9 Imem Abani Reitz

8/11/2019 9 Imem Abani Reitz

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ILASS Americas, 20thAnnual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 2007

_________________________________*Corresponding author, Graduate Research Assistantemail: [email protected]

A Model to Predict Spray-tip Penetration for Time-varying Injection Profiles

Neerav Abani* and Rolf D. ReitzEngine Research Center, University of Wisconsin-Madison

Abstract

A new model to predict spray-tip penetration for a time-varying injection profile has been formulated based on gas-

jet theory. The approach involves using an effective injection velocity for the spray tip based on a representative

spray response time . It is assumed that the instantaneous injection velocity affects the spray tip with an exponential

response function and that the response time is the particle residence time, consistent with the theory of translation

of jet vortex rings from Helmholtzs vortex motion analysis [11]. This Helmholtz theory is also shown to yield thewell-known velocity decay rate of turbulent gas jets. A Duhamel superposition integral is used to determine the

effective injection velocity for time-varying injection rates. The model is tested with different injection profiles and

different ambient densities. The results are also compared with numerical results from a CFD code that has been

calibrated for spray simulations. The comparisons agree very well and the new spray penetration model offers an

efficient method to predict penetrations. The model also can be used to predict equivalence ratio distributions for

combusting sprays and jets.

Keywords: Sprays, Droplet, Super-position integral, Vortex Motion, CFD

Introduction:

Spray models are extensively used in the modeling andsimulation of various combustion systems, such asInternal Combustion engines and Gas Turbine engines.Currently available zero-dimensional combustionmodeling codes us e spray penetration models which arebased on a quas i-steady assumption [1, 2]. These spraymodels are a direct extension of gas-jet theory based onsteady injections [3, 4, an d 5]. Such gas-jet theory-based models have been used extensively to determinethe spray-tip penetration for steady-state injections by

various researchers [3, 4]. The results have been foundto agree well with experiments and fine-mesh CFDcomputations. However, realistic injection profiles areusually time-varying and thus steady-state gas -jettheory cannot be applied directly. CFD modeling ofsprays and jets provides good predictions of tip-penetra tion, but as the mesh size is increased to becompatible with practical engine computations, theprediction accuracy becomes poorer. Hence, there is aneed for a better predictive model for spray penetrationthat can be used with practical time-varying injectionvelocity profiles and realistic engine ambient conditions.In this work, we present a new method, which is based

on jet-theory and a superposition integral formulation todetermine an effective injection velocity and hence, thespray-tip penetration.

There have been attempts to study non-stationary jetsand sprays. Measurements by Bore et al. [6], involveda study of a sudden decrease in injection velocity, andthey proposed a self-similar result based on a temporalscaling. They found that a time-scale of the form x/Uinj,2 , wherexis the position of the spray tip and Uinj,2is

the suddenly decreased injection velocity, leads to aself-similar result. However, their results and modelonly apply to a sudden decrease in injection velocityfrom an initial constant value to a constant lower value.Previous works pertaining to simple one-dimensionalspray models include the packet penetration model ofDesantes et al. [7], where an injected spray particleinstantly travels with a different momentum once it isovertaken by a speeding subsequently injected particle.This approach is an improvement over other quasi-steady state models in the literature, but their study didnot include a wide range of tested injection profiles.

Zhang et al. [8] studied the effects of flow accelerationon turbulent jets with linear, quadratic and exponentialinjection profiles using measurements. They found thatthe temporal evolution of the spray front follows thesame form as the forcing function at the nozzle. Wan etal. [9] also modeled spray penetration for evaporatingsprays by using scaling parameters. They showed theirmodel worked well for a linearly increasing injectionvelocity profile. However, they did not consider otherinjection velocity rate shapes. Breidenthal [10], in astudy of self-similar, turbulent jets postulated that if theflow is non-stationary or non-steady a choice of self-similarity function is the exponential function. This

self-similarity function can be thought of as theresponse function of a particle in the jet to a change ininjection velocity. Crowe et al. [11] studied theresponse function of a droplet to the surrounding gasvelocity. Their analysis also reveals an exponentialresponse function of time for the droplet to reach thesurrounding gas velocity.

In this work we propose a relatively simple explicitmodel that can be used to predict spray tip penetration.


