482 J. ENERGY VOL. 7, NO. 6

Aspects of Computer Simulation of Liquid-Fueled Combustors

A. D. Gosman* and E. loannidestImperial College, London, England

An existing "discrete droplet" model of liquid sprays has been extended to include a stochastic representationof turbulent dispersion effects. Applications to simple test cases, including the dispersion of single particles,produce reasonable agreement. However, two further applications involving volatile and combusting spraysshow that the turbulent dispersion effects are small in comparison to those due to uncertainties about the initialconditions of the spray.

CD

D

ikKmdmfu, mox,mp

MHdMUd,MVd,MWdNuPPrQLrRe

ScShTttrU,V,WU,V,W (u' ,v' ,wxreMeffP

Subscriptsde

Nomenclature= drag coefficient= specific heat at constant pressure= droplet diameter= mass transfer coefficient= enthalpy= heat of combustion=; stoichiometric oxygen/fuel ratio= kinetic energy of turbulence= thermal conductivity= droplet mass= mass fraction of fuel vapor, oxygen, and

products= droplet enthalpy= droplet U, V, ^momentum= Nusselt number= pressure= Prandtl number= latent heat of vaporization= radial coordinate= Reynolds number= combustion rate= radius of furnace= source term= droplet position vector= Schmidt number= Sherwood number= temperature= time= residence time of a droplet in an eddy= time-averaged velocity components= instantaneous velocity components= fluctuating velocity components= axial coordinate= diffusion coefficient= dissipation rate of turbulence= effective viscosity= density= general variable= droplet relaxation time

= droplet= eddy

Presented as Paper 81-0323 at the AIAA 19th Aerospace SciencesMeeting, St. Louis, Mo., Jan. 12-15, 1981; submitted June 29, 1981;revision received July 1, 1983. Copyright © American Institute ofAeronautics and Astronautics, Inc., 1983. All rights reserved.

* Reader in Fluid Mechanics, Mechanical Engineering Department.tResearcher, Mechanical Engineering Department (presently, Head

of Bearing Applications Department, SKF-ERC, Nieuwegein, theNetherlands). Member AIAA.

fu = fuelL = liquidm - means = droplet surfacev — vapor

Introduction

THE combustion of liquid-fuel sprays has numerousimportant applications in furnaces, diesel engines, gas

turbines, and other equipment. The increasing need for fueleconomy and emissions control has generated fresh interest inboth experimental and theoretical studies of spray flames.Recent experimental studies include Hiroyashu and Kadota,1

Shearer et al.,2 and Tishkoff et al.,3 on evaporating spraysand Onuma et al.,4 Owen et al.,5 and Found et al.,6 oncombusting sprays. On the theoretical side, variants of twomodels are currently employed, namely, the Williams7

statistical spray model used by Westbrook,8 Haselman andWestbrook,9 and Cliffe et al.10 and the "discrete droplet"model used by Crowe,11 Gosman and Johns,12 El-Banhawyand Whitelaw,13 Abbas et al.,14 Gosman et al.,15 Dukowicz,16

and O'Rourke and Bracco.17

The Williams7 statistical spray model considers ageneralized spray distribution function originally defined inan eight-dimensional space of droplet diameter, location,velocity, and time. Conservation principles yield a partialintegro differential equation for this function and the solutionof this equation, together with the gas conservationequations, provides the required model of the spray. Asapplied to date, however, this approach is costly in terms ofcomputer storage and time unless simplifications are in-troduced, such as the assumption of no slip between thedroplets or the gases or the representation of the spray by avery limited number of droplet sizes. Furthermore, due to thelimited resolution (especially in the vicinity of the atomizer) ofthe numerical methods generally employed for the solution ofthe spray equation, this method may introduce substantialspurious numerical diffusion into the calculation.

In the "discrete droplet" model, the spray is represented byindividual droplets rather than by a continuous distributionfunction. Because of the large number of actual dropletscontained in the spray, the representation is confined to astatistical sample. Therefore, each of these sample dropletscharacterizes a "parcel" of like numbers, all having the sameinitial size, velocity, and temperature. The motion, heating,and evaporation of each sample as it traverses the gas arecomputed by solving numerically the Lagrangian ordinarydifferential equations governing the mass, momentum, andenergy conservation. The effects of the droplets on the gasphase are introduced into the (Eulerian) calculations of thelatter by feeding in the local rates of heat, mass, andmomentum exchange deduced from the analysis of the droplet

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parcel trajectories. The overall solution is obtained byiterating between the calculations of the two phases.

