ILASS Americas 21st Annual Conference on Liquid Atomization and Spray Systems, Orlando, FL, May 2008

Modeling Volumetric Coupling of the Dispersed Phase using theEulerian-Lagrangian Approach

Ehsan Shams∗and Sourabh V. ApteSchool of Mechanical, Manufacturing and Industrial Engineering

Oregon State University204 Rogers Hall, Corvallis, OR 97331

AbstractThe Eulerian-Lagrangian approach is commonly used in modeling two-phase flows wherein liquid droplets,solid particles, or bubbles are dispersed in a continuum fluid of a different phase. Typically, the motion of thedispersed phase is modeled by assuming spherical, point-particles with models for added mass effects, drag,and lift forces. The effect of the dispersed phase on the fluid flow is modeled using reaction forces in the fluidmomentum equation. Such an approach is valid for dilute regions of the dispersed phase. For dense regions,however, the point-particle approach does not capture the interactions between the fluid and the dispersedphase accurately. In this work, the fluid volume displaced by the dispersed phase is taken into account tomodel the dense regions. The motion of the dispersed phase results in local, spatio-temporal variations of thevolume fraction fields. The resultant divergence in the fluid velocity acts as a source or sink displacing theflow due to dispersed phase and is termed as volumetric coupling. The size of the dispersed phase is assumedsmaller than the grid resolution and for the continuum phase. The variable–density, low–Mach numberequations based on mixture theory are solved using a co–located, finite volume scheme. The interphasemomentum exchange due to drag forces is treated implicitly to provide robustness in the dense regions.The volumetric coupling approach is first validated with analytical studies for flow induced by oscillatingbubbles and gravitational settling of particles. Simulations of Rayleigh-Taylor instability, particle-laden jetimpingement on a flat plate, and particle-laden jet in a cross are performed to test the robustness of thescheme.

∗Corresponding Author: shamssoe@engr.orst.edu

IntroductionMajority of spray systems in propulsion ap-

plications involve complex geometries and highlyunsteady, turbulent flows near the injector. Thenumerical models for spray calculations should beable to accurately represent droplet deformation,breakup, collision/coalescence, and dispersion dueto turbulence. In the traditional approach for spraycomputation, the Eulerian equations for gaseousphase are solved along with a Lagrangian model forparticle transport with two-way coupling of mass,momentum, and energy exchange between the twophases [1]. Typically simulations of spray sys-tems use DNS, LES or RANS for the carrier phasewhereas the motion of the dispersed phase is mod-eled. The ‘point-particle’ (PP) assumption is com-monly employed where forces on the dispersed phaseare computed through model coefficients. The effectof the particles1 on the carrier phase is representedby a force applied at the centroid of the particle.The disperse phase equations are typically solved ina Lagrangian frame by tracking a set of computa-tional particles or parcels [2] with models for dropletbreakup, collision/coalescence, evaporation, disper-sion, and deformation. Fully resolved simulationsinvolving comprehensive modeling of interfacial dy-namics are being developed [3, 4], however, are com-putationally expensive.

Several simulations of particle-laden flows havebeen performed with the carrier fluid simulated us-ing direct numerical simulation ([5, 6],[7],[8]), large-eddy simulation ([9, 10, 11, 12]), or Reynolds-averaged Navier Stokes equations [13], where the dis-persed phase is assumed subgrid (so dp < LK , theKolmogorov length scale, for DNS whereas dp < ∆,the grid size, in LES or RANS). However, modelingthe dispersed phase using point-particle approachdoes not always provide the correct results. Formoderate loadings and wall-bounded flows [11] haveshown that the point-particle approximation fails topredict the turbulence modulation compared to ex-perimental values. In addition, if the particle sizeis comparable to the Kolmogorov scale (for DNS)or the grid size (for LES/RANS), simple drag/liftlaws typically employed in PP do not capture theunsteady wake effects commonly observed in fullDNS studies ([14, 15]). These effects become evenmore pronounced in dense particulate regions. Inmany practical applications, the local particle sizeand concentrations may vary substantially. In liq-uid atomization process, e.g., the droplet sizes may

1In this paper, particle may mean solid particle, liquiddroplets, or bubbles depending upon the case being studied.

range from 1 mm to 1 µm with dense regions nearthe injector nozzle. The point-particle assumptionis invalid under these conditions.

