Analytical prediction of low-frequency fluctuations inside a one-dimensional shock

Abstract

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.

A python code for generating the bimodal NCCS PDFs

This appendix gives the code snippet for generating bimodal NCCS distribution. The values of code variables ‘R,’ ‘Ux1,’ ‘Ttr1,’ ‘beta1,’ ‘Ux2,’ ‘Ttr2,’ ‘beta2’ are 208.243, 3572.24, 710, $1.849 \times 10^{-3}$, 944.74, 12120.6, $4.451 \times 10^{-4}$, respectively. The values of code variables ‘Ux,’ ‘Ttr,’ ‘beta’ are obtained from the Mott-Smith velocity distribution and equal to 2012.8, 10149.8, $4.864\times 10^{-3}$, respectively, at probe P. The code variables ‘psi1’ and ‘psi2’ are Mott-Smith fractions and equal to 0.40649 and 0.59351, respectively, at probe P.