NumericalModelingofUnsteadyCavitatingFlowsarounda...

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2012, Article ID 215678, 17 pagesdoi:10.1155/2012/215678

Research Article

Numerical Modeling of Unsteady Cavitating Flows around aStationary Hydrofoil

Antoine Ducoin,1 Biao Huang,2 and Yin Lu Young1

1 Department of Naval Architecture and Marine Engineering, University of Michigan, 2600 Draper Drive, Ann Arbor, MI 48109, USA2 Department of Vehicle Engineering, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Yin Lu Young, [email protected]

Received 14 February 2012; Accepted 20 June 2012

Academic Editor: Moustafa Abdel-Maksoud

Copyright © 2012 Antoine Ducoin et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The objective of this paper is to evaluate the predictive capability of three popular transport equation-based cavitation modelsfor the simulations of partial sheet cavitation and unsteady sheet/cloud cavitating flows around a stationary NACA66 hydrofoil.The 2D calculations are performed by solving the Reynolds-averaged Navier-Stokes equation using the CFD solver CFX with thek-ω SST turbulence model. The local compressibility effect is considered using a local density correction for the turbulent eddyviscosity. The calculations are validated with experiments conducted in a cavitation tunnel at the French Naval Academy. Thehydrofoil has a fixed angle of attack of α = 6◦ with a Reynolds number of Re = 750,000 at different cavitation numbers σ . Withoutthe density modification, over-prediction of the turbulent viscosity near the cavity closure reduces the cavity length and modifiesthe cavity shedding characteristics. The results show that it is important to capture both the mean and fluctuating values of thehydrodynamic coefficients because (1) the high amplitude of the fluctuations is critical to capturing the extremes of the loads toensure structural safety and (2) the need to capture the frequency of the fluctuations, to avoid unwanted noise, vibrations, andaccelerated fatigue issues.

1. Introduction

In liquid flows, cavitation occurs if the local pressure dropsbelow the saturated vapor pressure, which leads to theformation of vaporous bubble(s) to relieve the negativepressure [1]. Cavitation is commonly observed when ahydraulic machine operates in high speeds or under off-design conditions, and is still a limiting factor in its designand operation. The occurrence of unsteady cavitation inmarine propulsion devices such as hydrofoils, propellers, andwaterjets can lead to problems such as pressure pulsations,sudden changes in loads, vibrations, noise, and erosion.Moreover, unsteady cavitation is usually the primary physicalphenomenon behind performance alterations in a hydraulicmachinery [2–4]. For this reason, it is crucial to be ableto accurately predict the development and evolution ofcavitation, and the resultant impact on the performance.

Phenomenologically, cavitation often involves complexinteractions between turbulent flow structures and phase-change dynamics with large variations in fluid density andpressure fluctuations [5]. These physical mechanisms arestill not well understood because of the complex, multiscale,multiphase phenomenon. Moreover, there are significantcomputational challenges associated with the accuracy, sta-bility, efficiency, and robustness of numerical algorithms forthe simulation of unsteady, turbulent cavitating flows.

In the numerical modeling of cavitating flows, the selec-tion of cavitation models plays a major role on the predictionof the onset, growth, breakup, and collapse of cavitationbubbles. In recent years, significant efforts have been madein the development of cavitation models; examples of recentreview articles can be found in [6–8]. Most cavitation modelsassume the flow to be homogenous and isothermal andapply either a barotropic equation of state or a transport

2 International Journal of Rotating Machinery

equation to solve for the variation of the mixture density inthe multiphase flow.

In barotropic cavitation models, the local mixture density(ρm) is assumed to depend only on the local pressure: ρm =f (p). Examples of barotropic cavitation models includethose proposed by [9, 10]. A recent experimental finding by[11] has shown that vorticity production is an importantaspect of cavitating flows, especially in the cavity closureregion. Specifically, this vorticity production is a conse-quence of the baroclinic torque:

∇ 1ρm×∇p. (1)

Clearly, if a barotropic equation of state, ρm = f (p), isused, the gradients of density and pressure are always paral-lel, which leads to zero baroclinic torque. Hence, barotropiccavitation models will not be able to properly predict thedynamics of cavitating flows [12], particularly for caseswith unsteady cavitation because of the importance ofcavitation in vorticity production.

The other popular approach to simulate cavitating flowsis transport equation models (TEMs), which solves anadditional transport equation for either the mass or volumefraction of vapor, with appropriate source/sink term(s) toregulate the mass transfer between the liquid and vaporphases. Different source/sink terms have been proposed,including the popular models presented in [6, 13–15].Kubota et al. [13] coupled the Rayleigh-Plesset equation tothe URANS equations to solve for the local void fractionbased on an assumed bubble radius, and hence it is appro-priate for modeling of unsteady cavitation. Merkle et al. [16]and Kunz et al. [14] employed the artificial compressibilitymethod with a special preconditioning formulation to solvethe multiphase URANS equations, comprised of the mixturevolume, mixture momentum, and constituent volume frac-tion equations. Singhal et al. [15] developed a “full-cavitationmodel” based on the phase-change rate derived from areduced form of the Rayleigh-Plesset equation for bubbledynamics and local flow conditions. The apparent advantageof the Singhal model comes from the convective characterof the equation. Senocak and Shyy [17] compared differentTEM models to develop an interfacial dynamics-basedcavitation model (IDM), which allows direct interpretationof the empirical parameters in existing transport equation-based models. In the TEM approach, the mixture density isa function of the transport process. Consequently, gradientsof the density and the pressure are not necessarily parallel,suggesting that TEM can accommodate baroclinic vorticitygeneration.

The objective of this study is to evaluate the predictivecapability of three popular transport-based cavitation mod-els for the simulation of quasisteady and unsteady cavitatingflow around a stationary hydrofoil. The aim is to improvethe understanding of the influence of multiphase flow onthe turbulent closure model, and the interplay betweenvorticity fields, cavity dynamics, and hydrodynamic perfor-mance. The numerical models are first presented. Threepopular transport-based cavitation models (named hereinas the Kubota model, the Merkle model, and the Singhal

model for simplicity) and the k-ω SST turbulence modelare summarized, followed by presentation of the densitycorrection model for the turbulent viscosity modification.The foil geometry, fluid mesh, and boundary conditions arepresented, followed by a brief summary of the experimentalsetup of [18] used for validation of the computations. Theresults are then analyzed for an NACA 66 hydrofoil at a fixedangle of incidence (α = 6◦) and Reynolds number (Re =750,000) for a case with quasisteady partial sheet cavitation(σ = 1.49), and a case with unsteady sheet/cloud cavitation(σ = 1.00). To study the influence of cavitation model and itsinteraction with turbulence modeling, results are comparedfor three different cavitation models and for differentturbulent viscosity modifications. Finally, the variation of thepredicted cavity length, lift and drag coefficients over arange cavitation numbers are compared with experimentalmeasurements.