In the present work, we extend the point-particleapproach by accounting for the volume displace-ments of the carried phase due to the motion of par-ticles or droplets. The disperse phase also affects thecarrier phase through mass, momentum, and energycoupling. The combined effect is termed as ‘volu-metric coupling’. This approach is based on the theoriginal formulation by Duckowicz [1] and later mod-ified by Joseph & Lundgren [16]. The approach isderived based on mixture theory that account forthe droplet (or particle) volume fraction in a givencomputational cell. This effect is important in densespray regimes, however, are typically ignored in thecontext of LES or DNS simulations [17, 12]. A sim-ilar formulation has been applied to bubbly flows atlow bubble concentrations (up to 0.02) to investigatethe effect of bubbles on drag reduction in turbulentflows [8, 18]. Several studies on laminar dense granu-lar flows [19, 20, 21] also use this approach. Recently,Apte etal. [22] have shown the effect of volumetricdisplacements on the carrier fluid in dense particle-laden flows. They compared the solutions for thecarrier phase and the particle dispersion obtainedfrom the point-particle assumption and by account-ing for volumetric displacements to show large dif-ferences. If the volume displaced by the dispersephase is taken into account, the velocity field is nolonger divergence free. This has a direct effect on thepressure Poisson equation, altering the pressure fieldthrough a local source/sink term. These effects maybecome important in dense regions of spray system.

However, computing dense spray systems by ac-counting for volume displacements due to dropletmotion could be numerically challenging. The tem-poral and spatial variations in fluid volume frac-tions could be locally large and causing numericalinstabilities. This is specifically true if the inter-phase coupling of mass, momentum, and energy istreated explicitly. In the present work, we focus onnon-reacting flows and only momentum exchangebetween the two-phases is considered. A numeri-cal approach based on co-located grid finite-volumemethod is developed with part of the momentumexchange terms treated implicitly. The approach issimilar to the fractional step algorithms for particle-in cell methods on staggered grids [19, 21, 20]. Im-plementation in co-located finite-volume formulationis discussed and is applicable to unstructured grids.

Governing EquationsThe formulation described below consists of the

Eulerian fluid and Lagrangian particle equations,and accounts for the displacement of the fluid by theparticles as well as the momentum exchange betweenthem ([16]). An Eulerian-Lagrangian framework isused to solve the coupled two-phase flow equations.The disperse phase equations are solved in a La-grangian frame with models for drag, buoyancy, andinter-particle collision forces.

Continuum-phase equationsIn the present formulation, both continuity and

momentum equations account for the local concen-tration of particles in the continuum phase. Thefluid mass for unit volume satisfies a continuity equa-tion,

∂

∂t(ρfΘf ) +5 · (ρfΘfuf ) = 0 (1)

where ρf , Θf , and uf are density, concentration,and velocity of the fluid phase, respectively. Localspatio-temporal variations of particle concentration,generate a non-divergence free velocity field in theflow. The non-zero velocity divergence can be shownby rearranging equation 1.

∇ · uf = − 1Θf

DΘf

Dt(2)

where DDt is the material derivative.

Fluid concentration is calculated as Θf = 1−Θp,where Θp is particle concentration. Lagrangianquantities, such as particle concentration, are inter-polated to the Eulerian control volumes.

Θp (xcv) =Np∑p=1

VpG∆ (xcv,xp) (3)

where xcv and xp are control volume and particlepositions, respectively, Vp is the particle volume, G∆

is the interpolation function, Np is the total numberof particles, and the summation is over all particles.

