mixing. Since both these processes are highly unsteadyconventional steady state methods cannot be used to elu-cidate the finer details. On the other hand, althoughunsteady mixing process can be studied quite accuratelyusing direct numerical simulation (DNS) (e.g., Poinsot,1996), application of DNS is limited to low to moderateReynolds numbers (Re) due to resolution requirements.This restriction limits the extension of conclusionsdrawn from DNS results to high Reynolds number com-plex flows typical in a gas turbine combustors. An alter-native approach called large-eddy simulation (LES) hasthe potential for application to high Re flows. However,the application of LES to reacting flows requires demon-strable ability to capture the effects of turbulent small-scale mixing and chemical reactions.In LES, the scales larger than the grid size arecomputed using a time- and space-accurate scheme,while the unresolved smaller scales, which are mostlyisotropic, are modeled using an eddy viscosity basedsubgrid model. Closure of the momentum and energytransport equations can be achieved using this methodsince 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 conventional LES approach.As a result, conventional subgrid eddy diffusivitymodels cannot be used to model these features.To address these issues, recently (Menon et al., 1993a;Menon and Calhoon, 1996; Calhoon and Menon, 1996,1997), a subgrid combustion model was developed andimplemented within the LES formulation. This modelseparately and simultaneously treats the physicalprocesses of molecular diffusion and small scaleturbulent convective stirring. This is in contrast toprobability density function closure whichphenomenologically treats these two processes by asingle model, thereby removing experimentallyobserved Schmidt number variation of the flow. Thecapabilities of this model have been demonstrated inthe above noted studies by carrying out quantitativecomparison with high-Re experimental data obtained in
reacting shear layers. Application of this method topremixed combustion has also been recentlydemonstrated (Menon et al., 1993b; Smith, Menon andChakravarthy, 1996; Chakravarthy and Menon, 1997.Results show that this method can capture thin, high-Re turbulent flames without any numerical dissipation.The predicted turbulent flame speed was also shown tobe in reasonable agreement with high-Re data.
The above methodology was originally developed tostudy gas phase combustion. To apply this method toinvestigate liquid fuel droplet vaporization and fuel-airmixing requires some additional modifications. Thispaper discusses the issues related to this developmentand presents some preliminary results for two-phaseflows in both non-reacting and reacting mixing layers.
2. Formulation of the Two-Phase LES Model
Both Eulerian and Lagrangian formulations have beenused to simulate two-phase flows in the past (e.g.,Mostafa and Mongia, 1983). Both methods have theirown merits and demerits; however, most state-of-the-art codes employ the Lagrangian form to capture thedroplet dynamics, while the gas phase is still 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), onlya limited range of droplet sizes are computed. Dropletsbelow an ad hoc cut-off size are assumed to vaporizeinstantaneously and to become fully mixed in the gasphase. This is a critical assumption and flawed, since(as noted above), even in pure gas phase flows it hasbeen determined that the small-scale mixing process isvery important for quantitative predictions. The finalstages of droplet vaporization and the subsequentmixing needs to be properly resolved for accurateprediction of the combustion process. Here, the gas-phase subgrid combustion methodology has beenextended to allow proper simulation of the final stagesof droplet evaporation and turbulent mixing.
The two-phase subgrid process is implemented withinthe 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 cutoff, a two-phase subgrid Eulerian model is employed to includethe effects of the small droplets within the LES cells.In the following, the details of the Lagrangian LESmodel and the new subgrid two-phase combustionmodel are briefly described.
2.1 Gas Phase LES Equations
Our present approach employs the incompressibleNavier Stokes equations in the zero Mach number limitsince most of the characteristic problems currentlyunder study are predominantly isobaric and can besolved efficiently using this approach. The extension tofully compressible flows (with acoustic wave motion)will be considered at a future date. No fundamentalproblems in extending the subgrid two-phase model tocompressible flows are expected. Zero-Mach numberapproach involves using a series expansion in terms ofMach number to remove the acoustic component fromthe compressible Navier-Stokes equations and is a wellestablished method (McMurtry et al., 1989;Chakravarthy and Menon, 1997). A key feature of thisexpansion is that the pressure field is split into twoparts: a kinematic pressure (p) related to the velocityfield and a thermodynamic pressure (p) that must bespecified.
