A universal scaling law for the evolution of granular gases

TL;DR

This paper uncovers a universal scaling law linking the evolution of dilute granular gases to the Mach number, enabling prediction of clustering behavior regardless of initial conditions or material properties.

Contribution

It introduces a universal quadratic relationship between density fluctuations and Mach number in granular gases, independent of restitution coefficient or initial velocity distribution.

Findings

Density fluctuations scale quadratically with Mach number.

Clustering becomes observable at Mach number around 1.

Universal behavior observed across different initial conditions.

Abstract

Dry, freely evolving granular materials in a dilute gaseous state coalesce into dense clusters only due to dissipative interactions. This clustering transition is important for a number of problems ranging from geophysics to cosmology. Here we show that the evolution of a dilute, freely cooling granular gas is determined in a universal way by the ratio of inertial flow and thermal velocities, that is, the Mach number. Theoretical calculations and direct numerical simulations of the granular Navier--Stokes equations show that irrespective of the coefficient of restitution, density or initial velocity distribution, the density fluctuations follow a universal quadratic dependence on the system's Mach number. We find that the clustering exhibits a scale-free dynamics but the clustered state becomes observable when the Mach number is approximately of $\mathcal{O}(1)$. Our results provide a method to determine the age of a granular gas and predict the macroscopic appearance of clusters.