Preferential concentration in the particle-induced convective instability

Heavy particles in turbulent flows have been shown to accumulate in regions of high strain rate or low vorticity, a process otherwise known as preferential concentration. This can be observed in geophysical flows, and is inferred to occur in astrophysical environments, often resulting in rapid particle growth which is critical to physical processes such as rain or planet formation. Here we study the effects of preferential concentration in a two-way coupled system in the context of the particle-driven convective instability. To do so, we use Direct Numerical Simulations and adopt the two-fluid approximation. We focus on a particle size range for which the latter is valid, namely when the Stokes number is $\lesssim O(0.1)$. For Stokes number above $\sim 0.01$, we find that the maximum particle concentration enhancement over the mean scales with the rms fluid velocity $u_{\rm{rms}}$, the particle stopping time $\tau_p$, and the particle diffusivity $\kappa_p$, as $u_{\rm{rms}}^2 \tau_p / \kappa_p$. We show that this scaling can be understood from simple arguments of dominant balance. We also show that the typical particle concentration enhancement over the mean scales as $(u_{\rm{rms}}^2 \tau_p/ \kappa_p)^{1/2}$. We finally find that the probability distribution function of the particle concentration enhancement over the mean has an exponential tail whose slope scales as $(u_{\rm{rms}}^2 \tau_p / \kappa_p)^{-1/2}$. We apply our model to geophysical and astrophysical examples, and discuss its limitations.