  In this paper we study the motion of a fluid with several dispersed particles
whose concentration is very small (smaller than $10^{-3}$), with possible
applications to problems coming from geophysics, meteorology, and oceanography.
We consider a very dilute suspension of heavy particles in a
quasi-incompressible fluid (low Mach number). In our case the Stokes number is
small and --as pointed out in the theory of multiphase turbulence-- we can use
an Eulerian model instead of a Lagrangian one. The assumption of low
concentration allows us to disregard particle--particle interactions, but we
take into account the effect of particles on the fluid (two-way coupling). In
this way we can study the physical effect of particle inertia (and not only
passive tracers), with a model similar to the Boussinesq equations. The
resulting model is used in both direct numerical simulations and large eddy
simulations of a dam-break (lock-exchange) problem, which is a well-known
academic test case. Keywords: Dilute suspensions, Eulerian models, direct and
large eddy simulations, slightly compressible flows, dam-break (lock-exchange)
problem.
