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International Journal for Multiscale Computational Engineering, 4(5&6)601–616(2006) Discrete Bubble Modeling of Unsteady Cavitating Flow Zhiliang Xu, 1,* Myoungnyoun Kim, 1,* Tianshi Lu, 1 Wonho Oh, 2,*,** James Glimm, 1,2,*,** Roman Samulyak, 1 Xiaolin Li, 2,** and Constantine Tzanos 3 1 Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973-5000 2 Department of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, NY 11794-3600 3 Reactor Analysis and Engineering, Argonne National Laboratory, Argonne, IL 60493 ABSTRACT A discrete vapor bubble model is developed to simulate unsteady cavitating flows. In this model, the mixed vapor-liquid mixture is modeled as a system of pure phase domains (vapor and liquid) separated by free interfaces. On the phase boundary, a numerical solution for the phase transition is developed for compressible flows. This model is used to study the effect of cavitation bubbles on atomization, i.e., the breakup of a high-speed jet and spray formation. The major conclusion is that a multiscale (three-scale) model is sufficient to achieve agreement with quantitative macroscale flow parameters, such as spray opening angle and spray volume fraction or density, or as a qualitative measure, the occurrence of spray formation. The authors believe this to be the first numerical study of the atomization process at such a level of detail in modeling of the related physics. * This research is supported in part by the U.S. Department of Energy, TSTT at BNL with Contract No. DE-AC02- 98CH10886, and TSTT at SB with Contract No. DEFC0201ER25461 ** Supported in part by the National Science Foundation Contract No. DMS0102480 0731-8898/06/$35.00 c 2006 by Begell House, Inc. 601 Begell House Inc., http://begellhouse.com Downloaded 2007-4-5 from IP 130.199.3.130 by Dr. Tianshi Lu (tianshilu)

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Page 1: Discrete Bubble Modeling of Unsteady Cavitating Flotlu/resume/diesel.pdf · Discrete Bubble Modeling of Unsteady Cavitating Flow ... cavitation bubbles on atomization, i.e., the breakup

International Journal for Multiscale Computational Engineering, 4(5&6)601–616(2006)

Discrete Bubble Modeling of UnsteadyCavitating Flow

Zhiliang Xu,1,∗ Myoungnyoun Kim,1,∗ Tianshi Lu,1

Wonho Oh,2,∗,∗∗ James Glimm,1,2,∗,∗∗ Roman Samulyak,1

Xiaolin Li,2,∗∗ and Constantine Tzanos3

1Computational Science Center, Brookhaven National Laboratory,Upton, NY 11973-5000

2Department of Applied Mathematics and Statistics, University at Stony Brook,Stony Brook, NY 11794-3600

3Reactor Analysis and Engineering, Argonne National Laboratory,Argonne, IL 60493

ABSTRACT

A discrete vapor bubble model is developed to simulate unsteady cavitating flows. In thismodel, the mixed vapor-liquid mixture is modeled as a system of pure phase domains (vaporand liquid) separated by free interfaces. On the phase boundary, a numerical solution for thephase transition is developed for compressible flows. This model is used to study the effect ofcavitation bubbles on atomization, i.e., the breakup of a high-speed jet and spray formation.The major conclusion is that a multiscale (three-scale) model is sufficient to achieve agreementwith quantitative macroscale flow parameters, such as spray opening angle and spray volumefraction or density, or as a qualitative measure, the occurrence of spray formation. The authorsbelieve this to be the first numerical study of the atomization process at such a level of detailin modeling of the related physics.

∗This research is supported in part by the U.S. Department of Energy, TSTT at BNL with Contract No. DE-AC02-98CH10886, and TSTT at SB with Contract No. DEFC0201ER25461∗∗Supported in part by the National Science Foundation Contract No. DMS0102480

0731-8898/06/$35.00 c© 2006 by Begell House, Inc. 601

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602 XU ET AL

NOMENCLATURE

c sound speed V volumeh bubble spacing v vaporE specific total energyEC critical energy Greek SymbolsE∞ energy translationJ nucleation rate γ adiabatic exponentkb Boltzmann’s constant ρ densityl liquid κ thermal conduction coefficientM mass flux ε specific internal energym mass of a molecule 4PC critical tensionN number density of the liquid σ surface tensionP pressure Σ Nucleation probabilityP∞ stiffening constant α Evaporation coefficientPv vapor pressure Γ Gruneisen coefficientPC negative pressure threshold ∆x Grid spacingQv heat of evaporizationRC critical radius AcronymsRe Reynolds numberr bubble radius AMR adaptive mesh refinementS specific entropy CFL courant, Friedrichs, and Lewy

conditions speed of the moving phase boundary EOS equation of stateT absolute temperature MUSCL monotone upstream-centered

schemes for conservation lawsTs interface temperature NS equations Navier-Stokes equationst time

1. INTRODUCTION

Arisen from high-speed nozzle flows, liquid cavita-tion, and jet atomization (i.e., jet breakup and sprayformation) are great challenges for computationalscience, as well as for theory. This fact is due tothe large range of spatial and temporal scales andthe complex flow regimes involved. Fluid mecha-nisms leading to spray formation are still debated.Fluid surface instabilities, parameters such as noz-zle shape, and the internal nozzle flow pattern (suchas turbulence and cavitation) all could be possi-ble mechanisms responsible for jet breakup and at-omization. For a literature review of jet atomiza-tion mechanisms (see [1–3]). Among these mech-

anisms, unsteady cavitation in fluid flow is a domi-nant effect in high-speed nozzle flows in many high-pressure injectors. Cavitation bubbles are formeddue to the sharp inlet corner of the nozzle and a”vena contracta,” i.e., the flow ”sticks” to the edgesof the opening, thus effectively reducing the size ofthe opening, inside the nozzle [4]. The simulation ofcavitation and the study of its role in the jet atom-ization is the main goal of the present paper.

Visual observation of cavitation in injectors is dif-ficult because of the small sizes of nozzles [5] andclouds around cavitation bubbles. To further under-stand cavitation in nozzles, many numerical mod-els have been proposed. Most of them employed acontinuum modeling or a single pseudofluid equa-

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 603

tion of state for multiphase (bubbly) flows [7]. Forthis class of methods, the key issue is to developa proper constitutive law for the mixture [8]. Of-ten the mixture is assumed to be homogeneous andbarotropic. Despite its simplicity, this class of meth-ods often lacks an ability to resolve detailed physics,such as drag, surface tension, phase transition, vis-cous friction between two phases, and other micro-physics phenomena. Moreover, cavitation leadingto the jet atomization has not been studied for thisclass of methods. A review of continuum model-ing and simulation of cavitating flows in nozzles isgiven in [9].