To obtain the LES equations, the low-Mach numberequations are filtered using a specified filter function toremove the contribution of the scales smaller than thegrid size. The filter function can be of any type,however, for the present finite-difference approach atop-hat function is employed. The resulting LES mass,momentum, energy and species equations (all LESterms are identified with a bar on top) can be written as:
______dx:
^i + n^i - <>£ + • 3T;..,
p
dx. p\' dxkdxk
_~
(D
(2)
(3)
dxkdxk
The above system of equations are supplemented by theequation of state for the thermodynamic pres-sure p = f>RT which_can_be used to obtain the tem-perature T. Here, p, ujt Ya and p are, respectively,the density, i-th velocity component, a -th species massfraction and the kinematic pressure. Also, v, X, D and
R are, respectively, the kinematic viscosity, the thermalconductivity, the mass diffusion (assumed constant andsame for all species here, but can be generalized) and thegas constant. In Eq. (4), cba is the LES filtered speciesproduction/destruction term which is typically a highlynon-linear term and very difficult to model. Also, in theabove equations, source terms p^, Fs, Qs , and Ssrepresent the volume-averaged rate of exchange of mass,momentum, energy and species between the gas-phaseand the liquid phase. These terms are computed, asdetailed elsewhere (Oefelein and Yang, 1996; Faeth,1983) and, therefore, omitted here for brevity. Further-more, note that eq. (3) is the equivalent energy equationin the zero-Mach number limit. In the absence of heatrelease and no phase change, this equation and eq. (1)will be identical.
In the above equations, the subgrid stress tensor-(UjUj - U;U:) andjhe species-velocity corre-5 • = -(YaUj-Yaiij) requires modeling. In
the present LES approach, the stress term t.. is modeledthe usual
, •';lations
using eddy viscosity approach asT(; = 2vfSjj where v( is the eddy viscosity and 5-
is the resolved rate-of-strain tensor. The subgrid eddyviscosity is obtained in terms of the grid scale A and thesubgrid kinetic energy, ksgs = (M(.M(.-M(.M(.) as:
vf - Cvksgs ' A . Here, ksgs is obtained by solvinga transport equation. More details are given elsewhere(Kim and Menon, 1995, 1996). The coefficient Cy inthe eddy viscosity model and the coefficients appearingin the k equation can be obtained using the dynamicprocedure as described elsewhere (Kim and Menon,1995; Menon and Kim, 1996).
It is worth noting here (and further discussed below),that the source terms p^ , Qs and Ss due to mass andenergy transfer from the liquid to the gas phase is mod-eled within the subgrid domain. Furthermore, the spe-cies transport equation (4) is not solved along with theother LES equations, since, it too is modeled using thesubgrid combustion approach. Thus, closure of Saj isnot needed. More details regarding the solution of thegas phase scalar equations is given elsewhere (Calhoonand Menon, 1996, 1997).
2.2 Liquid-phase LES equationsA Stochastic Separated Flow (SSF) formulation (Faeth,1983; Oefelein and Yang, 1996) is used to track thedroplets using Lagrangian equations of motion. Thegeneral equations of spherical droplets reduce to the fol-lowing form when the terms arising due to static pres-sure gradient, virtual-mass, Besset force and external
body-forces are neglected. The position and the velocityof the droplets are given by
dx
dt
(5)
(6)
where the droplet properties are denoted by subscript p,d is the droplet diameter and «(. is the instantaneousgas phase velocities computed at the droplet location.This gas phase velocity field is obtained using both thefiltered LES velocity field H(. and the subgrid kineticenergy ksgs (as in the eddy interaction model). Thedroplet Reynolds number is computedusing: Rep = -^ [ (K, - -u p t , . ) (« , - -u p t i ) ] l / 2
drag coefficient is modeled by (Faeth, 1983):and the
P
0.424
z6 P(7)
Re > 10"P
The conservation of the mass of the droplets results isgiven by: dm / dt = —nz where the mass transferrate for a droplet in a convective flow field is given as:
mpm. = 1 +
0.278^/V73
L232/RepSc V3(8)
Here, Sc is the Schmidt number and the subscriptRe = 0 indicates quiescent atmosphere when there
is no velocity difference between the gas and the liquidphase. The mass transfer under this condition is given as
«*«, = () = 2ltPA««yn(1 + V • Here' p<and Dsm are, respectively the gas mixture density andthe mixture diffusion coefficient at the droplet surfaceand BM is the Spalding number which is given as
BM = ( ¥,, F ~ Y~. F)/( 1 - Ys, F> • Here' YS, r ^
the fuel mass fraction at the surface of the droplet andcomputed using the procedure described in Chen andShuen (1993) whilethe ambient gas.
