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a b s t r a c tPopulation balance equations (PBE) for a number density function (NDF) arise in many applications of aerosol technology. Thus, there has been considerable interest in the development of numerical methods to find solutions to PBE, especially in the context of spatially inhomogeneous systems where moment realizability becomes a significant issue. Quadrature-based moment methods (QBMM) are an important class of methods for which the accuracy of the solution can be improved in a controlled manner by increasing the number of quadrature nodes. However, when a large number of nodes is required to achieve the desired accuracy, the moment-inversion problem can become ill-conditioned. Moreover, oftentimes pointwise values of the NDF are required, but are unavailable with existing QBMM. In this work, a new generation of QBMM is introduced that provides an explicit form for the NDF. This extended quadrature method of moments (EQMOM) approximates the NDF by a sum of non-negative weight functions, which allows unclosed source terms to be computed with great accuracy by increasing the number of quadrature nodes independent of the number of transported moments. Here, we use EQMOM to solve a spatially homogeneous PBE with aggregation, breakage, condensation, and evaporation terms, and compare the results with analytical solutions whenever possible. However, by employing realizable finite-volume methods, the extension of EQMOM to spatially inhomogeneous systems is straightforward.

Abstract. In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which include aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function NDF of particles/droplets which satisfies a population balance equation. (PBE, also called Williams equation in the spray community). In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinate, and its influence on moment dynamics results in a difficulty from both a modeling and a numerical point of view. We obtain in the present contribution a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description which will combine both the flexibility of Eulerian models for which the size phase space is discretized into "sections" (i.e. size intervals) and the efficiency of Direct Quadrature Method of Moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables which should lead to a low computational cost in multi-dimensional configurations.

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In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gasdroplet flows for which polydispersity description is of paramount importance. Starting from a kinetic description for point particles experiencing transport either at the carrier phase velocity for aerosols or at their own velocity for sprays as well as evaporation, we focus on an Eulerian high order moment method in size and consider a system of partial differential equations (PDEs) on a vector of successive integer size moments of order 0 to N , N > 2, over a compact size interval. There exists a stumbling block for the usual approaches using high order moment methods resolved with high order finite volume methods: the transport algorithm does not preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells coupled to the evolution algorithm can create N -dimensional vectors which fail to be moment vectors: it is impossible to find a size distribution for which there are the moments. We thus propose a new approach as well as an algorithm which is second order in space and time with very limited numerical diffusion and allows to accurately describe the advection process and naturally preserves the moment space. The algorithm also leads to a natural coupling with a recently designed algorithm for evaporation which also preserves the moment space; thus polydispersity is accounted for in the evaporation and advection process, very accurately and at a very reasonable computational cost. These modeling and algorithmic tools are referred to as the EMSM (Eulerian Multi Size Moment) model. We show that such an approach is very competitive compared to multi-fluid approaches, where the size phase space is discretized into several sections and low order moment methods are used in each section, as well as with other existing high order moment methods. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. Whereas the extension to unstructured meshes is provided, we focus in this paper on cartesian meshes and two 2D test-cases are presented: Taylor-Green vortices and turbulent free jets, where the accuracy and efficiency of the approach are assessed.

The scope of the present study is Eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation. The fundamental mathematical description is the Williams spray equation governing the joint number density function f (v, u; x, t) of droplet volume and velocity. Eulerian multi-fluid models have already been rigorously derived from this equation in Laurent et al. [22]. The first key feature of the paper is the application of direct quadrature method of moments (DQMOM) introduced by Marchisio and Fox [24] to the Williams spray equation. Both the multi-fluid method and DQMOM yield systems of Eulerian conservation equations with complicated interaction terms representing coalescence. In order to focus on the difficulties associated with treating size-dependent coalescence and to avoid numerical uncertainty issues associated with two-way coupling, only one-way coupling between the droplets and a given gas velocity field is considered. In order to validate and compare these approaches, the chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with sprays having various size distributions, ranging from smooth ones up to Dirac delta functions. The second key feature of the paper is a thorough comparison of the two approaches for various test-cases to a reference solution obtained through a classical stochastic Lagrangian solver. Both Eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the Lagrangian solver. The third key point of the study is a detailed description of the limitations associated with each method, thus giving criteria for their use as well as for their respective efficiency.

The accurate description and robust simulation, at relatively low cost, of global quantities (e.g. number density or volume fraction) as well as the size distribution of a population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, two types of methods are investigated for solving the population balance equation with aggregation, continuous particle size change (growth and size reduction), and nucleation: the extended quadrature method of moments (EQMOM) based on the work of Yuan et al.[52]and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al.[30]. For both methods, the closure employs a continuous reconstruction of the number density function of the particles from its moments, thus allowing evaluation of all the unclosed terms in the moment equations, including the negative flux due to the disappearance of particles. Here, new robust and efficient algorithms are developed for this reconstruction step and two kinds of reconstruction are tested for each method. Moreover, robust and accurate numerical methods are developed, ensuring the realizability of the moments. The robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebraic constraints thanks to a tailored overall strategy. EQMOM and TSM are compared to a sectional method for various simple but relevant test cases, showing their ability to describe accurately the fine-particle population with a much lower number of variables. These results demonstrate the efficiency of the modeling and numerical choices, and their potential for the simulation of real-world applications.

Abstract. We tackle the numerical simulation of reaction-diffusion equations modeling multiscale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. It considers a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Based on recent theoretical studies of numerical analysis such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by stability constraints of the diffusive steps, but only by the physics of the phenomenon. We aim thus at solving complete models including all time and space scales within a prescribed accuracy, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far, out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy to more complex configurations involving other physical phenomena as well as optimization capability on new computer architectures are finally discussed. Key words.Reaction-diffusion equations, multi-scale reaction waves, operator splitting, adaptive multiresolution AMS subject classifications. 33K57, 35A18, 65M50, 65M081. Introduction. Numerical simulations of multi-scale phenomena are commonly used for modeling purposes in many applications such as combustion, chemical vapor deposition, or air pollution modeling. In general, all these models raise several difficulties created by the high number of unknowns, the wide range of temporal scales due to large and detailed chemical kinetic mechanisms, as well as steep spatial gradients associated with very localized fronts of high chemical activity. Furthermore, a natural stumbling block to perform 3D simulations with all scales resolution is either the unreasonably small time step due to stability requirements or the unreasonable memory requirements for implicit methods. In this context, one can consider various numerical strategies in order to treat the induced stiffness for time dependent * This research was supported by a fundamental project grant from ANR (French National Research Agency -ANR Blancs) Séchelles (project leader S. Descombes -2009Descombes - -2013, by a CNRS PEPS Maths-ST2I project MIPAC (project leader V. Louvet -2009Louvet - -2010, and by a DIGITEO RTRA project MUSE (project leader M. Massot -2010Massot - -2014 problems. The most natural id...