ILASS Americas 21st Annual Conference on Liquid Atomization and Sray Systems, Orlando, FL, May 28 Modeling Volumetric Couling of the Disersed Phase using the Eulerian-Lagrangian Aroach Ehsan Shams and Sourabh V. Ate School of Mechanical, Manufacturing and Industrial Engineering Oregon State University 24 Rogers Hall, Corvallis, OR 97331 Abstract The Eulerian-Lagrangian aroach is commonly used in modeling two-hase flows wherein liquid drolets, solid articles, or bubbles are disersed in a continuum fluid of a different hase. Tyically, the motion of the disersed hase is modeled by assuming sherical, oint-articles with models for added mass effects, drag, and lift forces. The effect of the disersed hase on the fluid flow is modeled using reaction forces in the fluid momentum equation. Such an aroach is valid for dilute regions of the disersed hase. For dense regions, however, the oint-article aroach does not cature the interactions between the fluid and the disersed hase accurately. In this work, the fluid volume dislaced by the disersed hase is taken into account to model the dense regions. The motion of the disersed hase results in local, satio-temoral variations of the volume fraction fields. The resultant divergence in the fluid velocity acts as a source or sink dislacing the flow due to disersed hase and is termed as volumetric couling. The size of the disersed hase is assumed smaller than the grid resolution and for the continuum hase. The variable density, low Mach number equations based on mixture theory are solved using a co located, finite volume scheme. The interhase momentum exchange due to drag forces is treated imlicitly to rovide robustness in the dense regions. The volumetric couling aroach is first validated with analytical studies for flow induced by oscillating bubbles and gravitational settling of articles. Simulations of Rayleigh-Taylor instability, article-laden jet imingement on a flat late, and article-laden jet in a cross are erformed to test the robustness of the scheme. Corresonding Author: shamssoe@engr.orst.edu
Introduction Majority of sray systems in roulsion alications involve comlex geometries and highly unsteady, turbulent flows near the injector. The numerical models for sray calculations should be able to accurately reresent drolet deformation, breaku, collision/coalescence, and disersion due to turbulence. In the traditional aroach for sray comutation, the Eulerian equations for gaseous hase are solved along with a Lagrangian model for article transort with two-way couling of mass, momentum, and energy exchange between the two hases [1]. Tyically simulations of sray systems use DNS, LES or RANS for the carrier hase whereas the motion of the disersed hase is modeled. The oint-article PP) assumtion is commonly emloyed where forces on the disersed hase are comuted through model coefficients. The effect of the articles 1 on the carrier hase is reresented by a force alied at the centroid of the article. The diserse hase equations are tyically solved in a Lagrangian frame by tracking a set of comutational articles or arcels [2] with models for drolet breaku, collision/coalescence, evaoration, disersion, and deformation. Fully resolved simulations involving comrehensive modeling of interfacial dynamics are being develoed [3, 4], however, are comutationally exensive. Several simulations of article-laden flows have been erformed with the carrier fluid simulated using direct numerical simulation [5, 6],[7],[8]), largeeddy simulation [9, 1, 11, 12]), or Reynoldsaveraged Navier Stokes equations [13], where the disersed hase is assumed subgrid so d < L K, the Kolmogorov length scale, for DNS whereas d <, the grid size, in LES or RANS). However, modeling the disersed hase using oint-article aroach does not always rovide the correct results. For moderate loadings and wall-bounded flows [11] have shown that the oint-article aroximation fails to redict the turbulence modulation comared to exerimental values. In addition, if the article size is comarable to the Kolmogorov scale for DNS) or the grid size for LES/RANS), simle drag/lift laws tyically emloyed in PP do not cature the unsteady wake effects commonly observed in full DNS studies [14, 15]). These effects become