Interactions between oblique second mode and oblique waves in a high-speed boundary layer

Interactions between oblique second mode and oblique waves at a high-speed boundary at Mach 4.5 are studied using linear stability theory, nonlinear parabolized stability equations (NPSE) and direct numerical simulation (DNS). Parametric analysis based on the NPSE suggests that the oblique second mode can amplify both the oblique first and second modes, with the former experiencing a higher amplification. Our analysis reveals that the mean-flow distortion and difference mode contribute to this enhancement, with the latter exerting a key influence through the parametric resonance process. Kinetic energy transfer analysis demonstrates that the oblique waves gain energy from the mean flow, rather than from the oblique second mode. Furthermore, we find that the mechanism underlying the interaction between a pair of second oblique waves and a single oblique wave is similar to that between an oblique second mode and an oblique wave, as the steady modes generated by the pair of oblique second modes have a limited impact on the oblique wave. Finally, DNS confirms the validity of two transition paths proposed in this study based on the NPSE results. The first path suggests that a pair of low-amplitude second oblique waves alone are insufficient to cause oblique breakdown, but the introduction of a pair of low-amplitude damping first oblique modes could lead to boundary layer breakdown. The second path involves the formation of a domino-like effect through the combination of different types of oblique waves with the appropriate parameters. These two nonlinear paths can lead to a fully developed turbulent boundary layer.