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, $\text{St}<1$. We solve for the particles using both a Lagrangian and an Eulerian approach (with a shock-capturing scheme). In the latter, the particles are described by a dust-density field and a dust-velocity field. We find the following: the dust-density field in our Eulerian simulations has the same correlation dimension $d_2$ as obtained from the clustering of particles in the Lagrangian simulations for $\text{St}<1$; 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 falloff indicating the presence of voids; the energy spectrum of the dust velocity has a power-law range with an exponent that is the same as the gas-velocity spectrum except at very high Fourier modes; the compressibility of the dust-velocity field is proportional to ${\text{St}}^2$. We quantify the topological properties of the dust velocity and the gas velocity through their gradient matrices, called $\mathcal{A}$ and $\mathcal{B}$, respectively. Our DNS confirms that the statistics of topological properties of $\mathcal{B}$ are the same in Eulerian and Lagrangian frames only if the Eulerian data are weighed by the dust density. We use this correspondence to study the statistics of topological properties of $\mathcal{A}$ in the Lagrangian frame from our Eulerian simulations by calculating density-weighted probability distribution functions. We further find that in the Lagrangian frame, the mean value of the trace of $\mathcal{A}$ is negative and its magnitude increases with $\text{St}$ approximately as $exp(-C/\text{St})$ with a constant $C\approx 0.1$. The statistical distribution of different topological structures that appear in the dust flow is different in Eulerian and Lagrangian (density-weighted Eulerian) cases, particularly for $\text{St}$ close to unity. In both of these cases, for small $\text{St}$ the topological structures have close to zero divergence and are either vortical (elliptic) or strain dominated (hyperbolic, saddle). As $\text{St}$ 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 Lagrangian (density-weighted Eulerian) case has less outward spirals and more converging saddles. Inward spirals are the least probable topological structures in both cases.