Résumé — Modèles de spray dans les applications moteur, des corrélations aux simulationsnumériques directes — Les sprays sont parmi les principaux facteurs de qualité, dans la formation dumélange et la combustion, dans un grand nombre de moteurs (à combustion interne). Ils sont de toutepremière importance dans la formation de polluants et l’efficacité énergétique, bien qu’une modélisationadéquate soit encore en développement. Pour un grand nombre d’applications, la validation et lacalibration de ces modèles demeurent une question ouverte. Aussi, présentons-nous un aperçu desmodèles existants et proposons quelques voies d’amélioration. Les modèles sont classés en non-dimensionnels et dimensionnels allant de formules simples dédiées à des applications proches du tempsréel à des descriptions détaillées des premiers stades de l’atomisation.
Abstract — Spray Atomization Models in Engine Applications, from Correlations to Direct NumericalSimulations — Sprays are among the very main factors of mixture formation and combustion quality inalmost every (IC) engine. They are of great importance in pollutant formation and energy efficiencyalthough adequate modeling is still on development. For many applications, validation and calibration ofmodels are still an open question. Therefore, we present an overview of existing models and proposesome trends of improvement. Models are classified in zero dimensional and dimensional classes rangingfrom simple formulations aimed at close-to-real-time applications to complete detailed description ofearly atomization stages.
Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5802
INTRODUCTION
Spray applications range from large scale dispersion of insec-ticides to nanometer thin film deposits with ion sources.What they have in common is the transformation of a contin-uous phase (mostly liquids) into a number of separateddroplets created by a specific device, or atomizer. For charac-terization of sprays, the most commonly used qualities aresize/number distribution of droplets, penetration and sprayangle. Atomizers technologies are based on a number of prin-ciples that achieve break-up of continuous phase with surfaceor volume forces. The main forces used and their correspond-ing atomizers used in injection applications for engines arelisted non-exhaustively in Table 1.
TABLE 1
Main types of forces and corresponding atomizers
Force Atomizer type
Inertial Rotating, vibration, impinging, swirl type, etc.
Aerodynamic drag Pressure hole, swirl, two-fluid, air-assisted, etc.
Gas bubbles Pressure hole, cavitation, flash,
growth/collapse effervescent, void, etc.
The cost of calibration of the many parameters to beoptimized in modern engines renders real engine tests pro-hibitively high. It is therefore a need to develop effectivemodels able to give a description of mechanisms with a lesscost. Simple 0D models are currently integrated into com-plete phenomenological engine models able to find, throughoptimization algorithms, ensembles of parameter values thatcomply with pollutant regulations and fuel efficiency for anumber of functioning conditions. 1D to 3D models are usedin design tasks at every stage of engine development.
In the last years, intensive use of CFD (ComputationalFluid Dynamics) codes in engines has still increased and gen-eralized. Yet, among the features needed for more effectivemodels, there is the atomization and spray sub-models ofmany engine models. From pioneering work (Wakuri et al.,1960; Dent, 1971; Hiroyasu et al., 1978) on penetration cor-relations to recent efforts in LES and DNS modeling (Apte etal., 2003; De Villiers et al., 2004; Menard et al., 2005; Fusteret al., 2009; Lebas et al., 2009), spray models expandthrough a very large range of concepts. It is our aim to havean overview of existing tools for modeling and prospectiveinsight of future techniques. We focus our interest in the liq-uid and droplet treatment on a surrounding gas, putting asideevaporation and mixing models which we consider as out ofscope of this paper.
First we list non-dimensional models and correlations forspray penetration, spray angle, liquid length and characteris-tic drop size. In this category, we include atomization modelsfor primary and secondary break-up as their results do not
depend on space coordinates, although in order to obtainthose results dimensional equations must be solved (LeMoyne, 2010).
1 0D MODEL
Most of the investigations on sprays conclude with empiricalor semi-empirical laws, which predict the characteristics ofthe spray as a function of several parameters.
Liquidlength
Spray tip penetration
Sprayangle
SMDθ
Figure 1
Different spray parameters.
A spray can be roughly described by four parameters(Fig. 1): the spray tip penetration S, the spray angle θ, thebreak-up length Lb (or liquid length) and the global Sautermean diameter x–32. All together are related in the process ofdisintegration of the spray and can be modeled by 0D mod-els. Other parameters, like air entrainment which results fromthe spray, won’t be analyzed in this study. Also, accordingwith the data, the models are for hole injectors.
1.1 Spray Angle Models
Spray angle (defined in Fig. 1) is influenced by the injectorcharacteristics, fuel properties and ambient conditions.Basically during injection, the angle increases rapidly, reachesa maximum and then decreases to reach a constant value(Lefebvre, 1989). Many authors try to assess and calculate togive the constant value for the spray angle. The collected 0Dspray angle models are shown in Table 2.
1.1.1 Reitz and Bracco Model
Reitz and Bracco (1979) proposed a model for the sprayangle by employing the aerodynamic break-up model ofRanz (1958). It includes the ratio of Reynolds and Webernumbers of the liquid flow in the function f (γ):
(1)tan4π
0.5
θρ
ργ( )
⎛
⎝⎜
⎞
⎠⎟ ( )=
Afg
l
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F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 803
where ρg and ρl are the gas and liquid density, respectively. Ais a constant depending on the nozzle design and may beextracted from experiments or approximated by Equation (2):
(2)
where do is the nozzle diameter and lo the length of thenozzle hole. f (γ) is a weak function of the physical propertiesof the liquid and the injection velocity:
(3)
(4)
where Rel and Wel are the Reynolds number and the Webernumber respectively, based on liquid properties and nozzlediameter:
(5)
(6)
(7)V = CP
inj v
l
2Δρ
We =V d
ll inj o
l
ρ
σ
2
Re =V d
ll inj o
l
ρ
μ
γρρ
=Re
Wel
l
l
g
⎛
⎝⎜
⎞
⎠⎟
2
f = eγ γ( ) −( )−( )3
61 10
A = +l
do
o
3.0 0.28⎛
⎝⎜
⎞
⎠⎟
where Vinj is the injection velocity, Cv the velocity coefficient,μl the liquid dynamic viscosity, σl the surface tension ofliquid and ΔP the difference between injection pressure andambient pressure.
1.1.2 Reitz and Bracco Simplified Model
Heywood (1988) simplified Equation (3) into Equation (1)for cases with high-pressure sprays because the liquid warmsand his viscosity decreases. In these cases, f(γ) becomesasymptotically equal to 31/2/6 because of the rise of γ. Theseoperating conditions can be found in most of modern Dieselinjectors:
(8)
1.1.3 Ruiz and Chigier Model
Ruiz and Chigier (1991) examined the previous aerodynamicbreak-up model and suggested a modification based on injec-tion parameters atypical of modern Diesel engines:
(9)
1.1.4 Arai Model
Arai et al. (1984) gave the following equation in order todetermine the spray angle:
(10)
where ρg is the gas density, ΔP the difference between fuelinjector pressure and ambient gas pressure, do the nozzlediameter, and μg the gas dynamic viscosity.
1.1.5 Hiroyasu and Arai Model
Hiroyasu and Arai (1990) have proposed an empirical equa-tion for the spray angle which includes some characteristicsof the nozzle:
(11)
where lo is the length of the nozzle, do the diameter of thenozzle, and dsac the diameter of the sack chamber. All theseparameters are intrinsic characteristics of injector used.
1.1.6 Arrègle Model
Arrègle et al. (1999) used a simple equation with differentcoefficients which have been fitted with experimental resultson a Diesel spray. This equation depends only of the nozzle
2 83.50.22 0.15
θρ
ρ=
l
d
d
do
o
o
sac
g
l
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
⎛−
⎝⎝⎜
⎞
⎠⎟
0.26
θρ
μ=
Pdg o
g
0.0252
2
0.25Δ⎛
⎝⎜⎜
⎞
⎠⎟⎟
tan4π
0.5 0.2
θρ
ργ( )
⎛
⎝⎜
⎞
⎠⎟ ( )
⎛
⎝⎜
⎞
⎠⎟
−
=A
fRe
Weg
l
l
l
55
tan4π 3
6
0.5
θρ
ρ( )
⎛
⎝⎜
⎞
⎠⎟=
Ag
l
TABLE 2
Different correlations for spray angle
Model Correlation References
Reitz and BraccoReitz and
Bracco, 1979
Reitz and Bracco Heywood,
Simplified 1988
Ruiz and ChigierRuiz and
Chigier, 1991
AraiArai et al.,
1984
Hiroyasu Hiroyasu and
and Arai Arai, 1990
ArrègleArrègle et al.,
1999
Siebers Siebers, 1999
tan4
0.5
θπ ρ
ργ( )
⎛
⎝⎜
⎞
⎠⎟ ( )=
Afg
l
2 83.50.22 0.15
θρ
ρ=
l
d
d
do
o
o
sac
g
l
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
⎛−
⎝⎝⎜
⎞
⎠⎟
0.26
tan 0.508 0.00943 0.335θ ρ( )= d Po inj g
tan 0.00430.19 0
θρ
ρ
ρ
ρθ( )⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜⎜
⎞
⎠⎟⎟= C g
l
l
g
..5⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
θρ
μ=
Pdg o
g
0.0252
2
0.25Δ⎛
⎝⎜⎜
⎞
⎠⎟⎟
tan4π
0.5 0.2
θρ
ργ( )
⎛
⎝⎜
⎞
⎠⎟ ( )
⎛
⎝⎜
⎞
⎠⎟
−
=A
fRe
Weg
l
l
l
55
tan4π 3
6
0.5
θρ
ρ( )
⎛
⎝⎜
⎞
⎠⎟=
Ag
l
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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5
diameter, the injection pressure and the gas density. Equation(12) is the result:
(12)
where Pinj is the injection pressure. We can see that the sprayangle does not depend on injection pressure because theexponent is very small. This is in agreement with some othermodels.