An important component of any theoretical model is themanner in which the effects of turbulence on dropletdispersion are represented. In many studies (e.g., West-brook,8 Haselman and Westbrook,9 Crowe,11 Crowe et al.,18

El-Banhawy and Whitelaw,13 and Gosman et al.15) this effectis ignored. Some researchers, such as Abbas et al.,14 introduceit in a deterministic way by working out from somephenomenological model a "diffusion" velocity, which isadded to that obtained from the original Lagrangianequations of motion, thus modifying the parcel trajectory. Athird, and probably more correct, approach is the stochasticone developed by Dukowicz16 and later elaborated for thickspray applications by O'Rourke and Bracco,17 in which thegas turbulence is randomly sampled during each droplet'sflight and allowed to influence its motion, the gross spraybehavior being obtained by averaging over a statisticallysignificant sample of droplets.

The method to be described in this paper is a stochastic"discrete droplet" method in the spirit of Dukowicz.However, it differs in details of the treatment of the gas-phaseturbulence and the manner in which it is thought to interactwith the droplet motion. In particular, this interaction is heldto occur over a time interval that is the minimum of two timescales, one being a typical turbulent eddy lifetime and theother the residence time of the droplet in the eddy. These timescales, as well as the local turbulence intensity, are obtained inthe present study by solving for the time-averaged turbulenceenergy k and its dissipation rate e, using a version of the well-known "k-e" turbulence model of Launder and Spalding.19

The turbulence model also forms part of a conventionalEulerian finite-volume calculation procedure for the gasphase, the interactions between the droplets and gas beingdealt with in an iterative fashion as outlined earlier. Theaccuracy of the overall scheme has been checked by referenceto the known analytical solution for turbulence diffusion ofinertialess particles from a point source in a homogeneousisotropic turbulent flow (Hinze20) and by comparison with themeasured averaged dispersion rates of individual particlesintroduced into the turbulent gas flow downstream of a grid,as reported by Snyder and Lumley.21

In addition to these two cases, where the boundary con-ditions on both particles and gas are well-known and detailedinformation is available on the behavior of both phases,thereby allowing reasonably definitive assessment of themodeling, further calculations are presented for theevaporating spray reported by Tishkoff et al.3 and thecombusting swirling spray reported by Found et al.6 In bothof the latter cases information on the boundary conditionsand interior behavior is not sufficiently complete to allowdefinitive assessment. They are used here instead to explorethe sensitivity of the calculations of such flows to the tur-bulent dispersion modeling.

where £// is the velocity component in direction xi9 the impliedsummation being restricted to the axial and radial com-ponents; 4> represents any of the variables just mentionedapart from pressure. The terms S and Feff are, respectively,the "source" and the effective diffusion coefficient for entity,while Sd represents the particular sources due to the presenceof the droplets. The continuity equation is obtained by setting3> = 1 and Feff = 1. The particular expressions for 5, F, and Sdpertaining to each variable are given in Table 1.

In deriving Eq. (1), certain terms involving correlations ofthe fluctuating components arise that must be modeled. Theapproach employed here is similar to that described byJones22 for single-phase combusting flows, although it shouldbe said that at this stage that correlations involving fluc-tuating properties of the droplet field have been ignored forwant of _a_better method. Thus, the turbulent Reynoldsstresses pu-uj and scalar fluxes pw/c/>' are modeled by

dU,

X

(2)

(3)

where \it and F, are the turbulent viscosity and diffusivity,respectively, and k the time-mean kinetic energy of tur-bulence. The turbulent viscosity is obtained from

(4)

where e is the time-mean dissipation rate of k, CM is a con-stant, and Tt is calculated from

Ut,<t> (5)

Here at>(t) is the turbulent Prandtl/Schmidt number, also takenas a constant. Finally, the local values of k and e are deter-mined by solving two additional transport equations for themof the general form of Eq. (1). The associated transportcoefficients and source terms are shown in Table 1, togetherwith the values of the various empirical constants.

For the furnace application, a combustion model is em-ployed that assumes the droplets evaporate sufficientlyrapidly to form a cloud of vapor burning as a gaseous flame.However, it is necessary to recognize that as a result of thevariety of conditions occurring in furnaces this flame can beof a diffusion and/or premixed character (Faeth27) andtherefore the combustion model should be capable of han-dling both types. In the present study, the Magnussen andHjertager28 "eddy mixing control" model was used, whichpossesses this capability. This model assumes that the fuel andoxygen combine irreversibly in a single global reaction toform products, the time-averaged fuel consumption rate Rfuand species concentrations being linked by

The Mathematical FormulationThe Gas Field

The dependent variables characterizing the time-averagedaxisymmetric gas flow are the axial, radial, and cir-cumferential velocity components, denoted by U, V, and W,respectively, the static pressure P, the total enthalpy h, andthe mass fractions ra, of the chemical constituents. Thepartial differential conservation equations that govern theseare, for the case of a dilute spray in which the effects ofdisplacement of the gas by the droplets may be ignored, are ofthe following form:

k-

d

^(1)

(6)

where A and B are constants, the subscripts fu, ox, and prrefer to fuel, oxygen, and product, respectively, / is thestoichiometric combination ratio, and k/e is the time scale ofthe turbulent eddies, which is assumed to be much larger thanthe chemical kinetic time scales of the hydrocarbon reaction.The species mass fractions are therefore determined bysolving transport equations for each possessing the form ofEq. (1), in which, as indicated in Table 1, the reaction ratesare determined from Eq. (6) or an appropriate multiplethereof.

