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51-1
Efficient Numerical Calculation of Evaporating Spraysin
Combustion Chamber Flows
R. Schmehl, G. Klose, G. Maier and S. WittigLehrstuhl und
Institut für Thermische Strömungsmaschinen
Universität Karlsruhe (T.H.)76128 Karlsruhe, Germany
SummaryRepresenting two different conceptual approaches, either
Eule-rian continuum models or Lagrangian particle models are
com-monly applied for the numerical description of dispersed
twophase flows. Taking advantage of the positive features
inherentto each model, a combination approach is presented in this
studyfor the efficient computation of liquid fuel sprays in
combustorflows. In the preconditioning stage, Eulerian transport
equa-tions for gas phase and droplet phase are solved
simultaneouslyin a block-iterative scheme based on a coarse
discretization ofspray boundary conditions at the nozzle. Due to
the close cou-pling of both phases, the time expense of this
approximate flowfield computation is not much higher as for single
phase flows.In the refinement stage, Lagrangian droplet tracking is
appliedwith a detailed discretization of initial conditions. To
accountfor complete interaction between gas phase and droplets,
gasflow solution and droplet tracking are concatenated by an
iter-ative procedure. In this stage, the numerical description of
thespray is enhanced by additional modeling of droplet
breakup.Results of numerical simulations are compared with
measure-ments of the two phase flow in a premix duct of a LPP
researchcombustor.
1 Notation
cp specific heat capacityD droplet diameterD0.5 mass median
diameterD32 Sauter mean diameterD0.632 characteristic diameterf
body forceh enthalpyḢ enthalpy fluxH energy transfer rateI
momentum transfer ratek turbulent kinetic energyṁ mass fluxM mass
transfer rateOn Ohnesorge numberP pressurePr Prandtl numberQ̇
conductive heat fluxRe Reynolds numberS source termSc Schmidt
numberT temperatureTu degree of turbulence
U velocity componentWe Weber numberY mass fraction
Greek Symbolsα heat transfer coefficientαk liquid volume
fractionβ off axis angleε dissipation rate of kΓ diffusion
coefficientµ dynamic viscosityν kinematic viscosityρ densityτ shear
stress
Subscripts0 initial stateg, d gas, dropletint interfacet
turbulentvap vapor
2 IntroductionImproving modern gas turbine efficiencies by
increasing pres-sure and temperature levels of the combustion
process, essen-
tially requires sophisticated combustion concepts in order
tomeet todays strict limitations on pollutant emissions.
Funda-mental to these low emissions concepts is a characteristic
strat-egy to inject and mix the liquid fuel with the compressed
airflow, avoiding local stoichiometric combustion conditions as
faras possible. Two promising approaches in this context are
theconcepts of Lean-Premix-Prevaporize (LPP) and Rich-Quench-Lean
(RQL) combustion. In order to develop advanced com-bustor designs
with the required flow characteristics, a betterunderstanding of
the two phase flow physics is necessary. Twophase flow effects
typical for premix ducts of LPP combustorsor prefilming air blast
atomizers are summarized in Fig. 1.
EvaporationDispersion +Droplet
BreakupSpray-wallInteraction
Wall FilmFlowAtomization
Figure 1: Two phase flow effects in a LPP premix duct
Due to the enormous increase in computing performance,
Com-putational Fluid Dynamics (CFD) offers a promising potentialfor
efficient combustor design and optimization. In particularwhen
compared to experimental studies at elevated pressures,CFD analysis
may be employed to reduce turn-around timesand costs of combustor
design significantly. On the other hand,complex flow phenomena such
as turbulence, atomization orchemical reaction still represent some
of the most challengingtopics for CFD tools.Basically, two
different conceptual approaches may be em-ployed for the numerical
description of dispersed two phaseflows [3]. In analogy to single
phase gas flow, the Eulerian ap-proach is based on a continuum
model of the spray, resulting intransport equations describing the
propagation and evaporationof this droplet phase [28], [6]. In the
Lagrangian approach, thespray is modeled by superposition of
trajectories calculated forlarge numbers of representative
droplets. Each of the two basicapproaches is characterized by
specific advantages and restric-tions.In the Eulerian method, the
transport equations of the dropletphase are appended to the gas
phase transport equations, result-ing in a compact description of
the interacting two phase flowsystem. The essential advantage is a
simultaneous solution ofthe interacting flow fields of gas phase
and spray by a singlenumerical method. Applying a standard
block-iterative solverfor systems of linearized equations, the
information exchange
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51-2
between phases is realized on the level of the non-linear
itera-tions. As a consequence, computation times are generally
smallcompared to the Lagrangian approach. However, each
dropletinitial condition to be simulated requires the solution of
an in-dividual set of 6 transport equations. This conceptual
feature isa severe limitation for the discretization of complex
sprays withwide ranges of initial droplet size, injection angles
and veloci-ties. Furthermore, Eulerian methods are generally not
suited forthe modeling of complex two phase phenomena like
secondaryatomization or spray-wall interaction.In the Lagrangian
method, a large number of droplet trajecto-ries has to be tracked
to achieve a continuous distribution ofthe liquid phase. Due to the
stochastic simulation of turbulentspray dispersion by random
sampling of gas velocity fluctua-tions, identical droplet initial
conditions lead to different trajec-tories. As a consequence, the
liquid phase flow field is a statisti-cal quantity. To limit the
maximum field deviations, the numberof simulated droplet
trajectories has to be increased to values upto 104 - 106 for
typical combustor flows. Effects of the sprayback on the gas flow
such as aerodynamic dragging or evapo-ration cooling are recorded
in droplet source terms, describingthe local interfacial transfer
of mass, momentum and energy.Including these source terms in the
gas flow transport equa-tions, complete phase interaction is taken
into account by aniterative concatenation of gas flow computation
and Lagrangiandroplet tracking. Besides increased computation
times, this it-erative procedure entails an artificial decoupling
of gas flow andspray. In particular for flows with intense phase
interaction, thismay cause severe convergence problems, requiring
strong relax-ation of source term fields. Nevertheless, Lagrangian
methodsare commonly preferred for practical CFD analysis due to
sig-nificant advantages regarding complex spray discretization
andmodeling of flow phenomena such as secondary droplet breakupor
droplet-wall interaction.In this study, a new hybrid approach is
presented combiningEulerian and Lagrangian methods to take
advantage of the ca-pabilities inherent to both methods. In the
first stage of thisprocedure, the Eulerian method is used for an
efficient compu-tation of an approximate two phase flow field. A
coarse dis-cretization of spray boundary conditions at the nozzle
limitsthe size of the system of transport equations to a practical
di-mension. Good convergence rates are achieved due to the
closecoupling between gas flow and spray. In the refinement
stage,iterative cycles of single phase gas flow computation and
sub-sequent droplet tracking are employed to improve the qualityof
the preconditioned two phase flow field. Taking advantageof the
stochastical nature of the tracking approach, a fine
dis-cretization of polydisperse sprays is achieved by random
sam-pling of droplet initial conditions at the nozzle. The
numeri-cal description of the spray is enhanced by optional
modelingof secondary droplet breakup and spray-wall interaction.