The heat transfer rate of the droplet (assuming uniformtemperature in the droplet) is given by the followingrelation (Faeth, 1983):
dT^(9)
The heat transfer coefficient for a droplet in a convectiveflow field with mass transfer is modeled as
= 1+.0.278/?ep
1/2Pr1/3
l.232/RepPr 2/3 I/2 (10)
Here, Pr is the gas phase Prandtl number and the heattransfer coefficient for quiescent medium is given as
=0 /d where the NusseltnumbeY is obtained from:
NuRep= 0 =2}n(l+BM) Le
(11)(1 -1
Nu Re _ approaches a value of 2 in the case of zeromass t/ansfer and Le is the Lewis number. Only dropletsabove a cut-off diameter are solved using the aboveequations, while the droplets below the cut-off diameterare modeled using Eulerian formulation within the sub-grid.
In summary, the present LES approach for two-phaseflows involves the solution of the gas phase LES equa-tions (l)-(3) using a conventional high-accuracy finite-difference Eulerian scheme. Closure for the subgridstresses is achieved by using a localized dynamic modelfor the subgrid kinetic energy. Simultaneous to the solu-tion of the gas phase equations, the liquid phase equa-tions (5)-(ll) are solved using the Lagrangiantechnique. The range of droplet sizes tracked in thisscheme depends upon the computational constraints(which can be quite large if a large number of dropletsizes are to be tracked). The gas phase LES velocityfield and the subgrid kinetic energy are used to estimatethe instantaneous gas velocity at the droplet location.This essentially provides a coupling between the gas andliquid phase momentum transport. The gas phase scalarfield evolution is simulated in the subgrid domain as dis-cussed in the next section.
3. Subgrid Combustion Models
The principle difficulty in reacting LES simulations isthe proper modeling of the combustion related termsinvolving the temperature and species, for example, theconvective species fluxes such as 5a- due to subgridfluctuations and the filtered species mass production
rate cba . Probability density function methods whenapplied within LES either using assumed shape(Frankel et al., 1993) or evolution equation (Gao and
O'Brien, 1993) may be used to close cba , and, inprinciple, any scalar correlations. However, thetreatment of molecular mixing and small scale stirringusing phenomenological models as in pdf methodshave been only partially successful in predicting themixing effects. Problems have also been noted whengradient diffusion assumption/eddy viscosity model isused to approximate the species transport terms. Use ofthis type of assumption for reactive species is dubious,as noted earlier (Dimotakis, 1989; Pope, 1979). Frankelet al. (1993) attributed the use of this assumption as thesource of errors in the comparison of reacting LESsimulations with DNS data.
Earlier, Kerstein (1989, 1992) developed a mixingmodel termed the linear eddy mixing (LEM) model anddemonstrated its ability to separately treat themolecular diffusion and turbulent convective stirringprocesses at all relevant fluid mechanical length scalesof the flow. The scalar fields are simulated within a IDdomain which represents a stochastic slice through thelocal scalar field. Within the context of LES, the LEMmodel is used to represent the effect of only the smallunresolved scales on the scalar fields while the largerresolved turbulent scales of the flow are calculateddirectly from the LES equations of motion. Toaccomplish this, the mixing model is implementedwithin each LES cell. In the present investigation, theprocedure for coupling the LEM model to the LESequations is essentially identical to the methoddeveloped earlier for gas-phase combustion (Menon atal., 1993a; Menon and Calhoon, 1996; Calhoon andMenon, 1996, 1997) and, therefore, is onlysummarized here.
