The present contribution introduces a fourth-order moment formalism forparticle trajectory crossing (PTC) in the framework of multiscale modeling ofdisperse multiphase flow. In our previous work, the ability to treat PTC wasexamined with direct-numerical simulations (DNS) using either quadraturereconstruction based on a sum of Dirac delta functions denoted asQuadrature-Based Moment Methods (QBMM) in order to capture large scaletrajectory crossing, or by using low order hydrodynamics closures in theLevermore hierarchy denoted as Kinetic-Based Moment Methods (KBMM) in order tocapture small scale trajectory crossing. Whereas KBMM leads to well-posed PDEsand has a hard time capturing large scale trajectory crossing for particleswith enough inertia, QBMM based on a discrete reconstruction suffers fromsingularity formation and requires too many moments in order to capture theeffect of PTC at both small scale and large scale both to small-scaleturbulence as well as free transport coupled to drag in an Eulerian mesoscaleframework. The challenge addressed in this work is thus twofold: first, topropose a new generation of method at the interface between QBMM and KBMM withless singular behavior and the associated proper mathematical properties, whichis able to capture both small scale and large scale trajectory crossing, andsecond to limit the number of moments used for applicability in 2-D and 3-Dconfigurations without losing too much accuracy in the representation ofspatial fluxes. In order to illustrate its numerical properties, the proposedGaussian extended quadrature method of moments (Gaussian-EQMOM) is applied tosolve 1-D and 2-D kinetic equations representing finite-Stokes-number particlesin a known turbulent fluid flow.
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