Large eddy simulations of two-phase turbulent flows...Both Eulerian and Lagrangian formulations have been used to simulate two-phase flows in the past (e.g., Mostafa and Mongia, 1983)

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Large Eddy Simulations of Two-Phase Turbulent Flows"

S. Pannalat and S. Menon*School of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, GA 30332-0150

ABSTRACTA two-phase subgrid combustion model has been

developed for large-eddy simulations (LES). Thisapproach includes a more fundamental treatment of theeffects of the final stages of droplet vaporization,molecular diffusion, chemical reactions and small scaleturbulent mixing than other LES closure techniques.As a result, Reynolds, Schmidt and Damkohler numbereffects are explicitly included. This model has beenimplemented within an Eulerian-Lagrangian two-phaseformulation. In this approach, the liquid droplets aretracked using the Lagrangian approach up to a pre-specified cut-off size. The evaporation of the dropletslarger than the cut-off size and the evaporation andmixing of droplets smaller than the cut-off size aremodeled within the subgrid using an Eulerian two-phase model. It is shown that droplets with order unityStokes number disperse more than small droplets inagreement with earlier numerical and experimentalstudies. Conventional and the present approach agreevery well when droplets do not fall below the cut-off.However, the present approach gives consistentlybetter results when the cut-off is increased, thereby,demonstrating an important advantage of the newmethod. The limitations of the current methodologyare also highlighted and possible solutions arediscussed.

INTRODUCTIONCombustion efficiency, reduced emissions and

stable combustion in the lean limit are some of thedesirable features in the next generation gas turbineengines. To achieve these capabilities, current researchis focusing on improving the liquid fuel atomization

t. GRA, Student Member, AIAA$. Professor, Senior Member, AIAA* Copyright © 1997 by S. Pannala and S. Menon. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc., with permission

process and to increase fuel-air mixing downstream ofthe fuel injector. However, the structure of complexthree-dimensional, fuel-air mixing layers is verydifficult to resolve using current experimental andnumerical methods. Since, fuel atomization and fuel-air mixing are both highly unsteady, conventionalsteady state methods cannot be used to elucidate thefiner details. On the other hand, although the unsteadymixing process can be studied quite accurately usingdirect numerical simulation (DNS) (e.g., Poinsot,1996), the application of DNS is limited to low tomoderate Reynolds numbers (Re) due to resolutionrequirements and therefore, cannot be used for high Reflows of current interest.

In LES, the scales larger than the grid arecomputed using a time- and space-accurate scheme,while the unresolved smaller scales are modeled.Closure of momentum and energy transport equationscan be achieved using a subgrid eddy viscosity modelsince the small scales primarily provide a dissipativemechanism for the energy transferred from the largescales. However, for combustion to occur, the speciesmust first undergo mixing and come into molecularcontact. These processes occur at the small scaleswhich are not resolved in the conventional LESapproach. As a result, conventional subgrid eddydiffusivity models cannot be used to model thesefeatures.

To address these issues, recently (Menon et al.,1993; Menon and Calhoon, 1996; Calhoon and Menon,1996, 1997) a subgrid combustion model wasdeveloped and implemented within the LESformulation. This model separately and simultaneouslytreats the physical processes of molecular diffusion andsmall scale turbulent convective stirring. This is incontrast to probability density function closure whichphenomenologically treats these two processes by asingle model, thereby removing experimentallyobserved Schmidt number variations of the flow.


The gas-phase methodology was recentlyextended to two-phase flows (Menon and Pannala,1997) to accurately capture the process of phasechange and turbulent mixing. In the present paper, thisapproach has been further refined and used to studyvaporization and the subsequent chemical reactions.Both infinite and finite-rate kinetics have beeninvestigated and discussed in this paper.

FORMULATIONBoth Eulerian and Lagrangian formulations have

been used to simulate two-phase flows in the past (e.g.,Mostafa and Mongia, 1983). However, most state-of-the-art codes employ the Lagrangian form to capturethe droplet dynamics, while the gas phase is computedin the Eulerian form (e.g., Oefelein and Yang, 1996).In this formulation, the droplets are tracked explicitlyusing Lagrangian equations of motion, and heat andmass transfer are computed for each droplet. Due toresource constraints (computer time and memory),only a limited range of droplet sizes are computed.Droplets below an ad hoc cut-off size are assumed tovaporize instantaneously and to become fully mixed inthe gas phase. This is a flawed assumption, since evenin pure gas flows small-scale mixing process is veryimportant for quantitative predictions (Menon andCalhoon, 1996). Here, the gas-phase subgridcombustion methodology has been extended to allowproper simulation of the final stages of dropletevaporation and turbulent mixing.

