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International audience; Lagrangian simulations are today widely used for simulating aeronautical chambers. The way droplets are spatially distributed strongly affects the combustion, and accurate modeling and simulation strategies are required. The objective of the present contribution is to investigate how to correctly reproduce preferential concentration in Large Eddy Simulation (LES) of particleladen flows. Looking for a way to recover the DNS statistics, we highlight that stochastic models can fail in retrieving the tracer limit for non-inertial particles. We suggest a new strategy in the spirit of kinematic modeling of turbulence, which makes use of a random field with enforced divergence-free condition and spatial and temporal correlations. We show that the model can retrieve some Lagrangian statistics and in particular, particle segregation. We also suggest another approach for the LES particle model, based on retrieving not DNS statistics but filtered DNS statistics. We show that in this case, stochastic models can be relevant and appropriate.

International audience; The modulation of turbulence by sub-Kolmogorov particles has been thoroughly characterized in the literature, showing either enhancement or reduction of kinetic energy at small or large scale depending on the Stokes number and the mass loading. However , the impact of a third parameter, the number density of particles, has not been independently investigated. In the present work, we perform direct numerical simulations of decaying Homogeneous Isotropic Turbulence loaded with monodisperse sub-Kolmogorov particles, varying independently the Stokes number, the mass loading and the number density of particles. Like previous investigators, crossover and modulations of the fluid energy spectra are observed consistently with the change in Stokes number and mass loading. Additionally, DNS results show a clear impact of the particle number density, promoting the energy at small scales while reducing the energy at large scales. For high particle number density, the turbulence statistics and spectra become insensitive to the increase of this parameter, presenting a two-way asymptotic behavior. Our investigation identifies the energy transfer mechanisms, and highlights the differences between the influence of a highly concentrated disperse phase (high particle number density, limit behavior) and that of heterogeneous concentration fields (low particle number density). In particular, a measure of this heterogeneity is proposed and discussed which allows to identify specific regimes in the evolution of turbulence statistics and spectra.

International audience; A complete characterisation of the moment space corresponding to the Levermore basis is given here, through constraints on the moments. The necessary conditions are obtained thanks to classical tools, similar to Hankel determinants. In the mono-variate case, it is well-known that these conditions are sufficient. To generalize this result to multi-variate case, a non-classical constructive proof is given here reducing the problem to several mono-variate ones. However, it is also shown here on an example that the obtained multi-variate closure does not necessarily inherit of the good properties of the mono-variate closure.; Caractérisation de l'espace des moments correspondantà la base de Levermore. Une caractérisation complète de l'espace des moments correspondant à la base de Levermore est donnée ici,à travers des contraintes sur les moments. Les conditions nécessaires sont obtenues grâce à des outils classiques, similaires aux déterminants de Hankel. Dans le cas mono-varié, il est bien connu que ces conditions sont suffisantes. Pour généraliser ce résultat a un cas multi-varié, une preuve constructive non classique est donnée ici en se ramenant à des problèmes mono-variés. Cependant, il est également montré ici, sur un exemple, que la fermeture obtenue dans le cas multi-varié n'hérite pas nécessairement des bonnes propriétés de la fermeture mono-variée.

International audience; A hyperbolic two-fluid model for gas-particle flow derived using the Boltzmann-Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum, and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities Z = ρ f /ρ p. An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary Z is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the 1-D model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase.

International audience; The second-order realizable moment advection scheme developed in Laurent and Nguyen, (2017) is extended to the case of unstructured grids with cells of arbitrary shape. The necessary modifications to the scheme and the conditions under which the scheme ensures the realizability of the advected moment set are presented. The implementation of the scheme in the OpenFOAM ® CFD toolbox is verified comparing the results obtained in one-dimensional test cases involving moment sets well inside the moment space, and at the boundary of the moment space. Results obtained with the proposed scheme are compared to the corresponding analytical solution. The scheme is then tested considering two-dimensional cases of pure moment advection with an imposed irrotational velocity field. First, a quadrilateral grid is considered to determine the order of the scheme and compare it to the results reported in Laurent and Nguyen (2017) with the same grid resolution. Then, the accuracy of the scheme on two-dimensional triangular grids is determined.

International audience; 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.

In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization.

