CFD analysis of spray propagation and evaporation ...rschmehl.home.xs4all.nl/CV/publications/ijhff99_paper.pdf · tational unit. A second, separate program is based on Lagrangian

CFD analysis of spray propagation and evaporation including wall®lm formation and spray/®lm interactions

R. Schmehl *, H. Rosskamp, M. Willmann, S. Wittig

Institut fur Thermische Stromungsmaschinen, Universitat Karlsruhe, Kaiserstraûe 12, 76128 Karlsruhe, Germany

Abstract

Addressing the numerical simulation of complex two-phase ¯ows in gas turbine combustors, this study features a comprehensive

approach to the coupled solution of the interacting ¯ow ®elds of the gas phase, evaporating fuel spray and evaporating, shear-driven

fuel wall ®lm. The gas ¯ow and wall-®lm ¯ow are described in Eulerian coordinates and are both calculated in the same compu-

tational unit. A second, separate program is based on Lagrangian particle tracking to model spray dispersion and evaporation. To

account for interaction eects, an iterative procedure is applied considering mutual mass, momentum and energy transfer between

the three ¯ow regimes. For a realistic modeling of spray/wall and spray/®lm interaction, the droplet-trajectory computation com-

prises a set of droplet-impact models covering a broad range of impact conditions. The results of a two-phase ¯ow simulation in a

schematic LPP combustor premix duct demonstrate the eects of phase interaction as well as spray/wall and spray/®lm interac-

tion. Ó 1999 Elsevier Science Inc. All rights reserved.

Keywords: Sprays; Evaporation; CFD; Films; Droplet-trajectories

1. Introduction

Computational ¯uid dynamics of single- and two-phase¯ows in combustors has been a rapidly developing research

www.elsevier.com/locate/ijhInternational Journal of Heat and Fluid Flow 20 (1999) 520±529

Notation

a coecientcfm correction factor for mass transportcfh correction factor for energy transportcp speci®c heat capacity (J/(kg K))cW aerodynamic drag coecientD diameter (m)hf average ®lm thickness (m)h� non-dimensional ®lm thicknesshv evaporation enthalpy (J/kg)k turbulent kinetic energy m2=s2ks equivalent sandgrain roughness (m)La Laplace numberLe Lewis number_m�f fuel ®lm mass ¯ux per unit width (kg/(s m))

On Ohnesorge numberPr Prandtl numberRe Reynolds numberra radius of the duct (m)S splashing parameter, source termSc Schmidt numberTr droplet rate (1/s)U �� circumferential shear velocity scaleu; v;w velocity components (m/s)us shear stress velocity (m/s)Vc liquid volume concentration

W �� axial shear velocity scaleWe Weber numberY mass fractiony, z radial, axial coordinate (m)a impact angle (rad), relaxation factor ÿe turbulent dissipation rate m2=s3C general diusion coecientCim eective diusion coecient m2=sj mixing length constantg deposition ratel dynamic viscosity (Pa s)q density kg=m3r surface tension, variance (N/m)sw wall shear stress N=m2U equivalence ratiou circumferential coordinate/ general transport variable

Subscriptsb boilingg, f, d gas, ®lm, dropletL Leidenfrostm meanref references droplet surfacevap vaporw wall

* Corresponding author. E-mail: [email protected]

sruhe.de

0142-727X/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved.PII: S 0 1 4 2 - 7 2 7 X ( 9 9 ) 0 0 0 4 1 - 7

