On the behavior of pressure in a low Mach number flow

In our recent works, we proposed a theory of turbulence via the mean field effect of an intermolecular potential, which in part relies on an empirically observed "equilibrated" behavior of the pressure variable in a low Mach number flow -- that is, while other variables may exhibit considerable variations at low Mach numbers, the pressure is (nearly) constant. At the same time, conventional kinetic theory does not offer a satisfactory explanation for such a behavior of the pressure variable, instead leading to an isentropic flow in the form of the usual compressible Euler equations. In the current work, we introduce a novel correction term into the pair correlation function of the BBGKY closure for the collision integral, which adjusts the density of incident or recedent pairs of particles depending on the macroscopic compression or expansion rate of the gas. Remarkably, this term does not affect the density and momentum transport equations, and manifests solely in the pressure transport equation. We also find that the effect of the novel term on the pressure dynamics matches the observed behavior -- that is, it tends to attenuate the acoustic waves and smooth out the pressure solution. It appears that the missing piece of the low Mach number puzzle has finally been identified.