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NASA NTRS1994

A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

Yee, H. C.↗

NASA NTRS1977

The artificial compression method for computation of shocks and contact discontinuities. I - Single conservation laws

The paper discusses the use of the artificial compression method for the computation of discontinuous solutions of a single conservation law by finite difference methods. The single conservation law has either a shock or a contact discontinuity. Any monotone finite difference scheme applied to the original equation smears the discontinuity, while the same scheme applied to the equation modified by an artificial compression flux produces steady progressing profiles. If L is any finite difference scheme in conservation form and C is an artificial compressor, the split flux artificial compression method CL is a corrective scheme: L smears the discontinuity while propagating it; C compresses the smeared transition toward a sharp discontinuity. Numerical implementation of artificial compression is described.

Harten, A.↗

NASA NTRS1976

Two inviscid computational simulations of separated flow about airfoils

Two inviscid computational simulations of separated flow about airfoils are described. The basic computational method is the line relaxation finite-difference method. Viscous separation is approximated with inviscid free-streamline separation. The point of separation is specified, and the pressure in the separation region is calculated. In the first simulation, the empiricism of constant pressure in the separation region is employed. This empiricism is easier to implement with the present method than with singularity methods. In the second simulation, acoustic theory is used to determine the pressure in the separation region. The results of both simulations are compared with experiment.

Barnwell, R. W.↗

NASA NTRS1994

Spectral Methods for Computational Fluid Dynamics

As a tool for large-scale computations in fluid dynamics, spectral methods were prophesized in 1944, born in 1954, virtually buried in the mid-1960's, resurrected in 1969, evangalized in the 1970's, and catholicized in the 1980's. The use of spectral methods for meteorological problems was proposed by Blinova in 1944 and the first numerical computations were conducted by Silberman (1954). By the early 1960's computers had achieved sufficient power to permit calculations with hundreds of degrees of freedom. For problems of this size the traditional way of computing the nonlinear terms in spectral methods was expensive compared with finite-difference methods. Consequently, spectral methods fell out of favor. The expense of computing nonlinear terms remained a severe drawback until Orszag (1969) and Eliasen, Machenauer, and Rasmussen (1970) developed the transform methods that still form the backbone of many large-scale spectral computations. The original proselytes of spectral methods were meteorologists involved in global weather modeling and fluid dynamicists investigating isotropic turbulence. The converts who were inspired by the successes of these pioneers remained, for the most part, confined to these and closely related fields throughout the 1970's. During that decade spectral methods appeared to be well-suited only for problems governed by ordinary diSerential eqllations or by partial differential equations with periodic boundary conditions. And, of course, the solution itself needed to be smooth. Some of the obstacles to wider application of spectral methods were: (1) poor resolution of discontinuous solutions; (2) inefficient implementation of implicit methods; and (3) drastic geometric constraints. All of these barriers have undergone some erosion during the 1980's, particularly the latter two. As a result, the applicability and appeal of spectral methods for computational fluid dynamics has broadened considerably. The motivation for the use of spectral methods in numerical calculations stems from the attractive approximation properties of orthogonal polynomial expansions.

Zang, T. A.↗

NASA NTRS1993

Wing-Body Aeroelasticity Using Finite-Difference Fluid/Finite-Element Structural Equations on Parallel Computers

This paper presents a procedure for computing the aeroelasticity of wing-body configurations on multiple-instruction, multiple-data (MIMD) parallel computers. In this procedure, fluids are modeled using Euler equations discretized by a finite difference method, and structures are modeled using finite element equations. The procedure is designed in such a way that each discipline can be developed and maintained independently by using a domain decomposition approach. A parallel integration scheme is used to compute aeroelastic responses by solving the coupled fluid and structural equations concurrently while keeping modularity of each discipline. The present procedure is validated by computing the aeroelastic response of a wing and comparing with experiment. Aeroelastic computations are illustrated for a High Speed Civil Transport type wing-body configuration.

