1

Cav03-OS-1-007 Fifth International Symposium on Cavitation (CAV2003)

Osaka, Japan, November 1-4, 2003

ASSESSMENT OF MODELING STRATEGIES FOR CAVITATING FLOW AROUND A HYDROFOIL

Jiongyang Wu1, Yogen Utturkar1 & Wei Shyy1£

1 Department of Mechanical and Aerospace Engineering,

University of Florida, Gainesville, FL 32611, U.S.A. £ E-mail: wei-shyy@ufl.edu Phone: 1-352-392-0961

ABSTRACT The time-dependent cavitating flow around a hydrofoil has

been simulated by a pressure-based method and finite volume approach in framework of the Favre-averaged equations. Results from a transport equation-based cavitation model are compared with those obtained by alternate modeling strategies to gauge the modeling implications in the prediction. Furthermore, both the Launder and Spalding version and a recently developed filter-based modification of the k - ε model have been employed. In addition to presenting the instantaneous and time averaged results, salient flow structures have been distilled using Proper Orthogonal Decomposition (POD). The filter-based turbulence model, which significantly reduces viscous damping, leads to stronger time-dependency in flow structures, dynamic parameters such as lift coefficient, and cavity shape. The POD analysis shows that the solutions yielded by different turbulence and cavitation models share similar major modes for σ = 0.8, but noticeably different structures for the massively cavitating case, σ = 0.4.

INTRODUCTION

Several numerical studies on cavitating flows, mostly based on density based [1-2] or pressure based methods [3-4], are available in open literature. Despite these recent advancements, simulation of cavitating flow still remains a challenging proposition due to its multi-timescale nature, and because of the ever-increasing quest for adequate computational models.

The modeling task essentially comprises cavitation, compressibility and turbulence models. Investigation of currently known models and their mutual interaction can be a significant asset to improving their performance. Senocak and Shyy [5] evaluated the merits and demerits of existing cavitation models to develop a more fundamental cavitation model based on interfacial dynamics. Detailed derivation of the model and its performance is obtainable [6]. Vaidyanathan et al. [7] performed a sensitivity analysis on a transport equation based cavitation model to optimize the coefficients of its source terms. This effort led to significant improvement in the model performance. Furthermore, Senocak and Shyy [8] also examined two Speed of Sound (SoS) models in the biphasic mixture for two cavitation numbers. Their study was extended by Wu et al. [9] with correlation and spectral analysis to provide the quantitative

impact of the SoS modeling on cavitating flow in a control valve. Though a robust compressibility model for cavitating flows is still a research focus, the above studies certainly established better characteristics of one of the proposed SoS models over the other. Recently, Utturkar et al. [10] have proposed a reduced order approach to assess the modeling strategies. They have employed Proper Orthogonal Decomposition (POD) to extract key flow features that can highlight the impact of these numerical techniques. Thus, critical investigation of existing physical models has proven to be rewarding to progressive development of the models in terms of fidelity and robustness.

Besides cavitation and sound propagation, serious implications of turbulence modeling on cavitating flows were recently revealed by Wu et al. [9]. They reported that high viscosity of the original k - ε model dampens cavitation instabilities. Consequently, simulation of phenomena such as periodic cavity inception and detachment requires alternate modeling approaches. The widely pursued Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) methods are infeasible for multiphase cavitating flow computations. Alternatively, Johansen et al. [11] have very recently formulated a filter-based RANS turbulence model. The new model imposes a grid independent filter on the flow. While the imposed filter size prevents excessive dissipation of small scale motions like the original RANS model, it allows development of flow scales commensurate with the grid resolution. Thus, the resultant behavior of the model can be tuned between the limits of RANS-type and a hybrid RANS-LES model.

Noting the above advancement and issues, our interest in the current study is twofold. Firstly, the case study of cavitating flow around hydrofoil poses an excellent opportunity to assess the impact of filter-based model on flow and cavity structure in comparison to the RANS model. Secondly, uncertainty in existing cavitation models solicits a re-inspection in terms of their impact on the cavity shape and the interplay with the turbulence model. While examining the above effects on the hydrofoil geometry, we choose to account sound propagation with the better of the two SoS models [8-9]. In addition to qualitative comparison between two flow cases, the technique of

2

POD is employed to distill salient flow features thus enabling critical comparison of corresponding eigenmodes.

