stochastic modeling of atomizing spray in a complex swirl

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Proceedings of the Combustion Institute xxx (2009) xxx–xxx

www.elsevier.com/locate/proci

of the

CombustionInstitute

Stochastic modeling of atomizing spray in a complexswirl injector using large eddy simulation

Sourabh V. Apte a,*, Krishnan Mahesh b,Michael Gorokhovski c, Parviz Moin d

a School of Mechanical, Industrial and Manufacturing Engineering, Oregon State University, 204 Rogers Hall,

Corvallis, OR 97331, USAb Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA

c Laboratory of Fluid Mechanics and Acoustics, UMR 5509, CNRS, Ecole Centrale de Lyon, 69131 Ecully Cedex, Franced Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA

Abstract

Large-eddy simulation of an atomizing spray issuing from a gas-turbine injector is performed. The fil-tered Navier–Stokes equations with dynamic subgrid scale model are solved on unstructured grids to com-pute the swirling turbulent flow through complex passages of the injector. The collocated grid,incompressible flow algorithm on arbitrary shaped unstructured grids developed by Mahesh et al.(J. Comp. Phys. 197 (2004) 215–240) is used in this work. A Lagrangian point-particle formulation witha stochastic model for droplet breakup is used for the liquid phase. Following Kolmogorov’s concept ofviewing solid particle-breakup as a discrete random process, the droplet breakup is considered in theframework of uncorrelated breakup events, independent of the initial droplet size. The size and numberdensity of the newly produced droplets is governed by the Fokker–Planck equation for the evolution ofthe pdf of droplet radii. The parameters of the model are obtained dynamically by relating them to the localWeber number and resolved scale turbulence properties. A hybrid particle-parcel is used to represent thelarge number of spray droplets. The predictive capability of the LES together with Lagrangian dropletdynamics models to capture the droplet dispersion characteristics, size distributions, and the spray evolu-tion is examined in detail by comparing it with the spray patternation study for the gas-turbine injector.The present approach is computationally efficient and captures the global features of the fragmentary pro-cess of liquid atomization in complex configurations.� 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Sprays; LES; Complex geometries; Droplet breakup; Stochastic models

1. Introduction

Liquid spray atomization plays a crucial role inanalyzing the combustion dynamics in many

1540-7489/$ - see front matter � 2009 The Combustion Institdoi:10.1016/j.proci.2008.06.156

* Corresponding author. Fax: +1 541 737 2600.E-mail address: [email protected] (S.V. Apte).

Please cite this article in press as: S.V. Apte etj.proci.2008.06.156

propulsion related applications. This has ledresearchers to focus on modeling of droplet for-mation in numerical investigations of chemicallyreacting flows with sprays. In the traditionalapproach for spray computation, the Eulerianequations for gaseous phase are solved along witha Lagrangian model for droplet transport withtwo-way coupling of mass, momentum, and

ute. Published by Elsevier Inc. All rights reserved.

al., Proc. Combust. Inst. (2009), doi:10.1016/

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2 S.V. Apte et al. / Proceedings of the Combustion Institute xxx (2009) xxx–xxx

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energy exchange between the two phases [1]. Thestandard approach is to first perform spray patter-nation studies for the injector used in combustionchambers and measure the size distributions atvarious cross-sections from the injector. These dis-tributions are then used as an input to a numericalsimulation which then computes the secondaryatomization of the injected droplets. The second-ary atomization is typically modeled by standarddeterministic breakup models based on TaylorAnalogy Breakup (TAB) [2], or wave [3] models.However, this requires performance of experimen-tal tests for any new injector design which can bevery costly.

Development of numerical approaches fordirect simulations of the primary atomization ofa liquid jet or sheet is necessary. However, suchapproaches also require significant computationaleffort. Such numerical schemes capture the com-plex interactions and instabilities near the gas–liquid interface, formation of ligaments and theirdisintegration into droplets. Considerableadvances have been made in this area [4–6]. Thepredictive capability of such schemes may bestrongly influenced by the grid resolutions usedand capabilities for realistic injector geometriesare still under development.