The two-phase subgrid process is implementedwithin the framework of the Eulerian-Lagrangian LESapproach. Thus, droplets larger than the cut-off size aretracked as in the usual Lagrangian approach. However,once the droplets are smaller than the cut-off, a two-phase subgrid Eulerian model is employed to includethe effects of the small droplets within the LES cells.

Gas Phase LES EquationsThe incompressible Navier Stokes equations in the

zero Mach number limit are employed for the presentstudy. Zero-Mach number approach involves using aseries expansion in terms of Mach number to removethe acoustic component from the equations and is awell established method (McMurtry et al., 1989;Chakravarthy and Menon, 1997).

The LES mass, momentum, energy and speciesequations in the zero-Mach number limit are:

+ U:aV. at,, .

•^ + Fs,i (2)

3.x. Y pa2r

dxkdxk(3)

3p+ (D

dt

The above system of equations is supplemented bythe equation of state for the thermodynamicpressure/? = pRT which can_be_used to obtain thetemperature T. Here, p, uit Ya and p are,respectively, the density, i-th velocity component, a-th species mass fraction and the kinematic pressure.Also, v, A,, D and R are, respectively, the kinematicviscosity, the thermal conductivity, the mass diffusion(assumed constant and same for all species here, butcan be generalized) and the gas constant. In Eq. (4),cba is the LES filtered species production/destructionterm. Also, in the above equations, source termsPS. Fs, Qs, and Ss represent the volume-averaged rateof exchange of mass, momentum, energy and speciesbetween the gas-phase and the liquid phase. Theseterms are computed, as detailed elsewhere (Oefeleinand Yang, 1996; Faeth, 1983) and, therefore, omittedhere for brevity. Furthermore, note that Eq. (3) is theequivalent energy equation in the zero-Mach numberlimit. In the absence of heat release and no phasechange, this equation and Eq. (1) will be identical.

In the above equations, the subgrid stress tensorltj = -(UjUj-Ujiij) and _ the species-velocitycorrelations 5a;- = -(Y au j-Y au j) require modeling.In the present LES approach, the stress term t/y ismodeled as itj = 2v,S^ where vr is the eddyviscosity and 5/;- is the resolved rate-of-strain tensor.The subgrid eddy viscosity is obtained in terms of thegrid scale _A_ and the subgrid kinetic energy,ks = (Ujiij-iijUj) as: v, = Cv/Jics&. Here, ks isobtained by solving a transport equation (e.g., Menonand Km, 1996). The coefficient Cv in the eddyviscosity model and the coefficients appearing in theks equation can be obtained using the dynamic


procedure as described elsewhere (Menon and Kim,1996).

Liquid-phase LES equationsA Stochastic Separated Flow (SSF) formulation

(Faeth, 1983; Oefelein and Yang, 1996) is used totrack the droplets using Lagrangian equations ofmotion. The general equations of spherical dropletsreduce to the following form (here, terms arising due tostatic pressure gradient, virtual-mass, Besset force andexternal body-forces are neglected for simplicity):

dx.dt

du»

(5)

(6)

Dsm are, respectively, the gas mixture density and themixture diffusion coefficient at the droplet surfaceand BM is the Spalding number which is givenas BM = (Y^r-Y^rVd - YS>F) . Here, Y,iF is thefuel mass fraction at the surface of the droplet andcomputed using the procedure described in Chen andShuen (1993), while Y^ F is the fuel mass fraction inthe ambient gas.

The heat transfer rate of the droplet (assuminguniform temperature in the droplet) is given by thefollowing relation (Faeth, 1983):

dt = hpndp(T-Tp)-mpAhv (9)

The heat transfer coefficient for a droplet in aconvective flow field with mass transfer is modeled as

where the droplet properties are denoted by subscriptp, dp is the droplet diameter and w,- is theinstantaneous gas phase velocities computed at thedroplet location. This gas phase velocity field isobtained using both the filtered LES velocity field u{

and the subgrid kinetic energy ksss (as in the eddyinteraction model). The droplet Reynolds number is

computed using: Rep=^[(Uj-up ,-)(«,• -upi)]

and the drag coefficient is modeled by (Faeth, 1983):