Sincemodeling of spray-wall interaction has been described in
detailin Ref. [24], only secondary atomization of droplets is
consid-ered in this study.To demonstrate the performance and
accuracy of the numeri-cal methods discussed in this paper, the
evaporating spray inthe premix duct of a LPP research combustor is
simulated andassessed by measured droplet data. The experimental
investiga-tion of this premix section has been a focus of various
studies[15], [16], [13] and represents a valuable source of
experimen-tal data.
3 Eulerian approachThe Eulerian approach for the numerical
description of dis-persed two phase flows is based on the
assumption that the liq-
uid phase represents an additional continuum penetrating thegas
phase. In analogy to the continuum approach of singlephase flows,
each phase is described by a set of transport equa-tions for mass,
momentum and energy extended by interfacialexchange terms. This set
of transport equations can be recastedinto a universal formulation
which is discretized by a conser-vative Finite Volume method and
solved by a block-iterativescheme for systems of linearized
equations.
3.1 Transport equations of the two phase flow
3.1.1 Basic equations
Except for the near region of the atomizer, the volume
fractionof fuel in the flow field is low. In this dilute two phase
flowregime, interactions between fuel fragments can be
neglected.Starting from the basic Navier-Stokes equations, the
instanta-neous transport equations for gas and droplet phase are
derivedeither by spatial, temporal [9] or ensemble phase
averaging.
Gas phase:
∂
∂tαgρg +
∂
∂xjαgρgUg,j = Mint,g (1)
∂
∂tαgρgUg,i +
∂
∂xjαgρgUg,jUg,i =
−αg ∂∂xi
Pg +∂τi,j∂xj
+ αgρgfi + Iint,g,i (2)
∂
∂tαgρghg +
∂
∂xjαgρgUg,jhg =
− ∂q̇j∂xj
+ Sh,g + Hint,g (3)
Droplet phase:
∂
∂tαdρd +
∂
∂xjαdρdUd,j = Mint,d (4)
∂
∂tαdρdUd,i +
∂
∂xjαdρdUd,jUd,i =
−αd∂
∂xiPg + αdρdfi + Iint,d,i (5)
∂
∂tαdρdhd +
∂
∂xjαdρdUd,jhd = Hint,d (6)
The weighting factors αg and αd are a result of the averag-ing
process and represent the local volume fractions of gas andliquid
phases related by the following equation
αg + αd = 1 (7)
Dilute two phase flows are characterized by the conditions
αd � 1 ; αg ≈ 1 (8)
In this flow regime, the transport equations of the gas
phaseapproach the standard single phase transport equations
ex-tended by additional interfacial exchange terms Mint,g,
Iint,gand Hint,g .
3.1.2 Interfacial exchange terms
The interfacial exchange terms describe the local rates of
mass,momentum and energy transfer across the liquid-gas
interface.Assuming a spherical shape of the droplets and a uniform
inter-nal temperature distribution, the transfer rates may be
estimatedfrom Lagrangian single droplet physics. The following
modelexpressions are derived from Eq. (23) and the heat and
mass
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51-3
fluxes (28), (29) and (30) of an isolated droplet
Mint,d = −Mint,g =6αdπD3
ṁ∗vap (9)
Iint,d,i = −Iint,g,i =6αdπD3
�π
8D2ρgCD |Ug − Ud|
(Ug,i − Ud,i) + ṁ∗vap Ud,i � (10)Hint,d = −Hint,g =
6αdπD3
�Ḣ∗vap,s + Q̇
∗
cond,s � (11)3.1.3 Transport equation of the droplet
diameter
According to Ref. [6], a transport equation of the mean
dropletdiameter can be established by combining transport
equationsof droplet number and droplet mass (4), giving
∂
∂tαdρdD +
∂
∂xjαdρdUd,jD =
8 αdπD2
ṁvap (12)
For non evaporating sprays this equation is identical to the
con-tinuity equation.
3.1.4 Turbulence modeling
The transport equations derived so far are suited for the
numer-ical description of sprays in laminar gas flows. Since
combus-tors generally operate in the turbulent flow regime, the
systemof transport equations (1) - (6), (12) is extended by
introduc-ing turbulent fluctuations of the transport quantities
followedby Reynolds averaging of the equations. With respect to the
gasphase, the standard k-ε model is employed to model the
trans-port terms resulting from correlations of fluctuating
quantities.This procedure has been described in detail by several
authors[12], [21]. The turbulence terms in the droplet phase
trans-port equations are approximated by an algebraic model whichis
based on a Bousinesq approach.
∂
∂tαdρd +
∂
∂xjαdρdUd,j =
∂
∂xj
�µt,dSct,d
αd∂xj � + Mint,d (13)
∂
∂tαdρdUd,i +
∂
∂xjαdρdUd,jUd,i =
∂
∂xj � αdµt,d � ∂Ud,i∂xj + ∂Ud,j∂xi ��� +αdρdfi − αd
∂
∂xiPg + Iint,d,i (14)
∂
∂tαdρdhd +
∂
∂xjαdρdUd,jhd =
∂
∂xj
�αd
µt,dPrt,d
∂hd∂xj � + Hint,d (15)
∂
∂tαdρdD +
∂
∂xjαdρdUd,jD =
8 αdπD2
ṁvap +
∂
∂xj
�αd
µt,dPrt,d
∂D
∂xj � (16)Double and triple correlations involving fluctuations
of αd areneglected on the right hand side of Eq. (14). A value of
0.9for the turbulent Schmidt and Prandtl numbers, Sct,d, Prt,d
ischosen for the present calculations. Using this value, Eq.
(13)effectively is a transport equation of a scalar in the
asymptoticalcase of a vanishing droplet diameter. A fundamental
assumptionof this approach is the dependence of the turbulent
viscosity ofthe droplet phase µt,d on local mean flow properties
[17], [10]
νt,dνt,g
=µt,dµt,g
ρgρd
=1
1 +
�tdtg � 2 n (17)
Thus, the ratio of the kinematic viscosities of droplet and
gasphase is postulated to be a function of the characteristic
timescales of both phases. In Ref. [10], a value of 0.25 for
theempirical parameter n is suggested. The time scale td which
isdenoted as droplet relaxation time characterizes the ability of
adroplet to follow turbulent gas velocity fluctuations:
td =4
3
ρdρg
D2
CD Red νg. (18)
Originally, the characteristic time of the gas flow turbulence
tgis taken to be the dissipation time scale given by Eq. 25. Inthis
formulation, the droplet phase turbulence model fails todescribe
the crossing trajectory effect which has a significantinfluence on
turbulent droplet dispersion [27]. According tothe turbulence
modeling of the Lagrangian approach, the ex-tended version of the
model considers a second characteristictime scale. This crossing
time tc is the time required by adroplet to cross the current
coherent turbulence structure whichis estimated from Eq. 26.