Majority of spray systems in propulsion appli-cations involve complex geometries and highlyunsteady, turbulent flows near the injector. Thenumerical models for spray calculations shouldbe able to accurately represent droplet deforma-tion, breakup, collision/coalescence, and disper-sion due to turbulence. Simulations involvingcomprehensive modeling of these phenomena arerare. Engineering prediction of such flows reliespredominantly on the Reynolds-averagedNavier–Stokes equations (RANS) [7,8]. However,the large-eddy simulation (LES) technique hasbeen convincingly shown to be superior to RANSin accurately predicting turbulent mixing in simple[9], and realistic [10–12] combustor geometries. Itwas shown that LES captures the gas-phase flowphysics accurately in swirling, separated flowscommonly observed in propulsion systems.Recently, Apte et al. [13] have shown good predic-tive capability of LES in swirling, particle-ladencoaxial combustors. The particle-dispersion char-acteristics were well captured by the Eulerian–Lagrangian formulation.

In this work, LES together with a stochasticsubgrid model for droplet atomization is usedfor simulation of spray evolution in a real gas-tur-bine injector geometry. Modeling of the complex-ities of the atomization process is based on astochastic approach. Here, the details of theligament formation, liquid sheet/jet breakup inthe near injector region are not computed indetail, but their global features are modeled in astatistical sense. Following Kolmogorov’s conceptof viewing solid particle-breakup as a discrete


random process [14], atomization of liquid dropsat high relative liquid-to-gas velocity is consideredin the framework of uncorrelated breakup events,independent of the initial droplet size. Gorokhov-ski and Saveliev [15] reformulated Kolmogorov’sdiscrete model of breakup in the form of a differ-ential Fokker–Planck equation for the pdf ofdroplet radii. The probability to break each par-ent drop into a certain number of parts is assumedindependent of the parent-drop size. Using centrallimit theorem, it was pointed out that such a gen-eral assumption leads to a log-normal distributionof particle size in the long-time limit. Thisapproach was further extended in the context oflarge-eddy simulations of the gas-phase by Apteet al. [16] and validated for spray evolution in sim-plified diesel engine configuration.

In this work, the stochastic breakup model isapplied to simulate a sprayevolution from a realisticpressure-swirl injector to evaluate the predictivecapabilityofthemodel togetherwiththeLESframe-work. As the first step, cold flow simulation withstochastic model for secondary atomization isperformed. This study thus isolates the problem ofliquid atomization in pressure-swirl injectors typi-cally used in gas-turbine engines and serves as asystematic validation study for multiphysics, react-ing flow simulations in realistic combustors [12].

In subsequent sections, the mathematicalformulations for the large-eddy simulation of thegaseous-phase and subgrid modeling of the liquidphase are summarized. Next, the stochastic modelfor liquid drop atomization is discussed togetherwith a hybrid particle-parcel algorithm, basedon the original parcels approach proposed byO’Rourke and Bracco [17], for spray simulations.The numerical approach is then applied to com-pute unsteady, swirling flows in a complex injectorgeometry and the results are compared with avail-able experimental data on spray patternationstudies.

2. Mathematical formulation

The governing equations used for the gaseousand droplet phases are described briefly. Thedroplets are treated as point-sources and influencethe gas-phase only through momentum-exchangeterms [13].

2.1. Gas-phase equations

The three-dimensional, incompressible, filteredNavier–Stokes equations are solved on unstruc-tured grids with arbitrary elements. These equa-tions are written as

oui

otþ ouiuj

oxj¼ � o/

oxiþ 1

Reref

o2ui

oxjxj�

oqij

oxjþ _Si; ð1Þ


S.V. Apte et al. / Proceedings of the Combustion Institute xxx (2009) xxx–xxx 3

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where qij denotes the anisotropic part of the sub-grid-scale stress tensor, uiuj � uiuj, and the over-bar indicates filtered variables. The dynamicSmagorinsky model is used for qij [18]. Equation(1) is non-dimensionalized by the reference length,velocity, and density scales, Lref ;U ref ; qref , respec-tively. The reference Reynolds number is definedas Reref ¼ qref Lref U ref=lref . The source-term _Si inthe momentum-equations represent the ‘two-way’ coupling between the gas and particle-phasesand is given by

_Si ¼ �X

k

Gr x; xp

� � qkp

qref

V kp

dukp i

dt; ð2Þ

where the subscript p stands for the droplet phase.The

Pk is over all droplets in a computational

control volume. The function Gr is a conservativeinterpolation operator with the constraint

RV cv

Gr

ðx; xpÞdV ¼ 1 [13],where V cv is the volume of the

grid cell and V kp is the volume of the kth droplet.