In this paper, we follow a different approach inthe study of cavitating flows and the jet atomiza-tion: the heterogeneous method, or direct numer-ical simulation. The central part of the proposedmethod is the discrete vapor bubble model. Our nu-merical simulation of multiphase systems is basedon the front tracking method for explicit numeri-cal resolution of phase boundaries, and the Fron-Tier hydrodynamic code [10–13]. The front track-ing method has been used extensively to study fluidinterface instability problems, such as fluid mixing,and is capable of resolving complex fluid interfaces.These studies have achieved good agreements withexperiments. See also [10–13], and references citedthere. The direct numerical simulation is a powerfulmethod for studying multiphase problems based ontechniques developed for free surface flows and hasalready attracted the attention of many researchers..Welch [14] studied a single vapor bubble in a liq-uid, including interface tracking with mass trans-fer while the phase boundary was assumed to ex-ist in thermal and chemical (Gibbs potential) equi-librium. Juric and Tryggvason [15] simulated boil-ing flows using the incompressible approximationfor both liquid and vapor phases, and a nonequi-librium phase transition model with a parametercalled kinetic mobility whose value was measuredexperimentally. In contrast, we solve compressiblefluid dynamics equations for both liquid and va-por phases, as the cavitation phenomena dependson nonlinear waves (rarefaction waves) in the liq-uid. To describe cavitation, a discrete vapor bubblemode, introduced in [6] to describe the mixed phaseregion, is further developed. Using this model tostudy cavitation in a Mercury jet has achieved goodagreements with experiments [6]. In this model, theliquid-vapor mixture is described by pure phase do-

mains (vapor bubble and liquid) separated by freeinterfaces. The critical pressure necessary for theformation of a cavitation bubble, the initial radius,and parameters regulating the bubble populationare estimated by the homogeneous nucleation the-ory. The numerical phase boundary solution is ob-tained by solving the compressible Navier-Stokesequations, which includes the influence of thermalnonequilibrium and the discontinuity of state vari-ables across the phase transition boundary. The onlyprior information needed to obtain such a phasetransition solution is the phase exchange rate coef-ficient, which is set from experimental considera-tions. Matsumoto and Takemura [16] studied nu-merically the influence of internal phenomena onthe motion of a single gas bubble with compressibleNavier-Stokes equations and the interfacial dynam-ics of phase transitions. Their governing equationcontain more technical details, such as the concen-tration of noncondensible gas in the liquid and va-por phases. In [16], the temperature field was firstsolved from the simplified energy equations, andother fields were solved afterward. Our approachupdates all fields simultaneously. Therefore, the dy-namics of the phase boundary in our approach iscoupled to acoustic waves in the interior. Moreover,[16] only studied single bubble dynamics. Multiplebubble dynamics is studied in the present paper.

From a computational point of view, we ar-gue that cavitation and spray formation should betreated as a multiscale phenomena. Macroscopicflow parameters, such as the spray opening angleand the volume fractions of liquid within the spray,are on the scale of centimeter. The finite vapor bub-ble size, which is essential for the spray formation inour simulations, is on the scale of microns. The dy-namics of the moving phase boundary, with masstransfer across the phase boundary, depends on athermal diffusion layer, having width of the orderof nanometers. The model developed in the pa-per is multiscale and multiphysics. It is applica-ble to a range of problems involving unsteady cav-itating flows. The main conclusion of this paper isthat modeling of the mixed phase region in termsof finite-sized vapor bubbles in the liquid achievesmacroscopic flow descriptions, such as the sprayopening angle and the liquid mass flux within thespray, is in approximate agreement with experimen-tal measurements.

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604 XU ET AL

In our study of unsteady cavitating flows lead-ing to spray formation, we neglect some other lessimportant phenomena. For example, turbulence ap-pears not to play a significant role in the nozzle flow.Although the flow Reynolds number is above a crit-ical value for transition, the length of the nozzle isbelow the critical length and there is not enoughtime (or distance) for the transition into turbulence.Surface tension is also neglected. It is easy to esti-mate that the surface tension is not negligible onlyfor submicron-size bubbles for the liquid. Techni-cally, such bubbles are still beyond the current nu-merical resolution. They occur as nucleation sitesand eventually grow into the larger bubbles studiedhere.

The paper is organized as follows. In Section 2,numerical algorithms implemented in the discretevapor bubble model are described. To allow the cre-ation of vapor bubbles, a dynamic bubble insertionalgorithm is developed and a new type of Riemannproblem associated with liquid-vapor phase changeis solved. In Section 3.1, the internal nozzle flow isstudied. Section 3.2 contains the discrete vapor bub-ble model simulation results. Section 4 discusses thedifferences in the predicted results of two models.

2. NUMERICAL ALGORITHMS

Numerical simulations of the jet atomizationreported in this paper attempt to accuratelyreproduce experimental conditions. The schematicof the high-pressure chamber and nozzle are shownin Fig. 1. To simulate the formation of the jet,only inflow boundary conditions with time-dependent pressure are prescribed at the inletto the chamber. A flow-through boundary con-dition is used at the flow outlet, and a no slipcondition is used at the nozzle wall. The formu-lation of these boundary conditions follow [22].Flows in the high-pressure chamber and theformation of the jet are simulated by solvingthe Navier-Stokes (NS) equations for both theliquid and ambient gas, and the jet–ambient-gasinterface is explicitly resolved using the fronttracking method. The liquid is treated as vis-cous and heat conducting. Because the thermalconductivity and the viscosity are both small,the Navier-Stokes equations are solved with anexplicit algorithm. In the interior, for the con-vection terms of the NS equations, a monotone

FIGURE 1. Geometry of injection reservoir and nozzleleading to combustion chamber

upstream-centered scheme for conservation laws(MUSCL scheme) [23] is used, which is sec-ond order accurate in both space and time, andfor the diffusion terms, central differencing isemployed. To update a front solution, an operatorsplitting method is used, which divides into anormal front propagation step and a tangentialfront propagation step. In the normal front propa-gation step, a front is propagated to a new positionin its normal direction by solving a generalizedRiemann problem. The method of characteristicsis used to update the states on the two sidesof the front at this new position. In the tangentialfront propagation step, a first-order Lax-Friedrichsscheme is applied.

