Several new numerical methods for compressible shear-layer simulations

Abstract

An investigation is conducted of several numerical schemes for use in the computation of two-dimensional, spatially evolving, laminar, variable-density compressible shear layers. Schemes with various temporal accuracies and arbitrary spatial accuracy for both inviscid and viscous terms are presented and analyzed. All integration schemes make use of explicit or compact finite-difference derivative operators. Three classes of schemes are considered: an extension of MacCormack's original second-order temporally accurate method,a new third-order temporally accurate variant of the coupled space–time schemes proposed by Rusanov and by Kutler et al. (RKLW), and third- and fourth-order Runge–Kutta schemes. The RKLW scheme offers the simplicity and robustness of the MacCormack schemes and gives the stability domain and the nonlinear third-order temporal accuracy of the Runge–Kutta method. In each of the schemes, stability and formal accuracy are considered for the interior operators on the convection–diffusion equation U t + aU x = a v U xx , for which a and α v are constant. Both spatial and temporal accuracies are verified on the equation U t =[ b( x) U x ] x , as well as on U t + F x =0. Numerical boundary treat ments of various orders of accuracy are chosen and evaluated for asymptotic stability. Formally accurate boundary conditions are derived for explicit sixth-order, pentadiagonal sixth-order, and explicit, tridiagonal, and pentadiagonal eighth-order central-difference operators when used in conjunction with Runge–Kutta integrators. Damping of high wavenumber, nonphysical information is accomplished for all schemes with the use of explicit filters, derived up to sixth order on the boundaries and twelfth order in the interior. Several schemes are used to compute variable-density compressible shear layers, where regions of large gradients of flowfield variables arise near and away from the shear-layer centerline. Results indicate that in the present simulations, the effects of differences in temporal and spatial accuracy between the schemes were less important than the filtering effects. Extended MacCormack schemes were very robust, but were inefficient because of restrictive CFL limits. The third-order temporally accurate RKLW schemes were less dissipative, but had shorter run times. Runge–Kutta integrators did not possess sufficient dissipation to be useful candidates for the computation of variable-density compressible shear layers at the levels of resolution used in the current work.