Finally, radiation is calculated by way of a "four-flux"model similar to that employed by Gosman and Lockwood29;

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however, at this stage droplet and soot radiation are notmodeled. The local density is obtained from the equation ofstate for an ideal mixture.

The Droplet FieldThe model employed for the droplet calculations in the

present study is outlined in this section. The Lagrangianforms of the governing equations for the instantaneousmotion and rates of heating and evaporation used in thepresent study are as given below.

Droplet Motion and TrajectoryThe axial, radial, and tangential momentum equations are,

respectively, as follows:

dt

dt 4

dwd 3 p~dt ~ ~4 PdD

(7)

(8)

^ i ^ \CD(w-wd) \u-

where u and ud are the local gas and droplet velocity vectors,respectively. Integration of these equations provides thevelocity components of the droplets, from which the

trajectories may be obtained by a further integration of theequation for the position vector sd,

ds^dt (10)

The drag coefficient CD is obtained from the following ex-pressions, taken from Wallis30:

CD = 0.44 for Red>1000 (11)

CD = (l + 0.15Re°d687)/(Red/24) for Red<100Q (12)

where the droplet Reynolds number Red is defined by

Red = p\u-ud\D/n (13)

Droplet HeatingThe balance equation is (from Borman and Johnson31):

(14)•-7 dt

where

(15)

Table 1 Transport coefficients and source terms for the variable </>

VariableSource term S Droplet source term 5^

U

V

w

k

e

h

rrifu

mpr

d ( dU\ 1 d ( dV\ dP 1 ^Meff IT Ueff —— ) + - — — UeffrlT / IT ~ 77 LJ ( MUd0 ~ MUdj ) ,dx \ dx ' r dr \ dx / dx Ns k=] ° '

d ( dU\ 1 d ( dV\ 2^V 1 ^Meff T~ Ueff 7~ J+~7~Ueff / '~ ) ~ ~T ~ 77 2-f (MK^ ~MKrf/^dx \ dx / r dr \ dr / r Ns k=] ° '

pW2 dP' r2 dr

V J N

\ 2 r r dr / 7V5 ̂

^eff(Tgff A,

-̂ - (e/*)(C7G*-C2pe) 0aeff, e

7 N

Meff V fM// A///aeff,/2 ^5 A:=7

N^eff ^ V^

^eff./w Ns k=l

//? naeff , ox

Meff (7 I / ) /> 0^eff , pr

Notes:1) 5^ expressions are presented in the forms employed in the numerical calculations.2) Turbulence model constants are assigned the following values: CM = 0.09,C/ = 1.44,C2 = 1.92, ak --

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and the Nusselt number Nu is calculated from the Ranz andMarshall23 correlation

(16)

Droplet EvaporationThe evaporation rate is obtained from

^ = -TTDPvDvtn(l+By)Sh (17)

where By = (mfUjS -mfuoo)/(\ -mfUtS).The Sherwood number Sh is obtained from Eq. (16), with

Nu and Pr replaced by Sh and Sc, respectively.

Turbulent Dispersion of DropletsAs mentioned in the Introduction, the effect of the tur-

bulence on the droplet motion is simulated here by astochastic approach, one element of which is the evaluation ofthe instantaneous gas velocity u in the droplet equations ofmotion (14-16) from the time-averaged gas velocity U andturbulence energy k fields. For this purpose, the turbulence isassumed to be isotropic and to possess a Gaussian probabilitydistribution in the fluctuating velocity, whose standarddeviation an is given by

(18)

Random sampling of this distribution at appropriate pointsin the trajectory calculation then yields the estimatedprevailing fluctuating velocity field u' and hence the in-stantaneous field u = U+ u'.