High fidelity modeling and simulation of moderately dense sprays at relatively low cost is still a major challenge for many applications. For that purpose, we introduce a new multi-fluid model based on a two-size moment formalism in sections, which are size intervals of discretization. It is derived from a Boltzmann type equation taking into account drag, evaporation and coalescence, which are representative of the complex terms that arise in multi-physics environments. The closure of the model comes from a reconstruction of the distribution. A piecewise affine reconstruction in size is thoroughly analyzed in terms of stability and accuracy, a key point for a high-fidelity and reliable description of the spray. Robust and accurate numerical methods are then developed, ensuring the realizability of the moments. The model and method are proven to describe the spray with a high accuracy in size and size-conditioned variables, resorting to a lower number of sections compared to one size moment methods. Moreover, robustness is ensured with efficient and tractable algorithms despite the numerous couplings and various algebra thanks to a tailored overall strategy. This strategy is successfully tested on a difficult 2D unsteady case, which proves the efficiency of the modeling and numerical choices.

The extended quadrature method of moments (EQMOM) for the solution of population balance equations (PBE) is implemented in the open-source computational fluid dynamic (CFD) toolbox OpenFOAM as part of the OpenQBMM project. The moment inversion procedure was designed (Nguyen et al., 2016) to maximize the number of conserved moments in the transported moment set. The algorithm is implemented in a general structure to allow the addition of other kernel density functions defined on R + , and arbitrary kernels to describe physical phenomena involved in the evolution of the number density function. The implementation is verified with a set of zero-dimensional cases involving aggregation and breakage problems. Comparison to the rigorous solution of the PBE provides validation for these cases. The coupling of the EQMOM procedure with a CFD solver to address aggregation and breakage problems of non-inertial particles is validated against experimental measurements in a Taylor-Couette reactor from the literature.

International audience; Relying on two recent contributions by Massot et al. [SIAM J. Appl. Math. 70 (2010), 3203--3234] and Kah et al. [J. Comput. Phys. 231 (2012), 394--422], where a Eulerian Multi-Size Moment (EMSM) model for the simulation of polydisperse evaporating sprays has been introduced, we investigate the potential of such an approach for the robust and accurate simulation of the injection of a liquid disperse phase into a gas for automotive engine applications. The original model used a high order moment method in droplet size to resolve polydispersity, with built-in realizability preserving numerical algorithm of high order in space and time, but only dealt with one-way coupling and was restricted to fixed meshes. Extending the approach to internal combustion engine and fuel injection requires solving two major steps forward, while preserving the properties of robustness, accuracy and realizability: 1- the extension of the method and numerical strategy to two-way coupling with stable integration of potential stiff source terms, 2- the introduction of a moving geometry and meshes. We therefore present a detailed account on how we have solved these two issues, provide a series of verification of the proposed algorithm, showing its potential in simplified configurations. The method is then implemented in the IFP-C3D unstructured solver for reactive compressible flows in engines and validated through comparisons with a structured fixed mesh solver; it finally proves its potential on a free spray jet injection where it is compared to a Lagrangian approach and its reliability and robustness are assessed, thus making it a good candidate for realistic injection applications.

Wittgenstein’s notions of use and meaning are compared with the corresponding Heidegger’s ones. Unlike Jocelyn Benoist, we suggest that the analogy between Wittgenstein and Heidegger is not superficial. Heidegger’s metaphysics makes explicit some of the implicit presuppositions of Wittgenstein’s philosophy. Wittgenstein’s naturalistic pragmatism can be theorized. In particular, Wittgenstein’s notion of use, or language game, can be understood as both natural and normative rulegoverning practice.

International audience; To assist industrial engine design, 3D simulations are increasingly used as they allow evaluation of a wide range of engine configurations and operating conditions and bring a comprehension of the underlying physics comple-mentary to experiments. While the gaseous flow description has reached a certain level of maturity, the multiphase flow description involving the liquid jet fuel injected into the chamber still faces some major challenges. There is a pressing need for a spray model that is time efficient and accurately describes the fuel-particle cloud dynamics downstream of the injector, which is an essential prerequisite for predictive combustion simulations. Due to the highly unsteady nature of the flow following the high-pressure injection process and the complexity of the flow regimes from separated/dense compressible phases to fully developed turbulent spray with evaporating droplets, Eulerian-Eulerian descriptions of two-phase flows are seen as very promising approaches towards realistic and predictive simulations of the mixing process. However they require some effort in terms of physical modeling and numerical analysis related to the more complex mathematical structure of the system of equations and to the unclosed terms appearing in space/time-average equations. Among the various challenges faced, one critical as-pect is to capture spray polydispersity in this framework. A review of recent developments that have permitted key advances in the spray modeling community is proposed in this paper. It is divided into four parts. First, an introduction to automotive spray modeling is provided. Then the formalisms for the description of the disperse region of an engine spray are presented with particular emphasis on the pros and cons of classical Lagrangian par-ticle methods versus Eulerian approaches. The third part presents the motivation for and the recent developments of Eulerian high-order moment methods for size polydispersion. Finally, the extension to fully two-way coupled interactions with the gas phase and the implementation of such methods for variable-geometry applications in CFD codes is described in the fourth part. Using realistic direct injection conditions computed with the IFP-C3D solver, the application and efficiency of Eulerian approaches is illustrated.