We adopted the Berger-Colella [24] adaptivemesh refinement (AMR) to the front trackingmethod by merging FronTier with the Overturecode, the AMR package developed at LawrenceLivermore National Laboratory. All material in-terfaces are tracked. The tracked interfaces are allcovered by the finest level patches. We refer to[25] for additional details. For the simulationspresented in this paper, three levels of refinementwith a refinement factor 2 were used. The base levelhas 170 × 1000 cells on the 1.513 mm × 8.9 mm do-main, which is half of the injection reservoir andcombustion chamber as shown in Fig. 1. The finestgrid level has a mesh resolution of 2 µm. There-fore, the grid spacing ∆x of the finest grid level is∆x = 2 µm.

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 605

The equation of state (EOS) model for the pureliquid phase of the flow is given by a stiffened poly-tropic equation

P + γP∞ = (γ− 1)ρ(ε + E∞) (1)

For n-heptane used in the simulations, adiabatic ex-ponent is γ = 3.19, stiffening constant is P∞ =3000 bar and energy translation is E∞ = 4.85 ×103 erg/g. Here ρ is the density, P is the pressure,and ε is the specific internal energy. The thermalconduction of the liquid is κ = 1.3 × 104 erg/s g K.The equation of state for the ambient gas and vaporinside cavitation bubbles used in the simulations isγ-law. The thermal conduction of the n-heptane va-por is κ = 1.3× 103 erg/s g K.

2.1 Algorithms of the Discrete VaporBubble Model

The discrete vapor bubble model is at the techni-cal heart of this paper. We introduce, and resolve,two new length scales in order to achieve experi-mental agreement for macroscopic descriptions ofthe flow. The larger of the two new length scales isthe vapor bubble radius, resolved at the grid spac-ing level (microns). The subsequent dynamics ofthe bubbles depends on a thermal diffusion lengthscale at the vapor bubble (phase transition) bound-ary. This layer is measured in nanometeters, and isdescribed here through a subgrid model, as it is be-low the level of the computational grid.

2.1.1 Critical Bubble Radius

The discrete vapor bubble model accounts for thefinite-size effects of the vapor bubbles in the mixedphase flow regime. The mixed phase regime is mod-eled by vapor bubbles of finite size inserted into theliquid.

Vapor bubbles are formed by liquid vaporizationwhen the liquid pressure P fluctuates and falls be-low the saturated vapor pressure Pv at given tem-perature. The pressure fluctuation ∆P = Pv − P ,which is a positive quantity if Pv > P , is called ten-sion. Physically, a vapor bubble will appear when-ever it is thermodynamically favorable. Cavitationis the result of rapid growth of vapor nuclei that be-

come unstable due to a change in ambient pressure.If the maximum size of a nucleus is defined by theradius RC (critical radius), then at equilibrium, thecritical tension 4PC is given by [34,35]

4PC =2σ

RC(2)

where σ is the surface tension of the liquid, and thecritical radius of a cavitation bubble is

RC =2σ

4PC(3)

To create such a nucleus with critical radius RC , a

critical energy EC =16πσ3

34P 2C

[34,35] must be de-

posited into the liquid to break the barrier againstnucleation. This critical energy EC accounts onlyfor surface energy and the gain in volume energy.The energy needed to convert liquid to vapor (heatof vaporization) is neglected, because it is relativelysmall. One can write a nucleation rate J

J = J0expEC/(kbT ) (4)

per unit volume and per unit time. Here, kb is theBoltzmann’s constant, T is the absolute liquid tem-perature, and J0 is a factor of proportionality de-fined as

J0 = N

(2σ

πm

)1/2

(5)

where N is the number density of the liquid(molecules/m3) and m is the mass of a molecule.For N -heptane used in the present study, m = 1.66×10−22 g and N = 3.98 × 1027 molecules/m3. Thus,the nucleation probability Σ in a volume V during atime period t is [36]

Σ = 1− exp−J0V t exp[−EC/(kbT )](6)

Equations (3)–(6) can be used to compute the neg-ative pressure threshold PC needed to create cav-itation bubbles for a given critical radius and nu-cleation probability. For the nucleation probabilityΣ = 0.5, this relation is given by

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606 XU ET AL

PC∼= −

(16πσ3

3kbT ln(J0V t/ln2)

)1/2

(7)

We would like to note that Eq. (7) is applicableto only very clean fluids. It is well known thatreal fluids contain large amount of nucleation cen-ters that increase the cavitation probability, but ac-curate experimental measurements of critical cavita-tion parameters are not available. Since we are inter-ested in the simulation of real fluids, we assume that(7) gives the correct functional relation between thecritical pressure, volume, and the nucleation proba-bility, whereas the absolute value of the critical vol-ume at a specific value of the critical pressure has tobe calibrated through the comparison of simulationresults to experimental data. A phenomenologicalbubble-spacing parameter h is chosen by a series ofnumerical experiments. By calibrating this parame-ter, we also account for effects of initial growth ofsubmicron bubbles, as the initial growth stage ofthese submicron bubbles is not resolved by the cur-rent numerical resolution.

2.1.2 Dynamic Creation of Vapor Bubbles

The bubble spacing h, which is defined as the mini-mum distance between the centers of two newly cre-ated bubbles, is controlled at the time of bubble in-sertion to account for the distribution of nucleationcenters and model the effect of microbubble growth-induced pressure reduction on the nucleation prob-ability. For the present study, the default bubble ra-dius at the time of insertion is r = 2∆x = 4 µm,which meets the requirement for the minimum nu-merical resolution. The default bubble spacing ish = 4r. The choice of bubble spacing and bubbleradius will be discussed in Section 3.1.

The dynamic bubble creation algorithm proceedsas follows. In each cell at every time step, it ischecked whether the liquid pressure p is less than−10 bar. Here, −10 bar is the negative pressurethreshold PC estimated using relations given inSection 2.1.1 for the liquid n-heptane used in thepresent study. The properties of n-heptane is givenin Table 1. If there is a 2r × 2r block thats p < −10

bar centered in a larger h×h region, which does notcontain bubbles, a circular bubble of radius r is in-serted into this block. For the vapor bubble states,the temperature and velocity are set to be the av-erage temperature T and velocity U of the liquidthat occupied this block. From the static Clausius-Clapeyron relation, the initial vapor pressure is setto be Psat(T ), which is the saturated liquid-vaporpressure at a temperature T . The vapor density iscomputed from the EOS.