topic over the last years. Especially, design and optimization ofadvanced, low-emission combustors are based on numericalsimulations to a growing extent. In these combustor designs,performance and reliability signi®cantly depend on local ¯owphenomena. At this point, advanced CFD codes are a valuablecomplement to experimental investigations, since they allow adetailed local analysis of the ¯ow. Engineering ¯ow predictionof single-phase ¯ows is a standard application of CFD andwidely used nowadays. In the same way, the computation ofspray dispersion and evaporation has been developed into apracticable prediction tool (Tolpadi et al., 1995; Tolpadi, 1995;Kurreck et al., 1996). Even though advanced models have beendeveloped (Witting et al., 1991), the numerical prediction ofevaporating shear-driven wall ®lms has still been limited tospecial geometries or ¯ow cases (Baumann et al., 1989). Thenumerical simulation of complex, two-phase ¯ows in comb-ustors with evaporating fuel sprays and wall ®lms is a chal-lenge for many years now. However, attempts to consider theinteraction of gas ¯ow, spray and wall ®lm have been stag-nating due to a lack of practicable models for spray/wall andspray/®lm interaction as well as wall-®lm propagation. Basedon extensive experimental research, reliable and validatedmodels accomplishing these demands have recently been pro-posed by Rosskamp et al. (1997) and Samen®nk (1997).

2. Numerical approach

In the fuel/air mixing region the combustor ¯ow is deter-mined by phase interaction eects and in the case of premixingducts or air-blast nozzles by spray/wall and spray/®lm inter-action. Fig. 1 gives an overview of the two-phase ¯ow phen-omenain a schematic premix duct of a Lean PremixPrevaporize (LPP) combustor. A pressure atomizer ejects thedroplet spray into the hot compressor air ¯ow where it isdispersed and evaporated. The gas ¯ow in turn is aected by

evaporation cooling and aerodynamic acceleration due to thespray. The liquid wall ®lm is fed either directly onto the wall orit is built up by deposition of droplets. In both cases, gas ¯owand ®lm ¯ow are coupled by mass, momentum and energytransfer due to ®lm evaporation and surface shear stress. De-pending on the impact parameters, droplet impact on the ductwall or the wall ®lm may result in deposition, splashing, re-¯ection or spontaneous evaporation of the droplet. These walleects have a signi®cant in¯uence on the vapor concentrationin the near-wall ¯ow region. The numerical approach pre-sented in this study is based on an Eulerian description of thegas and ®lm ¯ow and a Lagrangian description of spray dis-persion and evaporation. The gas ¯ow and ®lm ¯ow are si-multaneously calculated in the program EPOS, applying a 3D®nite volume discretization and a linear system solver for thegas ¯ow and a 2D Finite Dierence scheme combined with aRunge±Kutta solver for the ®lm ¯ow. Based on the gas-phase¯ow ®eld and ®lm data, spray dispersion and evaporation iscalculated in a second individual program, Ladrop (see Fig. 2).The trajectory approach for the spray has been chosen, be-cause of its inherent suitability for the modeling of complextechnical sprays characterized by a broad range of dropletinitial conditions and droplet/wall interactions. Interactionbetween the ¯ow regimes' gas ¯ow, ®lm ¯ow and spray is re-alized by arranging the programs EPOS and Ladrop in an it-eration loop. The reaction of gas and ®lm ¯ow upon the spraydue to evaporation cooling, aerodynamic acceleration or ®lmbuild up by droplet deposition is taken into account by con-sidering droplet source term ®elds computed in Ladrop in thefollowing gas and ®lm ¯ow calculation by EPOS.

3. Gas ¯ow

The computation of the turbulent, three-dimensional gas¯ow ®eld is done with the computer program EPOS, developed

Fig. 1. Two-phase ¯ow phenomena in the premix duct.

R. Schmehl et al. / Int. J. Heat and Fluid Flow 20 (1999) 520±529 521

at the Institut fur Thermische Stromungsmaschinen. EPOS isbased on the Reynolds-averaged Navier±Stokes equationsrepresented by the general formulation for ¯ow variable /

div q~u/ div C/grad/ S/: 1In this equation, C/ is an eective diusion coecient and

