A molecular-kinetic hypothesis on the mechanics of compressible gas flow at low Mach numbers

TL;DR

This paper introduces a molecular-kinetic correction to explain the pressure behavior in low Mach number compressible flows, linking dissipation effects to turbulence creation and stability.

Contribution

It proposes a novel correction in the kinetic theory that accounts for pressure stabilization and turbulence onset at low Mach numbers, advancing understanding of compressible flow mechanics.

Findings

The correction introduces strong dissipation into the pressure equation.

Dissipation manifests as second viscosity, suppressing velocity divergence.

The ratio of viscosities and critical Reynolds number depend on packing fraction.

Abstract

In recent works, we proposed a theory of turbulence creation via the second coefficient of the virial expansion (i.e. the van der Waals effect). This theory relies, in part, on the empirically observed "equilibrated" behavior of pressure in compressible flows at low Mach numbers. However, a fundamental explanation for such a behavior of pressure does not currently exist, because the conventional kinetic theory leads instead to the adiabatic flow in the form of the usual compressible Euler or Navier-Stokes equations. To explain this behavior of pressure from the molecular-kinetic perspective, in the current work we introduce a novel correction into the pair correlation function in the closure of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. This correction matches the rate of change of the average distance between particles to the macroscopic compression or expansion rate of the gas. Remarkably, the novel correction introduces strong dissipation into the pressure equation at low Mach numbers, which stabilizes the pressure solution. At small scales, the novel dissipation effect manifests as the second viscosity in the momentum equation, which selectively suppresses the velocity divergence. As a result, the second viscosity governs the linear instability which creates turbulent dynamics, thereby setting the critical value of the Reynolds number. The ratio of the second and shear viscosities, together with the critical value of the Reynolds number, are proportional to the reciprocal of the packing fraction.