Combining both time scales, the gasphase time scale is now defined
as
tg = min[te, tc] (19)
The validation of this enhanced turbulence model is based onthe
basic experiment described in Ref. [27].
3.2 Discretization of polydisperse spraysTo complete the
numerical description of the spray, boundaryconditions of the
droplet phase have to be specified at the atom-izer nozzle.
However, most sprays of technical importance arecharacterized by a
broad variety of initial droplet diameters andvelocities. Since
each individual droplet phase boundary condi-tion theoretically
requires the numerical solution of a separateset of transport
equations (13) - (16), a polydisperse spray hasto be discretized by
a limited number of representative dropletclasses. In practice, the
computational effort increases at leastlinearly with the number of
droplet classes employed. As a con-sequence, the maximum number of
classes is restricted by theCPU time and memory capacity
available.For dilute sprays, direct interaction between droplet
classes canbe neglected although each class is coupled to the gas
phase bythe closure equation
Mint,g = −nc�
k=1
Mint,d,k (20)
Iint,g = −nc�
k=1
Iint,d,k (21)
Hint,g = −nc�
k=1
Hint,d,k (22)
4 Lagrangian approach4.1 Spray dispersionThe Lagrangian
simulation of dispersed two phase flow is basedon the tracking of
statistically significant droplet parcels in thegas flow. Each
parcel is represented by one droplet and is deter-mined by
discretization of the continuous spectra of droplet ini-tial
conditions in the near field of the atomizer. The tracking isbased
on the integration of the droplets equation of motion com-bined
with an empirical correlation for the aerodynamic dragcoefficient
CD,
d~uddt
= −34
ρgρd
CDD
|~ud − ~ug| (~ud − ~ug) (23)
CD = 0.36 + 5.48Re−0.573d +
24
Red(24)
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51-4
In order to simulate the effect of turbulent spray dispersion,
theturbulence structure of the gas flow field is modeled by a
ran-dom process along the droplet trajectories [5], [18]. In this
con-cept, the local turbulence structure is characterized by the
lengthscale le and dissipation time scale te of eddies representing
thecoherent flow structures
le = C1
2µ
k3
2
ε, (25)
In addition to the life time scale te, a crossing time scale tc
iscalculated from ���� ���� � tc
t0
(~ug − ~ud)dt���� ���� = le, (26)
taking into account the droplet dynamics. Each time the
smallerone of both time scales is elapsed, the droplet enters a
neweddy. Consequently, the random process generates a new ve-locity
fluctuation ~u′g from a Gaussian distribution which is de-termined
by
µ = 0 , σ = � 23k. (27)
This velocity fluctuation remains constant for the period
ofdroplet-eddy interaction and is added to the local value of
thegas flow velocity.
4.2 Spray evaporationIn this study, droplet evaporation is
simulated by means of theUniform Temperature model [4], [26], [2].
This computation-ally effective droplet model is based on the
assumption of ahomogeneous internal temperature distribution in the
dropletand phase equilibrium conditions at the surface. The
analyti-cal derivation of this model does not consider
contributions toheat and mass transport by forced convection by the
gas flowaround the droplet. Since diffusive time scales in the
surround-ing gas phase are much smaller than in the droplet fluid,
a quasistationary description of the gas phase is applied. Using
refer-ence values for variable fluid properties (1/3-rule), an
integra-tion of the radially symmetric differential equations
yields an-alytical expressions for the transport fluxes ṁvap,
Q̇cond,s andḢvap,s. At this point, convective transport is taken
into accountby two empirical factors [1] resulting in the corrected
fluxesṁ∗vap, Q̇
∗
cond,s and Ḣ∗
vap,s [23]
ṁ∗vap = cfm ṁvap, (28)
Q̇∗cond,s = πD2 α∗(Td − Tg), (29)
Ḣ∗vap,s = ṁ∗
vap cp,vap,ref (Td − Tg). (30)Vapor mass flux and heat transfer
coefficient are calculated asfollows
ṁvap = 2πD ρg,refΓim,ref ln1 − Yvap,g1 − Yvap,s
, (31)
α∗ = cfh
ṁvap cp,vap,refπD2
exp
�ṁvap cp,vap,ref
2πD λg,ref � − 1 , (32)cfm = 1 + 0.276 Re
1
2 Sc1
3 , (33)
cfh = 1 + 0.276 Re1
2 Pr1
3 . (34)
The balance equations of the droplet reduce to ordinary
differ-ential equations,
d
dtmd = −ṁ∗vap, (35)
d
dt(md hd) = −Q̇∗cond,s − Ḣ∗vap,s, (36)
which can be appended to the differential equations
describingthe droplet motion, Eq. 23 .
4.3 Secondary droplet breakupAt low relative velocities, the
spherical shape of the droplets ispreserved by the dominating
effects of surface tension and vis-cous forces in the liquid. With
increasing velocities, the desta-bilizing aerodynamic forces on the
droplet surface are intensi-fying, resulting in deformation,
oscillations and disintegrationof the droplets.
4.3.1 Classification of breakup mechanisms
A common practice to classify secondary droplet
atomizationprocesses is based on two characteristic groups of
parameters,
We =ρg u
2rel D
σd, On =
µd√ρd D σd
. (37)
The Weber number is a measure of the strength of
aerodynamicforces relative to surface tension forces, whereas the
Ohnesorgenumber assesses the damping effect of viscous friction in
thedroplet against surface tension. In the Weber number rangefrom
We = 1 up to a critical value We = Wec, non-destructivedroplet
deformation and oscillation is observed. As illustratedin Fig. 2,
three different mechanisms govern the breakup ofdroplets for
increased Weber numbers typical for flows in gasturbine combustors.
From these visualizations it is obvious that
We=70
ShearBreakup
We=20
MultimodeBreakup
We=10
BagBreakup
Figure 2: Breakup mechanisms of water droplets (Ref. [22])
a common feature of all three mechanisms is an initial
deforma-tion of the droplet into a disc shape. After this
deformation pe-riod, various complex flow phenomena lead to the
final dropletbreakup depending on the intensity of the aerodynamic
forces.Exceeding the critical Weber number, the first mechanism
ob-served is bag breakup. This process is characterized by the
for-mation of a thin hollow bag of droplet fluid stretching from
atoroidal rim. The thin film of this bag is eventually burstinginto
a cloud of tiny droplets, followed by a disintegration of
thetoroidal rim into significantly larger fragments. With
increasingaerodynamic forces, a transition to more complex bag
struc-tures is observed. In this multimode or stamen breakup
regimethe aerodynamic flow interaction is forming an additional
fluidfilament in the center of the bag structures which is
alignedwith the relative flow velocity. For even higher Weber
numbers,shear breakup is observed. This mechanism is
fundamentallydifferent to the preceding mechanisms and is
characterized by arapidly disintegrating film of fluid continuously
stripped off therim of the disc shaped droplet by shear forces.Fig.
