SEVERAL NEW NUMERICAL-METHODS FOR COMPRESSIBLE SHEAR-LAYER SIMULATIONS

An investigation is conducted of several numerical schemes for use in the computation of two-dimensional, spatially evolving, laminar, varia ble-density compressible shear layers. Schemes with various temporal a ccuracies 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 cla sses 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 M acCormack schemes and gives the stability domain and the nonlinear thi rd-order temporal accuracy of the Runge-Kutta method. In each of the s chemes, 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 alpha(v) are constant. Both spatial and temporal acc uracies are verified on the equation U(t) = [b(x)U(x)]x, as well as on U(t) + F(x) = 0. Numerical boundary treatments of various orders of a ccuracy are chosen and evaluated for asymptotic stability. Formally ac curate boundary conditions are derived for explicit sixth-order, penta diagonal sixth-order, and explicit, tridiagonal, and pentadiagonal eig hth-order central-difference operators when used in conjunction with R unge-Kutta integrators. Damping of high wavenumber, nonphysical inform ation 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 compre ssible shear layers, where regions of large gradients of flowfield var iables arise near and away from the shear-layer centerline. Results in dicate 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 robu st, 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-de nsity compressible shear layers at the levels of resolution used in th e current work.