Wave packets and supersonic second modes in a high-speed boundary layer

An initial-value problem is formulated for a wave packet in a two-dimensional compressible boundary layer flow. The problem is solved using the Fourier transform with respect to the streamwise and spanwise coordinates and eigenfunction expansion into modes of continuous and discrete spectra of temporal stability theory. The spectra of two-dimensional perturbations are analyzed numerically for a hypersonic flow with Mach number M = 6 and the wall-to-edge temperature ratio T_w/T_e = 0.5. An example of a two-dimensional temperature spot is considered. The inverse Fourier transform is obtained numerically for a portion of the wave packet comprised of the second Mack modes. Solution in the physical domain reveals existence of a satellite wave packet with an acoustic-like beam protruding into the outer flow as it was observed in earlier DNS results. The wave packet structure is attributed to branching of unstable discrete mode F at the point of synchronization with the slow acoustic mode leading to the appearance of a local maximum in the growth rate of supersonic second modes. Although there is a mechanism of energy transfer from a perturbation to the outer flow, the leading part of the wave packet is dominated by subsonic second modes confined inside the boundary layer.