A brief review of the numerical method is presented in the next section followed by results, discussion, summary and conclusions.

NOMENCLATURE : density; : velocity component

: co-ordinate axis; : pressure

: mass tranfer rate; : phase fraction

: dynamic viscosity; : velocity scale

: length scale; : velocity

: turbulent K.E.; : dissip

u

x p

m

U

L V

k

ρ

αµ

ε

�

th

th

ation

: filter size; : time; : normal

: stress; : kinematic viscosity

: Cavitation number; : lift coefficient

: drag coefficient; ( ) : n eigenmode

: n eigen value; : matrix inner produ

L

D n

n

t n

C

C x

τ νσ

ψλ

∆

��

�

th

ct

( ) : scaling coefficient of n mode

Subscripts:

: liquid; : vapor; : mixture

, , : axes direction; : turbulent

: inlet; : entropy; : interfacial

: normal component; : mean flow direction

Super

n t

l v m

i j k t

s I

n

φ

ξ∞

scripts:

+ production - destruction

' correction component

NUMERICAL METHOD The set of governing equations consists of the conservative

form of the Favre-averaged Navier-Stokes equations, the k ε− two-equation turbulence closure, and a volume fraction transport equation-based cavitation model. The mass continuity, momentum and cavitation model equations are given below:

( )0m jm

j

u

t x

ρρ ∂∂+ =

∂ ∂(1)

( )( )

2[( )( )]

3

m i jm i

j i

ji kt ij

j j i k

u uu p

t x x

uu u

x x x x

ρρ

µ µ δ

∂∂ ∂+ = − +∂ ∂ ∂

∂∂ ∂∂ + + −∂ ∂ ∂ ∂

(2)

)()( −+ +=

∂∂

+∂

∂mm

x

u

t j

jll��

αα(3)

The mixture density is given by: (1 )m l l v lρ ρ α ρ α= + − (4)

The present Navier-Stokes solver employs pressure-based algorithms and the finite volume approach. The governing equations are solved on multi-block structured curvilinear grids. PISO algorithm is used to march non-iteratively along time.

Detailed documentation of the numerical procedure can be readily obtained [4, 6, 12]. Cavitation modeling Physically, the cavitation process is governed by thermodynamics and kinetics of the phase change process. The liquid-vapor conversion associated with the cavitation process is modeled through +m� and −m� terms in Eq. (3), which represents, respectively, condensation and evaporation. The particular form of these phase transformation rates forms the basis of the cavitation model. Given below are the two approaches probed in the present study. Heuristic model of Kunz et al. [1] The liquid-vapor condensation rates for this particular model are given as:

2

2

Min(0, )

( / 2)

(1 )

dest v l v

l l

prod v l l

l

C p pm

U t

Cm

t

ρ αρ ρρ α α

ρ

−

∞ ∞

+

∞

−=

−=

�

�

(5)

where, 5100.9 ×=destC and 4100.3 ×=prodC are empirical

constant values obtained by sensitivity analysis. ∞U is chosen as

the inlet velocity value. The time scale in the equation is defined as the ratio of the characteristic length scale to the reference velocity scale (L/U). Interfacial model of Senocak and Shyy [6] The source terms in this model were derived by Senocak and Shyy by applying mass and momentum balance across the cavity interface. Their physical form is given below:

2, ,

2, ,

Min(0, )

( ) ( )

(1 )Max(0, )

( ) ( )

l l v

v v n I n l v

l v

v n I n l v

p pm

V V t

p pm

V V t

ρ αρ ρ ρ

αρ ρ

−

∞

+

∞

−=

− −− −

=− −

�

�

(6)

where,

, , lv n

l

V u n nαα

∇= ⋅ =

∇� � �

(7)

Though the interfacial cavitation model is not completely empiricism free, it certainly imparts physical understanding to the mass transfer rates ( ,dest prodC C ) in the source terms.

