Shock trapping and inertial escape: Dust-particle clustering in compressible turbulence

TL;DR

This study probes how inertial dust particles cluster in shock-dominated compressible turbulence using a minimalist 2D stochastically forced Burgers model. By varying the Stokes number, it reveals a clear crossover from shock trapping and near-singular clustering at small $St$ to inertial escape and quasi-ballistic motion at large $St$, with a regime around $St \sim \mathcal{O}(1)$ exhibiting strong intermittency and scale-free density statistics. The authors quantify this behavior via the density field $\Theta$, a mesoscale coarse-grained density $\rho_r$, and the correlation dimension $D_2$, finding $\Theta_{\rm rms} \sim St^{-3/2}$ in the intermediate regime and $D_2$ transitioning from 0 to 2 as $St$ increases. Their results offer a clean, physically transparent baseline for shock-driven particle concentration with potential applications to dust coagulation in protoplanetary discs, while highlighting limitations and avenues for extending to more realistic 3D, self-gravitating, and two-way coupled systems.

Abstract

We study the dynamics and clustering of dust particles with inertia in shock-dominated compressible turbulence using the two-dimensional, stochastically forced Burgers equation. At small Stokes numbers, shock trapping leads to extreme density inhomogeneities and nearly singular aggregation, with correlation dimensions approaching zero. With increasing inertia, particles undergo inertial escape and intermittently cross shock fronts, producing a sharp crossover from shock-dominated trapping to quasi-ballistic dynamics. This crossover is accompanied by a pronounced reduction in density fluctuations, a continuous increase of the correlation dimension from zero to the embedding dimension, and a power-law dependence of density fluctuations on the Stokes number over an extended intermediate regime. In this regime, particle distributions show scale-free coarse-grained density statistics arising from repeated trap--escape dynamics. This behaviour is qualitatively distinct from inertial-particle clustering in incompressible turbulence and is directly relevant to dust concentration in shock-rich regions of protoplanetary discs and other compressible astrophysical environments.

Figures (4)

• Figure 1: Representative snapshots of the coarse grained particle density fields $\Theta$ for (a) $St = 7.8$, (b) $St = 53.125$, (c) $St = 500$, and (d) $St = 1250$, overlaid on the divergence $\nabla \cdot {\bf u}$ field (shown in grey scale) of the carrier flow. These snapshots are taken after the non-equilibrium steady state has been reached for all the suspensions; an animation of the evolution of the particles is given in Ref. YT-movies.
• Figure 2: (Inset) Plots of the evolution of $\Theta_{\rm rms}$ --- measure of the inhomogeneity in particle distribution (see text) --- as a function of the non-dimensional time $\tau = t/\tau_{\rm \eta}$ for representative values of the Stokes number (see legend). The shaded region corresponds to the steady state distribution of particles from whence we calculate the mean value $\langle \Theta_{\rm rms} \rangle$ of $\Theta_{\rm rms}$ over this steady state for different $St$ numbers. In the main panel we plot $\langle \Theta_{\rm rms} \rangle$ vs $St$. The error bars on these are the standard deviation in values of $\Theta_{\rm rms}$ over the stationary regime (see inset). The two, dashed horizontal lines denote two different limiting behaviours. The upper one $\langle \Theta_{\rm rms} \rangle = \sqrt{N_p}$ corresponds to all particles collapsing to a singular point whereas the lower one $\langle \Theta_{\rm rms} \rangle = \sqrt{2}$ emerges as the asymptotic limit for a uniform distribution with Poisson fluctuations (see text).
• Figure 3: Loglog plots of the probability density functions of the particle density fields coarse grained at scales ${r} = 0.15$, for different Stokes numbers (see legend).
• Figure 4: (a) Loglog plots of the cumulative density function $P^< (r)$ vs $r/\eta$ for different Stokes numbers showing a clear power-law scaling. From a local slope analysis of these power laws we estimate the correlation dimension $D_2$ (and its error bars) which are shown, as a function of $St$, in panel (b). In panel (b) we also include several insets showing the particle positions which illustrate the corresponding fractal dimension $D_2$ of their areas.