2. Transport-Based Cavitation Models

2.1. Conservation of Mass and Momentum. The URANSequations, in their conservative form, for a Newtonian fluidwithout body forces and heat transfers are presented belowalong with the mass transport equation in the Cartesiancoordinates [19]:

∂ρm∂t

+∂(ρmuj

)

∂xj= 0, (2)

∂(ρmui

)

∂t+∂(ρmuiuj

)

∂xj

= − ∂p

∂xi+

∂

∂xj

[(μm + μT

)( ∂ui∂xj

+∂uj

∂xi− 2

3∂uk∂xk

δi j

)],

(3)

∂ρlαl∂t

+∂(ρlαluj

)

∂xj= m+ + m−, (4)

ρm = ρlαl + ρvαv, (5)

μm = μlαl + μvαv, (6)

where ρm is the mixture density, ρl is the liquid density, ρvis the vapor density, αv is the vapor fraction, αl is the liquidfraction, u is the velocity, p is the pressure, μm is the mixturelaminar viscosity, μl and μv are, respectively, the liquid andvapor dynamic viscosities, and μT is the turbulent viscosity.The subscripts (i, j, and k) denote the directions of theCartesian coordinates. The source term m+ and the sink termm− in (4) represent the condensation and evaporation rates,respectively.

2.2. Kubota Model. The Kubota model [13] assumes a con-stant nuclei density in the fluid domain. The growth andcollapse of the bubble clusters are governed by the simplifiedRayleigh-Plesset equation for single-bubble dynamics [20].The cavitation process is governed by the mass transfer

International Journal of Rotating Machinery 3

equation given in (4), and the source and sink terms aredefined as follows:

m− = −Ckdest

3αnuc(1− αv)ρvRB

(23pv − p

ρl

)1/2

, p < pv, (7)

m+ = Ck prod3αvρvRB

(23p − pvρl

)1/2

, p > pv, (8)

where αnuc is the nucleation volume fraction, RB is the bubblediameter, pv is the saturated liquid vapor pressure, and p isthe local fluid pressure. Ck dest is the rate constant for vaporgenerated from the liquid in a region where the local pressureis less than the vapor pressure. Conversely, Ck prod is the rateconstant for reconversion of vapor back into liquid in regionswhere the local pressure exceeds the vapor pressure. Asshown in (7) and (8), both the evaporation and condensationterms are assumed to be proportional to the square root ofthe difference between the local pressure and vapor pressurebecause the Kubota model was derived by assuming that thebubble dynamics are governed by the simplified Rayleigh-Plesset equation [20]. In this work, assumed values for themodel constants are αnuc = 5 × 10−4, RB = 1 × 10−6 m,Ck dest = 50, and Ck prod = 0.01, which are the default valuesin CFX [5] and are used because of their supposed generalapplicability.

It should be noted that (7) and (8) are different fromthe original model proposed by [13], and instead follow theform presented in [21]. Since the evaporation rate is muchhigher than the condensation rate, different coefficients areneeded for condensation and evaporation terms, as shownin (7) and (8). In addition, αv is replaced by αnuc (1 − αv)in the vaporization term to account for the decrease of thenucleation site density with the increase of the vapor volumefraction.

2.3. Merkle Model. Several researchers have adopted theMerkle model proposed by [16] (e.g., see [12]), whichhas been presented in both the volume fraction form andthe mass fraction form. It was derived primarily based ondimensional arguments for large-bubble clusters instead ofindividual bubbles. Consequently, the source and sink termsfor the Merkle model shown in (9) are directly related to thepressure difference, p − pv, instead of the square root of thepressure difference as in the Kubota model:

m− = Cm destρlρlMIN(p − pv, 0

)(1− αv)(

0.50ρlU2∞)ρv t∞

, p < pv,

m+ = Cm prodρlMAX(p − pv, 0

)αv(

0.50ρlU2∞)t∞

, p > pv.

(9)

In this work, the empirical factors are set to be Cm dest = 1and Cm prod = 80, which follows the constants used

by Senocak and Shyy [17] for simulation of quasisteadycavitation around a NACA 66(MOD) hydrofoil. The meanflow time scale is defined as t∞ = c/U∞ [12, 17].

2.4. Singhal Model. The Singhal mass transfer model pro-posed by [15] is originally known as the full-cavitationmodel. This model is also based on the reduced formof the Rayleigh-Plesset equation for bubble dynamics, butempirical constants have been added to take into account theinteraction between water and vapor phases. In the deriva-tion of the Singhal model, the bubble size is determinedby the surface tension force, and the characteristic velocity,which reflects the effect of the local relative velocity betweenthe liquid and vapor phases. Similar to the Kubota model,both evaporation and condensation terms are assumed tobe proportional to the square root of the difference betweenthe local pressure and vapor pressure. The Singhal modelassumes that the mass transfer rate is related to the localturbulence kinetic energy, k. Following [8], the equationsbelow represent a modified form of the original Singhalmodel, which consider the effects of noncondensable gas.The evaporation and condensation terms are defined asfollows:

m− = Cs dest

√k

γρlρv

[23pv − p

ρl

]1/2(1− αvρv

ρm

), p < pv,

m+ = Cs prod

√k

γρlρl

[23p − pvρl

]1/2αvρvρm

, p > pv.

(10)

In this work, the empirical factors, Cs dest and Cs prod

(resp., the destruction and the production coefficients for theSinghal model), have the following values: Cs dest = 0.02 m/sand Cs prod = 0.01 m/s, and the surface tension coefficient isassumed to be γ = 0.0717 N/m. fv is the vapor mass fraction,defined as fv = αv/ρm. It should be noted that in the presentform of the mass transfer equations, the empirical factorsCs dest and Cs prod have to be expressed with m/s units to beconsistent with the dimensions of m+and m− in (4), which isin units of kg/(m3s1).

2.5. Turbulence Model and Local Compressibility Correction.The numerical results shown in this paper are performedusing the commercial CFD code, CFX, to solve the URANSequations. The k-ω SST (shear stress transport) turbulencemodel is used, which combines the advantages of the originalk-ε and k-ω models by using the k-ω model near the wall,and the k-ε model away from the wall [6]. The k-ω SSThas been found to give good predictions of boundary layerdetachment characteristics, and it has been validated forthe prediction of subcavitating flow around the NACA66hydrofoil shown in this paper [22].

It should be noted that the original URANS models weredeveloped for fully incompressible single-phase flows andwere not intended for flow problems involving compressiblemultiphase mixtures. As shown in [23], standard two-equation models (such as k-ε and k-ω models) overpredictthe turbulent eddy viscosity in the rear part of the cavity,


0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

n = 1n = 3n = 10

ρ m∗f D

CM

(kg/

m3)

Water vapor fraction (αv)

Figure 1: Local compressibility modification of the mixture densitywith different n values according to (11).

which results in overprediction of turbulent stresses, causingthe reentrant jet in cavitating flows to lose momentum. Thus,the reentrant jet is not able to cut across the cavity sheet,which significantly modifies the cavity shedding behavior.For partial sheet cavitation in steady flow conditions,inability of the reentrant jet to reach the leading edge willlead to stabilization of the sheet cavity, which prevents thebreaking and shedding of the sheet cavity. For unsteady flowconditions, inability of the reentrant jet to fully reach theleading edge will lead to frequent partial shedding of thecavity near the trailing edge, which tends to increase thecavity shedding frequency.

To improve numerical simulations by taking into accountthe influence of the local compressibility effect of two-phasemixtures on turbulent closure models, Coutier-Delgosha[23] proposed to reduce the mixture turbulent viscositybased on the local liquid volume fraction αl by substitutingμT in (3) with μT mod:

f (n) = ρv + (1− αv)n(ρl − ρv

)

ρv + (1− αv)(ρl − ρv

) , μT mod = μt f (n).