= 1+-0.278J?ep'/2/V1/3

+ 1.232//?epPr2/3]1/2(10)

Here, Pr is the gas phase Prandtl number and theheat transfer coefficient for quiescent medium is givenas hRe _ 0 = kNuRe =0/dp where the Nusselt

number is obtained from:

NuRep= 0 =

,£.£•'

(ID

CD =0.424

Re <10

Rep>l03(7)

The conservation of the mass of the dropletsresults is given by: dmp/dt = —m where the masstransfer rate for a droplet in a convective flow field isgiven as:

0.278/?ep1/25c1/3

(8)

Here, Sc is the Schmidt number and the subscriptRep = 0 indicates quiescent atmosphere when there isno velocity difference between the gas and the liquidphase. The mass transfer under this condition is givenas ™Ren = a = 2^9sDsmd ln(l+BM) . Here, p, and

NuRe _ 0 approaches a value of 2 in the case of

zero mass transfer and Le is the Lewis number. Onlydroplets above a cut-off diameter are solved using theabove equations, while the droplets below the cut-offdiameter are modeled using Eulerian formulationwithin the subgrid.

In summary, the present LES approach solves onlythe momentum equations on the LES grid. Closure forthe subgrid stresses is achieved by using a localizeddynamic model for the subgrid kinetic energy.Concurrently, the liquid phase equations are solvedusing the Lagrangian technique. The range of dropletsizes tracked depends on the computational constraints.The gas phase LES velocity field and the subgridkinetic energy are used to estimate the instantaneousgas velocity at the droplet location. This essentiallyprovides a coupling between the gas and liquid phasemomentum transport. The mass conservation and the


gas phase scalar field equations are simulated in thesubgrid domain as discussed in the next section.

SUBGRID COMBUSTION MODELSThe principle difficulty in reacting LES

simulations is the proper modeling of the combustionrelated terms involving temperature and species, forexample, the convective species fluxes such as 5aj dueto subgrid fluctuations and the filtered species massproduction rate <ba . Probability density functionmethods when applied within LES either usingassumed_shape or evolution equation may be used toclose ti)a and, in principle, any scalar correlations.However, the treatment of molecular mixing and smallscale stirring using phenomenological models as in pdfmethods, have not been very successful in predictingthe mixing effects. Problems have also been notedwhen the gradient diffusion model is used toapproximate the species transport terms.

The linear eddy mixing (LEM) model treatsseparately molecular diffusion and turbulent mixingprocesses at all relevant length scales of the flow. Thescalar fields are simulated within a ID domain which,in the context of LES, represents a ID slice of thesubgrid flame brush. The subgrid model simulates onlythe effect of the small unresolved scales on the scalarfields while the larger resolved turbulent scales of theflow are simulated by the LES equations. The subgridLEM has several advantages over conventional LES ofreacting flows. In addition to providing an accuratetreatment of the small-scale turbulent mixing andmolecular diffusion processes, this method avoidsgradient diffusion modeling of scalar transport. Thus,both co- and counter-gradient diffusion can besimulated. More details of this approach (which isidentical to the method used for gas phase LES) isgiven elsewhere (Menon and Calhoon, 1996; Calhoonand Menon, 1996, 1997) and therefore, avoided here.

The Linear-Eddy Single Phase ModelIn the baseline LEM model (e.g., Kerstein, 1989,

1992; Menon et al., 1993) the exact reaction-diffusionequations are numerically solved using a finite-difference scheme in the local subgrid ID domainusing a grid fine enough to resolve the Kolmogorovand/or the Batchelor microscales. Consequently, theproduction rate cba can be obtained without anymodeling. Simultaneous to the deterministic evolutionof the reaction-diffusion processes, turbulentconvective stirring within the ID domain is modeledby a stochastic mapping process (Kerstein 1992). Thisprocedure models the effect of turbulent eddies on the

scalar fields and is implemented as an instantaneousrearrangement of the scalar fields without changing themagnitudes of the individual fluid elements, consistentwith the concept of turbulent stirring.

The implementation of the stirring processrequires (randomly) determining the eddy size / from alength scale pdf /(/) in the range r\ < I < 1LEM whereT| is the Kolmogorov scale and ILEM is me

characteristic subgrid length scale which is currentlyassumed to be the local grid resolution A. A keyfeature of this approach is that this range of scales isdetermined from inertial range scaling as in 3Dturbulence for a given subgrid Reynolds number:R£LEM = U<ILEM/V where, "' is obtained from ks.