International audience; The accurate description and robust simulation at relatively low cost of a size polydisperse population of fine particles in a carrier fluid is still a major challenge for many applications. For this purpose, moment methods, derived from a population balance equation, represent a very interesting strategy. However, one of the major issues of such methods is the realizability: the numerical schemes have to ensure that the moment sets stay realizable, i.e. that an underlying distribution exists. This issue is all the more crucial that some moment vectors can be at the boundary of the moment space for practical applications, corresponding to a population of particles with only one or a few sizes. It is then investigated here for the advection operator, for which it is particularly significant. Then second order realizable kinetic finite volume schemes are designed, with two strategies for the fluxes evaluation based on the work of Kah et al. (J. Comput. Phys., 231:394-422, 2012) and of Vikas et al. (J. Comput. Phys., 230:5328-5352, 2011), which are here completely revisited, extended to take into account the boundary of the moment space and any number of moments, analyzed and compared in a Cartesian mesh context. For a potential easiest generalization to un-structured meshes, simplified but still realizable versions of these schemes are also developed. The high accuracy of all the schemes is then numerically checked on 1D and 2D test cases, with Cartesian meshes, and their robustness is shown, even when some moments vectors are at the boundary of the moment space.

International audience; In this paper, we tackle the issue of the accurate simulation of evaporating and reactive polydisperse sprays strongly coupled to unsteady gaseous flows. In solid propulsion, aluminum particles are included in the propellant to improve the global performances but the distributed combustion of these droplets in the chamber is suspected to be a driving mechanism of hydrodynamic and acoustic instabilities. The faithful prediction of two-phase interactions is a determining step for future solid rocket motor optimization. When looking at saving computational ressources as required for industrial applications, performing reliable simulations of two-phase flow instabilities appears as a challenge for both modeling and scientific computing. The size polydispersity, which conditions the droplet dynamics, is a key parameter that has to be accounted for. For moderately dense sprays, a kinetic approach based on a statistical point of view is particularly appropriate. The spray is described by a number density function and its evolution follows a Williams-Boltzmann transport equation. To solve it, we use Eulerian Multi-Fluid methods, based on a continuous discretization of the size phase space into sections, which offer an accurate treatment of the polydispersion. The objective of this paper is threefold: first to derive a new Two Size Moment Multi-Fluid model that is able to tackle evaporating polydisperse sprays at low cost while accurately describing the main driving mechanisms, second to develop a dedicated evaporation scheme to treat simultaneously mass, moment and energy exchanges with the gas and between the sections. Finally, to design a time splitting operator strategy respecting both reactive two-phase flow physics and cost/accuracy ratio required for industrial computations. Using a research code, we provide 0D validations of the new scheme before assessing the splitting technique's ability on a reference two-phase flow acoustic case. Implemented in the industrial oriented CEDRE code, all developments allow to simulate realistic solid rocket motor configurations featuring the first polydisperse reactive computations with a fully Eulerian method.

International audience; Modeling the oxidation of soot particles in flames is a challenging topic both from a chemical point of view and regarding the statistical treatment of the evolution of the soot number density function (NDF). The method of moments is widely-used for the statistical modeling of aerosol dynamics in various applications, and a number of different moment methods have been established and successfully applied to the modeling of soot formation and growth. However, a shortcoming of existing moment methods is the lack of an accurate, numerically robust, and computationally efficient way to treat soot oxidation, especially regarding the prediction of the particle number density. In this work, the recently developed Extended Quadrature Method of Moments (EQMOM) is integrated with a physico-chemical soot model and combined with a treatment for particle removal by oxidation. This leads to a modeling framework for the simulation of coupled inception, growth, coagulation, and oxidation of soot in flames. In EQMOM, the moment equations are closed by reconstructing the soot NDF with a superposition of continuous kernel functions. Various standard distribution functions can be used as kernel functions, and the algorithm has been implemented here using gamma and lognormal distributions. It is shown that and discussed why gamma distributions are more suitable as kernel functions than lognormal distributions in order to accurately predict soot oxidation. The integrated model is validated by comparisons with analytical solutions for the NDF, results from Monte Carlo simulations of soot formation and oxidation in flames, and experimental data.