(11)

The variation of the modified effective density, ρm f (n),with the vapor volume fraction, αv, for different n values isshown in Figure 1.

In [23], they recommended n = 10 to better simulatethe reentrant jet behavior and vapor shedding process.In [24], the same model was applied for the simulationof supercavitating flow around a hydrofoil, and obtainedfavorable agreement between numerical and experimentalresults for n = 2. The same modification has also beenapplied by [25] for the NACA 66 hydrofoil considered in thispaper; they used n = 3 and obtained good agreements withexperimental measurements.

3. Experimental Setup and Description

To evaluate the predictive capabilities of the three differ-ent cavitation models with different local compressibilitycorrections, numerical computations are compared withexperimental measurements of the NACA 66 hydrofoilconducted at the cavitation tunnel at the Research Instituteof the French Naval Academy. The test section has a cross-sectional area of 0.192 m2 and length of 1 m. The inflowvelocity ranges between 3 m/s and 15 m/s, and the pressure inthe test section ranges between 30 m bar to 3 bars. The tunnelinflow turbulence intensity, defined as V∝rms/V∝ at the inletof the test section, is about 2%. The foil has a uniformcross-section with an NACA 66 thickness distribution witha maximum thickness-to-chord ratio of 12%, and an NACAa = 0.8 camber distribution with a maximum camber-to-chord ratio of 2%. The chord length is c = 0.15 m and thespan length is s = 0.191 m. The hydrofoil is made of stainlesssteel, and behaved like a 2D rigid hydrofoil even though it wasmounted using a cantilevered setup with a small gap (1 mm)between the free end of the hydrofoil and the test section wall.Pressure measurements were carried out using seventeenflush-mounted piezo-resistive transducers (Keller AG 2 MIPAA100-075-010) with a maximum pressure of 10 bars. Thetransducers locations were aligned along the chord on thesuction side of the hydrofoil at mid-span, starting from thefoil leading edge at a reduced coordinate of x/c = 0.1 to thetrailing edge at x/c = 0.9, with increments of 0.1 c. Lift anddrag were measured using a resistive gauge hydrodynamicbalance with a range up to 150 daN in lift and 15 daN indrag. Readers should refer to [18, 26] for additional detailsabout the rigid hydrofoil experimental setup and results. Theexperimental results presented in this paper are taken from[18] for cases with partial sheet cavitation and from [26] forcases with unsteady sheet/cloud cavitation.

4. Numerical Setup and Description

To demonstrate and validate the numerical model, results areshown for the rigid rectangular NACA 66 hydrofoil describedabove. All the results shown in this paper corresponds to thehydrofoil fixed at α = 6◦ and subject to a nominal free streamvelocity of V∞ = 5 m/s, which yields a moderate-to-highReynolds number of Re = V∞c/νl = 0.75× 106.

The density and dynamic viscosities of the liquid aretaken to be ρl = 999.19 kg/m3 and μl = ρlνl = 1.139 ×10−3 Pa·s, respectively, which correspond to fresh water at25◦C. The vapor density is ρv = 0.02308 kg/m3 and thevapor viscosity is μv = 9.8626 × 10−6 Pa·s. The vaporpressure of water at 25◦C is pv = 3169 Pa. The 2D fluiddomain is shown in Figure 2, which corresponds to theheight of the experimental test section at the French NavalAcademy. The computational domain has an extent of about5 c upstream and 10 c downstream of the foil to simulatenear-infinite boundary conditions at the inlet and outlet.A no-slip boundary condition is imposed on the hydrofoilsurface, and symmetry conditions are imposed on the topand bottom boundaries of the tunnel. The inlet velocity isset to be V∞ = 5 m/s and the outlet pressure is set to vary


Leadingedge

Symmetry

Symmetry

Inlet OutletP∝V∝

Figure 2: 2D fluid mesh and boundary conditions.

according to the cavitation number, defined as σ = (p∞ −pv)/(0.5ρlV 2∞), where p∞ is the tunnel pressure. A constantturbulent intensity of 2% is set at the inlet boundary and isequal to the experimentally measured turbulent intensity.

All cavitating runs have been initialized with a steady-state calculation using fully wetted models to avoid anyvapor fraction at the initial time step. The tunnel pressureis then decreased progressively until the particular cavitationnumber is reached. Only 2D results are shown in this work.The 2D fluid mesh (shown in Figure 2) is composed of120,000 elements with 50 structured elements across the foilboundary layer, which is selected to ensure y+ = yuτ/v1 = 1,where y is the thickness of the first cell from the foil surface,and uτ is the wall frictional velocity. The regions outsidethe boundary layer have been discretized with unstructuredtriangular elements. Mesh refinements are performed at thefoil leading edge, trailing edge, and in the wake region. Thetime integration scheme is a second-order backward Euleralgorithm, and the spatial derivatives are computed using asecond-order upwind scheme. Mesh convergence and timediscretization have been studied in detail for the case of therigid NACA 66 hydrofoil at a fixed angle of attack of α = 6◦

in steady flow; see [22] for more details.A temporal convergence study is carried out in the case

of partial sheet cavitation, at an angle of attack of α = 6◦

with σ = 1.49. Figure 3 shows temporal convergence ofthe predicted surface pressure coefficients, CP = (P −P∞)/(0.5ρV 2∞), along with the measured values. The errorbars represent the maximum and minimum pressure fluc-tuations. The numerical results were generated using theMerkle model with n = 3. The largest time step size used,Δt = 10−3 s, yielded a much stronger pressure gradient inthe cavity closure region compared to the measured values.The two finer time step sizes yielded very similar results andcompared better with the measured values. Hence, Δt =10−4 s is chosen for the computations from here on, whichgives an average CFL number of CFL = V∝ Δt/Δx = 1.

5. Results

5.1. Partial Sheet Cavitation. The influence of cavitationmodels and the effect of the turbulent viscosity are firstanalyzed for the case with quasisteady partial sheet cavitation(σ = 1.49). Figure 4 shows the predicted vapor volume

1.5

0 0.2 0.4 0.6 0.8 1

x /c

−2

−1.5

−1

−0.5

0

0.5

1

Cp

ExpΔt = 5∗ 10−5 s

Δt = 1∗ 10−4 s

Δt = 1∗ 10−3 s

Figure 3: Comparisons of the measured versus predicted pressurecoefficients obtained using different time step sizes, Δt. The linesrepresent the mean values and the error bars represent the level offluctuations. σ = 1.49, Re = 750,000, V∝ = 5 m/s, and α = 6◦.Merkle cavitation model, n = 3.

fraction and turbulent kinetic energy obtained using theKubota, Merkle, and Singhal cavitation models with differentn values for the turbulent viscosity modification (see (11)).The vapor fraction contours for the three different cavitationmodels with n = 1 and n = 3 are, respectively, shown inthe first and third rows in Figure 4; a vapor volume fractionof 0 (black) corresponds to pure water, whereas 1 (white)corresponds to pure vapor. The corresponding turbulentkinetic energy contours are shown in the second and fourthrows in Figure 4, where the lowest turbulent kinetic energy isshown in white, and the highest turbulent kinetic energy isshown in black.

