Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications

Jules W. Lindau Robert F. Kunz David A. BogerDavid R. Stinebring Howard J. Gibeling

(Penn State Applied Research Laboratory, University Park, Pennsylvania 16802)

AbstractUnsteady, high Reynolds number validation

cases for a multi-phase CFD analysis tool have beenpursued. The tool, designated UNCLE-M, has a widerange of applicability including flows of navalrelevance. This includes supercavitating and cavitatingflows, bubbly flows, and water entry flows. Thus far thetool has been applied to a variety of configurations.Axisymmetric sheet cavity flow-fields have beenmodeled. In particular, an attempt to validate theunsteady reliability of UNCLE-M with consideration ofthe effect of cavitation number, Reynolds number andturbulence model has been made. Analysis of themodeled unsteady flow-field is also made andconclusions regarding the causes of success andshortcomings in the computational results are drawn.

IntroductionThe ability to properly model unsteady

multiphase flows is of great importance, particularly innaval applications. Cavitation may occur in submergedhigh speed vehicles as well as rotating machinery,nozzles, and numerous other venues. Traditionally,cavitation has had negative implications associated withdamage and/or noise. However, for high speedsubmerged vehicles, the reduction in drag associatedwith a natural or ventilated cavity has great potentialbenefit. Yet, cavitation modeling remains a difficulttask, and only recently have full Reynolds-averaged,three-dimensional, multi-phase, Navier-Stokes toolsreached the level of utility that they might be appliedfor engineering purposes.

UNCLE-M (Kunz, 1999(I,II)) is a fullyimplicit, pre-conditioned, multi-phase, 3-D, fullygeneralized multi-block, parallel, Reynolds-averagedNavier-Stokes solver. The code was initially evolvedfrom a version of the single-phase UNCLE codedeveloped at Mississippi State University (Taylor,1995), and has undergone significant furtherdevelopment. UNCLE-M incorporates mixture volumeand constituent volume fraction transport/generation forliquid, condensable vapor and non-condensable gasfields. Mixture momentum and turbulence scalarequations are also solved. Flux limiting has beenapplied to the inviscid flux terms based on the localslope of the solution volume fraction. As a result, high-

order accurate solutions containing crisp, physicallyreasonable interfaces at the cavity boundary may beobtained with minimal nonphysical oscillations.

Non-equilibrium mass transfer modeling isemployed to capture liquid and vapor phasic exchange.The code can handle buoyancy effects and the presence/interaction of condensable and non-condensable fields.This level of modeling complexity represents the state-of-the-art in CFD analysis of cavitation. The restrictionsin range of applicability associated with inviscid flow,slender body theory and other simplifying assumptionsare not present. In particular, the code can plausiblyaddress the physics associated with high-speedmaneuvers, body-cavity interactions and viscous effectssuch as flow separation.

The principal interest here is in modeling highReynolds number, unsteady flow about bodies withrunning cavities. These cavities are presumed to besheet cavities amenable to a homogeneous approach. Inother words, it is presumed that the nonequilibriumdynamic forces of bubbles are of negligible magnitude.In the present work, the effect of surface tension is notincorporated, since interface curvatures are very smallfor the configurations considered. This assumption issupported by model results of sheet cavitation with afull two-fluid approach (Grogger and Alajbegovic,1998).

In previous work (Kunz, 1999(I)), the fidelityof UNCLE-M has been demonstrated for steady statefluid flows. However, due to the reentrant jet, cavitypinching, and other effects of turbulent separated flow,multi-phase flows of naval importance are generallyunsteady. In the work presented here, UNCLE-M willbe applied to several configurations of naval relevance.Each of these configurations presents an experimentallydocumented, unsteady fluid dynamic test case. Modelresults will be presented for several ballistic, cavitatorgeometries. Both the steady (averaged) and unsteady(time domain and spectral) behavior of the flow will bepresented and compared with data. In addition,interesting unsteady numerical results will be presentedin a field form for comparison with photographic data.By comparison of the numerical and measured results,the reliability of the unsteady capabilities of the codemay be understood.

1

Nomenclature

Symbols:C1, C2 turbulence model constantsCdest, Cprod mass transfer model constants

CP pressure coefficientCd drag coefficientD body diameterdm bubble diameter

f cycling frequency (Hz)gi gravity vectork turbulent kinetic energyL bubble length

, mass transfer ratesP turbulent kinetic energy productionPrtk,Prtε turbulent Prandtl numbers for k and εp pressureReD Reynolds number based on body diameter

Str Strouhal frequency

s arc length along configuration (also seconds)physical time, mean flow time scale, time step

U velocity magnitudeui Cartesian velocity componentsxi Cartesian coordinates

y+ dimensionless wall distance

α volume fraction, angle of attackβ preconditioning parameterτ pseudo-timeε turbulence dissipation rateµ molecular viscosityρ density

σ cavitation number ( )

Subscripts, Superscripts:D body diameterl liquidm mixtureng non-condensable gast turbulentv condensable vapor

free stream value

Physical ModelThe physical model equations solved here

have been described previously (Kunz 1999 (I,II)). Thebasis of the model is the incompressible multiphaseReynolds Averaged Navier Stokes Equations in ahomogeneous form. Each phase is treated as a new

species and requires the inclusion of a separatecontinuity equation. Three species, representing aliquid, a condensable vapor, and a noncondensable gas,are included. Mass transfer between the liquid andvapor phases is achieved through a differential model.Other researchers have applied similar models with asingle species approach. However, the multiple speciesmodel of multiphase flow is presented as a moreflexible physical approach. A high Reynolds numberform of two-equation models with standard wallfunctions provides turbulence closure.

The governing differential equations, cast inCartesian tensor form are given as Equation (1):

(1)

Where mixture density and turbulent viscosityhave been defined in Equation (2).