Momentum conservation is satisfied by solving

∂

∂t(ρfΘfuf ) +∇ · (ρfΘfufuf ) =

−∇ (Θfp) +∇ · (µfDc) + F (4)

where p is dynamic pressure, µf is the fluid viscos-ity, Dc = ∇uc + ∇uT

c is the deformation tensor ofthe mixture, uc = Θfuf + Θpus is the compositevelocity of mixture, and F is the reaction force fromthe particle phase on the fluid phase per unit massof fluid. Average particle velocity in control volume,

us is calculated using the interpolation function

Θpus =Np∑p=1

VpG∆ (xcv,xp) up (5)

where up is the particle velocity.

Particle-phase equations

Position and velocity of particles are calculatedby solving the ordinary differential equations of mo-tion,

ddt

(xp) = up (6)

mpddt

(up) =∑

Fp (7)

where xp and up are particle position and velocity,mp is the mass of particle, and

∑Fp = mpAp is the

total force acting on particle, and Ap is the particleacceleration. In this study, only the effect of drag,gravitational force, and the inter-particle collisionsare considered. For high density ratios between thedisperse phase and the carrier phase (typical of spraysystems), the lift forces, added mass, and historyforces are much smaller than the drag force and areneglected. The total particle acceleration is givenas:

Ap = Dp (uf,p − up)︸ ︷︷ ︸Adrag

+(

1− ρfρp

)g︸ ︷︷ ︸Agravity

+Acp

(8)where uf,p is the fluid velocity at the particle po-sition and Acp is the particle acceleration due tointer-particle collisions. The inter-particle force ismodeled by the discrete-element method of Cundall& Strack as described by [20]. The drag force iscaused by the motion of a particle through the gas.In the drag model Dp is defined as

Dp =38Cdρfρp

| uf,p − up |rp

(9)

where Cd is given by [23]

Cd =24

Rep

(1 + aRebp

)Θ−2.65f , Rep < 1000

Cd = 0.44Θ−2.65f , Rep > 1000(10)

where the particle Reynolds number is defined as

Rep =2ρfΘf | uf,p − up | rp

µf(11)

and

rp =(

3Vp4π

)1/3

(12)

is the particle radius.

ρun, ρvn

n

ρn+3/2, φn+3/2

ρn+1/2, φn+1/2

ρun+1, ρvn

n+1

xp

n+3/2,Θp

n+3/2

xp

n+1/2,Θp

n+1/2

time

up

n+1,Fn+1

up

n

tn+2

tn+3/2

tn+1/2

tn

tn+1

Fluid phase Particle phase

Figure 1: Staggering of variables of each phase

Numerical SchemeThe numerical scheme is based on a co-located

grid, fractional step, finite-volume approach. Thefluid flow is solved on a structured grid (generaliza-tion to unstructured grids are feasible [24] For thepresent volumetric coupling, fluid flow equations be-come similar to the variable-density low-Mach num-ber formulation [12]. The numerical scheme pre-sented here has the following important features: (i)a time-staggered, co-located grid based fractionalstep scheme, (ii) low-Mach number variable den-sity flow solver, (iii) accounting for volume displace-ment effect of the Lagrangian particles on the fluidflow, (iv) implicit coupling of particle-fluid momen-tum exchange in the numerical solution, and (v) us-ing Gaussian kernel for interpolation of Lagrangianquantities to the Eulerian grid.

In many particle-laden flow regimes, where theparticle loading is high, the effect of particle reactionforce on the flow is important. In regions of denseloading, the momentum coupling force could be verylarge, and its explicit treatment affects the robust-ness of the flow solver. An implicit treatment ofthe reaction force is thus necessary. In simulationsconsidered here only the inter-phase drag force istreated implicitly. Numerical solution of the govern-ing equations of continuum and particle phases arestaggered in time to maintain time-centered, second-order advection of the particle and fluid equations.Figure 1 shows staggering of variables of each phasein time. Denoting the time level by a superscriptindex, the velocities are located at time level tn andtn+1, and pressure, density, viscosity, and the colorfunction at time levels tn−1/2 and tn+1/2. Particlevelocity (up) and inter-phase coupling force (F) aretreated at times n and n + 1, whereas particle po-sition (xp) and concentration (Θp) are calculated attimes n+ 1/2 and n+ 3/2.