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The model formulates the effective injection velocityexperienced at the spray tip at any given instant todetermine the spray tip penetration. The model extendsthe isolated drop theory proposed by Crowe et al. [11]that a droplet requires time to adjust to a change insurrounding gas velocity. This change is based on aresponse function which is exponential in time and theresponse t ime is the ratio of a characteristic length scaleto the surrounding gas velocity. The same argument isextended here to sprays where the characteristic lengthscale is taken as the location of the particle from theinjector tip and velocity is the time-convolved injectionvelocity. This is derived from the classical work ofHelmholtzs vortex motion theory [12-15], wherein inthe translational velocity of a vortex ring is related tothe circulation of the vortex. The present proposedmodel is tested with different time -varying injectionprofiles and varying ambient densit ies. The resultscompare very well with CFD results. It is to be notedthat the choice of selecting CFD simulations as the

benchmark for comparison with the proposed analyticalmodel was based on the ability to test the model overwide ranges of injection profiles, which experimentallyis a difficult and time -consuming task. The selectedinjection profiles include smooth profiles, profilesfeaturing a sudden decrease and sudden increase invelocity, and their combinations.

This paper is organized as follows. A detailed analysisincluding the derivation of the new model is providedfirst. A short description of the CFD and spray model[16] and details of the computational domain follows.In the same section we also discuss modifications to

previous penetration correlations [1] as applied to non-stationary and non-steady jets and sprays. Analyticalexpressions for an injection profile with rising andfalling rates are then derived based on an assumedaverage response-time. The analytical expression is alsoextended to consider the case of a sinusoidal injectionprofile. Next, we present 13 different injection cases tovalidate the proposed model. Finally, discussion aboutthe use of the model for analyzing vaporizing spraysand the usefulness of the model is presented.

Theory:

The steady-state solution for the velocity profile withina turbulent gas jet can be obtained a using similarityanalysis of the 2D Navier-Stokes equations as [3, 4]:

+

=2

22t

22

eq

2

inj

t

2

eq

2

inj

inj

x256

rdU31x32

dU3,Umin)r,x(U

(1)

where Uinj is the injection velocity at the injector exit.xis the axial distance of a particle from the injector tipand ris the radial distance of the parcel from the spraysymmetry axis. deq is the equivalent or effectivediameter of the gas jet defined as [3]:

g

l

nozeqdd

= (2)

where dnozis the nozzle diameter and land gare theinjected and surrounding fluid densities, or in presentstudy, the liquid and gas-phase densities, respectively.?

In the case of a turbulent jettis the turbulent viscositygiven by [3]:

2/5.0 eqinjtt dUC = (3)

Ct is a constant, as reported by Abraham [3] andSchlichting[17] who used Ct= 0.0161. For evaluatingspray tip penetration, the centerline tip position can beevaluated by setting r=0in Eq. (1) and generalizing thevelocity decay rate at the centerline as:

( )0eq

eqinjxKd3x

xK

dU3

dt

dx)x(U === (4)

where K is the entrainment constant and is equal to16

0.5Ct. Schlichtings choice of Ct=0.0161 gives

K=0.457. If we considerxto be the spray tip position atall times, then integrating Eq. ( 4) provides spray tippenetration as a function of time. For a constantinjection velocity the analytical solution of spray tipposition as a function of time is given as :

( ) 02/12/1

eqinj

2/1

xxfortdUK

6)t(x

= (5)

where the entrainment constant,K isassumed to be 0.5in this study to give favorable agreement withexperimental data. For cases with time-varyinginjection velocity, an analytical expression for the spraytip penetration is derived below.

A model of an isolated droplet responding to a givensurrounding gas velocity was given by Crowe et al. [11].The equation of motion for a particle in a gas can beexpressed by the balance of drag force to the change ofmomentum of the particle as:


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3

( ) vuvu4

DC

2

1

dt

dvm

g

2

D =

(6)

where m is the mass of the particle, v the particlevelocity, CDthe drag coefficient, Dthe particle size, gthe surrounding gas density and uthe surrounding gasvelocity. The particle Renolyds number is :

vuDRe

g

P

= (7)

where is the dynamic viscosity of the surrounding gas.Crowe et al. [11] assumed a low Reynolds number flow,where the factor 24/CDRePapproaches unity. We retainthe same assumption to analyze the physics of jetmomentum and response times. Using Eq. (6) and Eq.(7) the particle momentum equation can be written as:

( )vu1

dt

dv

v

=

(8)

where v is the momentum response time, whichsignifies the time required for a particle released fromrest to achieve 63% of the free stream velocity as:

pD

2

g

vReC

24

18

D

= (9)

Assuming the momentum response time is constant, thesolution for constant uis:

( )v/te1uv = (10)

The above analysis reveals that a particle responds to achange in surrounding gas velocity exponentially.

