Proceedings of FEDSM02 ASME 2002 Fluids Engineering Division Summer Meeting

Montreal, Quebec, Canada, July 1418, 2002

ITA

C

k a

g, My ofle, Fu, w

have received a growing attention due to advances in computational capabilitieproblems. The cavitatinghigh Reynolds number alarge disparity between inception and concurrenstages of cavitation, supercavitation, is drivhydrodynamic pressure dno computational model all these flow regimes ba

Due to changes mechanisms across the computational algorithmexperience severe convcavitating flows. To remmethods have been propmethods, the artificial co

axisymmetric bodies and hydrofoils with comparable accuracy.

Downloaded From: http://proceeding1 Copyright 2002 by ASME

s and physical understanding for these flow problem is complicated due to nd multiphase nature of the flow with fluid properties of each phase. The

t development of a cavity into other namely sheet, cloud, vortex and en by the phase change due to rop and bubble dynamics [1]. To date, can offer comprehensive capabilities of sed on first principles. in fluid properties and physical

liquid-vapor boundaries, conventional s of single-phase incompressible flow ergence and stability problems for edy this situation, improved numerical osed. In the context of density-based mpressibility method has been applied

A common approach in cavitation modeling is to use the homogeneous flow theory. In this theory, the mixture density concept is introduced and a single set of mass and momentum equations are solved. Different ideas have been proposed to generate the variable density field; a review of such studies is given in Ref. [9]. Some of the existing studies solve the energy equation and determine the density through suitable equations of state [3]. Since most cavitating flows are isothermal, arbitrary barotropic equations have been proposed to supplement the energy consideration [10]. Another popular approach is the transport equation-based model (TEM) [2, 4-6, 8]. In TEM, a transport equation for either mass or volume fraction, with appropriate source terms to regulate the mass transfer between phases, is solved. Different source terms have been proposed by different researchers, which will be discussed in the next section. A recent experimental finding [11] helps assess the adequacy of the above-mentioned physical models. It EVALUATION OF CAV

NAVIER-STOKES

Inanc Senoca

Department of Aerospace EngineerinUniversit

Gainesvilinanc@aero.ufl.ed

ABSTRACT The predictive capability of three transport equation-based

cavitation models is evaluated for attached turbulent, cavitating flows. To help shed light on the theoretical justification of these models, an analysis of the mass and normal-momentum conservation at a liquid-vapor interface is presented. The test problems include flows over an axisymmetric cylindrical body and a planar hydrofoil at different cavitation and Reynolds numbers. Proper grid distribution for high Reynolds number cavitating flows is emphasized. Although all three models give satisfactory predictions in overall pressure distributions, differences are observed in the closure region of the cavity, resulting from the differences in compressibility characteristics handled by each model. Keywords: CFD, cavitation, turbulence

INTRODUCTION

Navier-Stokes computations of turbulent cavitating flows s.asmedigitalcollection.asme.org/ on 07/13/2015 TermFEDSM2002-31011 TION MODELS FOR

OMPUTATIONS

nd Wei Shyy

echanics and Engineering Science Florida L 32611 ss@aero.ufl.edu

with special attention given to the preconditioning technique [2-5]. The preconditioning technique is guided by examining the eigenvalues of system and may not be unique depending on the forms of the governing equations [6]. Following the spirit of the well-established SIMPLE algorithm [7], a pressure-based method for turbulent cavitating flows has been developed [8]. It is shown that the pressure correction equation for turbulent cavitating flows shares common features of high speed flows and a pressure-density coupling scheme is developed that results in a unified incompressible-compressible formulation. Upwinded density interpolation is adopted to improve mass and momentum conservation in cavitating regions [8]. High-resolution non-oscillatory convection schemes have been applied in both pressure-based and density-based methods [2-5, 8]. The use of such convection schemes is very important especially in the vicinity of sharp density gradients. Both methods, pressure-based and density-based, have been successful to compute turbulent cavitating flows around s of Use: http://asme.org/terms

adopted. The resulting system of equations for turbulent

Downlsection. A recent experimental finding [11] helps assess the adequacy of the above-mentioned physical models. It shows that vorticity production is an important aspect of cavitating flows, especially in the closure region [11]. Specifically, this vorticity production is a consequence of the baroclinic generation term of vorticity equation.

