ILASS Americas 28th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2016

Parametric Study of HRM for Gasoline Sprays

K. Saha1∗, S. Som1, and M. Battistoni2

1 Argonne National Laboratory, USA2 University of Perugia, Italy

AbstractFlash boiling is known to be a common phenomenon for Gasoline Direct Injection (GDI) engine sprays.Homogeneous Relaxation Model has been adopted in many recent numerical studies for predicting cavita-tion and flash boiling. Assessment of the Homogeneous Relaxation Model has been presented in this study.Sensitivity analysis of the model parameters has been documented to infer the driving factors for the flashboiling predictions. The model parameters have been varied over a range and the differences in predictionsfor the extent of flashing have been studied. Apart from flashing in the near-nozzle regions, mild cavita-tion is also predicted inside the gasoline injectors. The variation in the predicted time-scales through themodel parameters for predicting these two different thermodynamic phenomena (cavitation, flash) have beenelaborated in this study.

∗Corresponding Author: ksaha@anl.gov

Introduction

Comprehensive understanding of fuel spray for-mation is warranted for optimization of existingtechnologies, as well as future developments, in thefield of diesel or gasoline direct injection combustionengines. Gasoline Direct Injection (GDI) engineshave been preferred to Port Fuel Injection (PFI)engines for almost a decade. Experimental studieshave shown that GDI offers better fuel economy andsuperior performance, when compared to PFI [1, 2].The GDI engines become advantageous owing to el-egant technologies such as, intelligent piston design,swirl flows, high pressure fuel injection, various in-jection strategies, exhaust gas recirculation (EGR),closed-coupled catalyst etc.

Flash boiling, can happen in modern GDI en-gines at throttled conditions and significantly affectsspray formation [3]. Throttled operating conditionscreate low in-cylinder chamber pressures (≈ 30 kPa).Therefore, when heated gasoline-type fuel (around363 K) is injected to such in-cylinder environments,liquid fuel is subjected to a superheated condition.Based on thermodynamic definition, superheatedconditions arise when any liquid substance experi-ences ambient pressure lower than the saturationpressure, corresponding to its temperature. Flashboiling or flashing causes rapid bulk conversion ofliquid fuel to gaseous vapor leading to volume expan-sion and thus, affecting spray width and penetration.Consequently, flashing can either be advantageous ordisadvantageous depending on chamber design andinjection timing as well as injector orientation. As aresult flash boiling (or flashing) for GDI engines isbecoming a relevant topic of research.

Apart from flashing there are two more phenom-ena involving phase change of liquid fuel to gaseousvapor - vaporization and cavitation. Flashing is typ-ically a near-nozzle exit phenomenon. High temper-ature environment in combustion chamber can pro-vide convective heating of liquid fuel causing “va-porization”. The liquid fuel enters the injector holesthrough the space between moving needle and nee-dle sac. This resembles the flow from a large cross-sectional area to a much smaller cross-sectional area,where flow separation occurs near the entry of thesmaller cross-sectional portion. Inside fuel injec-tor holes, the local pressure in this flow separationregion may drop below the saturation pressure ofthe liquid fuel, depending on local temperature, in-jection pressure and injector geometry. Such phe-nomenon is termed as “cavitation”. Cavitation oc-curs almost instantaneously, since it is typically anear-equilibrium process involving negligible heattransfer [4]. Cavitation is usually prominent in diesel

injector holes, because injection pressure in dieselinjector (upto 300 MPa) can be at least one orderof magnitude higher than that of a gasoline injector.Flashing is usually perceived as non-equilbrium phe-nomenon, since heat transfer is non-negligible andrequires a finite time-scale (relative to cavitation)for the phase change to occur [5].

Some experimental studies have been under-taken to investigate flash boiling effects on GDIspray patterns. VanDerWege and Hochgreb [6]observed in the Planar laser-induced fluorescence(PLIF) and Mie-scattering images the transition of apressure-swirl injector spray from hollow-cone shapeto solid-cone shape with increasing the superheat ofthe liquid fuel. The authors concluded from the im-age intensity comparisons that the droplet diame-ter reduction due to flash boiling is in the order of40%. Based on the bubble-point calculation, theyalso suggested that a superheating by roughly 20K is required for the flash boiling to generate no-ticeable effects. Their experimental results demon-strated the potential advantages of using flash boil-ing to enhance spray atomization. Zeng et al. [7, 8]studied experimentally the spray from a multi-holeinjector using Mie-scattering and laser induced ex-ciplex flourescense (LIEF). The authors quantifiedthe macroscopic spray structure with penetrationlength, plume width and normalized plume distance.They also characterized the different macroscopicspray behaviors based on the ratio of ambient pres-sure to saturation pressure. A set of empirical ex-pressions were proposed to predict the macroscopicstructure of the spray under the effects of flash boil-ing. In the analysis, the authors focused more on themacroscopic spray structure rather than fundamen-tal bubble and droplet dynamics. Weber and Leick[3] carried out high-speed Mie-scattering for globalspray characterization and investigation of near noz-zle spray structures using high-speed shadowgraphy.Additionally, droplet velocities were estimated byshadow particle image velocimetry technique. Theyobserved that individual spray plume length reducesand the initial cone angle increases due to flash boil-ing. They noticed that during intense flash boilingsituations two adjacent spray plumes might inter-act with each other. In a recent study with a GDIinjector, similar to Spray G [9], a comprehensiveexperimental analysis was documented [10]. Theyused a Delphi Valve Covered Orifice (VCO) injectorwith iso-octane fuel. The injection pressure, the fueltemperature and the chamber pressure were varied,with the chamber temperature remaining at 296 K.They observed that sub-atmospheric chamber pres-sure and elevated fuel temperatures led to large at-

