s notandThe

ionalrateationstruc-acey forer a

Vineet AhujaResearch Scientist

Ashvin HosangadiPrincipal Scientist

Srinivasan ArunajatesanResearch Scientist

Combustion Research andFlow Technology,

Inc. (CRAFT Tech),P.O. Box 1150, Dublin, PA 18917

Simulations of Cavitating FlowsUsing Hybrid UnstructuredMeshesA new multi-phase model for low speed gas/liquid mixtures is presented; it doerequire ad-hoc closure models for the variation of mixture density with pressureyields thermodynamically correct acoustic propagation for multi-phase mixtures.solution procedure has an interface-capturing scheme that incorporates an additscalar transport equation for the gas void fraction. Cavitation is modeled via a finitesource term that initiates phase change when liquid pressure drops below its saturvalue. The numerical procedure has been implemented within a multi-element untured framework CRUNCH that permits the grid to be locally refined in the interfregion. The solution technique incorporates a parallel, domain decomposition strategefficient 3D computations. Detailed results are presented for sheet cavitation ovcylindrical head form and a NACA 66 hydrofoil.@DOI: 10.1115/1.1362671#

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IntroductionNumerical simulations of cavitating flows are very challengi

since localized, large variations of density are present at theliquid interface while the remainder of the flow is generally icompressible. Furthermore, the cavitation zone can detach duthe influence of a re-entrant jet and convect downstream. Tdetachment/collapse of the cavity in the pressure recovery recan lead to strong acoustic disturbances. These disturbancemanifest themselves as high frequency noise that has the poteto considerably alter the acoustic signal/signature and degperformance. In general, current cavitation models, which horiginated from incompressible formulations, deal with thesesues by defining ad-hoc closure models relating density and psure which are not very general. In our paper, we describe a cpressible, cavitation formulation which yields the correacoustical and thermodynamic behavior for multi-phase systeThe importance of being thermodynamically consistent is thatsame cavitation model can be used for simulations from mapropellers to cryogenic pumps. In particular, the system doesrequire user defined closure models for capturing the gas/liqinterface and is designed to perform efficiently in the nearlycompressible liquid regime. We give a brief review of the literture below and highlight the differences with our formulation.

Cavitation models in the literature may broadly be classifiinto two categories; interface fitting and interface capturing pcedures. Interface fitting procedures explicitly track and fit a dtinct gas/liquid interface which is an internal boundary. While thprocedure gets around the numerical difficulties of integratthrough the interface, its applications are limited to simpler prlems where the cavity can be described as a well-defined clovolume of pure gas. Interface capturing schemes, where theliquid interface is obtained as part of the solution procedure,more general in their applications and may be applied to bsheet cavitation as well as bubbly cavitation. Here, the thermonamic issues of integrating through gas/liquid mixtures with dsity and acoustic speed variations have to be dealt with in ordeprovide closure to the equation system.

Closure models reported in the literature are generally tailofor specific problem classes and do not address the multi-pphysics in a fundamental fashion. For instance, bubbly cavita

Contributed by the Fluids Engineering Division for publication in the JOURNALOF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering DivisioApril 5, 2000; revised manuscript received January 29, 2001. Associate EdY. Matsumoto.

Copyright © 2Journal of Fluids Engineering

gas/-e tohisioncan

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has been tackled by providing pseudo-density relations derfrom various modified forms of the Rayleigh-Plesset equat~Kubota et al.@1# and Chen and Heister@2#!. Here, no additionalequation for the convection of gas species is specified andformulations make assumptions about the bubble number denin the fluid. On the other hand, for sheet cavitation an ad-hpseudo-density relationship based on the local pressure isvided to close the system~Delaunay and Kueny@3# and Janssenset al. @4#! which can be quite restrictive since it is decouplcompletely from the local phase composition.

More generalized cavitation formulations have been presenin recent papers by Merkle et al.@5# and Kunz et al.@6#. Here, anadditional equation for the gas void fraction is solved for andlocal mixture density is obtained from the local phase compotion. The equations are cast in a compressible-like time marchformat with preconditioning to overcome numerical stiffneproblems. However, in both cases, the authors implicitly specia pressure-density relation which leads to an erroneous acospeed in the mixture, particularly in the interface region. The nphysical acoustic speeds are clearly not appropriate for unstesimulations. However, even for steady calculations, this may pnumerical difficulties at the gas/liquid interface since the physiacoustic speed behaves in a nonlinear fashion~as we shall discusslater! and has a very small magnitude; the local Mach numat the interface can be large leading to very different numercharacteristics.