1.1.7 Siebers Model
Siebers (1999) has suggested a model in which the sprayangle depends on the liquid and gas densities, and a constant:
(13)
where Cθ is a coefficient depending on the injector charac-teristics.
1.1.8 Analysis and Validation
In order to compare the previous models, experimental datawith high-pressure injectors has been used (from Sandia(2010)’s online database). The 0D results are compared tothe measured experimental data. The fuel is the Phillipsresearch grade DF2 fuel (ρl = 699 kg/m3 at Tl = 450 K).Experimental and simulated values are considered for threedifferent nozzle diameters, whose characteristics are shownin Table 3.
The ambient temperature Tg is set to 300 K for diameter0.330 mm and 450 K for diameters 0.241 and 0.185 mm(non-vaporizing experiments). The percentage of oxygen isset to 0%, in order to disable combustion.
Experimental data show that the injection pressure doesn’tseem to influence the spray angle value (Fig. 2). This hasbeen observed as well by Arrègle et al. (1999), resulting in atiny injection pressure exponent in equation 12 (0.00943,while exponent for nozzle diameter is 0.508 and exponent forambient density is 0.335, proving that the injection pressuredoes not really affect spray angle). Considering that, manyspray models don’t take into account the injection pressure intheir equations.
tan 0.00430.19 0
θρ
ρρρθ( )
⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜⎜
⎞
⎠⎟⎟= C g
l
l
g
..5⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
tan 0.508 0.00943 0.335θ ρ( )= d Po inj g
Some other models show an evolution in Figure 2 becausethe injection pressure is indirectly included in their equation.For example in Arai model with the term ΔP and in Ruiz andChigier and Reitz and Bracco models with Reynolds andWeber numbers.
As discussed in (Siebers, 1999), Appendix A, the onlyparameter that seems to have a clear influence on sprayangle, apart from the injector orifice characteristics, is theambient gas to fuel density ratio. Thus, for a given fuel, theparameter of interest is the ambient density. Experimentaldata (Fig. 3) show a rise of the spray angle when the ambientdensity increases.
It can be noted that Reitz and Bracco, and Reitz andBracco Simplified models are similar in the cases with lowambient densities but not after. In fact when the ambient den-sity increases, the first term (Rel/Wel)
2 in the gamma functionrises a bit because of the change of the injection velocity, butgamma value is much more influenced (in our case) by thesecond term ρl/ρg, which reduces faster. Consequently,
804
Spray angle vs injection pressuredo = 0.241 mm, ρg = 30 kg/m3, Tg = 450 K
1601401201008060S
pray
ang
le (
°)
20
2
18
16
14
12
10
8
6
4
Liquid injection pressure (Pa)
Reitz and Bracco
Experiment
Ruitz and Chigier
Hiroyasu and Arai
Arai
Siebers
Arrègle Reitz and Bracco simplified
Figure 2
Evolution of spray angle with the injection pressure.
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F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 805
gamma decreases and f(γ) is no longer equal to 31/2/6.Another variable which can change considerably the valueand the evolution of f(γ) is the fuel temperature. When itincreases, the fuel viscosity μl decreases a lot, and changesthe Reynolds number. The term (Rel/Wel)
2 increases andbecomes more significant than ρl/ρg. In this case (highfuel temperature), both models should be equal (becausef(γ) ≈ 31/2/6).
In Figure 4 are represented 27 points with at least onedifferent characteristic between each other (ambient density,injection pressure, nozzle diameter...). The coefficient ofdetermination R2 shows a good correlation between modelsand experimental data. Concerning Siebers model, our valueschosen for the constant Cθ don’t seem to be adapted to theinjectors used for this study, but the evolution on spray anglefor a selected injector is well represented (Fig. 3).
The coefficient of determination and the mean absolutepercentage error for each model are given in Table 4 (the bestvalues are in bold). In our case, Hiroyasu and Arai gives a
spray angle value closed to the measured one but its predic-tion on the evolution of the spray angle is not really good. Onthe contrary, Arai model has a good correlation with theexperimental data, but must be calibrated in order to givevalues close to the measured spray angle.
1.2 Spray Tip Penetration Models
The knowledge of spray tip penetration is important in designof Diesel and Gasoline Direct Injection (GDI) engines. The
TABLE 4
Comparison results between models and data
Spray angle Coefficient of Mean absolute
model determination percentage error
Reitz and Bracco 92.53% 23.91%
Reitz and Bracco simplified 90.47% 38.21%
Ruiz and Chigier 92.12% 87.85%
Arai 94.17% 106.57%
Hiroyasu and Arai 87.41% 7.15%
Arrègle 90.74% 67.69%
Siebers 56.47% 24.27%
Spray angle vs gas densitydo = 0.33 mm, Pinj = 140 Mpa, Tg = 300 K
50 200100 1500
Spr
ay a
ngle
(°)
45
0
40
35
30
25
20
15
10
5
Gas density (kg/m3)
Reitz and Bracco
Experiment
Ruitz and Chigier
Hiroyasu and Arai
Arai
Siebers
ArrègleReitz and Bracco simplified
Figure 3
Evolution of the spray angle with the ambient density.
Measured and predicted spray angle
10 200
Pre
dict
ed s
pray
ang
le (
°)
0
5
10
15
20
25
30
35
40
Measured spray angle (°)
Reitz and BraccoR2 = 92.53%
Experiment
Ruitz and ChigierR2 = 92.12%
Hiroyasu and AraiR2 = 87.41%
Arai R2 = 94.17%
Siebers R2 = 56.47%
Arrègle R2 = 90.74%
Reitz and Bracco simplified R2 = 90.47%
Figure 4
Comparison between measured and predicted spray angle.
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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5
spray penetration is a function of several engine operatingparameters such as nozzle geometry, spray angle, injectionconditions including the injection pressure, ambient densityand temperature. The collected 0D spray tip penetration mod-els are shown in Table 5.
1.2.1 Wakuri Model
Wakuri et al. (1960) used momentum theory to develop thefuel spray model by assuming that the relative velocitybetween fuel droplets and entrained air can be neglected andthe injected liquid droplet momentum is transferred to thehomogeneous fuel droplet-entrained air mixture for the den-sity ratio range of 40-60. Their model is expressed byEquation (14):
(14)
where Ca is the coefficient of contraction (Cd = 1 in thismodel), ΔP the difference between fuel injector pressure andambient gas pressure, ρg the gas density, do the nozzle diame-ter and θ the spray angle.
1.2.2 Dent Model
Equation (15) of spray penetration is the jet mixing model,based on gas jet mixing theory, proposed by Dent (1971).
S = CP d t
a
g
o1.189tan(
0.25
0.25
Δρ θ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜
⎞
⎠)⎟⎟
0.5
This model is different from other models for considering theambient temperature Tg effects by the term (294/Tg):
(15)
1.2.3 Hiroyasu Model
Hiroyasu and Arai (1990) proposed Equations (16) and (17),which are derived from data obtained during the investiga-tion (Hiroyasu et al., 1978) and from applying the jet disinte-gration theory by Levich (1962). They used two differentequations, one from the beginning of the injection to the jetbreak-up time, where the penetration is proportional to time.The other occurs at times exceeding the jet break-up time,where the penetration is proportional to the square root oftime:
(16)
(17)
where tb is the jet break-up time, which can be evaluated byEquation (18):
(18)
1.2.4 Schihl Model
Schihl et al. (1996) analyzed the existing spray penetrationmodels and proposed the following phenomenological conepenetration model:
(19)
1.2.5 Naber and Siebers Model
Equation (20) is a correlation given in Appendix C of Naberand Siebers (1996):
(20)
where S~
is the dimensionless penetration, t~ the dimensionlesstime and n a model constant (n = 2.2):
(21)
(22) �t =
t
t +
�S =
S
x+
�� �
S =t
+t
nn n
1 11
0.5
⎛
⎝⎜⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
−
S = CP d t
v
g
o1.414tan(
0.5
0.25
Δρ θ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜
⎞
⎠⎟
)
00.5
t =d
Pb
l o
g
28.650.5
ρ
ρ Δ( )
S =P
d t t > tg
o b2.95
0.25
0.5Δρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟ ( )
S =P
t < t < tl
b0.392
00.5
Δρ
⎛
⎝⎜
⎞
⎠⎟
S =P
d tTg
o
g
3.07294
0.25
0.5Δρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟ ( )
⎛
⎝⎜⎜
⎞
⎠⎟⎟
00.25
806
TABLE 5
Different correlations for spray tip penetration
Model Correlation References
WakuriWakuri
et al., 1960
Dent Dent, 1971
HiroyasuHiroyasu and
Arai, 1990
SchihlSchihl
et al., 1996
Naber and Naber and
Siebers Siebers, 1996
ArrègleArregle
et al., 1999
S = CP d t
a
g
o1.189tan(
0.25
0.25
Δ
ρ θ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜
⎞
⎠)⎟⎟
0.5
S =P
t < t < tl
b0.392
00.5
Δ
ρ
⎛
⎝⎜
⎞
⎠⎟
S =P
d t t > tg
o b2.95
0.25
0.5Δ
ρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟ ( )
S = CP d t
v
g
o1.414tan(
0.5
0.25
Δ
ρ θ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜
⎞
⎠⎟
)
00.5
�� �
S =t
+t
nn n
1 1
1
0.5
⎛
⎝⎜⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
−
S = d P to inj g0.307 0.262 0.406 0.568⋅ ⋅ ⋅−ρ
S =P
d tTg
o
g
3.07294
0.25
0.5Δ
ρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟ ( )
⎛
⎝⎜⎜
⎞
⎠⎟⎟
00.25
ogst100079_DosSantos 17/11/11 11:16 Page 806
where t+ and x+ are the time scale and length scale respec-tively, they are defined by Equations (23) and (24):
(23)
(24)
1.2.6 Arrègle Model
Arrègle et al. (1999) used a simple equation with differentcoefficients which have been fitted with experimental resultson a Diesel spray. This equation depends only on the nozzlediameter, the injection pressure, the gas density and the time.This is the result:
(25)
where Pinj is the injection pressure. We can observe that thespray penetration is proportional to a value close to thesquare root of the time, as in the other models.
