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: 1- ability to reproduce the physics of PTC, at least for a given range of particle inertia, 2- well-posedness of the resulting set of PDEs on the chosen moments as well as guaranteed realizability, 3- capability 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.

Kah et al. (2010) recently developed the Eulerian Multi-Size Moment model (EMSM) which tackles the modeling and numerical simulation of polydisperse multiphase flows. Using a high order moment method in a compact interval, they suggested to reconstruct the number density function (NDF) by Entropy Maximization, which leads to a unique and realizable NDF, potentially in several size intervals, thus leading to an hybrid method between Multifluid and high order. This reconstruction is used to simulate the evaporation process, by an evaluation of the flux of droplet disappearance at zero size, the fluxes of droplets between size intervals, and an accurate description of the size shift induced by evaporation (Massot et al. 2010). Although this method demonstrated its potential for evaporating polydisperse flows, two issues remain to be addressed. First, the EMSM only considers one velocity for all droplets, thus decoupling size from velocity, which is too restrictive for distributions with a large size spectrum. In most applications size-conditioned dynamics have to be accounted for. Second, the possibility to have separated dynamics for each size can lead to quasi-monodisperse distributions, which corresponds to a hard limiting case for the EM algorithm. So the behavior of the algorithm needs to be investigated, in order to reproduce the entire moment space with a reasonable accuracy. The aim of this paper is thus twofold. The EM and its related algorithm are enhanced by using a more accurate integration method in order to handle NDF close to the frontier of the moment space associated with an adaptive number of parameters to reconstruct the NDF accurately and efficiently, as well as tabulated initial guess to optimize the computational time. Then, a new model called CSVM (Coupled Size-Velocity Moments model) is introduced. Size-velocity correlations are addressed either in the evaporation and drag processes, or in the convective transport. To reach this goal, a velocity reconstruction for each size is suggested, using only one additional moment per dimension, and which can be directly applied to several size intervals. Thus, this method is a direct generalization of EMSM. To handle the convective transport, a flux splitting scheme is proposed, based on the underlying kinetic description of the disperse phase. Comparing to existing approaches, a main novelty of the CSVM is that our kinetic approach ensures built-in realizability conditions, no additional corrections of the moments being needed at each time step. The full strategy is first evaluated in 0D and 1D cases, which either demonstrates the ability to reproduce both evaporation, drag force and convection with size-velocity correlations, or the possible extension to several size intervals. Finally, the method is applied on 2D cases with only one section, showing the ability of the CSVM and its related algorithms to capture the main physics of polydisperse evaporating sprays with a minimal number of moments.

Accurate characterization of swirled flames is a key point in the development of more efficient and safer aeronautical engines. The task is even more challenging for spray injection systems. From one side, spray interacts with both turbulence and flame, eventually affecting the flame dynamics. On the other side, the structure of turbulent spray flame is highly complex due to equivalence ratio inhomogeneities caused by evaporation and mixing processes. The first objective of this work is to numerically characterize the structure and dynamics of a swirled spray flame. The target configuration is the experimental benchmark named MERCATO, representative of an actual turbojet injection system. Due to the complex nature of the flame, a detailed description of chemical kinetics is necessary and is here obtained by using a 24-species chemical scheme, which has been expressly developed for DNS of spray flames. The first LES of a swirled spray flame using such a detailed chemical description is performed here and results are analyzed to study the complex interactions between the spray, the turbulent flow and the flame. It is observed that this coupling has an effect on the flame structure and that flame dynamics are governed by the interactions between spray, precessing vortex core and flame front. Even if such a detailed kinetic description leads to an accurate characterization of the flame, it is still highly expensive in terms of CPU time. Tabulated techniques have been expressly developed to account for detailed chemistry at a reduced computational cost in purely gaseous configurations. The second objective is then to verify the capability of the FPI tabulated chemistry method to correctly reproduce the spray flame characteristics by performing LES. To do this, results with the FPI method are compared to the experimental database and to the results obtained with the 24-species description in terms of mean and fluctuating axial gas velocity and liquid phase characteristics (droplet diameter and liquid velocity). Moreover, the flame characterization obtained with the FPI approach is compared to the results of the 24-species scheme focusing on the flame structure, on major and minor species concentrations as well as on pollutant emissions. The potential and the limits of the tabulated approach for spray flame are finally assessed.