Thus, the range of eddy sizes and the stirring frequency(or event time) incorporates the fact that the smallscales are 3D. This feature is one of the major reasonsfor the past successes of LEM in gas phase diffusionflame studies (Menon and Calhoon, 1996; Calhoon andMenon, 1996, 1997).

The Linear-Eddy Two-Phase ModelFor two-phase flows, the LEM reaction-diffusion

equations have been modified to include two newfeatures: (a) the vaporization of the droplets tracked bythe Lagrangian method, and (b) the effects of dropletsbelow the cut-off so that the final stages of dropletvaporization and mixing is included. However, somechanges are required since droplet vaporization willchange the subgrid mass of the gas (primarily the fuel).Thus, in addition to the scalar reaction-diffusionequations, the two-phase mass conservation equationsmust be solved in the subgrid.

The droplets below the cutoff have been includedby assuming that the droplets act as a psuedo-fluid andtherefore, the overall effect of the droplets within eachLES cell can be modeled as a void fraction. Thisapproach is valid only when the droplets form only asmall fraction of the total volume. However, this is anacceptable assumption here since all droplets largerthan the cut-off are still tracked using the Lagrangianapproach. The present Eulerian two-phase approach isalso preferred (in terms of accuracy) when compared tothe Lagrangian approach when the droplets are verysmall and begin to behave more like a continuum fluid.

Mass conservation in both the phases in the LEMis given by: pg(p + p;(l -(p) = pavg , where subscriptg represents gas phase, / the liquid phase and <p is thevolume fraction of the gas phase (1 - void fraction ofthe liquid (A,)). The void fraction X or (p evolve duringthe subgrid evolution. Although, the liquid density is aconstant, the gas density p changes and needs to be


determined. The equations in the LEM are:

3 (12)

= 5,-5, (13)

Here, the source term Sl is the contribution of thesupergrid to the subgrid liquid phase when the dropletsize falls below the cutoff. SL is due to vaporization ofthe droplets tracked in the supergrid, while S2represents vaporization of liquid in the subgrid.

The gas phase species equation for any scalar massfraction (*P) in the subgrid can be written as

= D- + S (14)

Here, "s" indicates the ID domain of LEM. Also,5V is the source term (only in the fuel speciesequation) for production due to vaporization of theliquid phase. An equation for temperature must also besolved with the above equations since vaporizationrequires heat absorption and is followed by a drop intemperature. This is quite similar to the method used inthe earlier gas phase studies of heat release effect(Calhoon and Menon, 1997).

Note that, in Equations (12-14) the convectiveterms are missing. This is consistent with the LEMapproach, whereby, the convection of the scalar fieldsis modeled using the small-scale turbulent stirring andby the splicing process (briefly described below) asnoted earlier (Menon et al., 1993a).

Subgrid implementation of LEMSince the filtered species Ya is calculated directly

by filtering the subgrid Yk fields, there is no need tosolve the LES filtered equations (i.e., Eq. 4).Consequently, use of conventional (gradient diffusion)models is avoided. However, since the Yk subgridfields are also influenced_by large scale convection(due to the velocity field M, and the subgrid turbulentfluctuation estimated from ks), additional couplingprocesses are required.

The convection of the scalar fields by the LESfield across LES cell faces is modeled by a "splicing"algorithm (Menon et al., 1993; Menon and Calhoon,

1996; Calhoon and Menon, 1996). Details of thisprocess are given in the cited references. Given theinitial subgrid scalar fields and void fraction, dropletvaporization, reaction-diffusion, turbulent stirring, andlarge scale convection processes are implemented asdiscrete events. The epochs of these processes aredetermined by their respective time scales. This type ofdiscrete implementation is similar to the fractional stepmethod used to solve the differential equations.

The splicing algorithm transports subgrid fluidelements from one LES cell to another based on thelocal velocity field.JThe local velocity consists of theresolved velocity M, plus a fluctuating component(estimated from the subgrid kinetic energy). Thesplicing events are implemented discretely on theconvective time scale. Each splicing event involves (1)the determination of volume transfer between adjacentLES grid cells, (2) the identification of the subgridelements to be transferred, and (3) the actual transportof the identified fluid elements. The underlyingrationale for this procedure has been discussedelsewhere (Calhoon and Menon, 1996). The samealgorithm is used here except that now both the scalarfields and the void fraction are spliced at the same time.