1.2.7 Analysis and Validation
The data come from the same experiment that previously forthe spray angle. The start of injection is defined as the firstmass leaving the nozzle.
S = d P to inj g0.307 0.262 0.406 0.568⋅ ⋅ ⋅−ρ
t =
C d
V+
a ol
g
inj
0.5
tan
ρρ
θ( )
x =
C d+
a ol
g
0.5
tan
ρρ
θ( )
Figure 5 shows predicted and measured spray tip penetra-tions for one example. The shape of penetration is similar forall the models because they are all proportional to the squareroot of time. Concerning the break-up time tb used byHiroyasu, it has an influence on the first part of the penetration,until a value comprised between 4.10–5 and 4.10–4 seconds inour case.
In Figure 6 are represented 1 040 points (from 27penetrations) for each model. All the models seem to bepredictive.
The coefficient of determination and the mean absolutepercentage error for each model are given in Table 6 (the best
TABLE 6
Comparison results between models and data
Spray angle Coefficient of Mean absolute
model determination percentage error
Wakuri 95.79% 7.57%
Dent 92.60% 11.14%
Hiroyasu 96.96% 12.61%
Schihl 96.75% 8.77%
Naber and Siebers 96.60% 15.26%
Arrègle 94.45% 17.86%
F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 807
Comparaison between experiment and modelsdo = 0.241 mm, Pinj = 140 Mpa, Tg = 450 K, ρg = 14.8 kg/m3
1.21.00.80.60.40.20
Spr
ay ti
p pe
netr
atio
n (m
)
0
0.10
0.08
0.06
0.04
0.02
Time (s) x 10-3
Experiment
Wakuri
Dent
Hiroyasu
Schihl
Naber and Siebers
Arrègle
Figure 5
Example of 0D results for spray tip penetration.
0.05 0.100
Pre
dict
ed s
pray
pen
etra
tion
(m)
0
0.04
0.02
0.06
0.08
0.10
0.12
Measured spray penetration (m)
Measured and predicted spray tip penetration
Wakuri R2 = 95.79%
Experiment
Hiroyasu R2 = 96.96%
Naber and Siebers R2 = 96.6%
Schihl R2 = 96.75%
ArrègleR2 = 94.45%
Dent R2 = 92.6%
Figure 6
Comparison between measured and predicted spray tippenetration.
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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5808
values are in bold). In our case, Wakuri model gives a spraytip penetration value closed to the measured one. But the bestprediction is given by Hiroyasu model. Also, the additionalterm with the ambient temperature, in Dent model equation,seems to reduce its validity.
1.3 Liquid Length Models
The purpose of this study is to investigate the applicability ofthe existing 0D liquid length models for spray. The collectedspray angle models are show in Table 7.
1.3.1 Chehroudi Model
The more famous and simple equation for the liquid lengthexpresses the fact that the core length is dependent on theratio of liquid and gas density and that it is directly propor-tional to the nozzle hole diameter:
(26)
where do is the nozzle diameter, ρl the liquid density and ρg
the gas density. C is an empirical constant which expressesthe influence of the nozzle flow conditions and other effectsthat cannot be described in detail. Chehroudi et al. (1985)performed some experiments and proposed that the constantC should be in the range 7-16. However, for our experiments,a value of 16.3 gives the best result.
L = Cdb ol
g
ρρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
0.5
1.3.2 Beale and Reitz Model
Beale and Reitz (1999) proposed a modified equation of theprevious model:
(27)
where B1 is the well known breakup constant of KelvinHelmholtz model. According with Su et al. (1996), thisconstant can be set to 60.
1.3.3 Hiroyasu and Arai Model
The following equation was given by Hiroyasu and Arai(1990) for the liquid length from experimental data whichcovered a wide range of conditions:
(28)
where Lb is the liquid length, ro is the radius of inlet edge ofhole, Pg the ambient gas pressure, Vinj the injection velocity,lo the length of the nozzle hole and ρg the gas density.
This equation includes the effect of the inlet edge round-ing which shifts the inception of cavitation and turbulence tohigher injection pressures and increases the liquid length. Theinfluence of cavitation and turbulence is also included via theterm (Pg/ρlV
2inj).
1.3.4 Siebers Model
Siebers (1999) has developed a scaling law to predict theliquid length. The assumptions made include quasi-steadyflow with a uniform rate, uniform velocity, uniform fuel con-centration and uniform temperature profiles (perfect mixinginside the spray boundaries), and finally no velocity slipbetween the injected fuel and the entrained gas. The modelequation is the following:
(29)
where a and b are two constants (0.66 and 0.41 respectively)and θ the spray angle. B is the ratio of the fuel and ambientgas mass flow rates resulting in complete vaporization of thefuel:
(30)
where Z is the compressibility factor, M the molecularweights, P the pressure, T the temperature and h the enthalpy.The subscript g, l and s represents the ambient gas, the vapor-ized liquid and the saturated liquid vapor condition at theliquid length respectively.
B =Z T ,P P P M
Z T ,P P P M=g s g s s l
l s s g s g
−( ) × ×
( ) × −⎡⎣ ⎤⎦×
hh T ,P h T ,P P
h T h T ,Pg g g g s g s
l s l l g
( ) − −( )( ) − ( )
L =b
a
d C
B+b
l
g
o aρρ θ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜
⎞
⎠⎟
0.50.5
tan(
21
)
22
1−
L = d +r
d
P
Vb oo
o
g
l inj
7.0 1 0.42
0.0⎛
⎝⎜
⎞
⎠⎟⎛
⎝⎜⎜
⎞
⎠⎟⎟ρ
55 0.13 0.5
l
do
o
l
g
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
ρρ
L = B db ol
g
0.5 1
0.5
ρρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
TABLE 7
Different correlations for liquid length
Type of model Correlation References
ChehroudiChehroudi
et al., 1985
Beale and Beale and
Reitz Reitz, 1999
Hiroyasu Hiroyasu and
and Arai Arai, 1990
Siebers Siebers, 1999
Enhanced -
model
L = Cdb ol
g
ρρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
0.5
L = B db ol
g
0.5 1
0.5
ρρ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
L =b
a
d C
B+b
l
g
o aρ
ρ θ
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜
⎞
⎠⎟
0.50.5
tan(
21
)
22
1−
L =T
db
g
ol
g
93294
1.43 0.5⎛
⎝⎜⎜
⎞
⎠⎟⎟ ⋅
⎛
⎝⎜⎜
⎞
⎠⎟⎟
ρρ
L = d +r
d
P
Vb oo
o
g
l inj
7.0 1 0.42
0.0⎛
⎝⎜
⎞
⎠⎟⎛
⎝⎜⎜
⎞
⎠⎟⎟ρ
55
0.13 0.5
⋅⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
l
do
o
l
g
ρρ
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F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 809
The unknown, Ts, can be solved iteratively, given the fueland ambient gas properties and initial fuel and ambient gasconditions. Once determined, Ts defines B, as well as thepressures, temperatures, and enthalpies of the fuel and ambi-ent gas at the liquid length location.
The term B given by Equation (30) is analogous to themass and thermal transfer numbers used in droplet vaporiza-tion studies (Lefebvre, 1989). Furthermore, the method ofsolving for Ts is analogous to that used in determining thesurface temperature of a vaporizing liquid droplet (Lefebvre,1989).
1.3.5 Enhanced Model
Experimental data shows a correlation between liquid lengthand ambient temperature (Fig. 8). Previous models don’t takeinto account this phenomenon, except Siebers’s model.