The present contribution introduces a fourth-order moment formalism for particle trajectory crossing (PTC) in the framework of multiscale modeling of disperse multiphase flow. In our previous work, the ability to treat PTC was examined with direct-numerical simulations (DNS) using either quadrature reconstruction based on a sum of Dirac delta functions denoted as Quadrature-Based Moment Methods (QBMM) in order to capture large scale trajectory crossing, or by using low order hydrodynamics closures in the Levermore hierarchy denoted as Kinetic-Based Moment Methods (KBMM) in order to capture small scale trajectory crossing. Whereas KBMM leads to well-posed PDEs and has a hard time capturing large scale trajectory crossing for particles with enough inertia, QBMM based on a discrete reconstruction suffers from singularity formation and requires too many moments in order to capture the effect of PTC at both small scale and large scale both to small-scale turbulence as well as free transport coupled to drag in an Eulerian mesoscale framework. The challenge addressed in this work is thus twofold: first, to propose a new generation of method at the interface between QBMM and KBMM with less singular behavior and the associated proper mathematical properties, which is able to capture both small scale and large scale trajectory crossing, and second to limit the number of moments used for applicability in 2-D and 3-D configurations without losing too much accuracy in the representation of spatial fluxes. In order to illustrate its numerical properties, the proposed Gaussian extended quadrature method of moments (Gaussian-EQMOM) is applied to solve 1-D and 2-D kinetic equations representing finite-Stokes-number particles in a known turbulent fluid flow.

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox [J. Comput. Phys. 230 (22), 8216-8246 (2011)] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in the corresponding kinetic equation. The CQMOM closure for the flux leads to a weakly hyperbolic system of moment equations. In subsequent work [Chalons et al., Proceed. CTR Sum. Prog. 2010, 347-358 (2010)], the Dirac delta functions were replaced by Gaussian distributions, which make the moment system hyperbolic but at the added cost of dealing with continuous distributions. Here, a hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions. While the moment set employed for multi-Gaussian and conditional HyQMOM (CHyQMOM) are equivalent, the latter is able to access all of moment space whereas the former cannot (e.g. arbitrary values of the fourth-order velocity moment in 1-D phase space with two nodes). By making use of the properties of CHyQMOM in 2-D phase space, it is possible to control a symmetrical subset of the optimal moments [Fox, Indust. & Engng. Chem. Res. 48 (21), 9686-9696 (2009)]. Furthermore, the moment sets for 2-D problems are smaller for CHyQMOM than in the original CQMOM thanks to a judicious choice of the velocity abscissas in phase space.

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International audienceThe understanding of spray combustion processes is of primary importance, as it is encountered in a wide range of industrial applications. In the present work, mesoscale-resolved simulations of a 3D turbulent counterflow spray configuration are conducted. Primary focus is on examining the effect of the coupling between turbulence, evap- oration, mixing, and combustion. By considering different initial droplet diameters and through comparisons with turbulent and laminar configurations at the same operating condition, it is shown that preferential concentration can lead to conditions of locally high mixture-fraction composition. In addition, local variability in strain rate and droplet diameter introduces a bifurcation of the spray flame. This bifurcation consists of spray flame structures exhibiting single-reaction or double-reaction structures. It is shown that this bimodal behavior is linked to the existence of a hysteresis in the laminar spray flame structure for droplet diameter variations, as well as the occurrence of a bifur- cation for strain rate variations. These results have direct implications for flamelet-based tabulation methods, since identifying the appropriate flamelet structure in turbulent spray flames would require informations about boundary conditions and the flamelet history

The design of new low-emission systems requires the development of models providing an accurate prediction of soot production for a small computational cost. In this work, a three-equation model is developed based on mono-disperse closure of the source terms from a sectional method. In addition, a post-processing technique to estimate the particles size distribution (PSD) from global quantities is proposed by combining Pareto and log-normal distributions. After validation, the developed strategy is used to perform a large eddy simulation of soot production in a model combustor representative of gas turbine combustion chambers. It is shown that the three-equation model is able to provide a good estimation of soot volume fraction and information on PSD in complex geometries for a low computational time.