Turbulence modeling For the system closure, the original two-equation turbulence model with wall functions is presented as follows:

( )( )

[( ) ]

m jmt m

j

t

j k j

u kk

t x

k

x x

ρρ ρ ε

µµσ

∂∂+ = Ρ −

∂ ∂

∂ ∂+ +∂ ∂

(8)

2

1 2

( )( )P

[( ) ]

m jmt m

j

t

j j

uC C

t x k k

x x

ε ε

ε

ρ ερ ε ε ερ

µ εµσ

∂∂+ = −

∂ ∂

∂ ∂+ +∂ ∂

(9)

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The turbulence production, Reynolds stress tensor terms, and the Boussinesq eddy viscosity concept are defined as:

2P ; ; ( )

3ij ji i

t ij ij i j i j tj j i

k uu uu u u u

x x x

δτ τ ρ ν

∂∂ ∂′ ′ ′ ′= = − = − +∂ ∂ ∂

(10)

With the common backbone equations described above, the following two strategies are adopted in the current study. Launder and Spalding model [13]: The original parameters for this model namely 44.11 =εC � 92.12 =εC � 3.1=εσ � 0.1=kσ are

calibrated from equilibrium shear flows. The turbulent viscosity is defined as:

2

, 0.09mt

C kCµ

µ

ρµ

ε= = (11)

Filter-based model of Johansen et al. [11] This newly developed model is being applied first time to cavitating flows. Though the model employs same parameter values as above, the difference lies in the formulation of the turbulent viscosity:

2

, 0.09mt

C kF Cµ

µ

ρµ

ε= = (12)

where, F is the filter function defined in terms of filter size ( ∆ )as:

3 3/ 2Min[1, ]F c

k

ε∆= (13)

The proposed filter, as in Eq. (13), recovers the Launder and Spalding k ε− model for a coarse filter size. Furthermore, at near wall nodes the chosen filter returns F = 1 recovering use of wall functions. However, in the far field zone the filter produces a hybrid RANS-LES behavior by allowing development of length scales comparable to grid resolution. The value of c3 is chosen 1.0 for the present case. Speed of Sound (SoS) Model [8] Due to lack of dependable equation of state for the multiphase mixture, numerical modeling of sound propagation is still a topic of research. We refer the reader to past studies [6, 8, 9] for modeling options, impact and issues, and just outline the currently employed model below.

1 1

1 1

| |SoS ( ) ( )

| |i i

si i

CP P P Pρ ξ

ρ ρρ ρ + −

+ −

−∂ ∆= = = =∂ ∆ −

(14)

The above definition is a close approximation to the fundamental

definition of speed of sound with the path of the partial

derivative / Pρ∂ ∂ taken to be to be the mean flow direction( )ξ .

The speed of sound affects the numerical calculation via the

pressure correction equation, thus endowing it with a convective-

diffusive form.

p p pC Pρ ′ ′= (15)

RESULTS AND DISCUSSION Figure 1 illustrates the hydrofoil geometry in physical as well as in computational domain, which comprises 6 blocks. All the cases discussed below are computed on the same grid.