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omized cloud surrounding fuel jets.Significant effort has been already made in terms

of numerical studies on flash boiling phenomenon.They can be broadly classified into two-types: a)bubble dynamics based models, b) empirical coeffi-cients based thermodynamic rate equation models.Blinkov et al. [11] assumed mono-disperse bubblepopulation and their model consisted of 5 trans-port equations including vapor formation, interfaceheat transfer and nucleation. Blinkov et al. em-ployed their model to analyze flashing and chokingwater flows and inferred that bulk nucleation be-comes increasingly significant with increment in thevolume to surface ratio of the geometry. Chang andLee (2002) [12] modified Blinkov’s approach by con-sidering poly-disperse bubble population in a con-verging water nozzle and concluded from their pri-mary analysis that capturing the bubble size dis-tribution is important for multiple and distributedsources of bubble nucleation. Kawano et al. (2004)[13] incorporated bubble growth, bubble perturba-tion and breakup to model the flashing and non-flashing sprays. A similar approach was adoptedby Zeng and Lee (2001) [14] to study n-pentanesprays, subjected to different degrees of superheat.They concluded that more flashing can be correlatedwith increment in superheat, smaller drop sizes (∼50 % reduction in SMD, when flashing is consid-ered in the model), wider sprays and even plumemerging. In this work, Zeng and Lee studied theeffect of two competing mechanisms of secondarybreakup - bubble growth and aerodynamic force, ina hollow cone spray issuing out of a pintle injector.They inferred that for lower degrees of superheataerodynamic forces dominate, while bubble growthtakes over at higher degrees of superheat. Zuo etal. (2000) [15] provided analysis of the externalflashing gasoline sprays downstream of the nozzleexit for the case of pressure-swirl atomizer. Theyconsidered vaporization of droplets caused by bothflashing and high temperature convective conditions.Lu et al. [16] resorted to LES modeling of flashingsprays and included vaporization and condensationsub-models in their Eulerian-Lagrangian Spray At-omization (ELSA) framework and presented somepreliminary results of LES simulations without fullimplementation of ELSA approach, stating that thefull implementation of their approach was still un-der progress. Another type of flash boiling modelbased on empirical coefficients is the HomogeneousRelaxation Model (HRM) [17, 18, 19, 20] which rep-resents the phase transition by one empirical equa-tion, which estimates the time-scale of phase change.The time-scale evaluates the temporal extent of the

deviation of the local condition from thermal equi-librium.

HRM lies in between the two extremes ofthermodynamic two-phase models - homogeneousequilibrium model (HEM) and homogeneous frozenmodel (HFM)[18, 4]. In case of HEM, the two-phasesare assumed to be mixed perfectly homogeneouswith the heat transfer occurring spontaneously. In areal world scenario of two-phase flows, such as bub-bly flows, instantaneous heat transfer is not feasible.The other extreme, HFM assumes zero heat trans-fer or heat transfer time-scale to be infinitley long.HRM captures the in-between practical two-phaseflow scenarios. The vapor-liquid equilibrium prop-erties are necessary for HRM implementation andhave been estimated for pure as well as blended fu-els, for GDI applications [21, 5]. Recently Moulaiet al. [22] and Saha et al. [23] carried out numeri-cal studies of iso-octane flash boiling in a multi-holeGDI fuel injector (Spray G) with conditions speci-fied in the Engine Combustion Network (ECN) [9].These studies examined cases where outlet chamberis filled with only iso-octane as well as cases wherechamber is occupied solely by nitrogen gas. They ex-plored different flashing and non-flashing conditionsin their work.

