Caustics of finitely dense inertial particles

TL;DR

This work investigates caustics, i.e., extreme clustering events, of finitely-dense inertial particles in two-dimensional flows. It extends the velocity-gradient framework to finite density via the tensor $\mathbb{Z}$ and derives a generalized caustics condition for frozen particles in terms of the invariants $Q$ and $R$, highlighting how added-mass effects alter the role of strain. In 2D turbulence, the authors identify two particle classes—Caustic (C) and Survivor (S)—with distinct histories: C particles spend longer in high-compressive-strain regions and form caustics, while S particles cross extensional-strain regions quickly and avoid caustics, even at similar strain levels. The results show that caustics formation scales with density and strain type, generalizing insights from infinitely-dense particles and revealing a universal mechanism across density ratios. Limitations include neglect of Faxén and history forces, and applicability to higher Stokes numbers or three-dimensional flows awaits future work.

Abstract

Estimating collision rates is of immense importance in particle-laden flows. An economical way of doing this is to directly identify incidences of caustics, or extreme clustering, by tracking particle velocity gradients in the neighborhoods of individual particles. The objective of this work is two-fold. (i) We find conditions under which caustics form, in point-vortex flow and in two-dimensional turbulence. While caustics are known to form in regions of strain, we show that the type of strain is key. Particles must remain in compressional strain throughout the process to form caustics, whereas survivor particles: which visit high strain but do not form caustics, briefly go through extensional strain during the early part of the process. This enables survivor particles to attain significantly straighter paths, and to move faster, whereas caustics particles follow paths of high curvature and move slower. As a result, caustics particles stay longer in high-strain regions than survivors. (ii) We ask about the effect of finite particle density, where the particle is denser than the background fluid. We show that finite-density particles need to sample stronger background strain than infinite-density ones to trigger caustics, but our other findings are universal across particle density.

Figures (10)

• Figure 1: Number of real roots of Eq. \ref{['root-eq']} in the $QR$ plane. The hatched area satisfies Eq. \ref{['caus_cond']}, which is a sufficient condition for caustics, while there are some regions with zero roots even outside it. The solid black line is the Vieillefosse line following $Q^3 + 27/4 R^2 = 0$ (see chong1990 for details). The black dashed lines show $Q - 4R = -1/16$, the caustics condition for $\alpha = 1$.
• Figure 1: Fraction of Caustics, Survivors, and Avoiders among $314000$ particles at time $t = 125 \tau_\eta$ with $St = 0.225$.
• Figure 2: Caustics time (a) and radius (b) in point-vortex flow as a function of the initial radial location, for particles of different density and of any Stokes number. The dashed line shows the inner solutions $T_c = \mathcal{R}_0^2$, and $\mathcal{R}_c = \sqrt2\mathcal{R}_0$. In this scaling, caustics formation in the inner region is independent of particle density. The vertical dotted line indicates the location at which caustics time diverges for $\alpha=1$, and beyond which caustics are not possible.
• Figure 3: (a-c) Snapshots of the vorticity field $\omega$ overlaid with caustics C particles in cyan, and the remaining ones in black at $t = 333 \ St$. All particles centrifuge out of large vortices and live in strain regions. As expected from prior studies balkovsky2001calzavarini2008afiabane2012karchniwy2019petersen2019motoori2023, lowering the particle density weakens centrifugation. (d) PDF of the time-averaged $Q$ experienced by the particles (in color), as well as that of the background field (in black). Progressive emptying of high vorticity regions with increasing particle density is seen.
• Figure 4: (a-c) PDF of the $Q$ explored by C particles of different density parameter $\alpha$. (d-f) the corresponding PDF of $\delta = \mathop{\mathrm{Tr}}\nolimits(\mathbb{Z})$. (g) Minimum of the average $Q$ explored by particles on their way to caustics as a function of $\alpha$. (h) The mean of $\delta$ shows similar behaviour across density ratios with an inflection to large negative values around the time of minimum $Q$.