For n = 1 (no turbulent viscosity modification), the threecavitation models yielded very different levels of turbulentkinetic energy in the cavity and at the cavity closure region,which in turn modified the vapor volume fraction and cavityshape, as shown in the first two rows of Figure 4. The Merklemodel has the lowest level of turbulent kinetic energy in thevapor region, which leads to a longer sheet cavity comparedto the other two models. Compared to the Merkle model,the Kubota model has a higher level of turbulent kineticenergy in the pure vapor region, and a shorter cavity length.It is observed that the production and destruction terms inthe Merkle model (9) are related to the pressure difference,instead or the square root of the pressure difference for theKubota and Singhal models (resp., (7), (8), and (10)). Foran equivalent pressure level in subcavitating flow, the directdependence of the production and destruction terms on thepressure difference leads to a longer cavity for the Merklemodel. The Singhal model leads to the highest turbulentkinetic energy in the vapor region. This is because the vaporproduction and destruction terms for the Singhal model,as shown in (10), are directly related to the square root of


(a) Kubota model

Volume fraction, n = 1

Turbulent kinetic energy, n = 1

Volume fraction, n = 3

Turbulent kinetic energy, n = 3

(b) Merkle model

0 0.06

0.12

0.18

0.24

0.31

0.37

0.43

0.49

0.55

0.61

0.67

0.73

0.8

0.86

0.92

0.98

0 0.05

0.1

0.15

0.2

0.24

0.29

0.34

0.39

0.44

0.49

0.54

0.59

0.64

0.69

0.73

0.78

(c) Singhal model

Figure 4: Comparisons of the vapor fraction and turbulence kinetic energy contours for the three different cavitation models with n = 1and n = 3 for σ = 1.49, Re = 750,000, V∞ = 5 m/s, and α = 6◦.

the local turbulent kinetic energy, which creates a nonlinearfeedback mechanism that increases the pressure gradient inthe cavity closure region, and hence tends to destabilize thecavity.

For n = 3, the results for all three models in the bottomtwo rows of Figure 4 show a general decrease in theturbulent kinetic energy in the vapor region, and slightlymore unsteady behavior at the cavity trailing edge, comparedto n = 1. The results show that the density correction tothe turbulent viscosity has different degrees of impact on thethree cavitation models. The Merkle model predicts similarcavity lengths with and without the density modification,although the decrease of the turbulent kinetic energy allowsthe reentrant jet at the cavity trailing edge to move upstream,and hence increases the flow unsteadiness and leads to partialshedding of the cavity. The Kubota model shows a noticeableincrease in cavity length and more unsteadiness at the cavitytrailing edge with the modification. The Singhal modelappears to be the most sensitive to the turbulent viscositymodification. Again, this is because of the dependence onthe square root of the local turbulent kinetic energy in theproduction and destruction terms. When n = 3, the tur-bulent kinetic energy is completely suppressed in the cavityregion, which allowed the cavity to grow longer but also leadto unsteady partial shedding of the vapor structure near thecavity trailing edge where the pressure gradient is high.

The increase of unsteadiness with increasing n valuesfor (11) is due to the development of a reentrant jet as aresult of the reduced turbulent eddy viscosity level at thecavity closure. Figure 5 shows the predicted vapor fractioncontours and the velocity vectors at the cavity closure regionfor σ = 1.49 obtained using the Merkle model with n =1 (no modification), n = 3, and n = 10. With n = 1(Figure 5(a)), the turbulent eddy viscosity in the cavityclosure region is too high, which prevented the upstreammotion of the reentrant jet, and hence the cavity is fullystable. With n = 3 (Figure 5(b)), the reduction of theturbulent viscosity in the cavity closure region allows thereentrant jet to develop and move upstream, which lead topartial shedding of the cavity. With n = 10 (Figures 5(c)and 5(d)), the turbulent eddy viscosity is reduced to nearzero in the pure vapor region, which allowed the reentrantjet to reach the cavity leading edge. Subsequently, the cavitydetached from the leading edge (Figure 5(d)) and then sheddownstream as a cloud cavity. This complete breakup of thecavity from the leading edge and unsteady transformation tocloud cavitation obtained using n = 10 are different fromexperimental observations by [18], whom reported partialsheet cavitation with behavior similar to that obtained usingn = 3 at σ = 1.49.

As suggested in [7, 27], the baroclinic torque created bythe mixture density and pressure gradients in the cavitating


(a) n = 1, steady cavitation

Vap

or fr

acti

on

0.120.240.370.490.610.730.860.98

Reentrant jet

Partial cavity break

0

(b) n = 3, quasisteady cavitation

Reentrant jet

(c) n = 10, unsteady cavitation, t1

Vap

or fr

acti

on

0.120.240.370.490.610.730.860.98

0

Leading edge cavity detachment

(d) n = 10, unsteady cavitation, t1 + 0.03 s

Figure 5: Comparison of the vapor fraction contours and velocity vectors for different n values with the Merkle model at σ = 1.49, Re =750,000, V∞ = 5 m/s, and α = 6◦.

region are responsible for the alteration of the vorticity field,which is evident via the vorticity transport equation:

∂ω

∂t+∇× (ω × u) = ∇ρm ×∇p

ρm2+ (υm + υt)∇2ω, (12)

where ω and u are, respectively, the fluid vorticity andvelocity vectors. υm is the laminar kinematic viscosity and υtis the turbulent kinematic viscosity.

The first term on the left-hand side (LHS) of (12)represents the rate of change of vorticity at a fixed point. Thesecond term on the LHS represents the convection as wellas the stretching and tilting of the vorticity due to velocitygradients. The first term on the right-hand side (RHS) of(12) is called the baroclinic torque, which is nonzero onlyin a variable-density flow field. The baroclinic torque corre-sponds to the generation of vorticity from unequal accelera-tion caused by unaligned pressure and density gradients, forexample, the generation of a shear layer caused by the fasteracceleration of the lighter fluid (vapor) relative to the heavierfluid (water). The second term on the right-hand side of (12)represents the laminar and viscous diffusion of the vorticity.

To examine the influence of cavitation on the produc-tion/alteration of vorticity around the case with quasisteadypartial sheet cavitation, the strengths of (∇ × (ω × u)) and(∇ρm × ∇p/ρm2) are compared in Figure 6 for the differentcavitation models with different n values. As expected, thebaroclinic term modifies the vorticity field in regions withhigh density and pressure gradients, that is, along the liquid-vapor interface and near the cavity closure. The strength ofthe baroclinic torque increases with n, particularly near thecavity closure region because the reduction of the turbulenteddy viscosity tends to promote the unsteady shedding ofthe cavities. Consequently, the vorticity field is modified, asevident via the contour of the vorticity transport/stretchingterm. The baroclinic term is found to be the lowest for theMerkle model, which shows a smoother cavity with conse-quently lower density and pressure gradients. The baroclinicterm is higher for the Singhal and Kubota cavitation models,

where the magnitude of the baroclinic torque is comparableto the vorticity transport/stretching term near the cavityclosure, which will tend to increase the flow unsteadiness, asobserved earlier in Figure 4.