Table 1 quantifies cavity morphology and various aerodynamic parameters at two cavitation numbers for the two turbulence models. While the effect of cavitation number on cavity shape and location is straightforward, its influence on lift coefficient is worth noting. As observed from the table, halving the cavitation number to 0.4 leads to a 50-60% reduction in lift. Since lift coefficient at low AOA is predominantly governed by surface pressure distribution, its response to cavitation strength can be best understood from Figures 2 and 3. Figure 2 illustrates time-averaged surface pressure distribution for cases of no cavitation, σ = 0.8 and σ = 0.4 obtained by employing two different turbulence models. Figure 3 shows time-dependent lift and drag coefficients correspondingly. Thus, it is apparent that integral analysis on pressure curves in Figure 2 can provide a reasonable estimate of time-averaged CL value indicated by Figure 3 for each case. As observed from Figure 2, cavity formation reduces the sharp plunge of pressure on the top surface of the cavity thus explaining the reduction of lift force with cavitation number seen in Table 1. The drag coefficient in comparison to CL shows restrained sensitivity to cavitation effects and turbulence model. This is mainly because of its strong dependence on wall stress, which is essentially modeled by wall functions by both turbulence models. It is however more striking to observe the effect of filter-based model on the cavity pressure. Note that the mean pressure plot of filter-based model depicts significant pressure variations within the cavity as compared to the original Launder and Spalding k ε− model. The above effect is more conspicuous for the σ = 0.8 case. Besides response of mean aerodynamic parameters, Figure 3, especially for the σ = 0.8 case, illustrates a strong time-dependent behavior of lift force for the filter-based model. This can be elucidated by examining instantaneous pressure coefficients at different instances (Figure 4). It can be easily observed that filter-based modeling causes wide ranging pressure variations as compared to the Launder and Spalding k ε− model. This corroborates the large oscillations in CL for the particular case. More important to observe is the fairly constant pressure sustained in the cavity by both models at both the instants. This observation reveals an interesting fact that the pressure variation within the cavity observed in Figure 2 is mainly due to time-averaging process and can be misleading. Thus, the time-averaged pressure curves should only be employed to acquire an estimate of mean aerodynamic coefficients. At the cavitation number of 0.4, the Launder and Spalding k ε− model (Figure 3(c)) shows well-ordered oscillations while the filter-based model produces a multi-timescale behavior in time. Thus, the time-dependent results at both cavitation numbers are sensitive to turbulence modeling. The time-averaged liquid phase fractions in Figure 5, besides registering the sensitivity of pressure distribution to filter-based model; are in concert with the data in Figure 2. As mentioned before, the observations with the new turbulence model can largely be ascribed to the production of lower viscosity. Figure 6 portrays that the viscosity produced by the filter-based model is lower by an order of magnitude than that produced by the Lauder and Spalding model. Indeed, this lower

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viscosity prevents decay of the instabilities as confirmed by Figures 3 and 4. A recent study reported that the present filter-based turbulence model can improve the predictive capability for a single phase flow [11]. However, lack of experimental data on the current hydrofoil case restricts further appraisal of the model performance. Greater insights on the above time-dependent phenomenon can be extracted by the technique of POD. Since mathematical documentation and detailed formulation of POD is available from past studies [14-15], only the main points are highlighted below. POD essentially yields key flow features of the flow from an ensemble of data (time-dependent data in our case). Thus, useful information of the flow field is optimally unfolded in order of the associated energy content in each mode. This facilitates accurate assessment of important issues through a reduced order model. Mathematically, each solution can be expressed as a linear combination of the resultant eigenmodes as shown below:

1

( , ) ( ) ( ) where ( ) ( , ) ( )N

n n n nn

u x t t x t u x t xφ ψ φ ψ=

= = � �� (16)

The energy content of each eigenmode can be obtained by a simple eigenvalue analysis [14]. Figure 7 illustrates first four eigenmodes of velocity fields for the two turbulence models at σ = 0.8 with their respective energy content in the entire domain. Note that the two models produce fairly similar first three eigenmodes. The impact on the fourth mode is also limited. This can be explained by the time-dependent behavior observed in Figure 3 (b). At σ = 0.8, the dynamic behavior of both turbulence models is fairly characterized by a single dominant frequency but different amplitudes. This difference is likewise manifested into the POD modes. The solution for the two models, thus, shows consistent POD modes and the resultant difference is largely caused by their re-scaling. In comparison, Figure 8 shows that the filter-based model at σ = 0.4 produces noticeable differences into the second POD mode. The subsequent POD modes of the two turbulence models also show larger variations as compared to Figure 7. This variation in eigenmodes for the σ = 0.4 case is a result of the multi-scale dynamics observed for the filter-based model in Figure 3(c). It is thus worth noting that the response of POD modes to a particular model is closely related to its dynamic behavior. Figure 9 shows that the POD modes for the interfacial cavitation model are fairly consistent to those for the heuristic model (Figure 7) at σ = 0.8. Consequently, from above discussion it can be predicted that the dynamic behavior of the interfacial cavitation model (for σ = 0.8) differs from the heuristic model mainly in terms of the amplitude of oscillations. It is also expected that a lower cavitation number or use of filter-based model might impart a noticeable difference to the POD modes for the two cavitation models. SUMMARY AND CONCLUSIONS A pressure-based approach combined with a finite volume method has been employed to simulate cavitating flow around a hydrofoil. A heuristic cavitation model has been utilized for the simulations and its results are compared to those obtained from an alternate approach. A filter-based model has been explored