(2)

In the present work, the density of eachconstituent is taken as constant. Equation (1) representsthe conservation of mixture volume, mixturemomentum, liquid phase volume fraction and non-condensable gas volume fraction, respectively. Physicaltime derivatives are included for unsteadycomputations. The formulation incorporates pre-conditioned pseudo-time-derivatives ( terms),defined by parameter β, which provide favorableconvergence characteristics for steady state andunsteady computations, as discussed further below.

The formation and collapse of a cavity ismodeled as a phase transformation. Detailed modelingof this process requires knowledge of thethermodynamic behavior of the fluid near a phasetransition point and the formation of interfaces.Simplified models are presented here, resulting in theuse of empirical factors. Given as Equation (3), twoseparate models are used to describe the transformationof liquid to vapor and the transformation of vapor backto liquid. For transformation of liquid to vapor, ismodeled as being proportional to the product of theliquid volume fraction and the difference between the

m·-

m·+

fD( ) U∞⁄

t t∞ ∆t, ,

ρmyUt( ) µm⁄

p∞ pv–

1/2ρlU∞2

--------------------≡

∞

1

ρmβ2--------------

τ∂∂p

+xj∂

∂ujm·

++m·

- 1

ρl----- 1

ρv------–

=

t∂∂ ρmui( )+

τ∂∂ ρmui( )+

xj∂∂ ρmuiuj

( ) - xi∂

∂p+

xj∂∂ µm,t xj∂

∂ui

+ρmgi=

t∂

∂α l+

α l

ρmβ2--------------

τ∂∂p

+τ∂

∂α l+

xj∂∂ α luj

( ) m·+

ρl-------+

m·-

ρl------

=

t∂

∂αng+

αng

ρmβ2--------------

τ∂∂p

+τ∂

∂αng+

xj∂∂ αngu

j( ) 0=

ρm ρlα l ρvαv ρngαng+ +=

µm t,ρmCµk2

ε------------------=

∂/∂τ

m·-

2

o,ting

en.ns

owthntowasachelerly ofelyfort

ble

theidsde.

tioneenngiorrtal

l,geh

thatatgend arth

nnnt

dels,id

y

rldsun0

computational cell pressure and the vapor pressure.This model is similar to the one used by Merkle et. al.(1998) for both evaporation and condensation. Fortransformation of vapor to liquid, a simplified form ofthe Ginzburg-Landau potential is used for the masstransfer rate .

(3)

Cdest and Cprod are empirical constants. For allwork presented here, Cdest = Cprod = 100. Both masstransfer rates are non-dimensionalized with respect to amean flow time scale.

In this work, a high Reynolds number two-equation turbulence model with standard wall functionshas been implemented to provide turbulence closure.Either the k-ε or RNG k-ε (Orszag et al. 1993) modelare represented in Equation (4):

(4)

As with velocity, the turbulence scalars areinterpreted as being mixture quantities.

Numerical MethodThe baseline numerical method has been

evolved from the work of Taylor and his coworkers atMississippi State University (Taylor et. al. (1995), forexample). Primitive variable interpolant type Roe fluxdifference splitting is used for spatial discretization. Animplicit procedure is adopted with inviscid and viscousflux Jacobians approximated numerically. A block-symmetric Gauss-Seidel iteration is employed to solvethe approximate Newton system at each timestep.

The multi-phase extension of the code retainsthese underlying numerics but incorporates twoadditional volume fraction constituent transportequations. During flux formulation, a Jameson-style(Jameson 1981) flux limiter based on liquid volumefraction is applied to the primitive interpolants. A non-diagonal pseudo-time-derivative preconditioning matrixis also employed. While the time derivative termvanishes from the mixture continuity equation as thelimit of incompressible constituent phases isapproached, the effect of preconditioning is to reducethe associated stiffness. This preconditioner gives riseto a system with well-conditioned eigenvalues whichare independent of density ratio and local volume

fraction. This system is well suited to high density ratiphase-separated two-phase flows, such as the cavitasystems of interest here.

A temporally second-order accurate dual-timscheme was implemented for physical time integratioAt each time step, the turbulence transport equatioare solved subsequent to solution of the mean flequations. The multiblock code is instrumented wiMPI for parallel execution based on domaidecomposition. During unsteady time integration, obtain results presented here message passing applied after each symmetric Gauss-Seidel sweep. Einner iterate involved twenty symmetric Gauss-Seidsweeps, and each time step involved fifteen inniterations. This procedure was sufficient to reliabreduce the unsteady residual by at least two ordersmagnitude. However, a case by case examination likcould have reduced the expended computational efyielding results similar in solution fidelity. Furtherdetails on the numerical method and code are availain Kunz et. al. (1999(II)).

ResultsAxisymmetric sheet cavity flow-fields have

been modeled. In particular, an attempt to validate unsteady reliability of a multiphase, computational fludynamics tool with consideration of the affectReynolds number and turbulence model has been maSteady, average, measurements of relevant cavitaparameters for the shapes chosen have bdocumented by Rouse and McNown (1949). Stinebriet al. (1983) documented the unsteady cycling behavof several axisymmetric cavitators. Their repoincluded results for both ventilated and naturcavitation. The unsteady performance of a 45o (22.5o inprofile from centerline to outer edge) conicahemispherical, and 0-caliber ogival cavitators at a ranof cavitation numbers were documented. AlthougUNCLE-M has the capability to model ventilatedcavitation (Kunz 1999(I)), only natural cavitationresults have been included here. It should be noted the results of Rouse and McNown (1948) indicated thfor the cavitator types and flows at or above the ranof experimental Reynolds numbers reported ainvestigated here, the flow should be turbulent oversignificant portion of the forebody. Therefore, fosingle phase flow, particularly for geometrically smooshapes, this should serve to avoid the well knowchaotic, critical laminar separation and transitioregime. The numerical results employ a fully turbulemodel.

