On the Eulerian Large Eddy Simulation of Disperse Phase Flows: An Asymptotic Preserving Scheme for Small Stokes Number Flows

Abstract

In the present work, the Eulerian large eddy simulation (LES) of dilute disperse phase flows is investigated. By highlighting the main advantages and drawbacks of the available approaches in the literature, a choice is made in terms of modeling: a Fokker--Planck-like filtered kinetic equation proposed by Zaichik, Simonin, and Alipchenkov [L. I. Zaichik, O. Simonin, and V. M. Alipchenkov, J. Turbul., 10 (2009), N4] and a kinetic-based moment method based on a Gaussian closure for the number density function proposed by Vié, Doisneau, and Massot [A. Vié, F. Doisneau, and M. Massot, Comm. Comput. Phys., 17 (2015), pp. 1--46]. The resulting Euler-like system of equations is able to reproduce the dynamics of particles for small to moderate Stokes number flows, given a LES model for the gaseous phase, and is representative of the generic difficulties of such models. Indeed, it encounters strong constraints in terms of numerics in the small Stokes number limit, which can lead to a degeneracy of the accuracy of standard numerical methods. These constraints are (1) as the resulting sound speed is inversely proportional to the Stokes number, it is highly CFL constraining, and (2) the system tends to an advection-diffusion limit equation on the number density that has to be properly approximated by the designed scheme used for the whole range of Stokes numbers. Then, the present work proposes a numerical scheme that is able to handle both. Relying on the ideas introduced in a different context by Chalons, Girardin, and Kokh [C. Chalons, M. Girardin, and S. Kokh, SIAM J. Sci. Comput., 35 (2013), pp. A2874--A2902], a Lagrange projection, a relaxation formulation, and a Harten--Lax--van Leer-Contact scheme with source terms, we extend the approach to a singular flux as well as properly handle the energy equation. The final scheme is proven to be asymptotic-preserving on one-dimensional cases comparing to either converged or analytical solutions and can easily be extended to multidimensional configurations, thus setting the path for realistic applications.