even more ronounced in dense articulate regions. In many ractical alications, the local article size and concentrations may vary substantially. In liquid atomization rocess, e.g., the drolet sizes may range from 1 mm to 1 µm with dense regions near the injector nozzle. The oint-article assumtion is invalid under these conditions. In the resent work, we extend the oint-article aroach by accounting for the volume dislacements of the carried hase due to the motion of articles or drolets. The diserse hase also affects the carrier hase through mass, momentum, and energy couling. The combined effect is termed as volumetric couling. This aroach is based on the the original formulation by Duckowicz [1] and later modified by Joseh & Lundgren [16]. The aroach is derived based on mixture theory that account for the drolet or article) volume fraction in a given comutational cell. This effect is imortant in dense sray regimes, however, are tyically ignored in the context of LES or DNS simulations [17, 12]. A similar formulation has been alied to bubbly flows at low bubble concentrations u to.2) to investigate the effect of bubbles on drag reduction in turbulent flows [8, 18]. Several studies on laminar dense granular flows [19, 2, 21] also use this aroach. Recently, Ate etal. [22] have shown the effect of volumetric dislacements on the carrier fluid in dense articleladen flows. They comared the solutions for the carrier hase and the article disersion obtained from the oint-article assumtion and by accounting for volumetric dislacements to show large differences. If the volume dislaced by the diserse hase is taken into account, the velocity field is no longer divergence free. This has a direct effect on the ressure Poisson equation, altering the ressure field through a local source/sink term. These effects may become imortant in dense regions of sray system. However, comuting dense sray systems by accounting for volume dislacements due to drolet motion could be numerically challenging. The temoral and satial variations in fluid volume fractions could be locally large and causing numerical instabilities. This is secifically true if the interhase couling of mass, momentum, and energy is treated exlicitly. In the resent work, we focus on non-reacting flows and only momentum exchange between the two-hases is considered. A numerical aroach based on co-located grid finite-volume method is develoed with art of the momentum exchange terms treated imlicitly. The aroach is similar to the fractional ste algorithms for articlein cell methods on staggered grids [19, 21, 2]. Imlementation in co-located finite-volume formulation is discussed and is alicable to unstructured grids. 1 In this aer, article may mean solid article, liquid drolets, or bubbles deending uon the case being studied.
Governing Equations The formulation described below consists of the Eulerian fluid and Lagrangian article equations, and accounts for the dislacement of the fluid by the articles as well as the momentum exchange between them [16]). An Eulerian-Lagrangian framework is used to solve the couled two-hase flow equations. The diserse hase equations are solved in a Lagrangian frame with models for drag, buoyancy, and inter-article collision forces. Continuum-hase equations In the resent formulation, both continuity and momentum equations account for the local concentration of articles in the continuum hase. The fluid mass for unit volume satisfies a continuity equation, t ρ f Θ f ) + ρ f Θ f u f ) = 1) where ρ f, Θ f, and u f are density, concentration, and velocity of the fluid hase, resectively. Local satio-temoral variations of article concentration, generate a non-divergence free velocity field in the flow. The non-zero velocity divergence can be shown by rearranging equation 1. u f = 1 DΘ f Θ f Dt 2) where D Dt is the material derivative. Fluid concentration is calculated as Θ f = 1 Θ, where Θ is article concentration. Lagrangian quantities, such as article concentration, are interolated to the Eulerian control volumes. N Θ x cv ) = V G x cv, x ) 3) =1 where x cv and x are control volume and article ositions, resectively, V is the article volume, G is the interolation function, N is the total number of articles, and the summation is