S/ comprises local sources. Possible substitutions for / are 1,resulting in the continuity equation, or u; v;w; h and Yvap. Thestandard k±e turbulence model is included by solving twoadditional transport equations for the turbulent kinetic energyk and its dissipation rate e (Launder and Spalding, 1974). Thesystem of transport equations is formulated in a cylindricalcoordinate system and discretized by means of a ®nite-volumemethod. The coupled velocity±pressure ®eld is determinedusing the SIMPLEC pressure-correction algorithm on non-staggered grids as described by Van Doormal and Raithby(1984). The temperature distribution is derived from theenthalpy ®eld and the density is calculated from an equation ofstate for an ideal mixture. The ®nite-volume balance equationsare arranged in septdiagonal matrix form,

aP /P X

nb

anb/nb S/; aP X

nb

anb Sm;

nb E;W ;N ; S;H ; L;2

which are solved by an iterative solution procedure based on aconjugate gradient method (Noll et al., 1991). In the presenceof a liquid wall ®lm, special wall functions have to be appliedto represent the ¯ow in the near-wall region correctly. The ®lmsurface is considered to be a rough, moving wall with negligiblevelocity. Additionally, the curvature of the wall must be takeninto account for a precise prediction of wall shear stress andinterfacial shear stress, respectively. The wall shear stress is

computed in two components (aligned with the axis and alongthe curved wall, respectively) according to the followingequations (Kind et al., 1989)

wW �� 1

jln

yW ��

m

� � C; 3

uU �� 1�ÿ y

ra

�1

jln

yU ��

m1ÿ y=ra� ��

C�; 4

where the shear velocity scales are de®ned as

U �� s2w;uqsw

s; W ��

sw;zqsw

r: 5

C denotes a constant which has to be chosen according to theroughness of the wall or the ®lm surface,respectively. Threedierent regimes identi®ed in terms of the roughness Reynoldsnumber Reks usks=m have to be considered (Kays andCrawford, 1980, Himmelsbach et al., 1994)· Hydraulically smooth surface, Reks 6 5:0:

CReks 5:15: 6· Slightly rough surface, 5:0 < Reks 6 70:0:

CReks 1:5497 19:1 log Reks ÿ 14:4339 log2 Reks 3:30869 log3 Reks ÿ

1

jln Reks : 7

· Rough surface, Reks P 70:0:

CReks 8:5ÿ1

jln Reks : 8

For the computation of heat transfer, the analogy of mo-mentum and heat transfer is used. Mass exchange is also

Fig. 2. Coupled calculation of the two-phase ¯ow with EPOS and Ladrop.

522 R. Schmehl et al. / Int. J. Heat and Fluid Flow 20 (1999) 520±529

computed making use of this analogy. However, in the case ofrough surfaces the analogy suers from one defect: whereas themomentum exchange shifts from molecular mechanisms tomore macroscale drag forces with increasing roughness, theheat and mass transfer is always limited to molecular mecha-nisms close to the surface. This phenomenon is taken intoaccount by means of a correction term which has been pro-posed for shear driven ®lms by Sill (1982) and Himmelsbachet al. (1994).

4. Spray dispersion and evaporation

The Lagrangian method is based on the tracking of indi-vidual droplets in the gas-phase ¯ow ®eld by integrating theirequation of motion combined with an empirical correlation forthe aerodynamic drag coecient given by Stein (1973)

d~uddt ÿ 3

4

qgqd

cWDj~ud ÿ~ug j ~ud ÿ~ug;

cW 0:36 5:48Reÿ0:573d 24

Red:

9

In order to simulate the eect of turbulent spray dispersion,the gas velocity is randomly sampled along the trajectoriesaccording to the approach of Gosman and Ioannides (1983).In this concept, characteristic quantities of the turbulencestructure are determined from mean gas ¯ow properties. Spe-ci®cally, the length scale le and dissipation time scale te of theidealized eddies are calculated from local turbulence propertiesas follows

le C1=2lk3=2

e; te

3

2

rC1=2l

ke: 10

In addition to the life time scale te, the droplet dynamics aretaken into account in the following integral relation for thecrossing time scale tcZ tc

t0

~ug

�� ÿ~uddt�� le: 11

When the smaller of these time scales has elapsed, thedroplet is deemed to enter a new eddy. Consequently, therandom process generates a new velocity ¯uctuation~u0g from aGaussian distribution determined by