Results presented here are given in the mocomputational system (SI) units. For all computationthe free stream velocity was set to 1 (m/s), the liqudensity was 1000 (kg/m3), and the vapor density was 1(kg/m3). For most computations, the liquid viscositwas then set equal to 10-3 (Pa-s), and that of the gasphases was set to 10-5 (Pa-s). Then the body diametewas chosen to achieve the desired model Reynonumber. In the case of the hemispherical forebody rat a body diameter based Reynolds number of 1.36x17,the liquid kinematic viscosity was then set equal to 10-5

m·+

m·- Cdestρvα

lMIN 0 p pv–,[ ]

1/2ρlU∞2

t∞

------------------------------------------------------------------=

m·+ Cprodρvα l

21 α l–( )

t∞----------------------------------------------------=

t∂∂ ρmk( )

xj∂∂ ρmku

j( )+

xj∂∂ µm t,

Prtk------------

xj∂∂k

P ρε–+=

t∂∂ ρmε( )

xj∂∂ ρmεu

j( )+

xj∂∂ µm t,

Prtε------------

xj∂∂ε

C1P C2ρε–[ ]+εk--

=

3

(Pa-s), and that of the gas phases was set to 10-7 (Pa-s).The model body diameter for this case was thus, 0.136(m). Prior to initiating unsteady computations, forpurposes of computational expediency, a steady state,

, integration was carried out. At the completionof this integration, it was possible to determine if themodel solution was physically unsteady. In general,physically unsteady conditions were indicated bymarginally convergent, flat-lined steady-state residualhistories, themselves containing large amounts ofunsteadiness.

A photograph of a 0-caliber axisymmetriccavitator operating at conditions similar to thosemodeled here is given in Figure 1 (Stinebring 1976)Figure 2 contains a series of snapshots of the volumefraction field from an unsteady model computation offlow over a blunt cavitator. Here the Reynolds number(based on diameter) was 1.46x105 and the cavitationnumber was 0.3. The time history for this case is givenin model seconds, and at t=0, unsteady integration wasinitiated after obtaining a steady-state, , initialcondition. Thus it is expected that there was some start-up transient associated with initialization from anartificially maintained set of conditions. For the volumefraction contours, dark blue indicates vapor, a liquidvolume fraction of less than 0.005, and bright redindicates liquid, a volume fraction of one. Somesignificant numerical integration time parameters forthis case are the body diameter to free stream velocityratio, seconds, and the physicalintegration step size, seconds.

This result is presented over an approximatemodel cycle. The figure also includes the correspondingtime history of drag coefficient. Note that the spikes indrag near t=37.725 and t=38.925 seconds correspond toreductions in the relative amount of vapor near thesharp leading edge. This marks the progress of a bulkvolume of liquid from the closure region to the forwardend of the cavity as part of the reentrant jet process.Although far from regular, these spikes also delineatethe approximate model cycle. This picture serves toillustrate the basic phenomenon of natural sheetcavitation as it is best captured by UNCLE-M. Thisresult is notable for the spatial and temporally irregularnature of the computed flow field. Even aftersignificant integration effort, a clearly periodic result

Figure 1: Zero caliber ogive in water tunnel at Re(D)=2.9x105, σ=0.35 (approximate) (Stinebring,

1976).

∆t ∞=

∆t ∞=

D U∞⁄ 0.146=

∆t 0.001=

Figure 2: Modeled flow over a 0-caliber ogive. Liquidvolume fraction contours and corresponding drag history. UNCLE-M result. σ=0.3. ReD=1.46x105.

t=37.525 t=37.625

t=37.725 t=37.825

t=37.925 t=38.025

t=38.125 t=38.225

t=38.325 t=38.425

t=38.525 t=38.625

t=38.725 t=38.825

t=38.925

37 37.5 38 38.5 391

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

t (s)

Cd

t=39.025

4

are,r

tor ise 3y

helat

ghis of

heityatart- atcedsrmedsn.veitysultls,

dy

ge

hise

ng

had not emerged. Thus, to deduce the dominantfrequency with some confidence, it was necessary toapply ensemble averaging.

An examination of the flow pattern capturedsuggests qualitative validity. Note, in Figure 2. thatover a significant portion of the sequence, the leading,or formative, edge of the cavity sits slightlydownstream from and not attached to the sharp corner.In their experiments, Rouse and McNown (1948)observed this phenomenon. They suggested that thisdelay in cavity formation was due to the tightseparation eddy which forms immediately downstreamof the corner and, hence, locally increases the pressure.The corresponding evolution of cavitation furtherdownstream, at the separation interface, was proposedto be due to tiny vortices. These vortices, after sometime, subsequently initiate the cavity. Figure 3 shows asingle frame at t=37.8 seconds from the same modelcalculation (as shown in Figure 2). Here, to clarify whatis captured, the volume fraction contours have beenenhanced with illustrative streamlines. Note that theseare streamlines drawn from a frozen time slice.Nonetheless, if all of the details envisioned by Rouseand McNown were present, the streamlines shouldindicate smaller/tighter vortical flows. The current levelof modeling was unable to capture small vorticalstructures in the flow. However, the overallcomputation was apparently able to capture the grossaffects of these phenomena and reproduce a delayedcavity. In fact from examination of the cavity cycleevolution shown in Figure 2, and the streamlines shownin the snapshot, it appears that gross unsteadiness isdriven by a combination of a reentrant jet and sometype of cavity pinching (Brennan 1992). The pinchingprocess is particularly well demonstrated in Figure 2from t=38.125 to 38.325 seconds. However, rather thancomplete division and convection into the free stream,it should be noted that, in later frames of Figure 2, thepinched portion of the cavity appears to rejoin the maincavity region.