3 summarizes the results of various experimental stud-ies [11],
[19] in a breakup regime map, indicating the relevantmechanism
corresponding to a specific combination of Ohne-sorge and Weber
number. For On > 0.1 a significant influenceof viscosity is
observed. The transitions between the differentmechanisms are given
by the following functions
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51-5
10-3 10-2 10-1 100 101
On
10
20
30
40
50
We
Deformation
BagBreakup
MultimodeBreakup
ShearBreakup
Figure 3: Breakup regime map (◦ : Present flow simulation)
• Critical Weber number (Transition to bag breakup):
Wec = 12 (1 + 1.077 On1.6) (38)
• Transition to multimode breakup:
We = 20 (1 + 1.200 On1.5) (39)
• Transition to shear breakup:
We = 32 (1 + 1.500 On1.4) (40)
4.3.2 Deformation and breakup times
Basically, the breakup process can be subdivided in two
stages:Initial deformation and further deformation with
disintegration.It is convenient to express the relevant times of
the breakup pro-cess in terms of the characteristic time of shear
breakup
t∗ =D0urel
� ρdρg . (41)As stated in Ref. [8], the time of initial droplet
deformationtdef has a constant value independent of the specific
breakupmechanism
tdeft∗
= 1.6. (42)
However, the breakup time tb measured from begin of defor-mation
until final destruction strongly depends on the specificmechanism.
A fit to a large number of experimental data isgiven in Ref.
[19]
tbt∗
=
������ �����6 (We − 12)−0.25 12 < We < 18
2.45 (We − 12)0.25 18 < We < 4514.1 (We − 12)−0.25 45 <
We < 351
0.766 (We − 12)0.25 351 < We < 26705.5 2670 < We.
(43)
For On > 1, liquid viscosity is the dominating parameter of
thebreakup process resulting in the following correlation
tbt∗
= 4.5 (1 + 1.2 On0.74). (44)
4.3.3 Droplet drag
Deformation of the droplet prior to breakup leads to a
significantincrease of aerodynamic drag. Due to the resulting
acceleration,the droplet generally experiences substantial
displacement frominitial deformation until final breakup. With
respect to the de-formation period, several authors [8] report a
linear increase
of the droplet size from D0 up to a maximum value given by(On
< 0.1, We < 100)
DmaxD0
= 1 + 0.19√
We. (45)
The effect of higher Ohnesorge numbers is taken into accountby
using a corrected Weber number in the above and
followingequations.
Wecorr =We
1 + 1.077 On1.6. (46)
As suggested in Ref. [8], a linear transition of the drag
coeffi-cient from the sphere shape value to the disc shape value is
usedin the present study to model the aerodynamic properties of
theflattening droplet.In the following period of disintegration,
droplet drag dependson the specific breakup mechanism. As
illustrated in Fig. 3, thebag and filament structures observed
prior to breakup are verycomplex. According to Refs. [11], [22] the
toroidal rim evolv-ing in bag breakup is expanding to seven times
the initial dropletdiameter, whereas in multimode breakup a maximum
diameterof six times the initial diameter is reached (see Fig. 6).
Atthis time however, a major part of the droplet cross section
con-sists of a thin fluid film accelerated in direction of the
relativevelocity thus decreasing the aerodynamic drag. To bypass
thedifficulties of describing these opposing effects, the drag of
thedisintegrating droplet is calculated from the constant disc
statereached at the end of the deformation period. In shear
breakup,the size of the disc shaped droplet is continuously
decreasing toits final value at tb.
4.3.4 Secondary droplet sizes
Based on an extended experimental study covering the com-plete
range of breakup mechanisms, a single correlation for theSauter
mean diameter D32 has been derived in Ref. [8] for allthree
mechanisms (On < 0.1, We < 1000)
D32D0
= 6.2 On0.5 We−0.25 (47)
The exponents in this correlation have been determined by
theauthors from an approximate analysis of the droplet internalflow
during shear breakup, leaving only a constant factor as aparameter
for the fitting to experimental data. Using the Webernumber given
by Eq. 46 to account for viscosity effects, addi-tional fitting of
the exponents leads to an improved correlation
D32D0
= 1.5 On0.2 We−0.25corr . (48)
This correlation which is used for the flow simulations in
thepresent study and the experimental data is shown in Fig.
4.Although the above correlation is valid for the complete
range
0.1 0.2 0.3 0.4On0.2Wecorr
-0.25
0.1
0.2
0.3
0.4
0.5
D32
/D0
WaterGlycerol 42%Glycerol 63%n-HeptaneEthyl Alcohol1.5 On0.2
Wecorr
-0.25
Figure 4: Improved correlation for D32 (Data from Ref. [8])
of breakup conditions, the distribution function of the
droplet
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51-6
diameter is substantially different for the various mechanismsof
secondary breakup.Bag and multimode breakupConsidering bag or
multimode breakup, the volume distributionof the droplet fragments
is approximated by a root normal dis-tribution [25], given by the
following density function
f(D) =x
2√
2π σ Dexp � −1
2 � x − µσ � 2 � , (49)and the parameters
x = � DD0.5
, µ = 1, σ = 0.238 (50)
For a volume distribution with this distribution function,
themass median diameter D0.5 is related to the Sauter mean
di-ameter D32 by
D0.5D32
= 1.2 (51)
Shear breakupAccording to Ref. [8], the volume distribution
resulting fromshear breakup is characterized by a bimodal density
functionwith a maximum at small diameters and a second maximum
atlarge diameters. From the experimental data illustrated in Fig.5,
it is concluded that the fine fraction of the droplet
fragmentscorresponds to approximately 80% of the total cumulated
vol-ume. These droplets result from film fragments stripped off
the
0 1 2D/D0.5
0
20
40
60
80
100
Cum
ula
tive
Vo
lum
e%
distributiondistribution
distribution
ExperimentRoot normal
Figure 5: Cumulative droplet volume (Data from Ref. [8])
disc-shaped droplet by shear forces. The 20% in the large
di-ameter range not specified in Fig. 5 represent the
contributionof the core droplet fragment left by the stripping
process. Thediameter Dc of this core droplet is estimated from the
criticalWeber number, evaluating Eq. (38) at flow conditions at
theinstant of breakup. As illustrated by the curve in Fig. 5,
thevolume distribution of the fine fraction of the droplet
spectrumafter shear breakup can be approximated by a root normal
dis-tribution based on a reduced Sauter mean diameter
D32,red =4 D32 Dc
5 Dc − D32. (52)
In this equation, the Sauter mean diameter of the
completedroplet spectrum is evaluated from Eq. 48.