The subgrid LEM has several advantages overconventional LES of reacting flows. In addition toproviding a fundamentally accurate treatment of thesmall-scale turbulent mixing and molecular diffusionprocesses, this method avoids gradient diffusionmodeling of scalar transport. Thus, both co- andcounter-gradient diffusion can be simulated.
3.1 The Linear-Eddy Single Phase ModelThe details of the baseline gas phase LEM model havebeen reported earlier (e.g., Kerstein, 1989, 1992;Menon et al., 1993a). Briefly, the exact reaction-diffusion equations (i.e., equation 4 without any LESfiltering and without the convective term) arenumerically solved using standard finite-differencescheme in the local 1-D domain using a grid fineenough to resolve the Kolmogorov and/or theBatchelor microscales. Consequently, the speciesproduction rate cba can be expressed in terms of theraw temperature and species fields without anymodeling. This is a particularly attractive feature of thepresent model since it obviates the need for anymodeling of the highly non-linear production terms.
Simultaneous to the deterministic evolution of thereaction-diffusion processes, turbulent convectivestirring within the ID domain is modeled by astochastic mapping process (Kerstein 1992). Thisprocedure models the mixing effect of turbulent eddieson the scalar fields and is implemented as aninstantaneous rearrangement of the scalar fieldswithout changing the magnitudes of the individualfluid elements, consistent with the concept of turbulentstirring. An underlying assumption employed here isthat the subgrid turbulence is homogeneous andisotropic. Within the context of the LES, this is areasonable assumption and is generally invoked forsubgrid modeling.
The implementation of the stirring process requires(randomly) determining the eddy size / from a length
scale pdf /(/) in the range r\ <1<1LEM where, TJ
is the Kolmogorov scale and ILEM 's 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:
where, u' is obtained from= U'^LEM//V
'k' . Thus, the range of eddy sizes and the stirringfrequency (or event time) incorporates the fact that thesmall scales are 3D even though it is still implementedon the ID stochastic domain. This feature is one of themajor reasons for the past successes of LEM in gasphase diffusion flame studies (Menon and Calhoon,1996; Calhoon and Menon, 1996, 1997). The details ofthe method to determine these parameters and themapping procedure are given elsewhere (Kerstein
1992; Menon et al., 1993a) and, therefore, avoided herefor brevity.
3.2 The Linear-Eddy Two-Phase ModelIn the present formulation, the LEM reaction-diffusionequations have been modified to include the effects ofdroplets below a certain cut-off so that the final stages ofdroplet vaporization and mixing is included. As notedearlier, this approach overcomes the earlier limitationsof the SSF where the droplets below certain size areassumed to instantaneously vaporize and mix. Imple-mentation in the LEM requires a reevaluation of thebasic LEM approach since droplet vaporization willchange the subgrid mass of the gas (primarily the fuel).Thus, in addition to the scalar reaction-diffusion equa-tions, mass equations needs to be included.