l 0; r 2

3k

r: 12

The model constant Cl in Eq. (10) has a decisive in¯uenceon the dispersion behavior of the turbulent spray. The mostrealistic results have been obtained by choosing the value 0.09(Milojevi�c, 1990). With respect to spray evaporation, theUniform Temperature model is used in this approach. Thiscomputationally eective model is based on the assumptionof a homogeneous temperature distribution in the dropletand phase equilibrium conditions at the liquid/gas interface.In the gas phase around the droplet, diusive time scales aresmaller by orders of magnitude giving rise to a quasi-sta-tionary description of the diusive transport processes. Usingreference values for variable ¯uid properties (1=3-rule ofSparrow and Gregg, 1958), an integration of the radialsymmetric dierential equations yields analytical expressionsfor the diusive transport ¯uxes. Convective transport is ac-counted for by two empirical correction factors cfm and cfhresulting in the following balance equations for droplet massand temperature

dmddtÿ cfm 2pD qg;ref Cim;ref ln

1ÿ Yvap;g1ÿ Yvap;s ; 13

dTddtÿ 1

mdcp;d

dmddt

"cfh

cfmcp;vap;refTg ÿ Td:

� 1ÿ Yvap;g1ÿ Yvap;s

� �1=Le ÿ 1!ÿ1ÿ hv

35;cfm 1 0:276Re1=2 Sc1=3; cfh 1 0:276Re1=2 Pr

1=3

: 14In these equations, Yvap;s is the vapor concentration at the

droplet surface, whereas Tg and Yvap;g represent free streamvalues. Variable ¯uid properties are determined by correla-tions. The resulting system of coupled ordinary dierentialequations describing the motion and evaporation of individualdroplets is integrated by a Runge±Kutta±Fehlberg schemefeaturing an automatic time-step control. The computationand superposition of a large number of droplet trajectoriesresults in a smooth and continuous ¯ow ®eld of the disperseliquid phase in a statistical sense: Increasing the number oftrajectories leads to a reduction of the variance of the liquid-phase ¯ow variables.

5. Spray/wall and spray/®lm interaction

In many two-phase ¯ow applications, interactions betweendroplets and ¯ow boundaries have a decisive in¯uence on theoverall ¯ow behavior. In the case of the LPP premix duct,interaction between the spray and the wall and the wall ®lmsigni®cantly aects the fuel distribution in the ¯ow entering thecombustion zone. Thus, a comprehensive modeling of thedroplet impact process is indispensable for a realistic simula-tion of the two-phase ¯ow close to the wall. The mechanismsfor a dry wall were investigated experimentally by severalgroups in the past (Stow and Stainer, 1977; Walzel, 1980;Samen®nk, 1997; Coghe et al., 1995). For a detailed simula-tion, the interaction mechanisms have to be classi®ed withrespect to the conditions on the wall and the impact parame-ters of the droplet. The ®rst level of classi®cation distinguishesbetween a dry wall and a liquid wall ®lm. Droplet impact on adry wall is further divided into three characteristic wall tem-perature regimes: Cold wall ± a temperature well below the¯uid boiling temperature, moderately hot wall ± a transitionrange up to a modi®ed Leidenfrost temperature and hot wall ±a temperature well above the modi®ed Leidenfrost tempera-ture (see Fig. 3).

5.1. Dry wall with T61:05Tb

For wall temperatures up to approximately 1:05Tb, thecold±wall interaction mechanisms are dominating (Samen®nk,1997). In this temperature regime, the two possible interactionmechanisms are either splashing or complete deposition of thedroplet. In the case of splashing, a fraction of the droplet massis deposited on the wall whereas the remainder is ejected backinto the gas ¯ow decomposed into secondary droplets. Apossible approach to distinguish between splashing and com-plete deposition involves the impact Reynolds number and theLaplace number,

