A moment method for splashing and evaporation processes of polydisperse sprays (original) (raw)
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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.
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.
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.
Numerical Simulation of the Drop Size Distribution in a Spray
Springer Proceedings in Mathematics & Statistics, 2012
Classical methods of modeling predict a steady-state drop size distribution by using empirical or analytical approaches. In the present analysis, we use the maximum of entropy method as an analytical approach for producing the initial data; then we solve the coagulation equation to approximate the evolution of the drop size distribution. This is done by a quasi-Monte Carlo simulation of the conservation form of the equation. We compare the use of pseudo-random and quasi-random numbers in the simulation. It is shown that the proposed method is able to predict experimental phenomena observed during spray generation.
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.
Eulerian models and three-dimensional numerical simulation of polydisperse sprays
2010
Providing accurate simulations of polydisperse evaporating sprays dynamics 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 usual Lagrangian approaches developed in polydisperse unsteady configurations require tremendous computational costs and may lead to a low level of resolution if not enough numerical parcels are used. Besides, they induce coupling issues due to the different kind of description of the two phases that are involved. A large range of Eulerian models have been recently developed to describe the dispersed liquid phase with a lower cost and an easier coupling with a carrier gaseous phase. Among these models, the multi-fluid model allows a detailed description of polydispersity and size/velocity correlations of droplets of various sizes. It has been studied in depth from a mathematical and numerical point of view (see Laurent & al (2004); Laurent (2006); Massot & al (2009)) and validated through comparisons versus Lagrangian simulations in de Chaisemartin (2009) and experimental measurements in Fréret & al (2008) in 2D and 2D-axisymmetrical configurations. However, the validation in three-dimensional unsteady configurations still remains to be done. In this work, we study the non-evaporating droplet segregation in three-dimensional Homogeneous Isotropic Turbulence (HIT) using a reference Lagrangian spray model versus the Eulerian multi-fluid model. A spectral Direct Numerical Simulation solver is used to describe the evolution of the turbulent carrier phase, whose characteristic properties remain statistically stationary due to a semi-deterministic forcing scheme. We focus on the optimization via a parallel implementation of the multi-fluid model and dedicated numerical methods which demonstrates the ability of the Eulerian DNS model to be used in high performance computing for academic three-dimensional configurations. We provide qualitative comparisons between the Euler-Lagrange and the Euler-Euler descriptions for two different values of the Stokes number based on the initial fluid Kolmogorov time scale, St = 0.17 and 1.05. A very good agreement is found between the mesoscopic Eulerian and Lagrangian predictions. We go further with first quantitative comparisons of the segregation effect of the vortices on the spray mass density distribution showing the accuracy and ability of the multi-fluid model to be used in 3D configurations from the tracer limit (St → 0) to unity.
A second-order multi-fluid model for evaporating sprays
ESAIM: Mathematical Modelling and Numerical Analysis, 2005
The aim of this paper is to present a method using both the ideas of sectional approach and moment methods in order to accurately simulate evaporation phenomena in gas-droplets flows. Using the underlying kinetic interpretation of the sectional method [35] exposed in [25], we propose an extension of this approach based on a more accurate representation of the droplet size number density in each section ensuring the exact conservation of two moments (as opposed to only one moment used in the classical approach). A corresponding second-order numerical scheme, with respect to the space and the droplet size variables, is also introduced and can be proved to be positive and to satisfy a maximum principle on the velocity and the mean droplet mass under a suitable CFL-like condition. Numerical simulations have been performed and the results confirm the accuracy of this new method even when a very coarse mesh for the droplet size variable (i.e : a low number of sections) is used.
Influence of droplet collision modelling in Euler/Lagrange calculations of spray evolution
International Journal of Multiphase Flow, 2020
Simulated hollow cone spray structure with gas-phase velocity field (colour coded), spray droplet concentration field indicated by three contour lines (from outer to inner side 3, 5 and 10 g/m 3) and droplet collision locations for bouncing, coalescence, stretching and reflexive separation indicated by black points.
Prediction of the droplet size and velocity joint distribution for sprays
Fuel, 2001
This work addresses the development of a mathematical model to predict the joint distribution for both size and velocity of the droplets in sprays, based on the maximum entropy formalism. Using this joint distribution, models to obtain separated distributions for size and velocity of sprays are also presented. Correlations for the average velocity for both pressure jet and airblast atomisers, based on assumed pro®les in the atomiser gun, are obtained as a function of easily measurable parameters. Several distributions for different types of atomisers are then predicted. Agreement between available data for the velocity distribution and the corresponding predictions is satisfactory.