Topology of two-dimensional turbulent flows of dust and gas

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.