Figures 7, 8, and 9 show the comparison of predicted wallpressure coefficients, obtained at σ = 1.49, for the Kubota,Merkle, and Singhal cavitation models, respectively, alongwith the measured values from [18]. On each of the threefigures, the mean and fluctuations of the predicted pressurecoefficients obtained using n = 1, 3, and 10 are shown.The nearly constant pressure region with CP = −1.49 corre-sponds to the leading edge, partial sheet cavity on the suctionside, which is followed by a sudden pressure jump at thecavity closure due to transition back to subcavitating flow.In general, the pressure fluctuations increased, particularly atthe cavity closure region, with increasing n values because ofthe decrease of turbulent eddy viscosity and the additionalvorticity generation/modification by the baroclinic torque.The Kubota cavitation model (Figure 7) has very steady pres-sure behavior without modification (n = 1), which showeda cavitating region up to X/c = 0.2. The results are approxi-mately the same for n = 3 and 10, which showed a constantpressure region up to X/c = 0.3, and larger pressure fluctua-tions at the cavity closure. The Merkle model (Figure 8), onthe other hand, showed negligible difference in the predictedpressure coefficients between the different n values except forthe increase in pressure fluctuations at the cavity closure forn = 10. Moreover, the predicted pressure distributions bythe Merkle model are in good general agreement with exper-imental measurements. The Singhal model (Figure 9) showsthe largest difference between the different n values becauseof the explicit dependence on the turbulent kinetic energy,which changes with the turbulent eddy viscosity as wellas vorticity generation/distortion by the baroclinic torque.Consequently, as n increases, the cavity length (constantpressure region) predicted by the Singhal model decreases.A general unsteadiness is observed along the entire suctionside pressure distribution. Moreover, the predicted pressure


∇ρm ×∇p/ρm2 ∇× (ω × u)

(a) Kubota model, n = 1

∇ρm ×∇p/ρm2 ∇× (ω × u)

(b) Kubota model, n = 3

(c) Merkle model, n = 1 (d) Merkle model, n = 3

−1E

+06

−6E

+05

−2E

+05

2E+

05

6E+

05

1E+

06

(e) Singhal model, n = 1

−1E

+06

−6E

+05

−2E

+05

2E+

05

6E+

05

1E+

06

(f) Singhal model, n = 3

Figure 6: Comparisons of the baroclinic torque and vorticity transport/stretching term in (12) for the different cavitation models with n = 1and n = 3 for σ = 1.49, Re = 750,000, V∞ = 5 m/s, and α = 6◦.

0 0.2 0.4 0.6 0.8 1

x /c

1.5

−2

−1.5

−1

−0.5

0

0.5

1

Cp

n = 1n = 3

n = 10

Exp

Figure 7: Computed versus measured pressure coefficients fordifferent n values, Kubota cavitation model, σ = 1.49, Re = 750,000,V∞ = 5 m/s, and α = 6◦. The lines represent the mean values andthe error bars represent the level of fluctuations.

coefficients show greater discrepancy with experimentalmeasurements compared to the Kubota and Merkle models.In general, the results obtained with the Merkle cavitationmodel with n = 3 matched better with experiments.

0 0.2 0.4 0.6 0.8 1

x /c

1.5

−2

−1.5

−1

−0.5

0

0.5

1

Cp

n = 1n = 3

n = 10Exp

Figure 8: Computed versus measured pressure coefficients fordifferent n values, Merkle cavitation model, σ = 1.49, Re = 750,000,V∞ = 5 m/s, and α = 6◦. The lines represent the mean values andthe error bars represent the level of fluctuations.

5.2. Unsteady Sheet/Cloud Cavitation. To evaluate the pre-dictability of the three different cavitation models forunsteady sheet/cloud cavitation, results are shown in thissection for the case with σ = 1.00, Re = 750,000, and α = 6◦.


0 0.2 0.4 0.6 0.8 1

x /c

1.5

−2

−1.5

−1

−0.5

0

0.5

1

Cp

n = 1

n = 3

n = 10

Exp

Figure 9: Computed versus measured pressure coefficients fordifferent n values, Singhal cavitation model, σ = 1.49, Re = 750,000,V∞ = 5 m/s, and α = 6◦. The lines represent the mean values andthe error bars represent the level of fluctuations.

It should be noted that the Kubota model did not convergefor this case with Δt = 0.0001 s. The authors believe thatit is due to the high value of Cdest in (7), which makes thecalculation unstable. Hence, results are shown in this case forthe Merkle and Singhal models only.

At σ = 1.00, Re = 750,000, and α = 6◦, unsteadysheet/cloud cavitation was reported in [26]. Figures 10 and11 show the evolutions of the vapor fraction, turbulentkinetic energy, and vorticity for n = 1 and n = 3 with theMerkle and Singhal cavitation models, respectively. Globally,the results show significant reduction in the turbulent kineticenergy in the cavity region, particularly at the cavity closure,and longer cavity with increasing n values because of thereduction of the turbulent eddy viscosity. For the Merklemodel (Figure 10), it is observed that the level of vorticityis closely related to the vapor fraction contour and thedistribution of turbulent kinetic energy. For n = 3, whenthe cavity develops, the decrease of turbulent kinetic energylead to an increase of vorticity near the wall (Figure 10(b)),which allowed the reentrant jet to form and move upstream(t4) while allowing the cavity to expand to the foil trailingedge; subsequently, the cavity breaks and lifts off at the atthe leading edge (t6) to form a large cavitation cloud, whichthen convects downstream. For the Merkle model, the cavityshedding frequency at n = 3 is approximately 3.5 Hz, whichagrees with the measured value, as shown in Table 1. Onthe contrary, for n = 1, the cavity partially breaks aroundthe midchord (see the vapor fraction in Figure 10(a), at t4),which limited the maximum cavity length to Lc/c = 0.8,and increased the shedding frequency to about 4.0 Hz, seeTable 1. Moreover, because of the presence of cloud cavita-tion near the foil trailing edge (Figure 10(a) at t6 and t8),the overprediction of turbulent viscosity without the density

Table 1: Comparison of the measures and predicted cavityshedding frequencies for the Merkle and Singhal cavitation modelswith n = 1 and n = 3 for the case of unsteady sheet/cloud cavitation.σ = 1, Re = 750, 000, V∞ = 5 m/s, and α = 6o.

ModelsMerkle Singhal

Experimentsn = 1 n = 3 n = 1 n = 3

Shedding frequency(Hz)

4.0 3.5 22.5 20.5 3.5

correction (n = 1) suppresses the formation of the counter-clockwise trailing edge vortex, which changes the vorticitypatterns around the foil, as well as the convection of the cloudin the wake. When the modification is applied (n = 3), thelevel of turbulent kinetic energy in the cloud is lower, whichallows the counter-clockwise trailing edge vortex to form andinteract with the cloud cavity (Figure 10(b) at t6 and t8).

As shown in Figure 11, the Singhal cavitation modelis more sensitive to the turbulent viscosity modification.Without the modification (n = 1), the level of turbulentkinetic energy is very high near the cavity closure (seeFigure 11(a)), which limited the length of the sheet cavity toapproximately X/c = 0.6 (t4), and the cavity partially breaksat X/c = 0.4 (t6), followed by successive shedding of smallervapor structures (t8). The cavity appears to be more unstableand transient than that predicted by the Merkle cavitationmodel, as shown by the scattered vorticity distribution nearthe wall (see Figure 11(a)). When the modification is applied(n = 3), the Singhal model also predicted an increase inthe cavity length up to Lc/c = 1, although general cavitybehavior is much more unstable than that predicted by theMerkle model. The cavity breaks around X/c = 0.2. However,it does not shed as one big cavity; instead, the aft end of thecavity is broken into multiple little cloud cavities, as shown inFigure 11(b). Therefore, the shedding frequencies predictedby the Singhal model (about f = 22.5 Hz for n = 1 and f =20.5 Hz for n = 3) are much higher than those by the Merklemodel and did not agree with experimental measurements,as summarized in Table 1. Again, the turbulent viscositymodification allowed the trailing edge vortex to developearlier, which interacted with the cloud cavity as it sheds intothe wake.