The formulation presented in this paper is an acoustically acrate form of the compressible multi-phase equations~Ahuja et al.@7#! and is an extension of our earlier work in high pressure gliquid systems~Hosangadi et al.@8#!. The numerical algorithmfollows a similar time-marching philosophy~as that in Merkleet al. @5# and Kunz et al.@6#!. However, here the local speed osound in the two-phase mixture is a function of the local vofraction and mimics the two-phase acoustic speed relationfrom classical analytic theory. We note that the system presehere is closed and does not require additional equations to resthe gas/liquid interface. The model is general and may be appto both sheet and bubbly cavitation by providing approprisource terms for gas generation/reabsorption in cavitating regi

The numerical formulation has been implemented on a hybunstructured framework which permits tetrahedral/prismatic ceAn unstructured framework is particularly suitable for geomecally complex systems like marine propellers where the blades

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001 by ASME JUNE 2001, Vol. 123 Õ 331

t

u

tl

c

s

se

l

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T

oss

ndm

he.

skewed strongly. In addition, the ability for local grid refinemein a complex flow field provided a strong motivation for usinunstructured grids here. For instance, in cavitation simulatiothe region near the gas/liquid interface exhibits strong gradienflow properties and requires high local grid resolution which cbe achieved most economically with a grid adaption procedFor efficient computations of large 3D problems, a parallel framwork for distributed memory systems has been implemented.tails of the numerical formulation are provided in the followinsection followed by details of the unstructured framework. InResults section, we discuss details of our simulation for a cydrical headform and a NACA 66 hydrofoil. The details of thflowfield in the closing region of the cavity and the turbulencharacteristics of the wake are examined in depth. Surface psure comparisons with experimental data are shown for a rangcavitation numbers to validate the numerics.

Multi-Phase Equation SystemThe multiphase equation system is written in vector form a

]Q

]t1

]E

]x1

]F

]y1

]G

]z5S1Dv (1)

HereQ is the vector of dependent variables,E, F, andG are theflux vectors,S the source terms andDv represents the viscoufluxes. The viscous fluxes are given by the standard full comprible form of Navier Stokes equations~see Hosangadi et al.@9# fordetails!. The vectorsQ, E, andS are given below with a detaileddiscussion on the details of the cavitation source terms to follater:

Q5S rm

rmurmvrmwrgfg

rmkrm«

D E5S rmurmu21P

rmuvrmuwrgfgurmkurm«u

D S5S 0000Sg

Sk

S«

D (2)

Here,rm is the mixture density, andfg is the volume fraction orporosity for the gas phase. Note that in Eq.~2! the energy equationis dropped since each phase is assumed to be nearly incompible thereby decoupling the pressure work term. An additioscalar equation for mixture enthalpy may be solved coupled toequation set if the temperature effects become important.mixture density and gas porosity are related by the followinglations locally in a given cell volume:

rm5rgfg1rLfL (3)

15fg1fL (4)

where rg ,rL are the physical material densities of the gas aliquid phase respectively.

To modify the system in Eq.~1! to a well-conditioned form inthe incompressible regime requires a two-step process; an actically accurate two-phase form of Eq.~1! is first derived, fol-lowed by a second step of time-scaling or preconditioning totain a well-conditioned system. We begin by defining the acouform of density differential for the individual gas and liquid phaas follows:

332 Õ Vol. 123, JUNE 2001

ntgns,s inanre.e-

De-ghein-eeres-e of

:

ss-

ow

ress-al

thishe

re-

nd

ous-

b-tice

drg51

cg2 dP, dr l5

1

cL2 dP (5)

Here cg is the isothermal speed of sound (]P/]rg)T in the puregas phase, andcL is the corresponding isothermal speed of souin the liquid phase, which is a finite-value. The differential forof the mixture densityrm is obtained by differentiating Eq.~3!,and using the relationship given in Eq.~5! to obtain,

drm5~rg2rL!dfg11

cf2 dP

S 1

cf2 5

fg

cg2 1

fL

cL2 D (6)

Here, cf is a variable defined for convenience and is not tacoustic speed,cm , in the mixture which will be defined laterUsing Eq.~6!, Eq. ~1! may be rewritten as:

G]Qv

]t1

]E

]x1

]F

]y1

]G

]z5S1Dv (7)

where

G5

¨

1

cf2 0 0 0 ~rg2rL! 0 0

u

cf2 rm 0 0 ~rg2rL!u 0 0

v

cf2 0 rm 0 ~rg2rL!v 0 0

w

cf2 0 0 rm ~rg2rL!w 0 0

fg

cg2 0 0 0 rg 0 0

k

cf2 0 0 0 ~rg2rL!k rm 0

«

cf2 0 0 0 ~rg2rL!« 0 rm

©(8)

and,

Qv5@p,u,v,w,fg ,k,«#T (9)

The numerical characteristics of the Eq.~7! are studied by ob-taining the eigenvalues of the matrix,@G21(]E/]Qv)#. The fluxJacobian]E/]Qv is given as:

Transactions of the ASME

A5

¨

u

cf2 rm 0 0 ~rg2r l !u 0 0

u2

cf2 11 rm2u 0 0 ~rg2r l !u

2 0 0

vu

cf2 rmv rmu 0 ~rg2r l !vu 0 0

wu

cf2 rmw 0 rmu ~rg2r l !wu 0 0

fgu

cf2 rgfg 0 0 rgu 0 0

ku

cf2 rmk 0 0 ~rg2rL!ku rmu 0

©(10)

«u

cf2 rm« 0 0 ~rg2rL!«u 0 rmu

f

g

t

f

i

tT

y

achctive

thisn-

re-

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The eigenvalues of the system are derived to be:

L5~u1cm ,u2cm ,u,u,u,u,u! (11)

wherecm turns out to be the well-known, harmonic expressionthe speed of sound in a two-phase mixture and is given as:

1

cm2 5rmb fg

rgcg2 1

fL

rLcL2c (12)

The behavior of the two-phase speed of sound is plotted in Fias a function of the gas porosity; at either limit the pure singphase acoustic speed is recovered. However, away fromsingle-phase limits, the acoustic speed rapidly drops below eilimit value and remains at the low-level in most of the mixturegime. As a consequence, the local Mach number in the interregion can be large even in low speed flows.

We re-emphasize a critical observation at this point: the eqtion system~7!–~9! is completely defined and does not requad-hoc closure models for the variation of mixture density wpressure. In that respect alone this represents a significanvancement over most other cavitation models in the literature.acoustic speeds for individual phases are well-defined physquantities, which may be specified, and so is the case with phcal material densities (rg ,rL) for each individual phase. For lowpressure incompressible regimes, the material densities maassumed to be constant without significant error in the solutio

Fig. 1 Speed-of-sound in a two-phase gas-liquid mixture

Journal of Fluids Engineering

or

. 1le-theherreace

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ad-he

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bens.

However, in its most general form the material densities for ephase may be obtained from the pressure using their respephysical equations of state~e.g., ideal gas law for gases, etc.! ifthat is so desired. If temperature variations were significant,would involve solving an additional equation for the mixture eergy ~this formulation is to be presented in a future paper!.

To obtain an efficient time-marching numerical scheme, pconditioning is now applied to the system in Eq.~7!, in order torescale the eigenvalues of the system so that the acoustic spare of the same order of magnitude as the local convective veties. This is achieved by replacingG in Eq. ~7! by Gp .

Gp

]Qv

]t1

]E

]x1

]F

]y1

]G

]z5S1Dv (13)

where

Gp5

l

b

cf2 0 0 0 ~rg2rL! 0 0

bu

cf2 rm 0 0 ~rg2rL!u 0 0

bv

cf2 0 rm 0 ~rg2rL!v 0 0

bw

cf2 0 0 rm ~rg2rL!w 0 0

bfg

cg2 0 0 0 rg 0 0

bk

cf2 0 0 0 ~rg2rL!k rm 0

b«

cf2 0 0 0 ~rg2rL!« 0 rm

m

Here the parameterb has been introduced to precondition theigenvalues. The modified eigenvalues of the preconditionedtem are given as:

JUNE 2001, Vol. 123 Õ 333

ie

v

h

u

e

mit

-e

u

ori

y-s

ted

totic. In

nceands,

om-e-the

hasodewe

lti-ionec-re aes,me-

ur-

LP5S u

2 S 111

b D1cm8 ,u

2 S 111

b D2cm8 ,u,u,u,u,uD (14)

where

cm8 51

2Au2S 12

1

b D 2

14cm

2

b(15)

Equation ~15! indicates that by settingb5(cm2 /up

2) where up5max(u,0.01cm) the pseudo-acoustic speed is of the order ofu atall mixture composition values. We note that, a rigorous definitof the physical two-phase acoustic speedcm ~as has been donhere! is critical to obtaining noise-free propagation of pressuwaves across interfaces where the density and acoustic speevarying rapidly.

