Stable high-order schemes and DNS of boundary-layer stability on a blunt cone at Mach 8

Abstract

The objectives of this paper are two fold: 1) to present new high-order (12th or higher order) explicit and compact finite difference schemes with stable boundary closures for the DNS of transitional or turbulent flows; 2) to present results of numerical simulation of nonlinear boundary layer stability of Mach 7.99 axisymmetric flow over a blunt cone. The first part of this paper presents a way to stabilize high-order finite difference schemes with boundary closures. Current numerical methods used in most practical DNS studies of compressible flows are limited to 6th-order or lower in the interior and 4th-order or lower on the boundary because of the numerical instability of the boundary closure schemes. This paper shows that this numerical instability for high-order schemes based on uniform grids is due to the instability of polynomial interpolation based on uniform grids (the Runge phenomena). It is shown that the instability can be overcome for arbitrarily high-order finite difference schemes with stable boundary closure schemes if the schemes are derived directly on a non-uniform stretched grid. Explicit formulas for computing the coefficients of high-order compact (and explicit) schemes on nonuniform grids are derived. The second part of the paper is motivated by a project of NATO RTO Working Group 10 on Boundary Layer Transition to conduct numerical simulation of nonlinear boundary layer stability for blunt cone at Mach 7.99 corresponding to Stetson's experiment. The emphasis is on the nonlinear second mode instability of the hypersonic boundary layer observed in the experiments. The initial results of the first test case under the isothermal wall condition are presented and compared with the experimental results.