Algebraic-Closure-Based Moment Method for Unsteady Eulerian Simulations of Non-Isothermal Particle-Laden Turbulent Flows at Moderate Stokes Numbers in Dilute Regime

Abstract

To model unsteady non-isothermal dilute particle-laden turbulent flows,an algebraic-closure-based moment method (ACBMM) is developed. ACBMM is a Eulerian approach for the dispersed phase conceived to be coupled with direct numerical simulations (DNSs) of the turbulence when an accurate local description of the turbulent mixture is required. It is based on the combination of a conditional probability density function (PDF) approach, which provides local instantaneous Eulerian equations for the low-order moments of the PDF, and appropriate constitutive relations, as algebraic closures, which are necessary to close the set of conservation equations. The computed low-order moments are the mesoscopic particle number density, particle velocity and particle temperature and the unclosed higher order moments are the particle random uncorrelated motion (RUM) stress tensor and the RUM heat flux (RUM-HF) which appear in the particle momentum and enthalpy equations, respectively. The RUM stress tensor is closed by an additional transport equation for the trace of the tensor and a polynomial representation for tensor functions modeling its deviatoric part. The polynomial representation is used in the framework of an assumption of equilibrium of the RUM anisotropy and leads to an explicit algebraic stress model (2EASM). Similarly, the RUM-HF is modeled assuming equilibrium of the scaled heat flux and explicit self-consistent solutions(2EAHFM) are found by analogy with turbulent heat flux models. As 2EAHFM entails the computation of the RUM temperature variance, an additional transport equation is developed for it. By means of an a priori analysis, the algebraic closures developed by the present study are assessed against actual particle Eulerian fields which are extracted from particle Lagrangian simulations coupled with DNS of a temporal non-isothermal particle-laden turbulent planar jet, for various Stokes numbers. Results show that both 2EASM and 2EAHFM are successful in reproducing the unclosed moments up to moderate turbulent-macroscale Stokes numbers allowing the ACBMM to accurately predict the unsteady non-isothermal dispersed phase.