Cavitation Source TermsIn the present effort, the cavitation source term is simplified

a simplified nonequilibrium, finite rate form as follows:

Sg5K frLfL1Kbrgfg (16)

where the constantK f is the rate constant for vapor being geneated from liquid in a region where the local pressure is less tthe vapor pressure. Conversely,Kb is the rate constant for reconversion of vapor back to liquid in regions where the pressexceeds the vapor pressure. Here, the rate constants are speusing the form~given by Merkle et al.@5#! as follows:

Kb5 b 0 p,pv

1

tbS Q`

L`D F p2pv

1

2r`Q`

2 G p.pvc(17)

K f5F 0 p.pv

1

t fS Q`

L`D F p2pv

1

2r`Q`

2 G p,pvGpv5p`2

1

2r`Q`

2 * Cav. No.

t f5time constant for vapor formation

tb5time constant for liquid reconversion

Cav. No.5p`2pv

12r`Q`

2

We note that the nonequilibrium time scales have not bcorrelated with experimental data. For steady attached cavitathis simplified form may be adequate since the cavitation tiscales do not interact with the fluid time scales if the cavitatrate constants are fast enough. This point has been demonsby repeating a calculation for a given cavitation number at thdifferent rates~see Results section!. For unsteady cavitation, however, the details of how the nonequilibrium source term is spfied could be crucial since it may couple with transient presswaves. The development of a more rigorous nonequilibrisource term model is a topic of ongoing research.

Turbulence ModelsThe standard high Reynolds number form of thek-« equations

form the basis for all turbulence modeling in CRUNCH. Transpequations for the turbulent kinetic energy and its dissipationare solved along with the basic momentum and energy equatThese equations, with supplemental low Re terms, are as follo

334 Õ Vol. 123, JUNE 2001

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red are

ia

r-an

-recified

ention

eonratedree

ci-urem

rtateons.ws,

]rk

]t1

]

]xiS ruik2S m1

mT

skD ]k

]xiD5Pk2r«1Sk

]r«

]t1

]

]xiS rui«2S m1

mT

s«D ]«

]xiD5c1f 1Pk2C2f 2r«1S«

(18)

Pk5t i j

]ui

]xj, t i j 52

2

3rk12mT* S Si j 2

1

3

]uk

]xkd i j D ,

mT5Cm f mrk2

«

where,sk , s« , C1 , andC2 are the modeling constants, andf 1 ,f 2 , f m are empirical modeling functions to account for low Renolds number~near wall!. ~They equal unity in the high Reynoldnumber form.!

Low Reynolds number effects in the near wall are accounfor by using an extension of the near wall model of So et al.@10#.This model has been shown to reproduce the near wall asymprelations for the Reynolds stress and kinetic energy accuratelythis model, the damping functions,f 1 , f 2 , f m , are defined asfollows,

f 151.02expb2S Ret

40 D 2cf 2512

2

9expF2S Ret

6 D 2GS«5

1

4c3mLF S ]k1/2

]x D 2

1S ]k1/2

]y D 2

1S ]k1/2

]z D 2G (19)

where

Ret5rk2

mL«

f m5~114 Ret23/4!tanhS Rek

125Dwhere

Rek5rAky

m

The constants for this model are given as follows.

cm50.09, sk51.4, s«51.4,(20)

c151.44, c251.92, c352.9556

This model has been tested for a number of standard turbuletest cases to verify that it gives correct boundary layer growthreattachment length for recirculating flows. For cavitating flowwhere large density gradients are present due to multi-fluid cposition, the rigorous validation of the turbulence model will rquire further study; however, this is beyond the scope ofpresent paper.

Unstructured Crunch Code OverviewThe multi-phase formulation derived in the previous section

been implemented within a three-dimensional unstructured cCRUNCH; a brief overview of the numerics is given here andrefer the reader to Hosangadi et al.,@9,11# and Barth@12,13# foradditional details. The CRUNCH code has a hybrid, muelement unstructured framework which allows for a combinatof tetrahedral, prismatic, and hexahedral cells. The grid conntivity is stored as an edge-based, cell-vertex data structure whedual volume is obtained for each vertex by defining surfacwhich cut across edges coming to a node. An edge-based frawork is attractive in dealing with multi-elements since dual s

Transactions of the ASME

ra

l

o

T

n

l

rcrv

b

i

t

le

as

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face areas for each edge can include contributions from diffeelement types making the inviscid flux calculation ‘‘grid transpent.’’