Consistent with the suggestion of Breidenthal [7], wealso extend this response to sprays where the injectionvelocity changes with time. We assume that the spraytip responds to the change in injection velocity with anexponential response function with a characteristicmomentum response time that depends on the particleresidence time in the jet. The momentum response timefor the spray tip is modeled using the ratio of theparticle distance from the injector location to theinstantaneous injection velocity, Uin j, as depictedschematically in Figure 1. The schematic depicts thehead vortices at the jet tip. For a given spray angle, theeddy size at the tip position is assumed to be related to

the tip posi tion and the tangent of the half spray angle .Therefore, Leddy=x tan().

It is shown in the present paper that the above analysiscan also be derived from Helmholtz' vortex motiontheory [12 (see also Tait [13], Taylor [14] and Lamb[15]). If a jet is issued from a circular hole as depictedin Fig. 1, the counter rotating vortices are due to thevortex ring at the leading tip of the jet. If is the total

circulation around the core, a mean jet radius is Leddy,and the cross-sectional radius of the vortex ring is ra,then the velocity of the translation of the vortex ring,UVis expressed as [12]:

eddya

eddy

eddyV L

~4

1

r

L8ln

L4U

= (11)

According to Helmholtzs second vortex theorem [13],circulation is the same for all cross sections of thevortex tube (here vortex ring) and is independent oftime. If we assume that the vortex ring expands only inits cross-sectional area, then the circulation aroundthe core of the vortex at time tis equal to the circulationwhen the ring was generated at the injector tip at timet=0,and is given [13, 14] as:

nozinj

nozinj

INJ

eddydU~

dU4L)x(U4)x(

===

(12)where the circulation (x) is the circulation of the ringat locationxand INJis the circulation of the vortex atthe injector location, and it is written as a function ofthe injection velocity Uinj and the nozzle diameter dnoz.If we assume that the translational velocity of thevortex is given by a ratio of a characteristic length scale

(~

L ) and time-scale (~

T), then from Eqs. (11) and (12)we have:

( )

tanx

dU~

L~

T

L~U

nozinj

eddy

~

~

V (13)

Reitz and Bracco [18] predicted the variation of spray-half angle and concluded that tan() ~ (g/ l)/

1/2.Using the correlation of tan() and the effectivediameter deq, Eq. (13) can thus be re-written as:

=

inj

eqeqinj

~

~

V

Ux

d

x

dU~

T

L~U (14)

Note that this result, derived independently fromHelmholtz vortex theory, is identical in form to theequation for the gas jet velocity decay of Eq.(4), whenthe vortex translational velocity equals the jet tip

velocity. The only time scale that appears in the aboveequation is (x/Uinj)and hence it is reasonable to assumethat the response time of the eddy to a particularinjection velocity at the nozzle exit is of the order of theconvective time-scale or flow residence time scale, F =(x/Uinj). This is also valid in the case of an injectionvelocity that is changing with time. Crowe et al. [11]argued in the case of droplets, that the response time isalso proportional to the flow-time scale and the twotime scales were related by the Stokes number.


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Introducing the Stokes number, St = v/F, where Fis acharacteristic time of the flow, which in the spray caseis x/Uinj , following Crowes model [11], the responsetime of the particle, vis given as :

= injv Ux

St (15)

This form of the time response is also consistent withthat introduced in the experimental study of Bore et al.[6]. In the present study the Stokes number is taken anexperimental constant assumed to be St=3 and theeffective injection profile of the spray tip correspondingto an injection velocity change from U1to U2is thus:

( )

+=

v

birth

121eff,INJ

tt(exp1UUUU

(16)

where tbirth is the time at which the injection velocitychanges from U1to U2. This effective injection velocityis then used in Eq. (4) or (14) to determine the spray tippenetration. For a time -varying injection where Uinjchanges n times, the Duhamel superposition principle[19] provides the effective injection velocity for thespray tip position as a function of time as :