An application of the discrete vapor bubblemodel for the description of cavitation in mercuryhas been presented in [37]. Simulations were per-formed at conditions typical for the Muon Collidertarget experiments. Juric and Tryggvason [15] alsoused the discrete vapor bubble model to simulatefilm boiling.

2.1.3 Dynamic Phase Boundaries forCompressible Fluids

The phase boundary is modeled as a sharp interfacein the present paper. The viscosity and surface ten-sion on the interface are neglected because thermaleffects are normally dominant over viscous effectsin phase transitions. The phase transition is thengoverned by the compressible Euler equations withheat diffusion,

ρt + (ρu)x = 0 (8)

(ρu)t + (ρu2)x + Px = 0 (9)

(ρE)t + (ρEu + Pu− κTx)x = 0 (10)

where subscripts t and x are used to denote deriva-tives with respect to time and space, respectively,E = 1/2(u2) + ε is the specific total energy, ε is thespecific internal energy, P is the pressure, κ is thethermal conductivity, and T is the absolute temper-ature.

Integration of the governing equations across theinterface yields the jump conditions for the dynamicphase boundary

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 607

TABLE 1. The thermal properties of N -heptane and No. 2 diesel liquid fuel measured at 298 K

Liquid type Density Sound speed Dynamic viscosity Surface tensiong/cm3 cm/ms g/(cm.ms) g/ms2

N -heptane 0.66 12 0.4× 10−5 1.96× 10−5

No. 2 diesel 0.843 16.9 0.25× 10−4 3× 10−5

[ρu] = s[ρ] (11)

[ρu2 + P ] = s[ρu] (12)

[ρuE + Pu− κTx] = s[ρE] (13)

where s is the speed of the moving phase boundaryand the symbol [U ] = Uleft − Uright denotes a jumpacross the phase boundary. Equations (11)–(13) give

ρv(uv − s) = ρl(ul − s) (14)

ρv(uv − s)2 + Pv = ρl(ul − s)2 + Pl (15)

(ρvEv+Pv)(uv−s)−κvTv,x =(ρlEl+Pl)(ul−s)−κlTl,x (16)

and the change of energy during the phase transi-tion is

εv +Pv

ρv= εl +

Pl

ρl+ Qv (17)

where Qv is the heat of vaporization and subscriptsl and v denote liquid and vapor, respectively.

For the mass flux M = ρv(uv−s) = ρl(ul−s), themass and momentum balance equations give

M =uv − ul

τv − τl(18)

M2 = −Pv − Pl

τv − τl, τ =

(19)

(uv − s)(ul − s) =Pv − Pl

ρv − ρl(20)

A combination of these equations with the energybalance equation leads to the generalized Hugoniotrelation

εl−εv+Pl + Pv

2(τl−τv) =

1M

(κvTv,x−κlTl,x) (21)

Since the mass flux M is unknown, one moreequation is needed to close the system. The ki-netic theory of evaporation gives the evaporationrate with a coefficient determined experimentally.Denote the vapor and liquid equilibrium pressureat temperature T by psat(T ), which satisfies theClausius-Clapeyron equation. The net mass flux is

Mev = αPsat(T )− Pv√

2πRT(22)

with given absolute temperature T and pressure pv.Here R = kB/m, where m is the molecular massand kB is the Boltzmann constant. α is called theevaporation coefficient or sometimes the condensa-tion coefficient. Research has shown that the valueof α < 1 results from either impurity of substancesor the deficiency in derivation kinetic theory modelused to defined α. A formula similar to Eq. (22) wasderived by Alty and Mackay [38] for condensation.Reviews of experiments and theories regarding theevaporation coefficient are given in [40,41]. In thepresent study, a value of 0.4 is used for α. Differentvalues of α have been studied numerically. The con-clusion is that influences of α on the phase bound-ary dynamics is not significant unless it is very smallcompared to unity [39].

The numerical algorithm for a dynamical phasetransition proceeds as follows. The characteristicform of Eqs. (8)–(10) is used at the interface

dP

dλ++ ρc

du

dλ+= ΓκTxx (23)

dP

dλ−− ρc

du

dλ−= ΓκTxx (24)

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608 XU ET AL

dλ0+ p

dλ0=

Txx (25)

where c =√

(∂P/∂ρ)S is the sound speed, S is thespecific entropy, and Γ is the Gruneisen coefficient.The characteristic derivatives λ+, λ−, and λ0 are de-fined by

ddλ+

=∂

∂t+ (u + c)

∂x

ddλ−

=∂

∂t+ (u− c)

∂x

ddλ0

=∂

∂t+ u

∂x

The phase boundary conditions are (18)–(21) and(22). To determine the interface temperature, it ispostulated that the temperatures of liquid and va-por at the interface are continuous Ts = Tl = Tv.This approach was also used in [44]. Thus, the in-terface temperature Ts is coupled to the phase tran-sition equations.

To solve the characteristic system with the phaseboundary conditions, an iteration algorithm is de-veloped. Backward characteristics are traced froman estimated location of the phase boundary at thenext time step. States at the feet of these character-istics (at the current time level) are first obtained byinterpolation. The characteristic speeds are λ± =u ± c and λ0 = u, whereas the phase transition

interface moves at a speed s = ∆(ρu)/∆ρ. For asmall time step ∆t, the characteristics are approxi-mated by straight lines. Figure 2 is a schematic di-agram showing the backward characteristics. Theleft foot is (cl + ul − s)∆t away from the interface,while the right foot is (cr−ur + s)∆t away from theinterface. During the iteration, the characteristicsare obtained by explicit formulas, and are thus ob-tained once only per time step, external to the ther-modynamic iteration. This simplification of charac-teristics is valid because s ¿ c. Once the states atthe characteristic feet are obtained by interpolation,the characteristic equations are solved by integrat-ing along the characteristics.

