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Supersonic flow field computation for wing-body combinations by shock-capturing finite difference techniques, discussing improvement based on Runge-Kutta method
An attempt is made 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 wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis 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 the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems.
The development of robust, accurate, and efficient implicit shock-capturing schemes for multidimensional compressible Navier-Stokes equations in the hypersonic and real gas flow regimes is presently undertaken by extending a class of implicit total variation-diminishing (TVD) schemes suitable for transonic and supersonic, multidimensional Euler and Navier-Stokes equations to hypersonic computations. Numerical aspects of TVD schemes are identified which affect the convergence rate for hypersonic Mach numbers and real gas flows, but which have a negligible effect on low Mach number or perfect gas flows.
The numerical simulation of transitional and turbulent processes in hypersonic boundary layers often involves a physical process of a shock-disturbance wave interaction in complex two-dimensional and three-dimensional flow fields. For such simulations, it is required that there be a high order of accuracy in capturing both the shock and the small disturbances. The purpose of this paper is to evaluate the viability of using high order shock capturing schemes to track small disturbances in a multi-dimensional steady hypersonic flow. The numerical methods that are to be studied are the Total Variation Diminishing (TVD) scheme, and Essentially Non-Oscillatory (ENO) scheme. This paper shows that the presence of numerical oscillations in the flow field solution may drastically hinder any attempt at tracking the propagation of any physical disturbances. It has been found that the numerical oscillations that exist for shock capturing methods may be significant enough to pollute a flow field containing small physical disturbances. The effects of the refinement of the grid do not reduce the oscillations, but rather they decrease the wavelength of the oscillations. It is shown that by aligning the shock with the grid, the amplitude of these spurious oscillations may be greatly reduced.
The construction of a reliable, shock capturing finite difference method to solve the Euler equations for inviscid, supersonic flow past fighter and missile type configurations is highly desirable. The numerical method must have a firm theoretical foundation and must be robust and efficient. It should be able to treat subsonic pockets in a predominantly supersonic flow. The method must also be easily applicable to the complex topologies of the aerodynamic configuration under consideration. The ongoing approach to this task is described and for steady supersonic flows is presented. This scheme is the basic numerical method. Results of work obtained during previous years are presented.
Multi-dimensional generalizations of a one-dimensional finite element shock capturing scheme are proposed. A scalar model problem is used to emphasize that 'preferred directions' are important in multi-dimensional applications. Schemes are developed for the two-dimensional Euler equations. One, based upon characteristics, employs the Mach lines and streamlines as preferred directions.
A systematic evaluation is conducted of all extant numerical schemes for nonlinear scalar transport problems, and several advanced shock-capturing schemes are used to solve the nonlinear Burgers' equation in order to characterize their ability to resolve the sharp discontinuity, expansion zone, and propagation and collision features of shocks. For discontinuous functions, the Warming-Beam scheme generates preshock wiggles, while the Lax-Wendroff scheme generates postshock ones. Such limiters as the MUSCL or the superbee are more compressive than minimod or monotonic limiters. The performance of such TVD schemes as the upwind, the symmetric, and the Roe-Sweby, resemble each other.
A critical appraisal is presented of a method-of-characteristics (MOC) and a shock-capturing technique (SCT) for calculating three-dimensional supersonic flow. Detailed comparisons are made for pointed and blunt-nosed bodies with a thick delta wing at angles of attack from 5 to 15 deg. The results show excellent agreement between the methods and with experiment. It is established that the SCT code yields accurate shock locations and jump conditions. The SCT code typically requires about three times as many mesh points as the MOC code in order to resolve the bow shock, but the total computation time is about the same. On a point for point basis, the SCT code is about four times faster than the MOC code.
Recent developments which have improved the understanding of how finite difference methods resolve discontinuous solutions to hyperbolic partial differential equations are discussed. As a result of this understanding improved shock capturing methods are currently being developed and tested. Some of these methods are described and numerical results are presented showing their performance on problems containing shocks in one and two dimensions. A conservative difference scheme is defined. Conservation implies that, except in very special circumstances, shocks must be spread over at least two grid intervals. These two interval shocks are actually attained in one dimension if the shock is steady and an upwind scheme is used. By analyzing this case, the reason for this excellent shock resolution can be determined. This result is used to provide a mechanism for improving the resolution of two dimensional steady shocks. Unfortunately, this same analysis shows that these results cannot be extended to shocks which move relative to the computing grid. Total variation diminishing (TVD) finite difference schemes and flux limiters are introduced to deal with money shocks and contact discontinuities.
The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.
