Multigrid schemes for hypersonic viscous flows

R. C. Swanson

NASA Langley Research Center, MS 156, Hampton, VA 23681, USA

R. Radespiel

DLR, Braunschweig, Germany


Abstract

In the past several years multigrid has been used to accelerate the convergence of Navier-Stokes computations for a variety of flow problems at both subsonic and transonic speeds. More recently, multigrid methods with either central or upwind differencing have been applied to viscous hypersonic flows, achieving convergence rates approaching those obtained at lower Mach numbers and moderate Reynolds numbers (Re < 10^7). However, at the higher Re values that are experienced by high-speed flight vehicles, there is a dramatic slowdown in the convergence rate. The reason for this is the deterioration in the high-frequency damping of the multigrid driving scheme, due to the very high-aspect-ratio cells occurring in the computational mesh in order to resolve the thin boundary layers. In this paper we consider some new multigrid schemes for the computation of viscous hypersonic flows. We use Fourier analysis of two-level multigrid applied to the two-dimensional (2-D) advection equation in order to study the damping behavior of these schemes. The basic solution algorithm employs upwind spatial discretization with explicit multistage time stepping. A variable coefficient implicit residual smoothing procedure is used to extend the local stability range. Full coarsening and various semicoarsening multigrid strategies are examined. The capabilities of the multigrid methods are also assessed by solving two different hypersonic flow problems. Semicoarsening is shown to be quite effective in relaxing the stiffness that arises from the resolution of thin boundary layers. Moreover, good convergence rates are obtained for Reynolds numbers up to 200x10^6.