HRM has been successfully used for modelingcavitation in diesel injectors in the recent past bymany researchers [24, 25, 26, 27, 28, 29, 30]. HRMis a useful model for the industry, because of its util-ity in both diesel and gasoline direct injection sprays.It is vital to point out that the model rate equationand values of model constants in HRM have beensame for solving flashing as well as cavitating prob-lems. This raises a concern about the applicabilityof HRM for modeling two thermodynamically dif-ferent phenomena such as, cavitation and flashing.Therefore, the objective of the current study is toprovide in-depth understanding of HRM, pros andcons of the model setup and explore potential justi-fications of using HRM in flashing, non-flashing orcavitating conditions. The operating conditions andfuel injector considered in this study are based onthe baseline Spray G condition [9] and some para-metric variations of that baseline condition. The pa-per is organized in the following way. First, the nu-merical setup and the governing equations requiredwill be described, followed by simulation results in-dicating the effect of different boundary conditionson spray patterns and their thermodynamic impli-cations and classifications of non-flashing, moderateflashing and intense flashing. Finally results fromparametric study using HRM and sensitivities of themodel predictions with respect to the different model

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parameters will be discussed.

Model Formulation

The problem considered in this study is iso-octane fuel flowing through a 8-hole counter-boredGDI fuel injector, which is denoted as Spray G noz-zle in the ECN [9]. Details of the cases studied inthis work are summarized in Table 1. From priorwork on Spray G cases by our group [23] and otherresearch groups [22] it has been evident that SprayG condition is supposed to be non-flashing, since thechamber pressure is higher than the saturation pres-sure corresponding to iso-octane (liquid) fuel tem-perature. However, the chamber temperature be-ing higher than the fuel temperature, vaporizationof the liquid jets is expected due to convective heat-ing from the hot environment. In case of Spray G2flashing is expected because the chamber pressure islower than the corresponding saturation pressure ofthe fuel. For Spray G2 the chamber is consideredat room temperature, therefore, vaporization is notexpected. The Spray G3 condition has been intro-duced for the first time in this study. Spray G3 hasa higher fuel temperature and is subjected to cham-ber pressure of 1 atmosphere. Thus, the degree ofsuperheat is higher than that of Spray G2 indicat-ing possibility of more than mild/moderate flashing.The chamber temperature of Spray G3 is at roomtemperature, hence, there is no possibility of vapor-ization of liquid jets for Spray G3.

0.1 Nozzle Geometry and Computational Domain

The internal geometry of Spray G nozzle is notsymmetric since it has 8 holes and 5 dimples whichresults in uneven upstream flow passages leading tocounter-bored, i.e., stepped holes. The nominal di-ameters of the holes and the counter-bores are 165µm and 388 µm, respectively. The length to diam-eter ratio of the nozzle holes and counter-bores arearound 1 and 1.2, respectively. Therefore for real-istic predictions, full nozzle geometry has been sim-ulated in this study. The nominal geometry of theSpray G nozzle has been obtained through the ECN.Figure 1 shows the location of the 8 stepped holeswith respect to the 5 dimples. The upstream asym-metry in the nozzle geometry is evident from thisfigure. The information about the orientation of theholes is vital for cases where hole-to-hole variationsbecome conspicuous inside the nozzle and/or in thenear-nozzle regions. A hemispherical outlet domainis needed to ensure that the near-nozzle flow featuresare modeled accurately. It is difficult to have a cor-rect a priori estimation of the extent of the outletdomain. Results using outlet domain of 9 mm di-ameter have been reported in this study. Previous

Figure 1. (Left) Layout of the 8 stepped holes ofSpray G injector (bottom view), and (Right) thecomputational domain considered in this study

study [23] indicated that results using 9 mm outletdomain could be reasonable for internal and near-nozzle flow analysis at a physical time-stamp of 0.1ms.

Simulations are performed using the CON-VERGE code [31], which uses a cut-cell techniqueto generate the mesh automatically during the runtime. A vertical cut-plane showing the mesh with9 mm outlet domain and 17.5 µm as minimum gridsize is shown in Fig. 2. Adaptive Mesh Refinement

Figure 2. Vertical cut-plane showing the mesh with17.5 µm as minimum grid size and 9 mm outlet do-main.

(AMR) has not been utilized to ensure efficient load-balancing while executing parallel computing withmillions of cells in the computational domain. Fixedembedding has been used near the walls, inside theholes and in the near-nozzle regions, and inside thechamber to capture the sharp gradients in species,velocity, temperature, etc. Three levels of embed-ding have been used which means smallest cell sizehas a dimension 23, i.e., 8 times smaller than the

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Table 1. Cases StudiedParameters Spray G Spray G2 Spray G3

Inj. Press.(MPa) 20 20 20Chamber Press.(kPa) 600 53 100Chamber Temp.(K) 573 293 293

Fuel Temp.(K) 363 363 413Chamber Fluid N2 N2 N2

Degree of Superheat N/A 12.34 40.68(∆T ) (K)

Pressure Ratio (RP ) 0.13 1.48 2.83Thermodynamic Non-flashing, Moderate Intense

State Vaporizing Flashing Flashing

largest cell size (base grid size) in the domain. Inthis work, results using 140 µm as base grid i.e. 17.5µm minimum grid size has been presented, becausethis grid resolution have been shown to provide rea-sonable estimates with affordable wall-clock times[32].