over all articles. Momentum conservation is satisfied by solving t ρ f Θ f u f ) + ρ f Θ f u f u f ) = Θ f ) + µ f D c ) + F 4) where is dynamic ressure, µ f is the fluid viscosity, D c = u c + u T c is the deformation tensor of the mixture, u c = Θ f u f + Θ u s is the comosite velocity of mixture, and F is the reaction force from the article hase on the fluid hase er unit mass of fluid. Average article velocity in control volume, u s is calculated using the interolation function N Θ u s = V G x cv, x ) u 5) =1 where u is the article velocity. Particle-hase equations Position and velocity of articles are calculated by solving the ordinary differential equations of motion, d dt x ) = u 6) m d dt u ) = F 7) where x and u are article osition and velocity, m is the mass of article, and F = m A is the total force acting on article, and A is the article acceleration. In this study, only the effect of drag, gravitational force, and the inter-article collisions are considered. For high density ratios between the diserse hase and the carrier hase tyical of sray systems), the lift forces, added mass, and history forces are much smaller than the drag force and are neglected. The total article acceleration is given as: A = D u f, u ) + 1 ρ ) f g +A c }{{} Adrag ρ }{{} Agravity 8) where u f, is the fluid velocity at the article osition and A c is the article acceleration due to inter-article collisions. The inter-article force is modeled by the discrete-element method of Cundall & Strack as described by [2]. The drag force is caused by the motion of a article through the gas. In the drag model D is defined as D = 3 8 C ρ f u f, u d 9) ρ r where C d is given by [23] C d = 24 ) 1 + are b Θ 2.65 f, Re < 1 Re C d =.44Θ 2.65 f, Re > 11) where the article Reynolds number is defined as and Re = 2ρ f Θ f u f, u r µ f 11) r = is the article radius. ) 1/3 3V 12) 4π
time Fluid hase ρ n+3/2, φ n+3/2 ρu n+1, ρv n n+1 ρ n+1/2, φ n+1/2 ρu n, ρv n n t n+2 t n+3/2 t n+1 t n+1/2 t n Particle hase x n+3/2,θ n+3/2 u n+1,f n+1 x n+1/2,θ n+1/2 Figure 1: Staggering of variables of each hase Numerical Scheme The numerical scheme is based on a co-located grid, fractional ste, finite-volume aroach. The fluid flow is solved on a structured grid generalization to unstructured grids are feasible [24] For the resent volumetric couling, fluid flow equations become similar to the variable-density low-mach number formulation [12]. The numerical scheme resented here has the following imortant features: i) a time-staggered, co-located grid based fractional ste scheme, ii) low-mach number variable density flow solver, iii) accounting for volume dislacement effect of the Lagrangian articles on the fluid flow, iv) imlicit couling of article-fluid momentum exchange in the numerical solution, and v) using Gaussian kernel for interolation of Lagrangian quantities to the Eulerian grid. In many article-laden flow regimes, where the article loading is high, the effect of article reaction force on the flow is imortant. In regions of dense loading, the momentum couling force could be very large, and its exlicit treatment affects the robustness of the flow solver. An imlicit treatment of the reaction force is thus necessary. In simulations considered here only the inter-hase drag force is treated imlicitly. Numerical solution of the governing equations of continuum and article hases are staggered in time to maintain time-centered, secondorder advection of the article and fluid equations. Figure 1 shows staggering of variables of each hase in time. Denoting the time level by a suerscrit index, the velocities are located at time level t n and t n+1, and ressure, density, viscosity, and the color function at time levels t n 1/2 and t n+1/2. Particle velocity u ) and inter-hase couling force F) are treated at times n and n + 1, whereas article osition x ) and concentration Θ ) are calculated at times n + 1/2 and n + 3/2. u n The continuity equation of the fluid hase is discretized as ρ n+3/2 ρ n+1/2 + 1 g N ) n+1 A face = V cv faces of cv 13) where N stands for face-normal, face for face of a control volume cv), and g n+1 N = ρn+1 u n+1 N and ρ = ρ f Θ f. Particle velocity in the imlicit formulation is given by u n+1 u n = ) u n+1 u n+1 f, +A n+1 c + τ r 1 ρ f ρ where u n+1 f, at the article location. This gives, [ u n+1 = u n + ) g 14) is the interolated velocity of fluid hase 1 1 + τ r A n+1 c + τ r ) u n+1 f, + 1 ρ ) ] f g ρ 15). Note that for an isolated article, in the absence of any external forces, for an extremely heavy article τ r and we get u n+1 u n. Whereas for a massless article, τ r and we obtain u n+1 u n+1 f,. The numerical algorithm consists of the following stes: Ste 1 First obtain drag and collision forces at time n and udate the article osition exlicitly: x n+1 = x n+1/2 + u n+1 = x n+1/2 + u n + A n ) where A is the total article acceleration from 8. Based on the new article ositions, the interarticle acceleration due to collision is comuted at the new osition. Then set A n+1 c = A n+1/2 c + A n+3/2 c )/2. Ste 2 Comute the article and fluid volume fractions at x n+3/2 by interolating from the Lagrangian article ositions to the Eulerian grid cv centers. Set redictors for fluid hase density and face-normal velocity u N ) as ρ = ρ n+3/2 Θ n+3/2 f u N = u n+1 N
Ste 3 Advance the gas-hase momentum equation using the fractional ste method [25], ρ u i + 1 2V + 1 2V faces of cv faces of cv [ u n i,face + u i,face] g n+1/2 N A face = µ face u i,face x j 2 3 x j ρ n u i n + un i,face x j µ un k δ ij x k n x ) i A face ) + F i where g n+1/2 N = gn + gn N )/2 and g N = ρ u N. Using the article momentum equation 14), gives the imlicit formula for the fluid hase advancement. The reaction force Fi is written as Fi = T u i ) + T u n ),i + T A n+1 c,i + ) An+1 gravity,i where the oerator T is [ T = G x n+1 ) m /τ r 1 + τ r The first term on the right hand side of Fi in terms of u i. Ste 4 Ste 5 ]. 16) is imlicit Remove the old ressure gradient to obtain ĝ i = gi + n x i 17) Interolate the velocity field to the control volume faces and solve the Poisson s equations is solved for ressure, 2 ) = 1 V Ste 6 faces of cv ĝ i,face A face + ρn+3/2 ρ n+1/2 18) Comute the new face-velocities satisfying continuity equation 13 Ste 7 g n+1 N ĝ N = n+1 N Reconstruct the ressure gradient, n+1 x i = n+1 N )LS 19) 2) where ) LS stands for least-squares interolation used by [24]. Now udate the cv center velocities Ste 8 g n+1, i ĝ i = n+1 x i 21) Now advance the article velocity field using equation 14 and the interolated carrier-hase velocity field u n+1, i,f = g n+1, i /ρ n+1. m u n+1 dt ) u n u n+1, f, u n+1 = m + A n+1 c + 1 ρ g τ r ρ Ste 9 = F n+1 = m A n+1 ) ) g In general on non-uniform grids), the interolation oerator from the grid CVs to the article location and the inverse oerator from the article location to the grid CVs) may not commute. To obtain discrete momentum conservation between the two-hases, any residual force is alied to the carrier-hase velocity field in an exlicit form, [ u n+1, i u n+1, i,f ρ n+1 u n+1 i = ρ n+1 u n+1, i ] )m /τ r 1 + /τ r ]) )m /τ r G x n+1 [ G x n+1 1 + /τ r 22) The above correction is usually small and does not introduce time-ste restrictions comarable to fully exlicit interhase couling. Results The numerical scheme is alied to different test cases in order to evaluate its accuracy and robustness. These test cases are described below. Oscillating bubble First we show the imortance of volumetric dislacement effect on the flow field, caused by change in local concentration of articles. The variable density formulation used in these simulations accounts for changes in the density of mixture. This can be the result of article accumulation/scattering in the flow field due to inter-hase momentum exchanges, or size variation in a cavitating bubble due to hydrodynamic ressure of the flow, etc. Here we set u a very simle case of imosed oscillation on the radius of a bubble which causes a otential flow filed
