Effects of streak structure in oblique resonance on the instability in supersonic plate boundary layers

Streaks are large-scale coherent structures in flows and play an important role in the laminar–turbulent transition. In this paper, the effect of streaks in oblique resonance on the instabilities of the first mode with the same spanwise wavenumber as the streaks is investigated. The streaks are obtained by the nonlinear parabolized stability equations, and the instability is analyzed by the bi-global analysis tool. Streaks can be decomposed into the two-dimensional mean-flow distortion (MFD) and the three-dimensional streak modes (SMs). For odd modes, the effects of the MFD dominate, stabilizing low-frequency waves and destabilizing high-frequency waves at small streak amplitudes. While the opposite effects of the SMs dominate at large streak amplitudes. Even modes show similar responses, except for the stabilization of low-frequency waves. The results of the global energy balance indicate that the more negative work of pressure-dilatation terms is responsible for the stabilization of the high-frequency waves compared to low-frequency waves at large streak amplitudes. Due to the stronger concentration of streamwise velocity distribution and the more negative Reynolds stress work in the spanwise direction, the even modes are more stabilized compared to the odd modes. Additionally, streaks can modify the distribution of the Fourier components of the even mode pressure in the spanwise direction; thus, they can induce the spontaneous excitation of a radiating component to the far field.