By using Chehroudi’s model, an additional term can beadded. The resulting enhanced model is expressed inEquation (31):
(31)
where Tg is the ambient gas temperature. C1 and C2 are twoconstants to determine by a linear regression method. In ourcase a value of 93 and 1.43 is chosen, respectively.
1.3.6 Analysis and Validation
The experiments are different to the previous ones; the ambi-ent temperature Tg is increased (700 K to 1 300 K instead of450 K). The nozzles characteristics are listed in Table 8.
TABLE 8
Characteristics of nozzles
Orifice DischargeArea-contraction coefficient
Length-to-
diameter coefficient diameter
do (mm) Cd Ca for 72 Mpa Ca for 138 Mpa lo/do
0.100 0.80 0.91 0.86 4.0
0.180 0.77 0.85 0.82 4.2
0.251 0.79 0.88 0.79 2.2
0.246 0.78 0.89 0.81 4.2
0.267 0.77 0.89 0.82 8.0
0.363 0.81 - 0.85 4.1
0.498 0.84 0.94 0.88 4.3
The fuel is 2,2,4,4,6,8,8 heptamethylnonane (C16H34),with a density ρl = 689 kg/m3 for a temperature Tl = 436 K.The percentage of oxygen is set to 0%, in order to disablecombustion.
L = CT
db
g
C
ol
g
1
2 0.5
294⎛
⎝⎜⎜
⎞
⎠⎟⎟ ⋅
⎛
⎝⎜⎜
⎞
⎠⎟⎟
ρρ
All the models include a density ratio in their equations, sotheir prediction for a variation of density is roughly the same.But concerning the injection pressure, only one modelincludes it (Hiroyasu and Arai model, indirectly with theinjection velocity Vinj in the cavitation term (Pg/ρlV
2inj)).
Figure 7 shows that the liquid length decreases when theinjection pressure rises, and Hiroyasu and Arai model pre-dicts very well this phenomenon.
Concerning the ambient temperature, its effect is todecrease the liquid length when it rises. The phenomenon,shown in Figure 8, is well predicted by the enhanced model.
The coefficient of determination and the mean absolutepercentage error for each model are given in Table 9 (the bestvalues are in bold). The coefficient of determination is verylow for the first models because the ambient temperature isnot taken into account. The enhanced model is clearly morepredictive with its additional term.
TABLE 9
Comparison results between models and data
Spray angle Coefficient of Mean absolute
model determination percentage error
Chehroudi 54.78% 23.30%
Beale and Reitz 54.78% 70.41%
Hiroyasu and Arai 51.33% 61.51%
Enhanced model 98.06% 6.07%
Liquid length vs injection pressuredo = 0.246 mm, ρg = 14.8 kg/m3, Tg = 700 K
Liqu
id le
ngth
(m
)
0.05
0.04
0.03
0.02
0.01
0
Liquid injection pressure (Pa)
Experiment
Chehroudi
Beale and Reitz
Hiroyasu and Arai
Enhanced model
160 1801401201008060
Figure 7
Evolution of liquid length with the injection pressure.
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Experiments used ambient conditions similar to the ones inan engine combustion chamber, and the injector is also com-parable to the ones used in car engines. So our values of theconstant C1 and C2 should be a good start in order to modelthe liquid length in an internal combustion engine.
1.4 Global Sauter Mean Diameter Models
The global Sauter Mean Diameter (SMD or x–32) character-izes a single droplet with the same volume to surface arearatio as the ratio of the respective quantities integrated overthe whole droplet size distribution present in a real spray:
(32)
The SMD is a quantity characterizing the average dropletsize of a spray. Many studies and correlations describe theSMD depending on the distance to the nozzle or on thebreaking type or phase (Wu and Faeth, 1995; Faeth, 1996;Lee et al., 2007).
Some global SMD models are listed in Table 10, but theavailable data, mostly obtained by time-resolved pointmeasurements is not pertinent for comparison.
x =
d
d
i
i=
Ndrops
i
i=
Ndrops32
3
1
2
1
∑
∑
1.4.1 Hiroyasu Model
The following equation is an empirical expression for theSMD for typical Diesel fuel properties and for hole typenozzle given by Hiroyasu et al. (1989):
(33)
where ΔP is the difference between fuel injector pressure andambient gas, ρg the gas density and ml is the amount of fueldelivery.
1.4.2 Varde Model
Varde et al. (1984) related the SMD to the diameter of theinjection nozzle in the following equation:
(34)
where do is the nozzle diameter, Rel the Reynolds numberand Wel the Weber number.
1.4.3 Hiroyasu and Arai Model
Hiroyasu and Arai (1990) studied more in detail the effects ofvarious parameters on the SMD like ambient pressure, injec-tion pressure or the nozzle characteristics. Dimensionlessanalysis leaded to the following experimental equations for acomplete spray:
(35)x = max x ,xLS HS32 32 32⎡⎣
⎤⎦
x = d Re Weo l l32
0.288.7 ( )−
x = P mg l323 0.135 0.121 0.1312.33 10⋅ − −Δ ρ
TABLE 10
Correlation of SMD for different models
Type
of modelCorrelation References
HiroyasuHiroyasu
et al., 1989
VardeVarde et al.,
1984
Hiroyasu Hiroyasu and
and Arai Arai 1990
Merrington Merrington
and and Richardson,
Richardson 1947
Elkotb Elkotb, 1982
x = d Re Weo l l32
0.288.7 ( )−
x =d
Vo l
inj
32
1.2 0.2500 ν
x = Pl l l g l320.385 0.737 0.737 0.06 0.53.085ν σ ρ ρ Δ − 44
x = max x ,x
x = d Re We
LS HS
LS o l0.12
32 32 32
324.12
⎡⎣
⎤⎦
lll
g
l
g
Hx
−⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
0.75
0.54 0.18
32
μ
μ
ρ
ρ
SS o l0.25
ll
g
l= d Re We0.38 0.32
0.37
−⎛
⎝⎜⎜
⎞
⎠⎟⎟
μ
μ
ρ
ρgg
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−0.47
x = P mg l323 0.135 0.121 0.1312.33 10⋅ − −Δ ρ
Liquid length vs ambient temperaturedo = 0.246 mm, ρg = 7.3 kg/m3, Pinj = 136.27 Mpa
Liqu
id le
ngth
(m
)
0.08
0.09
0.07
0.06
0.05
0.04
0.03
0.02
0.01
Ambient temperature (K)140012001000800600400
Experiment
Chehroudi
Beale and Reitz
Hiroyasu and Arai
Enhanced model
Figure 8
Evolution of liquid length with the ambient temperature.
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(36)
(37)
where μl and μg are the dynamic viscosity of the liquid andthe gas respectively.
1.4.4 Merrington and Richardson Model
The disintegration of liquid jets injected into stagnant air wasstudied empirically (Merrington and Richardson, 1947).Drop-size distribution curves were obtained by a stationarynozzle directed vertically downward in an enclosed spraytower. The pressure supply was held constant during eachtest run by using a pressure-regulated gas cylinder.
(Lefebvre, 1989) gives a revision of the Merrington andRichardson SMD correlation including the effect of orificediameter on the resulting SMD. The correlation for SMD isprovided in Equation (38):
(38)
where νl is the kinematic viscosity of the liquid and Ving theinjection velocity.
1.4.5 Elkotb Model
Elkotb (1982) has developed a model that takes into accountnumerous properties of both liquid and gas. In the end, exper-imental formula provides:
(39)
where ΔP is measured in bar. SMD value is given in μm.
1.5 Conclusion on 0D Model
We have seen in this part some spray characteristics of sprayslike tip penetration or spray angle. Several 0D models havebeen considered and analyzed, and the results of the compar-isons with experimental data show that these characteristicscan be accurately predicted, and thus the spray correctlydescribed.
Despite not being as exact as multidimensional models,0D models offer important advantages like their reduced sim-ulation time as well as their low computational requirements.For that reason, these models can be useful when a largenumber of simulations is needed.
But like there is not a spatial discretization in these models,the interaction between spray and the flow or the walls cannot be modeled. Also the ambient conditions are fixed(volume, temperature...). In order to describe a spray evolving
x = Pl l l g l320.385 0.737 0.737 0.06 0.53.085ν σ ρ ρ Δ − 44
x =d
Vo l
inj
32
1.2 0.2500 ν
x = d Re WeHS o l0.25
ll
g32
0.32
0.37
0.38 −⎛
⎝⎜⎜
⎞
⎠⎟⎟
μμ
ρρρ
l
g
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−0.47
x = d Re WeLS o l0.12
ll
g32
0.75
0.54
4.12 −⎛
⎝⎜⎜
⎞
⎠⎟⎟
μμ
ρρρ
l
g
⎛
⎝⎜⎜
⎞
⎠⎟⎟
0.18 in a combustion chamber, multidimensional models shouldare needed.
2 MULTIDIMENSIONAL MODEL: RANS AND LES
In this part are listed the actual multidimensional models andmethods used to model sprays, some results are shown forthe most popular methods.
2.1 RANS; Liquid is Pure Lagrangian
2.1.1 Primary Break-Up
The task of a primary break-up model is to determine thestarting conditions of drops which penetrate into the chamber(initial drop size and its velocity components). These condi-tions are principally influenced by the flow conditions insidethe nozzle.