4.3.5 Secondary droplet velocities
Due to the dominating influence of aerodynamic forces on
smalldroplets, tiny breakup products are immediately dragged
withthe gas flow. So, an accurate modeling of initial velocities is
notnecessary in general. These considerations apply in particularto
the tiny droplet fragments generated by bursting of film bagsor by
shear induced film stripping. The motion of large droplets
in turn is dominated by inertia forces. As a consequence,
themodeling of initial velocities of large droplet fragments has
asignificant influence on the dispersion behavior of the spray.As a
first approximation, the fragments generated by dropletbreakup
inherit the velocity of the parent droplet due to momen-tum
conservation. In bag or multimode breakup, a transversevelocity
component of droplet fragments is observed inducedby the transverse
spreading motion of droplet fluid during theexpansion of the
toroidal rim. This transverse velocity compo-nent is responsible
for increased dispersion of sprays with sec-ondary atomization. For
an approximate estimation, the growthvelocity of the rim is
determined from time-resolved visualiza-tions of breakup processes.
Fig. 6 indicates that the ring has
0 0.5 1 1.5t/tb
0
2
4
6
8
10
Dr,
max
/D0
We=20
We=10We=15
D
r,maxD
r,max
Figure 6: Growth of the toroidal rim (Water droplet, [22])
an extension of about seven times the original droplet
diameterin bag breakup against six times in multimode breakup.
Theseobservations agree with the values reported in Ref. [11].
Basedon these results, the transverse velocity component is
estimatedas
vt =Dr,max − D02(tb − tdef )
. (53)
With respect to multimode breakup, a fraction of the
dropletfluid is concentrating on the axis of the disintegrating
droplet(see Fig. 2, We = 20). Due to the alignment of this
prolatefilament with the flow, the fragments of this structure have
notransverse velocity component. The volume fraction of the
fil-ament is estimated from a Weber number based
interpolationbetween the limiting values of 0% for bag breakup and
20%(core droplet) for shear breakup.
4.3.6 Stochastical simulation of droplet breakup
The period of droplet disintegration is specified by the
char-acteristic times tdef , and tb of the breakup process. In
shearbreakup mode, the secondary droplets are continuously
gener-ated in the time from tdef to tb, whereas in bag or
multimodebreakup mode significant generation of fragments is
observedduring the second half of this time period [22]. Instead of
focus-ing on a realistic simulation of each individual breakup
event,the computational implementation makes use of the
Lagrangiantrajectory superposition approach involving large numbers
ofdroplet parcels.During droplet deformation, the cross sectional
area of thedroplet and the drag coefficient are continuously
increased upto their maximum values at tdef . A certain time later,
theparent droplet trajectory is terminated and a fixed number
ofchild droplets is generated by random sampling of initial
con-ditions. Each secondary trajectory is assigned an equal
fractionof the volume flux. To limit the number of secondary
dropletsto be tracked, only 3 child droplets have been modeled
perbreakup event in the present flow simulation in which
10000parent droplets are injected per Lagrangian step. In
analogy
-
51-7
to the stochastical modeling of fuel atomization and gas
flowturbulence effects, the superposition of large numbers of
trajec-tories leads to continuous and thus realistic representation
ofsecondary atomization.Modeling a single shear breakup event, the
time of droplet dis-integration is determined as a random number
with uniformdistribution between tdef and tb. To model bag and
multi-mode breakup events, the time of disintegration is sampled
inthe second half of this interval. The initial size of the
childdroplets is determined as a random number with a root nor-mal
distribution or from stability criteria (core droplet in
shearbreakup). Droplets generated by rim fragmentation are
pro-vided with an additional transverse velocity component. In
bagbreakup mode, virtually all larger fragments possess a
trans-verse velocity component whereas in multimode breakup a
vol-ume fraction of up to 20% of the largest fragments is
startingwithout dispersive transverse momentum. The limiting value
of20% represents the transition to shear breakup and correspondsto
the volume fraction of the core droplet. In this breakupregime all
secondary droplets inherit the velocity of the parentdroplet since
no significant spreading motion is observed.
1mm
Formation of Bag
Breakup
Injection
Beginning Deformation
Figure 7: Breakup of a 60 µm droplet
Fig. 7 illustrates the computation of a deforming and
disin-tegrating droplet in the near field of the nozzle. Compared
tothe trajectory of a rigid spherical droplet (dashed line), a
sig-nificant deflection of the deforming droplet due to
increasedaerodynamic forces is observed. This single droplet
computa-tion clearly demonstrates that advanced modeling of
secondarybreakup has to take into account the time scales of the
breakupprocess since the droplets experience considerable
displace-ments before their final disintegration.
4.4 Iterative solution procedureTracking of a statistically
significant number of droplet parcelsand superposition of their
trajectories yields a flow field ap-proximation of the dispersed
liquid phase. However, a simpleLagrangian two step calculation
consisting of a gas flow com-putation and subsequent droplet
tracking does not take accountof spray effects on the gas flow such
as aerodynamic draggingor evaporation cooling. In particular for
evaporating sprays incombustor flows, these effects have a
significant influence onthe overall two phase flow field. To
establish mutual informa-tion exchange between both phases,
Lagrangian two step cyclesare concatenated in an iterative
procedure with droplet sourceterms being updated during each
tracking step. Representinglocal transfer rates of mass, momentum
and energy from sprayto gas flow, droplet source terms are included
in the gas flowcomputation of the following iteration cycle. This
iterative ap-proach is illustrated in the lower part of Fig. 8.The
separated computation of gas and liquid phase flow fieldsand the
iterative exchange of interfacial transfer data entails
anartificial decoupling of both phases. In particular for two
phaseflows with intense phase interaction, this effect leads to a
crit-ical overestimation of droplet source terms in the first
iteration
cycles. To achieve convergence of the iterative procedure,
arelaxation of the droplet source term fields is employed.
Recur-sive damping of the source terms on the level of the
iterationcycles is realized by the following equation
Si+1
d,φ = αφSid,φ + (1 − αφ)S
i
d,φ . (54)
According to Eq. 54, only a fraction of the source termsrecorded
during the previous droplet tracking, Sid,φ, is con-
tributing to the source term Si+1
d,φ included in the current gas gasflow solution. The second
contribution is calculated from thesource term included in the
previous gas flow computation. Forgas flows which are substantially
influenced by the fuel spray,strong relaxation of droplet source
terms may be necessary toenforce convergence of the iterative
procedure. In these cases anincreased number of two stage
iterations may be necessary forcomplete consideration of phase
interaction [24]. In the presentflow simulation, only weak
relaxation (αφ > 0.5) is required toachieve a convergent
solution for the two phase flow field within10 two stage
iterations.