for the first time to simulate cavitating flows. Its results, in turn, have been compared to the original Launder and Spalding k ε− model. Proper Orthogonal Decomposition (POD) has been utilized to extract coherent flow structures to aid comparisons between different modeling strategies. Time-averaged surface pressures give a reasonable estimate of mean lift coefficient under various flow conditions. The drag force shows a restrained sensitivity to cavitation strength and turbulence modeling techniques mainly due to the low AOA and similarity in near wall treatment for the two models. The filter-based turbulence model significantly reduces the predicted eddy viscosity in comparison to the Launder and Spalding turbulence model. Consequently, the filter-based model yields larger pressure variations for the σ = 0.8 case. It should be noted that fairly constant instantaneous pressure is sustained within the cavity by both turbulence models. The time-averaged pressure distribution, however, illustrates noticeable pressure gradients for filter-based model inside the cavity. This indicates that mean pressure plots for the filter-based model can be misleading due to substantial cyclic variations. The time-averaged plots of density contours corroborate plots of other time-averaged variables. POD analysis yields useful correlation between time-dependent behavior and the salient flow structures (POD modes). In general, a dynamic behavior between the two models which largely differs by the amplitude of oscillations results in consistent eigenmodes that re-scale themselves in the solution (σ = 0.8). However, distinct POD modes are obtained if the two models induce different timescale patterns (σ = 0.4 case). From above discussion, POD may potentially be employed to predict the dynamic behavior of a model. Detailed POD investigation on the present case and validation of the filter-based model with additional experimental data shall be a part of our future endeavor.

ACKNOWLEDGMENTS The authors are grateful to NASA and Air Force for financial support.

REFERENCES 1 Kunz, R.F., Boger, D.A., Stinebring, D.R., Chyczewski, T.S.,

Lindau, J.W., Gibeling, H.J., Venkateswaran, S. and Govindan, T.R., “A Pre-conditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Prediction,” Computers & Fluids, vol. 29, pp. 849-875, 2000.

2 Ahuja, V., Hosangadi, A. and Arunajatesan, S., “Simulations of Cavitating Flows Using Hybrid Unstructured Meshes,” J. of Fluids Engineering, vol. 123, pp. 331-340, 2001.

3 Senocak, I. and Shyy, W., “A Pressure-Based Method for Turbulent Cavitating Flow Computations,” J. of Comp. Physics, vol. 176, pp. 363-383, 2002.

4 Wu, J., Senocak, I., Wang G., Wu, Y. and Shyy, W., “Three-Dimensional Simulation of Turbulent Cavitating Flows in a Hollow-Jet Valve,” Journal of Comp. Modeling in Eng. & Sci., accepted, 2002.

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5 Senocak, I. and Shyy, W., “Evaluation of Cavitation Models for Navier-Stokes Computations,” FEDSM2002-31011, Proc. of 2002 ASME Fluids Engineering Division Summer Meeting Montreal, CA, 2002

6 Senocak, I. and Shyy, W., “A Pressure-Based Method for Turbulent Cavitating Flow Computations,” J. of Comp. Physics, vol. 176, pp. 363-383, 2002

7 Vaidyanathan, R., Senocak, I., Wu, J. and Shyy, W., “Sensitivity Evaluation of a Transport Equation Based Turbulent Cavitation Model,” AIAA, Paper 2002-3184.

8 Senocak, I. and Shyy, W., “Computations of Unsteady Cavitation with a Pressure-based Method,” FEDSM2003-45009, Proc. of FEDSM'03, 4th ASME_JSME Joint Fluids Eng. Conference, Honolulu, Hawaii, 2003.