The presence of droplets have been incorporated into theLEM by assuming that the droplets act as a pseudo-fluidThus, discrete droplets are not tracked as in theLagrangian LES model but the overall effect of thesmall droplets within each LES cell is modeled in theLEM using a void (volume) fraction approach. Thisapproach is valid only when the droplets form only asmall fraction of the total volume. However, this is anacceptable assumption here since all droplets larger thanthe cutoff 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.The conservation of mass in both the phases in the LEMis given by the following relation:
- <P) = Pav g (12)
where subscript g represents gas phase, / the liquidphase and (p is the volume fraction of the gas phase (1 -void fraction of the liquid (X)). The void fraction A, or cpcan be initially specified but evolves during the subgridevolution. Although, the liquid density is a constant, thegas density p is not since there is mass addition fromthe liquid phase. Thus, p also needs to be determined.The equations governing the conservation of mass of thegas and liquid phases in the LEM are:
Here, the source term Sl is the contribution of thesupergrid to the subgrid liquid phase when the dropletsize falls below the cutoff. SL is the term due to thevaporization of the droplets tracked in the supergrid,while S2 represents the contribution from the subgridvaporization of the liquid going into the gaseous phase.The droplets from the supergrid process contribute tothe subgrid, if and only if the droplet size is smaller thanthe cutoff. If initially there are no drops below the cut-off size, then the formulation is identical to the pure gas-phase flow. Equations (13) and (14) are used to deter-mine the gas density p and the volume fraction (p . Aninherent assumption in equations (13) and (14) is thatthe source terms are obtained under the assumption thatthe droplets are so small that there is negligible relativemotion between the liquid and gas phase. Although thisassumption sounds reasonable (since droplets are verydilute and the liquid phase is behaving as a pseudo-fluid,this assumption still needs further justification and is anissue of current research).
The conversion of the subgrid droplets into gas phase isincluded as a source term in the gas phase speciesequation (only the fuel species is effected). Thus, theequation governing any scalar mass fraction (*P) in thesubgrid can be written as
(15)
Here, "s" indicates that this equation is solved on the IDdomain of LEM. Also, cb is the production/destruc-tion of the species *F due to gas-phase chemical reac-tions and Sv is the source term for the production of thespecies due to vaporization of the liquid phase. Thus,S represents a source only in the fuel species equation.If S is neglected, the above equation is the same as inthe earlier gas phase LEM approach. For heat releaseand droplet vaporization cases, a ID temperature equa-tion also needs to simulated in the subgrid as discussedin Calhoon and Menon (1997).
Another issue to be noted in equations (13)-(15) is thatthe convective term due to fluid motion is missing. Thisis consistent with the LEM approach, whereby, the con-vection of the scalar fields is modeled using the small-scale turbulent stirring and by the splicing process(described in section 3.3), as noted earlier (Kerstein,1992; Menon et al., 1993a). An area for further investi-gation is the effect of stirring on the droplets. Currently,it is assumed that the droplets in the subgrid have no rel-ative motion with respect to the gas phase. Thus, therearrangement process used to model the effect of stir-
ring on the gas species is also implemented unchangedto stir the droplets. If relative motion is to be included,some modifications to the stirring process may berequired. This is an issue for future investigation.3.3 Subgrid implementation of LEM
Since the filtered species Ya is calculated directly by
filtering the subgrid Yk fields, there is no need to solvethe LES filtered equations (i.e., Eq. 4). Consequently,use of conventional (gradient diffusion) models isavoided. However, since the Yk subgrid fields (in Eq.15) are also influenced by large scale convection (due
to the LES-resolved velocity field «( and the subgrid
turbulent fluctuation estimated from ksgs), additionalsubgrid-supergrid coupling processes are required.Here, supergrid denotes the resolved scale field ascomputed by the LES equations.The convection of the scalar fields by the supergridfield (supergrid-to-subgrid coupling) across LES cellfaces is modeled by a "splicing" algorithm (Menon etal., 1993a; Menon and Calhoon, 1996; Calhoon andMenon, 1996). Details of this process is given in thecited references and, therefore, avoided here since themethod for two-phase flows is identical to the onedeveloped earlier for gas phase flows.Thus, the subgrid and supergrid processes involve thefollowing processes. Given the initial subgrid scalarfields and void fraction in each LES cell, dropletvaporization, molecular diffusion, chemical reaction,turbulent stirring, and large scale convection processesare implemented as discrete events occurring in time.The epochs of these processes are determined by theirrespective time scales. This type of discreteimplementation is similar to the fractional step methodused to solve differential equations.As the subgrid scalar fields evolve under the action ofthese processes, the resolved LES fields (both the gasand liquid phases) are solved concurrently on the LEStime scale(s). The resolved scales influence theevolution of the subgrid scalar fields via thespecification of the subgrid length scale, the subgrid Reand the changes in the void fraction due to dropletsvaporizing and becoming smaller that the cutoff size.The subgrid scalar fields in turn influence thedevelopment of the resolved scales (subgrid-to-supergrid coupling) by providing the filtered scalarfields, temperature-species correlations and the heatrelease calculated from the subgrid scalar fields.The splicing algorithm employed in the present study