Pr1 (1)

Clearly, if an arbitrary barotropic equation )(Pf=r is used, then the gradients of density and pressure are always parallel; hence the baroclinic torque is zero. This suggests that physical models that utilize a barotropic equation will fail to capture an experimentally observed characteristic of cavitating flows. Likewise, solving an energy equation will also experience the same situation if the flow is essentially isothermal. However in TEM approach the density is a function of the transport process. Consequently, gradients of density and pressure are not necessarily parallel, suggesting that TEM can accommodate the baroclinic vorticity generation.

Different modeling concepts have been introduced in TEM, with inconsistent numerical treatments such as grid resolution and no consideration of pressure-density dependency. As a result, there is a lack of clear consensus of the capability and relative merits of these models. The goal of the present study is to assess the predictive capability of existing transport equation-based models using a unified pressure-based method for turbulent cavitating flows [8], with an emphasis on grid resolution. Three models [(2, 12), 4, 13] are considered in this study. A common characteristic of these models is the use of user-defined empirical factors to regulate the mass transfer process. Although these empirical factors may seem ad-hoc, satisfactory results for different geometries and flow conditions have been obtained with consistent modeling parameters [8, 14]. Nevertheless, it is desirable to gain insight into the meaning of these empirical parameters based on the first principles. In this study, an analysis based on the interfacial dynamics is presented to shed light on the nature of the modeling parameters.

In what follows, each cavitation model is shortly summarized along with the empirical parameters used in the computations. Following this, a derivation, starting from the mass and momentum conservation at a liquid-vapor interface is presented to explain the modeling concept. The first set of results cover the empirical cavitation models tested for the cases of flow over an axisymmetric cylindrical object with hemispherical forehead and a NACA66MOD hydrofoil. NUMERICAL METHOD

The present Navier-Stokes solver employs a pressure-based algorithm and the finite volume approach. The governing equations are solved on multi-block structured curvilinear grids [15, 16]. To represent the cavitation dynamics the transport equation models described in the upcoming section are

oaded From: http://proceedings.asmedigitalcollection.asme.org/ on 07/13/2015 Tercavitating flows is solved using the pressure-based method developed by Senocak and Shyy [8]. As already mentioned, a key feature of this method is to reformulate the pressure correction equation into a convective-diffusive, instead of a pure diffusive, equation. This modification is achieved through the inclusion of a pressure-velocity-density coupling scheme into the pressure correction equation. For details of the numerical algorithm, the reader is referred to Ref. [8]. The original k-e turbulence model with wall functions is employed as the turbulence closure [17].

SUMMARY OF TRANSPORT EQUATION-BASED MODELS (TEM) Model-1 (from Refs. [2, 12])

Several researchers have adopted this model [2, 5, 12]. Both volume fraction and mass fraction forms of it have been adopted. Evaporation and condensation terms are both functions of pressure. The liquid volume fraction form is considered in this study.

( )

( )( )

--

+-=+

tUPPMAXC

tUPPMINCu

t

L

LVLprod

LV

LVLLdestL

L

)50.0(10,

)50.0(0,)(

2

2

ra

rraraa r

(2)

The empirical factors have the following values (Cdest=1.0, Cprod=8.0x10

1). These parameters have been non-dimensionalized with free stream values to have the correct dimensional form as the convective terms. The same parameters can be used for different geometries and flow conditions provided that they are non-dimensionalized with the free stream values. The discussion holds for the other cavitation models also. Model-2 (from Ref. [4])

The liquid volume fraction is chosen as the dependent variable in the transport equation. The evaporation term is a function of pressure whereas the condensation is a function of the volume fraction.

( )

( )

-

+-

=+

t

C

tUPPMINC

ut

L

LLprod

LL

LVLVdestL

L

r

aa

rrar

aa

1

)50.0(0,

)(

2

2

r

(3)

The empirical factors have the following values, which have been determined previously [8]. (Cdest=9.0x10

5, Cprod=3.0x104)

Model-3 (from Ref. [13]) The vapor mass fraction is the dependent variable in the

transport equation. Both evaporation and condensation terms are functions of pressure. The model equations that are presented here are slightly different from the ones in the original paper [13], which considers only a non-condensable gas. No such a limitation is imposed here.

