High-Resolution Shock- Capturing Schemes for lnviscid and Viscous Hypersonic Flows

i A class of implicit Total Variation Diminishing (TVD) type algorithms suitable for transonic and supersonic multidimensional Euler and Navier-Stokes equations has been extended to hypersonic computations. The improved conservative shock-capturing schemes are spatially secondand thirdorder, and are fully implicit. They can be firstor 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 conlplimented 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. I. Motivation and Objective Most shock-capturing methods are either inefficient for practical computations or only valid for transonic or supersonic perfect gas calculations. For hypersonic, perfect gas, equilibrium real gases or nonequilibrium flow% improvement and modification to existing methods are necessary. In addition, viscous hypersonic and nonequilibrium flow problems are generally stiff and implicit methods are generally preferred over explicit methods. Some of the numerical issues for steady inviscid hypersonic blunt-body flow computations were addressed in our earlier paper [1]. A semi-implicit method and a fully implicit method for steady-state nonequilibrium flows were discussed in Yee and Shirm [2]. A basic study on numerical methods for unsteady inviscid nonequilibrium flows was presented in LeVeque and Yee [3]. The objective of this research is to efficiently extend and improve as well as to present an unified formulation of the existing implicit high-resolution shock-capturing schemes [4-6,2] for multidimensional compressible Euler and Navier-Stokes equations in the hypersonic, perfect and equilibrium real gas flow regimes. tAn abbreviated version will appear in the proceedings of the BAIL V Conference, June 20-24, 1988, Shanghai, China. 2Research Scientist, Computational Fluid Dynamics Branch 3Research Scientist 4Research Scientist, Theoretical Aerodynamics Division, this work was performed while on leave as an Ames Associate at NASA Ames Research Center, Moffett Field, CA 94035 USA. The improved schemes are based on a classof implicitTotal VariationDiminishing (TVD) type algorithmsoriginally designedfortransonicand supersonicmultidimensionalELderand Navier-Stokes equations[4-6]. The extended conservativeshock-capturingschemes are spatiallysecond-and thirdorder,and are and fullyimplicit.They can be firstor second-orderaccurateintime and are suitable for eithersteady or unsteady calculations.In addition,the currentunifiedformulationallowsthe inclusion of the MUSCL-type approach [7] in conjunction with a local characteristic approach [24,6] or flux-vector splittings [8] (see section II for an explanation). For the present study, particular emphasis is placed on second-order implicit time-accurate high-resolution algorithms. The algorithms are formLdated in finite volume and pseudo finite volume forms which, for certain physical problems and grid distributions, can enhance stability and convergence rate for highly clustered or skewed grids and require only a slight modification from the form originally presented in Yee and Harten [5] for generalized geometries. It is emphasized here that the use of the term TVD-type schemes pertains to the property of the algorithm as applied to one-dimensional nonlinear scalar hyperbolic conservation laws or constant coefficient hyperbolic systems in a semidiscrete sense. Theoretical justification of the proposed fully discretized schemes on the preservation of TVD property for the general nonlinear scalar hyperbolic conservation laws is under investigation. Moreover, the high-resolution property of these schemes for multidimensional nonlinear systems of hyperbolic conservation laws is evaluated by numerical experiments. In particular the following numerical issues are addressed: 1. Some numerical aspects of TVD-type schemes that affect the convergence rate for hypersonic Mach numbers and real gas flows but have negligible effect on low Mach number or perfect gas flows are identified. 2. The performance of the various linearized implicit forms of the proposed schemes similar to the transonic flow study [4-6] is reexamined for hypersonic flows. 3. The behavior of the proposed schemes with various temporal, differencing but. similar spatial discretization for inviscid and viscous flows is investigated. Studies indicated that their behavior in terms of stability and convergence rate is quite different between viscous and inviscid flows. However, with the proper choice of the temporal discretizatlon and suitable implicit linearization, these schemes are fairly efficient and accurate for very complex two-dimensional hypersonic inviscid and viscous shock interactions. 4. The relative efficiency and accuracy of typical TVD-type schemes [9-11] for shock wave computations are examined. A comparative study on steady and unsteady flows reveals that the proposed class of TVD-type schemes, in particular, for equilibrium real gas and nonequllibrium flows, produces just as accurate shock resolution and yet requires less operations count than most other TVD schemes (e.g., higher-order Godunov [10], Osher & Chakravarthy [9], and TVD flux-vector splitting approaches [8]). In the following section, the generalization of Roe's approximate Riemann solver and flux-vector splitting for real gases are reviewed. A description of the two-parameter family implicit shockcapturing scheme, the various enhancements on numerical stability and convergence rate for hypersonic flows, and the behavior of the scheme for inviscid and viscous flows are discussed in the subsequent sections. To illustrate the performance of the schemes for complex hypersonic flows, some representative numerical examples are also discussed. II. Description of the Numerical Algorithm The conservationlawsforthe two-dimensionalNavier-Stokesequationscan be writtenin the form