  In this paper we present three multiphase flow models suitable for the study
of the dynamics of compressible dispersed multiphase flows. We adopt the
Eulerian approach because we focus our attention to dispersed (concentration
smaller than 0.001) and small particles (the Stokes number has to be smaller
than 0.2). We apply these models to the compressible ($\text{Ma} = 0.2,\,0.5$)
homogeneous and isotropic decaying turbulence inside a periodic
three-dimensional box ($256^3$ cells) using a numerical solver based on the
OpenFOAM$^{R}$ C++ libraries. In order to validate our simulations in the
single-phase case we compare the energy spectrum obtained with our code with
the one computed by an eighth order scheme getting a very good result (the
relative error is very small $4*10^{-4}$). Moving to the bi-phase case,
initially we insert inside the box an homogeneous distribution of particles
leaving unchanged the initial velocity field. Because of the centrifugal force,
turbulence induce particle preferential concentration and we study the
evolution of the solid-phase density. Moreover, we do an {\em a-priori} test on
the new sub-grid term of the multiphase equations comparing them with the
standard sub-grid scale term of the Navier-Stokes equations.