5 The Hybrid procedureAs indicated in the preceding sections,
the continuum descrip-tion of the Eulerian method has the advantage
of close cou-pling of gas and liquid phase in a single numerical
scheme.Thus, computation times are rather short as long as the
num-ber of droplet classes used for the discretization of the spray
issmall. Consequently, Eulerian methods are particularly suitedfor
an approximate but efficient computation of polydispersetwo phase
flows in combustors. Lagrangian methods in turnachieve a high
resolution discretization of complex spray struc-tures by tracking
large numbers of droplet parcels of variousinitial sizes and
velocities. However, the price to be payed for
SourcesDroplet
Gas Phase
Trajectories
FieldFlow
Droplet SourcesFlow Field,
Droplet PhaseGas Phase
Lagrangian Method
Eulerian Method
Figure 8: Structure of the Hybrid procedure
a realistic spray representation is high. Due to the artificial
de-
-
51-8
coupling of the two phase flow computation by separate solu-tion
schemes for each phase and iterative realization of
phaseinteraction, total computation times are rather large [24].
Forflow cases where strong relaxation of droplet source terms
isrequired, time expenses can grow to practically
unmanagableextents.For such two phase flows, a reduction of
computational effort isachieved by preconditioning the two phase
flow field by meansof an Eulerian method based on a coarse
discretization of thespray. This is in fact the basic idea of the
Hybrid procedure: Atwo stage combination of both methods in order
to reduce totalcomputation times by Eulerian preconditioning yet
maintainingthe detailed modeling of spray physics in a Lagrangian
refine-ment stage. This approach is schematically illustrated in
Fig. 8.Following an approximate computation of the two phase
flow,the flow field and the droplet source terms are passed to the
re-finement stage. Here, Lagrangian iteration cycles are based ona
fine discretization of droplet injection conditions and an
ad-vanced modeling of secondary droplet breakup. Since the twophase
flow field calculated by the Eulerian method already ac-counts for
spray effects on the gas flow, the droplet source termsrecorded
during subsequent tracking steps are rather close to thefinal flow
result. Consequently, the number of iterations is sig-nificantly
reduced compared to a standard Lagrangian simula-tion.
6 Simulation of a LPP premix duct flowThe performance and
accuracy of the numerical methods pre-sented is demonstrated by a
simulation of the two phase flowin the premix duct of a LPP
combustor. The experimental in-vestigation of this combustor has
been part of an extended re-search project on low emission
combustion concepts [15], [16],[13]. A detailed description of the
test rig and the measurementtechniques is presented in a parallel
study [20]. The combustorsection of interest for the present flow
simulation is illustratedin Fig. 9. Compressed air is supplied to
the cylindrical duct
Figure 9: Premix zone of the LPP research combustor
(l = 124mm, di = 44.6mm) by two coaxial annular ducts.The fuel
is injected into the gas flow by a pressure swirl atom-izer aligned
with the duct axis. The nozzle diameter is about1 mm. In order to
perform PDPA measurements, optical ac-cess to the flow is given by
circumferencial slits in the ductliner at various axial positions.
For spray visualizations, themetallic liner is substituted by a
quartz glass cylinder. Premixand reaction zones are separated by an
arrangement of swirlervanes acting as a flame stabilizer. The
complete configurationillustrated in Fig. 9 is enclosed in a
pressure casing with wa-ter cooled window ports. The outer coaxial
annular air flow(bypass air) is shielding the duct liner from
droplet impact andfilm formation. The highly accelerated inner flow
(atomizationair) is focused directly onto the conical fuel sheet
generated bythe pressure swirl atomizer. The fundamental idea of
this atom-ization concept is a further reduction of droplet sizes
by high
velocity aerodynamic interaction between spray and gas flow.The
operating point of the premix duct investigated in this studyis
specified by the flow parameters summarized in Table 1. The
Gas Flow (Air) Fuel (Diesel)
ṁg 213 g/s ṁfuel 6 g/s
Tg 753 K Tfuel 350 K
pg 4 bar vsheet 30 m/s
Tug 15 % βsheet 40◦
Table 1: Parameters at the inlet of the premix duct (z =
0mm)computational domain of the flow simulation is illustrated
inFig. 10 including (from left to right) intake section, coaxial
an-nular ducts, premix zone, swirler vanes, reaction zone,
dilutionholes and burnout zone. The axial coordinate is measured
from
Premix
zone
Figure 10: Computational domain (30◦-Segment)
the atomizer nozzle. To model the evaporation behavior of
thediesel spray, tetradecane is used as a single component
dieselsubstitute in the present flow simulation. A detailed
descrip-tion of the calculated non-reacting single phase gas flow
in theintake and premix zone is given in Ref. [20].
6.1 Discretization of the sprayIn order to derive droplet phase
boundary conditions for the Eu-lerian method and droplet initial
conditions for the Lagrangianmethod, the spray is visualized in the
near field of the atomizer(0 < z < 10 mm). A side view on the
three dimensional spraycone is given by the flashlight shadowgraph
in Fig. 11(a). Thepicture was taken under atmospheric, cold
conditions withoutbypass air flow, using water as a fuel
substitute. It is evidentthat the outer region of the spray is
dominated by larger fuelfragments. To get an impression of the
spray structure inside
Figure 11: (a) Flashlight shadowgraphy, (b) Laser light
sheet
the cone, a laser light sheet photograph is shown in Fig.
11(b),which was taken under real operation conditions of the
combus-tor. In contrast to the coarse structure of the outer spray
region,this cross view reveals a very fine droplet distribution in
thespray cone.
-
51-9
The spray visualizations indicate two basic processes that
gov-ern the atomization of the liquid fuel [15]. Due to its high
swirl,the fuel is leaving the nozzle as a conical sheet. According
toFig. 11(a), this sheet is completely disintegrating along a
dis-tance of 1 to 2 mm. In this immediate near zone of the noz-zle,
prompt atomization is the governing process. This mecha-nism is
controlled mainly by the internal sheet dynamics [14].Further
disintegration of sheet fragments is induced by aerody-namic
interaction with the high velocity gas flow which pene-trates the
spray. The resulting small secondary fragments aredragged by the
gas flow into the core flow as indicated in Fig.11(b).With respect
to the numerical simulation, it is obvious that arepresentation of
the complex two phase flow in the nozzle nearfield by
non-interacting droplets is a rather crude approximationof the
physical reality. But despite of this simplified modeling offuel
atomization, the simulation of spray dispersion and evapo-ration in
the premix duct agrees well with the experimental data.In the
following sections, two strategies are presented to derivedroplet
injection conditions at the nozzle. The Eulerian calcu-lation is
essentially based on droplet data measured at a down-stream
position of the atomizer. It is important, that
secondaryatomization has ceased completely and droplets are
spherical atthis position. Basically, the procedure starts from an
assumeddiscretization of the injection conditions of the droplet
phase. Insubsequent optimization iterations, the computed droplet
datais fitted to the corresponding measured data. In contrast to
thisapproach, the Lagrangian calculation is based on a rather
crudeapproximation of the fuel sheet disintegration in the nozzle
nearfield. Here, a far more physical description of the spray in
thesecondary atomization region is achieved by modeling
dropletdeformation and breakup. A similar strategy is described in
Ref.[7] where pressure swirl fuel injection into a diesel engine
com-bustion chamber is discussed.