( )

( )k

n

1k

keff,INJ

k

k

n

1k

keff,INJ

U)tt,x(A)0(U)t,x(U

tt

U)tt,x(A)0(U)t,x(U

=

=

+=

+=

(17)where x is the spray tip position, U(0) is the initialinjection velocity at time t=0, tkis the birth time of thenew injection velocity U(k), and (U)k = Uk-Uk-1, i.e.,the change in injection velocity at t=tk. The functionA(x,t-tk) accounts for the exponential response and isgiven as:

=

k,v

k

k

texp1)t,x(A

(18)

where the v,k is the response time associated with theeddy at the spray tip at time t=kand locationxand isgiven as:

k

k,v

U

xSt= (19)

Note that the summation in Eq. (17) can be replaced byintegrals in the case of a continuous function Uinj(t).The spray tip position is determined using the effectiveinjection velocity in Eq. (4) as:

( )0

eqeff,INJxx

x

d)t,x(U

K

3

dt

dx= (20)

Ricou and Spalding [20] found in their experiments thatthe entrainment constant, K, increases as the injectionReynolds number is reduced. However, for simplicity,we assume a constant value ofK=0.5. The above modelis tested below with different injection profi les. At eachstep during the integration, the effective injectionvelocity at the spray tip position is evaluated from Eq.(17) and the new spray tip position is evaluated byintegrating Eq. (20) with fourth-order Runge-Kuttaintegration.

Modification to Spray-tip Penetration Correlations:

Practical spray-tip penetration models in zero-dimensional or Multi-Zone Combustion models [1, 2]use correlations such as that of Hiroyasu and Arai [21]:

( )

( ) 5.0g

nozl

bb

5.0

noz

25.0

g

b

5.0

l

b

P

d65.28t:,t,

tdP

95.2S:tt0,

tP2

39.0S:tt0,

=

=


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Approximate Analytical Solutions for Simple

Injection Profiles:

The model presented above evaluates spray-tippenetration bas ed on Eqs . (17) to (20). Approximateanalytical expressions can be obtained by assumingquasi-steady average response times to evaluate thevelocity field. Consider an injection velocity profile ofthe form Uinj (t)=(a+mt), where ais the initial injectionvelocity and m is the slope of injection velocity rateshape. For simplicity it is assumed that the responsetime, V,avis constant. Simplifying Eq. (17) in integralform gives:

( ) ( )

( )( )

dUt

exp)t(U)t,x(U

dd

dUtexp1)0(U)t,x(U

t

0 av,V

eff,INJ

t

0 av,V

eff,INJ

=

+=

=

=

=

=

(24)

where U denotes the time differential and v,av is anaveraged response time modeled as:

( )

===

T

0

injnozl

T

0

2

injnozl

AV

AV

eqA

av,V

dtUA

dtUA

U;U

dStt

(25)where T is the injection duration and UAV is the mass-averaged injection velocity. StA is viewed as anadjustable constant chosen here as 50. This high valueis consistent with the constant used in the Hiroyasucorrelation Eq. (23). From Eq. (15), the length scalexis

the distance from the particle location to the injectornozzle, which is dynamically calculated over the time.Since a-priori information about the particle position inthe approximate analytical approach is unavailable, thelength scale chosen is the effective nozzle diameter.

Using U= m, Eq. (24) gives an effective injectionvelocity for a linear injection profile as:

( )

+=

av,V

av,Veff,INJ

texp1mtma)t,x(U

(26)Substituting Eq. (26) in Eq. ( 20) and solving forx gives

the spray-tip penetration as:

( )2/1

2

av,V

av,V

2

av,V

2

av,V

eq

m2t

expm2

mtmat2

K

D3)t(x

+

+

=

(27)where for a particular linear-injection profile, V,av isgiven from Eq. (25) as:

( )( )

+

+

=33eqAav,V amTa

mT2

1amT3

dStT,m,a

(28)

Note that Eq. (27) reduces to Eq. (4) if m=0 and a=Uin j(top hat injection profile).

