A hierarchy of Eulerian models for trajectory crossing in particle-laden turbulent flows over a wide range of Stokes numbers

With the large increase in available computational resources, large-eddy simulation (LES) of industrial configurations has become an efficient and tractable alternative to traditional multiphase turbulence models. Many applications involve a liquid or solid disperse phase carried by a gas phase (e.g., fuel injection in automotive or aeronautical engines, fluidized beds, and alumina particles in rocket boosters). To simulate such flows, one may resort to a number density function (NDF) for the disperse phase that satisfies a kinetic equation. Solving for the NDF can make use of Lagrangian Monte-Carlo methods, but such approaches are expensive, especially for unsteady flows, as the amount of numerical particles needed to control statistical errors is large. Moreover, such methods are not well adapted to high-performance computing because of the intrinsic spatial inhomogeneity of the NDF in most of the applications of interest. To overcome these issues, one can resort to Eulerian methods that solve for the moments of the NDF using an Eulerian system of conservation laws. In the context of direct-numerical simulation (DNS), Février et al. (2005) introduced the mesoscopic Eulerian formalism (MEF). It yields a statistical decomposition of the motion of particles into correlated and uncorrelated parts, the former being common to all particles at a specific location, and the latter being induced by the history of each particle as it crosses different vortices before reaching a specific location. This decomposition induces unclosed stresses in the moment conservation equations. In Kaufmann et al. (2008) and more recently in Masi et al. (2011), algebraic-closure-based moment methods (ACBMM) are used to close these stresses: while solving for the random uncorrelated energy (i.e., granular energy), they provide constitutive closures to model the second-order moments. These closures are efficient at moderate Stokes numbers (Dombard 2011; Sierra 2012), for which particle trajectory crossings (PTC) occur at small scales, which are efficiently reproduced by second-order moments. However, at high Stokes numbers, this description of PTC is not satisfactory, as the correlated part of the motion will encounter large-scale PTC, the accurate representation of which requires high-order moments methods. To solve for high-order moments, kinetics-based moment methods (KBMM) can be employed (Desjardins et al. The main idea behind KBMM is to provide a presumed velocity distribution at the kinetic level potentially conditioned on size, which has as many parameters as the required number of moments. The presumed profile must be chosen carefully and should lead to simple algorithms in order …