The continuity equation of the fluid phase is dis-cretized as

ρn+3/2 − ρn+1/2

∆t+

1Vcv

∑faces of cv

(gN )n+1Aface = 0

(13)where N stands for face-normal, face for face of acontrol volume (cv), and gn+1

N = ρn+1un+1N and ρ =

ρfΘf .Particle velocity in the implicit formulation is

given by

un+1p − unp

∆t= −

(un+1p − un+1

f,p

τr

)

+An+1cp +

(1− ρf

ρp

)g (14)

where un+1f,p is the interpolated velocity of fluid phase

at the particle location. This gives,

un+1p =

11 + ∆t

τr

[unp +

(∆tτr

)un+1f,p +

∆tAn+1cp + ∆t

(1− ρf

ρp

)g]

(15)

. Note that for an isolated particle, in the absenceof any external forces, for an extremely heavy par-ticle τr → ∞ and we get un+1

p → unp . Whereasfor a massless particle, τr → 0 and we obtainun+1p → un+1

f,p . The numerical algorithm consistsof the following steps:

Step 1

First obtain drag and collision forces at time nand update the particle position explicitly:

xn+1p = xn+1/2

p + ∆tun+1p

= xn+1/2p + ∆t

(unp + ∆tAn

p

)where Ap is the total particle acceleration from 8.Based on the new particle positions, the inter-particle acceleration due to collision is computedat the new position. Then set An+1

cp = (An+1/2cp +

An+3/2cp )/2.

Step 2

Compute the particle and fluid volume fractionsat xn+3/2

p by interpolating from the Lagrangian par-ticle positions to the Eulerian grid cv centers. Setpredictors for fluid phase density and face-normalvelocity (uN ) as

ρ∗ = ρn+3/2Θn+3/2f

u∗N = un+1N

Step 3Advance the gas-phase momentum equation us-

ing the fractional step method [25],

ρ∗ui∗

∆t+

12V

∑faces of cv

[uni,face + u∗i,face

]gn+1/2N Aface =

ρnuin

∆t− ∂pn

∂xi

+1

2V

∑faces of cv

µface

(∂u∗i,face

∂xj+∂uni,face

∂xj

)Aface

−23∂

∂xj

(µ∂unk∂xk

δij

)+ F ∗

i

where gn+1/2N = (g∗N +gnN )/2 and g∗N = ρ∗u∗N . Using

the particle momentum equation (14), gives the im-plicit formula for the fluid phase advancement. Thereaction force F ∗

i is written as

F ∗i = −T (u∗i ) + T

(unp,i)

+ ∆tT(An+1cp,i +An+1

gravity,i

)where the operator T is

T =∑p

[G∆(xn+1

p )mp/τr

1 + ∆tτr

]. (16)

The first term on the right hand side of F ∗i is implicit

in terms of u∗i .

Step 4Remove the old pressure gradient to obtain

gi = g∗i + ∆t∂p

∂xi

n

(17)

Step 5Interpolate the velocity field to the control vol-

ume faces and solve the Poisson’s equations is solvedfor pressure,

52(p∆t) =1V

∑faces of cv

gi,faceAface+ρn+3/2 − ρn+1/2

∆t(18)

Step 6Compute the new face-velocities satisfying con-

tinuity equation 13

gn+1N − gN

∆t= − ∂p

∂N

n+1

(19)

Step 7Reconstruct the pressure gradient,

∂pn+1

∂xi=(∂pn+1

∂N

)LS(20)

where ( )LS

stands for least-squares interpolationused by [24]. Now update the cv center velocities

gn+1,∗i − gi

∆t= −∂p

n+1

∂xi(21)

Step 8Now advance the particle velocity field using

equation 14 and the interpolated carrier-phase ve-locity field un+1,∗

i,f = gn+1,∗i /ρn+1.

mp

(un+1p − unp

)dt

= Fn+1p

= mpAn+1p

= mp

(un+1,∗f,p − un+1

p

τr+ An+1

cp +(

1− ρgρp

)g

)