( ) )()( +- +=+

mmuftf

VmVm &&rrr (4) 2 Copyright 2002 by ASME

ms of Use: http://asme.org/terms

21

DownloVL

VVLprod

VL

VVLdest

PPifPPU

Cm

PPifPPU

Cm

>

-=

Eq. (19) forms a fundamental cavitation model for Navier-

DownlStokes computations of cavitating flows. Unlike the existing transport equation-based models in the literature, the proposed model does not require empirical constants to regulate the mass transfer process. It is general to handle the time-dependency since the interface velocity (VI) is taken into account in the model. The model requires that an interface be constructed in order to compute the interface velocity for time-dependent computations, as well as the normal velocity of the vapor phase. However in sheet type of attached cavitation the cavity is often modeled in a steady-state computation and the assumption can be valid and does produce satisfactory results [4, 5, 8]. Hence the interface velocity (VI) is zero for steady-state computations and the normal velocity of the vapor phase can be computed by taking the gradient of the liquid volume fraction [19, 20]. The vapor phase normal velocity is the dot product of the velocity and the normal vector.

L

Ln

=

nuV nV

=, (20)

RESULTS Two flow configurations have been considered, namely, (i)

an axisymmetric cylindrical object with a hemispherical forehead (referred to as hemispherical object) at a Reynolds number of 1.36x105 and a cavitation number of 0.30, and (ii) the NACA66MOD hydrofoil at an angle of attack of 4 with Reynolds number of 2.0x106. It is experimentally observed that sheet (attached) cavitation occurs for both of the geometries under given conditions and time averaged experimental data of pressure distribution along the surface is available [21, 22]. Hence steady-state computations are carried out in this study.

Figure 2 compares the performance of the cavitation models for the hemispherical object at a cavitation number of 0.30. All three models match the experimental data satisfactorily. Differences in the performance are more noticeable in the closure region, where the vapor phase condenses. Figure 3 shows the corresponding density distribution along the surface. As seen from density plots, the liquid phase first expands and vapor phase appears uniformly inside the cavity, then the vapor phase compresses, in a shock like fashion, back to the liquid phase. The differences among the three models in density profiles are significant. This implies that each model generates different compressibility characteristics, although they produce very similar steady-state pressure distributions. This issue has important implications in time dependent problems; which model produces the correct compressibility is an open question and needs further investigation. Figure 4 shows the distribution of density throughout the cavity and the spanwise vorticity distribution. The interface is captured sharply with Model-2 and Model-3 compared to Model-1, especially in the downstream region of the cavity. As seen from the spanwise vorticity distribution, also given in Fig. 4, there is additional generation of vorticity at the

oaded From: http://proceedings.asmedigitalcollection.asme.org/ on 07/13/2015 Termclosure region and this is primarily due to the baroclinic torque, confirming the discussion given in Introduction.

The NACA66MOD hydrofoil case is computed at a Reynolds number of 2x106. The turbulent boundary layer is extremely thin at such high Reynolds numbers. Since the original k- turbulence model along with the wall function is adopted [17], it is important to offer spatial resolutions consistent with the modeling requirement [23]. This requires that the non-dimensional normal distance from the wall (y+), a representation of the local Reynolds number, should be in the log-law region. More details on the wall function formulation are given in Ref. [23]. Once a cavity occurs on the surface, the local Reynolds number decreases due to reduction in density and the first grid point away from the wall may not be positioned in the log layer. This issue has important implications in both accuracy and convergence. It is found that proper grid distribution is very important for satisfactory results and convergence of cavitating flow computations. However there is a trade-off between positioning the first grid point away from the wall in the log layer verses having enough points to discretize the cavity, especially for high Reynolds number cases.