The integral form of the conservation equations are writtena dual control-volume around each node as follows:

Gp

DQvV

Dt1E

]V i

F~Qv ,n!ds5EV i

SdV1E]V i

D~Qv ,n!ds

(21)

The inviscid flux procedure involves looping over the edgeand computing the flux at the dual face area bisecting each eA Riemann problem is solved for using higher order reconstrucvalues of primitive variables at the dual face. Presently, a secoorder linear reconstruction procedure~following Barth @13#! isemployed to obtain a higher-order scheme. For flows with strgradients, the reconstructed higher variables need to be limiteobtain a stable TVD scheme. We note that the inviscid fluxoutlined above is grid-transparent since no details of the elemtype are required in the flux calculation.

The Riemann problem at the dual face is computed with a Raveraged, flux-differenced solver which defines the flux,Fm , as:

Fm51

2 b F~Qm2 ,nW oi!1F~Qm

1 ,nW oi!1

DF2~QRoem ,nW oi!1DF1~QRoe

m ,nW oi!c (22)

wherenW oi is the vector area of the dual face crossing edgeeWoi ,and the subscript ‘‘Roe’’ denotes Roe-averaged variables.flux differencesDF2 andDF1 are given as

DF1~QRoem ,nW oi!5

12~RRoe

m ~L1uLu!RoeLRoem !~Qm

12Qm2!

(23)

DF2~QRoem ,nW oi!5

12~RRoe

m ~L2uLu!RoeLRoem !~Qm

12Qm2!

(24)

Here, R and L are, respectively, the right and left eigenvectmatrices, whileL is the diagonal eigenvalue matrix. The Roe aeraged velocities and scalars are defined as

yRoe5y1Ar11y2Ar2

Ar11Ar2, $yP~u,v,w,k,«!% (25)

while, the volume fraction,fg , is defined by doing a simplearithmetic average:

fg,Roe512~fg

11fg2! (26)

This average value of thefg is used to define the speed of souof the mixture at the dual face.

For efficient computation of large 3D problems a paralframework for distributed memory systems has been impmented, along with a time-marching implicit solution proceduThe sparse implicit matrix is derived by doing a Euler explilinearization of the first-order flux, and a variety of iterative spamatrix solvers, e.g., GMRES, Gauss-Seidel procedure, are aable in the code~see Hosangadi et al.@9# for details!. The parallelframework is implemented by partitioning the grid into sudomains with each subdomain residing on an independent prosor. The message passing between processors has beenmented using MPI to provide portability across platforms.

Results and DiscussionThe multi-phase system described in the previous sections

been applied extensively to both cavitating and non-cavitaproblems. Since the single-phase incompressible formulationsubset of the more general multi-phase system, the code wasvalidated for standard incompressible test cases in the litera~see Ahuja et al.@7#, for results!. In the present section, we wilfocus on the following sheet-cavitation problems which have bstudied extensively in the literature: 1! cylindrical headform; and2! the NACA 66 hydrofoil.

Journal of Fluids Engineering

entr-

for

istdge.tednd-

ngd toasent

oe-

he

orv-

d

elle-e.itseail-

-ces-mple-

hasingis afirst

ture

en

All the simulations presented here have been computedsteady-state calculations, using the two-equation (k-e) turbulencemodel with near-wall damping described earlier. The liquid to gdensity ratio was specified to be 100. The baseline forwardbackward cavitation rate termst f , tb , are specified to be 0.001unless otherwise specified. Furthermore, the calculations wperformed on parallel distributed memory platforms using 8 pcessors. We note that although the flowfields simulated hereessentially two-dimensional in character, the CRUNCH codethree-dimensional; the simulations performed here were thrdimensional with minimal resolution in thez coordinate. Hence,we do not anticipate any serious issues in computing a truly thrdimensional flowfield apart from the increased numerical cost

Cylindrical Headform Simulations. Simulations of theRouse and McNown@14# experiments for water flowing over ahemisphere/cylinder geometry are presented. The Reynolds nber per inch for this configuration is 1.363105. A hexahedral gridwith dimensions of 2213113 was used for all the cavitation numbers reported. The grid was clustered both radially near theface as well as axially at the bend where sphere meets withcylinder. Simulations for three cavitation numbers of 0.4, 0.3, a0.2 are presented. The grid is best suited for the highest cavitanumber of 0.4 since the cavity zone is completely containwithin the zone of high clustering. Hence the results for the 0case will be analyzed in-depth to study the flow features incavity closure region, as well as other sensitivity studies suchthe effect of the rate constant.