A second important element is the manner of determiningthe time interval t-mt over which the droplet interacts with therandomly sampled velocity field. Here, it is convenient toenvisage the latter as being associated with a turbulent eddy,in which case the interaction time is determined by one or theother of the following possible events: 1) the droplet movessufficiently slowly relative to the gas to remain within theeddy during the whole of its lifetime te or 2) the relative or"slip" velocity between the gas and droplet is sufficient toallow it to traverse the eddy in a transit time tR shorter thante. The interaction time scale will therefore be the minimumof the above (see Ref. 24), i.e.,

t-mi=min(te,tR) (19)

Estimates of the eddy and transit time scales are made underthe further assumption that the characteristic size of therandomly sampled eddy is the dissipation length scale le, givenby*

le = C»k3/2/e (20)

The eddy lifetime is then estimated as

te = le/ l i / ' l (21)

The transit time scale tR is estimated from the followingsolution of a simplified and linearized form of the equation ofmotion of the droplet:

tR = -rHn[LO-le/(r\u-ud I ) ] (22)

$In an earlier version of this paper a CM exponent of 3/4 waserroneously deduced (the authors are grateful to Prof. G. M. Faethfor drawing their attention to this). The results here are based on themore appropriate value of 1/2. The change has not altered the con-clusions.

where r is the droplet "relaxation" time, defined as

T=(4/3)PdD/(pCD\u-ud\) (23)

In circumstances where Ie/(r\u — ud\)>l9 Eq. (22) has nosolution. This can be interpreted as implying that the droplethas been "captured" by the eddy, in which case t-mt = te.

The above procedure may be repeated for as many in-teraction times as are required for the droplet to traverse therequired distance. Clearly, if a statistically significant numberof droplet samples is tracked in this way, the ensemble-averaged behavior should represent the turbulent dispersioninduced by the prevailing gas field.

Boundary ConditionsThe boundary conditions for the gas-phase calculations

must be specified on all surfaces enclosing the region of in-terest; details will be provided when the individual cases arepresented. As for the droplet calculation, this is essentially aninitial-value problem, but the necessary information re-garding the initial velocity, size, and temperature of thedroplets is rarely available, due to the complexity and theirregular character of the atomization process that producesthem. As a consequence, a strong element of guesswork andempiricism enters into spray calculations, which inevitablyhinders their evaluation. The particular practices employed inthe present study will be given separately for each case.

Numerical AnalysisThe Gas Field

The solution method used for the gas equations is thatdescribed by Caretto et al.25 and embodied in the TEACH-Tcode that was adapted for the present circumstances.Inasmuch as details of both the method and code have beenpublished elsewhere (see, e.g., Hutchinson et al.26), only anoutline will be provided here.

The method is of the finite-volume variety, in which thedifferential equations are cast into an algebraic form thatpreserves their conservation and boundedness properties, on acomputing mesh formed from cylindrical-polar coordinatesurfaces. The resulting algebraic system is solved iteratively ina sequence that extracts the velocities from the momentumequations and then employs a continuity-based equation tocompute the pressures. The calculation of the other gas fieldvariables is embedded in this sequence, during which the"sources" associated with the droplets are held constant.

The Droplet FieldThe ordinary differential equations governing the behavior

of each droplet are solved by forward numerical integration,starting from the injection location, at which the initialconditions are (stochastically) prescribed, and proceedinguntil it either leaves the calculation domain or evaporates to anegligible size. During this process, the gas field propertiesappearing in the equations are interpolated from theprevailing values at the nearest nodes of the computationalgrid.

Overall Solution ProcedureEach iteration cycle of the two-phase calculation involves

two stages. First, a chosen number of droplets (18 in thepresent case) is introduced and their trajectories are calculatedaccording to the above procedure. At each computational celltraversed by the droplets, the mass, momentum, and energyextracted or deposited are accumulated and averaged via the"droplet source" expressions appearing in Table 1. These areof the form for, e.g., the mass source,

(24)

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where Ns is the cumulative total number of computationaldroplets introduced over all iterations performed, TV thecumulative number which traverse the cell in question, andMdi and Mdo are the mass fluxes of the kth parcel entering andleaving, respectively, i.e.,

= (>K/6)pdD3knk (25)

Here nk is the number of "real" droplets issuing from thespray nozzle per unit time and per computational droplet.Similar interpretations apply to the remaining droplet sourceentries in Table 1 .

Second, the droplet sources are inserted into the gas-phaseequations and one iteration is performed. This cyclic processis repeated until the gas-phase calculation converges, by whichstage it follows that the sources derived from the stochasticdroplet treatment have attained statistically stationary values.It may, of course, be possible to accelerate this process byintroducing a more or less computational droplets per cycle,but this has not yet been explored.

Applications and AssessmentDiffusion from a Point Source in a Homogeneous Flow

It is well known that the spatial dispersion of "marked"fluid particles introduced at a constant rate from a pointsource into a uniform turbulent flow of the same fluid isamenable to exact analysis when the turbulence ishomogeneous and isotropic and long diffusion times areconsidered. The distribution of the concentration C(xlrx2,x3)of these particles at a point (x},x2,x3) is, according toHinze,20

C(x/,x2,x3)=S Q\p[-U(x22+x2

3)/(4Tt \ t \x, I

(26)

where F, is the (uniform) turbulent diffusivity and S thevolumetric source strength.

