Initial-value problem for hypersonic boundary-layer flows

An initial-value problem is analyzed for a two-dimensional wave packet induced by a local two-dimensional disturbance in a hypersonic boundary layer. The problem is solved using Fourier transform with respect to the streamwise coordinate and Laplace transform with respect to time. The temporal continuous spectrum is revisited, and the uncertainty associated with the overlapping of continuous-spectrum branches is resolved. It is shown that the discrete spectrum's dispersion relationship is nonanalytic because of the synchronization of the first mode with the vorticity/entropy waves of the continuous spectrum. However, the inverse Laplace transform is regular at the synchronism point. Characteristics of the wave packet generated by an initial temperature spot are numerically calculated. It is shown that the hypersonic boundary layer is highly receptive to vorticity/entropy disturbances in the synchronism region. The feasibility of experimental verification of this receptivity mechanism is discussed.