Re unDqdld

and La Drdqdl2d

: 15

The impact Reynolds number is based on a correcteddroplet velocity normal to the wall un ud sin a0:63. Withrespect to a Re±La map, an analysis of numerous droplet


impact experiments indicates that splashing is separated fromcomplete deposition by the function

Re 24La0:419; 16represented by the bold line in Fig. 4a. In order to quantify thesplashing process, it is necessary to establish a relation for thedeposition rate g characterizing the fraction of the depositeddroplet mass. Up to now, experimental studies reveal only littleinformation about the details of the splashing process and thedeposition rate. However, it is evident from several studies byWei� (1993), Stow and Stainer (1977) and Mundo et al. (1995),that the distance from the splashing/deposition separation linein the logarithmic Re±La plane has a dominating in¯uence onthe deposition rate in the splashing domain Analytically, thedistance from the splashing/deposition separation line can bedescribed by the splashing parameter S expressed by the re-lation

Re S 24La0:419; 17where S 1 represents zero distance. Because the splashingprocess is insuciently understood, a practical approach ischosen in this study which is based on an extrapolation ofexperiments on the interaction of droplets with thin wavy ®lmsby Samen®nk (1997)

g Sÿ0:6: 18Local sources of fuel mass, momentum and enthalpy for the

liquid wall ®lm are determined as follows

Sm gTrmd; S~u gTrmd~ud; ST gTr mdhd: 19In these expressions, Tr represents the droplet rate, deter-

mining the droplet number ¯ux assigned to an individual tra-jectory of the spray.

A second important part of the modeling of the splashingprocess addresses the determination of initial conditions forsecondary droplets. Secondary droplets have much smallerdiameters compared to their parent droplet. The diameterdistribution of the droplet cloud produced by splashing followsa log-normal distribution

PD 12pp

xrexp

"ÿ xÿ �x

2

2r2

#; x ln D: 20

Based on data extracted from the literature, a relation forthe diameter parameter �x ln Dm in the log-normal distribu-

tion can be derived in satisfying agreement with the experi-mental work by Samen®nk (1997) and Stow and Stainer (1977)

lnDmD0 ÿ2 ÿ D0

Drefÿ 0:05S; Dref 4066lm: 21

In this relation, D0 denotes the droplet diameter just beforeimpact. Obviously, this correlation is not based on physicalreasoning but rather achieves experimental agreement for thetotal range of gas-turbine applications. The width of the di-ameter distribution slightly increases with an increasingsplashing parameter S. Due to the limited available experi-mental data, the constant value 0:45 for the variance parameterr is assumed in this study. Ejection angles of the secondarydroplets are typically in the range of a 10° to a 15° relativeto the wall whereas initial velocities are about 60% of the im-pact value. In combustor applications, secondary droplets fromsplashing are very tiny and instantly follow the gas ¯ow in thenear-wall region. Therefore, a detailed modeling of initial ve-locity and position is not necessary for the secondary droplets.

5.2. Dry wall with 1:05Tb6T < T �L

Increasing the wall temperature from 1:05Tb up to a mod-i®ed Leidenfrost temperature, a number of various complexphenomena are observed. Surface roughness peaks penetratethe thin vapor ®lm between droplet and wall and enhance theheat transfer into the droplet. Several transition states fromsplashing, disruption into smallest secondary droplet frag-ments,instant evaporation to local boiling are identi®ed. Tosimplify the modeling of all these phenomena, instant evapo-ration is considered as the representative mechanism for thiswall-temperature range. The upper limit of this temperaturerange is derived from several measurements of droplet impactson a hot surface varying size, temperature and incident ve-locity of the droplets. An analysis of these experiments indi-cates the following modi®ed Leidenfrost temperature which isdependent on liquid properties and impact velocity

T �L ÿ TLTL

0:027Wed

p: 22

5.3. Dry wall with T > T �L

On very hot walls, the determining interaction mechanismis re¯ection. Several authors have derived equations for the

Fig. 3. Visualizations of droplet/wall interaction mechanism.