An important property of the splicing algorithm isthat the species convection is treated as in Lagrangianschemes. Thus, convection is independent of themagnitude or gradient of the species which aretransported and depends only on the velocity field.This property allows this algorithm to avoid falsediffusion associated with numerical approximation ofconvective terms in differential equations. By avoidingboth numerical and gradient diffusion, the splicingalgorithm allows an accurate picture of the small scaleeffects of molecular diffusion to be captured, includingdifferential diffusion effects.

RESULTS AND DISCUSSIONThe two-phase subgrid model has been

implemented into a 3D zero-Mach number codedeveloped earlier (Chakravarthy and Menon, 1997).Briefly, this code solves the LES equations on a non-staggered grid. Time integration is achieved using atwo-step semi-implicit fractional step method that issecond-order accurate. The spatial difference scheme isfifth-order for the convective terms and fourth-orderfor the viscous term. The Poisson equation for pressureis solved numerically using a second-order accurateelliptic solver that uses a four-level multigrid schemeto converge the solution. The Lagrangian tracking ofthe droplets is carried out using a fourth-order Runge-Kutta scheme.

Before simulating reacting flows, an attempt was


made to validate the Lagrangian approach of particletracking. Although quantitative comparison withearlier studies is difficult due to differences in the setup and/or initial conditions, qualitative comparison canbe carried out. For this purpose we simulated themixing layer studied by Lazaro and Lasheras (1992a,b)and simulated by Martin and Meiburg (1994) using a2D vortex method. Here, we employed a 3D approachand simulated a temporal mixing layer on a 32x32x32grid. The particles were injected in every cell of theupper stream with velocities equal to the local cellvalues. The total number of particles tracked is 13,500.This case is very similar (except for the high resolutionboth in grid and number of particles) to the directsimulation of Martin and Meiburg (1994). Using the0.9-0.1 level thickness (Lazaro and Lasheras, 1992b),particle dispersion was computed for a range of Stokesnumbers. Here, Stokes number is defined as:St = (pp^/lSjlJAf/XS^, where 8^ is the vorticitythickness and At/ is the velocity difference betweenthe upper and lower stream. Also, the 0.9-0.1 levelthickness (8L) is the difference between the cross-stream locations where the particle concentrations are90% and 10% of the reference value, respectively.

Figure la shows the dispersion of particles (interms of the dispersion thickness) with time for a rangeof Stokes numbers. It can be seen that particles oforder St=l exceeds that of droplets with St<l (i.e.,smaller droplets). This phenomena was observedearlier in both experimental (Lazaro and Lasheras,1992b) and numerical (Martin and Meiburg, 1994)studies and was attributed to the increased lateraldispersion of the particles when the aerodynamicresponse time is of the order of the characteristic flowtime. The present result agrees with this result. Theincreased particle dispersion leads to the formation ofstreaks for particles for St=2.5 as shown in Fig. Ib.These results also agree with earlier observations.

In the following, the discussion focuses primarilyon comparing the subgrid model with the conventionalmodel. For these studies, droplets were injected intothe core of the mixing layer at time t=0. The mixinglayer is initialized by a tangent hyperbolic meanvelocity along with the most unstable 2D (ofdimensional wavelength 2n) mode and randomturbulence (similar to that described in Metcalfe et al.,1987). Results shown here employed a grid resolutionof 32x32x32. Grid independence studies were alsocarried out for some of these cases using a 64x64x64grid and good agreement was obtained.

The mixing layer is initialized with the oxidizer inboth the upper and lower streams at 350 K and the fueldroplets are initially introduced in the mid-plane. Arange of droplets from 10-50 micron radius with an

initial temperature of 300 K was used for allsimulations with the droplet cut-off radius at 5 micron.A total of 2100 droplet groups were tracked (the effectof varying the droplet group has not yet been carriedout). For simplicity, the droplet groups were uniformlydistributed and the number of droplets in each group ischosen such that the overall mass loading is 0.5 and thecorresponding volume loading is 0.0005.