Step 9In general (on non-uniform grids), the interpo-

lation operator from the grid CVs to the particlelocation and the inverse operator (from the parti-cle location to the grid CVs) may not commute.To obtain discrete momentum conservation betweenthe two-phases, any residual force is applied to thecarrier-phase velocity field in an explicit form,

ρn+1un+1i = ρn+1un+1,∗

i

−∆t

(∑p

[un+1,∗i,fp

G∆(xn+1p )mp/τr

1 + ∆t/τr

]−

un+1,∗i

∑p

[G∆(xn+1

p )mp/τr

1 + ∆t/τr

])(22)

The above correction is usually small and does notintroduce time-step restrictions comparable to fullyexplicit interphase coupling.

ResultsThe numerical scheme is applied to different test

cases in order to evaluate its accuracy and robust-ness. These test cases are described below.

Oscillating bubbleFirst we show the importance of volumetric dis-

placement effect on the flow field, caused by changein local concentration of particles. The variable den-sity formulation used in these simulations accountsfor changes in the density of mixture. This can bethe result of particle accumulation/scattering in theflow field due to inter-phase momentum exchanges,or size variation in a cavitating bubble due to hy-drodynamic pressure of the flow, etc. Here we setup a very simple case of imposed oscillation on theradius of a bubble which causes a potential flow filed

around itself. This phenomenon when simulated us-ing only two-way coupling (no volumetric effects), noflow is generated, whereas with volumetric couplingpotential flow is obtained.

We put a single air bubble in a cube of water andimpose sinusoidal perturbation on the bubble radius.Bubble radius changes in time as R = R0 +ε sin(ωt),where R and R0 are the instantaneous and the ini-tial radius, respectively, ε is the perturbation magni-tude, ω is frequency and t is time. In this simulation,R0 = 0.01×D, where D is the domain size, and givesoverall concentration of 4×10−6, ε = 0.1×R0, ω=50[Hz]. Figure 2 shows the radial distribution of hy-drodynamic pressure around the bubble created bythe size variation at t∗ = 0.3 where t∗ = t/T andT = 2π/ω. We compare the pressure with analyt-

r [m]

Pre

ssur

e

-0.003 0 0.0030.9998

0.99985

0.9999

0.99995

1

Figure 2: Pressure distribution caused by volumedisplacement around the bubble, from two-way cou-pling (dashed line), volumetric coupling (solid line),and analytical solution (dots).

ical solution (dots), given by [26] and results withtwo-way coupling no volumetric effect (dashed line).The two-way coupling did not show any effect onthe pressure, however the volumetric coupling givesgood agreement with the analytical solution.

In another similar example we consider two bub-bles oscillating in tandem. Two similar bubbles areput in a box and their radius changes sinusoidallywith π [rad] phase shift. All properties are similarto the case of single bubble, except they are bothlocated D/6 away from the box center. The result isa doublet-like flow which is shown in figure 4. Againin this case we did not observe any effect on the flowusing only two-way coupling.

Gravitational SettlingWe simulate sedimentation of solid particles un-

der gravity in a rectangular box. Details of thiscase are given in Table 1. The initial parcel posi-

X

Y

0 0.002 0.004 0.006 0.008 0.010

0.002

0.004

0.006

0.008

Figure 3: Velocity vectors around the bubble causedby radius variations

X

Y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 4: Doublet generated by bubbles oscillatingin tandem

tions are generated randomly over the entire lengthof the box. These parcels are then allowed to settlethrough the gas-medium under gravity. The domi-nant forces on the particles include gravity and inter-particle/particle-wall collision. As the particles hitthe bottom wall of the box, they bounce back andstop the incoming layer of particles, and finally set-tle to a close pack limit (∼ 0.6). Figures 5(a-c) showthe time evolution of particle positions in the rect-angular box. The particles eventually settle downwith close-packing near the bottom wall. Figure 5dshows the temporal evolution of the interface. Thenumerical formulation for volumetric coupling pre-dicts the interface evolution similar to the analyticalestimate h = gt2/2. As the particles settle, they ac-celerate the fluid in the upward direction, however,the effect of the drag force on the particle motionwas found to be small. The volume fraction of theparticles reaches the theoretical maximum (0.6 is theclose pack limit) as they accumulate near the bot-

Table 1: Parameter description for gravity-dominated sedimentation.