The accurate simulation of polydisperse sprays strongly coupled to unsteady gaseous flows is a major issue (e.g., for solid rocket motor optimization). The Eulerian multifluid method has proven to account for polydispersity efficiently by describing continuously droplet sizes that are sorted into “fluids,” which are coupled to the gas through drag and heat source terms. The potential of this model to capture polydisperse two-way interactions has not been addressed. Such an interaction is described through two strongly coupled systems of equations, which involve a large spectrum of scales in both time and space and require specific numerical methods to reach accuracy and predictability with an acceptable computational cost. In this paper, we define the physics and key issues of polydisperse spray acoustics, identify physically relevant test cases, and investigate the abilities of multifluid systems. We also describe the numerical peculiarities related to strong coupling, at high mass loading, of polydisper...

Kah et al. (2010) [30,33] 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) [15]. 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.

International audience; The accurate simulation of polydisperse sprays undergoing coalescence in unsteady gaseous flows is a crucial issue. In solid rocket motors, the internal flow depends strongly on the alumina droplet size distribution, which spreads up with coalescence. Yet solving for unsteady two-phase flows with high accuracy on the droplet sizes is a challenge for both modeling and scientific computing. As an alternative to Lagrangian approaches, a wide range of Eulerian models have been recently developed to describe the disperse liquid phase at a lower cost, with an easier coupling to the gaseous phase and with massively parallel codes. Among these models, the Multi-Fluid model allows the detailed description of polydispersity and size/velocity correlations by separately solving fluids of size-sorted droplets, the so-called sections. The existing One Size Moment method, which describes the size distribution with one size moment per section, provides simple and fast resolution for coalescence. On the other hand, a Two Size Moment method has been suggested to reduce the number of sections but it lacks an efficient coalescence resolution method. After introducing a new strategy for two size moment coalescence, the two methods are compared on various configurations in a research code and an industrial-oriented code, in order to conclude on computational accuracy and cost. Then the paper aims at describing the most efficient approach for multi-dimensional unsteady and eventually coalescing rocket chamber simulations. Its objective is threefold : first, to validate the Two Size Moment method by comparing simulations to reference solutions and dedicated experimental measurements conducted at ONERA, second to study the efficiency and robustness of both methods, third, to draw some firm conclusions about the necessity to use the One Size Moment or Two Size Moment method to simulate solid propellant alumina sprays. We finally perform the first simulations of coalescence in realistic 2D boosters with the Two Size Moment method, implemented in the industrial-oriented code CEDRE.

International audience; The accurate description and robust simulation, at relatively low cost, of global quantities (eg 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. (J. Aerosol Sci., 51:1--23, 2012)and a hybrid method (TSM) between the sectional and moment methods, considering two moments per section based on the work of Laurent et al. (Commun. Comput. Phys., accepted, 2016). 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.

International audience; 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.

Predictive simulation of liquid fuel injection in automotive engines has become a major challenge for science and applications. The key issue in order to properly predict various combustion regimes and pollutant formation is to accurately describe the interaction between the carrier gaseous phase and the polydisperse evaporating spray produced through atomization. For this purpose, we rely on the EMSM (Eulerian Multi-Size Moment) Eulerian polydisperse model. It is based on a high order moment method in size, with a maximization of entropy technique in order to provide a smooth reconstruction of the distribution, derived from a Williams-Boltzmann mesoscopic model under the monokinetic assumption [O. Emre (2014) PhD Thesis , Ecole Centrale Paris; O. Emre, R.O. Fox, M. Massot, S. Chaisemartin, S. Jay, F. Laurent (2014) Flow, Turbulence and Combustion 93 , 689-722; O. Emre, D. Kah, S. Jay, Q.-H. Tran, A. Velghe, S. de Chaisemartin, F. Laurent, M. Massot (2015) Atomization Sprays 25 , 189-254; D. Kah, F. Laurent, M. Massot, S. Jay (2012) J. Comput. Phys. 231 , 394-422; D. Kah, O. Emre, Q.-H. Tran, S. de Chaisemartin, S. Jay, F. Laurent, M. Massot (2015) Int. J. Multiphase Flows 71 , 38-65; A. Vie, F. Laurent, M. Massot (2013) J. Comp. Phys. 237 , 277-310]. The present contribution relies on a major extension of this model [M. Essadki, S. de Chaisemartin, F. Laurent, A. Larat, M. Massot (2016) Submitted to SIAM J. Appl. Math.], with the aim of building a unified approach and coupling with a separated phases model describing the dynamics and atomization of the interface near the injector. The novelty is to be found in terms of modeling, numerical schemes and implementation. A new high order moment approach is introduced using fractional moments in surface, which can be related to geometrical quantities of the gas-liquid interface. We also provide a novel algorithm for an accurate resolution of the evaporation. Adaptive mesh refinement properly scaling on massively parallel architectures yields a precise integration of transport in physical space limiting both numerical dissipation as well as the memory trace of the solver. A series of test-cases is presented and analyzed, thus assessing the proposed approach and its parallel computational efficiency while evaluating its potential for complex applications.