The cavitation zone for a cavitation number of 0.4 is shownFig. 2 for the baseline source term rates given earlier. We nthat the gas/liquid interface is sharp and captured within a couof cells is most of the flow except at the tip on the top cornerthe back end where a marginal ‘‘pulling’’ or extension of thcavitation zone is observed. This is attributed to the strong shat this location as the flow turns around the cavitation zoneenclose it. The details of the recirculation zone in the cavitatclosure region are illustrated in Fig. 3 by plotting the streamlinof the flow. A sharp recirculation zone originating from the tip othe cavitation zone is observed where a re-entrant jet is obseto transport fluid back into the cavity. As we shall see from tsurface pressure distribution, the re-entrant jet generates a lpressure peak at the rear of the cavity thereby keeping this cashape stable.

Fig. 2 Resolution of the cavitation zone interface on the nu-merical grid

JUNE 2001, Vol. 123 Õ 335

336 Õ

Fig. 3 Flow streamlines depicting recirculation zone Õre-entrant jet in the cavity closure region

-

h

ri

h

er-uld

.4,ionsur-ata.reder-

forase,the

der

The turbulent viscosity levels~nondimensionalized by the laminar viscosity! generated in the flowfield are shown in Figs. 4~a!and 4~b!. In Fig. 4~a! we plot the overall global characteristics othe turbulent wake while Fig. 4~b! gives a close-up view of thecavitation zone itself. From Fig. 4~a! we observe the incomingboundary layer lifting off as the flow turns and jumps over tcavitation zone. The cavitation zone itself is observed to be v‘‘quiet’’ with turbulence present mainly in the interface regioand the shear layer above it. The plot depicts increasing tulence intensity levels near the cavity closure/re-entrant jet regand a fully turbulent wake ensues. We note that since the calation was performed with a steady RANS turbulence model,wake simulated corresponds to the time-averaged solution w

Vol. 123, JUNE 2001

f

eerynbu-on,lcu-theich

reproduces the mean mixing rate. If the calculation were pformed with an unsteady LES model, an unsteady wake woresult with vortices being shed off the cavity tip.

The characteristics of the flowfield at cavitation numbers 0f 00.3, and 0.2 are compared in Fig. 5 by plotting both the cavitatzone as well as the surface pressure distribution. Overall theface pressure profiles compare very well with experimental dIn particular, the cavity closure region appears to be captuwell; the location, the pressure gradient, and the ovpressurization magnitude in the wake are very close to the datathe two higher cavitation numbers of 0.4 and 0.3. For the 0.2 cthe cavity zone is very large and extends into the region wheregrid is not as finely clustered, and we attribute the slight un

Fig. 4 Turbulent viscosities „m t ÕmL… contours in the headform cavity flowfield

Transactions of the ASME

Jour

Fig. 5 Cavitation zone and surface pressure profiles at various cavitation numbers for hemisphere Õcylinder headform

o

f

gp

mv

t jetpat-

thet theortedob-the

dent

ve

t

prediction of the cavity length to grid quality. The coarser gridthe interface region is also reflected in the broader interface zIt should be noted that all three cases were computed withbaseline source term rates.

To evaluate the impact of the source term ratest f andtb , wecomputed the 0.4 cavitation case using the following three difent rates: 1! the baseline rate~t f50.001 s, tb50.001 s!; 2! afaster rate (baseline310); and, 3! a slower rate (baseline30.1).The surface pressure comparisons are shown in Fig. 6 for the tdifferent rates. The results for the baseline rate and the fasterare very close; however, as expected, the faster rate gives slibetter results in the wake with the cavity closure being sharThe slower rate (baseline30.1) gives a more diffused cavity particularly in the rear with the time scale for the nonequilibriusource term coupling with the convective time scale. As the ca

nal of Fluids Engineering

inne.the

er-

hreeratehtlyer.

-

ity

closure region gets more diffuse, the strength of the re-entranweakens, and consequently, the pressure does not exhibit thetern of over-pressurization and relaxation which is observed indata as well as the other two faster rate cases. We note thabaseline rates have been used to compute all the results rephere including the hydrofoil case. At least for steady-state prlems, the nonequilibrium source terms perform adequately andprecise value of the rate appears to be problem/grid indepenas long as the time scale is sufficiently fast enough~as defined byour baseline rate! in comparison to the characteristic convectitime scales.