Simi lar approximate solutions can also be found forother profiles such as for a sinusoidal injection profileof the form Uinj(t)=C sin(t) , where =2/, and isthe total duration of injection.Using U()=cos()and Eq. (24), the effective injection velocity ispredicted to be:

( )

( ) ( )

+

+

=

av,V

av,V2

av,V

2

av,r

eff,INJ

texptsintcos

1

C

tsinC)t,x(U

(29)where for a particular sinusoidal-injection profile, V,avis given by Eq. (25) as:

( ) ( )( )

( )

=

T2sin2

1TC

1Tcos2dStT,,C eqAav,V

(30)The spray-tip penetration for a sinusoidal injectionprofile is then obtained by substituting the effectiveinjection velocity obtained above in Eq. (20) andsolving forx, yielding:

( )

( )

( )

( )

2/1

2

av,V

2

av,V

av,V

2

av,V

2

eq

2

av,V

3

1

tsin

texp

tcos

K

D.C61tx

+

+

+

+=

(31)

Fuel-air distribution in Evaporating Sprays:

An additional application of the present model is to

consider spray combustion. For steady-injections underevaporating conditions, the equivalence ratio at eachaxial location in the spray is defined as the ratio of theair mass flux to the fuel mass flux at that location [5] tothe stoichiometric air/fuel ratio. For unsteady injectionsthe effective injection velocity can be used to determinethe equivalence ratio distribution. For example,assuming a stoichiometric air/fuel ratio of 15, the localequivalence ratio at positionxis:


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( ) ( )

( )

( )02

eq

eff,injinjl

g

xx)(tanx

Kd5

t,xUA

t,xUt,xA15)t,x(

=

=

(32)whereA(x,t)=(xtan)2is the local cross-sectional areaof the spray, Ainj is the nozzle hole area, and Uinj,eff isthe effective injection velocity. Substituting x from Eq.(20) into Eq. (32) gives the equivalence ratiodistribution as a function of time and space for time-varying injection profiles. The results for the cases oflinear and sinusoidal injection profiles, Eqs. (27) and(31) are useful to give approximate analyticalexpressions for the equivalence ratio distributions forthose injection profiles.

Computational Fluid Dynamics Model:

Spray simulations were carried out using the KIVA-3VCFD code [16] to assess the validity of the present jetmodels. The geometry chosen was an axi -symmetricchamber of 40 mm radius and 100 mm height.Computations were made using a 0.5 sector mesh withperiodic boundary conditions, as shown in Figure 2.The mesh was chosen to be very fine and uniform withdimensions of 0.25mm in the axial direction and 0.25mm in the radial direction, respectively (400 cells in theaxial and 160 cells in the radial directions, respectively).Such mesh resolution has been shown to be adequate byAbraham et al. [22]. Drop collisions, coalescences ,

evaporation and breakup were not considered in thepresent study so as to just isolate the two-waymomentum coupling effect of swarms of particles in aspray. The nozzle diameter was 254 m and theinjected particle size was the same as the nozzlediameter. The droplet momentum equation is Eq. (6),with the drag coefficient CDexpressed as a function ofparticle (or droplet) Reynolds number [16]. Thechamber gas was Nitrogen at 451K with densities of60.6 kg/m3and 30.2 kg/m3. The liquid diesel fuel had adensity of 700 kg/m3. The spray tip penetration wasdefined as the location with 95% of the total liquidmass at any given instant from the injector.

Details of Test Cases:

A total of 13 different injection profiles were chosenand the model was tested at the two different ambientgas densities. The injection profiles were selected ascombinations of smooth functions and profiles withsudden increases or decreases in injection velocity.Figure 3 shows the injection profiles with a descriptivetitle in each case for ease of discussion. For all cases the

injection velocities were the same for both ambientdensities.

Results and Discussion:

The spray-tip penetration results from the present zero-dimensional model in Eq. (20) are compared with theresults from the corresponding CFD simulations. Thecomparisons are presented in Figs. 4 to 29. Each figureshows a comparison of spray tip penetration predictedfrom the zero-dimensional model (solid line) and theCFD simulation (circles). The plots also show theactual injection velocity profile (dashed line) and theeffective inject ion velocity profile as experienced bythe spray tip (dotted line). The plots also show the errorin the penetration as a function of time. The error iscalculated from:

( )

CFD

2

CFDD0

S

SS100Error

= (33)

where S0D and SCFDare the spray tip penetration fromthe 0-D model and CFD simulations, respectively. Therelative error can be high initially when the absolutepenetration is low. However as the spray progresses, theerror reduces with time. Thus, the error is evaluatedstarting only after 0.2 ms from the beginning ofinjection to avoid the anomalous near nozzle region.Tables 1 and 2 show the mean percentage error for allthe cases at the two different ambient densities of 30.3and 60.6 kg/m3, respectively. It can be noted that theerror is low for most of the profiles and in most cases is

less than 5%.

