Linear stability analysis on the most unstable frequencies of supersonic flat-plate boundary layers

In current supersonic transition models, the timescale models of the first and second modes are required to reflect the contribution to the transition onset by the development of the unstable modes. However, the existing correlations of the timescales have problems in showing the Mach number effect and temperature effect correctly. In this paper, linear stability analyses are conducted to study the existing correlations of the timescales for the first and second modes, and the Mach number effect and the temperature effect are explicitly involved and discussed in the modeling of the characteristic timescales. It is found that the existing correlation between the timescale of the first mode and the Reynolds number based on the displacement thickness is not suitable for the supersonic situation. The frequency of the most unstable first mode in supersonic cases rises rapidly with the increasing Mach number while the current timescale model fails to predict this tendency, and the step phenomenon appears in the frequency-Reynolds-number curves when the Mach number is high. For the second mode, the Mach number effect on the timescale cannot be reflected entirely in the original correlation. Additionally, the temperature effect is proved to affect the timescales of unstable modes, which is left out by the current model. Considering the Mach number, the wall temperature and the total temperature effects, new correlation models for the characteristic frequency of the first and second mode with the Reynolds number are established, in order to provide guidelines for supersonic transition modeling. Some inconsistencies between the timescale-based transition models and the linear stability theory are also commented in this paper.