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 (eg, fuel injection in automotive or aeronautical engines, fluidized beds, and alumina particles in rocket boosters). Disciplines Aerospace Engineering | Biological Engineering | Chemical Engineering | Mechanical Engineering Comments This article is from Center for Turbulence Research Annual Research Briefs 2012: 193-204. Posted with permission. Authors Frederique Laurent, Aymeric Vie, Christophe Chalons, Rodney O. Fox, and Marc Massot This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/cbe_pubs/120 Center for Turbulence Research Annual Research Briefs 2012 193 A hierarchy of Eulerian models for trajectory crossing in particle-laden turbulent flows over a wide range of Stokes numbers By F. Laurent, A. Vié, C. Chalons, R. O. Fox AND M. Massot 1. Motivation and objective 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 MonteCarlo 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 secondorder 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. 2008; Kah et al. 2010; Chalons et al. 2010; Yuan & Fox 2011; Chalons et al. 2012; Vié et al. 2011, 2012b,a). 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 to reconstruct the NDF from the moments, but also has to be based on physical arguments. In the context of small-scale PTC, the anisotropic Gaussian (AG) closure designed by Vié et al. (2012a) 194 F. Laurent et al. can be used, which controls all second-order moments. A comparison between ACBMM and AG is provided in Vié et al. (2012b). For large-scale PTC, a sum of Dirac δ-functions can be used to account for the multi-modal velocity distribution at the crossing location. In 3-D, the conditional quadrature method of moments (CQMOM) of Yuan & Fox (2011) provides a fast inversion algorithm, and has demonstrated its ability in Taylor-Green flows. However, CQMOM has an important drawback: the related system of equations is weakly hyperbolic, meaning that it can generate unphysical δ-shocks. They refer to rapid accumulation of particles, when more than two trajectories cross, as mathematically analyzed by Chalons et al. (2012). To overcome this issue, Chalons et al. (2010) have proposed the multi-Gaussian (MG) quadrature. Instead of using Dirac δ-functions as a kernel, Gaussian functions are used. In 1-D, by assuming equal variances for each Gaussian node, these authors provide a direct algorithm for the inversion. The resulting system of equations has three main properties: (1) it is hyperbolic, (2) in the limit of one node it degenerates towards the Gaussian distribution, ensuring the description of small-scale and large-scale PTC, and (3) in the context of LES, it can account for the velocity dispersion induced by subgrid scales of the turbulence thanks to the Gaussian kernel. In 2-D and 3-D, the MG algorithm has been extended using the idea behind CQMOM (Vié et al. 2011). The final method has been evaluated in Taylor-Green flow, where it has shown its potential. The aim of this work is to apply the MG quadrature to more complex 2-D cases, to demonstrate its ability to avoid δ-shocks, and to capture additional dynamics as compared to standard KBMM using AG or CQMOM. We begin by introducing the kinetic equation and its related moment problem. Then we describe briefly the 2-D MG quadrature that is used to close the moment problem and the related algorithms used for moment evolution. Finally, the full method is applied on 2-D frozen homogeneous isotropic turbulence (HIT), and compared to AG, CQMOM and Lagrangian results. 2. The moment problem In this work, we solve for the NDF f(t, x, v), at time t as a function of droplet position x = (x, y) and velocity v = (u, v), using a 2-D kinetic equation: ∂tf + v · ∂xf + ∂v · (Ff) = 0, t > 0, x ∈ R, v ∈ R, (2.1) where ∂x = (∂x, ∂y) , ∂v = (∂u, ∂v) t and F = (Fu, Fv) t is the acceleration due to the drag force. As we consider Stokes drag, the acceleration due to drag is F = (vg − v)/τp, where vg = (ug, vg) t is the gas velocity and τp, the relaxation time. We define the bivariate moments of order (i, j) as Mi,j(t, x) = ∫ R uvf(t, x, v) dv. The associated governing equations are easily obtained from Eq. (2.1): ∂tMi,j + ∂xMi+1,j + ∂yMi,j+1 = 1 τp [iMi−1,jug + jMi,j−1vg − (i + j)Mi,j ] . (2.2) There are two main issues with this type of model: the closure of the moment hierarchy and the numerical scheme. Indeed, for any given set of moments, there are always some fluxes that are not in closed form. To close the system, we use a reconstruction of the NDF at the kinetic level starting from a finite moment set. The choice of the reconstruction imposes the number of moments needed and induces the hyperbolicity or weak hyperbolicity property of the system. Moreover a kinetic finite volume scheme is used for convection, which makes use of the reconstruction to compute the unknown fluxes. For Eulerian models for particle trajectory crossing 195 the drag force terms, all moments are closed. However, the reconstruction is still useful to derive a quadrature approximation of these terms in case of arbitrary drag laws. 3. 2-D extended quadrature-based reconstruction Treatment of particle-laden flows with LES over a wide range of Stokes numbers requires a quadrature reconstruction that can treat both large-scale and small-scale PTC. In this work we employ a 2-D extended version of the MG quadrature developed by Chalons et al. (2010), which can degenerate into two simpler quadratures, namely CQMOM (Yuan & Fox 2011) and anisotropic Gaussian (AG) (Vié et al. 2012a). 3.1. MG quadrature The 2-D MG extended quadrature is constructed using the conditional moment algorithm described by Yuan & Fox (2011) and is found by conditioning, for example, the v velocity on the u velocity, or more precisely, conditioning a linear combination v−a0−a1u on u†: