A Kinetic Approach to Studying Low-Frequency Molecular Fluctuations in a One-Dimensional Shock

TL;DR

This study introduces a kinetic framework to analyze low-frequency molecular fluctuations in one-dimensional shock waves, revealing their bimodal PDF nature and correlation with stress fluctuations, with implications for flow instability modeling.

Contribution

It presents a novel two-bin model capturing the reduced-order dynamics of molecular fluctuations in shock regions, validated against DSMC simulations, and identifies a Mach number-independent Strouhal number for fluctuations.

Findings

The bimodal PDF characterizes molecular distributions inside shocks.

The two-bin model accurately predicts fluctuation frequency differences.

Low-frequency fluctuations are described by a Mach number-independent Strouhal number.

Abstract

Low-frequency molecular fluctuations in the translational nonequilibrium zone of one-dimensional strong shock waves are characterised for the first time in a kinetic collisional framework in the Mach number range $2\le M\le 10$. Our analysis draws upon the well-known bimodal nature of the probability density function (PDF) of gas particles in the shock, as opposed to their Maxwellian distribution in the freestream, the latter exhibiting an order of magnitude higher dominant frequencies than the former. Inside the (finite-thickness) shock region, the strong correlation between perturbations in the bimodal PDF and fluctuations in the normal stress suggests introducing a novel two-bin model to describe the reduced-order dynamics of a large number of collision interactions of gas particles. Our model correctly predicts the order-of-magnitude difference in fluctuation frequencies in the shock versus those in the freestream and is consistent with the small-amplitude fluctuations obtained from the highly resolved Direct Simulation Monte Carlo (DSMC) computations of the same configuration. The variation of low-frequency fluctuations with changes in the conditions upstream of the shock revealed that these fluctuations can be described by a Strouhal number, based on the bulk velocity upstream of the shock and the shock-thickness based on the maximum density-gradient inside the shock, that remains practically independent of Mach number in the range examined. Our results are expected to have far-reaching implications for boundary conditions employed in the vicinity of shocks in the framework of flow instability and laminar-turbulent transition studies of flows containing both unsteady and nominally stationary shocks.