The paper presents computational assessment of advanced numerical schemes for nonlinear acoustic problems related to combustion instabilities in liquid rocket engines. Several time-accurate, shock capturing schemes have been evaluated on a benchmark, closed-end resonant pipe flow problem. It involves the numerical solution of inviscid, compressible gas dynamics equations to predict acoustic wave propagation, wave steepening, formation of shocks, acoustic energy dissipation and wave-wall reflection for several hundred wave cycles. It was demonstrated that high accuracy TVD type schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time. The selected scheme was then applied to analyze the acoustic responses of resonant pipe-resonator, radial acoustic modes and hub-baffle configurations. Interesting observations of wave shape and damping characteristics have been drawn from presented computational studies.
The numerical performance of a second-order upwind-based total variation diminishing (TVD) scheme and that of a uniform second-order essentially non-oscillatory (ENO) scheme for shock capturing are compared. The TVD scheme used is a modified version of Liou, using the flux-difference splitting (FDS) of Roe and his superbee function as the limiter. The construction of the basic ENO scheme is based on Harten, Engquist, Osher, and Chakravarthy, and the 2-D extensions are obtained by using a Strang-type of fractional-step time-splitting method. Numerical results presented include both steady and unsteady, 1-D and 2-D calculations. All the chosen test problems have exact solutions so that numerical performance can be measured by comparing the computer results to them. For 1-D calculations, the standard shock-tube problems of Sod and Lax are chosen. A very strong shock-tube problem, with the initial density ratio of 400 to 1 and pressure ratio of 500 to 1, is also used to study the behavior of the two schemes. For 2-D calculations, the shock wave reflection problems are adopted for testing. The cases presented in this report include flows with Mach numbers of 2.9, 5.0, and 10.0.
Hypersonic computations are presently conducted with an extension of a class of high-resolution implicit TVD algorithms suited to transonic multidimensional Euler and Navier-Stokes equations. These conservative shock-capturing schemes, which are spatially second- and third-order, may be first- and second-order accurate in time and suitable for either steady or unsteady calculations. Attention is given to the enhancement of hypersonic flows' convergence rate and stability; accuracy and efficiency is achieved by these means for very complex two-dimensional hypersonic viscous and inviscid shock interactions.
A class of implicit Total Variation Diminishing (TVD) type algorithms suitable for transonic and supersonic multidimensional Euler and Navier-Stokes equations was extended to hypersonic computations. The improved conservative shock-capturing schemes are spatially second- and third-order, and are fully implicit. They can be first- or second-order accurate in time and are suitable for either steady or unsteady calculations. Enhancement of stability and convergence rate for hypersonic flows is discussed. With the proper choice of the temporal discretization and suitable implicit linearization, these schemes are fairly efficient and accurate for very complex two-dimensional hypersonic inviscid and viscous shock interactions. This study is complimented by a variety of steady and unsteady viscous and inviscid hypersonic blunt-body flow computations. Due to the inherent stiffness of viscous flow problems, numerical experiments indicated that the convergence rate is in general slower for viscous flows than for inviscid steady flows.
Some numerical aspects of finite-difference algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogenous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed provided that the Jacobian of the soruce terms possesses certain properties. The proposed semi-implicit method can be viewed as a variant of the Bussing and Murman point-implicit scheme with a more appropriate numerical dissipation for the computation of strong shock waves. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a better choice. The situation is complicated by problems that are higher than one dimension, and the presence of stiff source terms further complicates the solution procedures for alternating direction implicit (ADI) methods. Several alternatives are discussed. The primary motivation for constructing these schemes was to address thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be simplified, thus providing a more efficient solution procedure than one might have anticipated.
Some numerical aspects of finite-difference algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogeneous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed provided that the Jacobian of the source terms possesses certain properties. The proposed semi-implicit method can be viewed as a variant of the Bussing and Murman point-implicit scheme with a more appropriate numerical dissipation for the computation of strong shock waves. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a better choice. The situation is complicated by problems that are higher than one dimension, and the presence of stiff source terms further complicates the solution procedures for alternating direction implicit (ADI) methods. Several alternatives are discussed. The primary motivation for constructing these schemes was to address thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be simplified, thus providing a more efficient solution procedure than one might have anticipated.
Topics covered include: Low-dispersion scheme for nonlinear acoustic waves in nonuniform flow; Computation of acoustic scattering by a low-dispersion scheme; Algorithmic extension of low-dispersion scheme and modeling effects for acoustic wave simulation; The accuracy of shock capturing in two spatial dimensions; Using high-order methods on lower-order geometries; and Computational considerations for the simulation of discontinuous flows.
Lunar and Mars return conditions are examined using the LAURA flowfield code and the LORAN radiation code to assess the effect of radiative coupling on axisymmetric thermochemical nonequilibrium flows. Coupling of the two codes is achieved iteratively. Special treatment required to couple radiation in a shock-capturing method is discussed. Results indicate that while coupling effects are generally the same as occur in equilibrium flows, under certain conditions radiation can modify the chemical kinetics of a nonequilibrium flow and thus alter relaxation processes. Coupling effects are found to be small for all cases considered, except for a five meter diameter aerobrake returning from Mars at 13.6 km/sec.