Computational domain, 0.2× 0.6× 0.0275 mGrid 10× 30× 5Fluid density 1.25 kg/m3

Particle Density 2500 kg/m3

Number of Parcels 1000Particles per parcel 3375Diameter of particles 500 µmInitial concentration 0.2

X

Y

0 0.20

0.2

0.4

0.6t=0

X0 0.2

t=0.2

X0 0.2

t=0.4

time (s)

Hei

ght(

m)

0 0.2 0.4 0.60

0.3

0.6Vol. CouplingAnalytical

Figure 5: Temporal evolution of particle distributionduring gravity-dominated sedimentation: a) t = 0,b) t =, c) t =, d) Height from bottom wall comparedwith theory (H = H0 − 0.5gt2)

tom wall. The numerical scheme was able to handlelarge variations in the volume fraction.

Rayleigh-Taylor instabilityWe consider the sedimentation case generating

Rayleigh-Taylor instability similar to that studiedby Snider [21]. A set of heavy particles are initiallyarranged uniformly above a light fluid and the initialconcentration is approximately 0.038. The interfacebetween the particles and the fluid is perturbed bya cosine wave initially which causes an exponentialgrowth in the mixture. The parameters in this studyare presented in table 2. The computational domainis [1 4] and the grid resolution used is 64× 256. Slipwall conditions are used for the top and bottom wallsand periodic conditions are used in the x direction.

Particle radius [µm] 3250Fluid density [kg/m3] 0.1694Particle density [kg/m3] 1.225Initial particle volume fraction 3.77 ×10−2

Number of particles 131,000Gravity in y direction [m/s2] -9.81

Table 2: Fluid and particle properties in Rayleigh-Taylor instability.

Figure 6: Time evolution of Rayleigh-Taylor insta-bility

Figure shows the time evolution of the parti-cle volume fraction. The initial perturbation growsas the particles are accelerated downward by grav-ity. In the central region, the falling particles pushthe fluid downward which rises from the edges ofthe computational domain pushing the particles up-ward in Rayleigh-Taylor instability. Particles fallat a higher rate than their terminal velocity. Thisis due to the effect of upward flow generated bythis motion near the edges of the computationaldomain. The circulation caused by this motion isshown in figure 7. As presented by [21], this height(H) is a function of Atwood number, defined asA = (ρp−ρf )/(ρp+ρf ), gravity, and time. Initially,it grows exponentially in time until the interface de-forms.

X

Y

-0.4 -0.2 0 0.2 0.4

-0.4

-0.2

0

0.2

0.4

Figure 7: Circulation generated by particles

Particle-laden jet impingementThis test case is similar to that studied by

Snider [21] and evaluates the robustness of the cur-rent numerical approach for dense particle systems.A jet of particles from a 1.5 cm tube is directed ontoa flat plate at high velocity. Particles are fed at aninitial volume fraction of 0.3 at a velocity of 25 m/s.Table 3 shows the flow parameters used. At the topand bottom boundaries of the domain, no-slip condi-tions are applied. The left and right boundaries areconsidered outflow. A stable solution is obtained forlarge variations in the particle volume fractions inthis dynamic problem. An explicit drag force re-sulted in blow-up of the flow solver.