Blob MethodThe blob method, developed by Reitz and Diwakar (1987), isthe most popular of the models because of its simplicity. It isassumed that a detailed description of the atomization andbreakup processes within the primary breakup zone of thespray is not required. This method creates big sphericaldroplets (Fig. 9) with the same diameter (usually equal to thenozzle hole diameter), which are then subject to secondarybreak-up. The number of drops injected per unit time isdetermined from the mass flow rate and the conservation ofmass gives the injection velocity of the blobs.
Kuensberg Sarre et al. (1999) suggested an enhancedversion of the blob method. This method allows calculatingan effective injection velocity and an effective injection parti-cle diameter taking into account the reduction of the nozzlecross section due to cavitation (by decreasing the initial blobsize and estimating a more realistic initial velocity).
Uinj D
Nozzle holeBlob
Secondary break-up
Figure 9
Blob method.
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Altogether, the blob methods are a good way to define theinitial starting conditions for the liquid entering the chamber.But these methods do not represent a detailed physical andsatisfying modeling of the relevant processes during primarybreak-up.
Distribution FunctionsThis method assumes that the fuel is already fully atomizedat the nozzle exit and that the distribution of drop sizes can bedescribed by mathematical functions. In this case, a distribu-tion of droplet sizes is injected.
According to Levy et al. (1998), the droplet’s diameter Dat the nozzle exit should be sampled from a X2-law in orderto get a good agreement between measured and simulateddownstream drop size distributions:
(40)
where D–
= SMD/6. Various authors have used distributionfunctions in Diesel spray modeling like Long et al. (1994)and Lefebvre (1989).
It is also possible to predict the distribution function byusing the Maximum Entropy Formalism (MEF) instead of amathematical distribution function. Due to the use of a physi-cally based criterion, the entropy of the distribution given byShannon (1948), it is possible to estimate the most probabledistribution function. Cousin and Desjonquères (2003) haveused this method in order to predict drop size distributions insprays from pressure-swirl atomizers.
Kelvin-Helmholtz Break-Up Model (KH Model)The Kelvin-Helmholtz model (KH model or WAVE, Fig. 10)has been proposed by Reitz (1987). The model is based on afirst order linear analysis of a Kelvin-Helmholtz instabilitygrowing on the surface of a cylindrical liquid jet that is pene-trating into a stationary, inviscid and incompressible gas witha relative velocity.
P D =D
D e DD( ) −1
6 43
From the solution of a general dispersion equation, maximumgrowth rate ΩKH and corresponding wave length ΛKH aregiven by the following equations:
(41)
(42)
where r0 is the initial radius of the droplet, Z = √⎯Wel /Rel and
T = Z√⎯Weg. The new radius of droplet rnew is given by:
(43)
The new droplet continuously looses mass while penetra-tion into the gas, its radius r is expressed in Equation (44):
(44)
(45)
where τbu is the characteristic time span.
Turbulence-Induced Break-UpHuh and Gosman (1991) have published a model of turbu-lence-induced atomization for full-cone Diesel sprays. Theyassume that the turbulent forces within the liquid emergingfrom the nozzle are the producers of initial surface perturba-tions, which grow exponentially due to aerodynamic forcesand form new droplets.
The droplets break up with a characteristic atomizationlength scale LA and time scale τA. The characteristic atomiza-tion length is proportional to the turbulent length scale Lt:
LA = C1Lt = C2Lw (46)
where Lw is the wavelength of surface instability, determinedby turbulence. According with the authors, C1= 2 andC2= 0.5. The characteristic atomization time scale τA can becalculated under the assumption that the time scale of atom-ization is a linear combination of the turbulence time scale τt
and the wave growth time scale τw:
(47)
where C3 = 1.2 and C4 = 0.5. τspn and τexp indicate sponta-neous wave growth time and exponential growth time.
This model shows good agreement with availableexperimental data for the spray cone angle of steady-flowsingle-hole experiments, but the effects of cavitation are notincluded.
τ τ τ τ τA t w spn exp= C +C = +3 4
τbu
KH KH
= Br
3.788Ω1 Λ
�r =
r rnew
bu
−τ
r =new KH0.61Λ
ΛKH
gr
=+ Z + T
+ We0
0.5 0.7
19.02
1 0.45 1 0.4
1 0.87
( ) ( )..67 0.6( )
Ω0.34 0.38
1 1 1.03 0.5 1.5
KHl gr
=+ We
+Z +
ρσ
⎡
⎣⎢
⎤
⎦⎥
( ) 44 0.6T( )
Urel2 r
2.B0.Λ
Λ
Figure 10
Illustration of the KH model.
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Cavitation-Induced Break-UpArcoumanis and Gavaises (1997) have presented a primarybreak-up model that takes into account cavitation, turbulence,and aerodynamic effects. The initial droplet diameter is setequal to the effective hole diameter (blob method), and thefirst break-up of these blobs is modeled using the Kelvin-Helmholtz mechanism in the case of aerodynamic-inducedbreak-up, the previous model of Huh and Gosman (1991) forturbulence-induced break-up, and a new phenomenologicalmodel in the case of cavitation-induced break-up.
The cavitation bubbles are transported to the blob surfaceby the turbulent velocity inside the liquid and either burst onthe surface or collapse before reaching it, depending of thecharacteristic time scale. The radius of the bubbles is givenby the following equations:
(48)
(49)
where rcav is the radius of bubble, reff the effective radius andAeff is the effective area. The characteristic time scales forcollapse τcoll and bursting of bubbles τburs are:
(50)
(51)
where uturb is the turbulent velocity (uturb = √⎯2k /3, k is the
turbulent kinetic energy). The smaller characteristic timescale causes the break-up.
τburstcav
turb
=r r
u0 −
τρ
ρcoll cavl
g,bubble
= r0.9145
r =A
effeff
π
r = r rcav eff02 2−
Cavitation and Turbulence-Induced Break-UpNishimura and Assanis (2000) have proposed a cavitationand turbulence-induced primary break-up model for full-coneDiesel sprays. During the injection period, discrete fuelparcels enter in the chamber with an initial diameter equal tothe nozzle hole diameter. Each parcel contains bubbles (seeFig. 11), according to the volume fraction and size distributionat the hole exit, computed from a phenomenological cavitationmodel inside the injector.
The authors assume that the velocity fluctuations insidethe cylinder induce a deformation force on its surface:
(52)
It breaks up if the sum of Fturb and the aerodynamic dragforce Faero is no longer compensated by the surface tensionFsurf:
(53)
(54)
The diameter of the original cylinder is reduced untilFturb + Faero = Fsurf again.
2.1.2 Secondary Break-Up
Secondary break-up is the disintegration of already existingdroplets into smaller ones due to the aerodynamic forces.These forces are induced by the relative velocity betweendroplet and surrounding gas. The result is an unstable growthof waves on the droplet surface which finally leads to the dis-integration of the droplet into new ones.
F = Dsurf π σ
F =d
uaeroo
g relπ2
30.5
22ρ
F = = DD
uturbl
tursurface dynamic pressure π2
3 2
ρbb
2
Cylindrical shapeparent parcel
(contain one droplet)
Nozzlehole
Cavitationbubble
Pressure gradient force to sidesurface of parent parcel
Child parcel
No cavitation and noturbulence in child parcel
Child droplet size determined by wave length,and all turbulence energy in child parcel
is transformed to tangential velocity
Parent parcel continuesto generate child parcel until
bubbles collapse totally
When breakup force becomes biggerthan surface tension, parent parcel emits
excess volume as one child parcelSurface vibrates by
turbulent energy
Figure 11
Primary break-up model of Nishimura and Assanis (2000).
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Taylor-Analogy Break-Up Model (TAB Model)The Taylor Analogy Break-up model was proposed byO’Rourke and Amsden (1987). It is based on an analogybetween a forced oscillating spring-mass system and anoscillating and distorting droplet (see Tab. 11).
TABLE 11
Comparison of a spring-mass system to a distorting droplet
Spring-mass system Distorting and oscillating droplet
Restoring force of spring Surface tension forces
External force Droplet drag force
Damping force Droplet viscosity forces
The deformation of the droplet is given by:
(55)
where Cf = 1/3, Cb = 0.5, Ck = 8 and Cd = 5. urel is the relativevelocity between droplet and gas and σ is the surface tension.
Tanner (1997) showed that the TAB model under predictsthe droplets sizes of full cone sprays and has developed anenhanced version of this model (ETAB). The difference isthat an initial oscillation is chosen for the spherical dropsemerging from the nozzle (against no oscillation for TABmodel).
Droplet Deformation and Break-Up Model (DDB Model)Ibrahim et al. (1993) presented a droplet deformation andbreak-up model. It assumes that the liquid droplet isdeformed from the initial spherical shape into an ellipsoidalone (see Fig. 12).
