Lagrangian–Eulerian methods for multiphase flows

1 lagrangian eulerian methods for multiphaseflowsShankar Subramaniam Department of Mechanical engineering ,Iowa State UniversityAbstractThis review article aims to provide a comprehensive and understandable account ofthe theoretical foundation, modeling issues, and numerical implementation of theLagrangian eulerian (LE) approach for multiphase flows. The LE approach is basedon a statistical description of the dispersed phase in termsof a stochastic point pro-cess that is coupled with an eulerian statistical representation of the carrier fluidphase. A modeled transport equation for the particle distribution function alsoknown as Williams spray equation in the case of sprays is indirectly solved usinga lagrangian particle method. Interphase transfer of mass,momentum and energyare represented by coupling terms that appear in the eulerian conservation equa-tions for the fluid phase.

2 This theoretical foundation is then used to develop LEsubmodels for interphase interactions such as momentum transfer. Every LE modelimplies a corresponding closure in the eulerian - eulerian two fluid theory, and thesemoment equations are derived. Approaches to incorporate multiscale interactionsbetween particles (or spray droplets) and turbulent eddiesin the carrier gas thatresult in better predictions of particle (or droplet) dispersion are described. Nu-merical convergence of LE implementations is shown to be crucial to the success ofthe LE modeling approach. It is shown how numerical convergence and accuracy ofan LE implementation can be established using grid free estimators and computa-tional particle number density control algorithms.

3 This review of recent advancesestablishes that LE methods can be used to solve multiphase flow problems of prac-tical interest, provided submodels are implemented using numerically convergentalgorithms. These insights also provide the foundation forfurther development ofLagrangian methods for multiphase flows. Extensions to the LE method that canaccount for neighbor particle interactions and preferential concentration of particlesin turbulence are words:particle laden flow, gas solid flow, multiphase flow theory,spray,droplet, spray theory, lagrangian - eulerian , spray modeling, two-phase flow,numerical , submitted to Elsevier Science8 March 20121 Introduction and ObjectivesThis paper describes the use of the lagrangian eulerian (LE) approach tocalculate the properties of multiphase flows such as sprays or particle ladenflows that are encountered in many energy applications.

4 The LE approachis used to denote a family of modeling and simulation techniques whereindroplets or particles are represented in a lagrangian reference frame whilethe carrier phase flow field is represented in an eulerian frame. This paperprimarily focuses on the use of the LE approach as a solution method forthe transport equation of the droplet distribution function (ddf) or numberdensity function (NDF), which is also known as Williams spray a recent review article [1], Fox notes that the NDF representation of theparticle phase constitutes a mesoscopic approach that offers a clear separationbetween physical and mathematical approximations. Since the LE approachis widely used to simulate multiphase flows, a comprehensivedescription ofthis approach can be of use to theoreticians, model developers and end usersof order for any simulation methodology such as the LE approach to be apredictive tool, it must be based on(i) a mathematical representation that is capable of representing the physicalphenomena of interest,(ii) accurate and consistent models for the unclosed terms that need to bemodeled, and(iii) a numerically stable and convergent are challenges in each of these areas that must be surmounted in orderto develop such a predictive LE simulation methodology for multiphase , this paper addresses key issues related to the LEapproach in theareas of.

5 (i) mathematical representation, (ii) physics based modeling, and(iii) numerical implementation. Considerable progress has been made in ad-dressing many of these challenges since the inception of theLE approach. Thisarticle attempts to summarize these advances and also outline opportunitiesfor further development of the LE flows in energy applications are often also turbulent, reactive the field of turbulent reactive flows is a mature research area with manyauthoritative reviews [2 4], this work will focus mainly onthe multiphase Corresponding authorEmail Subramaniam).2aspects of the flow with some reference to turbulence interactions. There isalso a wide range of physico-chemical phenomena that are encountered innonreacting multiphase flows alone, and these are highly dependent on theparticular application area.