Particle radius [µm] 100Fluid density [kg/m3] 1Particle density [kg/m3] 2760Initial particle volume fraction 0.3Gravity in y direction [m/s2] -9.81Time step [s] 5× 10−5

Computational Domain [cm] 27× 27× 17Grid 24× 24× 14

Table 3: Fluid and particle properties in particle-laden jet impingement case

Particle-laden jet in cross flowFinally, we simulate the effect of particle-laden

jet on a laminar channel flow. The inlet flow is aplane Poiseuille flow and the particles are injected atx = 0.01[m] away from the inlet. Table 4 shows theparameters of this simulation. Particles are addedcontinuously in the form of a round circular jet. Themaximum particle volume fraction inside the jet isapproximately 0.2. Figure 9 shows the velocity vec-

Channel height [m] 0.02Channle length [m] 0.05Maxium flow velocity at inlet [m/s] 5Particle vertical velocity componentat injection [m/s] 1.25Fluid density [kg/m3] 1Particle density [kg/m3] 1000Particle diameter [µm] 40Maximum particle volume fraction 0.2Number of particles 106

Table 4: Simulation settings for particle jet in crossflow

tors of the channel flow under the influence of theparticle jet. The velocity vectors show a circula-

(a) (b) (c)

5 7 8 10 12 14 15 17 19 20 22 24 25 27 29Up

(c)(b)(a)

0.01 0.10 0.19 0.28 0.37 0.46 0.55 0.64 0.73

Θp

(a) (c)(b)

6 7 8 9 10 11 12 13 14 15 16 16 17 18 19

Umag

(c)(b)(a)

-300 75 450 825 1200 1575 1950 2325 2700

P

Figure 8: Time evolution of particle-laden jet im-pinging on a flat plate: (i) particle evolution, (ii)fluid volume fraction, (iii) fluid velocity magnitude,and (iv) pressure.

tion region generated behind the jet. Figure 10 alsoshows evolution of vorticity contours due to presenceof the particle jet. Both figures show that the com-bined effect of upward momentum from particles tothe the continuum phase and volumetric displace-ment effect due particles, generate a strong circula-tion in the flow field.

Summary and ConclusionA numerical formulation based on time-

staggered, co-located grid, finite volume approachis developed for simulation of dense particle-ladenflows. The original formulation for spray systemsby Duckowiz [1] was used to discretize the governingequations for a non-reacting, incompressible fluid,laden with particles on structured grids. This for-mulation takes into account the fluid displaced by

-0.01

-0.008

-0.006

-0.004

-0.002

0

t = 3.0 x 10 -2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.0-0.01

-0.008

-0.006

-0.004

-0.002

0

t = 5.5 x 10 -2

Figure 9: Velocity vectors generated by jet in crossflow at different times

the particle motion (‘volumetric coupling’). In ad-dition, the interphase momentum coupling is mod-eled through a reaction force exerted by the dis-perse phase on the carrier fluid. For dense regionsof particle concentrations, the interphase drag forceis treated implicitly in a fractional step method toimprove robustness of the approach. Several testcases are considered to evaluate the accuracy androbustness of the numerical scheme. First, the ef-fect of a single bubble undergoing forced periodicoscillations is computed by considering the presentapproach as well as the standard ‘two-way’ couplingbased on point-particle method to show large vari-ations in the predicted flow field. The results arecompared with analytical solutions to validate thenumerical approach for volumetric coupling. A testcase with two bubbles undergoing forced oscillationsin tandem is also investigated. The doublet-like flowpattern is well predicted by the present approach.Next, standard test cases of (i) gravitational settling,(ii) Rayleigh-Talor instability, (iii) particle-laden jetimpingement on a flat plate, and (iv) particle-ladenjet in a cross flow are simulated to test the robust-ness of the numerical scheme under dense loading.The numerical approach is fully three-dimensionaland applicable to structured or unstructured grids.