2.2. Liquid-phase equations

Droplet dynamics are simulated using aLagrangian point-particle model. It is assumed that(1) the density of the droplets is much greater thanthat of the carrier fluid, (2) the droplets are dis-persed, (3) the droplets are much smaller than theLES filter width, (4) droplet deformation effectsare small, and (5) motion due to shear is negligible.Under these assumptions, the Lagrangian equa-tions governing the droplet motions become [19]

dxp

dt¼ up;

dup

dt¼ 1

sp

ug;p� up

� �þ 1�

qg

qp

!g; ð3Þ

where xp is the position of the droplet centroid, up

denotes the droplet velocity, ug;p the gas-phasevelocities interpolated to the droplet location, qp

and qg are the droplet and gas-phase densities,and g is the gravitational acceleration. The dropletrelaxation time scale (sp) is given as [19]

sp ¼qpd2

p

18lg

1

1þ aRebp

; ð4Þ

where dp is the diameter and Rep ¼ qgdp j ug;p

�up j =lg is the droplet Reynolds number.The above correlation is valid for Rep 6 800.

The constants a ¼ 0:15; b ¼ 0:687 yield the dragwithin 5% from the standard drag curve. Notethat some of the above assumptions for thepoint-particle approach are not valid very closeto the injector. The droplets may undergo defor-mation [20], collision, and coalescence. However,as a first step these effects are not consideredand further investigations are needed to evaluatetheir influence.


2.3. Stochastic modeling of droplet breakup

As the physics of primary and secondary atom-ization are not well understood even in simpleand canonical flow configurations, a heuristicapproach based on stochastic modeling is fol-lowed in order to reduce the number of tuningparameters in an atomization model. A stochasticbreakup model capable of generating a broadrange of droplet sizes at high Weber numbershas been developed [6,15,16]. In this model, thecharacteristic radius of droplets is assumed to bea time-dependent stochastic variable with a giveninitial size-distribution. For very large Webernumbers, there is experimental evidence indicatingthe fractal nature of atomization process [21,22]wherein large droplets can directly disintegrateinto tiny droplets. The stochastic nature of thisprocess is modeled by the present approach. Thebreakup of parent drops into secondary dropletsis viewed as the temporal and spatial evolutionof this distribution function around the parent-droplet size according to the Fokker–Planck(FP) differential equation

oT ðx; tÞot

þ mðnÞ oT ðx; tÞox

¼ 1

2mðn2Þ o

2T ðx; tÞox2

; ð5Þ

where the breakup frequency (m) and time (t) are

introduced. The moments hni ¼R 0

�1 nSðnÞdn and

hn2i ¼R 0

�1 n2SðnÞdn are the two parameters ofthe model that need closure. Here, T ðx; tÞ is thedistribution function for x ¼ lnðrÞ, and r is thedroplet radius. Breakup occurs when t > tbu ¼1=m and r > rcr, the critical radius of the droplet.Following the arguments of scale similarity analo-gous to the turbulence cascade behavior at largeReynolds numbers, Gorokhovski and Saviliev[15] looked at the long-time behavior of thedroplet breakup. They showed that the initialdelta-function for the logarithm of radius of thejth primary droplet evolves into a steady state dis-tribution that is a solution to the Fokker–Planckequation [15,16]

T jðx; t þ 1Þ ¼ 1

21þ erf

x� xj � hniffiffiffiffiffiffiffiffiffiffiffi2hn2i

q0B@

1CA

264

375: ð6Þ

This long time behavior of the distribution ischaracterized by the dominant mechanism ofbreakup. Improvements to the model, whereinpresence of a liquid core near the injector is takeninto account [23], have been proposed, however,in the present work an initial dirac-delta functionis assumed at the injector surface.

The value of the breakup frequency and thecritical radius of breakup are obtained by the bal-ance between the aerodynamic and surface ten-sion forces. The critical (or maximum stable)radius for breakup is then given as



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rcr ¼ Wecrr=ðqgu2r;jÞ where jur;j j is the relative

velocity between the gas and droplet, r the surfacetension coefficient, Wecr the critical Weber num-ber, which is assumed to be on the order of sixover a wide range of Ohnesorge numbers. Forhighly turbulent flows, however, the instanta-neous value of Kolmogorov scale (g) is often lessthan the droplet size and the entire spectrum ofturbulent kinetic energy can contribute to thestretching and disintegration of the droplet. Inthis case, the critical radius should be obtainedas a balance between the capillary forces and tur-bulent kinetic energy supplied to the liquid drop-let. Accordingly, the relative droplet-to-gasvelocity is estimated from the mean viscous dissi-pation and Stokes time scale (sst) as j u2