Two grids have been tested. Model-2 is used in the computations. Grid-A is a three-block domain with approximately 6.6x104 nodes whereas Grid-B is a six-block domain with approximately 9.8x104 nodes. The grid resolution is further clustered close to the suction side of the hydrofoil in Grid-B compared to Grid A. Close up views of the same region in both grids are shown in Fig. 5 to demonstrate the spatial resolution. The extend of the cavity region is also highlighted on the grids to give an idea of how many grid points have been used to discretized the vapor cavity. As shown in Fig. 6 the predictions with Grid-A are very poor compared to the case with Grid-B. A very short cavity has been captured with Grid-A as a result of poor resolution. The vapor cavity is so thin that Grid-A can only accommodate two grid points in this region. In Fig. 7 the y+ distribution of the first grid points away from the wall is plotted along the surface. The law of the wall is also highlighted on this plot. As can be observed in Fig.5, special attention has been given during grid generation to place the points of the Grid-B in the log layer. A higher resolution grid is also utilized and it is found that if the y+ values of the cavitating region are in the viscous sublayer, the computation is not stable and convergent. This may be because in the wall function formulation a linear velocity profile is imposed in the viscous sublayer and such a profile may not be suitable to represent the phase change dynamics. It should be emphasized that the original k- turbulence model [17] is based on the equilibrium assumption, and the rate of turbulence production and dissipation are balanced in the log-layer, hence it is advisable to place the near wall region in the log layer [24]. If higher resolution is required a low Reynolds number two-equation turbulence model should be the choice. Nevertheless Grid-B is used for all other computations of NACA66MOD hydrofoil case. 4 Copyright 2002 by ASME

s of Use: http://asme.org/terms

DownloIn Fig. 8 and Fig. 9 the performances of the models have been assessed for cavitation numbers of 0.91 and 0.84 respectively. For cavitation number of 0.84, the vaporous cavity covers about 60% of the hydrofoil surface. All three models produce satisfactory results at both cavitation numbers. Differences in predictions are more pronounced, similar to the hemispherical object case, at the closure region, which is due to the different compressibility characteristics imposed by the cavitation models. Model-2 and Model-3 do relatively a better job to capture the pressure distribution at the closure region as compared to the predictions of Model-1.

CONCLUSIONS An improved pressure-based method for turbulent

cavitating flows is employed to evaluate the existing transport equation-based cavitation models in literature. Three versions of the transport equation-based model (TEM) are considered. To help understand these models, a theoretical derivation is presented starting from mass and normal momentum conservation at a liquid-vapor interface. Unlike the existing empirical cavitation models, the new model needs no user defined empirical factors to regulate the mass transfer process. It helps interpret the implication of the empirical parameters required by the existing models.

Two flow configurations have been tested, including a hemispherical projectile and the NACA66MOD hydrofoil. For both geometries, all three models produce comparable and satisfactory results in wall pressure distributions. However, different density distributions, especially in the closure region are predicted by the models, implying that the compressibility effects are handled differently. These differences can significantly impact the time dependent flow computations. It is also shown that a wall-function consistent grid is required to improve predictive capability for high Reynolds number cases. The present results have further validated the generality of the pressure-based algorithm for turbulent cavitating flows. The numerical method have successfully accounted for the compressibility effects induced by different cavitation models.

The present study suggests that the existing versions of the TEM have merits and limitations. Further research is needed to address the modeling uncertainties. Direct utilization of the interfacial dynamics, such as that presented here, should be further pursued to alleviate the need for empirically adjusting modeling parameters with a sound theoretical foundation. We will further pursue this path and report our findings in the future.

ACKNOWLEDGMENTS This study has been supported partially by ONR and NSF.

The first author would like to thank Marianne Francois for helpful discussions on computation of normal velocities.

REFERENCES [1] Knapp, R.T., Daily, J.W. and Hammitt, F.G., 1970,

Cavitation, McGraw-Hill, New York. 5

aded From: http://proceedings.asmedigitalcollection.asme.org/ on 07/13/2015 Terms of [2] Merkle, C.L., Feng, J. Z. and Buelow, P.E.O., 1998, Computational modeling of the dynamics of sheet cavitation, Proc. Third Intern. Symp. on Cavitation, Grenoble, France.

[3] Edwards, J.R. Franklin, R.K. and Liou, M.S., 2000, Low-diffusion flux-splitting methods for real fluid flows with phase transitions, AIAA J. 38(9), pp. 1624-1633.

