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.

Scale dependence of multiplier distributions for particle concentration, enstrophy, and dissipation in the inertial range of homogeneous turbulence

“…Particles of St η ∼ 1 are now unique in that they do not experience the usual impedance mismatch with faster eddy forcing going to decades of eddy scale r for a given St, centered on the combined parameter St r ⇠ 0.3, presumably the regime where history e↵ects in particle velocities play the dominant role. While our results support the idea that concentration is generically due to "eddies on the scale of ⌘St 3/2 ⌘ .." [13,24], we think a more refined description one could infer from the U-curve is that clustering is the cumulative result of a history of interactions with the flow of energy as it cascades over eddies ranging over two decades in size, driving particles ever deeper into a concentration "attractor" even in the inertial range [1,3,4]. The other "universal curve" of Zaichik and Alipchenkov [11] (their figure 1) and their improved model [2, their figure 3] reproduced in figure 13 also has this sense.…”

“…More recent evidence that clustering arises even in random, irrotational flows suggests that, while vorticity still plays a role, the dominant role is played by socalled "history effects", in which inertial particle velocity dispersions at any location carry a memory of particle encounters with more remote flow regimes which have larger characteristic velocity differences [8][9][10]. These history effects lead to spatial gradients in particle random relative velocities, and these gradients in turn generate systematic flows or currents which can outweigh dispersive effects and produce zones of highly variable particle concentration [2][3][4]11]. * Also, NASA Ames Research Center, Moffett Field, CA 94035, USA; thomas.hartlep@nasa.gov…”

Relative velocity distribution of inertial particles in turbulence: A numerical study

“…Whether the collision will be elastic, nonelastic, sticky, or such that breakup of the particles will occur depends sensitively on the speed at which the particles collide [3]. Factors influencing the relative velocities are the Stokes number St (ratio of particle relaxation time τ p = 2ρ p r 2 /(9ρ f ν) and the Kolmogorov time scale of the flow τ η = (ν/ ) 1/2 ), the mean dissipation rate , and the Reynolds number Re [6][7][8][9][10][11][12][13][14][15][16]. ν represents the viscosity of the carrier fluid, ρ p the density of the particles, and ρ f the density of the fluid.…”

Focus on dynamics of particles in turbulence

“…As a natural consequence of being a topic crossing several fields, many theories and mechanisms have been proposed to explain the dynamics of inertial particles in turbulence. It is very welcome to see the illustration of the differences and similarities between several leading theories on the spatial distribution [11] and the relative velocities between inertial particles [12], and the probing of the equivalence between several clustering scenarios [13].…”

“…Despite all considering only the linear Stokes drag on particles, the available theories differ in appearance, partly due to the different assumptions made and partly due to the complicated derivation involved. In two companion articles, Bragg and Collins [11,12] analyzed several popular theories on the spatial distribution of small inertial particles in homogeneous and isotropic turbulence and the relative velocities between them. They illustrated clearly the similarities and differences between these theories.…”