0.2 Governing Equations

The mixture multiphase model in the Eulerianframework has been used for this study. The numer-ical setup for flash boiling requires solving the setof governing equations of mass, momentum, speciesand energy conservation, additional scalar equationsfor turbulence as well as constitutive equations forliquid and vapor phases to account for variable den-sity due to pressure changes. The governing equa-tions adopted in this study are available in the liter-ature [[31, 25, 33, 26, 27]] and are provided here aswell for the convenience of the readers. The CON-VERGE code is used for the simulations and themass, momentum and energy conservation equationsare as follows:

∂ρ

∂t+∂ujρ

∂xj= S (1)

∂ρui∂t

+∂ρuiuj∂xj

= − ∂p

∂xi+∂σij∂xj

+ Si (2)

∂ρeui∂t

+∂ρeuj∂xj

= −p∂uj∂xj

+ σij∂uj∂xj

+∂

∂xj

(ρΣmDhm

∂Ym∂xj

)+

∂

∂xj

(k∂T

∂xj

)+ Se (3)

where,

σij =∂

∂xj

[(µ+ µt)

(∂ui∂xj

+∂uj∂xi− 2

3

∂ui∂xi

)](4)

and µ is the molecular viscosity and µt is the turbu-lent viscosity which is modeled as:

µt = Cµρk2

ε(5)

using the standard k−ε model. The mixture densityis calculated from:

ρ =∑

ρmαm (6)

and

αm = Ymρ

ρm(7)

where ρ is the mixture density and ρm stands forthe density of any constituent species of the mixture.The species considered are - liquid fuel, vapor fuel,N2 and O2. The densities of all the gaseous specieshave been estimated using the Ideal Gas Equation.αm and Ym represent the volume fraction and massfraction of the individual species. The individualmass fractions are determined from the species con-servation equations:

∂ρYm∂t

+∂ujρYm∂xj

=∂

∂xj

(ρD

∂Ym∂xj

)+ Sm (8)

where Sm represents the source or sink term. Thereare no source or sink terms for non-condensablegases (N2 and O2). Sv and Sl are the source termsfor vapor fuel and liquid fuel transport equations.Sv provides the rate of phase change for cavitating,flashing and vaporizing conditions. Consequently,we have Sv = −Sl. The relevant details for estimat-ing Sv are provided in the “Homogeneous RelaxationModel” sub-section.

0.3 Turbulence Model

The turbulent kinetic energy k and the turbu-lent dissipation rate ε are determined from the tur-bulence model. The standard k−ε turbulence modelhas been used. The conservation equations are:

∂ρk

∂t+∂ρujk

∂xj= τij

∂ui∂xj

+∂

∂xj

µ

Prk

∂k

∂xj− ρε (9)

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∂ρε

∂t+∂ρujε

∂xj=

∂

∂xj

(µ

Prε

∂ε

∂xj

)+ Cε3ρε

∂uj∂xj

+

(Cε1

∂ui∂xj

τij − Cε2ρε)ε

k

(10)

where Reynold’s stress is:

τij = −ρu′iu′j = 2µtSij −2

3δij

(ρk + µt

∂uj∂xj

)The model constants are as follows:

Cµ = 0.09; Cε1 = 1.44; Cε2 = 1.9; Cε3 =−1.0; 1/Prk = 1.0; 1/Prε = 0.77

0.4 Homogeneous Relaxation Model

The source term Sv in the vapor species con-servation equation is evaluated through the HRMformulation. The rate of change of local vapor qual-ity, (DxDt ) provides the estimate of Sv. HRM enables

to express DxDt as:

Dx

Dt=x− xθ

(11)

The time-scale θ is calculated as:

θ = θ0α−0.54ψ−1.76 (12)

where,

θ0 = 3.84×10−7; ψ =|psat − p|pcrit − psat

; α = αv+αN2+αO2

The values of the parameters specified above, haveknown to be effective from engineering perspectivefor flow problems with pressure higher than 10 bar[18, 5]. The injection pressure for the GDI cases in-vestigated in this study is 20 MPa (200 bar). There-fore HRM is used to capture any kind of liquid-vaporphase change such as: flash-boiling, cavitation or va-porization. It is vital to understand the relation be-tween Sm and Dx

Dt . [29] elaborated this relationshipin their work on effect of non-condensable gases oncavitation. If x0 is the computed current local vaporquality of a computational cell and x1 is the corre-sponding value for the following time-step, with atime-step size of ∆t, then x1 can be computed as -x1 = x−

(x− x0

)e−∆t/θ. Therefore,

Sv =(x1 − x0)(mvap +mliq)

V∆t= (x1−x0)ρ

(Yv + Yl)

V∆t(13)

where, V is the volume of the computational celland, mvap and mliq are mass of vapor and liquidphases respectively in that computational cell. Thevapor quality is mathematically represented as x =Yv

Yv+Yl=

mvap

mvap+mliq.