[4] Kunz, R.F., Boger, D.A., Stinebring, D.R., Chyczewski, T.S., Lindau, J.W., Gibeling, H.J., Venkateswaran, S. and Govindan, T.R., 2000, A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction, Computer and Fluids, 29, p. 849.

[5] Ahuja, V., Hosangadi, A. and Arunajatesan, S., 2001, Simulations of cavitating flows using hybrid unstructured meshes, ASME J. of Fluids Engineering, 123, pp. 331-340.

[6] Venkateswaran, S., Lindau, J.W., Kunz, R.F and Merkle, C.L., 2001, Preconditioning algorithms for the computation of multi-phase mixture flows, AIAA 39th Aerospace Sciences Meeting & Exhibit, AIAA-2001-0125.

[7] Patankar, S.V., 1980 Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington DC.

[8] Senocak, I. and Shyy, W., 2002, A pressure-based method for turbulent cavitating flow simulations, J. of Computational Physics, 176, pp. 1-22

[9] Wang, G., Senocak, I., Shyy, W., Ikohagi, T., and Cao, S., 2001, Dynamics of Attached Turbulent Cavitating Flows, Progress in Aerospace Sciences, 37, pp. 551-581.

[10] Delannoy, Y. and Kueny, J.L., 1990, Cavity flow predictions based on the Euler equations, ASME Cavitation and Multi-Phase Flow Forum.

[11] Gopalan, S. and Katz, J. 2000 Flow structure and modeling issues in the closure region of attached cavitation, Physics of Fluids, 12(4), p. 895.

[12] Singhal, A.K., Vaidya, N. and Leonard, A.D., 1997, Multi-dimensional simulation of cavitating flows using a PDF model for phase change, ASME Fluids Engineering Division Summer Meeting, ASME Paper FEDSM97-3272.

[13] Singhal, A.K., Li, N. H., Athavale, M. and Jiang, Y., 2001, Mathematical basis and validation of the full cavitation model, ASME Fluids Engineering Division Summer Meeting, ASME Paper FEDSM2001-18015.

[14] Senocak, I., and Shyy, W. 2001, "Numerical Simulation of Turbulent Flows with Sheet Cavitation," CAV2001 4th International Symposium on Cavitation, Paper No. CAV2001A7.002.

[15] Shyy, W., Thakur, S. S., Ouyang, H., Liu, J., and Blosch, E., 1997, Computational Techniques for Complex Transport Phenomena, Cambridge University Press, New York.

[16] Shyy, W., 1994, Computational Modeling for Fluid Flow and Interfacial Transport, Elsevier, Amsterdam, The Netherlands, (revised printing 1997).

[17] Launder, B. E., and Spalding, D. B., 1974, The Numerical Computation of Turbulent Flows, Computational Methods in Applied Mechanics and Engineering, 3, pp. 269-289.

[18] Wallis, G.B., 1969, One-dimensional Two-phase Flow, McGraw-Hill, New York.

[19] Brackbill, J.U., Kothe, D.B. and Zemach, C., 1992, A Continuum Method for Modeling Surface Tension, J. Of Computational Physics, 100, pp. 335-354.

[20] Francois, M., 1998, A Study of the Volume of Fluid Method for Moving Boundary Problems, M.Sc. Thesis, Copyright 2002 by ASME

Use: http://asme.org/terms

Embry Riddle Aeronautical University, Daytona Beach FL, USA.

[21] Rouse, H., and McNown, J. S., 1948, Cavitation and Pressure Distribution, Head Forms at Zero Angle of Yaw, Studies in Engineering, Bulletin 32, State University of Iowa.

[22] Shen, Y., and Dimotakis, P., 1989, The Influence of Surface Cavitation on Hydrodynamic Forces, Proceedings of 22nd ATTC, St. Johns, pp.44-53.

0.4

0.6

0.8

1

dens

ity

Hemispherical, Re=1.36x105, =0.30

Model-1Model-2Model-3

0.4

0.6

0.8

1

dens

ity

Hemispherical, Re=1.36x105, =0.30

Model-1Model-2Model-3

Downlo[23] He, X., Senocak, I., Shyy, W., Gangadharan, S.N. andThakur S., 2000, Evaluation of Laminar-TurbulentTransition and Equilibrium Near Wall Turbulence Models,Numerical Heat Transfer, Part A, 37, pp.101-112.