Figure 12 shows the baroclinic torque and the vorticitytransport/stretching terms, at t = t6 for both the Merkle andSinghal models. For both models, the baroclinic torque con-centrates along the regions where the cavity breaks, althoughthe magnitudes seem to be small compared to the vorticitytransport/stretching term. Nevertheless, the influence of thecavity on the vorticity distribution is clearly evident by thesignificant modification of the vorticity transport/stretchingterm (Figure 12) and vorticity contours (Figures 10 and11). Hence, the interdependency between cavitation gener-ation/production and vorticity generation/modification dueto the baroclinic torque is not negligible, particularly in termsof its effect on the cavity stability and shedding frequency.

Based on available experimental data presented in [26],an analysis of suction side loading can be calculated by


Vapor fraction Turbulent kinetic energy Vorticity

t4 = 0.46 s

t6 = 0.5 s

t8 = 0.54 s

(a) Merkle model, n = 1

t4 = 0.46 s

t6 = 0.5 s

t8 = 0.54 s

0 0.06

0.12

0.18

0.24

0.31

0.37

0.43

0.49

0.55

0.61

0.67

0.73

0.8

0.86

0.92

0.98

Vapor at 25 C. volume fraction

0 0.05

0.1

0.15

0.2

0.24

0.29

0.34

0.39

0.44

0.49

0.54

0.59

0.64

0.69

0.73

0.78

Water at 25 C. turbulence kinetic energy

(m2 s−2)

−500

−439

−316

−255

−194

−133

−71

−10

51 112

173

235

296

357

418

480

Vapor at 25 C. vorticity Z(s−1)

−378

(b) Merkle model, n = 3

Figure 10: Comparison of the vapor fraction (left), turbulent kinetic energy (middle), and vorticity (right) contours for the Merklecavitation models with (a) n = 1 and (b) n = 3, in the case of unsteady sheet/cloud cavitation with σ = 1.00, Re = 750,000, V∞ = 5 m/s, andα = 6◦.

summing the pressure coefficients along the suction side. Theapproximate suction side lift coefficient can be written as:

Cl+(t) =

10∑

i=1

CP

(xic,t

)Δ(xic

), (13)

where CP(xi/c,t) is the pressure coefficient at location xi/c,and Δ(xi/c) = 0.1 is the nondimensional distance betweentwo consecutive pressure transducers. A total of 10 pressuretransducers were placed along the suction side at the mid-span of the foil [22]. For comparison purposes, the same

procedure is applied to calculate the numerically derivedsuction side lift coefficients. It should be noted that since theflow is mostly attached along the pressure side, the changes tothe total lift coefficient should be dominated by the suctionside dynamics, which is represented by the suction side liftcoefficient.

As shown in Figure 13, a fair comparison is observedbetween the experimental measurements and numericalcomputations obtained using the Merkle model with n = 3.The peaks generally correspond to the time instance whenthe maximum cavity length is observed (t4 in Figure 10(b)),


t4 = 0.46 s

t6 = 0.47 s

t8 = 0.48 s

(a) Singhal model, n = 1

0.73

0 0.06

0.12

0.18

0.24

0.31

0.37

0.43

0.49

0.55

0.61

0.67

0.8

0.86

0.92

0.98


t4 = 0.46 s

t6 = 0.47 s

t8 = 0.48 s

0 0.05

0.1

0.15

0.2

0.24

0.29

0.34

0.39

0.44

0.49

0.54

0.59

0.64

0.69

0.73

0.78

Water at 25 C. turbulence kinetic energy

(m2 s−2)

Vapor at 25 C. vorticity Z(s−1)

51 112

173

235

296

357

418

480

−500

−439

−316

−255

−194

−133

−71

−10

−378

(b) Singhal model, n = 3

Figure 11: Comparison of the vapor fraction (left), turbulent kinetic energy (middle), and vorticity (right) contours for the Singhalcavitation models with (a) n = 1 and (b) n = 3, in the case of unsteady sheet/cloud cavitation with σ = 1.00, Re = 750,000, V∞ = 5 m/s, andα = 6◦.

and the unloading phase corresponds to the shedding ofthe cavity. The results show that the rate of the unloadingphase predicted by the Merkle model is faster than themeasured value. This difference could be due to inaccurateprediction of the convection/destruction of vapor in thecloud, which is by definition very transient and sensitiveto the production and destruction rate of vapor, and theinteraction between the cavity and the vorticity field throughthe baroclinic torque. Nevertheless, the shedding frequency

predicted by the Merkle model with n = 3 matched well withthe measured frequency of f = 3.5 Hz. As shown in Table 1,the cavity shedding frequency is higher for n = 1 becausethe cavity is shorter and undergoes partial shedding at itstrailing edge. Hence, it is important to take into account theinfluence of mixture density variation in the calculation ofthe turbulent viscosity in and near the cavity.

As shown in Figure 14, the Singhal model with n = 3shows a much higher shedding frequency, f = 20.5 Hz,


∇ρm ×∇p/ρm2 ∇× (ω × u)

(a) Merkle model, n = 1

(b) Merkle model, n = 3

(c) Singhal model, n = 1

−1E

+06

−6E

+05

−2E

+05

2E+

05

6E+

05

1E+

06−1

E+

06

−6E

+05

−2E

+05

2E+

05

6E+

05

1E+

06

(d) Singhal model, n = 3

Figure 12: Comparison of the baroclinic torque (left) and vorticity transport/stretching term (right) in (12) at t6 for the Merkle and Singhalmodels with n = 1 and 3 for σ = 1, Re = 750,000, V∞ = 5 m/s, and α = 6◦.

0.40.60.8

11.21.41.6

0 0.2 0.4 0.6 0.8 1

Time (s)

Experiments

CL

+

t4

t6

t8

Computations, n = 3

Figure 13: Comparisons of the measured and the predicted suctionside lift coefficients obtained using the Merkle cavitation modelwith n = 3. σ = 1, Re = 750,000, V∞ = 5 m/s, and α = 6◦.

than the measured frequency of 3.5 Hz. The high frequencypredicted by the Singhal model is because of the contin-uous presence of vapor along the hydrofoil surface (seeFigure 11(b)), which significantly perturbs the wall pressuredistributions and leads to the generation of vorticity throughthe baroclinic torque (see Figure 12(d)). As a consequence,the predicted suction side lift coefficient shows a much highershedding frequency and much lower amplitude compared to

0 0.2 0.4 0.6 0.8 1

Time (s)

ExperimentsComputations, n = 3

0.40.60.8

11.21.41.6

CL

+

Figure 14: Comparisons of the measured and the predicted suctionside lift coefficients obtained using the Singhal cavitation modelwith n = 3. σ = 1, Re = 750,000, V∞ = 5 m/s, and α = 6◦.

the measured values. The results shown in Table 1 confirmsthat the shedding frequency is higher for the Singhal modelif n = 1, without the turbulent viscosity modification.