The distortion of droplet that causes break-up is governedby the following equation:
(56)
Ky +
N
Re yy +
Wey y
g g
�� �4 1 27π
161 2
3π
42
2 6
−⎛
⎝⎜
⎞
⎠⎟
⎡
⎣
−
⎢⎢⎢
⎤
⎦⎥⎥
=3
8
�� �y =C
C
u
rC
ry C
ryF
b
g
l
relk
l
d
l
ρ
ρσρ
μρ
2
2 3 2− −
where K is the liquid-gas density ratio and N is the liquid-gasviscosity ratio. The droplets start with zero deformation(y = 4/3π). The break-up happens when the deformation ofdroplet reaches the value given by:
(57)
where a is the major semi-axis of the ellipsoidal cross sectionof the oblate spheroid.
The DDB and TAB model have a very similar breakupmechanism, so the DDB model can be regarded as an alter-native to the TAB model.
Rayleigh-Taylor Break-Up Model (RT Model)The Rayleigh-Taylor model (RT model, Fig. 13) is based onthe theoretical work of Taylor (1950), who investigated theinstability of the interface between two fluids of differentdensities in the case of an acceleration (or deceleration) nor-mal to this interface.
From the assumption of linearized disturbance growthrates and negligible viscosity, the frequency of the fast-grow-ing RT wave is given by:
(58)
The corresponding wavelength ΛRT and wave number KRT
are given by:
(59)
(60)
where a is the acceleration of the droplet and CRT is aconstant (CRT = 0.3).
K =a
RTl gρ ρ
σ
−( )⎛
⎝⎜⎜
⎞
⎠⎟⎟3
0.5
ΛRTRT
RT
=C
K
2π
Ω2
3 3
3 20.5
RT
l g
l g
=a
+σ
ρ ρ
ρ ρ
−( )⎡⎣
⎤⎦
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
/
a
r=
Weg
6π
Urel
b
a
Y
Figure 12
Schematic diagram of the deforming half drop.
Urel
a
Back Front
Λ
Figure 13
Illustration of the RT model.
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F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 815
If the corresponding wavelength is smaller than thedroplet diameter, the instability of the RT wave increases.While the wave is growing, the wave growth time is tracked,and then compared with the break-up time. In RT break-up,the break-up time τRT and the radius after break-up rnew aredefined by the following equations:
(61)
(62)
Combined ModelsA single break-up model is usually not able to describe allbreak-up processes and break-up regimes of engine sprays. It
τ τRT
RT
=C
Ω
r =C
KnewKH
RT
π
is necessary to use combined models (usually a primary anda secondary break-up) in order to improve the accuracy ofprediction.
Patterson and Reitz (1998) suggested the KH-RT hybridbreakup model with the concept that the competition betweenKH instability and RT instability causes the droplet breakup.Park et al. (2003) investigated the prediction accuracy ofvarious hybrid models for high-speed Diesel fuel sprays andreported that the results of KH-RT model, KH-DDB model,and Turbulence-DDB model agree well with the experimentalresults.
2.1.3 Calculation
Figure 14 shows an example of utilization of the KH-RTmodel in order to model a Diesel high pressure spray. Thesoftware used is OpenFOAM1.5-dev which includes nativelythe KH-RT model. The Lagrangian particles are injected inan Eulerian RANS flow using the k-ε turbulence model.Figure 15 is the penetration curve from 3D model com-pared to correlation and experimental data, showing a goodagreement.
2.2 RANS; Liquid is Pure Eulerian
Wan and Peters (1999) developed an Eulerian approach fromdroplet equations, integrating over the radial directionto model Diesel sprays with a 1D model. Their ICAS(Interactive Cross sectional Averaged Spray) model is effec-tive in simulating the vicinity of the nozzle up to the zonewhere the gas velocity becomes the predominant mixingmechanism. Vallet et al. (2001) and Beheshti and Burluka(2004) proposed a fully Eulerian spray atomization model bygeneralizing Kolmogorov hypothesis on turbulence to char-acteristic scales of spray. The principles of such modeling arecurrently used for the development of Eulerian-Lagrangecoupled codes for spray simulation. The compressible Favre-averaged flow equations are applied to a single fluid withvariable properties. The mass and momentum conservation
Comparaison between experiment and modelsdo = 0.241 mm, Pinj = 140 Mpa, Tg = 450 K, ρg = 58.5 kg/m3
2.52.0
Experiment
Wakuri 0D
KH-RT 3D
1.51.00.50 3.0
Spr
ay ti
p pe
netr
atio
n (m
)
0.01
0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Time (s) x 10-3
Figure 14
Lagrangian particles (color scale for diameter in meters) and Eulerian flow (color scale for velocities in m/s).
Figure 15
Penetration results for KH-RT Lagrangian particles inEulerian flow.
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equations consider the contributions from one phase onanother. The model is completed by k-ε equations for thetwo-phase turbulent kinetic energy (which includes kineticenergy in gas, liquid and the mean velocity gradient of thetwo phases) and dissipation rate, and then closed by anAlgebraic Stress Model (ASM) of the Reynolds stress tensor.Specific treatment of atomizing liquid consists of conserva-tion equation of liquid mean mass fraction:
(63)
with:
(64)
and a state law:
(65)
along with a transport equation of mean interfacial area perunit volume:
(66)
Diffusion term is written classically with a diffusioncoefficient Ds that depends on turbulence and liquid frac-tion gradient. The production consists of two mechanisms,first is the action of velocity gradients that, according to aKelvin-Helmholtz type instability, will stretch liquid-gasinterface and make the surface larger. The author’s choiceis to give it a characteristic time A–1 equivalent to the one ofturbulent kinetic energy production. The second is theaction of turbulence which can also stretch and augmentliquid-gas interface, with a characteristic time a–1 equiva-lent to the integral scale of turbulence. Last, the destructionterm allows compensating production in such a way that atsmall scales, inertial forces be of the same order as surfacetension: We = 1. The destruction term exponent beinggreater than 1, value of 2 is given to include the possibilityof having two interfaces interacting inside a given volume.
In this approach, oriented towards large Reynolds andWeber number flows with a main velocity direction (holeinjectors of Diesel type), surface tension, and viscosity areonly effective at small scales, and where curvature andvelocity gradients are important. At large scales, the charac-teristics of the flow are dependent only on fluid density. Itis the generalization to atomization of Kolmogorov’s resultsfor turbulence. Also, it is possible to define in the same
∂∂+∂∂
=
∂∂∂
⎛
⎝⎜
⎞
⎠⎟
∂+ +( ) −
Σ Σ
Σ
Σ Σt
u
x
Dx
xA a Vi
i
s
i
i
a
� 2
== diffusion + production destruction−( )
pY r T
Yg g
l
=−( )−
ρ
ρρ
1
1
�
�
1 1
ρ ρ ρ= +
−( )� �Y Y
l g
∂∂
+∂∂
= −∂∂
ρ ρ ρ� � �Y
t
u Y
x
u Y
xi
i
i
i
' '
way, a critical scale of atomization where inertial forcesbalance surface tension:
(67)
On the condition that balance between different processes ofdroplet formation of radius rc (coalescence, break-up) andturbulent diffusion is obtained. Such a model includes someconstants that can be tuned for solution. A correct calibrationof the model can therefore lead to good simulation efficiencybut the physical mechanisms of atomization are not explicitlyrepresented resulting in the potential need for calibration ofconstants for each different case and hence limited generality.
2.3 RANS; Liquid is Eulerian-Lagrangian Represented
The Eulerian multiphase method can also be used for model-ing the liquid in the near nozzle dense spray region combinedto the Lagrangian method for zones where the spray is suffi-ciently diluted (far away from the nozzle). This method iscalled ELSA (Eulerian-Lagrangian Spray Atomization) andhas been used by Demoulin et al. (2007) and Lebas et al.(2009).
2.4 LES; Liquid Fuel is Pure Lagrangian
Bharadwaj et al. (2009) have modeled an non-evaporativeDiesel spray using the Lagrangian method with large eddysimulation. Their results show a good agreement. It wasshown that a high speed Diesel spray can create significantenergy at the sub-grid scale in the near nozzle region. Thissub-grid kinetic energy is important in the models of sub-gridshear stress and droplet turbulent dispersion.
2.5 LES; Liquid Fuel is Pure Eulerian
Some studies have been to model a spray in full Eulerian inLES, like De Villiers et al. (2004). Their approach combinesmultiphase Volume-Of-Fluid (VOF) and large eddy simula-tion methodologies. It is used to perform quasi-direct tran-sient fully three dimensional calculations of the atomizationof a high-pressure Diesel jet, providing detailed informationon the processes and structures in the near nozzle region,which is difficult to obtain by experimentation.
This methodology allows separate examination of diverseinfluences on the breakup process and is expected in duecourse to provide a detailed picture of the mechanisms thatgovern the spray formation. It is a powerful tool for assistingin the development of accurate atomization models for practi-cal applications.
This approach has been used by Bianchi et al. (2007),who are also interested by the flow inside the nozzle whichcan influence the processes of atomization. In the case of
We =r u
cc c cρσ
2
1≈
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F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 817
high-pressure injectors, cavitation can occur in the nozzle andchanges the rest of the spray, Peng Karrholm et al. (2007)has worked on this phenomenon and has performed somecalculations in order to model cavitation, with LES.