6.1.1 Eulerian method, droplet phase boundary conditions
Since secondary droplet breakup is observed in a spray regionup
to 50 mm downstream of the atomizer, measured dropletdata at z = 55
mm is used for an optimization of droplet phaseboundary conditions.
The droplet size distribution of the sprayis discretized by means
of 3 diameter classes. The volume fluxfraction (or normalized
volume flux) of each class is describedby a Rosin-Rammler
distribution evaluated at the representativeclass diameter Di
V̇i
V̇= 1 − exp � − � DiD0.632 � n � , D0.632 = D0.50.693 1n .
(55)
Values of D0.5 = 46µm for the mass median diameter andn = 4.8
for the spreading parameter are determined as a goodfit to the
reference droplet data at z = 55 mm. The mean injec-tion velocity
of the droplet phase of 30 m/s is derived from themean velocity of
the liquid sheet. The remaining parameterswith significant
influence on the spray structure are the injec-tion angles of the
droplet phases. Introducing 3 angle classesper diameter class leads
to a final discretization of the spray bymeans of 9 droplet phases
with different boundary conditionsat the nozzle. Table 2 summarizes
the correlations betweendroplet phase diameter, normalized volume
flux and injectionangle employed for the Eulerian two phase flow
simulation inthe present study.
6.1.2 Lagrangian method, droplet initial conditions
In the Eulerian calculation, the boundary conditions of
thedroplet phase at the nozzle have to account implicitly for
sec-ondary atomization in an extended downstream flow region.
���[ ��� ] V̇i/V̇ [1] βi [◦]
0 - 41.2 4/15 341/30 71/30 21
41.2-50.7 4/15 301/30 101/30 20
50.7-∞ 4/15 271/30 13.51/30 20.5
v f3 0 m / s
Table 2: Droplet phase boundary conditions
It is evident that small secondary droplets originating
frombreakup of larger sheet fragments in outer flow regions may
notbe reproduced by a spray representation as described in the
pre-vious section. The approximation by rigid spherical particlesof
different size injected into the contracting, high velocity gasflow
leads to a separation of droplet sizes. As a consequence,tiny and
small droplets are captured in the axis region of thecore flow
unless they are injected with unphysically large radialvelocity
components.In the Lagrangian calculation, secondary breakup of
dropletsis taken into account during trajectory integration. Thus,
onlyprompt atomization of the conical sheet has to be considered
forthe formulation of droplet initial conditions. The basic idea is
toinject most of the fuel in form of large droplets with sizes
simi-lar to the characteristic sheet thickness of about 100 to 200
µm.Due to this coarse primary spray structure and the high
rela-tive velocities in the nozzle near zone, the critical
conditionsof droplet breakup are significantly exceeded as
indicated bythe data points mapped in Fig. 3. Modeling of delayed
dropletdeformation, drag increase and breakup results in a fine
sec-ondary spray contribution to the core flow region in the
premixduct. Very good agreement to the experimental droplet data
is
Droplet diameter:
• 10 size classes equally spaced from 0 to 200 µm• Rosin-Rammler
distribution of V̇i/V̇• D0.5 = 88.5 µm, n = 3
Droplet velocities:
• Sampled, Gaussian distribution• v = 30 m/s, σv = 5 m/s
Injection angle:
• Sampled, Gaussian distribution• β = 40◦, σβ,i = 30◦, . . . ,
5◦
• βmax = 45◦ (clipping value)
������������������������������
������������������������������
βsheet
������
�
�
�
�
�
�
vsheet
������������������������������������
Table 3: Droplet initial conditions
achieved by using the spray discretization summarized in Table3.
Droplet velocity and injection angle are sampled as randomnumbers
with Gaussian distributions. From Fig. 11(b) it is ob-vious that
the disintegration of the conical sheet is responsiblefor a fine
primary contribution to the spray in the core flow. Tomodel this
effect approximately, the variance σβ,i of the injec-tion angle β
is correlated with droplet size resulting in valuesof 30◦ for the
smallest droplets (D from 0 to 20µm) up to 5◦
for the largest droplets (D from 180µm to 200µm)
-
51-10
6.2 Results
To illustrate the calculated two phase flow in the premix
duct,contour plots of the Eulerian flow simulation are discussed
first.Comparing the axial gas velocities of the single phase and
thetwo phase flow calculations from Fig. 12 indicates that
spray-induced deceleration of the gas flow is limited to the core
flowof the duct. In particular in the nozzle near zone, the axial
gasvelocity is decreased by up to 40 m/s due to the
aerodynamicacceleration of the droplet phase.The influence of fuel
evaporation is indicated by the gas flowtemperature and fuel vapor
concentration in Figs. 13 and 14.Although the gas phase experiences
a substantial temperaturedrop across the whole core flow region,
significant concentra-tions of fuel vapor are not calculated in the
first half of theduct. This delay in vapor generation is a
consequence of lowevaporation rates in the transient heating phase
of the droplets.This conclusion is confirmed by an analysis of
Lagrangian sin-gle droplet computations, which indicate that heatup
and prop-agation time scales of typical droplets are of the same
order.In total, 28% of the injected fuel is evaporated in the
Euleriansimulation, in contrast to a value of 42% in the Lagrangian
sim-ulation. The difference is caused by the secondary
atomizationmodeling in the tracking algorithm, resulting in
considerablenumbers of small, rapidly evaporating droplet
fragments. Atthis point it should be noted that the total fraction
of evaporatedfuel substantially depends on the D32-correlation used
for thesecondary breakup modeling. Summarized over all
calculatedbreakup events, Eq. 47, which is actually not used, leads
to afragment mass median diameter of D0.5 = 53µm, whereas Eq.48
results in a value of D0.5 = 38µm.To compare the calculated axial
volume flux density αd Ud withPDPA measurements, it is weighted by
the annular area andnormalized by the total axial volume flux
V̇r
V̇=
2πr+0.5mm�
r−0.5mm
αd Ud r dr
2π24mm�
0
αd Ud r dr
. (56)
This normalized volume flux is illustrated in Fig. 15 and
repre-sents the fraction of the total liquid volume flux which
passes anannular fraction of the duct cross section. The contour
plot in-cludes mean trajectories determined by integration of the
meanaxial velocity of the droplet phase. The trajectories are
evalu-ated for the two limiting size classes and clearly
demonstratethe influence of initial droplet momentum on the
propagation ofthe droplet phase.The second point of discussion is
concerning the comparisonof experimental droplet data with Eulerian
and Lagrangian twophase flow simulations. In this context, the
Hybrid procedure isused as a computational tool to accelerate the
Lagrangian cal-culation. The overall reduction of computation time
achievedis about 30% of the time required for a standard
Lagrangianflow simulation without Eulerian preconditioning. Radial
pro-files of number averaged two phase flow variables are
presentedat axial positions z = 20, 55 and 90 mm. With respect to
themean axial velocities of the droplets shown in Figs. 16 and
17,both numerical methods predict a maximum in the core flow(r <
6 mm) which is not observed in experiment. In partic-ular at z = 55
and 90 mm the axial droplet velocities in thecore flow region are
overestimated by the Lagrangian calcula-tion. Basically, this
effect is a result of an insufficient spray-induced flow
deceleration. This conclusion is supported by theunderestimated
liquid volume flux in the core flow region at thecorresponding
axial positions as shown in Fig. 19. With re-spect to the Eulerian
result shown in Fig. 18, it is evident that
the unavailability of secondary breakup models requires
smallinjection angles of the droplet phase to meet the radial
volumeflux profile in the second half of the duct flow.