Assuming that the vapor is on the right side, thealgorithm can be described as follows:

1. Substituting the vapor pressure at S0+ forPv in Eq. (22) and discretizing (21), we havetwo equations that we solve for the two un-known variables Ts and M . The function Psat

is a nonlinear function. Since this step itselfis in the iteration of Pv and ∆τ, the Clausius-Clapeyron equation is linearized at a referencetemperature Tv and solved for Ts and M whilepreserving the convergence of S∗l and S∗r inFig. 2 through the iteration. For the first iter-ation step, Tv can be chosen to be equal to Ts atthe beginning of the time step if T0− = T0+. Forsubsequent iteration steps, Tv is simply the Ts

obtained in the last iteration. On linearization,we obtain solutions

FIGURE 2. Stencil for the phase boundary propagation. The new front states S∗l and S∗r are calculated

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 609

Ts =κlT−1+κvT1+

α√2πRTv

[Qv−τ̄(Pv−Pl)]∆x

[Pv−Psat(Tv)+

dpsat

dT(Tv)Tv

]

κl+κv+α√

2πRTv

[Qv−τ̄(Pv−Pl)]∆xdPsat

dT(Tv)

(26)

and

Mev =α√

2πRTv

×[Psat(Tv)− Pv +

dPsat

dT(Tv)(Ts − Tv)

](27)

wheredPsat

dT(Tv) is the slope of phase coexis-

tence curve at Tv by Clausius-Clapeyron equa-

tion. Here, τ̄ is τ̄ =τv + τl

2.

2. Having obtained the temperature Ts and massflux Mev , the characteristic equations (23) and(24) are solved with the Rankine-Hugoniot con-ditions (18) and (19). For the first iteration, the∆τ in (18) and (19) is set to τv0 − τl0. The loca-tion of the backward characteristic during theiteration steps is not changed because s ¿ c.The density ρ∗l and ρ∗v are determined fromtheir EOS’s with the pressure and temperatureobtained in the iteration.

3. Convergence of the iteration is controlled by Pv

and ∆τ. The last step in the iteration is to com-pare the newly obtained pv and ∆τ to the val-ues from the previous the iteration. The rel-ative error is defined to be the difference be-tween a value from the current iteration stepand that from the previous iteration divided bythe value from the current iteration. If the rel-ative errors of both Pv and ∆τ are smaller thanthe given tolerances, the iteration is over. Oth-erwise, we update the Pv and ∆τ for next it-eration and go back to step 1. In the presentstudy, the tolerances for Pv and ∆τ have thevalue 10−9.

The phase transition problem depends on a sub-grid model to describe the thermal layer at the phasetransition boundary. Because the width of the ther-mal layer of the liquid is proportional to

√κt/ρcp,

and the time step is restricted by the Courant,Friedrichs, and Lewy (CFL) condition dt = dx/c,

which states that the domain of dependence of hy-perbolic PDEs must lie within the domain of depen-dence of the finite difference scheme at each meshpoint to guarantee the stability of an explicit finitedifference scheme, it usually requires 103 ∼ 104

steps for the liquid thermal layer to expand to amicron-scale grid cell. If the thermal layer is thin-ner than a grid cell in the finite difference scheme,the temperature profile takes the form

T ≈ Tp + (T−1 − Tp)(

x√4νt

)(28)

where Tp is the phase boundary temperature, T−1 isthe temperature one grid cell away from the phase

boundary, and ν =κt

ρcp. Thus the temperature gra-

dient at the interface is approximated by

∂T

∂x≈ T−1 − Tp√

πνt(29)

The interfacial heat flux in the form ofκ∆T

∆xshould

be replaced by the above approximation. When thethermal layer is wider than a grid cell, the conven-tional finite difference approximation of the temper-ature gradient at the interface gives satisfactory re-sults.

This is a new description of the Riemann problemassociated with a phase transition in a fully com-pressible fluid. Mass transfer across an interfacedue to the phase change is allowed. Thus, the in-terface motion also depends on the phase changeunder nonequilibrium thermodynamic and hydro-dynamic conditions. The details of the numericalstudy of phase boundaries are reported in [39].

3. SIMULATION RESULTS

For studies presented in this paper, fluid and jet noz-zle parameters have been chosen to be in the atom-ization regime, which is typical of diesel jet fluidinjection, following experiments performed at Ar-gonne National Laboratory [17] (see Fig. 1). The

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610 XU ET AL

nozzle diameter is 0.178 mm, and its length is 1mm. A finite pulse of diesel fuel is injected into achamber of SF6 (a heavy, inert gas chosen to em-ulate the density of compressed air in a diesel en-gine). In 0.3 ms, the pressure of injected fuel riseslinearly from 1 to 500 bar, then it is maintained atthis level for 0.4 ms, and subsequently, it drops lin-early to 1 bar over 0.1 ms. Synchrotron x-ray imag-ing of fuel flow parameters, such as the spray open-ing angle, the mass distribution of fuel, and the jettip velocity evolution [17–20], provide important in-formation for the validation of numerical experi-ments. In this paper, n-heptane is used for calcu-lations as a substitute for diesel fuel used in [17–20] because data for the thermodynamic properties,such as the vapor-liquid saturation, were not avail-able for diesel fuel. The thermodynamic propertiesof n-heptane, the major component of this fuel, aredescribed in [21]. In Sec. 2.1.3, the vapor-liquid satu-ration is critical in formulating a numerical solutionfor interfacial phase change. The thermal and me-chanical properties of these two fuels are comparedin Table 1.

3.1 Flow in the Nozzle

In this section, we present numerical simulation re-sults of a flow in the nozzle. We first identify thatcavitation is an important process. Then, we ob-tain an estimate on initial bubble radius and bub-ble spacing. Because of the symmetry, only 1/2 ofthe nozzle was simulated, which has a 0.088 mm ×1.0 mm domain. We thus afford simulations on afiner grid. This domain is discretized into 80 × 910cells, with ∆x = 1 µm. A constant pressure flow-through boundary condition is used at the nozzleexit. The exit pressure is set to be 1 bar.