Byun, Chansup↗

NASA NTRS1997

Continued Development of the Discontinuous Galerkin Method for Computational Aeroacoustic Applications

The formulation and the implementation of boundary conditions within the context of the quadrature-free form of the discontinuous Galerkin method are presented for several types of boundary conditions for the Euler equations. An important feature of the discontinuous Galerkin method is that the interior point algorithm is well behaved in the neighborhood of the boundary and requires no modifications. This feature leads to a simple and accurate treatment for wall boundary conditions and simple inflow and outflow boundary conditions. Curved walls are accurately treated with only minor changes to the implementation described in earlier work. The 'perfectly matched layer' approach to nonreflecting boundary conditions is easily applied to the discontinuous Galerkin. The compactness of the discontinuous Galerkin method makes it better suited for buffer-zone-type methods than high-order finite-difference methods. Results are presented for wall, characteristic inflow and outflow, and nonreflecting boundary conditions.

Atkins, H. L.↗

NASA NTRS1996

Wing-Body Aeroelasticity on Parallel Computers

This article presents a procedure for computing the aeroelasticity of wing-body configurations on multiple-instruction, multiple-data parallel computers. In this procedure, fluids are modeled using Euler equations discretized by a finite difference method, and structures are modeled using finite element equations. The procedure is designed in such a way that each discipline can be developed and maintained independently by using a domain decomposition approach. A parallel integration scheme is used to compute aeroelastic responses by solving the coupled fluid and structural equations concurrently while keeping modularity of each discipline. The present procedure is validated by computing the aeroelastic response of a wing and comparing with experiment. Aeroelastic computations are illustrated for a high speed civil transport type wing-body configuration.

Guruswamy, Guru P.↗

NASA NTRS2017

A Non-Intrusive Algorithm for Sensitivity Analysis of Chaotic Flow Simulations

We demonstrate a novel algorithm for computing the sensitivity of statistics in chaotic flow simulations to parameter perturbations. The algorithm is non-intrusive but requires exposing an interface. Based on the principle of shadowing in dynamical systems, this algorithm is designed to reduce the effect of the sampling error in computing sensitivity of statistics in chaotic simulations. We compare the effectiveness of this method to that of the conventional finite difference method.

Blonigan, Patrick J.↗

NASA NTRS1978

Computational advances in fluid dynamics

An office size digital computer with capabilities exceeding a CDC 7600 can be constructed for less than 300 K and spectral methods offer a factor ten improvement over finite difference methods. Integration of these facts can produce two orders of magnitude reduction in cost of fluid dynamic computations and permit solution of 3-D problems which were not previously feasible.

Taylor, T. D.↗

NASA NTRS1981

Pseudospectral method for problems with nonperiodic boundary conditions

The capability of the pseudospectral technique and the time integration scheme for the time dependent nonperiodic problem is demonstrated. Significant improvements are shown in the accuracy for calculation of the two dimensional time dependent advection diffusion equation. The pseudospectral method is a viable alternative to the finite difference methods for the equation of the spatial deriviations.

Lee, H. N.↗

NASA NTRS1988

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for high Reynolds number laminar flows

A velocity-pressure integrated, mixed interpolation, Galerkin finite element method for the Navier-Stokes equations is presented. In the method, the velocity variables were interpolated using complete quadratic shape functions and the pressure was interpolated using linear shape functions. For the two dimensional case, the pressure is defined on a triangular element which is contained inside the complete biquadratic element for velocity variables; and for the three dimensional case, the pressure is defined on a tetrahedral element which is again contained inside the complete tri-quadratic element. Thus the pressure is discontinuous across the element boundaries. Example problems considered include: a cavity flow for Reynolds number of 400 through 10,000; a laminar backward facing step flow; and a laminar flow in a square duct of strong curvature. The computational results compared favorable with those of the finite difference methods as well as experimental data available. A finite elememt computer program for incompressible, laminar flows is presented.