The low frequency mode apparent in most ofthe experimental 0-caliber results appears to have beencaptured at the lowest cavitation number (σ=0.3), asshown in Figure 2, and is evidenced in the testphotograph (Figure 1). In Figure 4, the drag coefficienthistory for a 40 model second interval from the same

computation as in Figure 2 is shown. Here, a clepicture of the persistence, over a long integration timof the irregular flow behavior is documented. At highecavitation numbers, the current set of 0-caliber cavitaresults indicate a more regular periodic motion. Thiscontrary to the experimental data. However, as Figurindicates, the ability to capture this motion at ancavitation number may not necessarily require texplicit capture of the finer flow details of the vorticaflow structure. This is encouraging and suggests thwith increased computational effort, without alterinthe current physical model, the representation of tphenomenon could be improved over a greater rangecavitation numbers.

Figure 5 presents the spectral content of tresult given in Figure 4. This power spectral densplot is based on four averaged Hanning windowed dblocks of the time domain result. To eliminate the staup transient effect, the record was truncated, startingt=10 seconds and, to tighten the resulting confidenintervals, more time domain results, after t=40 seconwere included. As is typical of highly nonlineasequences, the experience of this unsteady tiintegration demonstrated that, additional time recormerely enrich the power spectral density functioHowever, the additional records do serve to improthe confidence intervals, and, therefore, add reliabilto the numerical convergence process. The model reused, was, as indicated by the confidence intervasufficient for a comparison to experimental, unstearesults.

Figure 6 contains a time record of dracoefficient during modeled flow over a 0-caliber ogivat a Reynolds number of 1.46x105 and cavitationnumber of 0.35. The Strouhal frequency based on tresult is 0.0909. Here it is apparent that thcomputational modeling was incapable of reproduci

Figure 3: Snapshot of modeled flow over a 0-caliber ogive. Liquid volume fraction contours and selected

streamlines. UNCLE-M result. σ=0.3. ReD=1.46x105.

Figure 4: Model time record of drag coefficient for flowover a 0-caliber ogive at ReD=1.46x105 and σ=0.3. In model units, (s), physical time step,

(s).

0 10 20 30 401

1.2

1.4

1.6

1.8

2

Cd

t (s)

D U∞⁄ 0.146=

∆t 0.001=

5

elesenag to ofdlnd

inaghe

ehalofr

st a

hetheuregd

what should have been a lower frequency result withflow around the forebody dominated by a moreirregular cavity. It is supposed that the correct result, incomparison with the experimental data in the Strouhalfrequency plot (Figure 22) would have been similar innature to the results presented for a cavitation numberof 0.30 in Figure 4. In addition to lacking the richfrequency content of the result for lower cavitationnumbers, it appears that the amplitude of theunsteadiness present is an order of magnitude lower.

Figure 7 contains a series of snapshots fromthe unsteady model computation of a hemisphericalcavitator at a Reynolds number (based on diameter) of1.36x105 and a cavitation number of 0.2. This result ispresented over a period slightly longer than the

approximate model cycle. In this case the modStrouhal frequency is 0.0326. There are ten frampresented, and the first (or last) nine of those tconstitute an approximate model cycle. The drhistory trace in Figure 8 demonstrates how, relativethe modeled flow over the blunt forebody, the patternflow over the hemispherical forebody is regular anperiodic. This is consistent with experimentaobservations made (for example) by Rouse a

Mcnown (1948). Note the evolution of flow shown inFigure 7 as it compares to the drag history shownFigure 8. As would be expected, the large spike in drcorresponds to the minimum in vapor shown near tmodeled t=1.6 seconds.

The next three figures demonstrate thexpected and captured dependence of Stroufrequency on cavitation number. Here the trend increasing cycling frequency with cavitation numbeduring flow over a hemispherical forebody ireproduced. The result has been demonstrated aReynolds number of 1.36x105. This Reynolds numberwas intended to scale the problem properly with tdata available. Here, the magnitude of the drag and amplitude of the unsteadiness may be examined. Fig9 contains a time record of drag coefficient durinmodeled flow over a hemispherical forebody an

Figure 5: UNCLE-M result. 0-caliber ogive at ReD=1.46x105 and σ=0.3. Power spectral density function with 50% confidence intervals shown.

Figure 6: UNCLE-M result. Time record of drag coefficient for flow over a 0-caliber ogive at ReD=1.46x105 and σ=0.35. In model units,

(s), physical time step, (s).

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

f (Hz)

Cd

Am

plit

ude

0 5 10 15 20 251.1685

1.169

1.1695

1.17

1.1705

1.171

Cd

t (s)

D U∞⁄ 0.146= ∆t 0.001=

Figure 7: Liquid volume fraction contours. Modeled flow over a hemispherical forebody and cylinder.

UNCLE-M result. σ=0.2, Re(D)=1.36x105.

t=0.1 t=0.6

t=1.1 t=1.6

t=2.6 t=3.1

t=3.6 t=4.1

t=4.6 t=5.1

6

gry

galoferk

h- aofp,

l, it

cylinder at a Reynolds number of 1.36x105 andcavitation number of 0.25. The Strouhal frequencybased on this result is 0.0484. Figure 10 contains asimilar time record of drag coefficient during modeledflow over a hemispherical forebody and cylinder at aReynolds number of 1.36x105 and cavitation number of0.30. The Strouhal frequency based on this result is0.0622. Figure 11 contains a time record of dragcoefficient during modeled flow over a hemisphericalforebody and cylinder at a Reynolds number of1.36x105 and cavitation number of 0.35. The Strouhalfrequency based on this result is 0.0933. In addition, thehigher σ, higher frequency results contain smallercavities. In these situations, cavities drive the overallunsteadiness of the flow, and the problem of sufficientgrid points to define an unsteady cavity becomesapparent. Thus, by pushing the limits of reasonablediscretization, the limits of effective modeling also aretested.