In the first simulation, the flow in the nozzlewas assumed to be pure liquid (the bubble inser-tion algorithm was turned off). With a liquid den-sity 0.66 g/cm3, mean velocity 100 m/s, and dy-namic viscosity 0.004 g/(cm s), the Reynolds num-ber Re = 3.3× 104 greatly exceeds the critical valuefor transition to turbulence. Strong vorticity was ob-served only near the nozzle boundary layer, whichwas essentially laminar in nature (see Fig. 3). How-ever, the development of boundary layer separationor turbulence was not observed. The distance fromthe pipe entrance to the location where the transi-tion to turbulent flow is first fully developed con-

FIGURE 3. Vorticity in the nozzle

stitutes the theoretical initial length of the laminarflow and its magnitude is, approximately,

l = 0.03d× Re (30)

where d is the pipe diameter [26]. For Re ∼= 104, itis about 300 pipe diameters. On the experimentalside, according to the measurements performed byKirsten [27] and by Szablewski [28], it ranges from25 to 100 pipe diameters. Because the length of thenozzle simulated here is five pipe diameters, transi-tion to fully turbulent flow is not expected.

The simulation showed regions with large val-ues of “negative pressure” (Fig. 4), which indicated

FIGURE 4. The pressure field of the flow at consecutivetimes. (The unit of pressure is bar. Pressure above 0 barsis represented as white color)

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 611

the likely regions of cavitation. Figure 4 showssnap shots of pressure field at consecutive times,from the top frame to the bottom, t = 2.54 × 10−5 s,2.66× 10−5 s, and 2.92× 10−5 s. The negative pres-sure first appears at the upstream corner. Increasinginlet pressure causes the negative pressure to extendthroughout the nozzle.

To simulate cavitation, the discrete vapor bubbleapproach to cavitation was then used. Usually, cavi-tation parameters, such as the nuclei density for thebubble growth, need to be determined through ex-periments. For the present problem we study, thesedata are not available. We have performed a se-ries of numerical simulations with various valuesof the bubble spacing parameter and critical bubbleradius. Figure 5 includes snapshots of flow inter-face plots that show simulations with different crit-ical bubble radii, respectively, at 3 × 10−5 s. Thisset of simulations displays similar bubble distribu-tion, and void fraction values are approximately thesame. Here, the void fraction is defined as the ratioof the vapor volume to the volume of the nozzle. Weconclude that the critical bubble radius is not a sen-sitive parameter. Figure 6 shows snapshots of flowinterface plots with varied bubble spacing param-eter h and a fixed critical bubble radius r = 2 µmat 3 × 10−5 s. The simulations of flow in the noz-zle exhibited some sensitivity to the bubble spacingparameter in terms of the void fraction of the flowinside the nozzle. The bubbles display similar dis-tribution. However, the void fraction varies in thesesimulations. The void fraction is larger for smaller

FIGURE 5. The flow interface plots with different criti-cal bubble radius. Above: r = 2 µm. Below: r = 3 µm

FIGURE 6. The flow interface plots with different bub-ble spacing. r = 2 µm in these simulations. Above:h = 3r. Middle: h = 4r. Below: h = 5r

bubble spacing. Although we are not able to studythe dependence of the parameters on the grid forthe jet atomization due to the limits of the computa-tional resources, the numerical convergence for thenozzle flow is studied. In this study, the fixed val-ues of critical bubble radii and the bubble spacingparameter are used. We set r = 4 µm, and h = 4r.Figure 7 shows the flow interface plots on a mesh

FIGURE 7. The flow interface plots with bubble radiusr = 4 microns on different meshes. h = 4r. Above: gridspacing ∆x = 1 micron. Below: grid spacing ∆x = 2microns

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612 XU ET AL

with the grid spacing ∆x = 1µm and a mesh withthe grid spacing ∆x = 2µm, respectively. Theydisplay similar bubble distribution and void frac-tion. Based on this study, we conclude that the meshwith the grid spacing ∆x = 2µm is sufficient toresolve the dynamics of this bubbly flow. The so-lution is convergent in the sense of the predictionsof the similar bubble distribution and the void frac-tion. The mesh with the grid spacing ∆x = 2 µmis feasible for the study of jet atomization from thecomputational point of view. On the other hand,jet atomization, which is the breakup of the down-stream flow, was always predicted for various spac-

ing parameters (see Fig. 8). Simulations with dif-ferent h values exhibited quantitatively similar be-havior on the average and were all within the scopeof the experimental regime. More details of simu-lations of jet atomization will be given in Sec. 3.2.Similar studies were performed in [6] for the caseof cavitation of mercury under large external ener-gies (the interaction with an intense proton pulse).Because of the large external energies, the results of[6] were insensitive to the bubble spacing and ini-tial radius, and the presence of cavitation was itselfsufficient for obtaining the correct evolution of theflow.

(a) (b)

FIGURE 8. Plot of jet interface at late time. (a) is bubble spacing = 3r. (b) is bubble spacing = 4r

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 613

3.2 Jet Atomization

The algorithms described in Sec. 2 were used tosimulate cavitation, jet breakup, and spray forma-tion. In the simulations, vapor bubbles were cre-ated inside the nozzle. These bubbles then werecarried downstream outside the nozzle by the flow,and new bubbles continued to form in the nozzle.The disturbance of the cavitation bubbles producedupstream significantly impact the downstream jetbreakup. These vapor bubbles grew and broke thejet surface, forming spray, and droplets. We did sim-ulations using two different bubble spacing param-eter, namely, h = 3r and h = 4r, respectively. Wefound both simulation results agreed with the ex-periment. Figure 8 shows the snapshot of the spraydevelopment simulated using different spacing pa-rameters. Figures 8(a) and 8(b) display similar smallopening angles for the bubble spacing parameterh = 3r and h = 4r, respectively. Both are withinthe experimental range [20] (see also Fig. 2 in [20]).

The simulation predictions were compared to ex-perimentally measured quantities [17–20], such asthe fuel mass along the central axis of the spray andthe tip velocity of the jet. These experiments mea-sured mass vs. time in a 0.55 mm wide observationwindow, which is centered 1 mm from the nozzleexit. Aswith the experiments, the simulations alsopredicted a peak mass. But the width of the peak isless than that of the experimentally measured peak.After the peak, the mass predicted by the discretevapor bubble model exhibits some oscillations nearexperimental values (see Fig. 9).

0

1

2

3

4

5

6

7

8

0.17 0.18 0.19 0.2 0.21 0.22 0.23

Mas

s in

bea

m [1

0- 6g]

Time [msec]

Experiment4r spacing3r spacing

FIGURE 9. Plot of mass through a narrow window lo-cated 1 mm from the nozzle exit

Figure 10 shows the simulated jet tip velocitiescompared to experimental data. Both simulationsagreed with the experimental value on average.The simulation with h = 4r exhibited oscillations,whereas the simulation with h = 3r is quite smooth.It should be noted that the experimental data havebeen averaged over 100 injection cycles to removefluctuations.