0.5 Boundary and Initial Conditions

Pressure is identified at the inlet and outlet ofthe nozzle, and no-slip boundary condition is consid-ered at the walls. No-slip condition is also assumedat the interface of liquid and vapor for the mixturemultiphase approach. Turbulence boundary condi-tions are specified by the turbulent kinetic energyand turbulence dissipation rate at the inlet and theoutlet.

A liquid fuel cannot be 100% pure, therefore, itis reasonable to consider the presence of very smallamount (≈ 10−5 mass fraction) of non-condensablegases [28, 34], which acts like nucleation-sites forcavitation initiation. The volume fraction of thegaseous phase obtained from the solution includesboth the vapor and non-condensable gases. Ini-tially no vapor is present in the computational do-main. However, non-condensable gases are presentin the liquid fuel in very small mass fractions (N2 =10−5; O2 = 10−5). The chamber is initialized withchamber pressure, chamber temperature and onlynitrogen. The 8 holes along with counter-bores, areinitialized with chamber pressure, fuel temperature,and 99.998% liquid fuel and trace amounts of non-condensable gases as mentioned before. The needlesac and the upstream of sac region upto the inlet areinitialized with a pressure of fuel injection pressure,fuel temperature and 99.998 % liquid fuel and traceamounts of non-condensable gases. Simulations aretransient, and all calculations were run until boththe inflow and the outflow mass flow rates are sta-bilized.

0.6 Solution Procedure

The commercial software, CONVERGE v2.2, isused in the current study to solve the governingequations using the finite volume approach. Rhie-Chow algorithm has been adopted to have all thetransport variables collocated at the cell center.Pressure implicit with splitting of operator (PISO)algorithm is used for pressure-velocity coupling,since it is expected to work well for unsteady com-pressible flow problems [31]. For momentum equa-tion second order central differencing is employed,while for other transport equations first order up-wind scheme has been applied. Successive Over-Relaxation (SOR) algorithm has been observed tooffer better numerical stability. The flash boilingproblem studied in this work is a two-phase com-pressible flow where sharp or nearly sharp interfacebetween liquid and vapor phases is not expected.As a result, both piecewise-linear interface calcu-lation (PLIC) and high-resolution interface captur-ing (HRIC) schemes have been de-activated for the

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present set of simulations. PLIC is known to be ef-fective for only incompressible flows and HRIC isexpected to be useful for very sharp density changessuch as compressible cavitating flows in high pres-sure diesel injector [25]. Flash boiling conditionswill have more smeared two-phase distribution witha diffused interface [23, 32].

0.7 MATERIAL PROPERTIES

Majority of the material properties, utilized inthis study, are obtained from the material databaseof the code [31]. Modifications were required for cap-turing liquid density changes with changes in pres-sure and temperature. For implementing liquid com-pressibility, reference pressure, reference density andbulk modulus information are incorporated using thedata available in the literature [35].

1 Results and Discussions

In this section, first the effect of thermodynami-cally different boundary conditions will be demon-strated through HRM predictions. Then, resultsfrom parametric study will be presented.

1.1 Boundary & Operating Conditions

In the current study, we do not show any vali-dation of the simulation setup. Our previous stud-ies [23, 32] show rigorous validation and hence isnot shown here for the sake of brevity. The differ-ent conditions studied are Spray G, Spray G2 andSpray G3 and are shown in Table 1. The expectedoutcomes for these 3 conditions have been alreadystated prior to this section. Figure 3 presents thevapor mass fraction contours at 0.1 ms. The pre-dicted results are in accordance with prior estima-tion based on thermodynamic analysis. Spray G cal-culation indicates that the periphery of the liquidjets are vaporizing due to convective heating in thehigh temperature chamber. Spray G2 is moderatelyflashing, since the degree of superheat is not veryhigh. Degree of superheat is very high for SprayG3 and hence there is abundant vapor formation,i.e., intense flashing in the chamber and significantplume-to-plume interaction, as shown in the simu-lation result. It is vital to point out that HRM isused to model any sort of phase change in the sim-ulations. Moreover, HRM has not been proven inthe literature to be effective for vaporization causedby high ambient temperature. Therefore, vapor for-mation prediction in Spray G may not necessarily bean accurate representation of the actual non-flashingbut vaporizing scenario.

The above result and corresponding discussiononly depicts how the mass fraction contours vary,when subjected to different thermodynamic condi-

tions. It is vital to keep in mind that density ofvapor fuel is order of magnitude lower than that ofliquid fuel. Therefore, the volume fraction of va-por fuel is going to be significantly higher than themass fraction values. Figure 4 presents the massfraction and volume fraction contours of vapor iso-octane for Spray G2 case. As a result, it is clearthat, even though not considerable mass of vaporis formed at moderate flashing, the vapor phase oc-cupies substantial volume in the counter-bores andfarther downstream along the spray plumes.