Figure 12 contains a time record of dragcoefficient during modeled flow over a hemisphericalforebody and cylinder at a Reynolds number of1.36x106 and cavitation number of 0.3. The Strouhalfrequency based on this result is 0.0614. Figure 13contains a time record of drag coefficient duringmodeled flow over a hemispherical forebody andcylinder at a Reynolds number of 1.36x107 andcavitation number of 0.3. The Strouhal frequency basedon this result is 0.133. Here, the standard trend ofincreased turbulent cycle frequency with increasedReynolds number appears to have been presented.

Figure 14 contains a time record of dragcoefficient during modeled flow over a conicalforebody and cylinder at a Reynolds number of1.36x105 and cavitation number of 0.2. The Strouhalfrequency based on this result is 0.0383. As anticipated,due to the expected stability of cavities about this

shape, this model flow exhibited very regular cyclinwith little additional strong components from secondamodes.

Figure 15 contains a time record of dracoefficient during modeled flow over a hemisphericforebody and cylinder at a Reynolds number 1.36x105 and cavitation number of 0.25. This is anothUNCLE-M result; however, rather than the standard −ε model, the RNG k-ε turbulence model has beenapplied. For the hemispherical forebody witcylindrical afterbody, when using the standard kεmodel, to obtain, during a complete dual time cycle,reduction in the unsteady residual of two orders magnitude, it was sufficient to apply a time ste∆t=0.0025 seconds. However, with the RNG k-ε model,to obtain the same reduction in the unsteady residua

Figure 8 Unsteady drag coefficient. Flow over a hemispherical forebody and cylinder. UNCLE-M result.

σ=0.2, Re(D)=1.36x105. In model units, (s), physical time step, (s).

0 5 10 15 20 250.4

0.405

0.41

0.415

0.42

0.425

0.43

Cd

t (s)

D U∞⁄ 0.136= ∆t 0.001=

Figure 9: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody andcylinder at ReD=1.36x105 and σ=0.25. In model units,

(s), physical time step, (s).

Figure 10: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody andcylinder at ReD=1.36x105 and σ=0.3. In model units,

(s), physical time step, (s).

0 5 10 15 200.43

0.435

0.44

0.445

0.45

time

Cd

D U∞⁄ 0.136= ∆t 0.0025=

0 5 10 15 200.46

0.465

0.47

0.475C

d

t (s)

D U∞⁄ 0.136= ∆t 0.0025=

7

ithd, ofhethenad

tsavewnerarnd a

r

was necessary to run a physical time step of 0.001seconds. Note that in this time history trace, there is agreat deal of unsteadiness. The result appears far lesscoherent than the standard k-ε result given in Figure 9.The Strouhal frequency based on this result is 0.1855.Based on the measured data (Stinebring 1983), thisfrequency is far too high. When applied for a highercavitation number, σ=0.30, the RNG k-e based modelagain required a smaller time step (0.001 units) andpredicted a Strouhal frequency of 0.068. Here the valueis nearly the same as that predicted by the model usingthe k-ε turbulence model. Clearly, the trend based onthese results is incorrect. It appears that the currentimplementation of the RNG model has yielded resultsconsistent with the k-ε model at one cavitation number,σ=0.30, but at a lower value, the cycle frequency is fargreater than the standard k-ε modeled result or themeasured data. It seems probable that with finer time

and space discretization, the current RNG k-ε modelimplementation would achieve results comparable wthe k-ε model at all cavitation numbers. As expectethe RNG model increased the overall unsteadinessthe results. However, the computational cost of tcurrent results is already significant, and based on UNCLE-M solutions obtained thus far, and comparisoto experimental data, little benefit appears to be hfrom the current application of the RNG k-ε model.

Where applicable, for the unsteady resulpresented here, the arithmetically averaged results hbeen compared to the results of Rouse and McNo(1948). Figure 16 contains a comparison for flow ovthe 0-caliber cavitator, Figure 17 contains a similcomparison for flow over a hemispherical cavitator, aFigure 18 contains a similar comparison for flow over

Figure 11: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at ReD=1.36x105 and σ=0.35. In model units,

(s), physical time step, (s).

Figure 12: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at ReD=1.36x106 and σ=0.3. In model units,

(s), physical time step, (s).

0 5 10 15 200.49

0.492

0.494

0.496

Cd

t (s)

D U∞⁄ 0.136= ∆t 0.0025=

0 50 100 1500.463

0.4635

0.464

0.4645

Cd

t (s)

D U∞⁄ 1.36= ∆t 0.025=

Figure 13: UNCLE-M result. Time record of drag coefficient for flow over a hemispherical forebody andcylinder at ReD=1.36x107 and σ=0.3. In model units,

(s), physical time step, (s).

Figure 14: UNCLE-M result. Time record of drag coefficient for flow over a conical forebody and cylinde

at ReD=1.36x105 and σ=0.3. In model units, (s), physical time step, (s).

0 5 10 15 200.46

0.4605

0.461

0.4615

0.462

0.4625

0.463

0.4635

0.464

Cd

t (s)

D U∞⁄ 0.136= ∆t 0.0025=

0 10 20 30 400.366

0.3665

0.367

0.3675

0.368

0.3685

0.369C

d

t (s)

D U∞⁄ 0.136= ∆t 0.0025=

8

ofrst

ey

talseofter-d,

the

conical cavitator. In each of these figures, the overallperformance of the code seems to generally agree withthe data. Clearly as the cavitation number is reduced,the UNCLE-M result tends further from the data. Forboth the numerical and experimental results, theaverage initiation and termination point of the cavitymay be deduced from this figure. Accordingly, theability of the code to properly model the average cavityis well represented in these figures. The averagedperformance over the 0-caliber cavitator appears betterthan that of either of the others. The performance overthe conical shape is the worst. It is clear that theformation point of the average cavity should be welldefined in the axisymmetric shapes with discontinuous

profile slopes. Thus it is not clear why the prediction termination of the cavity should, on average, be wofor the conical shape.

