- Published 2017

The aim of the paper is to introduce and validate a Monte-Carlo algorithm for the prediction of an ensemble of colliding solid particles, or coalescing liquid droplets, suspended in a turbulent gas flow predicted by Reynolds Averaged Navier Stokes approach (RANS). The new algorithm is based on the direct discretization of the collision/coalescence kernel derived in the framework of a joint fluid–particle pdf approach proposed by Simonin et al. (2002). This approach allows to take into account correlations between colliding inertial particle velocities induced by their interaction with the fluid turbulence. Validation is performed by comparing the Monte-Carlo predictions with deterministic simulations of discrete solid particles coupled with Direct Numerical Simulation (DPS/DNS), or Large Eddy Simulation (DPS/LES), where the collision/coalescence effects are treated in a deterministic way. Five cases are investigated: elastic monodisperse particles, non-elastic monodisperse particles, binary mixture of elastic particles and binary mixture of elastic settling particles in turbulent flow and finally coalescing droplets. The predictions using the new Monte-Carlo algorithm are in much better agreement with DPS/DNS results than the ones using the standard algorithm. Introduction Discrete Particle Simulation (DPS) can be either coupled with Turbulent gas flows carrying dispersed solid or liquid phases are extensively found in industrial and environmental processes. Some typical examples are fuel spray injection in combustion chamber, solid rocket boosters with alumina droplets, pulverized coal combustion chamber, sediments transport, or rain droplet growth. In such particleor droplet-laden flows many complex physical phenomena take place such as turbulent dispersion, particle–particle collisions, particle–wall rebounds/impingement, or turbulence modulation by the particles. For example, in a combustion chamber the droplet and gas phase mixing governs the quality and efficiency of the combustion and, consequently, the pollutant emissions. In the near liquid injection zone, droplet coalescence may influence the droplet size distribution and must be accounted for in mathematical models and numerical simulation tools. The Lagrangian tracking of particles, or droplets, is widely used for the numerical simulation of particle-laden turbulent flows. Direct Numerical Simulation (DPS/DNS), Large Eddy Simulation (DPS/LES) or Reynolds Averaged Navier–Stokes approach (DPS/RANS) (see for example Balachandar and Eaton (2010), Riber et al. (2009), Fox (2012), and Sommerfeld (2001)). When the collisions are handled by a deterministic algorithm (Discrete Element Method) the DPS/DNS can be considered as a deterministic simulation because no stochastic model for both particle turbulent dispersion and particle collision are needed. For the DPS/LES, a dispersion model to reconstruct the subgrid fluid fluctuating velocity along the particle trajectory can be necessary if the particle relaxation time is of the same order, or smaller than, the characteristic time of the subgrid fluid turbulence (Fede and Simonin, 2006). The DPS/RANS can be stated has stochastic because even if the collisions are not taken into account a stochastic model has to be used for the turbulence induced particle dispersion. In practical applications, due to the huge number of real particles involved, the simultaneous computation of all individual particle trajectories is generally not yet possible. To overcome this difficulty, in the framework of a statistical approach, only a restricted number of numerical particles (also called parcels) may be tracked, each parcel representing a given number of real particles. To account for the collisions, stochastic algorithms are then used instead of deterministic ones. Stochastic algorithms were first derived for the collision of molecules in rarefied gases (Bird, 1969). A few decades ago, in the framework of DPS/RANS approach these algorithms were directly used for taking into account the particle–particle collisions in gas–solid turbulent flow (O’Rourke, 1981; Tanaka and Tsuji, 1991). However, Berlemont et al. (1995) proved that when applied to turbulent two-phase flow the standard stochastic collision algorithms lead to a decrease in particle kinetic energy although the particle collisions are elastics. They identified that this spurious phenomenon is induced by the so-called ‘‘molecular chaos’’ assumption that destroys the fluid–particle covariance and therefore decreases the production of particle fluctuating kinetic energy due to the interaction with fluid turbulence. Sommerfeld (2001) and Berlemont et al. (2001) proposed stochastic approaches where one single particle is tracked and successive random processes are applied to generate fictitious partners of collision accounting for the particle–particle velocity correlations induced by their interaction with fluid turbulence. Their two algorithms differ in the way used for the sampling of the fictitious colliding partner velocity. In order to validate both approaches, simulations have been carried out for homogeneous isotropic flow and they have been compared with DPS/LES from Laviéville et al. (1995) and Gourdel et al. (1998). Even if these single-particle algorithms have been successfully