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NASA NTRS1984

A full-implicit-continuous-Eulerian (FICE) scheme for multidimensional transient magnetohydrodynamic (MHD) flows

A full implicit continuous Eulerian (FICE) scheme is developed for solving multidimensional transient MHD flow problems. The physical system under consideration is a general case of a transient MHD flow in which an initial steady state is subject to a finite amplitude disturbance. The governing equations are described, their finite difference formulation is presented and the FICE algorithm is given. The boundary conditions are treated by classifying them into physical and computational ones. The usefulness of the FICE algorithm is demonstrated using a physical example concerning the dynamical response of the static solar atmosphere due to a representative photospheric disturbance.

Wu, S. T.↗

NASA NTRS1985

An implicit finite-difference code for a two-equation turbulence model for three-dimensional flows

An implicit finite difference code was developed which solves the transport equations for the turbulence kinetic energy and its dissipation rate in generalized coordinates in three dimensions. The finite difference equations are solved using the Beam-Warming algorithm. The kinetic energy-dissipation code, KEM, provides the closure; i.e., the turbulent viscosity for calculation of either compressible or incompressible flows. Turbulent internal flow over a backward-facing step has been calculated using the present code in conjunction with the Incompressible Navier-Stokes Code, INS3D. The results are in good agreement with experiments and two dimensional computations of other researchers.

Kaul, U. K.↗

NASA NTRS1985

A fast efficient implicit scheme for the gasdynamic equations using a matrix reduction technique

An efficient implicit finite-difference algorithm for the gasdynamic equations utilizing matrix reduction techniques is presented. A significant reduction in arithmetic operations is achieved without loss of the stability characteristics generality found in the Beam and Warming approximate factorization algorithm. Steady-state solutions to the conservative Euler equations in generalized coordinates are obtained for transonic flows and used to show that the method offers computational advantages over the conventional Beam and Warming scheme. Existing Beam and Warming codes can be retrofit with minimal effort. The theoretical extension of the matrix reduction technique to the full Navier-Stokes equations in Cartesian coordinates is presented in detail. Linear stability, using a Fourier stability analysis, is demonstrated and discussed for the one-dimensional Euler equations.

Barth, T. J.↗

NASA NTRS1985

Implicit flux-split schemes for the Euler equations

Recent progress in the development of implicit algorithms for the Euler equations using the flux-vector splitting method is described. Comparisons of the relative efficiency of relaxation and spatially-split approximately factored methods on a vector processor for two-dimensional flows are made. For transonic flows, the higher convergence rate per iteration of the Gauss-Seidel relaxation algorithms, which are only partially vectorizable, is amply compensated for by the faster computational rate per iteration of the approximately factored algorithm. For supersonic flows, the fully-upwind line-relaxation method is more efficient since the numerical domain of dependence is more closely matched to the physical domain of dependence. A hybrid three-dimensional algorithm using relaxation in one coordinate direction and approximate factorization in the cross-flow plane is developed and applied to a forebody shape at supersonic speeds and a swept, tapered wing at transonic speeds.

Thomas, J. L.↗

NASA NTRS1985

Implicit lambda methods for three-dimensional compressible flow

This paper derives the three dimensional lambda-formulation equations for a general orthogonal curvilinear coordinate system and provides various block-explicit and block-implicit methods for solving them, numerically. Three model problems, characterized by subsonic, supersonic and transonic flow conditions, are used to assess the reliability and compare the efficiency of the proposed methods.

Napolitano, M.↗

NASA NTRS1986

Implicit multigrid algorithms for the three-dimensional flux split Euler equations

The full approximation scheme multigrid method is applied to several implicit flux-split algorithms for solving the three-dimensional Euler equations in a body fitted coordinate system. Each uses a variation of approximate factorization and is implemented in a finite volume formulation. The algorithms are all vectorizable with little or no scalar computations required. The flux vectors are split into upwind components using both the splittings of Steger-Warming and Van Leer. Results comparing pressure distributions with experimental data using both splitting types are shown. The stability and smoothing rate of each of the schemes are examined using a Fourier analysis of the complete system of equations. Results are presented for three-dimensional subsonic, transonic, and supersonic flows which demonstrate substantially improved convergence rates with the multigrid algorithm. The influence of using both a V-cycle and a W-cycle on the convergence is examined. Using the multigrid method on both subsonic and transonic wing calculations, the final lift coefficient is obtained to within 0.1 percent of its final value in a few as 15 cycles for a mesh with over 210,000 points. A spectral radius of 0.89 is achieved for both subsonic and transonic flow over the ONERA M6 wing while a spectral radius of 0.83 is obtained for supersonic flow over an analytically defined forebody. Results compared with experiment for all cases show good agreement.