transports subgrid fluid elements from one LES cell toanother based on the local velocity field. The local
velocity consists of the resolved velocity «. plus afluctuating component (estimated from the subgridkinetic energy). The splicing events are implementeddiscretely on the convective time scale (which is alsothe time-step for the gas phase LES). Each splicingevent involves (1) the determination of volume transferbetween adjacent LES grid cells, (2) the identificationof the subgrid elements to be transferred, and (3) theactual transport of the identified fluid elements. Theunderlying rationale for this procedure has beendiscussed elsewhere (Menon et al., 1993a; Calhoon andMenon, 1996). The same splicing algorithm is usedhere except that now both the scalar fields and the voidfraction in the subgrid field are spliced at the same time.An important property of the splicing algorithm is thatthe 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. Asa result, subgrid elements are transported withoutchanging their species and temperature magnitudes.This property allows this algorithm to avoid difficultyof false numerical diffusion associated with thenumerical approximation of convective terms indifferential equations. By avoiding both numerical andgradient diffusion, the splicing algorithm allows anaccurate picture of the small scale effects of moleculardiffusion to be captured, including differentialdiffusion effects.4. Results and Discussion
The two-phase LES model has been successfully imple-mented into a 3D zero-Mach number code developedearlier for gas phase combustion (Menon and Chakra-varthy, 1996; Chakravarthy and Menon, 1997). Briefly,this code solves the gas phase LES equations on a non-staggered grid using a high-order finite differencescheme. Time integration is achieved using a two-stepsemi-implicit fractional step method that is second-orderaccurate. The spatial difference scheme is fifth-order forthe convective terms and fourth-order for the viscousterm. The Poisson equation for pressure is solvednumerically using a second-order accurate elliptic solverthat uses a four-level multigrid scheme to converge thesolutions.The Lagrangian tracking of the droplets in the LESdomain is carried out using a fourth-order Runge-Kuttascheme. The number of droplets tracked in the LES isdetermined at present by the computational constraints.
However, more detailed studies are planned in the futureto determine an optimum cut-off size.In the following, some preliminary results are discussedfocussing primarily on the subgrid model since theLagrangian LES approach is conventional (e.g. Oefeleinand Yang, 1996) and well established. First, using anexact solution of a decaying vortex, the trajectories ofthe injected droplets is simulated. Then, simulationswere carried out in a 3D temporal mixing layer withdroplet continuously injected in the core of the mixinglayer. Subsequently, the initial droplet void fraction ineach LES cell is determined based on a specified dropletcut-off size. Using this initial conditions, droplet vapor-ization, turbulent mixing and infinite rate chemical reac-tions (with no heat release) are studied within thesubgrid. No coupling between the supergrid and subgridis included here. Coupling (via the splicing technique) iscurrently available and can be included without anymodifications to the code. Fully coupled LES-LEM willbe discussed at a later date (Pannala and Menon, 1998).Infinite rate kinetics in the subgrid is studied withoutany direct influence from the Lagrangian LES part(other than providing the initial conditions). Thus, thesource term Sl in equation (14) is reflected as the initialvoid fraction of the fuel and term SL in equation (13)and (15) goes to zero. The source term for the subgridvaporization is given as: 52 = "„"'„ » where n isthe number of droplets (determined initially from thedroplet cut-off size and the initial volume fraction(1 -cp) of the droplets in the subgrid). Once n is
determined for each LES cell, it is assumed to be con-stant. However, the droplet size is still allowed todecrease due to vaporization. This assumption will berelaxed in the future especially when the large-scaleconvection via splicing is included.The mass transfer rate rh is determined from the rela-tion: mp = 2itpgDgd Jn(l + BM) , which is sim-ilar to the mass rate used in the Lagrangian LES for thequiescent case (i.e., for no relative motion between gasand droplets). Here, the subscript "g" indicates the gasproperty at the droplet surface and the droplet diameteris determined from the void fraction by the relation:
dn = 281/3
(16)
Here, 8 is the subgrid resolution (determined based onthe requirement that the Kolmogorov eddy must beresolved). Finally, the scalar equation (15) is replacedfor infinite rate kinetics by equations for the fuel and
oxidizer mass fractions. The source term S^ is non-zeroonly for the fuel species equation and is given as:S = n m . Also, for present, equal diffusivity for
the fuel and oxidizer is assumed.