9 Wu, J., Utturkar, Y., Senocak, I., Shyy, W. and Arakere, N., “Impact of Turbulence and Compressibility Modeling on Three-Dimensional Cavitating Flow Computations,” AIAA, Paper 2003-4264.

10 Utturkar, Y., Shyy, W. and Arakere, N., “Proper Orthogonal Decomposition for Probing Turbulent Cavitating Flow Modeling,” Unpublished Technical Report, University of Florida, 2003.

11 Johansen, S., Wu, J. and Shyy, W., “Filter-based Unsteady RANS Computations,” Unpublished Technical Report, University of Florida, 2003.

12 Thakur, S., Wright, J. and Shyy, W., "STREAM: A Computational Fluid Dynamics and Heat Transfer Navier-Stokes Solver. Theory and Applications." Streamline Numerics, Inc., and Computational Thermo-Fluids Laboratory, Department of Mechanical and Aerospace Engineering Technical Report, Gainesville, Florida, 2002.

13 Launder, B.E. and Spalding, D.B., “The Numerical Computation of Turbulent Flows,” Comp. Meth. Appl. Mech Eng., vol. 3, pp. 269-289, 1974.

14 Ahlman, D., Soderlund, F., Jackson, J., Kurdilla, A. and Shyy, W., “Proper Orthogonal Decomposition for Time-Dependent Lid-driven Cavity Flows,” Numerical Heat Transfer, Part B, vol. 42, pp. 285-306, 2002.

15 Zhang, B., Lian, Y. and Shyy, W., “Proper Orthogonal Decomposition for Three-dimensional Membrane Wing Aerodynamics,” AIAA, Paper 2003-3917.

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Table 1 Summary of the cavitation averaged-parameters: Heuristic cavitation model

Cavitation number

Turbulence model

310dl × ( / )dl c

1max 10l ×

max( / )l c

2max 10t ×

max( / )t c

max 10tl ×

max( / )tl c 110

LavgC

×

310

DavgC

×

St maxf

( )l

V Launder-Spalding

7.46 6.01 6.50 4.52 5.21 8.24 0.09 #1

0.8σ = Filter 0.2C∆ =

7.36 6.57 7.54 4.86 4.58 7.49 0.08

Launder-Spalding

9.85 10.10 12.52 7.13 2.56 9.00 0.21 #2

0.4σ = Filter 0.2C∆ =

8.55 10.60 12.29 7.2 2.08 7.49 0.22

XY

-0.2 -0.1 0 0.1 0.2

-0.2

-0.1

0

0.1

0.2

60×70

80 × 60

100 × 6080× 60

100 × 60 70 × 60

Figure 1 Geometry sketch and grid (2-point skip in both x & y directions)

x/c

Cp

-0.5 -0.25 0 0.25

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

LSM, upper CpLSM, lower CpFBM, upper Cp

FBM, lower Cp

x/c

Cp

-0.25 0 0.25-1

-0.5

0

0.5

1

1.5

LSM, upper Cp

LSM, lower CpFBM, upper Cp

FBM, lower Cp

x/c

Cp

-0.25 0 0.25-1

-0.5

0

0.5

1

1.5

LSM, upper Cp

LSM, lower CpFBM, upper Cp

FBM, lower Cp

(a) Heuristic cavitation model: No cavitation (b) Heuristic cavitation model: 0.8σ = (c) Heuristic cavitation model: 0.4σ = Figure 2 Time-averaged pressure coefficients Cp of different turbulence models. LSM: Launder & Spalding, FBM: Filter-based model

t

CL,

CD

0 0.05 0.1 0.15 0.2

0

0.02

0.04

0.06

0.08LSM, CL

LSM, CDFBM, CL

FBM, CD

t

CL,

CD

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.02

0.04

0.06

0.08

LSM, CLLSM, CD

FBM, CLFBM, CD

t

CL,

CD

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.02

0.04

0.06

0.08 LSM, CLLSM, CD

FBM, CLFBM, CD

(a) Heuristic cavitation model: No cavitation (b) Heuristic cavitation model: 0.8σ = (c) Heuristic cavitation model: 0.4σ = Figure 3 Lift and drag coefficients CL & CD of different turbulence models. LSM: Launder & Spalding, FBM: Filter-based model