1.2 HRM Parametric Study

The constants of the HRM have been optimizedfor cases where water flowing through a nozzle isflashing [17, 18]. In the current study, the fluid, op-erating and boundary conditions are different from[18] wherein these model constants were optimizedthrough curve-fitting with measured data for wa-ter. Hence, analyzing the sensitivity of the modelpredictions with respect to the model constants iswarranted in studies involving different fluids withHRM. The time-scale constant θ0 and exponents ofα and ψ from Eq. 12 are varied for these sensitivitystudies. θ0 has been varied from the order of 10−6

to 10−8. Figure 5 presents the effect of changing θ0

on vapor formation for Spray G2 condition. Lowervalue of θ0 seems to trigger more flashing which isexpected since θ0 considerably lowers the equilib-rium time-scale, leading to significant increment ofDxDt throughout the run-time. Therefore, it is evi-dent that θ0 is a very influential factor, at least forthe case of moderate flashing. Intense flashing is ex-pected when pressure ratio (RP ) is greater than 2 or3 [3]and for Spray G2 RP < 2. Therefore, Spray G2should be moderately flashing and θ0 ≈ 10−7 wouldstill seem to be reasonable for Spray G2 condition.

Exponent of α has been varied from -0.27 to -0.81 with the default value being -0.54. The effectof variation is shown in Fig. 6 . There is hardly anychange in the prediction even with ±50% variationabout the default value of -0.54. It should be notedthat for this Spray G2 condition the chamber is filledwith non-condensable gases and α comprises of bothvapor fuel and non-condensable gases. Unlike θ0, αis a variable that approaches a constant value (≤ 1)as the fluid travels through the nozzle and is con-vected downstream. Hence, along the fluid path linethe influence of α on θ diminishes. Therefore, α isnot going to be a dominant factor for the cases ofinterest in this study.

The parameter ψ, in Eq. 12, incorporates theeffect of fuel physical properties in the time-scale es-timation, as well as, the essence of local thermody-

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Figure 3. Effect of operating and boundary conditions on two-phase flow through Spray G injector: G, G2and G3. Vapor mass fraction contours are shown through a cut-plane showing 2 of the 8 holes.

Figure 4. Mass fraction and volume fraction contours of vapor iso-octane on two-phase flow through SprayG injector under G2 condition from Table 1.

Figure 5. Effect of changing θ0 on fuel vapor mass fraction contour for Spray G injector under G2 conditionfrom Table 1.

.

Figure 6. Effect of changing the exponent of α on fuel vapor mass fraction contour for Spray G injectorunder G2 condition from Table 1.

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Figure 7. Effect of changing the exponent of ψ on fuel vapor mass fraction contours for Spray G injectorunder G2 condition from Table 1.

namic condition. With increase in temperature forany given fuel, the corresponding saturation pres-sure increases and so does the value of ψ. This isdue to the fact that with increase in psat, denomina-tor of ψ decreases and numerator increases, providedthe local pressure remains the same. Exponent of ψhas been varied from -0.88 to -2.64 with the defaultvalue being -1.76. ψ affects significantly the vaporformation, as depicted in the Fig. 7 . ψ can varyby order of magnitude depending on local pressurevalues. Consequently, in flashing prone zones, ψ isgoing to have significantly high values and it con-siderably influences the vapor formation. As previ-ously mentioned, Spray G2 condition is supposed tomoderately flash and hence, default value of -1.76as exponent of ψ should be reasonable for Spray G2condition.

In the code there is another crucial parametercalled ψmin. For cases where local pressure is equalto the saturation pressure, value of DxDt in Eq. 11 willblow to infinity, because θ becomes zero. To negatethis issue, ψmin has been used as a thresholding pa-rameter, such that when value of ψ is below ψmin

the value of ψmin will be used instead of ψ. Typi-cally the value of this parameter is set as 10−5. Thevalues have been varied by few orders, from 10−3 to10−6. Similar to α, ψ varies during the run-time andψmin becomes useful only when local pressure is veryclose to saturation pressure. However, it is impera-tive from Fig. 8 that prediction of vapor formationis not highly sensitive to ψmin values.

On the basis of results presented at this point,it can be inferred that time-scale θ is strongly influ-enced by θ0 and ψ as well as its exponent. Never-theless, it is not very clear how θ affects the vaporproduction in order to capture the phase changesunder various thermodynamic conditions. A sim-ple mathematical exercise can help in visualizing thequalitative trend of θ. It is known that local pres-sure can vary from tens of Pa inside the injectorholes to few hundreds of kPa in the near-nozzle re-