r;j j� �sst

[24]. Using this relative velocity, the critical radiusof breakup becomes

rcr ¼9

2

Wecrrmlam

�ql

� �1=3

; ð7Þ

where mlam is the kinematic viscosity, ql is the li-quid density, and � is the viscous dissipation rate.In the present LES study, the viscous dissipationcan be obtained dynamically form the resolvedscale energy flux. The breakup frequency is ob-tained following the analogy with expressionsused for aerodynamic breakup and utilizing therelative velocity (jur;j j) from above

tbu ¼ Bffiffiffiffiffiql

qg

rrj

ur;j

�� ; ð8Þ

where rj is the radius of parent drop and

B ¼ffiffiffiffiffiffiffiffi1=3

p[2,25].

If the breakup criterion (t > tbu and r > rcr) fora parent droplet is satisfied, secondary dropletsare sampled from the analytical solution (Eq. 6)corresponding to the breakup time-scale. Theparameters encountered in the FP equation (hniand hn2i) are computed by relating them to thelocal Weber number for the parent drop, therebyaccounting for the capillary forces and turbulentproperties. Apte et al. [16] assumed that in theintermediate range of scales between the parentdrop element (large Weber number) and the max-imum stable droplet (critical Weber number) thereexists no preferred length scale, following the frac-tal nature of atomizing spray [22]. This closelyresembles the inertial range of the energy cascadeprocess in homogeneous turbulence at high Rey-nolds numbers. Analogously, assuming u3

r;j=rj ¼u3

r;cr=rcr, one obtains

rcr

rj¼ Wecr

Wej

� �3=5

) hln ai � hni ¼ K lnWecr

Wej

� �;

ð9Þwhere ur;cr is the relative velocity at which disrup-tive forces are balanced by capillary forces (simi-lar to turbulent velocity scale of the smallest


eddies) and the constant K is of order unity(� 0:6). This gives expression for one of theparameters hni.

Furthermore, from the Einstein’s theory ofBrownian motion, the diffusion coefficient in theFokker–Planck equation is known to be theenergy of Brownian particles multiplied by theirmobility. The drift velocity is presented in theform of drag force times the mobility. The ratioof diffusion to drift velocity is given by the ratioof energy to drag force. In the breakup process,the energy in Einstein’s theory is associated withthe disruptive energy while the force is associatedwith the capillary force on the droplet. Normal-ized by the length scale of the parent drop, thisratio is characterized by the Weber number. Con-sidering the Fokker–Planck equation (Eq. 5), thediffusion to drift velocity ratio is scaled by�hn2i=hni. Then it is assumed that

� hnihn2i� � hln aihln2ai

¼ We�1j : ð10Þ

This relationship gives the maximum dispersion ofnewly produced droplet sizes. Thus, both theparameters in the Fokker–Planck equation areobtained dynamically by computing the local va-lue of Wej, and knowing Wecr.

Once new droplets are created, the productdroplet velocity is computed by adding a factorwbu to the primary drop velocity. This additionalvelocity is randomly distributed in a plane normalto the relative velocity vector between the gas-phase and parent drop, and the magnitude isdetermined by the radius of the parent drop andthe breakup frequency, j wbu j¼ rm. This modifica-tion of newly formed droplets follows the physicalpicture of parent droplets being torn apart byaerodynamic forces giving momentum to thenewly formed droplets in the direction normal tothe relative velocity between the gas-phase andparent drops [2].

As new droplets are formed, parent dropletsare destroyed and Lagrangian tracking in thephysical space is continued till further breakupevents. In the present work, the liquid spray isinjected at atmospheric pressure and tempera-tures. The rates of evaporation are very smalland droplet evaporation is neglected.

2.4. Subgrid scale modeling

In LES of droplet-laden flows, the droplets arepresumed to be subgrid, and the droplet-size is smal-ler than the filter-width used. The gas-phase velocityfield required in Eq. (3) is the total (unfiltered) veloc-ity, however, only the filtered velocity field is com-puted in Eq. (1). The direct effect of unresolvedvelocity fluctuations ondroplet trajectoriesdependson the droplet relaxation time-scale, and the subgridkinetic energy. Considerable progress has been



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made in reconstructing the unfiltered velocity fieldby modeling the subgrid scale effects on droplet dis-persion. Bellan [26] provides a good review on thistopic in the context of spray modeling. Majority ofthe works related to subgrid scale effects on dropletmotion have been performed for dilute loadings,whereinthedropletsareeitherassumedsmaller thanthe LES filter size or the Kolmogorov length scale.For dense spray systems, droplet dispersion anddroplet interactions with subgrid scale turbulencearenotwellunderstood.Inaddition,Inrealisticcon-figurations the droplet sizes very close to the injectorcan be on the order of the grid size used for LEScomputations.

