Multivariate Gaussian extended quadrature method of moments for turbulent disperse multiphase flow

The present contribution introduces a fourth-order moment formalism for particle trajectory crossing (PTC) in the framework of multiscale modeling of disperse multiphase flow. In our previous work, the ability to treat PTC was examined with direct-numerical simulations using either quadrature reconstruction based on a sum of Dirac delta functions denoted as quadrature-based moment methods (QBMM) in order to capture large scale trajectory crossing, or by using low-order hydrodynamics closures in the Levermore hierarchy denoted as kinetic-based moment methods (KBMM) in order to capture small-scale trajectory crossing. Whereas KBMM leads to well-posed PDEs and has a hard time capturing large-scale trajectory crossing for particles with enough inertia, QBMM based on a discrete reconstruction suffers from singularity formation and requires too many moments in order to capture the effect of PTC at both the small scale and the large scale both to small-scale turbulence as well as free transport coupled to drag in an Eulerian mesoscale framework. The challenge addressed in this work is thus twofold: first, to propose a new generation of method at the interface between QBMM and KBMM with less singular behavior and the associated proper mathematical properties, which is able to capture both small-scale and large-scale trajectory crossing, and second to limit the number of moments used for applicability in two-dimensional (2-D) and 3-D configurations without losing too much accuracy in the representation of spatial fluxes. In order to illustrate its numerical properties, the proposed Gaussian extended quadrature method of moments is applied to solve 1-D and 2-D kinetic equations representing finite-Stokes-number particles in a known turbulent fluid flow.