gion in GDI engines, where flashing usually occurs.Using the mathematical expression of θ in Eq. 12,the pattern of variation of θ can be investigated fora range of α, and ψ i.e., local pressure, for a givenfuel temperature. For fuel temperatures of 300 Kand 360 K, Fig. 9 presents variation of θ with αand local pressure. For a fixed fuel temperature,saturation pressure is fixed and therefore, variationof local pressure over a wide range would encom-pass the possible range of variation of ψ. The localpressures have been selected such that the qualita-tive pattern of variation of θ becomes apparent. Thepeak values are occurring typically around the sat-uration pressure for both 300 K (psat = 7,320 Pa)and 360 K (psat = 70,900 Pa). The peak value of θ,however, will definitely change depending on selec-tion of local pressure, close to the saturation pres-sure. The variations of θ indicates again, that thelocal pressure or to be precise, the ψ parameter ismore dominant factor than α for influencing θ. Forboth the cases, it is seen that θ decreases when αapproaches unity and is more evident from the mag-nified view of the 360 K case. Under typical phasechange conditions the tendency to produce vapor issupposed to decrease as the gas phase approaches itspeak concentration. HRM was originally developedwith measured data for mixture of liquid and vaporphases only. Moreover, in computational cells wheregaseous phase (vapor fuel + non-condensable gases)is predominant, chance of local vapor quality beingclose to equilibrium is high. As a result, behavior ofθ with change in α appears to be logical.

Since ψ is more dominant factor, it is importantto focus on its effect on θ. Figure 10 helps in clarify-ing the role of ψ in influencing θ. Close to saturationpressure for a given fuel temperature, ψ drops downdrastically, leading to an abrupt rise in θ, as seen inFig. 9. Another noteworthy aspect is that variationof ψ (and as a consequence the same of θ) in Fig. 9about the saturation pressure is symmetric. In thisregard, HRM suffers from the same fundamental in-

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Figure 8. Effect of changing the ψmin on fuel vapor mass fraction contour for the Spray G injector underG2 condition from Table 1.

Figure 9. Variation of θ with α and local pressure (kPa) for fuel temperatures of 300 K and 360 K.Zoomed-in view highlights the extent of variations when θ is approaching the peak values for fuel at 360 K.

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Figure 10. Variation of ψ with local pressure (kPa) for fuel temperatures of 300 K and 360 K.

adequacy, as the Linear Rayleigh Equation basedcavitation models used in single-fluid (mixture orVOF) and two-fluid (Eulerian-Eulerian) models forstudying in-nozzle analyses of diesel injectors [34].Fundamentally, the process of bubble growth andcollapse are not same. Nevertheless, the time-scalerequired to capture complex bubble collapse is toosmall to be considered for multi-dimensional simu-lation involving cluster of bubbles or cavities [34].As a result effectiveness of HRM, in terms of vaporcondensation, for engineering calculations is not in-ferior compared to the commonly adopted, simplebubble-based approaches.

The next aspect to observe is that the peakvalues of θ are similar for both the fuel tempera-tures. However, for chamber pressures in the orderof 50 kPa and higher, θ values are larger for 360 Kcompared to 300 K. As explained previously, non-equilibrium phenomenon, i.e., flashing tends to oc-cur at elevated fuel temperatures. Additionally, θ for360 K drops down noticeably with further decreasein local pressure. The local pressure inside the GDIholes are in the order of few hundred Pascals. Conse-quently, the equilibrium time-scales inside the holesare smaller than those outside the injector, in thechamber. As a result phase changes inside the holescan be considered cavitation as the non-equilibriumeffects are low due to small values of θ. Keepingin mind chamber pressure of Spray G2 condition is53 kPa, the predictions of HRM are very realisticin physical sense for GDI applications. However, for300 K case, θ peaks at a very low local pressure, sinceat low fuel temperature the saturation pressure willalso be low. Cavitation, being a near equilibriumphenomenon, should be represented by small equi-librium time-scales, which is clearly not getting sat-

isfied. This is because of the bound at 0 Pa, whichbasically limits the minimum value of theta for verylow local pressures. Therefore, for simulation of cav-itation in high pressure diesel injectors with low fueltemperatures, θ predictions could not be realistic.This indicates predictions of HRM is more meaning-ful for high fuel temperatures. It can thus be inferredthat HRM is a phenomenological model, because it isnot derived from the first principles. Therefore, de-spite having some physical basis, θ variation mightnot always be 100 % fundamentally correct. Ideally,a phase change sub-model should consider the ef-fect of local pressure, local temperature and physicalproperties of the fuel. Inclusion of non-dimensionalnumbers like Jakob number in the model formula-tion, thus, could be useful for physically meaningfulsub-models. Nevertheless, such endeavors constitutescope for future work.

Eventually it will be interesting to see how theabove analysis can be reflected in the typical sit-uations of GDI engine. For Spray G, G2 and G3conditions, Fig. 11 shows the vapor concentrations,as well as the θ values in different computationalcells in the scatter plots of local pressure versus lo-cal temperature. The blue lines correspond to thesaturation pressure versus saturation temperaturecurve for iso-octane. The cells belong to one of the8 holes and the cells in the corresponding counter-bore are also included. It is evident that cells hav-ing local pressure lower than the saturation pressurecomprise the major portion of vapor fuel occupy-ing cells. The cells with local pressure close to thesaturation pressure have the peak θ values. Inter-estingly, few cells with local pressure above the sat-uration pressure also have some fuel vapor, owing tothe mathematical nature of θ, which feeds in to the

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Figure 11. Scatter plot of local pressure and local temperature in the different computational cells in oneof the 8 holes (including the counterbore) of the Spray G injector under Spray G, G2 and G3 conditions,colored by θ and vapor mass fraction (blue line corresponds to the saturation pressure versus saturationtemperature curve for iso-octane.)