used to simulate two-phase flows, they have several limitations. First, both approaches need an a priori model to sample the fictitious partner velocity, given the velocity of the tracked particle. Second, in contrast with theory and DPS/DNS results, these algorithms do not preserve fluid–particle velocity covariance. To overcome this issue Berlemont et al. (2001) proposed a multiple particle collision algorithm based on the simultaneous tracking of several particles combined with an approximate method to enforce the fluid–particle velocity covariance conservation. This ad-hoc method consists in changing the seen fluid velocity of each colliding particle after a collision in order to ensure that the fluid–particle velocity covariance is conserved. But, as pointed out by Berlemont et al. (2001), this method is not satisfactory because the distribution of collision angles remains the same whatever the particle inertia since no specific correlation is accounting for before the collision. Finally for single-particle methods the conservation of momentum and of the particle kinetic energy (for elastic collision) cannot be exactly ensured because the collision partner is fictitious. In fact, such algorithms are statistically conservative only if the number of parcels is very large. In the present paper, we propose a rigorous approach to derive a Monte-Carlo algorithm which allows to overcome all the previously mentioned limitations in the framework of DPS/RANS approach. This algorithm can be interpreted as a direct discretization of the collision kernel introduced in Laviéville et al. (1995, 1997) and Simonin et al. (2002) for taking into account the velocity correlations induced by the interactions of particles with turbulence. For the sake of completeness the paper first introduces the full derivation of the joint fluid–particle Number Density Function (NDF) kinetic equation. The case of solid particles, and liquid droplets, are both addressed. The standard and the new Monte-Carlo algorithms for solving the kinetic equation are described in the third section. In this section an analysis of the effect of the numerical parameters introduced by the novel Monte-Carlo algorithm is performed. Section ‘Monodisperse solid particles’ shows the results obtained by considering a monodisperse elastic, and non-elastic, particles suspended in homogeneous isotropic turbulent flow. Section ‘Binary mixture of colliding particles’ is dedicated to binary mixture of particles suspended, and settling, in homogeneous isotropic turbulence and Section Coalescing droplets shows the application of the modified Monte-Carlo algorithm for coalescing liquid droplets transported in a homogeneous isotropic turbulent flow. Statistical description of binary collision between particles transported by a turbulent flows Statistical description The statistical description of a dispersed phase, composed of solid particles or droplets, transported by a turbulent fluid flow relies on the analogy with the thermal motion of molecules as described by the kinetic theory of rarefied gases (Chapman and Cowling, 1970). In this framework, the dispersed phase statistical properties are described by the particle number density function f pðcp;lp;x; tÞ defined such that f pðcp;lp;x; tÞdcpdlpdx is the mean probable number of particles at time t with a centre of mass located in the volume 1⁄2x;xþ dx , having a mass mp in 1⁄2lp;lp þ dlp and a translation velocity up in 1⁄2cp; cp þ dcp . By analogy with statistical approaches for single-phase turbulent flows, the statistical average associated to the definition of f p may be defined as an ensemble average on an infinite number of realizations of a given gas–particle flow (Buyevich, 1971, 1972). The Number Density Function (NDF), also called in the literature the probability density function (PDF), obeys to a Boltzmann-like kinetic equation. However, in contrast to the thermal motion of molecules or to dry granular flows, the particle, or droplet, motion is driven by the fluid turbulence, and the influence of the fluid turbulent flow on the particle dynamics must be taken into account. The closure of the term representing the forces acting on the particles is the topic of many studies for taking into account the turbulent dispersion by the fluid turbulence (Derevich and Zaichik, 1988; Reeks, 1992, 1993). Simonin (1996) proposed an original statistical description taking into account the instantaneous fluid velocity seen by the particles. A joint fluid–particle distribution f fpðcp; cf ;lp;x; tÞ is then introduced, which is defined such that f fpðcp; cf ;lp;x; tÞdcpdcf dlpdx is the mean probable number of particles at time t with a centre of mass located in the volume 1⁄2x;xþ dx , and having a mass mp in 1⁄2lp;lp þ dlp , a translation velocity up in 1⁄2cp; cp þ dcp and ‘‘seeing’’ a fluid velocity uf@p in 1⁄2cf ; cf þ dcf . In the framework of Gatignol (1983) or Maxey and Riley (1983) approach the fluid velocity uf@p represents the local undisturbed fluid velocity at the particle position introduced to model the fluid–particle momentum transfer. In the case without mass transfer meaning that the change of any particle mass is only due to collision, Simonin (1996) proposed the following Boltzmann-like transport equation for the joint fluid–particle