New insights from comparing statistical theories for inertial particles in turbulence: I. Spatial distribution of particles

Abstract

In this paper, we contrast two theoretical models for the spatial clustering of inertial particles in isotropic turbulence, one by Chun et al (2005 J. Fluid Mech. 536 219) and the other by Zaichik et al (2007 Phys. Fluids 19 113308). Although their predictions for the radial distribution function are similar in the regime ≪ St 1, they appear to describe the physical mechanism responsible for the clustering in quite different ways. We demonstrate why the theories generate such similar results in the regime ≪ St 1 by showing that the clustering mechanism in the Chun et al theory captures the leading order effects of the clustering mechanism in the Zaichik et al theory for ≪ St 1. However, outside of this regime, the similarity between the predictions of the theories breaks down, and we consider the sources of the differences as well as the physical meaning and implications of the differences. Using DNS data we then show that the clustering mechanism described by the Zaichik et al theory accurately describes the clustering up to ≈ St 1, and we identify a possible source of error for some of the slight quantitative discrepancies at larger St. We then compare these theories with others in the literature and attempt to reconcile as many of the physical explanations for clustering as we can. Finally, we consider the relationship between clustering in isotropic turbulence and the near-wall accumulation of Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. inertial particles in a turbulent boundary layer, and how they scale with the Stokes number in the weak inertia limit. New J. Phys. 16 (2014) 055013 A D Bragg and L R Collins 2 particle inertia. Here and throughout, τ τ = η St pwhere τ p is the particle response time and τ η is the Kolmogorov time scale. There are two aspects of clustering that are important to keep in mind. First, clustering peaks at Stokes numbers near unity and rapidly falls off for smaller or larger values. Second, clustering continues to increase at separation distances below the Kolmogorov length scale (the smallest scale of motion of the turbulence), implying no inner cutoff of the mechanism driving the particles together.Theories that describe and predict inertial particle clustering are simultaneously under development. Some, based on a phenomenological description of the process, capture clustering reasonably well [28,29], but do not yield new physical insights into the phenomenon. Others make assumptions about the turbulence that cannot be fully justified, such as white noise in time [30] or finite-time correlations that are significantly shorter than in real turbulence [31]. Two theories by Chun et al [32] and Zaichik et al [33] (hereafter CT and ZT, respectively) are based on similar descriptions of the turbulence, focus on sub-Kolmogorov scale clustering, and for small particles yield similar predictions in ...

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.…”

“…For the third and final method, method C, we use a least-square solver to determine the minimum-norm solution of (A. 3), that is the solution that is closest to equally distributed concentrations. Clearly, this causes a bias towards the least intermittent spatial distribution.…”

“…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…”

Phoresis in turbulent flows

“…In the absence of phoretic effect ( 0 a = ), the term u d is responsible for turbulent particle dispersion. This term averages to zero when the colloids are released, as the particle positions are a priori not correlated with the flow velocity field [22]. At small times, this term contributes only a small correction to the constant term, in particular when…”

Settling of heated particles in homogeneous turbulence

“…Particles of St η = 1 also display maximum dispersion rates and maximum time rate of change of their temperature fluctuations (Wetchagarun & Riley 2010). In the last decade, experimental, numerical and theoretical studies have deepened our understanding of the origins of turbulence clustering (Goto & Vassilicos 2008;Salazar et al 2008;Gibert, Xu & Bodenschatz 2012;Bragg & Collins 2014), its consequences on the statistics of particle motion (Ayyalasomayajula et al 2006;Bec et al 2006) and on the particle collision rate (Sundaram & Collins 1997;Wang, Wexler & Zhou 2000).…”