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale 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 large spatial gradients in the reactive 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. 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 restrictive diffusive step stability limits, but only by the physics of the phenomenon. We aim thus at solving accurately complete models including all time and space scales of the phenomenon, considering large simulation domains with conventional computing resources. The efficiency of the method is evaluated through 2D and 3D numerical simulations of a human ischemic stroke model, conducted on a simplified brain geometry, for which a simple parallelization strategy for shared memory architectures was implemented, in order to reduce computing costs related to "detailed chemistry" features of the model.

To assist the industrial engine design process, 3-D computational fluid dynamics simulations are widely used, bringing a comprehension of the underlying physics unattainable from experiments. However, the multiphase flow description involving a liquid fuel jet injected into the chamber is still in its early stages of development. There is a pressing need for a spray model that is time efficient and accurately describes the cloud of fuel and droplet dynamics downstream of the injector. Eulerian descriptions of the spray are well adapted to this highly unsteady configuration. The challenge is then to capture accurately both the evaporating spray polydispersity and its two-way mass, momentum and energy interactions with the surrounding gas phase in this framework. The Eulerian Multi-Size Moment (EMSM) model, a high-order (in size) moment model has proved to be well adapted for the treatment of polydispersity. Moreover, it requires less computational effort relative to competing methods, with a single section for the size phase space. Academic test cases have demonstrated its great potential for industrial applications using one-way coupling. Recent modeling and numerical development efforts have resulted in an extension to two-way coupling with an unconditionally stable and accurate numerical scheme, while preserving the size moment space. All these developments have been successfully verified through injection simulations in the context of laminar flow. In the present contribution, in preparation for comparisons with experimental data, an extension to two-way coupling to account for the droplet-gas turbulence interactions is developed and validated for academic cases using physical data for realistic internal combustion engine conditions.

An Eulerian multi-fluid model is validated by comparison with experimental measurements in the test case of laminar spray counterflow diffusion flames. Special attention is devoted, both from the modelling and experimental point of view, to the treatment of the droplet distribution tail, characterized by the rare occurrence of relatively large droplets carrying a non-negligible amount of mass. The Eulerian multi-fluid approach is shown to capture the dynamics, evaporation and heating of the droplets with a limited number of sections and, thus, at a modest cost. This simplification will be essential for the use of multi-fluid methods in multi-dimensional problems.

In this paper, we present a new Eulerian multi-fluid modeling for dense sprays of evaporating liquid droplets which is able to describe droplet coalescence and size polydispersion as well as the associated size-conditioned dynamics. It is an uncommon feature of Eulerian spray models which are required in a number of non-stationary simulations because of the optimization capability of a solver coupling a Eulerian description for both phases. The chosen framework is the one of laminar flows or the one of direct numerical simulations since no turbulence models are included in the present study. The model is based on a rigorous derivation from the kinetic level of description (p.d.f. equation) and can be considered as a major extension of the original sectional method introduced by Tambour et al. We obtain a set of conservation equations for each "fluid": a statistical average of all the droplets in given size intervals associated to a discretization of the size phase space. The coalescence phenomenon appears as quadratic source terms, the coefficients of which, the collisional integrals, can be pre-calculated from a given droplet size discretization and do not depend on space nor time. We validate this Eulerian model by performing several comparisons, for both stationary and non-stationary cases, to a classical Lagrangian model which involves a stochastic algorithm in order to treat the coalescence phenomenon. The chosen configuration is a self-similar 2D axisymmetrical decelerating nozzle with evaporating sprays having various size distributions, ranging from smooth ones up to Dirac delta functions through discontinuous ones. We show that the Eulerian model, if the discretization in the size phase space is fine enough (the problem is then 3D unstationary, 2D in space and 1D in size), is able to reproduce very accurately the non-stationary coupling of evaporation, dynamics and coalescence. Moreover, it can still reproduce the global features of the behavior of the spray with a coarse size discretization, which is a nice feature compared to Lagrangian approaches. The computational efficiency of both approaches are then compared and the Eulerian model is provided to be a good candidate for more complex and realistic configurations.