Several parameters of relevance in thcharacterization of cavitation bubbles include boddiameter, D, bubble length, L, bubble diameter, dm, andform drag coefficient associated with the cavitator, Cd.Some ambiguity is inherent in both the experimenand computational definition of the latter three of theparameters. Bubble closure location is difficult tdefine due to unsteadiness and its dependence on abody diameter (which can range from 0 [isolatecavitator] to the cavitator diameter). Accordinglybubble length is often, and here, taken as twice

Figure 15: UNCLE-M/RNG k-ε turbulence model result. Time record of drag coefficient for flow over a hemispherical forebody and cylinder at ReD=1.36x105

and σ=0.25. In model units, (s), physical time step, (s).

Figure 16: Flow over a 0-caliber cavitator (s/d=arc length over diameter). Averaged unsteady pressure

computations and measured data (Rouse and McNown 1948).

0 2 4 6 8 100.42

0.425

0.43

0.435

0.44

0.445

0.45C

d

t (s)

D U∞⁄ 0.136=

∆t 0.001=

0 2 4 6 8 10

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Cp

s/d

computation, σ=0.3data, σ=0.3computation, σ=0.4data, σ=0.4

computation, σ=0.3data, σ=0.3computation, σ=0.4data, σ=0.4

Figure 17: Flow over a hemispherical cavitator (s/d=arc length over diameter). Averaged unsteady pressure

computations and measured data (Rouse and McNown 1948).

Figure 18: Flow over a conical cavitator (s/d=arc length over diameter). Averaged unsteady pressure

computations and measured data (Rouse and McNown 1948).

0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Cp

computation, σ=0.2data, σ=0.2computation, σ=0.3data, σ=0.3

s/d

0 1 2 3 4 5 6−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Cp

computation, σ=0.3data, σ=0.3computation, σ=0.4data, σ=0.4

computation, σ=0.3data, σ=0.3computation, σ=0.4data, σ=0.4

s/d

9

distance from cavity leading edge to the location ofmaximum bubble diameter (see Figure 19). The formdrag coefficient is taken as the pressure drag on anisolated cavitator shape. For cavitators with afterbodies,such as here, the pressure contribution to Cd associatedwith the back of the cavitator is assumed equal to thecavity pressure (≅ pv). For the model computations, dmis determined by examining the αl = 0.5 contour anddetermining its maximum radial location.

In Figure 20, the quantity is plotted

against cavitation number for experimental data setsassembled by May (1975). Arithmetically averagedUNCLE-M results are included for ten unsteadycomputations made with three cavitator shapes. The

correlation between and σ has been long

established (see Reichardt (1946), Garabedian (1958),for example). Despite the significant uncertaintiesassociated with experimental and computationalevaluation of L and CD, the data and simulations docorrelate well, close to independently of cavitatorshape.

Another parameter that has been established tobe well correlated with cavitation number is thefineness ratio, L/dm. May (1975) noted that thisparameter is particularly independent of ambientpressure, vapor pressure, free stream velocity, andwhether the cavity was filled with vapor or a mixture ofvapor and air. Once again, May assembled a largequantity of experimental results for this parameter.Figure 21 contains a comparison of the fineness ratio,L/dm, for averaged unsteady UNCLE-M computationsand data.

As a blanket observation, the spread of databetween the experiments and computations in Figure 22appears to be rather large. However, there are severalencouraging items to be reviewed. It is clear that (for agiven cavitation number) the computational results arebounded by the experimental data, and the propertrends (rate of change of Strouhal frequency withcavitation number) are well captured. More insight intothe physical relevance of the data requires examinationof specific results.

When run at similar cavitation numbers, theextremely low frequencies observed in the 0-caliberogive testing was not captured by the model. However,considering only model results at a cavitation numberof 0.3 (see Figure 4), it appears that the observed ofbehavior was captured.

Figure 22 contains a large survey of unsteadycomputational and experimentally obtained data(Stinebring 1983). The numerical results in this figuresummarize this validation effort. Here, Strouhalfrequency is shown over a range of cavitation numbers.Computational results are given for hemispherical, 1/4-caliber, conical, and 0-caliber forebodies. Unsteadyexperimental data is included for the hemispherical,conical and 0-caliber shapes. Computational results forthe hemisphere, 1/4-caliber and conical forebodies,were obtained at a Reynolds number based on diameter

Figure 19: Definition used to determine bubble length, L, and diameter, dm.

Figure 20: Dimensionless drag to bubble length parameter and cavitation number. Flow over

axisymmetric cavitators. Arithmetically averaged, unsteady UNCLE-M results and data (May 1975).

L/ DCd1/2( )

L/ DCd1/2( )

dmL/2D

10−2

10−1

100

100

101

102

103

UNCLE-M hemisphereUNCLE-M 0-caliberUNCLE-M conedata spheredata stagnation cupdata cone

UNCLE-M hemisphereUNCLE-M 0-caliberUNCLE-M conedata spheredata stagnation cupdata cone

σ

L/(

DC

d1/2 )

Figure 21: Cavity fineness ratio and cavitation index. Flow over axisymmetric cavitators. Arithmetically

averaged, unsteady, UNCLE-M results and data (May 1975).

