Surface instabilities in laminar compressible boundary layers with sublimation

Abstract

Surface patterns on ablating materials are observed in high-speed ground and flight tests, but the mechanisms behind their formation are not known. In this paper, the origin of surface patterns is investigated via a local linear stability analysis of compressible laminar boundary layers over a flat camphor plate. The effects of sublimation and conjugate heat transfer are included both on the baseflow and the linear fluctuations. This framework identifies one mode that fully characterizes the stability of the surface, which becomes unstable when the wall temperature exceeds that of an adiabatic wall, $T_{ad}$. These findings are consistent with experimental observations, where laminar flow conditions at adiabatic wall temperatures are found to be stable. The analysis also reveals that the nature of this surface mode varies as a function of the oblique angle $\psi = \tan^{-1}{\beta/\alpha}$, where $\alpha$ and $\beta$ are the streamwise and spanwise wavenumbers. Specifically, for baseflow temperatures below $\approx 1.15~T_{ad}$, the surface mode is most unstable at $\psi = 0$. Conversely, above $\approx 1.15~T_{ad}$ the surface is most unstable near the sonic angle $\psi_s = \cos^{-1}(1/M_e)$, which is the angle at which the normal Mach number equals one. Finally, a critical wavenumber is identified (i.e., one at which the temporal growth rate reaches a maximum) that is in good agreement with available experimental observations of turbulent flows.