Using the exact solution of a decaying vortex, the dis-persion of the droplets is first investigated. Five differentsized particles are injected in the core of the vortex inthe x-direction with a specified velocity. The subsequenttrajectories of these particles is tracked. Figure la showsthe velocity vector field and Fig. Ib shows the trajecto-ries of the injected droplets. It can be seen that thesmaller particles follow the vortex streamlines while thelarger particles follow independent paths due to theirinertia. This qualitatively confirms that the Lagrangianparticle tracking model in the LES code is implementedcorrectly. To obtain quantitative verification, an experi-mental study is underway at Georgia Tech to obtain datafor code validation (currently no such data exists).
To evaluate the LES and subgrid two-phase LEM model,a 3D temporal mixing layer is simulated using a wellestablished procedure. The mixing layer (slip walls inthe transverse and periodic in streamwise and spanwisedirections) is initialized using a tangent hyperbolic meanvelocity along with the most unstable 2D mode and a 3Dmode (this initialization is similar to that described inMcMurtry et al., 1989). A grid resolution of 49x49x33 isused for the LES. As noted earlier, the goal here is pri-marily to investigate the subgrid model given initialstates in the LES cells. To accomplish this, the 3D mix-ing layer is initialized by injecting droplets randomly inthe mid spanwise plane with velocity magnitude ran-domly varying in the range 0-30 m/s. The droplets inthree groups: 60, 20 and 7.5 \lm are injected continu-ously in this mid spanwise plane. As the mixing layerevolves in time and coherent 2D spanwise structuresevolve, the droplet distribution is tracked using theLagrangian method.
Figures 2a and 2b show, respectively, two typical stagesof evolution of the mixing layer in terms of the vorticitycontours. The corresponding droplet distribution at thisstage are shown in Figs 3a and 3b, respectively. As canbe seen, the smaller particles are entrained into the coreof the vortex while the larger particles are not.
The contribution of the Lagrangian droplets to the sub-grid void fractions in the LES cells is then determinedby assuming a cut-off size of 10 microns. Figure 4shows the computed void fraction in the LES cells alongthe streamwise direction at the center mid plane. Sincemost of the smaller droplets are concentrated in the core,the volume fraction of the fuel is much higher there than
at other locations. These values of void fraction thenbecome the initial conditions for the subgrid evolution ineach LES cell. Figure 5 shows the typical distribution ofthe void fraction over the entire domain. For the LEScase studied here only a small fraction of the domaincontains droplets smaller that the Lagrangian cut-offsize. However, note that in an actual reacting LES thereis likely to be more cells containing non-zero void frac-tion. In any event, due to unequal distribution of the voidfraction of the fuel in the subgrid, the subgrid LEM isnot required at every LES grid point. This implies thatproper dynamic load balancing needs to be incorporatedwhen attempting parallel simulations of this LESapproach.
The initial subgrid void fractions are very small. How-ever, note that since the liquid density is very large, theoverall mass of the liquid droplets is quite significant.Since we are not simulating the coupled LES-LEMmodel, only the initial states from the LES is used forthe LEM simulations. Also, the subgrid processes evolveindependently between consecutive LES time steps.Here, the evolution of the subgrid Eulerian field as afunction of some relevant parameters is investigated.These parameters are: the initial subgrid temperature,the initial volume fraction and the subgrid turbulence.Some characteristic results are summarized below.
