On the Anisotropic Gaussian velocity closure for inertial-particle laden flows

The accurate simulation of disperse two-phase flows, where a discrete particulate condensed phase is transported by a carrier gas, is crucial for many applications; Eulerian approaches are well suited for high performance computations of such flows. However when the particles from the disperse phase have a significant inertia compared to the time scales of the flow, particle trajectory crossing (PTC) occurs i.e. the particle velocity distribution at a given location can become multi-valued. To properly account for such a phenomenon many Eulerian moment methods have been recently proposed in the literature. The resulting models hardly comply with a full set of desired criteria involving: 1ability to reproduce the physics of PTC, at least for a given range of particle inertia, 2well-posedness of the resulting set of PDEs on the chosen moments as well as guaranteed realizability, 3capability of the model to be associated with a high order realizable numerical scheme for the accurate resolution of particle segregation in turbulent flows. The purpose of the present contribution is to introduce a multi-variate Anisotropic Gaussian closure for such particulate flows, in the spirit of the closure that has been suggested for out-of-equilibrium gas dynamics and which satisfies the three criteria. The novelty of the contribution is three-fold. First we derive the related moment system of conservation laws with source terms, and justify the use of such a model in the context of high Knudsen numbers, where collision operators play no role. We exhibit the main features and advantages in terms of mathematical structure and realizability. Then a second order accurate and realizable MUSCL/HLL scheme is proposed and validated. Finally the behavior of the method for the description of PTC is thoroughly investigated and its ability to account accurately for inertial particulate flow dynamics in typical configurations is assessed. AMS subject classifications: 76T10, 76P05, 82C21, 65M08, 35L40, 52B55 ∗Corresponding author. Email addresses: [email protected], [email protected] (A. Vié, presently at CTR, Stanford University, formerly postdoctoral fellow at EM2C Laboratory (2011-2013)), [email protected] (F. Doisneau), [email protected] (M. Massot, Visiting Professor at Stanford University, CTR (2011-2012)).