Pressure gradient effects on the secondary instability of Mack mode disturbances in hypersonic boundary layers

Abstract

In hypersonic boundary layers, Mack mode disturbances play an important role in laminar–turbulence transition. Understanding the secondary instability mechanism of Mack mode disturbances will provide physical insight into turbulence generation for the researchers, which is also meaningful for transition control. According to the previous studies over straight cones and flared cones, it seems that a pressure gradient may affect the primary instability and secondary instability of Mack mode disturbances obviously. In this paper, we are trying to make it clear that what the pressure gradient effect on the secondary instability of Mack modes is and what the influence rule is. Four hypersonic flat plate cases with various pressure gradients at Mach 6 are analyzed through linear stability theory, non-linear parabolized stability equations, and spatial secondary instability theory methods. We found that the essence of the pressure gradient influence on the secondary instability mechanism is by affecting the primary amplitude of Mack modes, rather than other routes or factors. An adverse pressure gradient can enlarge both the primary instability and secondary instability growth rates and advance the transition. Moreover, an adverse pressure gradient will form a larger primary amplitude of the Mack mode, leading to a fundamental resonance dominated secondary instability. In contrast, the favorable pressure gradient will suppress the primary amplitude so that the subharmonic resonance may dominate the secondary instability. Therefore, it is very meaningful and valuable for transition prediction and turbulence generation to conduct the present study of pressure gradient effects on the secondary instability of Mack mode disturbances.