A comparison of concentration profiles extracted from Eq.(26) and predictions from the present dispersion model forcorresponding circumstances (i.e., very small particles of thesame density as the fluid) is shown in Fig. 1 . In the numericalcalculations, fixed values were ascribed to (7, k, and e and thetrajectories of some 800 particles were computed and thenprocessed to obtain C. The analytical solution was obtainedwith F/ evaluated as C^pk2 /e, which is consistent with thediffusion model on which Eq. (26) is based. Both profiles arenormalized by the analytically derived Cmax .§

Agreement between the two sets of results is reasonable andcould, of course, have been improved by introducing moreparticles. The irregularities in the calculated curve are in-dicative of the stochastic nature of the model and of thedifferences that occur when a different random numbersequence is used.

Dispersion of Single ParticlesSnyder and Lumley21 performed an interesting experiment

in which single spherical solid particles of various densitiesand sizes were isokinetically introduced into the uniformturbulent flow downstream of a grid and their trajectorieswere photographed. The present dispersion model was ap-plied to this problem, making use of the fact that grid-generated turbulence has been well studied in the past20 andthe decay rates of both the turbulence kinetic energy and itsdissipation rate are known. Moreover, it was (reasonably)assumed that both the turbulence and the mean flow areunlikely to have been appreciably disturbed by the singleparticles. As in the experimental study, the trajectories of

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Fig. 1 Analytical and numerical solutions for the transversedistributions from a point source at 2.6 m downstream.

— o hollow glass,D=46.5um,p=0.26g/cc^—--A corn pollen,D-87 ym,P=2.5g/ccX"—• glass,D=87MmsP=lg/cc_-—A copper,D=46.5pm,p=8."

100 200 300 400Time from the first station (ms) —-

Fig. 2 Measurements21 (—, —, ---) and calculation ( o , A , •,A ) of particle dispersion.

§The calculated curves in this and the next application representaverages over radial intervals of 0.005 m.

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5r (cm) ————

Fig. 3 Measurements21 and calculations for copper particledistributions at 1.95 m downstream.

some 800 examples of each particle type were calculated andthen processed to yield the dispersion profiles displayed inFig. 2, along with the data. The predictions for the heavierparticles (i.e., copper and glass) are reasonably good, butagreement deteriorates in the case of the light ones (hollowglass and corn pollen). (The diameters of the copper and glassparticles are such that they possess nearly identical relaxationtimes, which is why they exhibit nearly identical dispersionrates.)

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Figures 3 and 4 show examples of the predicted andmeasured number density probability distributions in a cross-stream plane for heavy (copper) and light (hollow glass)particles, respectively. According to Ref. 21, themeasurements are well represented by the Gaussiandistributions shown. (The profiles are normalized in eachinstance by the predicted Cmax values.)

These graphs reinforce the conclusions drawn from Fig. 2,i.e., the numerical prediction agrees fairly well with the heavy-particle distribution of Fig. 3, but less with the light-particlecase of Fig. 4, where the calculated distribution has about theright width, but is more concentrated at the edges, due in partto the smaller samples of particles that reach these positions.

The reason for the larger errors observed for light particlesis not entirely understood, although it is probably connectedwith the fact that their interaction time is invariably the eddylifetime, whereas with the heavier particles it is the transittime. The fact that good agreement was obtained for thediffusing particles of the previous example, whose interactiontime was also the eddy lifetime, suggests that theinhomogeneity of the turbulence prevailing in the present casemay be inadequately allowed for in the model, although theremay be other explanations.

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Fig. 4 Measurements21 and calculations for hollow glass particledistributions at 2.6 m downstream.

Experiment— photograph—-- shadowgraph— •- scattering

20-

x ( mm )

Fig. 5 Measured3 and computed spray boundaries (computations:A no random entry, no droplet dispersion; o no random entry,

droplet dispersion; A random entry, no droplet dispersion; • ran-dom entry, droplet dispersion).

Q_ -MI O

13 U-

30

2Q

10

10 20 30 40 50r (mm) ——».

40

30

OLO> oa,^ 20

10

Station at 50 mm

experiment

b)10 20 30

r(mm) ——•40 50

c) 10 20 30 40 50r(mm) ——»-

40

10 20d) r (mm) ——•-

Fig. 6 Measured3 and calculated radial variation of liquid volumefraction at two stations: a) and b) o calculation without dropletdispersion, • calculation with droplet dispersion; c) and d) ocalculation without random entry, • calculation with random entry.

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a)

measured at 25 mm

measured at 50mm

b)10 20 30 40

r (mm) ———•*-50

Fig. 7 Measured and calculated vapor concentrations: a) ncalculation without droplet dispersion, station 25 mm; • calculationwith droplet dispersion, station 25 mm; o calculation without dropletdispersion, station 50 mm; • calculation with droplet dispersion,station 50 mm; b) n calculation without random entry, station 25mm; • calculation with random entry, station 25 mm; o calculationwithout droplet dispersion, station 50 mm; • calculation with dropletdispersion, station 50 mm.