Since there is no data to validate the subgrid vaporiza-tion model discussed here, we first compare the currentpredictions with results obtained earlier by McMurtry etal. (1993) who employed the LEM in a stand-alonemode to study decay of a non-reactive scalar field. Here,using very similar initialization, the decay of a scalarwas investigated in the presence of droplet vaporization.A range of initial values of the void fraction was usedfor these simulations. As shown in Fig. 6, as the dropletevaporates and the void fraction tends to zero (or
cp—» 1), the scalar variance approaches the value pre-dicted by McMurtry et al. (1993) in the absence of drop-lets.
In Fig. 7, the product mass fraction evolution in time(subgrid time between two LES time steps) is shown fora range of initial values of void fraction under otherwiseidentical conditions. The product formation increases intime and with increase in the initial void fraction of fuel.However, since the vaporization process is endothermicand non-linear (initially very high but levels off in timeas temperature fall), the product increase is also non-lin-ear (product formation and vaporization have a directcorrespondence). Figure 8 shows the product mass frac-tion increases with initial liquid and gas temperature. Atpresent, it is assumed that both gas and liquid is at the
same temperature (or that there is infinite conductivityin the subgrid). However, this assumption can berelaxed. An increase of 100 K in the temperatureincreases the initial vaporization rate considerably andthis results in much larger amount of product formed.
Finally, Fig. 9 shows the effect of changing the subgridRe (or the turbulent mixing rate) on the product forma-tion. A physical reasoning for the observed decrease inproduct formation with increase in subgrid Re is notavailable at present. Decrease in product formation isdirectly due to a decrease in the vaporization rate. In thecurrent vaporization model, the estimate for the Spald-ing number BM plays a direct role in the estimate of thevaporization rate. This parameter depends on the pro-cess of diffusion of fuel from the droplet surface to thesurrounding gas. Current equal diffusivity approachneeds to be reevaluated to address this effect. Also, thereis likely to be some subtle effect of the rearrangementprocess used to model the mixing process. Increase inthe subgrid Re changes both the typical eddy size andthe event frequency used to carry out small scale mix-ing. How this impacts the vaporization process is anissue under study.
5. Conclusions
In this study, a gas phase subgrid combustion modeldeveloped earlier has been extended for application intwo-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 otherLES closure techniques. As a result, Reynolds,Schmidt and Damkohler number effects are explicitlyincluded. This model has been implemented within anEulerian-Lagrangian two phase large-eddy simulation(LES) formulation. In this approach, the liquid dropletsare tracked using the Lagrangian approach up to a pre-specified cut-off size. The evaporation and mixing ofthe droplets smaller than the cutoff size is 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 along withsome preliminary results that demonstrate itscapabilities.
Acknowledgments
This work was supported in part by the Army ResearchOffice Multidisciplinary University Research Initiative
grant DAAH04-96-1-0008 and by the Air Force Officeof Scientific Research Focussed Research Initiative con-tract F49620-95-C-0080 monitored by General ElectricAircraft Engine Company, Cincinnati, Ohio.
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Figure 1 a) The velocity vector field around thecore of a decaying vortex.
480 urn384 urn307 urn245 urn196 urn
Figure 1 b) Trajectories of the particles injected intoabove decaying vortex. Smaller particles followingthe vortex while the bigger particles tend to retain theirinitial path.
Figure 2. 3D perspective of mixing layer. Iso-surface corresponding of abs(co) = 0.707. Also shownare the z-vortcity contours at three planes, a) T = 5 & b) T = 15. Here t is normalized time by theinitial [8 u/8y]"1
1.0 rI o°
0.8
0.6
0.4
0.2 r OOO
ooo °0 o o o* -
o o o o _u an o <b - o
0.0 ! o o 00o o o
0.0 0.2a)
0.4 0.6x/xl=n
0.8
Figure 3. Spatial distribution in the (X-Y) plane of particles injected into the center (mid span plane)of the mixing layer. As can be seen the smaller particles are entrained into the vortex core whencompared to the larger particles, a) T = 5 & b) T = 15. The X,Y distances are normalized by respectivedomain lengths.
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