10−2

10−1

100

10−1

100

101

102

UNCLE-M hemisphereUNCLE-M 0-caliberUNCLE-M conedata conedata stagnation cupdata spheredata disk

L/d

m

σ

10

of 1.36x105. For the 0-caliber ogive, computations weremade at a Reynolds number of 1.46x105. In addition,for the hemisphere, results are included for Reynoldsnumbers of 1.36x106 and 1.36x107. The experimentalresults included in the figure were obtained at Reynoldsnumbers ranging from 3.5x105 to 1.55x106.

For the hemispherical forebody results, as maybe seen in Figure 22, there is a significant but almostconstant offset between the measured unsteady data andthe modeled results both of which appear to follow alinear trend over the range presented. An interestingresult occurs in the model data for the hemisphericalforebody with a Reynolds number of 1.36x107

(pentagrams in Figure 22). Here the numerical resultsappear to agree quite well with the experimental datafor hemispherical forebodies. The experiments weretaken at an order of magnitude lower Reynolds number,but the agreement is apparent in both cases wheremodel results have been obtained. For design purposes,this may suggest an avenue towards model calibration.

Another result found in the Str versus σ plot(Figure 22) is the tendency of the modeled flows tobecome steady at higher cavitation numbers. For the 0-caliber or the conical cavitators, this is the reason modelresults are not included for cavitation numbers greaterthan 0.4. For the modeled hemisphere, the upper limitof cavitation number to yield unsteady model resultswas found to be Reynolds number dependent. At aReD=1.36x105, the maximum cavitation numberyielding an unsteady result was , atReD=1.36x106, that number was , and at

ReD=1.36x107, the maximum cavitation number forunsteady computations was not determined. This result

may indicate a limit of the computational grid appliedto the problems rather than a limit of the level ofphysical modeling. In addition, physically in the modeof unsteadiness present, a transition does occur fromcavity driven to separated, turbulent, but single phasedriven flow.

For the conical forebody, the datum shown inFigure 22 suggests that the cycling frequency should behigher, 0.123. It is worth considering that the Reynoldsnumber of the experimental flow was 3.9x105 and thatthe general trend with increasing Reynolds numbers isto increased frequency. However, based on the standardlevel of dependence of Strouhal frequency (seeSchlicting 1979 for example) on Reynolds number forbluff body flows, it would seem unlikely that the rate ofchange in frequency with Reynolds number (at

) would be as high as three to two. Inaddition, compared to shapes with geometricallysmooth surfaces, the nature of unsteady flow over aconical shape is not expected to be nearly so dependenton Reynolds number. In the case of a cone, at lowvalues of cavitation number (i.e. σ=0.3), the separationlocation, and, hence, the likely forward location of thecavity, is rarely in question.

A trend that is captured in the model resultsbut not represented in the experimental data includedhere, is the tendency for the Strouhal frequency of agiven cavitator shape to exhibit two distinct flowregimes. The first regime exists at moderate cavitationnumbers and is indicated by a low Strouhal frequencywhere the value of Str will have an apparent lineardependence on σ. The second regime tends towardmuch higher cycling frequencies. Here the dependentStrouhal frequency appears to asymptotically approacha vertical line with higher cavitation number, just priorto the complete elimination of the cavity. This isdocumented in Stinebring (1983) and demonstrated inFigure 22 for the modeled hemisphere atReD=1.36x106. Based on the model results, it appearsthat this is characteristic of a change from a flow modedominated by a large unsteady cavity to one dominatedby other, single-phase, turbulent, sources ofunsteadiness.

During this investigation, some effort towardsthe establishment of temporal and spatial discretizationindependence was made. As a requirement of themodel, to accommodate the use of wall functions, forregions of attached liquid flow, fine-grid near-wallpoints were established at locations yielding10<y+<100. Temporal convergence was established bythe successive reduction of time integration step for aselected few cases. Figure 23 contains a comparison ofthe spectral content of results for flow over ahemispherical forebody and cylindrical afterbody, withReD=1.36x105 and σ=0.3, for three, successivelysmaller, integration step sizes. Here, the computed flowresulted in a Strouhal frequency, Str=0.600 with aphysical time step, , Str=0.0622 with aphysical time step, seconds, andStr=0.0680 with a physical time step, seconds. As demonstrated in the figure, for the smallertwo integration step sizes, over the range of relevant

Figure 22: Axisymmetric running cavitators with cylindrical afterbodies. Strouhal frequency and

cavitation number. UNCLE-M results (open symbols) and data reported in Stinebring (1983).

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6hemisphere ReD=(0.35-1.55)x106

0-cal ReD=(0.96-1.46)x105

cone ReD

=3.9x105

model 1/4 cal ReD

=1.36x105

model 0 caliber ReD

=1.46x105

model hemisphere ReD

=1.36x105

model hemisphere ReD

=1.36x106

model hemisphere ReD=1.36x107

model cone ReD=1.36x105

σ

Str

σ 0.35≈σ 0.45≈

ReD 105≈

∆t 0.005=∆t 0.0025=

∆t 0.001=

11

(shown) frequencies, there was very similar modalbehavior. Unfortunately only the fine-grid modelstended to provide unsteady results. Thus time andspatial fidelity were judged independently. Ademonstration of the steady-state spatial convergence ofthe modeled conical forebody and cylindrical afterbodyis given in Figure 24.

It should be noted that during thisinvestigation, steady state results (time integrationsbased on ) using UNCLE-M have been found tobe quite consistent with arithmetically averaged time-dependent results. This result is expected to be useful inexpediting the future interpretation of complex three-dimensional flows. In addition, real single phase flows,at the Reynolds numbers considered, over theseaxisymmetric bodies are in fact unsteady. However,with the grids and level of modeling applied here, theUNCLE-M solutions tended to be steady. It seemspossible that increased resolution and incorporation oflow Reynolds number turbulence modeling wouldresolve this issue.

