New insights from comparing statistical theories for inertial particles in turbulence: II. Relative velocities

Abstract

Part I of this two-part series compared two theories for the radial distribution function (RDF), a statistical measure of the clustering of inertial particles in isotropic turbulence. In Part II, we will contrast three theoretical models for the relative velocities of inertial particles in isotropic turbulence, one by Zaichik et al (2009 New. J. Phys. 11 103018), the second by Pan et al (2010 J. Fluid Mech. 661 73) and the third by Gustavsson et al (2011 Phys. Rev. E. 84 045304). We find that in general they describe the relative velocities in qualitatively similar ways, capturing the influence of the non-local dynamics on the formation of caustics and non-smooth scaling behavior in the dissipation range. We then compare the theoretical predictions with direct numerical simulation data and find that although they capture the qualitative behavior of the data consistently, they differ quantitatively, and we discuss the possible sources of error in each of the theories. Finally, we consider how the Zaichik et al theory predicts that the formation of caustics modifies the form of the RDF, and show that the theory describes the behavior of the RDF for particles whose response time scales with the inertial range time scales of the turbulence.