3 MULTIDIMENSIONAL MODEL: DNS
As classic simulations carry problematic issues concerninggrid dependency, in order to correctly represent the spray,alternative solutions must be considered. The simplest is tocarry direct numerical solutions. This of course can involvecostly computer features if trying to solve the integral spacedomain. We propose here an alternative consisting on simu-lating only the very close vicinity of the spray and consider-ing only the main processes in that region. For example,given the length and time scales, we do not consider evapora-tion (although some marginal and not precisely estimatedamount of numerical “evaporation” is hard to avoid with theinterface capturing techniques employed), Tomar et al.(2010). The idea is to carry independent simulations withvarying parameters that can furnish adequate initial condi-tions (velocity, position and size of droplets) to spray models.Thus, DNS is carried up to the point where droplets reach asize small enough to be represented by Lagrangian particles.The principle has been studied and implemented by Tomar etal. (2010) based on GERRIS code by Popinet (2003), whichhas been applied to spray simulations (Fuster et al., 2009).When droplets reach a size small enough they are simplytaken out from the fluid domain and implemented asLagrangian particles. Balance of exchanges between dropletsand gas is made for each cell, thus modifying gas flow.
Probably the main interest of DNS applied to spray consistson the possibility of modeling unsteady complex processesfor which no validated models exist, for example the beginningof jet injection.
Simulations presented here are intended to explore theactual possibilities of the methodology. They are carried withan adaptive 3D mesh of a maximum of 29(512) cells perdimension when detailed aspects of liquid interface arerequired, or 27(128) when global quantities like penetrationare estimated. With 29 cells the equivalent size of the smallestcells is 2.3 μm for an initial jet diameter of 0.2 mm, with 27
cells it is of 9.4 μm. The Reynolds and Weber number ofexperiments and simulations are theoretically 30 103 and 75103 respectively although the effects of numerical viscosityand surface tension are still to be measured. Simulation resultsare therefore still to be validated and we only show here acomparison with spray penetration and qualitative aspects.Density ratio (gas/liquid) of simulation and experiments is .Results are presented in non-dimensional variables S* and t*:
(68)S =S
do
*
(69)
Figure 16 shows an example of the results obtained for thesimulation of a pressure-hole type of spray at time t* = 20with the minimum mesh size, along with details of the adap-tive mesh used by GERRIS code. The traced value is the iso-surface 0.5 for VOF tracer (0 is gas only, 1 is liquid only).Note that surface instabilities grow from inlet in a symmetricway due to meshing of the boundary conditions. They laterdevelop into ligaments that ultimately form the droplets thatdetach and sub-divide. Although this can be considered asnon-physical because dependent on meshing characteristics,surface instabilities growth are proven to be responsible foratomization. The real origin of such perturbations can beattributed to turbulence, roughness and other causes.Enhancement of such perturbations differs in a less iso-tropi-cal distribution but growth depends on flow characteristicsand relatively low on meshing. Globally, droplet and liga-ment formation is realistic. For computing time reasonsdroplets below a certain size have been removed. They canbe treated in a Lagrangian way for full-spray calculations.
t =t
d
Vo
inj
*
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Figure 16
View of a pressure-hole type spray simulated by DNS withGERRIS code (top: VOF tracer 0.5 iso-surface), details ofmesh in the cross section (bottom left) and of the squaredzone (bottom right). Minimum cell size is 1/29.
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In Figure 17, the same spray is represented at time t* = 80for a coarser mesh. Note that details on ligaments have notbeen lost although they are less refined. For values like pene-tration, the mesh change from 29 to 27 induces variations ofless than 5%, for the time span we have tested. For laterdevelopment of spray, as the liquid core vanishes, this obser-vation may not remain valid.
In Figures (18-20), we show a comparison of simulationwith images issued from experiments in similar conditions.Imaging is obtained through LIF of spray droplets with afour-hole injector (do = 0.2 mm) in a pressure vessel usingiso-octane doped with fluoranthene as described in work byLe Moyne et al. (1997). Only one of the 4 sprays is correctlyilluminated by the laser sheet intersecting it in its axis. A PIVset-up with a micro-telescope equipped camera is used.Figures 18 to 20 concern times t* at 20, 40, and 80. In theexperimental reference they correspond to 200, 400 and 800microseconds after start of injection. The start of injectiont = 0, is detected experimentally by the appearance of liquidat the nozzle hole.
Figure 21 shows a comparison of penetration for experi-mental and simulation data issued from a single simulation,showing good agreement. Low dispersion of this parameterhas been verified for a limited set of simulations. Note thateven for very early steps of spray development, penetrationfollows a power law. In this case:
(70)S t* * 0.6772∝ ( )
This is contrary to some assumptions made previouslyin literature, as no experimental data was available for sosmall times.
Simulation is carried removing all small droplets, so onlythe liquid core is represented here. Globally the generalaspects of the liquid core are well captured by the simulation.Qualitatively the length scales of ligaments and droplets cor-respond to those of experiment images. As no effectivemethod to measure droplet sizes in this very dense region ofspray is available, a quantitative comparison of droplets sizesbetween simulation and experiments is not possible. Tworepresentations of spray are shown with a cross section viewof VOF tracer value and a 3D view of iso-surface at value 0.5of VOF tracer. One can see that differences with experimentsexist although no model for atomization or turbulence isimplemented. One reason is that numerical aspects may notallow capturing the very complex interface. Another reasonmay be that boundary conditions for the simulation are notrealistic.
Indeed, a parabolic velocity profile is injected into thesimulation domain and break-up of droplets is not effectivein the simulation until surface instabilities have grownenough. One can notice from experiment images that someatomization is visible from the very nozzle exit and thatthe smooth cylinder of simulations is not apparent inexperiments.
The precocity of this atomization may be due to fastestgrowing instabilities or intra-hole phenomena. As the code is
Figure 17
Spray aspect for non-dimensional time t* = 80 (top: VOF tracer 0.5 iso-surface), corresponding mesh (middle) and detail of mesh (below).Minimum cell size is 1/27.
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incompressible, and with no phase change for the time being,it is not possible to model cavitation-like mechanisms.However, as shown in Figure 22, by solving the 3D flowinside the injector it is possible to induce early atomization,namely from t* ~ 3. This suggests that the velocity profileinduced by complex flow inside the injector with vorticitysources, induced by rugosity for example, may play a role atleast on early stages of atomization. Figure 23 shows a view
of the spray at a very early stage of injection. The injectionpressure has probably not reached its steady value as injectorneedle may still be moving. Expansion in the lateral direc-tions and the corrugated aspect of the interface suggests thathigh levels of turbulence prevail from spray inception.
Further work is needed to validate the results and assump-tions made, but as the DNS allows to capture the finestdetails of atomization it is, with the growing power ofprocessors, a possible way to explore the spray formationmechanisms.
4 DISCUSSION
All the models cited in the previous sections show all thediversity of the results concerning spray formation. As the
Figure 18
Spray image experiment and simulation at t* = 20.
20 10040 60 80
Data
Simu
0
S*
40
0
35
30
25
20
15
10
5
t*
Figure 20
Spray image experiment and simulation at t* = 80.
Figure 21
Non-dimensional penetration versus non-dimensional timefor experiments and simulation.
Figure 19
Spray image experiment and simulation at t* = 40.
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process of atomization is of singular complexity the efforts toresume to few equations the effects of some parameters onstrongly non-linear mechanisms give limited agreement withexperiments. In the sense that when applied to different con-ditions to those where model constants and hypothesis werevalidated one often obtains non physical results. Everyknown spray model has a limited range of validity as quitedifferent atomization regimes exist with a single device(Rayleigh regime, wind induced, atomization, flash boiling,effervescent atomization, etc.). Moreover, the experimentaltechniques available do not always allow access to necessarydata like drop size and velocity in dense regions of spray. Asa whole, when confronted to the need of simulating combus-tion processes in engines one often finds difficulties in choos-ing the correct spray sub-model and setting its several con-stants. Very often the final objective is completely reversedas a specific set of experiments is needed to calibrate thespray model. The semi-empirical correlations and 0D modelshave been extensively used and have proven their utility andlimits. The widely spread use of Eulerian-Lagrangian modelswith RANS equations for gas phase has given relatively lowvalid results as the constants and parameters in spray modelslack of a proven methodology for correct adjustment.Moreover, the issue of grid dependency, when addressed,asks difficult numeric questions at a certain computer cost.Finally, great hope is put on the LES and DNS simulations asthe possibilities now at range allow starting to address thedifficult points of spray modeling.
ACKNOWLEDGMENTS
Authors would like to thank Profs. Guibert and Zaleski,Danielson Engineering and Conseil Régional de Bourgognefor continuous support.
REFERENCES
Apte S.V., Gorokhovski M., Moin P. (2003) LES of AtomizingSpray with Stochastic Modeling of Secondary Breakup, Int. J.Multiphase Flow 29, 1503-1522.
Arai M., Tabata M., Hiroyasu H., Shimizu M. (1984) DisintegratingProcess and Spray Characterization of Fuel Jet Injected by a DieselNozzle, SAE International, SAE paper 840275.
Arcoumanis C., Gavaises M. (1997) Effect of Fuel InjectionProcesses on the Structure of Diesel Sprays, SAE International, SAEpaper 970799.