Figures 4, 5 and 6 show the flat injection velocityprofiles for the chamber density of 60.6 kg/m3. It can benoted that the error for intermediate and high injectionvelocities is larger than for the low injection velocity,although the mean error is less 5%. Also, the effectiveinjection profile at the tip coincides with the actualinjection profile, (see Eq. (17)). Figure 7 corresponds tothe case with a peak in the injection profile . Note thatthe zero-dimensional model follows the trend of thespray tip penetration during the change in injectionvelocity. The injection velocity does not increase

suddenly, as would occur with a quasi-steady statemodel, but instead gradually increases following therise in injection velocity. A similar trend is observedduring the decreasing portion of the velocity profile.Figure 8 shows the sinusoidal injection profile and thepenetration appears to be very close to the numericalresult with reduced error as time progresses. Figures 9,10 and 11 show the results from the STEP, STEP3 andSTEPR3 injection profiles. The penetration is seenagain to compare well with the numerical results, but


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with slight differences towards the end. It can be notedthat the slopes of the penetration curves are similar tothose obtained from the CFD results.

Figure 12 shows case PEAK2, where the peak exists fora shorter duration but with a higher injection velocityduring the rise as compared to case PEAK. As can beseen, a more severe high peak also agrees well with theCFD predicted penetration. Figures 13, 14 and 15represent pulsed injection cases (PULSE1, PULSE2and PULSE3), respectively. In the case of multiplepulses of equal pulse duration and equal pulse height(viz., PULSE1 and PULSE3), the effective injectionvelocity oscillates about a particular velocity. Figure 14is a one pulse case. The tip penetration from the newmodel compares well with the spray-tip penetrationfrom the CFD simulation and demonstrates anexponentially reducing the effective injection velocityat the spray-tip. Finally , the triangular injection profilecase is shown in Figure 16. The penetration again

compares well with the CFD prediction.

Similar results were obtained with a lower chamberdensity of 30.2 kg/m3, and the comparison of the spraytip penetration with the zero -dimensional model and theCFD simulations agreed well for most of the cases.Only cases PEAK2 and SINE shown in Figs. 17 and 18had significant errors in penetration. The SINE caseshows that the penetration is under predicted fromabout 0.7 ms after the start of injection. This show thatchoice of the St=3, after 0.7 ms of injection gives arelatively slower response-time and hence, a relativelylower effective injection velocity in the first half which

also effect the penetration in the second half ofinjection. In case of PEAK2, the injection velocitychanges from about 280 m/s to 690 m/s. Thepenetration compares well with the CFD result duringthe duration when injection velocity is increasing.However during the period when the injection velocityis decreasing the penetration is under-predicted whichis due to a relatively lower effective injection velocity.This show that the choice of St=3 results in a relativelyfaster time response during this period of injection.Hence, these two cases show that for some particularinjection cases, a variable St depending upon themagnitude of the injection velocity change could help

in improving result. The agreement could also havebeen improved by considering a variable entrainmentconstant, as suggested by Ricou and Spalding [13], orwith a variable Stokes number for the dropletmomentum response. Such tuning of the model couldbe considered as a future improvement for wider rangesof injection conditions, such as for the case of thesudden large increase in injection velocity of Fig. 12.

It is also of interest to explore results obtained with theanalytical spray tip penetrations derived for the linearand sinusoidal injection profiles in Eqs. (27) and (31).Spray-tip penetration plots and the effective injectionvelocity profile for the line-shape injection cases areshown in Figs. 19 and 20 for positive and negativeslopes U

inj=a +mt, where a is the injection velocity at

t=0 and m is rate of change of injection velocity,respectively.