Anderson, W. K.↗

NASA NTRS1986

An implicit flux-difference splitting scheme for three-dimensional, incompressible Navier-Stokes solutions to leading edge vortex flows

A new, implicit finite-difference scheme designed to solve the conservative, flux-difference split Navier-Stokes equations is used to compute incompressible vortex flows around delta wings. The completely vectorizable hybrid algorithm is constructed in delta form for steady state solutions independent of the time-step sizes. The scheme combines approximate factorization in crossflow planes with a symmetric planar Gauss-Seidel relaxation in the remaining spatial direction. The governing equations are solved in curvilinear, body-fitted coordinates for treating complex geometries. The computed flow field results are compared with other theoretical and experimental data.

Hartwich, P.-M.↗

NASA NTRS1986

Parametric effects of CFL number and artificial smoothing on numerical solutions using implicit approximate factorization algorithm

An implicit approximate factorization algorithm is employed to quantify the parametric effects of Courant number and artificial smoothing on numerical solutions of the unsteady 3-D Euler equations for a windmilling propeller (low speed) flow field. The results show that propeller global or performance chracteristics vary strongly with Courant number and artificial dissipation parameters, though the variation is such less severe at high Courant numbers. Candidate sets of Courant number and dissipation parameters could result in parameter-dependent solutions. Parameter-independent numerical solutions can be obtained if low values of the dissipation parameter-time step ratio are used in the computations. Furthermore, it is realized that too much artificial damping can degrade numerical stability. Finally, it is demonstrated that highly resolved meshes may, in some cases, delay convergence, thereby suggesting some optimum cell size for a given flow solution. It is suspected that improper boundary treatment may account for the cell size constraint.

Daso, E. O.↗

NASA NTRS1986

An LU implicit scheme for high speed inlet analysis

A numerical method is developed to analyze the inviscid flowfield of a high speed inlet by the solution of the Euler equations. The LU implicit scheme in conjunction with adaptive dissipation proves to be an efficient and robust nonoscillatory shock capturing technique for high Mach number flows as well as for transonic flows.

Yoon, S.↗

NASA NTRS1986

Implicit hybrid schemes for the flux-difference split, three-dimensional Navier-Stokes equations

Implicit hybrid algorithms employing symmetric planar Gauss-Seidel (SPGS) relaxation and either block-tridiagonally structured coefficient matrices (AF-SPGS) or block-triangular coefficient matrices (LU-SPGS) are derived to solve the flux-difference-split Navier-Stokes equations for three-dimensional incompressible flow in an upwind scheme. The physical basis of the approach is discussed, and results for problems involving vortex flow around a thin delta wing at Reynolds numbers 900,000 and 10,000 are presented graphically. It is found that AF-SPGS converges faster on vector computers which depend on long vector lengths to achieve optimum performance, whereas LU-SPGS is preferable on sequentially operating machines and vector computers using shorter vector lengths.

Hartwich, P.-M.↗

NASA NTRS1986

Implicit methods for computing chemically reacting flow

The backward Euler scheme was used to solve a large system of inviscid flow and chemical rate equations in three spatial coordinates. The flow equations were integrated simultaneously in time by a conventional ADI factorization technique, then the species equations were solved by either simultaneous or successive techniques. The methods were evaluated in their efficiency and robustness for a hypersonic flow problem involving an aerobrake configuration. It was found that both implicit methods can effectively reduce the stiffness associated with the chemical production term and that the successive solution for the species was as stable as the simultaneous solution. The latter method is more economical because the computation time varies linearly with the number of species.