To test the importance of compressibility andstructure of the multiphase mixture in the jet, weperformed simulations using the gamma law gasEOS model (easily compressible) for the liquid aswell as the stiffened EOS model (practically in-compressible) that describes a single phase liquid.Figures 11(a) and 11(b) are front plots of the jet pro-duced from simulations using the gamma law EOSmodel and the stiffened EOS model, respectively.From these two figures, we see that the jet simula-tion using the gamma law EOS exhibited expansion,whereas the jet obtained from the simulation usingthe stiffened EOS model did not exhibit expansion.This comparison shows the dependence of the sim-ulation prediction of the opening angle of the jet onthe EOS model. The bubbly flow mixed phase EOSis intermediate between the gamma law gas EOS,and the stiffened gamma law gas EOS, i.e., the bub-bly liquid, is intermediate between the gas and theliquid. We conclude that the compressibility of thebubbly liquid in the jet is critical for obtaining thecorrect opening angle of the jet. In the discrete va-por bubble model, this compressibility is achievedthrough the expansion of discrete bubbles. The het-erogeneous structure of the multiphase flow is nec-

0

20

40

60

80

100

120

140

160

180

0 0.5 1 1.5 2 2.5 3 3.5

Spe

ed [m

/sec

]

Distance to Nozzle Exit [mm]

Experiment4r spacing3r spacing

FIGURE 10. Plot of jet tip velocity

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614 XU ET AL

(a) (b)

FIGURE 11. (a) The front plot of the jet from the gammalaw EOS model. (b) The front plot of the jet from the stiff-ened EOS model

essary for obtaining the jet breakup and atomiza-tion.

4. CONCLUSIONS

A discrete vapor bubble model to describe multi-phase (cavitating) flows was developed. In the dis-crete vapor bubble model, the vapor-liquid mixtureis modeled as two pure phase domains describingvapor bubbles of finite size inserted into the liq-uid and separated from the liquid by interfaces.A dynamic bubble creation algorithm was formu-lated to allow vapor bubble insertion, and a new de-scription of the Riemann problem associated with aphase transition was developed. The phase transi-tion Riemann problem depends on a subgrid modelto describe the thermal layer at the phase transitionboundary.

The discrete vapor bubble model was applied tothe direct numerical simulations of cavitating flows.The simulations show jet breakup and atomizationin agreement with experiments.

The discrete vapor bubble model predicted cavi-tating regions made of many microbubbles that ledto jet breakup. It displays the influence of the distur-bances brought by the finite-size cavitation bubblesthat causes atomization. The authors believe this tobe the first direct numerical simulation of the impactof cavitation bubbles on atomization. The results in-dicate that the importance of resolving at a detailedlevel of multiscale science the physics of unsteady

cavitating flows. The discrete vapor bubble modeldeveloped in the present paper contains free param-eters, such as critical bubble radii and bubble spac-ing. The choice of parameters for the discrete va-por bubble model must be carefully examined withrespect to different applications. Finally, this workwas mainly focused on the simulation of the two-phase mixture resulting from cavitation. The influ-ence of other parameters on spray formation is asubject of further research.

REFERENCES

1. Reitz, R. D., and Bracco, F. V., Mechanism of at-omization of a liquid jet, Phys. Fluids. 25:1730–1742, 1982.

2. Reitz, R. D., and Bracco, F. V., Mechanisms ofbreakup of round liquid jets, The Encyclopedia ofFluid Mechanics. Gulf Pub. lishing, Houston,Texas, 1986.

3. Lin, S. P., and Reitz, R. D., Drop and spray for-mation from a liquid jet, Annu. Rev. Fluid Mech.30:85–105, 1998.

4. Bergwerk, W., Flow pattern in diesel nozzlespray holes, Proc. Inst. Mech. Eng. 173(25):655–660, 1959.

5. Chaves, H., Knapp, M., Kubitzek, A., and Ober-meier, F., Experimental study of cavitation inthe nozzle hole of diesel injectors using trans-parent nozzles. SAE Paper No. 950290:199–211,1995.

6. Samulyak, R., Lu, T., Prykarpatskyy, Y.,Glimm, J., Xu, Z., and Kim, M. N., Comparisonof heterogeneous and homogeneous numericalapproaches to cavitation modeling, Int. J.Multiscale Comp. Eng. (in press).

7. Schmidt, D. P., Rutland, C. J., Corradini, M. L.,Roosen, P., and Genge, O., Cavitation in two-dimensional asymmetric nozzles, SAE Interna-tional Congress, SAE Paper No. 1999-01-0518,1999.

8. Ventikos, Y., and Tzabiras, G., A numericalmethod for the simulation of steady and un-steady cavitating flows, Comput. Fluids. 29:63–88, 2000.

9. Schmidt, D. P., and Rutland, C. J., and Cor-radini, M. L., A fully compressible, two-dimensional, model of small, high speed, cav-

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DISCRETE BUBBLE MODELING OF UNSTEADY CAVITATING FLOW 615

itating nozzles, Atomization Spray. 9(3):255–276,1999.

10. Bukiet, B., Gardner, C. L., Glimm, J.,Grove, J. W., Jones, J., McBryan, O.,Menikoff, R., and Sharp, D. H., Applica-tions of front tracking to combustion, surfaceinstabilities and two-dimensional Riemannproblems. In Transactions of the Third Army Con-ference on Applied Mathematics and Computing,ARO Report No. 86–1, 223–243, 1986.

11. Glimm, J., Graham, M. J., Grove, J. W.,Li, X.-L., Smith, T. M., Tan, D., Tangerman, F.,and Zhang, Q., Front tracking in two andthree dimensions, Comput. Math. Appl. 35(7):1–11, 1998.

12. Glimm, J., Grove, J. W., Li, X.-L.,Shyue, K.-M., Zhang, Q., and Zeng, Y., Threedimensional front tracking, SIAM J. Sci. Comp.19:703–727, 1998.

13. Glimm, J., Grove, J. W., Li, X.-L., and Tan, D. C.,Robust computational algorithms for dynamicinterface tracking in three dimensions, SIAM J.Sci. Comput. 21:2240–2256, 2000.