.8.6 Y.4.2.2.4.6.8.1 Figure 3: Velocity vectors around the bubble caused by radius variations 1 1.8 Pressure.99995.9999 Y.6.4.99985.2.9998 -.3.3 around itself. This henomenon when simulated using only two-way couling no volumetric effects), no flow is generated, whereas with volumetric couling otential flow is obtained. We ut a single air bubble in a cube of water and imose sinusoidal erturbation on the bubble radius. Bubble radius changes in time as R = R +ɛ sinωt), where R and R are the instantaneous and the initial radius, resectively, ɛ is the erturbation magnitude, ω is frequency and t is time. In this simulation, R =.1 D, where D is the domain size, and gives overall concentration of 4 1 6, ɛ =.1 R, ω=5 [Hz]. Figure 2 shows the radial distribution of hydrodynamic ressure around the bubble created by the size variation at t =.3 where t = t/t and T = 2π/ω. We comare the ressure with analytr [m].2.4.6.8 1 Figure 2: Pressure distribution caused by volume dislacement around the bubble, from two-way couling dashed line), volumetric couling solid line), and analytical solution dots). ical solution dots), given by [26] and results with two-way couling no volumetric effect dashed line). The two-way couling did not show any effect on the ressure, however the volumetric couling gives good agreement with the analytical solution. In another similar examle we consider two bubbles oscillating in tandem. Two similar bubbles are ut in a box and their radius changes sinusoidally with π [rad] hase shift. All roerties are similar to the case of single bubble, excet they are both located D/6 away from the box center. The result is a doublet-like flow which is shown in figure 4. Again in this case we did not observe any effect on the flow using only two-way couling. Gravitational Settling We simulate sedimentation of solid articles under gravity in a rectangular box. Details of this case are given in Table 1. The initial arcel osi- Figure 4: Doublet generated by bubbles oscillating in tandem tions are generated randomly over the entire length of the box. These arcels are then allowed to settle through the gas-medium under gravity. The dominant forces on the articles include gravity and interarticle/article-wall collision. As the articles hit the bottom wall of the box, they bounce back and sto the incoming layer of articles, and finally settle to a close ack limit.6). Figures 5a-c) show the time evolution of article ositions in the rectangular box. The articles eventually settle down with close-acking near the bottom wall. Figure 5d shows the temoral evolution of the interface. The numerical formulation for volumetric couling redicts the interface evolution similar to the analytical estimate h = gt 2 /2. As the articles settle, they accelerate the fluid in the uward direction, however, the effect of the drag force on the article motion was found to be small. The volume fraction of the articles reaches the theoretical maximum.6 is the close ack limit) as they accumulate near the bot-
Table 1: Parameter descrition for gravitydominated sedimentation. Comutational domain,.2.6.275 m Grid 1 3 5 Fluid density 1.25 kg/m 3 Particle Density 25 kg/m 3 Number of Parcels 1 Particles er arcel 3375 Diameter of articles 5 µm Initial concentration.2 t=.6.4 Y.2 t=.2 t=.4.2.2.2 Height m).6.3 Vol. Couling Analytical.2.4.6 time s) Figure 5: Temoral evolution of article distribution during gravity-dominated sedimentation: a) t =, b) t =, c) t =, d) Height from bottom wall comared with theory H = H.5gt 2 ) tom wall. The numerical scheme was able to handle large variations in the volume fraction. Rayleigh-Taylor instability We consider the sedimentation case generating Rayleigh-Taylor instability similar to that studied by Snider [21]. A set of heavy articles are initially arranged uniformly above a light fluid and the initial concentration is aroximately.38. The interface between the articles and the fluid is erturbed by a cosine wave initially which causes an exonential growth in the mixture. The arameters in this study are resented in table 2. The comutational domain is [1 4] and the grid resolution used is 64 256. Sli wall conditions are used for the to and bottom walls and eriodic conditions are used in the x direction. Figure 6: Time evolution of Rayleigh-Taylor instability Figure shows the time evolution of the article volume fraction. The initial erturbation grows as the articles are accelerated downward by gravity. In the central region, the falling articles ush the fluid downward which rises from the edges of the comutational domain ushing the articles uward in Rayleigh-Taylor instability. Particles fall at a higher rate than their terminal velocity. This is due to the effect of uward flow generated by this motion near the edges of the comutational domain. The circulation caused by this motion is shown in figure 7. As resented by [21], this height H) is a function of Atwood number, defined as A = ρ ρ f )/ρ + ρ f ), gravity, and time. Initially, it grows exonentially in time until the interface deforms. Y.4.2 Particle radius [µm] 325 Fluid density [kg/m 3 ].1694 Particle density [kg/m 3 ] 1.225 Initial article volume fraction 3.77 1 2 Number of articles 131, Gravity in y direction [m/s 2 ] -9.81 -.2 -.4 -.4 -.2.2.4 Table 2: Fluid and article roerties in Rayleigh- Taylor instability. Figure 7: Circulation generated by articles