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source term equation of fuel vapor species equation.The temperature variations for different conditionsare caused by differences in the chamber tempera-tures. Previous works [22, 23, 32] have indicatedthat there is backflow of the chamber gas in to thecounter-bores. Spray G is exposed to high chambertemperature (573 K), while for Spray G2 and G3conditions the chambers are at room temperature(293 K). The fuel temperature being 363 K for G andG2, and 413 K for G3, the distributions of the cells inthe scatter plots make sense for all the three condi-tions. In Spray G condition, the fuel heating processoccurs entirely at pressure levels above the satura-tion curve. Cavitation is clearly shown by those cellswith isothermal pressure drop and recover. In SprayG2 , the low chamber temperature induces some fuelcooling, while simultaneously vapor is being formedbecause of the pressure being below the saturationlevel. In Spray G3, the same process occurs, but athigher fuel temperatures, leading to a larger amountof fuel vapor formation.

2 Summary And Concluding Remarks

A numerical parametric study has been carriedout for a typical GDI fuel injector. The HRM isshown to be capable of predicting non-flashing yetvaporizing, moderately flashing and intensely flash-ing scenarios. The set of parametric studies lead tofollowing inferences:

1. ψ varies over several orders of magnitude, bothspatially and temporally, in a simulation. Thus,local pressure and physical properties signifi-cantly affect HRM predictions, since ψ turnedout to be a dominant factor for vapor produc-tion.

2. The time scale constant (θ0) is another domi-nant factor affecting the vapor distribution con-siderably. Lower values of θ0 seems to triggermore flashing, the reason for this was explained.Reasonable values of θ0 were suggested.

3. Void fraction, i.e., α did not appear to be an in-fluential variable in the HRM prediction. Typ-ically α can only vary from 0 to 1. For SprayG conditions the chamber is initially filled withN2 gas and hence, α does not vary much duringthe run-time, especially for flashing conditions.

4. HRM is effective for simulating phase changesfor GDI spray applications, despite the fact thatit is a phenomenological model with tunable co-efficients. This is because, at typical GDI enginechamber pressures, HRM is capable of predict-ing larger equilibrium time-scales for high fuel

temperatures when compared to those for lowfuel temperatures. Thus, HRM provides realis-tic depiction of the flashing scenario.

Overall, HRM even with its caveats is a desir-able option for internal combustion engines, whetherit is diesel or gasoline fuel. Future modifications inHRM could include effects of Jakob number to fur-ther improve its fundamental basis.

Acknowledgment

This research was partially funded by DOE’s Of-fice of Vehicle Technologies, Office of Energy Effi-ciency and Renewable Energy under Contract No.DE-AC02-06CH11357. The authors wish to thankGurpreet Singh and Leo Breton at DOE, for theirsupport.

The authors would like to acknowledge RonaldGrover at General Motors and Lyle Pickett at San-dia National Laboratory for providing the nominalgeometry of Spray G nozzle. The authors want toexpress their gratitude to Yanheng Li and Eric Pom-raning at Convergent Sciences Inc. for providingbetter understanding of volume-averaging methodfor VOF simulations.

The authors are grateful to LCRC computingresources at Argonne National Laboratory.

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Nomenclature

CFL Courant Friedrichs Lewy numberD Mass diffusivity (m2/s)e Specific internal energy (J/kg)hlv Latent heat of vaporization (kJ/kg)hm Specific enthalpy of mth species (kJ/kg)Ja Jakob Numberk Turbulent kinetic energy (m2/s2)

mvap,mliq Mass of vapor and liquid phases in a cell (kg)p Local pressure (Pa)

Pch, Pchamber Chamber pressure (kPa)pcrit Critical pressure (Pa))Pinj Injection pressure (MPa)

Psat, psat Saturation pressure (Pa)RP Pressure ratio

S, Si, Se, Sm Source term in conservation equationsSij Strain rate (1/s)t Time (s)T Local cell temperature (K)Tfuel Fuel temperature (K)Tsat Saturation temperature (K)u Local cell velocity (m/s)ui Advecting mean velocity (m/s)uj Advected mean velocity (m/s)V Volume of a cell (m3)x Local cell vapor qualityx Local cell equilibrium qualityYm Mass fraction of mth species

Greekα Void fraction (vapor and non-condensable gases)ε Turbulence dissipation rate (m2/s3)θ,θ0 Equilibrium time-scale and Empirical time-constant(s)µ,µt Dynamic and Turbulent viscosity coefficient (kg/m.s)

ρ, ρv, ρl, ρg Density of mixture, vapor, liquid and gas (kg/m3)σij Strain rate tensor (1/s)τij Reynold’s stress (Pa)

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