  A new fluid-dynamic model is developed to numerically simulate the
non-equilibrium dynamics of polydisperse gas-particle mixtures forming volcanic
plumes. Starting from the three-dimensional N-phase Eulerian transport
equations for a mixture of gases and solid particles, we adopt an asymptotic
expansion strategy to derive a compressible version of the first-order
non-equilibrium model, valid for low concentration regimes and small particles
Stokes $St<0.2$. When $St < 0.001$ the model reduces to the dusty-gas one. The
new model is significantly faster than the Eulerian model while retaining the
capability to describe gas-particle non-equilibrium. Direct numerical
simulation accurately reproduce the dynamics of isotropic turbulence in
subsonic regime. For gas-particle mixtures, it describes the main features of
density fluctuations and the preferential concentration of particles by
turbulence, verifying the model reliability and suitability for the simulation
of high-Reynolds number and high-temperature regimes. On the other hand,
Large-Eddy Numerical Simulations of forced plumes are able to reproduce their
observed averaged and instantaneous properties. The self-similar radial profile
and the development of large-scale structures are reproduced, including the
rate of entrainment of atmospheric air. Application to the Large-Eddy
Simulation of the injection of the eruptive mixture in a stratified atmosphere
describes some of important features of turbulent volcanic plumes, including
air entrainment, buoyancy reversal, and maximum plume height. Coarse particles
partially decouple from the gas within eddies, modifying the turbulent
structure, and preferentially concentrate at the eddy periphery, eventually
being lost from the plume margins due to the gravity. By these mechanisms,
gas-particle non-equilibrium is able to influence the large-scale behavior of
volcanic plumes.