Figure 19 shows the case with positive slopes. Figure19(a) shows the comparison of penetration predictionsfrom the zero-dimensional model, CFD model and theanalytical model. The penetration predictions fromanalytical model agree well with both the CFD modeland the zero-dimensional model. Figure 19(b) comparesthe prediction of the effective injection velocity profileat the spray tip from the zero-dimensional model andanalytical model with that of the actual injectionvelocity profile. The effective velocity profile from the

analytical model is observed to be greater than that ofthe zero-dimensional model. However its effect on thepenetration prediction is less, as observed from Fig.19(a). Similar plots are shown for the case withnegative slope of the linear injection profile in Figs.20(a) and 20(b). From Fig. 20 (b) it can be observedthat the prediction of the effective injection velocity atthe tip from the analytical model matches well with thatof the CFD and zero-dimensional models. Thepenetrations also match very well as shown in Fig.20(a). Figures 21(a) and 21(b) show similar plots forthe sinusoidal case and show a good match with theCFD results and the zero-dimensional model. Again it

can be observed from Fig. 21(b) that the effectiveinjection velocity at the spray tip from the analyticalmodel is under-predicted compared to that of the zero-dimensional model, but the penetrations agree well, asshown in Fig. 21(a). Hence, it be concluded that for thelinear and sinusoidal injection profiles, the approximatemodel to predict the effective injection velocity profilecan provide good predictions of spray tip penetration.

Finally, Fig. 22 shows a plot of the variation ofequivalence ratio at the spray tip with time for fourdifferent cases, viz., the linear injection profile withpositive and negative slopes, and the sinusoidal cases as

discussed above analytically, plus a constant injectionvelocity case having an average velocity approximatelyequivalent to the former three cases (385 m/s). Theequivalence ratio is analytically calculated from Eq.(32). Figure 22 shows the comparison of the four cases.It can be observed that the constant injection case andthe linearly decreasing case exhibit similar trends ofvariation of . The plot also shows a change in slope inthe near nozzle area, which is due to the change in slopeof the penetration, x in the near nozzle area. This is


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consistent with the observations of Naber and Siebers[5] and with Eqs. (21) where in the near nozzle area,penetration is proportional to t and in the far-fieldproportional to t1/2. This trend is not visible for the othertwo cases of the linearly increasing profile and thesinusoidal profile. This is because these profiles startwith a very low injection velocity, which is notsignificant enough to reveal the changes in the gradientin the near-field. The plot also shows the time when theequivalence ratio at the spray tip reaches unity. For thelinear profile with negative and positive slopes , the timeto reach = 1.0 is m-=0.05 ms and m+=0.26 ms,respectively. Similarly for constant injection andsinusoidal injection profiles, the time to reach =1 is

const= 0.07 ms and sine=0.37 ms, respectively. Thus, m-~ const


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Philosophical magazine and Journal of Science,Vol. 33, Series 4, 485-512, 1867.

14. Taylor G. I., Formation of a Vortex Ring byGiving an Impulse to a Circular Disk and thenDissolving it Away, Journal of Applied Physics,Vol. 24, Number1, 1953.

15. Lamb, Hydrodynamics, Cambridge UniversityPress, Cambridge, 6thEdition, Sec 163, p. 241.

16. Amsden,A.A., KIVA-3V: A block structuredKIVA program for engines with vertical or cantedvalves, Technical report No. LA-13313-MS, LosAlamos National Laboratory, July, 1997.

17.

Schlichting, H., Boundary Layer Theory, McGraw-Hill, New York, 1976.

18. Reitz , R.D., and Bracco, F. B., On theDependence of Spray Angle and Other Spray

Parameters on Nozzle Design and OperatingConditions, SAE Paper 790494, 1979.

19. Hilderband, F. B., Advanced Calculus forApplications, Prentice-Hall Inc., pp. 451-453,1962.

20. Ricou. F. P., and Spalding. D. B., Measurementsof entrainment by axisymmetrical turbulent jets, JFluid Mech., 11, 21-32, 1961.

21. Hiroyasu, H., and Arai, M., Fuel SprayPenetration and Spray Angle of Diesel Engines,Trans. Of JSAE, Vol. 21, pp.5-11, 1980. SAE-930612.

22.

Abraham, J., 1997, "What is Adequate Resolutionin the Numerical Computations of Transient Jets?"SAE Trans., 106, No. 3, pp. 141155.

Table 1: Error in predicted spray tip penetrationfor g=60.6 kg/m3.

Table 2: Error in predicted spray tip penetration

g=30.6 kg/m3.

