated from a quasi one dimensional interpolation procedure. Shock, contact, and expansion waves and thermochemical nonequilibrium phenomena are captured by the Lax-Friedrichs technique. Relaxation techniques were developed to enhance their effectiveness in dealing with spatial and temporal stiffness associated with the physical problems. Both explicit and implicit smoothers were implemented into the standard multigrid time stepping method. Unsteady and steady scalar problems are discussed. A perfect gas and equilibrium air shock tube problem is investigated. Numerical schemes and techniques are compared for the problems of shock and boundary layer interaction and three dimensional viscous, nonequilibrium flow encompassing an aerobrake. The results are comparable in accuracy against other high order non-oscillatory techniques. The multigrid methods are assessed using a Mach 8 flow over a complete planar body, a sphere, and a blunt delta wing at 20 deg incidence. Applying an implicit multigrid method on a nested grid of 128 by 64 nodes the reduction factor is 0.25. The central processing unit reduction factor is 2.2 after both the single and multigrid Runge-Kutta solutions converged to machine zero on a grid of 37 by 41 by 73 nodes.

Li, C. P.↗