ConclusionsThe effect of Reynolds number on the results

for the hemispherical cavitator was not anticipated. Itwas assumed that with the appropriate application ofthe high Reynolds number turbulence model at the wall,the inviscid external flow would dominate the flow-field, determining cavity shape and size (i.e. surfacepressure). However, it appears that strong flow-fieldinteractions due to the highly turbulent separatedclosure region are important to determining theunsteady mode. To some extent, based on the averageresults, these phenomena are being accurately captured.However, there are shortcomings in the currentlyemployed level of single-phase turbulence modeling.

The validation cases examined havedemonstrated the capabilities of UNCLE-M over arange of important flow conditions. The mostprominent result for validation is that the unsteadyfrequencies obtained in numerical results appear to bebounded by the experimental data of Stinebring (1983)for all the modeled cases. Other qualitative observationsmade regarding the modeled case of the 0-calibercavitator at a cavitation number of 0.3 suggest thatUNCLE-M has the ability to correctly represent theoverall nature of unsteady, complex, multiphase flowswithout necessarily capturing some of the finer flowdetails. This in itself is a validation of the approachtaken here. Validated modeling based on parametersrelated to profile drag, cavity length, cavity shape, andtrends of Strouhal frequency with cavitation numberhas been accomplished. It seems clear that with higherfidelity turbulence and mass transfer modeling andsubsequent improved modeling of the closure region, abenefit to the modeling of unsteady cavitating flowswould be obtained. However, the current approach hasallowed rendering of unsteady multiphase flows atReynolds numbers relevant to engineering applicationsin a modeling method amenable to complex geometriesand design applications.

The authors continue to develop thecapabilities of UNCLE-M. This includes the pursuitrelevant validation cases for complex three-dimensionalflows. In addition, new levels of physical modeling,such as compressible phases via isothermal and fullenergy modeling, will be incorporated. These newcapabilities, in addition to the already incorporatedabilities to model buoyancy and ventilation, are critical

Figure 23: Spectral comparison of effect of physical integration time step size on Cd history. UNCLE-M

result. Flow over a hemispherical forebody with cylindrical afterbody. ReD=1.36x105. σ=0.3.

Figure 24: Comparison of predicted surface pressure distributions for naturally cavitating axisymmetric flow over a conical cavitator with cylindrical afterbody, σ =

0.3. Coarse (65x17), medium (129x33) and fine (257x65) mesh solutions are plotted.

0 2 4 6 8 1010

−8

10−6

10−4

10−2

∆t=0.005∆t=0.0025∆t=0.001

f

Cd

Am

plit

ude

0 1 2 3 4 5s/d

-0.4

-0.2

0

0.2

0.4

0.6

0.8

CP

Fine GridMedium GridCoarse Grid

∆ t ∞=

12

to a current research goal, the full configurationmodeling of a high speed supercavitating vehicleundergoing maneuvers.

AcknowledgmentsThis work is supported by the Office of Naval

Research, contract #N00014-98-0143, with Dr. KamNg as contract monitor.

ReferencesBrennan, C.E., Cavitation and Bubble Dynamics, Oxford University Press, New York, 1995.Garabedian, P.R., Calculation of Axially Symmetric Cavities and Jets, Pac. J. of Math 6, 1958.Grogger, H.A. & Alajbegovic, A., Calculation of the Cavitating Flow in Venturi Geometries Using Two Fluid Model, ASME Paper FEDSM 98-5295 1998.Jameson, A., Schmidt, W., & E. Turkel, Numerical Solutions of the Euler Equations by Finite Volume Meth-ods Using Runge-Kutta Time-Stepping Schemes, AIAA Paper 81-1259, 1981.Kunz, Robert F., et al., Multi-Phase CFD Analysis of Natural and Ventilated Cavitation about Submerged Bodies, ASME Paper FEDSM99-7364, 1999(I).Kunz, Robert F., et al., A Preconditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Predication, AIAA Paper 99-3329, 1999 (II) to be published in Computers and Fluids.May, A., Water Entry and the Cavity-Running Behav-iour of Missles, Naval Sea Systems Command Hydroballistics Advisory Committee Technical Report 75-2, 1975.Merkle, C.L., Feng, J., & Buelow, P.E.O., Computa-tional Modeling of the Dynamics of Sheet Cavitation, 3rd International Symposium on Cavitation, Grenoble, France, 1998.Orszag, S.A. et al., Renormalization Group Modeling and Turbulence Simulations, Near Wall Turbulent Flows, Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1993.Reichardt, H., The Laws of Cavitation Bubbles at Axi-ally Symmetrical Bodies in a Flow, Ministry of Aircraft Production Volkenrode, MAP-VG, Reports and Transla-tions 766, Office of Naval Research, 1946.Rouse, H. & McNown, J. S., Cavitation and Pressure Distribution, Head Forms at Zero Angle of Yaw, Studies in Engineering Bulletin 32, State University of Iowa, 1948.Schlichting, H., Boundary-Layer Theory, McGraw-Hill, New York, 1979.Stinebring, D.R., Billet, M.L., & Holl, J.W., An Inves-tigation of Cavity Cycling for Ventilated and Natural Cavities, TM 83-13, The Pennsylvania State University Applied Research Laboratory, 1983.

Stinebring, D.R., Scaling of Cavitation Damage, M.S. Thesis, The Pennsylvania State University, University Park, Pennsylvania, August 1976.Taylor, L. K., Arabshahi, A., & Whitfield, D. L., Unsteady Three-Dimensional Incompressible Navier-Stokes Computations for a Prolate Spheroid Undergoing Time-Dependent Maneuvers, AIAA Paper 95-0313, 1995.

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