A moment method for splashing and evaporation processes of polydisperse sprays (original) (raw)
High Order Moment Model for Polydisperse Evaporating Sprays Towards Interfacial Geometry
HAL (Le Centre pour la Communication Scientifique Directe), 2018
In this paper we propose a new Eulerian model 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 relies on the use of fractional droplet surface moments and their ability to predict some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. Evaporation is evaluated by using a Maximum Entropy (ME) reconstruction. The use of fractional moments introduces some theoretical and numerical difficulties. First, relying on a study of the moment space, we extend the ME reconstruction to the case of fractional moments. Then, we propose a new high order and robust algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. It involves some negative order fractional moments for which a novel treatment is introduced. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as computational cost, but also provides an additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in order to describe atomization.
A high order moment method simulating evaporation and advection of a polydisperse liquid spray
Journal of Computational Physics, 2012
In this paper, we tackle the modeling and numerical simulation of sprays and aerosols, that is dilute gas-droplet 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 Eulerian Multi Size Moment (EMSM) 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.
High Order Moment Model for Polydisperse Evaporating Sprays towards Interfacial Geometry Description
SIAM Journal on Applied Mathematics, 2018
In this paper we propose a new Eulerian model 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 relies on the use of fractional droplet surface moments and their ability to predict some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. Evaporation is evaluated by using a Maximum Entropy (ME) reconstruction. The use of fractional moments introduces some theoretical and numerical difficulties. First, relying on a study of the moment space, we extend the ME reconstruction to the case of fractional moments. Then, we propose a new high order and robust algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. It involves some negative order fractional moments for which a novel treatment is introduced. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as computational cost, but also provides an additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in order to describe atomization.
Eulerian Quadrature-Based Moment Models for Dilute Polydisperse Evaporating Sprays
Flow Turbulence and Combustion, 2010
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 droplet size intervals, called sections, 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.
Journal of Computational Physics, 2013
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.
Eulerian Multi-Fluid Models for Polydisperse Evaporating Sprays
CISM International Centre for Mechanical Sciences, 2007
In this contribution we propose a presentation of Eulerian multi-fluid models for polydisperse evaporating sprays. The purpose of such a model is to obtain a Eulerian-type description with three main criteria: to take into account accurately the polydispersity of the spray as well as size-conditioned dynamics and evaporation; to keep a rigorous link with the Williams spray equation at the kinetic, also called mesoscopic, level of description, where elementary phenomena such as coalescence can be described properly; to have an extension to take into account non-resolved but modeled fluctuating quantities in turbulent flows. We aim at presenting the fundamentals of the model, the associated precise set of related assumptions as well as its implication on the mathematical structure of solutions, robust numerical methods able to cope with the potential presence of singularities and finally a set of validations showing the efficiency and the limits of the model.
Experimental Investigations of the Binary Interaction of Polydisperse Sprays
Particle & Particle Systems Characterization, 1998
Experimental investigations of the interaction of two polydisperse semi-hollow cone sprays are presented. The process, although of considerable signi®cance for the chemical industry and applications like¯ue gas cleaning, has not been well-covered in the existing literature. This may be due to dif®culties in getting general results from experiments involving particular geometries, like conical sprays, with ®xed spray angle and geometrical arrangement of the nozzles. The present work develops a representation of the effects of the spray interaction on the spray drops in the resulting two-phase¯ow. The measurement technique used is phase-Doppler anemometry (PDA), which provides information about the size and two velocity components of the drops at each measurement position in the sprays. A factorial design of the experiments allows the in¯uence of the intersection angle and the liquid¯owrate of the sprays on an integral mean drop size in a spray cross section to be quanti®ed. For varying values of these parameters, the downstream evolution of the interacting sprays is quanti®ed in terms of the smoothness of pro®les of the number-mean drop size. The collisional interaction of the spray drops is identi®ed as the reason for the observed increase of the mean drop size caused by the spray interaction.
International Journal of Multiphase Flow, 2015
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.
International Journal of Fluid Mechanics Research
In many technical processes, liquid sprays impinge on a solid surface, producing a liquid film on the impacted surface and a secondary atomization in the form of smaller secondary droplets detaching from the wetted surface. One of the main purposes of the present work, it is to obtain an empirical model to describe the splashed flux and the velocity and diameter probability distribution functions of the secondary droplets as a function of the impact parameters (film thickness, impacting droplet velocity and diameter, liquid properties). In this work, an experimental set-up to obtain such a result is presented and a first application of the empirical model to a polydispersed spray is given.
Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, 2016
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.