  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.