Particle-laden jet imingement This test case is similar to that studied by Snider [21] and evaluates the robustness of the current numerical aroach for dense article systems. A jet of articles from a 1.5 cm tube is directed onto a flat late at high velocity. Particles are fed at an initial volume fraction of.3 at a velocity of 25 m/s. Table 3 shows the flow arameters used. At the to and bottom boundaries of the domain, no-sli conditions are alied. The left and right boundaries are considered outflow. A stable solution is obtained for large variations in the article volume fractions in this dynamic roblem. An exlicit drag force resulted in blow-u of the flow solver. Particle radius [µm] Fluid density [kg/m3 ] Particle density [kg/m3 ] Initial article volume fraction Gravity in y direction [m/s2 ] Time ste [s] Comutational Domain [cm] Grid 1 1 276.3-9.81 5 1 5 27 27 17 24 24 14 U 5 7 8 1 12 14 15 17 19 2 22 24 25 27 29 a) c) b) Θ.1.1.19.28.37.46 a).55.64.73 c) b) U mag 6 7 8 9 1 11 12 a) 13 14 15 16 16 17 18 19 c) b) P Table 3: Fluid and article roerties in articleladen jet imingement case Particle-laden jet in cross flow Finally, we simulate the effect of article-laden jet on a laminar channel flow. The inlet flow is a lane Poiseuille flow and the articles are injected at x =.1[m] away from the inlet. Table 4 shows the arameters of this simulation. Particles are added continuously in the form of a round circular jet. The maximum article volume fraction inside the jet is aroximately.2. Figure 9 shows the velocity vecchannel height [m] Channle length [m] Maxium flow velocity at inlet [m/s] Particle vertical velocity comonent at injection [m/s] Fluid density [kg/m3 ] Particle density [kg/m3 ] Particle diameter [µm] Maximum article volume fraction Number of articles.2.5 5 1.25 1 1 4.2 16 Table 4: Simulation settings for article jet in cross flow tors of the channel flow under the influence of the article jet. The velocity vectors show a circula- -3 75 a) 45 825 12 1575 b) 195 2325 27 c) Figure 8: Time evolution of article-laden jet iminging on a flat late: i) article evolution, ii) fluid volume fraction, iii) fluid velocity magnitude, and iv) ressure. tion region generated behind the jet. Figure 1 also shows evolution of vorticity contours due to resence of the article jet. Both figures show that the combined effect of uward momentum from articles to the the continuum hase and volumetric dislacement effect due articles, generate a strong circulation in the flow field. Summary and Conclusion A numerical formulation based on timestaggered, co-located grid, finite volume aroach is develoed for simulation of dense article-laden flows. The original formulation for sray systems by Duckowiz [1] was used to discretize the governing equations for a non-reacting, incomressible fluid, laden with articles on structured grids. This formulation takes into account the fluid dislaced by
-.2 -.2 -.4 -.4 -.6 -.6 -.8 t = 3. x 1-2 -.8 t = 3. x 1-2 -.1 -.1 -.2 -.4 -.6 -.8 t = 5.5 x 1-2 -.1.5.1.15.2.25.3.35. -.2 -.4 -.6 -.8 ω z,max = 1 4 [Hz] t = 5.5 x 1-2 -.1.5.1.15.2.25.3.35. Figure 9: Velocity vectors generated by jet in cross flow at different times Figure 1: Vorticity contours generated by jet in cross flow at different times the article motion volumetric couling ). In addition, the interhase momentum couling is modeled through a reaction force exerted by the diserse hase on the carrier fluid. For dense regions of article concentrations, the interhase drag force is treated imlicitly in a fractional ste method to imrove robustness of the aroach. Several test cases are considered to evaluate the