14. Welch, W. J., Local simulation of two-phaseflows including interface tracking with masstransfer, J. Comp. Phys. 12:142–154, 1995.

15. Juric, D., and Tryggvason, G., Computations ofboiling flows, Int. J. Multiphase Flow. 24(3):387–410, 1998.

16. Matsumoto, Y., and Takemura, F., Influenceof internal phenomena on gas bubble motion,JSME Int. J. B37:288–296,1994.

17. MacPhee, A. G., Tate, M. W., and Powell, C. F.,Yue, Y., Renzi, M. J., Ercan, A., Narayanan, S.,Fontes, E., Walther, J., Schaller, J., Gruner, S. M.,and Wang, J., X-ray imaging of shock wavesgenerated by high pressure fuel sprays, Science.295:1261–1263, 2002.

18. Powell, C. F., Yue, Y., Gupta, S., McPherson, A.,Poola, R., and Wang, J., Development of a quan-titative measurement of a diesel spray coreusing synchrotron X-rays, Eighth InternationalConference on Liquid Atomization and Spray Sys-tems, Pasadena, CA, 2000.

19. Powell, C. F., Yue, Y., Poola, R., and Wang, J.,Time resolved measurements of supersonic fuelsprays using synchrotron x-rays, J. Synchrotron

Rad. 7:356–360, 2000.20. Powell, C. F., Yue, Y., Poola, R., Wang, J., Lai, M.,

and Schaller, J., Quantitative X-ray measure-ments of a diesel spray core. In Proc. 14th An-nual Conference on Liquid Atomization and SpraySystems (ILASS), Dearborn, MI, 2001.

21. Starling, K. E., Fluid Thermodynamic Propertiesfor Light Petroleum Systems, Gulf Pub. lishing,Houston, 1973.

22. Poinsot, T. J., and Lele, S. K., Boundary condi-tions for direct simulations of compressible vis-cous flows, J. Comput. Phys. 101:104–129, 1992.

23. Colella, P., A direct Eulerian MUSCL schemefor gas dynamics, SIAM J. Comput. 6(1):104–117,1985.

24. Berger, M., and Colella, P., Local adaptive meshrefinement for shock hydrodynamics, J. Comput.Phys. 82:64–84, 1989.

25. Glimm, J., Li, X.-L., and Xu, Z.-L., Front track-ing algorithm using adaptively refined meshes.In Proceedings of Chicago Workshop on AdaptiveMesh Refinement Theory and Applications, Sept. 3-5, 2003, Springer, Lecture Notes in Computa-tional Science and Engineering, Vol. 41, Plea,Tomasz; Lined, Timor; Weirs, V., Gregory (eds.),2005.

26. Schlichting, H., Boundary Layer Theory,Mcgraw-Hill, New York, 1960.

27. Kirsten, H., Experimentelle Unter-suchungen der Entwicklung derGeschwindigkeitsverteilung der turbulen-ten Rohrstromung, Ph.D. thesis, Leipzig,1927.

28. Szablewski, W., Der einlauf einer tubulentenrohrstromung, Ing. Arch. 21:323–330, 1953.

29. Wallis, G., One-Dimensional Two-Phase Flow,McGraw-Hill, New York, 1969.

30. Samulyak, R., and Prykarpatskyy, Y.,Richtmyer-Meshkov instability in liquidmetal flows: Influence of cavitation and mag-netic fields, Math. Comput. Simul. 65:431–446,2004.

31. Glimm, J., Li, X. L., Oh, W., Marchese, A.,Kim, M.-N., Samulyak, R., and Tzanos, C., Jetbreakup and spray formation in a diesel engine.In Proceedings of 2nd MIT Conference on Compu-tational Fluid and Solid Mechanics. Cambridge,

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616 XU ET AL

MA, SUNY Stony Brook Preprint No. susb-ams-02-20, 2003.

32. Miles, J. W., On the disturbed motion of a planevortex sheet. J. Fluid Mech. 4:538–552, 1958.

33. Foglizzo, T., and Ruffert, M., An analytic studyof bondi-hoyle-lyttleton accretion, Astron. As-trophys. 347:901–914, 1999.

34. Brennen, C. E., Cavitation and Bubble Dynamics,Oxford University Press, New York, 1995.

35. Landau, L., and Lifshitz, E., Statistical Physics,Pregamon Press, Oxford, 1980.

36. Balibar, S., and Caupin, F., Metastable liquids, J.Phys. Condensed Matter. 15:s75–s82, 2003.

37. Samulyak, R., Lu, T., Prykarpatskyy, Y.,Glimm, J., Xu, Z., and Kim, M. N., Compari-son of direct and homogeneous numerical ap-proaches to cavitation modeling, Accepted Int. J.Multiscale Comput. Eng. 4:377–389, 2006.

38. Alty, T., and Mackay, C. A., The accommoda-tion coefficient and the evaporation coefficientof water, Proc. R. Soc. London, Ser. A, 149:104–116, 1935.

39. Lu, T., and Xu, Z. L., and Samulyak, R.,

and Glimm, J., and Ji, X. M., Dynamic PhaseBoundaries for Compressible Fluids Siam J. onScientific Computing, Submitted. Stony Brookpreprint No. sunysb-ams-06-07, 2006. (in print)

40. Hagen, D. E., Schmitt, J., Trueblood, M., andCarstens, J., White, D. R., and Alofs, D. J., Con-densation coefficient measurement for water inthe umr cloud simulation chamber, J. Atmos. Sci.46:803–816, 1989.

41. Carey, V. P., Liquid-Vapor Phase-Change Phenom-ena. Hemisphere Publishing Corporation, NewYork, NY, 1992.

42. Caginalp, G., and Socolovsky, E. A., Computa-tion of Sharp Phase Boundaries by Spreading:The Planar and Spherically Symmetric Cases, J.Comput. Phys. 95:85–100, 1991.

43. Kim, Y.-T., Provatas, N., Goldenfeld, N., andDantzig, J., Universal dynamics of phase-fieldmodels for dendritic growth, Phys. Review E.59(3):R2546–R2549, 1999.

44. Huang, A., and Joseph, D. D., Instability of theequilibrium of a liquid below its vapor betweenhorizontal heated plates, J. Fluid Mech. 242:235–247, 1992.

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