Classical Euler-Euler two-fluid models based on the kinetic theory of the granular flow assume the particle phase to be dominated by collisions, even when the particle volume fraction is low and hence collisions are negligible. This leads to erroneous predictions of the particle-phase flow patterns and to the inability of such models to capture phenomena like particle trajectory crossing for finite Stokes numbers. To correctly predict the behaviour of dilute gas-particle flows a more fundamental approach based on solving the Boltzmann equation is necessary to treat non-zero Knudsen numbers and finite Stokes numbers. In this work a dilute collision-less (i.e. infinite Knudsen number) gas-particle flow with mono-dispersed finite-Stokes particles in a lid-driven cavity is studied by means of numerical simulations. An Eulerian quadrature-based moment method for the direct solution of the Boltzmann equation (Fox, 2008) is adopted to describe the particle phase, and it is fully coupled with an Eulerian fluid solver to account for the two-way coupling between the phases. Various Stokes numbers in the range of 0.03-3 are considered to show the capability of the Euler-Euler approach to correctly capture the particle-phase behaviour and the correct segregation phenomena. Results are compared with the predictions of both a classical two-fluid model, whose limitations are pointed out, and of Euler-Lagrange simulations, showing good agreement with the latter.

Fox, R. O., A quadrature based third-order moment method for dilute gas-particle flows, Journal of Computational Physics, 227, 6313 – 6350, 2008.