Figure 2a shows the spanwise vorticity contours inthe mixing layer at the roll-up stage for a case in whichthe particles are passively transported upon insertion(i.e., no vaporization included and hence, there is nocoupling between the two phases). It can be seen thatthe shear layer rolls into coherent structures as seen inpure gas phase flows. However, when dropletvaporization is included, as shown in Fig. 2b, theassociated heat absorption results in major changes inthe shear layer. The formation of the coherentspanwise vortices is inhibited due to vaporization (andmass addition to gas phase). Although the extent of themixing layer appears to be large, the peak value of thespanwise vorticity is substantially lower for thevaporizing case. Analysis shows that, in thevaporization case, significant 3D vorticity is generatedand this plays a major role in the inhibiting thespanwise coherence.

The enhancement of streamwise vorticity can bevisualized by comparing Figs. 3a and 3b which show,respectively, the streamwise vorticity for the passiveand vaporization cases. The 3D nature of the shearlayer has been enhanced in the vaporization case(notice that the same contour interval is employed fordirect comparison). Further analysis shows thatvaporization causes significant production of thebaroclinic torque (the spanwise component of thebaroclinic torque, VpXVp/p is shown in Fig. 4a)outside the vortex core. This production plays a majorrole in redistributing the vorticity in the mixing layer.This can be confirmed by calculating the various termsin the 3D vorticity transport equation. For example, theexpansion term 0&(V • t>) in vorticity equation isshown in Fig. 4b. Comparison with Fig. 4a indicatesthat baroclinic term dominates in this case.Interestingly, this behavior is quite similar to the casewhen heat is released (McMurtry et al., 1989) exceptthat, in the present case, heat is absorbed and thetemperature is decreasing.

The droplet distribution for the above two cases isshown in Figs. 5a and 5b, respectively. The Stokesnumber for all the droplets tracked is in the range of0.0004-0.01. As expected, the droplets follow the fluidmotion and this behavior qualitatively agrees with theresults obtained by Ling et al. (1997). Differences existdue to modulation of the vortex structures as a result of


vaporization (as noted above).LES using the new subgrid model were then

compared to identical conventional LES. For thesestudies, a subgrid resolution that captured the effect ofturbulent stirring by the largest small scales (here, forsimplicity, a subgrid eddy size 60% of the gridresolution is used), was used to reduce thecomputational cost. Subgrid resolution was doubledand similar results were obtained indicating that thepresent tests captures the effect of the largest subgridscales quite accurately.

The spanwise and streamwise vorticity for theconventional and subgrid approach are shown in Figs.6a and 6b, respectively, and correspond to Figs. 2b and3b for the conventional case. The effect of vaporizationon shear layer is qualitatively similar in nature but themagnitude is much higher for the subgrid approach.This can be explained by noting that in the LEMapproach phase change occurs in the subgrid.Similarity in the droplet distribution is more apparentand can be observed by comparing Figs. 6c and 5b.However, this is not too surprising since the largerdroplets are still tracked using the Lagrangian methodin both the LES.

Note that, in the subgrid case, if the droplet cut-offis chosen such that no droplet drops below the cut-off,then the void fraction is zero. In this case, the subgridand the conventional approaches should agreereasonably well. This has been confirmed using infinitekinetics when the vaporized fuel mixes with theoxidizer and reacts. The product mass fractionpredicted by the conventional and the Subgridapproaches are compared in Fig. 7a. It can be seen thatthere is very good agreement thereby confirming thevalidity of the subgrid approach. The predictedtemperature of the gas phase (Fig. 7b) also shows goodagreement.

If the cut-off size is large, then it is expected thatthe conventional LES will be in significant error sinceit assumes that all droplets below cut-offinstantaneously vaporize. However, if the new subgridapproach can deal with this increased cut-off size (bythe subgrid void fraction approach) then it will reducethe computational cost of the Lagrangian trackingconsiderably. To determine this, two cut-off sizes of 10and 20 microns were used in otherwise identicalsimulations. Since the droplet distribution is lost oncethe drops are in the subgrid, it is assumed that all thedroplets are at the average diameter between the cut-off and zero. This is not an accurate assumption and inthe future, droplet distribution information will becarried to the subgrid. Another source of error is that,currently, the liquid void fraction is passivelytransported across LES cells based on the volume

transfer of the gas phase. However, the liquid phasetransport should be based on the liquid volume transferfrom the LES cell. A method to deal with this transporthas been developed and will be used in future studies.

