The Stability of a Compressible Rayleigh Layer

The aim of this work is to determine the linear stability of a compressible Rayleigh layer and to ascertain what role unsteady effects play. A Rayleigh layer is formed when an infinite flat plate is impulsively set in motion in its own plane with constant velocity beneath an initially quiescent fluid. When the fluid is compressible there is a motion both parallel and normal to the plate. The classical boundary-layer scaling is employed to determine solutions which are expressed in terms of a similarity variable and are valid for a large range of Mach, Prandtl and Reynolds numbers. Solutions are presented for both an adiabatic and iso-thermal temperature boundary condition at the plate. The temporal stability of the flow is considered by solving an Orr–Sommerfeld system: here the underlying flow is calculated at a certain time and the instantaneous stability to viscous travelling waves is determined. The stability is seen to be altered by changing the Mach number (an increase of which decreases the stability of the flow), and also by cooling and heating the wall. These results are limited by the fact that the growth of the layer in time is not taken into account. To include this we consider the large Reynolds number limit and use a triple-deck structure to determine the modes characteristics. The triple-deck approach is used to determine an asymptote to the lower branch of the neutral curve and unsteady effects can be included in a consistent manner. For the upper branch, however, a five-deck structure is required due to the fact that the critical layer is now distinct from the viscous sublayer. The upper-branch stability is only calculated to the first order which is sufficient to give an insight into the stability characteristics.