Consequently,substantial deviations are observed in the upstream
flow regionwhere secondary atomization occurs. As illustrated by
Figs. 20and 21, Eulerian and Lagrangian flow simulations predict
rathersimilar radial profiles of the Sauter mean diameter.
Althoughthe calculated values are deviating from experimental data
byup to 20 µm, both flow simulations reproduce the trend in
theevolution of the droplet size spectrum.
7 ConclusionsEvaporating fuel sprays in combustor flows are
characterizedby high rates of mass, momentum and enthalpy transfer
be-tween spray and gas flow. Fuel atomization, spray dispersionand
evaporation are often complicated by additional physicaleffects
such as droplet-wall interaction, shear driven evaporat-ing wall
films or secondary breakup of droplets.The primary objective in
this study is the design of a compu-tational tool for an efficient
numerical simulation of combus-tor two phase flows including
advanced modeling of secondaryatomization physics. Compared to
state of the art Lagrangianspray simulations, a significant
reduction of computation timeis achieved by the presented Hybrid
procedure. Basically, thiscomputational strategy is a two stage
combination of an Eule-rian and a Lagrangian method. The Eulerian
method is usedas an efficient preconditioner for the interacting
two phase flowfield, in order to reduce the number of subsequent
Lagrangiangas flow computation - droplet tracking iterations. In
this re-finement stage, advanced modeling of secondary breakup
ofdroplets including bag, multimode and shear mechanisms isused to
improve the physical description of the spray.To assess the
accuracy of the two fundamental numerical ap-proaches, Eulerian and
Lagrangian flow simulations are com-pared with droplet data
measured in the two phase flow of a LPPcombustor premix duct. The
atomization concept employed inthis premix duct achieves a
substantial improvement of fuel at-omization by aerodynamic breakup
of droplets in an extendedflow region downstream of the nozzle.
With respect to the Eule-rian flow simulation, extrapolation of
measured droplet data tothe point of injection with implicit
consideration of secondaryatomization effects results in a rather
crude approximation ofthe spray structure. Since secondary breakup
of droplets is mod-eled in detail by the tracking algorithm, the
numerical descrip-tion of the spray structure in the Lagrangian
flow simulation issignificantly improved.It is evident that the
computational acceleration achieved by theHybrid procedure
significantly depends on the structure of thetwo phase flow. With
respect to the rather uncritical premix ductflow presented in this
study, it is certainly questionable whetherthe moderate time
savings are worth the additional complexityof the CFD program.
However, technical practice offers a suf-ficient number of critical
two phase flow applications, wherestrong relaxation of droplet
source terms requires an excessivenumber of Lagrangian coupling
iterations [24]. In these flowcases, the Hybrid procedure has an
increased potential for sub-stantial speedup of flow
simulations.
AcknowledgementThis work has been founded by the Department of
Education,Science, Research and Technology of Germany under
contractNo. 50-807642. The support is gratefully acknowledged.The
authors would like to thank Mr. Michael Willmann, BMWRolls-Royce,
Germany, for his support concerning the develop-
-
51-11
ment of droplet breakup models.
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Karlsruhe,http://itsnova.mach.uni-karlsruhe.de/∼schmehl, 1998.
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[27] W. Snyder and J. L. Lumley. Some Measurements ofParticle
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-
51-12
0.0 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0Ugas [m/s] :
Figure 12: Axial gas velocity: Single phase calculation (top
half) and two phase calculation (bottom half)
460.00 490.00 520.00 550.00 580.00 610.00 640.00 670.00 700.00
730.00 760.00Tgas [K] :
Figure 13: Calculated gas temperature
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045
0.050Cfuel [-] :
Figure 14: Calculated vapor concentration
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50VolFlux
[-] :
33 mµ57 mµ
Figure 15: Normalized volume flux and mean trajectories
-
51-13
40
60
80
100
ud
[m/s
]
z=90 mm
40
60
80
100
ud
[m/s
]z=20 mm
z=90 mm
40
60
80
100
u d[m
/s]
z=55 mm
Gas Phase, CalculationDroplets, ExperimentDroplets,
Calculation
0 5 10 15 20r [mm]
Figure 16: Mean axial velocities (Euler)
0 5 10 15 2040
60
80
100
u d[m
/s]
z=55 mm
0 5 10 15 2040
60
80
100
ud
[m/s
]
z=20 mm
Gas Phase, Calculation
Droplets, CalculationDroplets, Experiment
0 5 10 15 20r [mm]
40
60
80
100
ud
[m/s
]
z=90 mm
Figure 17: Mean axial velocities (Lagrange)
0
0.1
0.2
0.3
0.4
Vo
lFlu
x
z=90 mm
0
0.1
0.2
0.3
0.4
Vol
Flu
x
z=55 mm
0
0.1
0.2
0.3
0.4
Vol
Flu
x
z=20 mm
0 5 10 15 20
0 5 10 15 20
0 5 10 15 20r [mm]
Figure 18: Normalized volume flux (Euler)
0 5 10 15 200
0.1
0.2
0.3
0.4
Vo
lFlu
x
z=55 mm
0 5 10 15 200
0.1
0.2
0.3
0.4
Vol
Flu
x
z=20 mm
0 5 10 15 20r [mm]
0
0.1
0.2
0.3
0.4
Vo
lFlu
x
z=90 mm
Figure 19: Normalized volume flux (Lagrange)
0 5 10 15 20r [mm]
0 5 10 15 200
25
50
75
100
D32
[µm
]
z=55 mm
0
25
50
75
100
D32
[µm
]
z=20 mm
0
25
50
75
100
D32
[µm
]
z=90 mm
z=55 mm
Figure 20: Sauter mean diameter (Euler)
0 5 10 15 200
25
50
75
100
D32
[µm
]
z=20 mm
0 5 10 15 200
25
50
75
100
D32
[µm
]
z=55 mm
0 5 10 15 20r [mm]
0
25
50
75
100
D32
[µm
]
z=90 mm
Figure 21: Sauter mean diameter (Lagrange)