The product mass fraction obtained by the twodifferent LES methods is compared in Fig. 8 for thevarious cut-off sizes. The results predicted by theconventional LES are in gross error for cut-off radiusof 10 and 20 micron. However, the presentmethodology captures most of the trends andmagnitudes. Ideally, the subgrid model should predictidentical results for a range of cut-off sizes. Theobserved differences in the magnitude and the spreadare due to the lower vaporization rates in the coreregion. Analysis shows that this is primarily due toabove noted problem with the LES transport of theliquid phase. It is expected that the correctiondeveloped for this will result in more consistentpredictions.

Finite rate cases for three different Damkohlernumbers (Da) were also simulated. The product massfractions for different Da are shown in Fig. 9a and thetemperatures are compared in Fig. 9b. When thechemical time scale decreases (Da increases), theproduct mass fraction increases in agreement withearlier results and physics. Due to finite-rate effects,there is a considerable amount of unreacted fuel in thegaseous form. In addition to the droplet temperatureand the surrounding oxidizer concentration, the amountof fuel present in the gaseous form dictates the liquidto gaseous fuel phase change. Thus, the vaporizationrate is coupled to the rate of chemical kinetics, heatrelease and the other processes such as convection. Thedemonstration that the present method can capture Da-effects needs to be reevaluated in the presence of heatrelease. Furthermore, since the present method candeal with differential diffusion, Lewis number effectscan also be captured. This will be demonstrated in thenear future.

CONCLUSIONSIn this study, an earlier developed gas phase

subgrid combustion model has been extended for two-phase flows. This approach includes a morefundamental treatment of the effects of the final stagesof droplet vaporization, molecular diffusion, chemicalreactions and small scale turbulent stirring than other*LES closure techniques. As a result, Reynolds, Lewisand Damkohler number effects are explicitly included.This model has been implemented within an Eulerian-Lagrangian two phase large-eddy simulation (LES)formulation. In this approach, the liquid droplets aretracked using the Lagrangian approach up to a pre-


specified cut-off size. However, the vaporization of theLagrangian droplets and the evaporation and mixing ofthe droplets smaller than the cut-off size are modeledwithin the subgrid using an Eulerian two-phase modelthat is an extension of the earlier gas-phase subgridmodel. The issues (both resolved and unresolved)related to the implementation of this subgrid modelwithin the LES are discussed in this paper.

The present results on the increased dispersion ofnon-vaporizing droplets at intermediate sizes in forcedmixing layers agrees well with trends seen in earlierexperimental and numerical studies. Whenvaporization was included, modifications of the vortexstructure was found to be due to the production ofbaroclinic torque. This result is quite similar to the heatrelease effect seen earlier. The effect of varying theDamkohler number was also captured correctly.Finally, it has been shown that when the droplet cut-offsize is increased, the conventional method giveserroneous results while the current methodologycaptures most of the trends within the limitation of thecurrent implementation.

ACKNOWLEDGMENTSThis work was supported in part by the Army

Research Office Multidisciplinary University ResearchInitiative grant DAAH04-96-1-0008 and by the AirForce Office of Scientific Research Focused ResearchInitiative contract F49620-95-C-0080 monitored byGeneral Electric Aircraft Engine Company, Cincinnati,Ohio. Computations were carried out under the DoDHPC Grand Challenge Project at ARC, Huntsville.

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Menon, S. and Chakravarthy, V. K. (1996) "Large-Eddy Simulations of Premixed Flames in CouetteFlow," AIAA 96-3077, 32nd AIAA/ASME/SAE/-ASEE Joint Prop. Conference.

Menon, S. and Kirn, W.-W. (1996) "HighReynolds Number Flow Simulations using theLocalized Dynamic Subgrid-Scale Model," AIAA 96-0425, 34th AIAA Aerospace Sciences Meeting.

Menon, S. and Calhoon, W. (1996) "SubgridMixing and Molecular Transport Modeling for Large-Eddy Simulations of Turbulent Reacting Flows,"Symp. (International) on Combustion, 26.

Metcalfe, R. W., Orszag, S. A., Brachet, M. E.,Menon, S., and Riley, J. J. (1987) "SecondaryInstability of a Temporally Growing Mixing Layer," J.Fluid Mech., Vol. 184, pp. 207-243.

Mostafa, A. A. and Mongia, H. C. (1983) "On theModeling of Turbulent Evaporating Sprays: Eulerianversus Lagrangian Approach," Int. J. of Heat MassTransfer, Vol. 30, pp. 2583-2593.

Oefelein, J. C. and Yang, V. (1996) "Analysis ofTranscritical Spray Phenomena in Turbulent MixingLayers," AIAA 96-0085, 34th AIAA AerospaceSciences Meeting, Reno, NV, Jan. 15-18.

