On the instability of boundary layers on heated flat plates

Abstract

The stability of a boundary layer on a heated flat plate is investigated in the linear regime. The flow is shown to be unstable to longitudinal vortex structures which develop in a non-parallel manner in the streamwise direction. Solutions of the non-parallel equations are obtained numerically at O(1) values of the appropriate stability parameter, i.e. the Grashof number. We investigate the particular cases in which instability is induced by localized or distributed wall roughness or non-uniform wall heating. The case when the vortices are induced by free-stream disturbances is also considered. We then investigate the high-Grashof-number limit and the fastest growing mode. The fastest growing mode is found to be governed by a quasi-parallel theory and occurs at high wavenumbers. The wavenumber and growth rate of the fastest growing mode are found in closed form. At low wavenumbers the vortex instability is shown to be closely related to Tollmein–Schlichting waves; the effect of wall heating or cooling on the latter type of instability is discussed.