Li, C. P.↗

NASA NTRS1986

Implicit time-marching solution of the Navier-Stokes equations for thrust reversing and thrust vectoring nozzle flows

An implicit finite volume method is investigated for the solution of the compressible Navier-Stokes equations for flows within thrust reversing and thrust vectoring nozzles. Thrust reversing nozzles typically have sharp corners, and the rapid expansion and large turning angles near these corners are shown to cause unacceptable time step restrictions when conventional approximate factorization methods are used. In this investigation these limitations are overcome by using second-order upwind differencing and line Gauss-Siedel relaxation. This method is implemented with a zonal mesh so that flows through complex nozzle geometries may be efficiently calculated. Results are presented for five nozzle configurations including two with time varying geometries. Three cases are compared with available experimental data and the results are generally acceptable.

Imlay, S. T.↗

NASA NTRS1986

A locally implicit method for fluid flow problems

The fluid flow inside the space shuttle main engine (SSME) traverses through a complex geometrical configuration. The flow is compressible, viscous, and turbulent with pockets of separated regions. Several computer codes are being developed to solve three dimensional Navier-Stokes equations with different turbulence models for analyzing the SSME internal flow. The locally implicit scheme is a computationally efficient scheme which converges rapidly in multi-grid modes for elliptic problems. It has the promise of providing a rapidly converging algorithm for steady-state viscous flow problems.

Reddy, K. C.↗

NASA NTRS1987

Description of an aeronautical geometry conversion package: Wave-drag to Langley Wireframe Geometry Standard (LaWGS) to Supersonic Implicit Marching Potential (SIMP)

Documented is an aeronautical geometry conversion package which translates wave-drag geometry into the Langley Wireframe Geometry Standard (LaWGS) format and then into a format which is used by the Supersonic Implicit Marching Potential (SIMP) program. The programs described were developed by Computer Sciences Corporation for the Advanced Vehicles Division/Advanced Concepts Branch at NASA Langley Research Center. Included also are the input and output from a benchmark test case.

Wiese, Michael R.↗

NASA NTRS1987

On implicit Runge-Kutta methods for parallel computations

Implicit Runge-Kutta methods which are well-suited for parallel computations are characterized. It is claimed that such methods are first of all, those for which the associated rational approximation to the exponential has distinct poles, and these are called multiply explicit (MIRK) methods. Also, because of the so-called order reduction phenomenon, there is reason to require that these poles be real. Then, it is proved that a necessary condition for a q-stage, real MIRK to be A sub 0-stable with maximal order q + 1 is that q = 1, 2, 3, or 5. Nevertheless, it is shown that for every positive integer q, there exists a q-stage, real MIRK which is I-stable with order q. Finally, some useful examples of algebraically stable MIRKs are given.

Keeling, Stephen L.↗

NASA NTRS1987

A diagonal implicit multigrid algorithm for the Euler equations

A multigrid implementation of the Alternating Direction Implicit algorithm has been developed to solve the Euler equations of inviscid, compressible flow. The equations are approximated using a finite-volume spatial approximation with added dissipation provided by an adaptive blend of second and fourth differences. For computational efficiency, the equations are diagonalized by a local similariity transformation so that only a decoupled system of scalar pentadiagonal systems need be solved along each line. Results are computed for transonic flows past airfoils and include pressure distributions to verify the accuracy of the basic scheme and convergence histories to demonstrate the efficiency of the method.

Caughey, David A.↗

NASA NTRS1987

Implicit flux-corrected transport scheme for low-speed flow

A new high-order nonlinear Eulerian convection algorithm, the barely implicit correction (BIC)-FCT method, is presented. The BIC-FCT makes it possible to perform accurate, multidimensional, slow-flow calculations in which the fluid expansion is important. Owing to its low cost even in two dimensions, fairly detailed chemistry models and other physical processes can be included. Areas of BIC-FCT applicability include premixed flames, diffusion flames, and turbulent jet flames.

Patnaik, G.↗

NASA NTRS1987

Analysis of implicit local linearization techniques for upwind and TVD algorithms

An attempt is made to investigate local time linearization techniques for implicit flux-difference splitting and flux-vector splitting schemes in the simplest settings (i.e., first-order spatial schemes and one-dimensional Euler flows). It is noted that first-order spatial schemes provide the simplest examples of schemes which are collective extensions of scalar TVD schemes. Simple analytical results concerning the local linearizations are highlighted and subsequently verified using a numerical fixed-point analysis on selected problems. It is noted that while primary emphasis is on asymptotic behavior, many of the results have implications for time-accurate calculations as well.

Barth, Timothy J.↗