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. It is shown that the solution can be presented as an expansion in the biorthogonal eigenfunction system. This provides a compact and robust formalism for theoretical and numerical studies of excitation and evolution of wave packets generated by local sources. The temporal continuous spectrum is revisited, and the uncertainty associated with the overlapping of continuous-spectrum branches is resolved. It is shown that the behavior of the discrete spectrum's dispersion relationship is non-analytic due to its relationship with the synchronization of the first or second mode with the vorticity/entropy waves of the continuous spectrum. The characteristics of the wave packet are numerically calculated using an expansion to the biorthogonal eigenfunction system, which comprises modes of discrete and continuous spectra. 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.