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A new fluid-dynamic model is developed to numerically simulate thenon-equilibrium dynamics of polydisperse gas-particle mixtures forming volcanicplumes. Starting from the three-dimensional N-phase Eulerian transportequations for a mixture of gases and solid particles, we adopt an asymptoticexpansion strategy to derive a compressible version of the first-ordernon-equilibrium model, valid for low concentration regimes and small particlesStokes $St<0.2$. When $St < 0.001$ the model reduces to the dusty-gas one. Thenew model is significantly faster than the Eulerian model while retaining thecapability to describe gas-particle non-equilibrium. Direct numericalsimulation accurately reproduce the dynamics of isotropic turbulence insubsonic regime. For gas-particle mixtures, it describes the main features ofdensity fluctuations and the preferential concentration of particles byturbulence, verifying the model reliability and suitability for the simulationof high-Reynolds number and high-temperature regimes. On the other hand,Large-Eddy Numerical Simulations of forced plumes are able to reproduce theirobserved averaged and instantaneous properties. The self-similar radial profileand the development of large-scale structures are reproduced, including therate of entrainment of atmospheric air. Application to the Large-EddySimulation of the injection of the eruptive mixture in a stratified atmospheredescribes some of important features of turbulent volcanic plumes, includingair entrainment, buoyancy reversal, and maximum plume height. Coarse particlespartially decouple from the gas within eddies, modifying the turbulentstructure, and preferentially concentrate at the eddy periphery, eventuallybeing lost from the plume margins due to the gravity. By these mechanisms,gas-particle non-equilibrium is able to influence the large-scale behavior ofvolcanic plumes.

ASHEE: a compressible, equilibrium-Eulerian model for volcanic ash plumes