6 For instance, in the area of sprays one can findmany comprehensive reviews of single droplet behavior and spray atomiza-tion and vaporization [5 7]. In light of the wide variety of physico chemicalphenomena in multiphase flows, this review will only consider these genericcharacteristics of a dispersed two way coupled, two phaseflow that need tobe incorporated in the LE nonlinear and multiscale interactions in multiphase flow result in a richvariety of flow phenomena spanning many flow regimes. One of the primaryfeatures of multiphase flow that distinguish it from advection and diffusionof chemical species in multicomponent flows is the inertia ofdispersed phaseparticles or droplets. Particle inertia results in a nonlinear dependence of par-ticle acceleration on particle velocity outside the Stokesflow regime, and thisnonlinearity is important in many applications where the particle Reynoldsnumber is finite.

7 Also in many multiphase flows one must consider the influ-ence of the dispersed phase on the carrier phase momentum balance, and thistwo way coupling is a source of nonlinear behavior in the of the dispersed phase particles or droplets introduces a rangeof length and time scales. Interactions of these polydisperse particles withcarrier phase turbulence that is inherently multiscale innature presents fur-ther modeling challenges. Furthermore, it is not uncommon to encounter awide variation in dispersed phase volume fraction in the same multiphaseflow, ranging from dilute to dense. For example, in a fluidizedbed the particlevolume fraction can range from near close packed at the baseof the bed toless than 5% in the riser. The particle volume fraction in conjunction with thelevel of particle fluctuating velocity (that can be characterized by the particleMach number) determines the relative importance of advective transport tocollisional effects.

8 Since unlike molecular gases not all multiphase flows arecollision dominated, it is possible for the probability density function (PDF)of velocity to depart significantly from the equilibrium Maxwellian distribu-tion. These nonlinearities, multiscale interactions and nonequilibrium effectslead to the emergence of new phenomena such as preferential concentrationand clustering that have a significant impact in multiphase flow the LE simulation approach as a numerical solution to the ddf (orNDF) evolution equation reveals the specific advantages of this mesoscopic [1]mathematical representation underlying the LE approach for capturing thesenonlinear, multiscale interactions and nonequilibrium effects in multiphaseflow. Williams [8] introduced the ddf in his seminal 1958 paper, and its coun-terpart in the kinetic theory of gas solid flow is the number density functionor one particle distribution function (see Koch 1990 [9] for example).

9 The ddf3or NDF is an unnormalized joint probability density of droplet (or particle)size and velocity as a function of space and time. Since the ddf (or NDF)contains the distribution of droplet (or particle) sizes itnaturally capturesthe size dependence of drag and vaporization rate in closedform, whereasother approaches such as the eulerian eulerian (EE) two fluid theory [10 12]that only represent the average size and average velocity ofdroplets (or par-ticles) must rely on approximate closure models. One of the major challengesin the two fluid averaged equation approach that is based onaverage size isthe incorporation of the range of droplet (or particle) sizes, and the nonlineardependence of interphase transfer processes on droplet (orparticle) size.

10 Thetwo-fluid EE approach referred to here is not to be confused with the Eulerianmoment equations that can be derived from the ddf, although those momentequations also contain less information than the ddf. A complete discussioncan be found in Pai and Subramaniam [13].Similarly, because the ddf (or NDF) contains the velocity distribution ofdroplets (or particles), it also captures the nonlinear dependence of parti-cle drag on particle velocity in closed form. Furthermore, particle velocityfluctuations, whose statistics are characterized by the granular temperatureand higher moments of particle velocity, are also easily modeled in the LEframework [14,15]. As pointed out by Fox and co-workers, theLE approach aswell as the quadrature method of moments (QMOM) developed byFox [16]lead to physically correct solutions to the problem of crossing particle jets [17 19], whereas the eulerian two fluid theory leads to anomalous results for thisproblem.