diameter change due to the formation of a vapor ®lm (Li et al.,1995; Wachters and Westerling, 1966). For ¯ow conditions incombustors, this diameter decrease is of minor importance andis consequently neglected. Because of a very short interactiontime and an isolating eect of the vapor cushion, the energytransport to the droplet during the interaction time has nosigni®cant eect. The change of the wall normal velocitycomponent due to the re¯ection can be described by the fol-lowing equation

un;1un;0

1ÿ 37We0:25 On

p: 23

5.4. Wall covered with liquid ®lm

Basically, the mechanisms of interaction do not change inthe presence of a liquid ®lm. By analogy with the cold dry wallcase described above, the separation line dividing the splashingand deposition domain is described by Eq. (16). For typicalthin ®lms in combustor ¯ows (hf < 150 lm), ®lm thickness hasno signi®cant in¯uence on the splashing/deposition border.However, a decisive in¯uence is observed on the deposition

rate g. The following correlation can be deduced from theexperimental data of Samen®nk (1997)

1ÿ gfilm 1ÿ gdrywall expÿh�; h� hfD0: 24

From this correlation, it is obvious that a thicker ®lm leadsto an enlarged deposition rate by damping the splashing pro-cess as illustrated in Fig. 4b.

5.5. Computation of secondary droplets

In order to simulate a spray generated by an atomizer, onlya limited number of individual droplets with dierent initialconditions are tracked. The larger this number and the ®nerthe discretization of the spectrum of initial conditions, themore realistic the simulation of the atomization process. In asimilar way, a splashing event is simulated only by a limitednumber of representative secondary droplets. Consistent withthe stochastic concept of the Lagrangian approach, the initialconditions of the secondary droplets are determined by arandom procedure. In the present ¯ow calculation, ten totwenty secondary droplets have been generated to simulate asplashing droplet.

6. Wall ®lm propagation and evaporation

The calculation of the ®lm ¯ow on the duct wall is based ona boundary layer description (Rosskamp et al., 1997). In thisapproach, the wavy wall ®lm is developed in a plane and de-scribed in a time-averaged manner. Fig. 5a shows a schematicmodel ®lm which represents all important integral character-istics of a shear driven ®lm. The wavy ®lm surface structure istaken into account by introducing an equivalent sand grainroughness ks 2hf W (Himmelsbach et al., 1994). In this re-lation, W is a correction factor taking into account the changein surface structure as the shear stress at the liquid/gas inter-face increases. For many liquids with a viscosity and surfacetension similar to water, alcohols or hydrocarbon fuels thisfactor is calculated from W 0:735 0:092551=Pass. Theinternal temperature pro®le of the model ®lm is determinedfrom a superposition of a rapid mixing zone and a laminarsublayer zone (Rosskamp et al., 1997). This temperature pro-®le is used to calculate variable thermo-physical ¯uid proper-ties. Conservation of energy is implemented in terms of anenthalpy balance on the control volume indicated in Fig. 5b.Based on these modeling assumptions, a set of three dierentialequations can be derived for mass transport, ®lm surface speedand ®lm temperature. In a similar way to the integration of thedroplet trajectories, this system of equations is integrated inthe ®lm-propagation direction by means of a Runge±Kuttamethod featuring a self-adapting step size.

7. Coupled solution of gas ¯ow, spray and ®lm ¯ow

As described above, the ¯ow regimes of gas phase, wall ®lmand spray are interacting by means of mass, momentum andenergy transfer. With respect to Fig. 2 the computation of the®lm ¯ow is implemented directly into the pressure based, block-iterative solution procedure of the gas-phase ¯ow ®eld in theprogram EPOS. In the present simulation, individual ®lm-¯owsolutions are calculated every 50th non-linear iteration of thegas ¯ow solution procedure, each time returning updates forthe source term ®elds due to ®lm ¯ow and evaporation. In thisway, a simultaneous solution of the gas-¯ow ®eld and wall-®lm¯ow ®eld is obtained. Designed as a post processing-processing

Fig. 4. (a) Re±La plane for the cold wall. (b) Deposition rate g for wall®lm.


unit to EPOS, the program Ladrop computes dispersion andevaporation of the spray including secondary droplets fromsplashing. As a result, Ladrop returns droplet source term ®eldsfor the gas and ®lm ¯ow solution in the next EPOS/Ladropiteration. Because of the intense interdependence of spray andgas ¯ow in the present calculation, a special relaxation strategyhas to be applied for the gas phase droplet source ®elds

Sid;/ aSid;/ 1ÿ aS

iÿ1d;/ ; 25

with the source term Sid;/ calculated by Ladrop and the sourceterm S

id;/ actually used in the gas ¯ow calculation according to

Eq. (2). In particular, the enthalpy source ®elds caused byevaporation are prone to excessive oscillations during the it-eration procedure. Eectively, a relaxation factor a 0:05 hasto be chosen in order to assure a stable iteration procedure. As

a consequence, 130 EPOS/Ladrop iterations have been neces-sary to completely account for the droplet source terms. Beinga characteristic integral quantity, the total evaporated fuelmass fraction is used as a criterion to assess the convergence ofthe solution.

8. Results

8.1. Example geometry

The test case is a schematic arrangement of an atomizer in apremix duct basically similar to the sketch in Fig. 1. It waschosen as a representative two-phase ¯ow case with a signi®-cant in¯uence of the evaporating liquid phase on the overall

Fig. 5. (a) Model for the the shear driven wall ®lm. (b) Control volume for mass and enthalpy ¯uxes.


¯ow structure. Table 1 summarizes the most important testcase parameters. A hollow-cone pressure atomizer with 100°

spray cone angle is used to atomize tetradecane, a diesel fuelsubstitute. The initial diameter spectrum of the spray is rep-

resented by ten diameter classes and follows a Rosin±Rammlerdistribution of D63 60lm and n 1:5.

The gas-¯ow plot in Fig. 2 illustrates the plain gas-phasesolution, resulting from a computation with 8370 ®nite

Fig. 6. (a) Streamlines and liquid volume concentration. (b) Streamlines and fuel vapor concentration.

Table 1

Parameters of the premix duct test case

ra (m) lduct (m) _mair (kg/s) _mfuel (kg/s) Tair (K) pair (Pa) Tfuel (K) Uglobal ÿ0.023 0.11 0.25 0.00866 658 500 000 300 0.5


volumes. It is taken as a reference solution to analyze the in-¯uence of the two-phase ¯ow eects and to demonstrate theperformance of the iterative two-phase ¯ow approach pre-sented in this study. The atomizer on the axis of the premixduct is creating a short recirculation zone with a rather smalleect on the main ¯ow. As illustrated in Fig. 1 the duct liner issurrounded by hot discharge air from the compressor andrepresents an adiabatic boundary for the gas ¯ow.

8.2. Spray calculation without wall ®lm and spray/wall interac-tion

In order to demonstrate the eect of gas ¯ow/spray inter-action, a ¯ow ®eld is calculated in a ®rst step considering onlyspray dispersion and evaporation in addition to the gas ¯ow.During the ®rst 30 EPOS/Ladrop iterations, the two-phase¯ow ®eld is changing signi®cantly due to aerodynamic accel-eration and evaporation cooling of the gas ¯ow by the spray.Due to a total fuel evaporation of 26.5% in the ¯ow ®eld (seeFig. 7), the temperature of the gas ¯ow on the duct axis isdecreased by 100 K. The eect on the ¯ow separation regionbehind the atomizer is illustrated in Fig. 6. The 100°-spraycone is clearly indicated by the contour plot of the liquidvolume concentration. Due to the massive aerodynamic dragof the spray in the injection region, the simple recirculationzone of the plain air ¯ow case (Fig. 2) is now split into threeindividual vortex structures (Fig. 6). The high vapor concen-tration in the inner vortex zone between atomizer and spraycone is constantly fed by vapor and the smallest droplets fromthe fuel nozzle. Due to its complete separation from the main¯ow by the two other vortices,the vapor accumulates in thisvortex zone. A moderately high vapor concentration extendsfrom the remaining part of the ¯ow separation region down-

stream of the ¯ow in the premix duct. In general, smallerdroplets are dragged with the main ¯ow, whereas largerdroplets with a dominating inertia penetrate the main ¯ow andhit the duct liner. Since no wall interaction and ®lm formationis considered in this calculation, fuel evaporation and con-centration are signi®cantly underestimated in the near-wallregion (see Fig. 7).