[24] Versteeg, H.K. and Malalasekera, W., 1995, AnIntroduction to Computational Fluid Dynamics, p. 73,Longman, London.

U L

V

m

Figure 1. Representation of a vaporous cavity in homogeneous flow theory.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

s/D

Cp

Hemispherical, Re=1.36x105, =0.30

Model-1 Model-2 Model-3 Exp. data

0 0.5 1 1.5 2 2.5 3 3.5 4-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

s/D

Cp

Hemispherical, Re=1.36x105, =0.30

Model-1 Model-2 Model-3 Exp. data

Figure 2. Comparison of surface pressure distribution obtained from empirical cavitation models. Experimental data is from Ref. [21]

aded From: http://proceedings.asmedigitalcollection.asme.org/ on 07/13/2015 Ter0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

s/D0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.2

s/D Figure 3. Comparison of surface density distribution obtained from empirical cavitation models. Experimental data is from Ref. [21].

Model-3

Model-2

Model-1

Model-3

Model-2

Model-1

Figure 4. Detailed view of the cavitating region. On the left is the density distribution; on the right is the spanwise vorticity distribution.

X

Y

-0.23 -0.225 -0.22 -0.215 -0.210.02

0.025

0.03

0.035

0.04

X

Y

-0.23 -0.225 -0.22 -0.215 -0.210.02

0.025

0.03

0.035

0.04

Cavity edge

GRID-BGRID-A

X

Y

-0.23 -0.225 -0.22 -0.215 -0.210.02

0.025

0.03

0.035

0.04

X

Y

-0.23 -0.225 -0.22 -0.215 -0.210.02

0.025

0.03

0.035

0.04

Cavity edge

GRID-BGRID-A

Figure 5. Visual comparison of the grid distributions. 6 Copyright 2002 by ASME

ms of Use: http://asme.org/terms

1.5NACA66MOD, Re=2.0x106, =0.91

1.5NACA66MOD, Re=2.0x106, =0.91

0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

X/C

-Cp

NACA66MOD, Re=2.0x106, =0.84

Model-1 Model-2 Model-3 Exp. data

0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

X/C

-Cp

NACA66MOD, Re=2.0x106, =0.84

Model-1 Model-2 Model-3 Exp. data

Figure 9. Comparison of surface pressure distribution obtained from empirical cavitation models. Experimental data is from Ref. [22].

Downl0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

X/C

-Cp

Grid-A Grid-B Exp. data

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

X/C

-Cp

Grid-A Grid-B Exp. data

Figure 6. Effect of grid distribution on predictions. Model-2 is used. Experimental data is from Ref. [22].

0 0.2 0.4 0.6 0.8 110

0

101

102

103

104

X/C

Y+

NACA66MOD, Re=2.0x106, =0.91

Grid-AGrid-B

y+=11.63

viscous sublayer

log layer

defect layer

y+=400.0

0 0.2 0.4 0.6 0.8 110

0

101

102

103

104

X/C

Y+

NACA66MOD, Re=2.0x106, =0.91

Grid-AGrid-BGrid-AGrid-B

y+=11.63

viscous sublayer

log layer

defect layer

y+=400.0

Figure 7. Distribution of the y+ values of the first grid points away from the wall along the hydrofoil surface.

0 0.2 0.4 0.6 0.8 1-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X/C

-Cp

NACA66MOD, Re=2.0x106, =0.91

Model-1 Model-2 Model-3 Exp. data

0 0.2 0.4 0.6 0.8 1-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X/C

-Cp

NACA66MOD, Re=2.0x106, =0.91

Model-1 Model-2 Model-3 Exp. data

Figure 8. Comparison of surface pressure distribution obtained from empirical cavitation models. Experimental data is from Ref. [22].

oaded From: http://proceedings.asmedigitalcollection.asme.org/ on 07/13/2015 Term7 Copyright 2002 by ASME

s of Use: http://asme.org/terms