It should be noted that the importance of capturing andcomparing both the mean and fluctuating values is evidentin Figures 13 and 14. As shown in the figures, althoughthe mean suction side lift coefficient predicted by both the


Merkle and Singhal models are approximately the same, andagreed well with experimental measurements, the amplitudeand frequency of the fluctuations predicted by the Singhalmodel are very different from the Merkle model, whichcompared much better with experimental measurements.

Figure 15 shows the comparisons between the predictedvapor fraction contours obtained using the Merkle cavitationmodel with n = 3 and photographs from the experimentspresented in [26] for σ = 1, Re = 750,000, and α = 6◦. Thewhite vapor fraction represents the pure vapor and the blackrepresents the pure water. The predicted flow streamlines arealso shown to facilitate visualization of the flow dynamics.Similar to the comparison of the suction side loading, thenumerical predictions obtained using the Merkle model withn = 3 are in good agreement with the experiments. Thecavity expanded up to the foil trailing edge at X/c = 1 (t1 tot4), followed by breakup near the foil leading edge due to thereentrant jet at t5. An interaction between the bubbly cavitymixture and the trailing edge vortex is observed at t6 and t7,followed by complete convection of the cloud cavity into thewake at t8.

5.3. Comparisons of the Measured and Predicted Hydrody-namic Load Coefficients and Cavity Lengths. Figures 16, 17,and 18 show the measured and predicted normalized cavitylengths (Lc/c), lift coefficient (CL), and drag coefficient (CD)over a range of cavitation numbers (σ). The numerical resultsfor all three models are obtained using n = 3, whichcompared the best with the experimental measurements, asshown in the previous section.

Figure 16 shows the predicted maximum and minimumcavity lengths, along with the measured maximum cavitylengths. Cavitation inception is observed at σ ≈ 2.4 forboth computations and experiments. For 1.5 < σ < 2.0,the minimum and maximum values are equal, which suggeststable cavitation. It should be noted that all three cavitationmodels give very similar results for stable cavitation, andall agree well with the experimental measurements (starsymbols). For σ < 1.5, the cavity becomes unstable, asindicated by the diverging maximum and minimum values ofthe cavity length, and differences could be observed betweenthe three cavitation models. The results suggest that thecavity breaks directly at the leading edge for the Kubotamodel, as indicated by the near-zero value of the minimumlength represented by the open circles in Figure 16 forσ < 1.5. However, experimental observations suggest theexistence of partial sheet cavitation between 1.3 < σ < 1.6,which is characterized by a partial break of the cavity witha stable vapor region at the cavity leading edge [18]. Asshown in Figure 16, the Singhal and Merkle models agreewell with the experimental observations, but not the Kubotamodel. As the cavitation number further decreases, the breakof the cavity moves closer to the foil leading edge. At σ =1.00, unsteady sheet/cloud cavitation occurs, as shown in thenumerical and experimental results reported in the previoussection. It should be noted that the measured maximumcavity lengths agree well with the numerical values for allthree cavitation models considered. However, as evident viathe results shown in the previous section and in Table 1,

even if the minimal and maximal values of the cavity lengthagree well between the Singhal and the Merkle cavitationmodels, the shedding frequencies are very different. It shouldbe noted that the Kubota model did not converge due toexcessive fluctuations for σ < 1.2, and hence results are onlyshown for σ > 1.2 in Figures 16–18.

Figure 17 compares the measured and predicted liftcoefficients (CL = F/(0.5ρV 2∝c), where F is the lift force)for different cavitation numbers. The lift coefficient increasesslightly with the development of small, leading edge, andpartial sheet cavitation (1.3 < σ < 1.6). The increase in liftis attributed to the suction effect created by the small leadingedge cavitation bubble caused by the high pressure gradient.As the cavity grows with decreasing cavitation number (i.e.,σ < 1.2, LCmax/c > 0.8), the lift coefficient decreases. Boththe slight increase in the lift coefficient for 1.3 < σ < 1.6and subsequent drop in lift coefficient for σ < 1.2 arecaptured by the Merkle and Singhal models. The numericalpredictions show large fluctuations in the lift coefficientsfor σ < 1.2, as indicated by the error bars, due to theunsteady cavity dynamics. The numerical results show goodagreement with the experimental measurements for σ > 1.6,in the quasisteady cavitation and fully wetted flow regimes.The predicted lift coefficients by all three models are slightlyless than the measured values for 1.3 < σ < 1.6, in the partialsheet cavitation flow regime. In that range, the differencesbetween the three cavitation models are relatively small,with the Merkle model showing the most favorable, andthe Singhal model showing the least favorable comparisons.The Kubota model also predicts much higher fluctuationscompared to the other two models in the unsteady cavitationflow regime, and the fluctuations grow with decreasingcavitation number.

Comparisons of the predicted and measured dragcoefficients (CD = D/(0.5ρV 2∝c), where D is the dragforce) are shown in Figure 18. All three cavitation modelsunderpredicted the drag coefficients, but all three trendsagree well with the experiments. It should be noted that theunderprediction of the drag may be attributed to 3D effects,where downwash and wall boundary layer effects tend toincrease the drag. The results show that the developmentof cavitation significantly increases the drag, particularly forunsteady cavitation. The combination of decreases in lift andincreases in drag leads to a decrease in the efficiency, whichis a classical issue for cavitation. It should be noted that themagnitudes of the fluctuations in the lift and drag coefficientsare higher for the Merkle model than the Singhal modelat σ = 1.00, which is consistent with the results shown inFigures 13 and 14.

6. Conclusions

The predictive capability of three popular cavitation modelsis investigated for the prediction of cavitation flow around aNACA 66 hydrofoil at a fixed angle of incidence (α = 6◦)at Re = 750,000, for a case with partial sheet cavitation(σ = 1.49), and a case with unsteady sheet/cloud cavitation(σ = 1.00). Comparisons of the predicted and measured


t1 = 0.38 s

t2 = 0.4 s

t3 = 0.42 s

t4 = 0.46 s

t5 = 0.48 s

t6 = 0.5 s

t7 = 0.52 s

t8 = 0.54 s

(a) Merkle model, n = 3 (b) Experiments

0 0.06

0.12

0.18

0.24

0.31

0.37

0.43

0.49

0.55

0.61

0.67

0.73

0.8

0.86

0.92

0.98


0 0.06

0.12

0.18

0.24

0.31

0.37

0.43

0.49

0.55

0.61

0.67

0.73

0.8

0.86

0.92

0.98


Figure 15: Comparisons of the predicted vapor fraction contours and flow streamlines predicted using the Merkle cavitation model withn = 3 on the left with experimental observations on the right. σ = 1.00, Re = 750,000, V∞ = 5 m/s, and α = 6◦.


0.8 1.2 1.6 2 2.4 2.80

0.2

0.4

0.6

0.8

1

1.2

Steady cavitation

Subcavitating flow

σ

Unsteadysheet/cloudcavitation

Partialsheet

cavitation

Lc/c

Lc/c-num-Kubota, max

Lc/c-num-Merkle, max

Lc/c-num-Singhal, max

Lc/c-exp, max

Lc/c-num-Kubota, minLc/c-num-Merkle, minLc/c-num-Singhal, min

Figure 16: Comparison of the predicted and measured normalizedmaximum and minimum cavity lengths (Lc/c) at various cavitationnumbers. Re = 750,000, V∞ = 5 m/s, α = 6◦, and n = 3.

