Development of an algebraic-closure-based moment method for unsteady Eulerian simulations of particle-laden turbulent flows in very dilute regime

• We modeled particles by an unsteady Eulerian model coupled with DNS of fluid.
• We developed closures for the particle kinetic stress tensor.
• The most fruitful closures are explicit algebraic stress models.
• They are based on a self-similarity assumption of the stress tensor.
• These closures enable the approach to model up to large Stokes numbers in dilute regime.

An algebraic-closure-based moment method (ACBMM) is developed for unsteady Eulerian particle simulations, coupled with direct numerical simulations (DNSs) of fluid turbulent flows, in very dilute regime and up to large Stokes numbers StK (based on the Kolmogorov timescale) or moderate Stokes numbers St (based on the turbulent macroscale seen by the particles). The proposed method is developed in the frame of a conditional statistical approach which provides a local and instantaneous characterization of the dispersed-phase dynamic accounting for the effect of crossing between particle trajectories which becomes substantial for StK > 1. The computed Eulerian quantities are low-order moments of the conditional probability density function (PDF) and the corresponding governing equations are derived from the PDF kinetic equation in the general frame of the kinetic theory of gases. At the first order, the computation of the mesoscopic particle number density and velocity requires the modeling of the second-order moment tensor appearing in the particle momentum equation and referred to as random uncorrelated motion (RUM) particle kinetic stress tensor. The current work proposes a variety of different algebraic closures for the deviatoric part of the tensor. An evaluation of some effective propositions is given by performing an a priori analysis using particle Eulerian fields which are extracted from particle Lagrangian simulations coupled with DNS of a temporal particle-laden turbulent planar jet. Several million-particle simulations are analyzed in order to assess the models for various Stokes numbers. It is apparent that the most fruitful are explicit algebraic stress models (2ΦEASM) which are based on an equilibrium assumption of RUM anisotropy for which explicit solutions are provided by means of a polynomial representation for tensor functions. These models compare very well with Eulerian–Lagrangian DNSs and properly represent all crucial trends extracted from such simulations.