Recently, Pozorski and Apte [27] performed asystematic study of the direct effect of subgrid scalevelocity on particle motion for forced isotropic tur-bulence. It was shown that, in poorly resolvedregions, where the subgrid kinetic energy is morethan 30%, the effect on droplet motion is more pro-nounced. A stochastic model reconstructing thesubgrid-scale velocity in a statistical sense wasdeveloped [27]. However, in well resolved regions,where the amount of energy in the subgrid scalesis small, this direct effect was not strong. In the pres-ent work, the direct effect of subgrid scale velocityon the droplet motion is neglected. However, notethat the droplets do feel the subgrid scales throughthe subgrid model that affects the resolved velocityfield. For well-resolved LES of swirling, separatedflows with the subgrid scale energy content muchsmaller than the resolved scales, the direct effectwas shown to be small [13].

3. Numerical method

The computational approach is based on a co-located, finite-volume, energy-conserving numeri-cal scheme on unstructured grids [10] and solvesthe incompressible Navier–Stokes equations. Thevelocity and pressure are stored at the centroidsof the control volumes. The cell-centered veloci-ties are advanced in a predictor step such thatthe kinetic energy is conserved. The predictedvelocities are interpolated to the faces and thenprojected. Projection yields the pressure potentialat the cell-centers, and its gradient is used to cor-rect the cell and face-normal velocities. A noveldiscretization scheme for the pressure gradientwas developed by Mahesh et al. [10] to providerobustness without numerical dissipation on gridswith rapidly varying elements. This algorithmwas found to be imperative to perform LES athigh Reynolds numbers in realistic combustorgeometries and is essential for the present config-uration. This formulation has been shown to pro-vide very good results for both simple andcomplex geometries [10–12].

In addition, for two-phase flows the particle cen-troids are tracked using the Lagrangian framework.


The particle equations are integrated using third-order Runge–Kutta schemes. Owing to the dispar-ities in the flowfield time-scale and the droplet relax-ation time (sp) subcycling of the droplet equationsmay become necessary. After obtaining the newdroplet positions, the droplets are relocated, drop-lets that cross interprocessor boundaries are dulytransferred, boundary conditions on droplets cross-ing boundaries are applied, source terms in the gas-phase equation are computed, and the computationis further advanced. Solving these Lagrangianequations thus requires addressing the followingkey issues: (i) efficient search for locations of drop-lets on an unstructured grid, (ii) interpolation ofgas-phase properties to the droplet location forarbitrarily shaped control volumes, (iii) inter-pro-cessor droplet transfer. An efficient Lagrangianframework was developed which allows trackingmillions of droplet trajectories on unstructuredgrids [13,16].

3.1. Hybrid droplet-parcel algorithm for spraycomputations

Performing spray breakup computations usingLagrangian tracking of each individual dropletgives rise to a large number of droplets (�20–50million) in localized regions very close to the injec-tor. Simulating all droplet trajectories gives severeload-imbalance due to presence of droplets ononly a few processors. On the other hand, correctrepresentation of the fuel vapor distributionobtained from droplet evaporation is necessaryto capture the dynamics of spray flames. In theirpioneering work, O’Rourke and Bracco [17] useda ‘discrete-parcel model’ to represent the spraydrops. A parcel or computational particle repre-sents a group of droplets, N par, with similar char-acteristics (diameter, velocity, and temperature).Typically, the number of computational parcelstracked influences the spray statistics predictedby a simulation.

The original work of O’Rourke and co-work-ers [2,17] inject parcels from the injector, resultingin much fewer number of tracked computationalparticles. In this work, the parcels model is furtherextended to a hybrid particle-parcel scheme [16].The basic idea behind the hybrid-approach is asfollows. At every time step, droplets of the sizeof the spray nozzle are injected based on the fuelmass flow rate. New droplets added to the compu-tational domain are pure drops (N par ¼ 1). Thesedrops are tracked by Lagrangian particle trackingand undergo breakup according to the stochasticmodel creating new droplets of smaller size. Asthe local droplet number density exceeds a pre-scribed threshold, all droplets in that control vol-ume are collected and grouped into binscorresponding to their size. The droplets in binsare then used to form a parcel by conserving mass.Other properties of the parcel are obtained by



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mass-weighted averaging from individual dropletsin the bin. The number of parcels created woulddepend on the number of bins and the thresholdvalue used to sample them. A parcel thus createdthen undergoes breakup according to the abovestochastic sub-grid model, however, does not cre-ate new parcels. On the other hand, N par isincreased and the diameter is decreased by mass-conservation.