Arrègle J.M., Pastor J.V., Ruiz S. (1999) The Influence of InjectionParameters on Diesel Spray Characteristics, SAE International, SAEpaper 1999-01-0200.
Example of DNS simulation for inter-injector flow (insideinjector view left t* = 3 and t* = 20; and out-coming liquidright for t* = 20).
Figure 23
Early view of spray at t* = 3.
ogst100079_DosSantos 17/11/11 11:16 Page 820
F Dos Santos and L Le Moyne / Spray Atomization Models in Engine Applications, from Correlations to Direct Numerical Simulations 821
Beheshti N., Burluka A. (2004) Eulerian Modelling of Atomisationin Turbulent Flows, 19th Annual Meeting of the Institute for LiquidAtomization and Spray Systems (Europe), Nottingham, 6-8September, 207-212.
Bharadwaj N., Rutland C., Chang S. (2009) Large Eddy SimulationModelling of Spray-Induced Turbulence Effects, Int. J. Engine Res.10, 97-119.
Bianchi G. Minelli F., Scardovelli R., Zaleski S. (2007) 3D LargeScale Simulation of the High-Speed Liquid Jet Atomization, SAEInternational, SAE paper 2007-01-0244.
Chehroudi B., Chen S.-H., Bracco F.V., Onuma Y. (1985) On theIntact Core of Full-Cone Sprays, SAE International, SAE paper850126.
Cousin J., Desjonquères P. (2003) A New Approach for theApplication of the Maximum Entropy Formalism on Sprays,ICLASS 2003, Sorrento, Italy, 13-17 July.
De Villiers E., Gosman A., Weller H. (2004) Large EddySimulation of Primary Diesel Spray Atomization, SAEInternational, SAE paper 2004-01-0100.
Demoulin F.X., Beau P.A., Blokkeel G., Mura A., Borghi R. (2007)A New Model for Turbulent Flows with Large DensityFluctuations: Application to Liquid Atomization, AtomizationSprays 17, 315-345.
Dent J.C. (1971) Basis for the Comparison of Various ExperimentalMethods for Studying Spray Penetration, SAE International, SAEpaper 710571.
Elkotb M. (1982) Fuel Atomization for Spray Modelling, Progr.Energ. Combust. Sci. 8, 61-90.
Faeth G. (1996) Spray Combustion Phenomena, Symp. Int.Combust. 26, 1593-1612.
Fuster D., Bagué A., Boeck T., Moyne L.L., Leboissetier A.,Popinet S., Ray P., Scardovelli R., Zaleski S. (2009) Simulation ofPrimary Atomization With an Octree Adaptive Mesh Refinementand VOF Method, Int. J. Multiphase Flow 35, 550-565.
Heywood J. (1988) Internal Combustion Engine Fundamentals,McGraw-Hill.
Hiroyasu H., Arai M. (1990) Structures of Fuel Sprays in DieselEngines, SAE International, SAE paper 900475.
Hiroyasu H., Arai M., Tabata M. (1989) Empirical Equations for theSauter Mean Diameter of a Diesel Spray, SAE International, SAEpaper 890464.
Hiroyasu H., Kadota T., Tasaka S. (1978) Study of the Penetrationof Diesel, JSME International Journal 44, 3208-3219.
Huh K.Y., Gosman A.D. (1991) A Phenomenological Model ofDiesel Spray Atomization, Proceedings of International Conferenceon Multiphase Flows, Tsukuba, Japan, 24-27 September.
Ibrahim E.A., Yang H.Q., Przekwas A.J. (1993) Modeling of SprayDroplets Deformation and Breakup, J. Propuls. Power 9, 651-654.
Kuensberg Sarre C., Kong S.C., Reitz R.D. (1999) Modeling theEffects of Injector Nozzle Geometry on Diesel Sprays, SAEInternational, SAE paper 1999-01-0912.
Le Moyne L. (2010) Trends in atomization theory, Int. J. SprayCombustion Dynamics 2, 49-84.
Le Moyne L., Maroteaux F., Guibert P., Murat M. (1997) Modeland Measure of Flows at the Intake of Engines, J. Phys. III France7, 1927-1940.
Lebas R., Menard T., Beau P., Berlemont A., Demoulin F. (2009)Numerical simulation of primary break-up and atomization: DNSand modelling study, Int. J. Multiphase Flow 35, 247-260.
Lee K., Aalburg C., Diez F.J., Faeth G.M., Sallam K.A. (2007)Primary breakup of turbulent round liquid jets in uniform cross-flows, AIAA J. 45, 1907-1916.
Lefebvre H.A. (1989) Atomization and Sprays. Combustion: AnInternational Series, Hemisphere, New-York, 434 p.
Levich V. (1962) Physicochemical Hydrodynamics, Prentice-HallInc., pp. 639-650.
Levy N., Amara, S., Champoussin J.C. (1998) Simulation of aDiesel Jet Assumed Fully Atomized at the Nozzle Exit, SAEInternational, SAE paper 981067.
Long W., Hosoya H., Mashimo T., Kobayashi K., Obokata T.,Durst F., Xu T. (1994) Analytical Functions to Match SizeDistributions in Diesel-Sprays, International Symposium COMODIA,Yokohama, Japan, 11-14 July.
Menard T., Beau P., Tanguy S., Demoulin F., Berlemont A. (2005)Primary break-up: DNS of liquid jet to improve atomization model-ling, WIT Transactions on Engineering Sciences 55, 343-352.
Merrington A.C., Richardson E.G. (1947) The break-up of liquidjets, Proc. Phys. Soc. 59, 1.
Naber J.D., Siebers D.L. (1996) Effects of Gas Density andVaporization on Penetration and Dispersion of Diesel Sprays, SAEInternational, SAE paper 960034.
Nishimura A., Assanis D.N. (2000) A Model for Primary DieselFuel Atomization Based on Cavitation Bubble Collapse Energy,ICLASS 2000, Pasadena, CA, 16-20 July, pp. 1249-1256.
O’Rourke P.J., Amsden A.A. (1987) The Tab Method forNumerical Calculation of Spray Droplet Breakup, SAEInternational, SAE paper 872089.
Park S.W., Lee C.S., Kim H.J. (2003) Investigation of AtomizationCharacteristics and Prediction Accuracy of Hybrid Models forHigh-Speed Diesel Fuel Sprays, SAE International, SAE paper2003-01-1045.
Patterson M.A., Reitz R.D. (1998) Modeling the Effects of FuelSpray Characteristics on Diesel Engine Combustion and Emissions,SAE International, SAE paper 980131.
Peng Karrholm F., Weller H., Nordin N. (2007) Modelling injectorflow including cavitation effects for Diesel applications,Proceedings of FEDSM2007 5th Joint ASME/JSME FluidsEngineering Conference, San Diego, CA, USA, 30 July - 2 August.
Popinet S. (2003) Gerris: a tree-based adaptive solver for the incom-pressible Euler equations in complex geometries, J. Computat.Phys. 190, 572-600.
Ranz W.E. (1958) Some Experiments on Orifice Sprays, Can. J.Chem. Eng. 36, 175-181.
Reitz R. (1987) Modeling atomization processes in high pressurevaporizing sprays, Atomization Sprays 3, 309-337.
Reitz R.D., Bracco F.B. (1979) On the Dependence of Spray Angleand Other Spray Parameters on Nozzle Design and OperatingConditions, SAE International, SAE paper 790494.
Reitz R.D., Diwakar R. (1987) Structure of High-Pressure FuelSprays, SAE International, SAE paper 870598.
Ruiz F., Chigier N. (1991) Parametric Experiments on Liquid JetAtomization Spray Angle, Atomization Sprays 1, 23-45.
Sandia National Laboratories (2010) online database,http://www.sandia.gov/ecn/cvdata/dsearch.php.
Schihl P., Bryzik W., Altreya A. (1996) Analysis of Current SprayPenetration Models and Proposal of a Phenomenological ConePenetration Model, SAE International, SAE paper 960773.
ogst100079_DosSantos 17/11/11 11:16 Page 821
Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 66 (2011), No. 5822
Shannon C.E. (1948) A Mathematical Theory of Communication,Bell Syst. Tech. J. 27, 379-423, 623-656.
Siebers D.L. (1999) Scaling Liquid-Phase Fuel Penetration in DieselSprays Based on Mixing-Limited Vaporization, SAE International,SAE paper 1999-01-0528.
Su T.F., Patterson M.A., Reitz R.D., Farrell F.V. (1996)Experimental and Numerical Studies of High Pressure MultipleInjection Sprays, SAE International, SAE paper 960861.
Tanner F.X. (1997) Liquid Jet Atomization and Droplet BreakupModeling of Non-Evaporating Diesel Fuel Sprays, SAEInternational, SAE paper 970050.
Taylor G.I. (1950) The Instability of Liquid Surfaces WhenAccelerated in a Direction Perpendicular to their Planes, Proc. R.Soc. London, 192-196.
Tomar G., Fuster D., Zaleski S., Popinet S. (2010) MultiscaleSimulations of Primary Atomization, Comput. Fluids 39, 1864-1874.
Vallet A., Burluka A.A., Borghi R. (2001) Development of aEulerian Model for the “Atomization” of a Liquid Jet, Atomizationand Sprays 11, 24.