NACA 66 Hydrofoil Simulations. In this section, we presenresults for sheet cavitation on a NACA 66 hydrofoil~Shen andDimotakis @15#!. The freestream flow is at a 4 degree angle-of-

JUNE 2001, Vol. 123 Õ 337

.

a

i

b

4,ion

andavi-sheon

arel

attack, with a Reynolds number of 23106 based on chord lengthThe hybrid grid used for the calculations is shown in Fig. 7; it ha combination of approximately 200,000 tetrahedral and prismcells with clustering around the surface of the hydrofoil. Calcutions for two different cavitation numbers of 0.84 and 0.91 adiscussed in the following paragraphs. Note that the calculatshown here were computed with a wall function procedure.

The pressure contours for the flowfield at a cavitation numof 0.84 are plotted in Fig. 8. We observe that the pressure ctours cluster around the cavitation boundary where the dengradient is very large and the flow turns around the cavitatbubble. The gas void fraction contours at the two cavitation nu

Fig. 6 Sensitivity of cavitation solution to the cavitationsource term rate

Fig. 7 The prismatic Õtetrahedral grid used to capture cavita-tion on the NACA 66 modified hydrofoil

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bers are shown in Fig. 9. At the lower cavitation number of 0.8the cavity extends up to 50 percent of chord. As the cavitatnumber increases the gas bubble region decreases in lengthcomes closer to the surface; the void fraction contours at a ctation number of 0.91~Fig. 9~b!!, indicate that the bubble extendto only 30 percent of chord. The thin confinement region of tgas bubble highlights the potential for using local grid adaptiwithin our unstructured framework.

The surface pressure profiles at the two cavitation numbersplotted in Figs. 10~a! and 10~b! for comparison the experimenta

Fig. 8 A representative pressure distribution on the NACA 66hydrofoil at 4 degrees angle of attack and cavitation number of0.84

Fig. 9 Cavitation bubbles indicated by void fraction contourson the NACA 66 modified hydrofoil at cavitation numbers of „a…0.91 and „b… 0.84

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data points as well as results from the numerical study by Sinet al. @16# are also shown. We note that the simulation by Sinet al. @16# was computed on a structured hexahedral grid usinuser specified equation of state, and had a pdf based modecavitation. We make the following observations; 1! Both the nu-merical simulations are very similar to each other despite theferences in the numerics and the formulation. The good compson between the two numerical results is an important validaof the more fundamental formulation presented here which didrequire a user specified closure model for the equation of stat!While the comparisons with the experimental data are reasongood, the numerical results underpredict the length of the caand overpredict the aft recovery pressure. A preliminary invegation indicates that this may be related to a combination of logrid resolution in the interface region, as well as the details ofnear wall turbulence model. However, for the purposes of illtrating the applicability of the new formulation the current resuwere deemed to be acceptable.

Fig. 10 Surface pressure distribution on the NACA 66 hydro-foil using wall-function procedure

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Conclusions

A multi-phase model for low speed gas-liquid mixtures hbeen developed by reducing the compressible system of equato an acoustically accurate form that performs efficiently in tincompressible regime. In particular, the equation system doesrequire ad-hoc density-pressure relations to close the systemyields the correct acoustic speed as a function of local mixtcomposition. For efficient steady-state solutions, the physical mture acoustic speed is preconditioned to obtain good convergeThe solution procedure has been implemented within a hybmulti-element unstructured framework which operates in a palel, domain-decomposed environment for distributed memory stems.

Detailed results are presented for steady-state sheet cavitin an hemisphere/cylinder geometry as well as a NACA 66drofoil at various cavitation numbers. Good comparison is otained with experimental data, and in particular the details ofcavity closure, and the re-entrant jet are captured well in the silation. Examination of the turbulence characteristics indicatesturbulent kinetic energy associated with the upstream boundlayer jumps over the cavity, and the interior of the cavity itselfnearly laminar. However, the recirculation zone in the re-entrjet region generates a fully turbulent wake which stabilizescavity zone and generates a pressure pattern of over-pressurizand gradual relaxation in the wake. Sensitivity of the cavitatzone to the finite rate source terms was examined. The soluwas found to be relatively insensitive to the source term ratelong as the rate was high enough. As the source term rate droby an order of magnitude, coupling between the convective tscale and the source term was observed leading to a more dicavity in the closure region and poorer pressure predictioClearly, for unsteady detached cavitation problems, the modeof the source term needs to be examined carefully and this iarea of ongoing work.