distribution @f fp @t þ @ @xi cp;if fp þ @ @cp;i dup;i dt cp; cf f fp þ @ @cf ;i duf ;i dt cp; cf f fp 1⁄4 @f fp @t coll ð1Þ where h jcp; cf i is the ensemble average conditioned by the particle and fluid velocity seen by any particle with a centre of mass at position xp 1⁄4 x : up 1⁄4 cp and uf@p 1⁄4 cf (with up and uf@p being the particle and the fluid velocity seen by the particle in physical space). The term on the right-hand side of Eq. (1) represents the change of f fp due to collision, coalescence or break-up. The third term on the left-hand side of Eq. (1) accounts for the effects of the particle acceleration on the NDF. Considering that the forces acting on the particles are only the gravity and the drag force, the particle acceleration writes dup dt 1⁄4 Fp mp 1⁄4 up uf@p sp þ g ð2Þ where uf@p is the locally undisturbed fluid velocity at the particle position, or the so-called fluid velocity ‘‘seen’’ by the particle, g the gravity and sp the local instantaneous particle response time. Compared to the standard approach based on f pðcp;x; tÞ, the particle acceleration term (the third term of Eq. (1)) is directly closed because the instantaneous fluid velocity is known through the joint fluid–particle NDF. However, the price-to-pay is the presence of the fourth term representing the acceleration of the fluid velocity along the solid particle trajectory. Several models can be found in the literature dedicated to the Lagrangian modelling of the fluid turbulence along fluid elements (Haworth and Pope, 1986; Pope, 1994, 2002) or along inertial particle trajectories (Simonin et al., 1993; Pascal and Oesterlé, 2000; Minier and Peirano, 2001; Minier et al., 2004; Pialat et al., 2007; Tanière et al., 2010). As the purpose of the present paper is the numerical treatment of collision, the Lagrangian prediction of the fluid velocity along a particle trajectory will not be longer detailed. In the present paper, the fluid flow, for stochastic simulations, is predicted by a Langevin equation proposed by Simonin et al. (1993) that is an extension of the model proposed by Pope (1994) (see Appendix B). Collision/coalescence kernel The collision/coalescence operator may be split in two contributions @f fp @t coll 1⁄4 Kðcp; cf@p;lp;x; tÞ K ðcp; cf@p;lp;x; tÞ ð3Þ where Kðcp; cf@p;lp;x; tÞ represents the apparition rate and K ðcp; cf@p;lp;x; tÞ the vanishing rate. Both terms require the knowledge of the velocity and mass for the two colliding particles. These information are given by the pair NDF f ð2Þ fp defined such that f ð2Þ fp ðcp; cf@p;lp; cq; cf@q;lq; xp;xq; tÞdxpdxqdcpdcqdcf@pdcf@qdlpdlq is the mean probable number of particle pairs with centres of mass located in the volume 1⁄2xp;xp þ dxp and 1⁄2xq;xq þ dxq and having masses mp in 1⁄2lp;lp þ dlp and mq in 1⁄2lq;lq þ dlq , and translation velocities up in 1⁄2cp; cp þ dcp and uq in 1⁄2cq; cq þ dcq , and viewing fluid velocities uf@p in 1⁄2cf@p; cf@p þ dcf@p and uf@q in 1⁄2cf@q; cf@q þ dcf@q , respectively. From this definition the vanishing rate of droplets writes K ðcp;cf@p;lp;x;tÞ1⁄4 Z wpq :kpq<0 f ð2Þ fp ðcp;cf@p;lp;cq;cf@q;lq;x;xþdpqkpq;tÞ jwpq:kpqjd2pqdcqdcf@qdlqdkpq ð4Þ where the particle–particle relative velocity is defined as wpq 1⁄4 cq cp. The impact vector is defined by the particle position kpq 1⁄4 ðxq xpÞ=dpq and the collision diameter dpq 1⁄4 ðdp þ dqÞ=2. The rate of particle apparition is then given by Kðcp;cf@p;lp;x;tÞ1⁄4 Z wpq :kpq<0 Rðcm;cf@m;lm;cn;cf@n;ln;kmn ! cp;cf@p;lp;x;tÞ f ð2Þ fp ðcm;cf@mlm;cn;cf@n;ln;x;xþdmnkmn;tÞ jwmn:kmnjd2mndcndcf@ndlndkmndcmdcf@mdlm ð5Þ where the transition probability by particle–particle collision Rðcm; cf@m; lm; cn; cf@n; ln; kmn ! cp; cf@p; lp; x; tÞ 1⁄4 Pðcp; cf@p; lpjcm; cf@m;lm; cn; cf@n;ln;kmn;x; tÞ is introduced. This function is the probability density of having one particle located at x with a velocity cp and mass lp resulting from the collision of a particlem located in xm 1⁄4 xwith a velocity cm, and a mass lm, with a particle n located in xn 1⁄4 xþ dmnkmn with a velocity cn and a mass ln. Considering the hard-sphere collision model, the transition probability density reads: Pðcp; cf@p;lpjcm; cf@m;lm; cn; cf@n;ln;kmn;x; tÞ 1⁄4 d lp lm h i d cp cm Pðcf@pjcf@m; cf@nÞ ð6Þ where cm is the m-particle velocity after an inelastic collision with any n-particle. Assuming, frictionless hard-sphere collision, these velocities are given by: cm 1⁄4 cm þ mn mm þmn ð1þ ecÞðwmn:kmnÞkmn; ð7Þ cn 1⁄4 cn mm mm þmn ð1þ ecÞðwmn:kmnÞkmn ð8Þ where ec is the particle–particle restitution coefficient (Walton, 1993). When considering the collision of two droplets several phenomena may occur as pure coalescence, elastic bouncing, or production of satellite droplets (Ashgriz and Poo, 1990; Crowe et al., 1998). In the present study, only the permanent coalescence regime where a droplet–droplet collision leads to a new droplet is considered (Villedieu and Simonin, 2004). The mass and momentum conservation leads to the two following relations

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@inproceedings{Fede2017MonteCarloSO,
title={Monte-Carlo simulation of colliding particles or coalescing droplets transported by a turbulent flow in the framework of a joint fluid–particle pdf approach},
author={Pascal Fede and Olivier Simonin and Philippe Villedieu},
year={2017}
}