Resume Cette Note concerne l'analyse numerique d'une methode multi-fluide Eulerienne pour la description de sprays polydisperses qui s'evaporent et ce, sur une configuration stationnaire 1D, sans effet dynamique ni thermique ou seul l'aspect evaporation subsiste. L'espace des phases se reduit alors a la position et la taille des gouttes. On donne les conditions sous lesquelles la methode est d'ordre 1 en le pas de discretisation de la taille de goutte. D'autre part, la stabilite du θ-schema pour la discretisation en espace ainsi que la positivite des variables necessitent des conditions CFL precisees ici. On donne ainsi le « bon choix » de la variable taille de goutte et de la discretisation en cette variable. Pour citer cet article : F. Laurent, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 417–422.

A first attempt at deriving a fully Eulerian model for polydisperse evaporating sprays was developed by Tambour et al with the so-called sectional approach. However, the complete derivation of the sectional ‘multi-fluid' conservation equations from the Boltzmann-type spray equation was never provided, neither was the set of underlying assumptions nor the comparison with the classical Lagrangian model: the sampling method. In this paper, we clarify the set of assumptions necessary in order to derive the multi-fluid sectional model from the spray equation at the ‘kinetic level' and provide the derivation of the whole set of conservation equations describing the dispersed liquid phase. Whereas the previous derivation is conducted in any space dimension, we restrict ourselves to one-dimensional stationary flows where the droplets do not turn back and derive a Eulerian sampling model which is equivalent in this context to the usual Lagrangian particle approach. We then identify some situations, even within this restrictive framework, where the sectional approach fails to reproduce the coupling of the vaporization and dynamics of the spray, the sampling method then being required. In the domain of applicability of the sectional approach, the two methods are then compared numerically in the configuration of counterflow spray diffusion flames. The two methods, if refined enough, give quite similar results, except for some small differences, the origin of which is identified. It is proved that the sampling method is more precise even if it generates oscillations due to the intrinsic representation of a continuous function by Dirac delta functions. We thus provide a comprehensive analysis of the sectional approach from both the modelling and numerical points of view.

Resume L'effet de la rhizosphere sur la mineralisation de l'azote organique du sol est quantifie par comparaison de la mineralisation de l'azote dans un sol rhizospherique (sol cultive en ray-grass, plante a forte densite racinaire) et dans un sol non rhizospherique (sol sans vegetation). En utilisant trois approches differentes, on montre que la plante induit une augmentation de la mineralisation nette, de la mineralisation brute (au moins 35 %) et de la quantite d'azote disponible (50 %). Cet effet rhizospherique pourrait contribuer fortement a la fourniture d'azote aux plantes cultivees en region tropicale.

International audience; Solid rocket motor flows depend strongly on alumina granulometry, which spreads up with coalescence. Solving droplet sizes in unsteady flows is a modeling and numerical challenge, specially when high loadings generate retrocoupling. As an alternative to Lagrangian approaches, the Eulerian Multi-Fluid model describes polydispersity with sizesorted droplet fluids called sections. A two size moment method, that spares sections, is implemented in the CEDRE code. A new transport and coupling method allows to capture moderately dense spray structures with low diffusion. The aim is threefold: compare CEDRE's Lagrangian and new Eulerian two-phase approaches, prove the feasibility of unsteady and complex motor flow simulations, and show the effect of coalescence on instabilities.

International audience; In this work, we are interested in the modeling of spray polydispersion in size as well as size-velocity correlations, which may greatly influence the evaporation and the dynamics of the disperse phase. Vié et al. 2011 proposed a new model called Coupled Size-Velocity Moment method (CSVM), which handles the polydispersion using the NDF reconstruction proposed in Kah at al. 2011 for size distribution, a velocity reconstruction and adapted numerical methods for moments evolution. Here, the CSVM is evaluated in a Frozen Homogeneous Isotropic Turbulence. Results shows the ability of the method to capture the statistical information of a turbulent flow with a minimal number of moments. As soon as a numerical scheme for unstructured grid is provided, the CSVM would be an interesting approach for complex simulations in industrial codes.

International audience; A new Eulerian model is derived for polydisperse moderately-inertial sprays with coalescence: an Anisotropic Gaussian Velocity Closure conserves all the second order moments to render the spatial structure of the spray after Droplet Trajectory Crossing. Polydispersity is treated with a high order in size Multi-Fluid method where coalescence terms are directly computed from the velocity reconstructions. The approach is qualified on a crossing jet case against a semi-analytical solution and a Lagrangian Discrete Particle Simulation. The new model is ready for structured grid computations and promising for industrial computations, as it is flexible and of a high order in size and velocity.

