Statistics of the relative velocity of particles in turbulent flows : monodisperse particles

TL;DR

This study uses direct numerical simulations to analyze the joint probability density function of relative velocity and distance of inertial particles in turbulent flows, confirming theoretical asymptotic regimes and exploring their dependence on particle inertia.

Contribution

It provides the first detailed numerical validation of the scale-invariant distribution regimes and quantifies the correlation dimension and scaling parameters as functions of Stokes number.

Findings

Distribution is scale invariant at small scales

Confirmed two asymptotic regimes for relative velocities

Quantified dependence of $D_2$ and $z^*$ on Stokes number

Abstract

We use direct numerical simulations to calculate the joint probability density function of the relative distance $R$ and relative radial velocity component $V_R$ for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, $D_2$. It was argued [1, 2] that the scale invariant part of the distribution has two asymptotic regimes: (1) $|V_R| \ll R$ where the distribution depends solely on $R$; and (2) $|V_R| \gg R$ where the distribution is a function of $|V_R|$ alone. The probability distributions in these two regimes are matched along a straight line $|V_R| = z^\ast R$. Our simulations confirm that this is indeed correct. We further obtain $D_2$ and $z^\ast$ as a function of the Stokes number, ${\rm St}$. The former depends non-monotonically on ${\rm St}$ with a minimum at about ${\rm St} \approx 0.7$ and the latter has only a weak dependence on ${\rm St}$.