8.3. Spray calculation including wall and ®lm interaction

Based on the preceding calculation, the models for spray/wall and spray/®lm interaction are now applied to simulatedroplet impacts on a propagating and evaporating liquid wall®lm. As a consequence, the ¯ow ®eld is modi®ed signi®cantlyin the region close to the duct wall. Representing a decisivefraction of the liquid fuel mass, the larger droplets of thespray penetrate the main ¯ow and hit the liquid ®lm on theduct wall. Depending on their impact parameters, they de-posit on the ®lm and/or produce clouds of tiny secondarydroplets. The instantly dragged secondary droplets as well asthe liquid ®lm intensify the fuel evaporation in the near-wallregion, increasing fuel concentration and decreasing gastemperature. As a result, the overall fuel evaporation is nowraised to 30.8% of the injected fuel mass. Due to this addi-tional region of intense fuel evaporation, the homogeneousdistribution of fuel vapor in the premix duct is signi®cantlyimproved. It is obvious, that the quality of the present ¯owsimulation decisively depends on the advanced modeling of®lm propagation and evaporation in combination with so-phisticated spray/wall and spray/®lm interaction models. Ofparticular importance is a realistic description of the deposi-tion rate and the diameter distribution of secondary dropletsdue to splashing (see Fig. 8).

Fig. 7. History of the iterative solution procedure.

Fig. 8. Droplet trajectories and fuel vapor concentration in the premix duct.


9. Summary and conclusions

A new approach to the coupled solution of the interde-pendent ¯ow ®elds of gas phase, spray and wall ®lm has beenpresented in this paper. The iterative procedure comprisesthree individual sub-units. In a ®rst step, the gas phase andliquid wall-®lm ¯ow are simultaneously solved by means of a®nite-volume method and a ®nite-dierence method, respec-tively. In a second step, dispersion and evaporation of the fuelspray is computed using a Lagrangian particle tracking ap-proach. The individual calculations of gas-phase ¯ow, sprayand ®lm ¯ow communicate by exchanging source ®elds forliquid and gaseous fuel. Due to the intense interaction betweenliquid and gas phase in the reported ¯ow simulation, relax-ation of the iterative solution procedure is required. To im-prove the quality of the ¯ow simulation in the wall region ofthe ¯ow, advanced spray/wall and spray/®lm interactionmodels are implemented in the particle tracking method. Basedon various experimental studies, this paper provides a com-plete classi®cation and description of droplet/wall and droplet/®lm interaction models necessary for the development of acomputer code. Results of a ¯ow simulation in a schematicpremix duct demonstrate the eect of the phase interaction andemphasize the importance of a coupled solution of the two-phase ¯ow. In addition, the test case illustrates the in¯uence ofrealistic spray/wall and spray/®lm interaction modeling on the¯ow simulation in the wall region of the duct.

Acknowledgements

This work combines the results of recent developmentswithin the ``Low Emission Combustor Technology Program''of the European Community, within the ``GraduiertenkollegEnergie- und Umwelttechnik'' and within the ``Son-derforschungsbereich 167 ± Hochbelastete Brennraume''. The®nancial support for these programs by the European Com-munity (Contract AER2-CT-92-0036) and the Deutsche For-schungsgemeinschaft (DFG) is gratefully acknowledged.

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CFD analysis of spray propagation and evaporation ...rschmehl.home.xs4all.nl/CV/publications/ijhff99_paper.pdf · tational unit. A second, separate program is based on Lagrangian