Std. Deviation Mean

LOWFLAT 0.7 1.1

FLAT 2.7 6.3

HIGHFLAT 2.7 3.3

PEAK 1.9 2.5

SINE 2.1 3.5

STEP 2.2 2.4

STEP3 2.1 3.3

STEPR3 1.8 2.8

PEAK2 3.3 4.2PULSE1 2.1 3.5

PULSE2 0.9 4.2

PULSE3 2.4 8.1

TRIANGULAR 5.0 6.4

% Error (g= 60.6 kg/m3)

Std. Deviation Mean

LOWFLAT 1.6 1.6

FLAT 1.6 1.6

HIGHFLAT 2.4 4.2

PEAK 2.1 2.1

SINE 7.6 8.6

STEP 1.9 2.2

STEP3 3.3 6.1

STEPR3 1.2 0.8

PEAK2 5.6 4.9PULSE1 3.3 4.6

PULSE2 1.7 7.0

PULSE3 1.9 9.2

TRIANGULAR 6.9 7.3

% Error (g= 30.2 kg/m3)


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Figure 1: Schematic describing the momentum response time for the spray-tip eddy to adjust to theinjection velocity

Figure 2: Details of computational domain and mesh size for CFD simulations.

Injector Location

=0.5

100 mm; Nx=400

40 mm; Nr=160

Injector Location

=0.5

100 mm; Nx=400

40 mm; Nr=160

Leddy = X tan()X

Injectorlocation

Jet showing counter rotating vorticesat the spray tip and Vortex Ring

Uinjv = X/Uinj

Radius of cross

section = ra

Leddy = X tan()X

Injectorlocation

Jet showing counter rotating vorticesat the spray tip and Vortex Ring

Uinjv = X/Uinj

Radius of cross

section = ra


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Figure 3: Nomenclature of the 13 injection profiles tested.

1. LOWFLAT 2. FLAT 3. HIGHFLAT

4. PEAK 5. SINE 6. STEP

7. STEP3 8. STEPR3 9. PEAK2

1. LOWFLAT 2. FLAT 3. HIGHFLAT

4. PEAK 5. SINE 6. STEP

7. STEP3 8. STEPR3 9. PEAK2

10. PULSE1 11. PULSE2 12. PULSE3

13. TRIANGULAR

10. PULSE1 11. PULSE2 12. PULSE3

13. TRIANGULAR


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Figure 4: Spray Tip Penetration for LOWFLATcase, g~60.6 kg/m3.

Figure 5: Spray Tip Penetration for FLAT case,g~60.6 kg/m3.

Figure 6: Spray Tip Penetration forHIGHFLAT case, g~60.6 kg/m3.

Figure 7: Spray Tip Penetration for PEAKcase, g~60.6 kg/m3.


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Figure 8: Spray Tip Penetration for PEAK case,

g~60.6 kg/m3.

Figure 9: Spray Tip Penetration for STEP case,

g~60.6 kg/m3

.

Figure 10: Spray Tip Penetration for STEP3

case, g~60.6 kg/m3.

Figure 11: Spray Tip Penetration for STEPR3

case, g~60.6 kg/m3

.


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Figure 12: Spray Tip Penetration for PEAK2case, g~60.6 kg/m3.

Figure 13: Spray Tip Penetration for PULSE1case, g~60.6 kg/m

3.

Figure 14: Spray Tip Penetration for PULSE2case, g~60.6 kg/m3.

Figure 15: Spray Tip Penetration for PULSE3case, g~60.6 kg/m

3.


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Figure 16: Spray Tip Penetration for

TRIANGULAR case, g~60.6 kg/m3.

Figure 17: Spray Tip Penetration for SINE case,g~30.2 kg/m3.

Figure 18: Spray Tip Penetration for PEAK2 case,

g~30.2 kg/m3.

(a) (b)Figure 19: Spray Tip Penetration for line-shapedcase (positive slope), g~60.6 kg/m3.


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(a) (b)

Figure 20: Spray Tip Penetration for line-shaped case (negative slope), g~60.6 kg/m3.

(a) (b)Figure 21: Spray Tip Penetration for sinusoidal-shaped case, g~60.6 kg/m3.

Figure 22: Equivalence Ratio at Spray-tip positionfor linear and sinusoidal-shaped injection rate cases,g~60.6 kg/m3.

m- m+const sinem- m+const sine

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