References@1# Kubota, A., Kato, H., and Yamaguchi, H., 1992, ‘‘Cavity Flow Prediction

Based on the Euler Equations,’’ J. Fluid Mech.,240, pp. 59–96.@2# Chen, Y., and Heister, S. D., 1996, ‘‘Modeling Hydrodynamic Nonequilibriu

in Cavitating Flows,’’ ASME J. Fluids Eng.,118, pp. 172–178.@3# Delaunay, Y., and Kueny, J. L., 1990, ‘‘Cavity Flow Predictions based on

Euler Equations,’’ASME Cavitation and Multi-Phase Flow Forum, Vol. 109,pp. 153–158.

@4# Janssens, M. E., Hulshoff, S. J., and Hoeijmakers, H. W. M., 1997, ‘‘Callation of Unsteady Attached Cavitation,’’ AIAA-97-1936, 13th AIAA CFDConferences, Snowmass, CO, June 29–July 2.

@5# Merkle, C. L., Feng, J. Z. and Buelow, P. E. O., 1998, ‘‘Computational Moeling of the Dynamics of Sheet Cavitation,’’Proceedings of the 3rd Interna-tional Symposium on Cavitation, Grenoble.

@6# Kunz, R. F., et al., 1999, ‘‘A Preconditioned Navier-Stokes Method for TwPhase Flows with Application to Cavitation Prediction,’’ 14th AIAA CFDConference, Norfolk, VA, June 28–July 1.

@7# Ahuja, V., Hosangadi, A., Ungewitter, R., and Dash, S. M., 1999, ‘‘A HybrUnstructured Mesh Solver for Multi-Fluid Mixtures,’’ AIAA-99-3330, 14thAIAA CFD Conference, Norfolk, VA, June 28–July 1.

@8# Hosangadi, A., Sinha, N., and Dash, S. M., 1997, ‘‘A Unified HyperboInterface Capturing Scheme for Gas/Liquid Flows,’’ AIAA-97-2081, 13AIAA CFD Conferences, Snowmass, CO, June 29–July 2.

@9# Hosangadi, A., Lee, R. A., Cavallo, P. A., Sinha, N., and York, B. J., 19‘‘Hybrid, Viscous, Unstructured Mesh Solver for Propulsive ApplicationsAIAA-98-3153, AIAA 34th JPC, Cleveland, OH, July 13–15.

@10# So, R. M. C., Sarkar, A., Gerodimos, G., and Zhang, J., 1997, ‘‘A DissipatRate Equation for Low Reynolds Number and Near-Wall Technique,’’ TheComput. Fluid Dyn.,9, pp. 47–63.

@11# Hosangadi, A., Lee, R. A., York, B. J., Sinha, N., and Dash, S. M., 19‘‘Upwind Unstructured Scheme for Three-Dimensional Combusting FlowsJ. Propul. Power,12, No. 3, May–June, pp. 494–503.

@12# Barth, T. J., 1991, ‘‘A 3-D Upwind Euler Solver for Unstructured MeshesAIAA-91-1548, 10th AIAA CFD Conference, Honolulu, HI, June.

@13# Barth, T. J., and Linton, S. W., 1995, ‘‘An Unstructured Mesh Newton So

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tion for Compressible Fluid Flow and Its Parallel Implementation,’’ AIAA-950221, 33th AIAA Aerospace Sciences Meeting at Reno, NV, Jan 9–14.

@14# Rouse, H., and McNown, J. S., 1948, ‘‘Cavitation and Pressure DistributHead Forms at a Zero Angle of Yaw,’’ Technical Report: State UniversityIowa Engineering Bulletin No. 32.

@15# Shen, Y. T., and Dimotakis, P., 1989, ‘‘The Influence of Surface Cavitation

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on:of

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Hydrodynamic Forces,’’ 22nd American Towing Tank Conference, St JohNF, August 8–11.

@16# Singal, A. K., Vaidya, N., and Leonard, A. D., 1997, ‘‘Multi-DimensionaSimulation of Cavitating Flows Using a PDF Model for Phase ChangeFEDSM97-3272, 1997 ASME Fluids Engineering Division Summer MeetinVancouver, British Columbia, Canada, June 22–26.

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