Steady Conical Evaporating SprayThe n-heptane spray measurements reported by Tishkoff et

al.3 form the first testing ground for the fully coupled gasmotion and droplet model. The evaporating solid-cone spraywas injected into a coflowing airstream of almost uniformvelocity and detailed measurements were made at two planes25 and 50 mm downstream of the nozzle of the following: thedroplet-size distributions of the axis, edge, and middle of thespray; the liquid-phase volume fraction; and vapor volumeconcentration. In addition, photographs of the sprayboundary were taken from which such global characteristicsas the loci could be deduced. Predictions for one of the severalinjection pressures examined (1.48 MPa) were generated usingestimated initial conditions for the spray. The practiceadopted for this was to subdivide the measured initial sprayangle of about 50 deg into three equal-angle bands and thenestimate the initial conditions for each (in an admittedly crudemanner) from the available measurements in the plane nearestto the injector by assuming that all droplets evaporated in avapor-free environment according to the so-called d2 lawduring their travel from the injector to this plane. Differentassumptions were made for the initial axial velocities andtemperatures of the droplets, which were assigned uniformvalues calculated from the nozzle exit conditions. Within eachband, the initial proportion of the total mass flow and thedroplet size distribution were estimated in this way. The latterwas randomly sampled during the calculations, as was the

radial velocity compdnent of the droplet between upper andlower limits corresponding to the included angle of theparticular band considered.

In order to assess the separate influence of the variousmethods employed in the droplet modeling, three additionalcalculations were performed in which, independently: 1) therandom radial entry velocity treatment was suppressed and alldroplets in a particular band were ascribed the median entryangle; 2) the turbulent dispersion model was suppressed; and3) both of the foregoing practices were invoked simulta-neously, which effectively makes the approach a deterministicone in the mold of the earlier discrete-droplet models of thiskind mentioned in the Introduction.

The spray boundaries obtained from each method areshown in Fig. 5 and indicate that droplet dispersion has only amarginal effect on the spray angle, especially in comparisonwith that of the random radial entry velocity treatment. Asexpected, the latter causes some droplets to reach larger radii.The spray boundary demarcated by these is in good agreementwith the outer band of the experimental measurements.

Further details of the predictions are shown in Fig. 6, whichshows the radial variations in the liquid volume fraction at thetwo axial measurement locations. Figures 6a and 6b containresults using the stochastic entry treatment with and withoutdispersion modeling, while Figs. 6c and 6d show the effect ofretaining the latter and dispensing with the former. FromFigs. 6a and 6b it can be seen that the influence of turbulentdispersion in these circumstances is predicted to be small, withthe most pronounced effects occurring near the axis where, aswould be expected, dispersion gives rise to smaller radialgradients. By contrast, the influence of the initial conditionsascribed to the droplets is very pronounced, as is illustrated inFigs. 6c and 6d, where the stochastic treatment tends tosubstantially reduce the radial gradients. It is also clear thatnone of the predictions is in acceptable agreement with themeasurements, for in all cases the calculated liquid volumefraction distribution diminishes rapidly away from the axisand increases toward the edge of the spray to only a fractionof the measured values. These errors are probably the con-sequence of inaccuracies in estimating the initial conditions ofthe spray (further calculations are in progress that point inthis direction) and suggest the need for measurements in,and/or a model of, the atomization zone.

Figures 7a and 7b show similar plots of the radial variationsof the vapor concentration in the two measurement planes.The two sets of predictions in Fig. 7a relating to the differentdispersion treatments are again close, showing that here toothe effect is small at all but a few locations where the con-centration is itself low. According to Fig. 7b the vapordistribution is less sensitive to the different entry treatmentsthan was the liquid, but the considerable differences betweenall of the predictions and the experimental data near the outerperiphery of the spray at the downstream plane (which areconsistent with the underprediction of the liquid con-centrations there) reinforce the above comments about theneed for a better inlet treatment. These calculations wereperformed on a 20x20 computational grid and required 35.4and 43.5 s of CDC 7600 CPU time without and with thedispersion time, respectively.

Combusting SprayThe final application of the method described here is to the

liquid-fueled furnace experiments reported by Founti et al.,6in which a hollow-cone swirling kerosene spray was axiallyinjected into a cylindrical combustion chamber. The initialconditions of the spray were estimated from knowledge of theatomizer geometry, low-resolution photographs of the nearbyregion under cold and burning conditions, and measurementsof the cold-flow droplet size distribution a short distancedownstream. Details of the estimated conditions are given inan earlier publication by the present authors.15 Reference 15also presents calculations based on the deterministic version

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No droplet dispersion

W i t h droplet d ispers ionFig. 8a Streamlines in a plane through the axis of the furnace.