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reaction waves. This type of problem 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. The paper 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 restricted by neither 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 two-dimensional spiral waves and three-dimensional 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 discussed.

Accepted for Publication; International audience; 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.

International audience; Population 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.

The accurate simulation of polydisperse sprays strongly coupled to unsteady gaseous flows is a major issue for solid rocket motor optimization and a challenge for both modeling and scientific computing. The Eulerian Multi-Fluid method (MF) has proven to account for polydispersity efficiently, considering conservation equations for moments of the spray integrated on droplet size intervals. This describes droplet size sorted "fluids" which are all coupled to the gas through drag and heat transfer source terms. The potential of this model to deal with polydisperse acoustics has not been addressed, which is an issue regarding both physics and code quality evaluation. The resulting system is indeed strongly coupled and requires dedicated numerical methods which one wants to control as regards cost and accuracy. In this paper, we define the key issues of polydisperse acoustics and the ability of MF systems to tackle them. We then describe the numerical difficulties of strong coupling. The case of small droplets is particularly harsh and carefully studied. We finally introduce, study and adapt to an industrial-oriented code a numerical strategy for polydisperse moderately dense sprays that can be adapted to accuracy needs. The method is tested on an unsteady polydisperse solid rocket motor case to prove the feasibility of the approach.

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale 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 large 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. 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 restrictive diffusive step stability limits, but only by the physics of the phenomenon. We thus aim at solving accurately complete models including all time and space scales of the phenomenon, 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 are finally discussed. Estudiamos la simulación numérica de ecuaciones de reacción-difusión que modelan ondas de reacción multi-escala. Este tipo de problemas induce dificultades peculiares y problemas de condicionamiento potencialmente grandes que surgen del amplio espectro de escalas temporales en el término no-lineal de la fuente química y de la presencia de grandes gradientes espaciales en el frente de reacción, muy localizados espacialmente. En este trabajo, introducimos una nueva estrategia de resolución basada en el particionamiento del operador temporal y en la multiresolución adaptiva espacial. Sobre la base de estudios teóricos recientes en el campo de análisis numérico, esta estrategia conduce a un paso de tiempo que no está restringido por las escalas más veloces en el término fuente ni por límites de estabilidad en la etapa de difusión, sino sólo por la propia física del fenómeno en estudio. De esta forma pretendemos resolver de forma precisa modelos completos que incluyen todas las escalas espaciales y temporales del fenómeno, teniendo en cuenta dominios de simulación amplios utilizando recursos computacionales convencionales. La eficiencia se evalúa a través de la simulación numérica de configuraciones que hasta el momento estaban fuera del alcance de los métodos estándar en el campo de la dinámica química no lineal para ondas espiral 2D y ondas scroll 3D. Extensiones futuras de la estrategia se discuten al final.

PDF available at http://www.stanford.edu/group/ctr/Summer/SP10/index.html; The use of robust and accurate Eulerian/Eulerian formulations in the modeling of reactive two-phase flow would be a major step forward in the framework of turbulent combustion modeling with massively parallel supercomputers. Therefore, the ability of the Eulerian multi-fluid model to capture all stages of turbulent spray combustion has been tested and compared with a usual Lagrangian formulation. The multi-fluid model and related dedicated schemes and algorithms are able to characterize properly the spray dispersion and segregation as well as the vaporization dynamics leading to the fuel mass fraction topology. Eventually, flame propagation and structure in 2D and 3D forced isotropic homogeneous turbulence have been characterized showing the capacity of the multi-fluid model to simulate such reactive flows with results with the same level of accuracy as a baseline solution obtained with Lagrangian droplet tracking.

International audience; 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 some 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.

Special Issue dedicated to S. B. Pope; International audience; Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the Williams-Boltzmann equation, containing terms for spatial transport, evaporation and fluid drag. The most common method for simulating the Williams-Boltzmann equation uses Lagrangian particle tracking wherein a finite ensemble of numerical ``parcels'' provides a statistical estimate of the joint surface area, velocity number density function (NDF). An alternative approach is to discretize the NDF into sections corresponding to different droplet sizes and to neglect velocity fluctuations conditioned on droplet size, resulting in an Eulerian multi-fluid model. In comparison to Lagrangian particle tracking, multi-fluid models contain no statistical error (due to the finite number of parcels) but they cannot reproduce the particle trajectory crossings observed in Lagrangian simulations of non-collisional kinetic equations. Here, in order to overcome this limitation, a quadrature-based moment method is used to describe the velocity moments. When coupled with the sectional description of droplet sizes, the resulting Eulerian multi-fluid, multi-velocity model is shown to capture accurately both particle trajectory crossings and the size-dependent dynamics of evaporation and fluid drag. Model validation is carried out using direct comparisons between the Lagrangian and Eulerian models for an unsteady free-jet configuration with mono- and polydisperse droplets with and without evaporation. Comparisons between the Eulerian and Lagrangian instantaneous number density and gas-phase fuel mass fraction fields show excellent agreement, suggesting that the multi-fluid, multi-velocity model is well suited for describing spray combustion.