accuracy and robustness of the numerical scheme. First, the effect of a single bubble undergoing forced eriodic oscillations is comuted by considering the resent aroach as well as the standard two-way couling based on oint-article method to show large variations in the redicted flow field. The results are comared with analytical solutions to validate the numerical aroach for volumetric couling. A test case with two bubbles undergoing forced oscillations in tandem is also investigated. The doublet-like flow attern is well redicted by the resent aroach. Next, standard test cases of i) gravitational settling, ii) Rayleigh-Talor instability, iii) article-laden jet imingement on a flat late, and iv) article-laden jet in a cross flow are simulated to test the robustness of the numerical scheme under dense loading. The numerical aroach is fully three-dimensional and alicable to structured or unstructured grids. The resent numerical aroach is caable of simulated dense regions of sray systems near the injector. Two issues need further investigation: i) This aroach, requires that the grid used for fluid flow solver be coarser than the size of the articles or drolets). As the comutational grid becomes comletely occuied by the diserse hase, a con- tinuum formulation should be used for the disersed hase. A hybrid aroach combining discrete and continuum aroaches is necessary to model the different regimes of dense sray systems, ii) Equations, closure models, and numerical techniques for volumetric couling in the context of large-eddy simulation are needed for dense article/drolet-laden systems. The fluid volume fraction Θ f ) together with the fluid density ρ f can be used to define an effective fluid density ρ = ρ f Θ f ) and density-weighted Favre-averaging can be used to derive filtered equations for the resolved scales in LES. Models for closure of the subgrid-scale terms involving fluid volume fraction need to be develoed, and will be the focus on future work. References [1] John K. Dukowicz. Journal of Comutational Physics, 352):229 253, 198. [2] PJ O Rourke and FV Bracco. Proc. of the Inst. of Mech. Eng, 9:11 16, 198. [3] S. Tanguy and A. Berlemont. Int. J. Mult. Flow, 31:115 135, 25. [4] M. Gorokhovski and M. Herrmann. Ann. Rev. Fl. Mech., 4:343 366, 28. [5] S. Elghobashi. Flow, Turbulence and Combustion, 524):39 329, 1994. [6] Walter C. Reade and Lance R. Collins. Physics of Fluids, 121):253 254, 2.
[7] D.W.I. ROUSON and J.K. EATON. Journal of Fluid Mechanics, 428:149 169, 21. [8] JIN U, M.R. MAEY, and G.E.M. KARNI- ADAKIS. Journal of Fluid Mechanics, 468:271 281, 22. [9] Qunzhen Wang and Kyle D. Squires. Physics of Fluids, 85):127 1223, 1996. [1] S. V. Ate, K. Mahesh, P. Moin, and J. C. Oefelein. International Journal of Multihase Flow, 298):1311 1331, 23. [24] K. Mahesh, G. Constantinescu, and P. Moin. Journal of Comutational Physics, 1971):215 24, 24. [25] K. Mahesh, G. Constantinescu, S. Ate, G. Iaccarino, F. Ham, and P. Moin. Journal of Alied Mechanics, 73:374, 26. [26] R.L. Panton. Incomressible Flow. Wiley- Interscience, 26. [11] J.C. Segura, J. K. Eaton, and J. C. Oefelein. PhD thesis, STANFORD UNIVERSITY, 25. [12] P. Moin and S.V. Ate. acceted for ublication in AIAA J., 25. [13] M. Sommerfeld, A. Ando, and D. Wennerberg. ASME, Transactions, Journal of Fluids Engineering, 1144):648 656, 1992. [14] T.M. Burton and J.K. Eaton. Journal of Fluid Mechanics, 545:67 111, 25. [15] P. Bagchi and S. Balachandar. Physics of Fluids, 15:3496 3513. [16] DD Joseh, TS Lundgren, R. Jackson, and DA Saville. International journal of multihase flow, 161):35 42, 199. [17] S. V. Ate, M. Gorokhovski, and P. Moin. International Journal of Multihase Flow, 299):153 1522, 23. [18] A. Ferrante and S. Elghobashi. Physics of Fluids, 152):315 329, 23. [19] MJ Andrews and PJ O Rourke. International Journal of Multihase Flow, 222):379 42, 1996. [2] N. A. Patankar and D. D. Joseh. International Journal of Multihase Flow, 271):1685 176, 21. [21] D. M. Snider. Journal of Comutational Physics, 172):523 549, 21. [22] S. V. Ate, K. Mahesh, and T. Lundgren. International Journal of Multihase Flow, 343):26 271, 28. [23] D. Gidasow. Multihase Flow and Fluidization: Continuum and Kinetic Theory Descritions. Academic Press, 1994.