No droplet d ispers ion

W i t h dropTet 'dispersionFig. 8b Calculated vapor mass fraction contours in a plane throughthe axis of the furnace.

193017501570139012101030

850670490

W i t h droplet d ispers ionFig. 8c Calculated temperature contours in a plane through the axisof the furnace.

of the method, which were found to produce poor agreementwith the measurements.

In the further calculations reported here, the deterministicspecification of the initial conditions was retained but thestochastic turbulent dispersion treatment was introduced, theobjective being to determine the sensitivity of the predictionsto this facet of the modeling. An overall impression of thesensitivity may be obtained from Figs. 8a-8c, which show,respectively, the streamlines, fuel concentration contours, andisotherms, in each case with and without the dispersiontreatment. Evidently the effect is small in both cases, and itmay therefore be concluded that the absence of turbulentdispersion modeling is unlikely to have been responsible forthe poor agreement obtained in our earlier calculations.

Although other sources of error are known to exist in thepresent version of the method, notably in the treatment ofcombustion and radiation, it is believed that the overridingfactor is once again inadequate knowledge of the initialconditions of the spray. In this instance, the CPU times on a20 x 20 grid were 300 s with the dispersion modeling and some50% less without.

ConclusionsThe present study has demonstrated that it is possible to

calculate turbulent dispersion effects on droplets and particlesto a tolerable degree of accuracy using a stochastic discretedroplet approach, provided that the initial conditions are welldefined, as they were in the simple test cases. It is true that theaccuracy deteriorated for the lighter particles, whose in-teraction times were invariably the eddy lifetime, suggestingthat refinement of the approach is required for these cir-cumstances. Nevertheless, the current performance is believedto be adequate enough for the purposes of evaluating theimportance of turbulent dispersion in the much more complexcircumstances of real sprays.

The initial impression gained from the two such evaluationsperformed here is that the dispersion effects are small ascompared with the major source of uncertainty, which ap-pears to be inadequate knowledge of the initial sizedistributions, mean velocities, and other properties of thedroplets. This impression should be tempered by theacknowledgment that there are other dispersion-dependentphenomena occurring in sprays that were not considered inthe present study, notably droplet proximity andcollision/coalescence effects. Some progress in these has beenmade recently by O'Rourke and Bracco.17 However, in thecontext of furnace and gas turbine applications, these effectsare likely to be confined to the immediate vicinity of theatomizer and may therefore be regarded as part and parcel ofthe initial-condition problem.

It should also be noted that the extent of dispersion isdependent on the ratio of the length of the droplet trajectoryto the average eddy size, which tended to be small in the casesexamined here, due to either the limited region considered(evaporating case) or the rapid disappearance of the volatiledroplets (combusting case). In other circumstances, notablythose of pulverized coal combustion, dispersion would beexpected to play a much more important role.

References^iroyasu, H. and Kadota, T., "Fuel Droplet Size Distribution in

Diesel Combustion Chamber," Bulletin of the Japan Society ofMechanical Engineers, Vol. 19, 1976, pp. 1064-1072.

2Shearer, A. J., Tamura, H., and Faeth, G. M., "Evaluation of aLocally Homogeneous Flow Model of Spray Evaporation," Journalof Energy, Vol. 3, Sept.-Oct. 1979, pp. 271-278.

3Tishkoff, J. M., Hammon, D. C. Jr., and Chraplyvy, A. R.,"Diagnostic Measurements of a Fuel Spray Dispersion," ASMEPaper 80-WA/HT-35, 1980.

Onuma, Y., Ogasawara, M., and Inoue, T., "Further Ex-periments on the Structure of a Spray Combustion Flame," 16thSymposium (International} on Combustion, The Combustion In-stitute, Pittsburgh, Pa., 1977, pp. 561-568.

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5Owen, F. K., Spadaccini, L. J., Kennedy, J. B., and Bowman, C.T., "Effects of Inlet Air Swirl and Fuel Volatility on the Structure ofConfined Spray Flames," 17th Symposium (International) onCombustion, The Combustion Institute, Pittsburgh, Pa., 1978, pp.467,473.

6Founti, M., Hutchinson, P., and Whitelaw, J. H.,"Measurements and Calculations of a Kerosene-Fueled Flow in aModel Furnace," Mechanical Engineering Dept., Imperial College,London, Rept. FS/79/19, 1979.

7Williams, A., Combustion of Sprays of Liquid Fuels, ElekScience, London, England, 1976.

8Westbrook, C. K., "Three-Dimensional Numerical Modeling ofLiquid Fuel Sprays," 16th Symposium (International) on Com-bustion, The Combustion Institute, Pittsburgh, Pa., 1976, pp. 15IT-1525.