International audience; The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows the detailed description of polydispersity and size/velocity correlations for droplets of various sizes. The purpose of the present study is two-fold. First, we extend the multi-fluid model in order to cope with droplet trajectory crossings by using the quadrature method of moments in velocity phase space conditioned by size. We identify the numerical difficulties and provide dedicated numerical schemes in order to preserve the velocity moment space. Second, we conduct a comparison study and demonstrate the capability of such an approach to capture the dynamics of an evaporating polydisperse spray in a two-dimensional free jet configuration. We evaluate the accuracy and computational cost of Eulerian models and related discretization schemes versus Lagrangian solvers. It shows that, even for finite Stokes number, the standard Eulerian multi-fluid model is accurate at reasonable cost.

International audience; In the present contribution, we consider acoustically pulsated free jets with a polydisperse previous termspraynext term injection in an axisymmetrical configuration. The acoustic excitation creates periodical large vortical structures, which are representative of the dynamics of gaseous flows in more complex configurations, and a strong interaction with the injected previous termspray.next term In this context, we provide a series of detailed experimental measurements using laser diagnostics; we analyze droplet size distributions, associated size-conditioned dynamics and evaporation, for three liquids, leading to various droplet preferential concentrations due to the vortices. Besides the achievements in terms of diagnostics, we use the Eulerian multi-fluid model and dedicated robust numerical schemes in order to conduct simulations in the chosen configuration. Detailed comparisons between numerical simulations and experimental measurements are provided in terms of previous termspraynext term velocity and number density. Size-conditioned preferential concentration of both non-evaporating and evaporating previous termspraysnext term are observed and reproduced by the numerical simulations, which eventually yields a validation of the proposed model.

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. [F. Laurent, M. Massot, P. Villedieu, Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays, J. Comput. Phys. 194 (2004) 505-543]. The first key feature of the paper is the application of direct quadrature method of moments (DQMOM) introduced by Marchisio and Fox [D.L. Marchisio, R.O. Fox, Solution of population balance equations using the direct quadrature method of moments, J. Aerosol Sci. 36 (2005) 43-73] 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 purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays. First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model. 5 (2001) 537-572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue. The classical multi-fluid method consists then in a discretization of the droplet size variable into cells called sections. This analysis provides a justification of the "right" choice for this discretization to obtain a first order accurate and monotone scheme, with no restrictive CFL condition. This result leads to the development of a class of methods of arbitrary high order accuracy through the use of moments on the droplet surface in each section and a Godunov type method. Moreover, an extension of the two moments method is proposed which preserves the positivity and limits the total variation. Numerical results of the multi-fluid methods are compared to examine their capability to accurately describe the mass density in the spray with a small number of variables. This is shown to be a key point for the use of such methods in realistic flow configurations.

Resume Dans cette Note, on considere un modele de type thermo-diffusif pour decrire la propagation d'une flamme 1D dans un milieu gazeux avec une chimie complexe de type graphe ouvert. On redemontre l'existence d'ondes progressives pour ce type de systeme en utilisant une nouvelle approche et l'on etudie l'effet du passage a la limite d'une diffusion massique nulle : on montre la continuite de la vitesse de l'onde ainsi que des profils de temperature et de fraction massique. Pour citer cet article : F. Laurent et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 405–410.

In solid rocket motors, the distributed combustion of aluminum droplets is suspected to be a driving mechanism of hydrodynamic and thermoacoustic instabilities. The accurate simulation of evaporating bi-component polydisperse sprays, strongly coupled to unsteady gaseous flows, appears as a determining step for future solid rocket motor optimization. In this contribution, we propose a new Eulerian Multi-Fluid to model reactive bi-component sprays that has been designed to follow accurately the evolution of the composition and the complex thermodynamics of the droplets. The objective of this work is twofold. The first one consists in the derivation of this new Multi-Fluid model and its hypotheses, and the way it is conditioned by the droplet composition. Then we present a time splitting integration strategy particularly efficient to deal with unsteady two-phase flow computations.