This strategy effectively reduces the total num-ber of computational particles in the domain.Regions of low number densities are captured byindividual droplet trajectories, giving a more accu-rate spray representation.

4. Computational details

Figure 1 shows a schematic of the computa-tional domain used for the spray patternationstudy of a realistic Pratt and Whitney injector.The experimental data set [28] (R. McKinney,R.K. Madabhushi, S. Syed, private communica-tion, 2002) was obtained by mounting the actualinjector in a cylindrical plenum through whichgas with prescribed mass-flow rate was injected.Figure 1b shows a cut through the symmetryplane (Z=Lref ¼ 0) of the computational domainalong with the mesh and boundary conditionsused. For this case, 3:2M grid points are used withhigh resolution near the injector. The grid ele-ments are a combination of tetrahedra, prisms,wedges, and hexahedra to represent complex geo-metric passages inside the injector. Grid refine-ment study for LES of single phase flow hasbeen performed for different cases in complex con-figurations [10,11]. The grid resolution for thepresent case was decided based on these validationstudies.

Air from the inlet plenum goes through thecentral core, guide, and outer swirlers to create

a

Fig. 1. The computational domain: (a) schematic of entire resymmetry plane.


highly unsteady multiple swirling jets. The domaindecomposition is based on the optimal perfor-mance of the Eulerian gas-phase solver on 96 pro-cessors. Brankovic et al. [28] provide details of theexperimental measurement techniques and inflowconditions for a lower pressure drop across thefuel nozzle. The inflow conditions in the presentstudy are appropriately scaled to a higher pressuredrop providing the air mass flow rate of0:02687 kg=s. The flow Reynolds number basedon the inlet conditions is 14; 960. A uniform meaninflow velocity was specified at the inlet withoutany turbulent fluctuations. In the present case,the downstream cylindrical plenum is open toatmosphere. The air jet coming out of the nozzlethus entrains air from the surrounding. Entrainedflow along the surface of the downstream plenumwas modeled as a radially inward velocity alongthe entire plenum surface. The experimental dataprofiles at different cross-sections were integratedat each station to obtain the total flow rate atthose locations. Knowing the net inflow rate, theentrained mass at each of the entrainment bound-aries was estimated and assigned to the calcula-tion. This modeling approach for entrained flowis subject to experimental verification, however,was shown to have little impact on the predictedflowfield [28]. No-slip conditions are specified onthe wall. Convective boundary conditions areapplied at the exit section by conserving the globalmass flow rate through the computationaldomain. and experimentally measured radialentrainment rate is applied on the cylindrical sur-face of the computational domain downstream ofthe injector.

Liquid fuel is injected through the filmer sur-face which forms an annular ring near the outerswirler. In the symmetry plane this is indicatedby two points on the edge of the annular ring.The ratio of the liquid to air mass flow rates atthe inlet is fixed at 0:648. The liquid film at the fil-

x

y

-5 0 5-6

-4

-2

0

2

4

6

inlet

entrainment

entrainmentwall

coreswirler

outer swirler

guide swirler

wall

outlet

b

gion and (b) unstructured grid near the injector in the


Fig. 2. Instantaneous snapshots of near-injector axialvelocity field and spray droplets: (a) axial velocity inZ=Lref ¼ 0, (b) axial velocity in X=Lref ¼ 1:1, (c) spraydroplets in Z=Lref ¼ 0, and (d) spray droplets inX=Lref ¼ 1:1. Also shown is the color scale for normal-ized axial velocity.


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mer surface is approximated by injecting uniformsize large drops of the size of the annular ringthickness. These drops are convected downstreamby the swirling air and undergo breakup accord-ing to the stochastic model. The velocity of eachdroplet is specified based on the velocity of theliquid film. A large number of droplets are createdin the vicinity of the injector due to breakup. Thelocation of droplet injection around the annularring is chosen using uniform random distribution.This discrete representation of the film near theinjector surface may not represent the physics ofligament formation and film breakup. However,the statistical nature of droplet formation furtheraway from the injector is of interest in the presentstudy and is well captured by the stochastic modeltogether with LES of the air flow.