9Haselman, L. C. and Westbrook, C. K., "A Theoretical Modelfor Two-Phase Fuel Injection in Stratified Charge Engines," SAEPaper 780138, 1978.

f°Cliffe, K. A., Lever, D. A., and Winters, K. H., "A Finite-Difference Calculation of Spray Combustion in Turbulent, SwirlingFlow," Paper presented at International Conference on NumericalMethods in Thermal Problems, Swansea, U.K., July 1979.

HCrowe, C. T., "A Computational Model for the Gas-DropletFlow Field in the Vicinity of an Atomiser," Proceedings of the llthJANNAFSymposium, 1974, Paper 74-23.

. 12Gosman, A. D. and Johns, R., "Computer Analysis of Fuel-AirMixing in Direct-Injection Engines," SAE Paper 800091, 1980.

13El-Banhawy, Y. and Whitelaw, J. H., "The Calculation of theFlow Properties of a Confined Kerosene-Spray Flame," AIAA Paper79-7020, 1979.

14Abbas, A. S., Koussa, S. S., and Lockwood, F. C., "ThePrediction of the Particle Laden Gas Flows," Proceedings of the 18thSymposium on Combustion, Combustion Institute, Waterloo,Canada, 1981, pp. 1427-1438.

15Gosman, A. D., loannides, E., Lever, D. A., and Cliffe, K. A.,"A Comparison of Continuum and Discrete Droplet Finite-Difference Models Used in the Calculation of Spray Combustion inSwirling Turbulent Flows," AERE Harwell Rept. TP 865, 1980.

16Dukowicz, J. K., "A Particle-Fluid Numerical Model for LiquidSprays," Journal of Computational Physics, Vol. 35, 1980, pp. 229-253.

17O'Rourke, P. J. and Bracco, F. V., "Modeling of Drop In-teraction in Thick Sprays and a Comparison with Experiments,"Paper presented at IME Stratified Charge Automotive EnginesConference, London, 1980.

18Crowe, C. T., Sharma, M. P., and Stock, D. E., "The Particle-Source-in-Cell (PSI-cell) Model for Gas-Droplet Flows, ASME PaperNo. 75-WA/HT-25, 1975.

19Launder, B. E. and Spalding, D. B., Mathematical Models ofTurbulence, Academic Press, London, 1972.

20Hinze, J. O., Turbulence, McGraw-Hill Book Co., New York,1975.

21Snyder, W. H. and Lumley, J. L., "Some Measurements ofParticle Velocity Autocorrelation Functions in a Turbulent Flow,"Journal of Fluid Mechanics, Vol. 48, Pt. 1, 1971, pp. 41-47.

22Jones, W. P., " Prediction Methods for Turbulent Flows,"Lecture Series 1979-2, Von Karman Institute for Fluid Dynamics,Brussels, 1979.

23Ranz, W. E. and Marshall, W. R. Jr., "Evaporation fromDrops," Chemical Engineering Progress, Vol. 48, 1952, p. 173.

24Brown, D. J. and Hutchinson, P., "The Interaction of Solid orLiquid and Turbulent Fluid Flow Fields—A Numerical Simulation,"Journal of Fluids Engineering, Vol. 101, 1979, pp. 265-269.

25Caretto, L. S., Gosman, A. D., Patankar, S. V., and Spalding,D. B., "Two Calculation Procedures for Steady Three-DimensionalFlows with Recirculation," Proceedings of the 3rd InternationalConference on Numerical Methods in Fluid Mechanics, Vol. 2, 1972,p. 60.

26Hutchinson, P., Khalil, E. E., and Whitelaw, J. H., "Ex-perimental Investigation of Flows and Combustion in AxisymmetricFurnaces," Journal of Energy, Vol. 1, 1977, pp. 212-219.

27Faeth, G. M., "Current States of Droplet and Liquid Com-bustion," Progress in Energy Combustion Science, Vol. 3, 1977, pp.191-224.

28Magnussen, B. F. and Hjertager, B. H., "On MathematicalModeling of Turbulent Combustion with Special Emphasis on SootFormation and Combustion," 16th Symposium (International) onCombustion, The Combustion Institute, Pittsburgh, Pa., 1976, pp.719-728.

29Gosman, A. D. and Lockwood, F. C., "Prediction of Influenceof Turbulent Fluctuation of Flow, Heat Transfer Surfaces," 14thSymposium (International) on Combustion, The Combustion In-stitute, Pittsburgh, Pa., 1973, p. 661.

30Wallis, G. B., One Dimensional and Two Phase Flow, McGraw-Hill Book Co., New York, 1969.

31Borman, G. L. and Johnson, J. H., "Unsteady VaporizationHistories and Trajectories of Fuel Drops into Swirling Air," SAEPaper 598C, Oct. 1962.

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