Abstract 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.

中文翻译：

---

超音速平板边界层最不稳定频率的线性稳定性分析

摘要 在当前的超音速跃迁模型中，需要第一阶和第二阶模态的时间尺度模型来反映不稳定振型的发展对跃迁开始的贡献。然而，现有的时间尺度相关性在正确显示马赫数效应和温度效应方面存在问题。在本文中，通过线性稳定性分析来研究第一和第二模式时间尺度的现有相关性，并在特征时间尺度的建模中明确涉及和讨论马赫数效应和温度效应。发现现有的基于位移厚度的第一模态时标与雷诺数之间的相关性不适用于超音速情况。超音速情况下最不稳定的第一模态频率随着马赫数的增加而迅速上升，而目前的时间尺度模型无法预测这种趋势，并且当马赫数较高时，频率-雷诺数曲线中出现阶跃现象。对于第二种模式，马赫数对时间尺度的影响不能完全反映在原始相关性中。此外，温度效应被证明会影响不稳定模式的时间尺度，而当前模型忽略了这一点。考虑马赫数、壁温和总温度效应，建立了第一和第二模态特征频率与雷诺数的新关联模型，为超音速跃迁建模提供指导。