0.8 1.2 1.6 2 2.4 2.80

0.2

0.4

0.6

0.8

1

1.2

1.4

CL

CL-num-KubotaCL-num-Merkle

CL-num-Singhal

CL-exp

Figure 17: Comparisons of the measured and predicted liftcoefficients (CL) for various cavitation numbers. Re = 750,000,V∞ = 5 m/s, α = 6◦, and n = 3.

cavity lengths and hydrodynamic coefficients over a rangeof cavitation numbers are also shown. In addition to theapplication of different cavitation models, different levels ofdensity correction to the turbulent viscosity are applied toinvestigate the interaction between cavitation and turbulentflows. The aim of this density correction is to take intoaccount the local compressibility effects in the cavitatingregion, especially near the cavity closure, where standard tur-bulence models tend to overpredict the turbulent viscosity,which will modify the cavity shedding dynamics.

0.8 1.2 1.6 2 2.4 2.80

0.05

0.1

0.15

0.2

0.25

0.3

σ

CD

CD-num-Kubota

CD-num-MerkleCD-num-Singhal

CD-exp

Figure 18: Comparison of the measured and predicted dragcoefficients (CD) for various cavitation numbers. Re = 750,000,V∞ = 5 m/s, α = 6◦, and n = 3.

The general conclusions of the paper are as follows:(i) the density modification helps to reduce the turbulent

viscosity in the cavitating region, which decreases theturbulent kinetic energy in the cavity, as well as aroundthe cavity closure. With the reduction of the turbulentviscosity, the reentrant jet is allowed to develop underneaththe cavity and move upstream while allowing the cavityto expand toward the foil trailing edge. Consequently, thecavities tend to be slightly longer and more unsteady withthe density modification. This lead to a better comparisonwith experimental measurements and observations for boththe quasisteady cavitation case and the unsteady sheet/cloudcavitation case;

(ii) the density correction to the turbulent viscosity hasvarying degrees of impact on the different cavitation models.The Singhal model is the most sensitive to the turbulentviscosity modification, which already overpredicted thegeneral unsteadiness due to the explicit dependence of thesource and sink terms on the turbulent kinetic energy, theaddition of the density correction will further increase theoverall unsteadiness, and hence lead to greater discrepanciescompared to experimental measurements. On the otherhand, the Merkle model seems to be the least sensitive tothe turbulent viscosity modification, and tends to be morestable than the Singhal and Kubota models, particularly forunsteady cavitating flows. Addition of the density correctionfor the Merkle model helps to avoid overprediction of theturbulent eddy viscosity, which allows the reentrant jet toproperly develop, and hence leads to improved comparisonsbetween predictions and measurements;

(iii) the baroclinic torque is responsible for the gen-eration of additional vorticity due to shear flow createdby the unequal acceleration between the lighter vapor andheavier liquid phases. Hence, in regions with high, unaligned


density and pressure gradients, for example, along the liquid-vapor interface and the cavity closure region, the vorticityfield is modified by the cavitation, which in turn changesthe cavitation dynamics. The interdependency betweenthe cavity and vorticity dynamics through the baroclinictorque is not negligible, particularly for unsteady cavitatingflows. The magnitude of the baroclinic torque tends toincrease with the power factor n in the density correctionfor the turbulent viscosity (11), which in turn increasesthe flow unsteadiness, particularly near the cavity closure.New detailed experimental measurements of the mean andfluctuating velocity fields in the cavitating region are neededto investigate the validity of the density correction, andthe importance of the baroclinic torque on the cavitatingresponse of hydrofoils;

(iv) the results show that while predictions by allthree cavitation models compare well with experimentalmeasurements in the quasisteady partial sheet cavitationrange, the predicted cavity shedding frequencies and loadfluctuations predicted by the three cavitation models arevery different in the unsteady sheet/cloud cavitation range.Moreover, the numerical stabilities of the cavitation modelsare very sensitive to the dependency of the condensation andevaporation rates on the power to the pressure differenceterm (1.0 in the Merkel model and 0.5 in the Kubota andSinghal models), which is probably attributed to the inter-dependency between the cavitation and vorticity dynamicsthrough the baroclinic torque. The Singhal model is differentfrom the Kubota and Merkel models in that it explicitlydepends on the square root of the turbulent kinetic energy(see (10)), which increases the coupling effect between thevorticity distribution in the turbulent flow field and thecavity dynamics, and hence tends to destabilize the cavity.Consequently, the predicted cavity shedding frequencies wasmuch higher than the measured values, and the amplitudeof the load fluctuation was much lower than the measuredvalues. Physically, as noted in [28], transition to turbulencewill sweep away attached cavities. Hence, explicit dependenceof the sink (vaporization) term on the turbulent kineticenergy does not agree with experimental observations pre-sented in [28], nor with experimental measurements shownin this paper. Nevertheless, additional detailed experimentalmeasurements of the mean and fluctuating velocity fields, aswell as the time histories of the vapor fraction distributionin the cavity is needed to truly quantify the validity of thevarious cavitation models;

(v) in general, the Merkel model with a density correctionof n = 3 shows the best agreement with experimental mea-surements and observations for both steady and unsteadycavitating flows, particularly in terms of the cavity sheddingfrequency. The predicted shedding for unsteady sheet/cloudcavitation obtained using the Merkle model with n = 3matches with the measured value of 3.5 Hz. The Kubotamodel is not able to converge for the case of unsteadysheet/cloud cavitation, and the Singhal model predicts muchhigher cavity shedding frequency and much lower amplitudeof fluctuations compared to the measured values;

(vi) the analysis showed that the destruction and produc-tion coefficients in the Singhal model are not dimensionless

parameters, but instead should have units of velocity to beconsistent with the units of the mass transport equation;

(vii) the results demonstrate the importance of capturingboth the mean and fluctuating values of the hydrodynamiccoefficients and cavity lengths for the case of unsteadysheet/cloud cavitation because (1) the fluctuations are inthe same order as the mean values, and hence knowledgeof the fluctuations is critical to capturing the extremes ofthe loads to ensure structural safety; (2) the need to capturethe frequency of the fluctuations, which are critical to avoidunwanted noise, vibrations, and accelerated fatigue issues.Moreover, it is also critical to compare both the amplitudeand frequencies of fluctuations in addition to the meanvalues when evaluating the accuracy of numerical models.

This paper helps to improve the understanding of thepredictive capability of the three popular cavitation models.The impact of the density correction term on the differentcavitation models and its influence on the cavitationdynamics, vorticity field, and the force coefficients havebeen investigated. Additional research is needed to betterunderstand the influence of the local compressibility andbaroclinic torque on the cavitation and wake dynamics,as well as turbulent vorticity fields, particularly for caseswith spatially/temporally varying inflow, and for caseswith rigid and/or elastic body motions. Such research isimportant because accurate prediction of the cavity andload fluctuations is critical when analyzing the hydroelasticstability, fatigue, noise, vibration, and erosion characteristicsof hydraulic machineries.

Acknowledgments

The authors gratefully knowledge the Office of NavalResearch (ONR) and Dr. Ki-Han Kim (program manager)for their financial support through Grant no. N0014-09-1-1204.

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