With the hybrid approach, the total number ofcomputational particles tracked at stationary stateis around 3:5 M and includes around 150; 000parcels. Together these represent approximately13 M droplets.

The computations were performed on the IBMcluster at the San Diego Supercomputing center.

5. Results and discussion

Figure 2a and b shows the instantaneous snap-shots of the axial velocity contours in theZ=Lref ¼ 0 symmetry plane and in cross-sectionX=Lref ¼ 1:1. Figure 2c and d shows the corre-sponding snapshots for spray droplets (whitedots). The swirling air jet from the core swirlerenters the dump region and forms a recirculationzone. Jets from guide and outer swirlers interactwith the core flow. The swirling air jets enteringthe sudden expansion region create radiallyspreading conical jets with a large recirculationregion just downstream of the injector. A complexvortex break down phenomenon is observed andits accurate prediction is necessary to correctlyrepresent the injector flow. The swirl strengthdecays further away from the injector due to vis-cous dissipation. The scatter plot of the spraydroplets show dense spray regimes close to theinjector which become dilute further away. Theparent droplets are injected at the edge of theannular ring. These droplets are carried by theswirling flow and form a conical spray. The con-centration of the spray droplets is high on theedge of the recirculation region. The strong rela-tive motion between the large inertial dropletsnear the injector and the fluid flow leads tobreakup and generation of smaller droplets. Thedroplets spread radially outward and swirl aroundthe injector axis as they move downstream.

Figure 3a and b compares the LES predictionsto the available experimental data of radialvariations of mean axial and swirl velocity at dif-ferent axial locations. The numerical results are


azimuthally averaged. The predictions from oursimulation are in close agreement with the exper-imental data (R. McKinney, R.K. Madabhushi,S. Syed, private communication, 2002). The sizeand evolution of the recirculation region is wellcaptured as indicated by the axial velocity predic-tions. The swirl strength decays downstream ofthe injector. Small disagreement at X=Lref ¼ 2:1is partly related to the coarse grid resolution usedaway from the injector. It should be noted that theamount of swirl generator by the injectors deter-mines the size of the recirculation zone. Goodagreement of the axial and swirl velocities indicatethat LES with dynamic sgs model can capture thevortex break-down phenomenon accurately incomplex geometries. Also shown are the corre-sponding predictions using the standard k � �model on the present grid. The unsteady RANSsolutions are in agreement with the LES andexperimental data very close to the injector, how-ever, degrade rapidly further away, showing limi-tations of the turbulence model. RANSpredictions of the flow through the same injectorat different conditions [28] show similar trends.Improved predictions using advanced RANSmodels can be obtained, however, the superiorityof LES is clearly demonstrated. Any artificial dis-sipation or inaccurate numerics gives faster decayof the swirl velocities and incorrect size of therecirculation region, further emphasizing theimportance of non-dissipative numerical schemesfor LES.

Figure 4a and b compares the radial variationof liquid mass-flowrates using LES and the sto-chastic model to the experimental data at two dif-ferent cross-sections. The flow rates are presented


R/Lref

u x/U

ref

-2 -1 0 1R/Lref

-2 -1 0 1 2R/Lref

u x/U

ref

-1 0 1 2

R/Lref

⎯uθ

/Ure

f

-2 -1 0 1 2

R/Lref

⎯uθ

/Ure

f

-2 -1 0 1 2

R/Lref

u θ/U

ref

-2 -1 0 1 2

a b c

d e f

Fig. 3. Radial variation of normalized mean axial and swirl velocity at different axial locations, � experimental data (R.McKinney, R.K. Madabhushi, S. Syed, private communication, 2002), — LES,- - - - RANS: (a and d) X=Lref ¼ 0:4, (band e) X=Lref ¼ 1:1, (c and f) X=Lref ¼ 2:1.


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as the ratio of the effective to the integrated flowrate. The effective flow rate is defined as the flowrate the patternator would record if the fuel fluxwas uniform at the local value. This normalizationinherently carries the ratio of the total cross-sec-tional area to the area of the local patternatorholes. The LES results are generally in goodagreement with the experiments. Average dropletsizes at two axial location from the injector wallhave been measured using the Malvern line ofsight technique (R. McKinney, R.K. Madabhu-shi, S. Syed, private communication, 2002). TheSauter mean diameters averaged over the cross-section at these two axial locations are predictedwithin 5% of the experimental values.

