On the instability of hypersonic flow past a flat plate

Abstract

The instability of hypersonic boundary-layer flow over a flat plate is considered. The viscosity of the fluid is taken to be governed by Sutherland's formula, which gives a more accurate representation of the temperature dependence of fluid viscosity at hypersonic speeds than Chapman's approximate linear law. A Prandtl number of unity is assumed. Attention is focused on inviscid instability modes of viscous hypersonic boundary layers. One such mode, the ‘vorticity’ mode, is thought to be the fastest growing disturbance at high Mach numbers, M [Gt ] 1; in particular it is believed to have an asymptotically larger growth rate than any viscous instability. As a starting point we investigate the instability of the hypersonic boundary layer which exists far downstream from the leading edge of the plate. In this regime the shock that is attached to the leading edge of the plate plays no role, so that the basic boundary layer is non-interactive. It is shown that the vorticity mode of instability operates on a different lengthscale from that obtained if a Chapman viscosity law is assumed. In particular, we find that the growth rate predicted by a linear viscosity law overestimates the size of the growth rate by O((log M)½). Next, the development of the vorticity mode as the wavenumber decreases is described. It is shown, inter alia, that when the wavenumber is reduced to O(M-3/2) from the O(1) initial, ‘vorticity-mode’ scaling, ‘acoustic’ modes emerge.Finally, the inviscid instability of the boundary layer near the leading-edge interaction zone is discussed. Particular attention is focused on the strong-interaction zone which occurs sufficiently close to the leading edge. We find that the vorticity mode in this regime is again unstable. The fastest growing mode is centred in the adjustment layer at the edge of the boundary layer where the temperature changes from its large, O(M2). value in the viscous boundary layer, to its O(1) free-stream value. The existence of the shock indirectly, but significantly, influences the instability problem by modifying the basic flow structure in this layer.