Linear instability of high-speed boundary layers is routinely examined assuming quiescent edge conditions, without reference to the internal structure of shocks or to instabilities potentially generated in them. Our recent work has shown that the kinetically modeled internal nonequilibrium zone of straight shocks away from solid boundaries exhibits low-frequency molecular fluctuations. The presence of the dominant low frequencies observed using the direct simulation Monte Carlo (DSMC) method has been explained as a consequence of the well-known bimodal probability density function (PDF) of the energy of particles inside a shock. Here, PDFs of particle energies are derived in the upstream and downstream equilibrium regions, as well as inside shocks, and it is shown for the first time that they have the form of the noncentral Chi-squared (NCCS) distributions. A linear correlation is proposed to relate the change in the shape of the analytical PDFs at a specified upstream number density and temperature as a function of Mach number, within the range \(3 \le M \le 10\), with the DSMC-derived average characteristic low-frequency of shocks, as computed in our earlier work. At a given Mach number \(M=7.2\) and upstream number density \(n_1=10^{22}\,\hbox {m}^{-3}\), it is shown that the variation in DSMC-derived low frequencies is correlated with the change in most-probable-speed inside shocks at the location of maximum bulk velocity gradient for upstream translational temperature in the range \(\sim 90 \le T_{tr,1}/(K) \le 1420\). Using the proposed linear functions, average low frequencies are estimated within the examined ranges of Mach number and input temperature and a semi-empirical relationship is derived to predict low-frequency oscillations in shocks. Our model can be used to provide realistic physics-based boundary conditions in receptivity and linear stability analysis studies of laminar-turbulent transition in high-speed flows.

高速边界层的线性不稳定性通常在假设静止边缘条件的情况下进行检查，而不参考冲击的内部结构或其中可能产生的不稳定性。我们最近的工作表明，远离固体边界的直接冲击的动力学建模内部非平衡区表现出低频分子波动。使用直接模拟蒙特卡罗 (DSMC) 方法观察到的主要低频的存在已被解释为众所周知的冲击内部粒子能量的双峰概率密度函数 (PDF) 的结果。在这里，在上游和下游平衡区域以及内部冲击中导出粒子能量的 PDF，并且首次表明它们具有非中心卡方 (NCCS) 分布的形式。提出了一种线性相关性，将指定上游数密度和温度下分析 PDF 的形状变化作为马赫数的函数，在范围内\(3 \le M \le 10\)，具有 DSMC 衍生的平均特征低频冲击，如我们早期工作中计算的那样。在给定的马赫数\(M=7.2\)和上游数密度\(n_1=10^{22}\,\hbox {m}^{-3}\) 下，表明 DSMC 导出的变化低频与上游平移温度范围内最大体积速度梯度位置的最可能速度内部激波的变化相关联\(\sim 90 \le T_{tr,1}/(K) \le 1420 \). 使用所提出的线性函数，在马赫数和输入温度的检查范围内估计平均低频，并推导出半经验关系来预测冲击中的低频振荡。我们的模型可用于在高速流动中层流-湍流转变的接受度和线性稳定性分析研究中提供基于物理的现实边界条件。