Kim, Sang-Wook↗

NASA NTRS1987

Development of computational methods for unsteady aerodynamics at the NASA Langley Research Center

The current scope, recent progress, and plans for research and development of computational methods for unsteady aerodynamics at the NASA Langley Research Center are reviewed. Both integral equations and finite difference methods for inviscid and viscous flows are discussed. Although the great bulk of the effort has focused on finite difference solution of the transonic small perturbation equation, the integral equation program is given primary emphasis here because it is less well known.

Yates, E. Carson, Jr.↗

NASA NTRS1984

Improved finite-difference vibration analysis of pretwisted, tapered beams

An improved finite difference procedure based upon second order central differences is developed. Several difficulties encountered in earlier works with fictitious stations that arise in using second order central differences, are eliminated by developing certain recursive relations. The need for forward or backward differences at the beam boundaries or other similar procedures is eliminated in the present theory. By using this improved theory, the vibration characteristics of pretwisted and tapered blades are calculated. Results of the second order theory are compared with published theoretical and experimental results and are found to be in good agreement. The present method generally produces close lower bound solutions and shows fast convergence. Thus, extrapolation procedures that are customary with first order finite-difference methods are unnecessary. Furthermore, the computational time and effort needed for this improved method are almost the same as required for the conventional first order finite-difference approach.

Subrahmanyam, K. B.↗

NASA NTRS2009

Ablative Thermal Response Analysis Using the Finite Element Method

A review of the classic techniques used to solve ablative thermal response problems is presented. The advantages and disadvantages of both the finite element and finite difference methods are described. As a first step in developing a three dimensional finite element based ablative thermal response capability, a one dimensional computer tool has been developed. The finite element method is used to discretize the governing differential equations and Galerkin's method of weighted residuals is used to derive the element equations. A code to code comparison between the current 1-D tool and the 1-D Fully Implicit Ablation and Thermal Response Program (FIAT) has been performed.

Dec John A.↗

NASA NTRS1993

A total variation diminishing finite difference algorithm for sonic boom propagation models

It is difficult to accurately model the rise phases of sonic boom waveforms with traditional finite difference algorithms because of finite difference phase dispersion. This paper introduces the concept of a total variation diminishing (TVD) finite difference method as a tool for accurately modeling the rise phases of sonic booms. A standard second order finite difference algorithm and its TVD modified counterpart are both applied to the one-way propagation of a square pulse. The TVD method clearly outperforms the non-TVD method, showing great potential as a new computational tool in the analysis of sonic boom propagation.

Sparrow, Victor W.↗

NASA NTRS1976

Large deflections of shallow conical membrane

Large deflections of a shallow elastic conical membrane fixed at the outer edge and loaded by either uniform or hydrostatic pressure are investigated. The overning equations were solved by the method of matched asymptotic expansions and by a finite difference method. Agreement between the two methods was excellent for the small values of the perturbation parameter.

Chang, W. H.↗

NASA NTRS1977

Compressible laminar boundary layers with suction on swept and tapered wings

In this paper we present a numerical method for solving the compressible laminar boundary-layer equations with suction on swept and tapered wings. The method employs an efficient two-point finite-difference method to solve the governing equations, and a very convenient similarity transformation which removes the wall normal velocity as a boundary condition and places it into the governing equations as a parameter. In this way the awkward nonlinear boundary condition which couples all the variables is avoided. To test and demonstrate the method, we present a sample calculation for a typical laminar-flow-control (LFC) wing.

Kaups, K.↗

NASA NTRS1981

Numerical solution of compressible viscous flows at high Reynolds numbers

A new numerical method which was used to reduce the computation time required in fluid dynamics to solve the Navier-Stokes equations at flight Reynolds numbers is described. The method is the implicit analogue of the explicit finite different method. It uses this as its first stage, while the second stage removes the restrictive stability condition by recasting the difference equations in an implicit form. The resulting matrix equations to be solved are either upper or lower block bidiagonal equations. The new method makes it possible and practical to calculate many important three dimensional, high Reynolds number flow fields on computers.

Maccormack, R. W.↗