Figure 5a and b compares the mass-based size-distribution function compared with the experi-mental data at two different cross-sections fromthe injector. It is observed that the predicted dis-tribution functions agree with experimental obser-vations for large-size droplets. However, the

Normalized Radius

Nor

mal

ized

Axia

lMas

sFl

owra

te

0 0.5 1 1.5 20

2

4

6

8

Normalized Radius0 0.5 1 1.5 2

0

1

2

3

4

5

6

a b

Fig. 4. Comparison of radial variation of normalizedliquid axial mass flux at two axial locations, � — �experimental error bar (R. McKinney, R.K. Madabhu-shi, S. Syed, private communication, 2002), — LES: (a)X=Lref ¼ 1:1 and (b) X=Lref ¼ 2:1.


simulations predict larger mass of small size drop-lets compared with the experimental data. This isattributed to the lack of collision/coalescencemodels in the present simulation. Also, small sizedroplets can easily evaporate even at low temper-atures and the present simulations do not considerthis effect. In addition, the initial droplet size atthe injector nozzle is assumed to be a constant,whereas it may vary depending on the local condi-tions governing primary atomization. Models tak-ing into account the presence of a liquid core nearthe injector can be incorporated to better capturethe recirculation regions. A dirac-delta functionwas used to inject large drops from the injectorsurface and a better representation of these initialconditions can improve the predictions [23]. Fur-ther improvements to the model can also beobtained by modeling the primary breakup regimevery close to the injector. An investigation withinclusion of collision models as well as using a sizedistribution at the inlet should be performed in

Dp/Lref

50 1000

10

20

30

40

50

60x 10-3

Dp/Lref

1/m

pdm

p/d(d

p)

50 1000

10

20

30

40

50

60x 10-3

a b

Fig. 5. Comparison of normalized droplet mass-distri-bution at different axial locations, � experiments (R.McKinney, R.K. Madabhushi, S. Syed, private commu-nication, 2002), — LES: (a) X=Lref ¼ 1:1 and (b)X=Lref ¼ 2:1.



ARTICLE IN PRESS

order to investigate uncertainties in model predic-tions. However, the overall predictions of the LESmethodology together with a simple stochasticbreakup model of liquid atomization are in goodagreement with experiments. The dispersion ofdroplets in an unsteady turbulent flow is well rep-resented when the flowfield is computed usingLES.

6. Summary and conclusions

A large-eddy simulation of an atomizing sprayissuing from a gas-turbine injector is performedcorresponding to the spray patternation studyof an injector used in a Pratt and Whitney com-bustor. The filtered Navier–Stokes equations withdynamic subgrid scale models are solved onunstructured grids to compute the swirling turbu-lent flow through complex passages of the injec-tor. A Lagrangian point-particle formulationwith stochastic models for droplet breakup isused for the liquid phase. The atomization pro-cess is viewed as a discrete random process withuncorrelated breakup events, independent of theinitial droplet size. The size and number densityof the newly produced droplets is governed bythe Fokker–Planck equation for the evolutionof the pdf of droplet radii. The parameters ofthe model are obtained dynamically by relatingthem to the local Weber number and resolvedscale turbulence properties. It is assumed thatfor large Weber numbers there exists no preferredlength scale in the intermediate range of scalesbetween the parent drop element and the maxi-mum stable droplet, following the fractal natureof atomizing spray [21,22]. A hybrid particle-par-cel approach is used to represent the large num-ber of spray droplets. The swirling, separatedregions of the flow in this complex configurationare well predicted by the LES. The droplet massfluxes and size distributions predicted are withinthe experimental uncertainties further away fromthe injector. The present approach, however,overpredicts the number density of small sizedroplets which can be attributed to the lack ofcoalescence modeling. In addition, the primarybreakup regime very close to the injector wasnot simulated. Models taking into account thepresence of a liquid core near the injector [23]can be incorporated to better capture the recircu-lation regions. However, with present stochasticapproach the droplet dispersion characteristicsare well captured. The global features of the frag-mentary process of liquid atomization resulting ina conical spray are well represented by the pres-ent LES in realistic injector geometry. This sto-chastic modeling approach has been used toperform full scale simulations of turbulent spraycombustion in a real Pratt and Whitney combus-tion chamber [12].


Acknowledgments

Support for this work was provided by theUnited States Department of Energy under theAdvanced Scientific Computing (ASC) program.The computer resources at San Diego Supercom-puting Center are greatly appreciated. We are in-debted to Dr. Gianluca Iaccarino and thecombustor group at Pratt and Whitney.

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stochastic modeling of atomizing spray in a complex swirl

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