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

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 <= M <= 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) 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 similar to 90 <= T-tr,T-1/(K) <= 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.

Automated summary

The study focuses on the low-frequency molecular fluctuations within straight shocks and proposes a linear correlation between the shape of energy probability density functions and characteristic low frequencies of shocks. By deriving noncentral Chi-squared distributions of particle energies inside shocks and analyzing changes in most-probable-speed, the research aims to predict and estimate low-frequency oscillations in shocks for different Mach numbers and input temperatures. This model provides a physics-based approach to boundary conditions in the analysis of laminar-turbulent transition in high-speed flows, offering insights into receptivity and linear stability studies.