The present numerical approach is capable ofsimulated dense regions of spray systems near theinjector. Two issues need further investigation: (i)This approach, requires that the grid used for fluidflow solver be coarser than the size of the particles(or droplets). As the computational grid becomescompletely occupied by the disperse phase, a con-

-0.01

-0.008

-0.006

-0.004

-0.002

0

t = 3.0 x 10 -2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.0-0.01

-0.008

-0.006

-0.004

-0.002

0

t = 5.5 x 10 -2

ωz,max = 104 [Hz]

Figure 10: Vorticity contours generated by jet incross flow at different times

tinuum formulation should be used for the dispersedphase. A hybrid approach combining discrete andcontinuum approaches is necessary to model the dif-ferent regimes of dense spray systems, (ii) Equations,closure models, and numerical techniques for volu-metric coupling in the context of large-eddy simula-tion are needed for dense particle/droplet-laden sys-tems. The fluid volume fraction (Θf ) together withthe fluid density ρf can be used to define an effec-tive fluid density (ρ = ρfΘf ) and density-weightedFavre-averaging can be used to derive filtered equa-tions for the resolved scales in LES. Models for clo-sure of the subgrid-scale terms involving fluid vol-ume fraction need to be developed, and will be thefocus on future work.

References[1] John K. Dukowicz. Journal of Computational

Physics, 35(2):229–253, 1980.

[2] PJ O’Rourke and FV Bracco. Proc. of the Inst.of Mech. Eng, 9:101–106, 1980.

[3] S. Tanguy and A. Berlemont. Int. J. Mult.Flow, 31:1015–1035, 2005.

[4] M. Gorokhovski and M. Herrmann. Ann. Rev.Fl. Mech., 40:343–366, 2008.

[5] S. Elghobashi. Flow, Turbulence and Combus-tion, 52(4):309–329, 1994.

[6] Walter C. Reade and Lance R. Collins. Physicsof Fluids, 12(10):2530–2540, 2000.

[7] D.W.I. ROUSON and J.K. EATON. Journal ofFluid Mechanics, 428:149–169, 2001.

[8] JIN XU, M.R. MAXEY, and G.E.M. KARNI-ADAKIS. Journal of Fluid Mechanics, 468:271–281, 2002.

[9] Qunzhen Wang and Kyle D. Squires. Physicsof Fluids, 8(5):1207–1223, 1996.

[10] S. V. Apte, K. Mahesh, P. Moin, and J. C.Oefelein. International Journal of MultiphaseFlow, 29(8):1311–1331, 2003.

[11] J.C. Segura, J. K. Eaton, and J. C. Oefelein.PhD thesis, STANFORD UNIVERSITY, 2005.

[12] P. Moin and S.V. Apte. accepted for publicationin AIAA J., 2005.

[13] M. Sommerfeld, A. Ando, and D. Wennerberg.ASME, Transactions, Journal of Fluids Engi-neering, 114(4):648–656, 1992.

[14] T.M. Burton and J.K. Eaton. Journal of FluidMechanics, 545:67–111, 2005.

[15] P. Bagchi and S. Balachandar. Physics of Flu-ids, 15:3496–3513.

[16] DD Joseph, TS Lundgren, R. Jackson, andDA Saville. International journal of multiphaseflow, 16(1):35–42, 1990.

[17] S. V. Apte, M. Gorokhovski, and P. Moin.International Journal of Multiphase Flow,29(9):1503–1522, 2003.

[18] A. Ferrante and S. Elghobashi. Physics of Flu-ids, 15(2):315–329, 2003.

[19] MJ Andrews and PJ O’Rourke. Interna-tional Journal of Multiphase Flow, 22(2):379–402, 1996.

[20] N. A. Patankar and D. D. Joseph. InternationalJournal of Multiphase Flow, 27(10):1685–1706,2001.

[21] D. M. Snider. Journal of ComputationalPhysics, 170(2):523–549, 2001.

[22] S. V. Apte, K. Mahesh, and T. Lund-gren. International Journal of Multiphase Flow,34(3):260–271, 2008.

[23] D. Gidaspow. Multiphase Flow and Fluidiza-tion: Continuum and Kinetic Theory Descrip-tions. Academic Press, 1994.

[24] K. Mahesh, G. Constantinescu, and P. Moin.Journal of Computational Physics, 197(1):215–240, 2004.

[25] K. Mahesh, G. Constantinescu, S. Apte, G. Iac-carino, F. Ham, and P. Moin. Journal of Ap-plied Mechanics, 73:374, 2006.

[26] R.L. Panton. Incompressible Flow. Wiley-Interscience, 2006.