Topology of two-dimensional turbulent flows of dust and gas

TL;DR

This study uses direct numerical simulations to analyze the clustering, spectral, and topological properties of dust particles in two-dimensional turbulent flows, revealing how these properties depend on the Stokes number and differ between Eulerian and Lagrangian perspectives.

Contribution

It provides a detailed comparison of Eulerian and Lagrangian approaches in studying dust in turbulence, highlighting the topological and statistical properties of dust density and velocity fields for Stokes numbers less than one.

Findings

Dust-density field correlation dimension matches clustering in Lagrangian simulations.

Void regions in dust density follow a power-law distribution.

Dust velocity spectrum mirrors gas velocity spectrum at most scales.

Abstract

We perform direct numerical simulations (DNS) of passive heavy inertial particles (dust) in homogeneous and isotropic two-dimensional turbulent flows (gas) for a range of Stokes number, ${\rm St} < 1$, using both Lagrangian and Eulerian approach (with a shock-capturing scheme). We find that: The dust-density field in our Eulerian simulations have the same correlation dimension $d_2$ as obtained from the clustering of particles in the Lagrangian simulations for ${\rm St} < 1$; The cumulative probability distribution function of the dust-density coarse-grained over a scale $r$ in the inertial range has a left-tail with a power-law fall-off indicating presence of voids; The energy spectrum of the dust-velocity has a power-law range with an exponent that is same as the gas-velocity spectrum except at very high Fourier modes; The compressibility of the dust-velocity field is proportional to ${\rm St}^2$. We quantify the topological properties of the dust-velocity and the gas-velocity through their gradient matrices, called $\mathcal{A}$ and $\mathcal{B}$, respectively. The topological properties of $\mathcal{B}$ are the same in Eulerian and Lagrangian frames only if the Eulerian data are weighed by the dust-density -- a correspondence that we use to study Lagrangian properties of $\mathcal{A}$. In the Lagrangian frame, the mean value of the trace of $\mathcal{A} \sim - \exp(-C/{\rm St}$, with a constant $C\approx 0.1$. The topology of the dust-velocity fields shows that as ${\rm St} increases the contribution to negative divergence comes mostly from saddles and the contribution to positive divergence comes from both vortices and saddles. Compared to the Eulerian case, the density-weighed Eulerian case has less inward spirals and more converging saddles. Outward spirals are the least probable topological structures in both cases.