7

x/c

Cp

-0.25 0 0.25

-0.5

0

0.5

1

1.5

Max CL, upper Cp

Max CL, lower Cp

Min CL, upper Cp

Min CL, lower Cp

x/c

Cp

-0.25 0 0.25-1

-0.5

0

0.5

1

1.5

Max CL, upper CpMax CL, lower CpMin CL, upper Cp

Min CL, lower Cp

(a) Launder & Spalding model (b) Filter-based model

Figure 4 Instantaneous Cp at maximum and minimum C� of different turbulence models. Heuristic cavitation model: 0.8σ =

X

Y

-0.05 0 0.05

-0.04

-0.02

0

0.02

0.04

0.90.80.70.60.50.40.30.20.1

X

Y

-0.05 0 0.05

-0.04

-0.02

0

0.02

0.04

0.90.80.70.60.50.40.30.20.1

X

Y

-0.05 0 0.05

-0.04

-0.02

0

0.02

0.04

0.90.80.70.60.50.40.30.20.1

X

Y

-0.05 0 0.05-0.06

-0.04

-0.02

0

0.02

0.04

0.90.80.70.60.50.40.30.20.1

(a) 0.8σ = : Launder-Spalding (Left)&Filter-based (Right) (b) 0.4σ = : Launder-Spalding (Left)&Filter-based (Right) Figure 5 Time-averaged liquid volume fractions of different turbulence models, Heuristic cavitation model

X

Y

-0.2 -0.1 0 0.1 0.2

-0.1

0

0.1

0.216.096215.023213.950212.877211.804210.73119.658138.585117.51216.439085.366074.293063.220042.147031.07402

X

Y

-0.2 -0.1 0 0.1 0.2

-0.1

0

0.1

0.216.096215.023213.950212.877211.804210.73119.658138.585117.51216.439085.366074.293063.220042.147031.07402

(a) Launder-Spalding model: maximum CL (Left) and Minimum CL (Right)

X

Y

-0.05 0 0.05 0.1

-0.05

0

0.05

1.65541.54511.43481.32451.21421.10390.99360.88330.77300.66280.55250.44220.33190.22160.1113

X

Y

-0.05 0 0.05 0.1

-0.05

0

0.05

1.65541.54511.43481.32451.21421.10390.99360.88330.77300.66280.55250.44220.33190.22160.1113

(b) Filter-based model: maximum CL (Left) and Minimum CL (Right)

Figure 6 Instantaneous viscosity at maximum and minimum CL of different turbulence models. Heuristic cavitation model: 0.8σ =

8

99.83% of total energy 45.07% of fluctuating energy 72.20% of fluctuating energy 81.79% of fluctuating energy

99.57% of total energy 50.21% of fluctuating energy 78.85% of fluctuating energy 83.67% of fluctuating energy

Figure 7 POD modes showing difference to the flow structure imparted by turbulence modeling at σ=0.8 HM: Heuristic cavitation model; LSM: Launder and Spalding model; FBM: Filter-based model

99.87% of total energy 39.88% of fluctuating energy 60.40% of fluctuating energy 71.66% of fluctuating energy

99.77% of total energy 22.39% of fluctuating energy 39.33% of fluctuating energy 55.11% of fluctuating energy

Figure 8 POD modes showing difference to the flow structure imparted by turbulence modeling at σ=0.4

HM: Heuristic cavitation model; LSM: Launder and Spalding model; FBM: Filter-based model

99.99% of total energy 49.59% of fluctuating energy 71.57% of fluctuating energy 84.02% of fluctuating energy

Figure 9 POD modes showing of streamlines for σ=0.8, with Interfacial cavitation model and Launder and Spalding model.

Streamlines; σ=0.8; HM; FBM

Streamlines; σ=0.4; HM; FBM

Streamlines; σ=0.4; HM; LSM

Streamlines; σ=0.8; HM; LSM