Stability of compressible boundary layers with a velocity overshoot

Abstract

The stability of compressible boundary layers is advanced in the literature, however, boundary layers with a velocity overshoot are yet to be considered. These boundary layers can arise when the surface over which the fluid flows is heated, and an accelerating pressure gradient is applied. The analytic behaviour of solutions of the governing equations was investigated in order to demonstrate the existence of velocity overshoot. An analysis of the structure of these solutions in the large Mach number limit is given. The local maximum in the streamwise velocity and the presence of multiple generalised points of inflection in these boundary layers, indicate that the stability properties of the ow are likely to be affected. A numerical study of the inviscid stability behaviour of a class of boundary layers with overshoot, reveals a new mode of instability, in addition to the classical first- and higher-mode solutions previously described in the literature. The circumstances under which this new unstable mode can attain higher growth rates than the classical modes are explored together with its role in laminar-turbulent transition. To support, and validate, the numerical results an analytic solution of the new mode in the small wavenumber limit was derived from which it is demonstrated that this mode propagates at a wavespeed equal to the maximum boundary-layer velocity. The inviscid stability results were extended by demonstrating that they match with the viscous stability behaviour in the large Reynolds number limit. In cases where the velocity overshoot and/or the Mach number are sufficiently large, the inflectional and the new non-inflectional neutral modes may propagate at a wavespeed that is no longer subsonic relative to the free-stream velocity; this is shown to render the basic boundary-layer ow stable to the first-mode and the newly discovered new-mode disturbance.