Initial-value problem for three-dimensional disturbances in a compressible boundary layer

An initial-value problem is formulated for a three-dimensional wave packet in a compressible boundary layer flow. The problem is solved using a Laplace transform with respect to time and Fourier transforms with respect to the streamwise and spanwise coordinates. The solution can be presented as a sum of modes consisting of continuous and discrete spectra of temporal stability theory. Two discrete modes, known as mode S and mode F, are of interest in high-speed flows since they may be involved in a laminar-turbulent transition scenario. The continuous and discrete spectrum are analyzed numerically for a hypersonic flow with Mach number M=5.6. The following features are revealed: (1) The synchronism of mode S with acoustic waves at a streamwise wave number α1;→0 is primarily two-dimensional; (2) at high angles of disturbance propagation, mode F is no longer synchronized with entropy and vorticity waves; (3) at high angles of disturbance propagation, the synchronism between mode S and mode F is not accompanied by a mode S instability, and at even higher angles of disturbance propagation, mode S and mode F are not synchronized.