Poinsot, T. (1996) "Using Direct NumericalSimulations to Understand Premixed TurbulentCombustion," 26th Symp. (Intn.) on Combustion, pp.219-232.


5.0

4.0

3.0

2.0

1.0

0.00.0

=&-— "Si-snow

10.0 20.0 30.0

Figure la) Evolution of the 0.9-0.1 particle concen-tration level thickness for various St.

Figure Ib) Projection of the droplet distribution in the X-Y forparticles of St. = 2.5 at t=28.

min = -1.17196max =0.02867increment = 0.0374

A

Figure 2a) Spanwise vorticity in the mixing layer withoutcoupling from liquid phase. Nondimensionalized by U(/L0

min = -0.741s max = 0.0274"tecjrement = 0.03.74-r/' .._... — •——.——.

Figure 2b) Spanwise vorticity in the mixing layer with cou-pling from liquid phase (Conventional LES).

min = -0.0608max = 0.0435increment = O.Q3,

min = -0.3803max = 0.4036™increment = 0.03

Figure 3a) Streamwise vorticity in the mixing layer without Figure 3b) Streamwise vorticity in the mixing layer with cou-coupling from liquid phase. pling from liquid phase (Conventional LES).


min = -0.0235max = 0.03508

Figure 4a) Instantaneous value of spanwise component of thebaroclinic term in vorticity equation in mid-span plane.

0.80

0.60 -

0.40 -

0.20

rpo 40nm < rp < 30)lin•j 3Q\un < ip < 20nm» 20|jm < rp < lOum• rp< 10itm

'«

* *f

#9 *%>•«*.•» $*,

*•

%ft*% O

*

*i»- "̂?*

^

0.4 0.6 0.8 1.00.0 0.2

Figure 5a) Projection of the droplet distribution in the X-Yplane without coupling from liquid phase.

min = -0.0063max = 0.0097increment = 0.0005

Figure 4b) Instantaneous value of spanwise component of theexpansion term in vorticity equation in mid-span plane.

0.80

0.70

O SOflm < rp < 40)uno 40|im < rp < 30m

rp <20(Un < rp <

0.20

Figure 5b) Projection of the droplet distribution in the X-Yplane with coupling from liquid phase (Conventional LES).

min = -1.408^max. = 0.252 ,-increment = 0,03X4 -- ' ' , , ;

min = -0.736max = 0.843

Figure 6a) Spanwise vorticity in the mixing layer with cou- Fi.gure ̂ Streamwise vorticity in the mixing layer with cou-pling from liquid phase (LES/LEM). Plin8 from li(Vid Phase (LES/LEM).

10


0.80

0.70

0.20

Figure 6c) Projection of the droplet distribution in the X-Yplane with coupling from liquid phase (LES/LEM).

380.0

360.0

= 340.0

320.0

300.0

280.0

LEM/LESConventional

0.0 0.2 0.4 0.6 0.8 1.0Y/Y

0.25

0.20

& 0.15u.

0.10

0.05

0.00 u-0.0 0.2 0.4 0.6 0.8 1.0

Y/Y

Figure 7b) Variation of X-Z plane averaged temperatureacross the mixing layer for infinite rate.

Figure 7a) Variation of X-Z plane averaged product massfraction across the mixing layer for infinite rate.

LES (Cut-off Sum)LES (Cut-off 20nm)LES (Cut-off 10(im)LEM (Cut-off Sum)

»--<>LEM (Cut-off 10|im)LEM (Cut-off 20|im)

Figure 8) Variation of X-Z plane averaged product densityacross the mixing layer for different cut-offs.

0.20

0.15

0.10

0.05

o.oo

a——o Da=l»-—» Da =10........... Da = 100«——» Infinite Rate

0.0 0.2 0.4 0.6 0.8 1.0

Figure 9a) Variation of X-Z plane averaged product mass frac-tion across the mixing layer for different Damkohler numbers.

360.0

340.0 -

S 320.0 -

300.0 -

280.0

Figure 9b) Variation of X-Z plane averaged temperatureacross the mixing layer for different Damkohler numbers.

11

Documents

Large eddy simulations of two-phase turbulent flows...Both Eulerian and Lagrangian formulations have been used to simulate two-phase flows in the past (e.g., Mostafa and Mongia, 1983)