1

Impact of Turbulence and Compressibility Modeling on Three-Dimensional Cavitating Flow Computations

Jiongyang Wu1, Yogen Utturkar1, Inanc Senocak2, Wei Shyy3 and Nagaraj Arakere4

Department of Mechanical and Aerospace Engineering University of Florida, Gainesville, FL, U.S.A

Abstract

The large density ratio, up to 1000 in water, between liquid and vapor, turbulence with complicated interface dynamics and fast and multiple time scales make the computation of cavitating flows difficult. Utilizing a pressure-based algorithm extended for such flows, the present study focuses on the non-equilibrium and non-stationary aspects in the turbulence model, and the compressibility effects in the cavitation model. Assessment of several modeling concepts has been made in the context of the Favre-averaged Navier-Stokes equations, along with a transport equation-based cavitation model and the ε−k two-equation turbulence model. The three-dimensional turbulent cavitating flow in a hollow-jet valve is adopted as the physical focus. While the non-equilibrium and non-stationary modification to the turbulence closures do not seem to influence the qualitative characteristics of the simulation, the compressibility modeling can cause the result to vary substantially.

1 Introduction Cavitation appears in a wide variety of fluid

machinery like pumps, inducers, turbine blades, propellers and immersed hydrofoils. Pressure fluctuations, noise, power loss, vibrations and erosion are some of the pronounced effects of the cavitation phenomenon. It thus becomes imperative to gain insight into this phenomenon, and as a result, has been a topic of interest for many researchers. Athavale et al. [1] adopted a vapor transport equation for the vapor phase coupled with the turbulent N-S equations and reduced Rayleigh-Plesset equations to study of cavitation in pumps and inducers. To simulate the cavitating flows in rocket turbopump elements, Athavale and Singhal [2] presented a homogeneous two-phase approach with a transport equation for vapor, again using the reduced Rayleigh-Plesset equations for bubble dynamics and phase change rates. Medvitz et al. [3] used multi-phase computations to analyze the performance of centrifugal pumps under cavitating conditions. Duttweiler & Brennen [4] experimentally investigated a previously unrecognized instability phenomenon in cavitating flow in a water tunnel with a propeller. The cavitation on the blades and in the tip vortices was investigated through visual observation. A comprehensive review on the topic has been presented by Wang et al. [5].

One of the main computational difficulties for

cavitation is the large density ratio between the liquid and the vapor phases. Moreover, cavitating flows are usually turbulent and the interfacial dynamics involves complex interactions between vapor and liquid phases. The fast time scales associated with turbulent cavitation also pose substantial challenges to experimental observations. Multi-scale modeling and computational treatments typically utilize techniques such as averaging, matching, patching, or asymptotes. Detailed description on multi-scale problems in different physical and modeling contexts can be found in several references [6-11].

The present study focuses on the cavitation phenomenon in flow-control valves due to its serious implications on their safe and sound operation. A limited number of experimental studies have been conducted on this topic, such as those by Oba et al. [12] and Tani et al. [13, 14]. Wang [15] used high-speed cameras and Laser Light Sheet (LLS) to observe the cavitation behavior in a hollow-jet valve under various cavitation conditions and for different valve openings. However, complex geometries and inaccessible regions of occurrence restrain experimental investigations in cavitation. Furthermore, to the best of our knowledge, to date, no comprehensive numerical study has been conducted on this topic.

To simulate turbulent cavitating flows, we adopt a transport equation model for the volume or mass fraction, which allows coexistence of the liquid and vapor phases. Several recent studies have employed modeling concepts and computational approaches based on the same approach. Singhal et al. [16] employed a mass fraction equation with pressure dependent source terms. Kunz et al. [17] utilized the artificial compressibility method with special focus on the preconditioning formulation. Ahuja et al. [18] developed an algorithm to account for the compressibility effects in context of the artificial compressibility methods with use of adaptive unstructured grids. Senocak and Shyy [19] introduced a pressure-based algorithm to correct the density fields, which in turn was coupled with the ε−k turbulence equations. Senocak and Shyy also [20] developed an interfacial-dynamics-based model and appraised different cavitation models for N-S computation. They found that these alternative models gave qualitatively comparable and satisfactory predictions for the pressure distributions but quantitatively produced differences in properties, such as phase fractions and pressure, in the closure region of the cavity.

In SIMPLE-type of the pressure-based methods [21, 22], the equations are typically solved successively by

1 Graduate Student Assistant, Student Member AIAA 2 Post-doctoral Scholar, Member AIAA 3 Professor and Dept. Chair, Fellow AIAA 4 Associate Professor

33rd AIAA Fluid Dynamics Conference and Exhibit23-26 June 2003, Orlando, Florida

AIAA 2003-4264

Copyright © 2003 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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employing iterations. In cavitating flow computations, the typical relaxation factors used in the iterative solution process are smaller than the ones used in single-phase flows, and hence smaller time steps are needed to study the cavitation dynamics. Issa [23] developed a method called the Pressure-Implicit with Splitting of Operator (PISO) for the solution of unsteady flows. The splitting of pressure and velocity makes the solution procedure sequential in time domain and enables the exact solution of the discretized equations to have a formal order of accuracy at each time step eliminating the need for severe under-relaxation in SIMPLE type algorithm. Bressloff [24] extended the PISO for high-speed flows by adopting the pressure-density coupling procedure in all-speed SIMPLE type of methods. Oliveira and Issa [25] followed up the previous PISO work to combine the temperature equation to simulate buoyancy-driven flows. Thakur and Wright [26] have developed approaches using curvilinear coordinates with suitability to all speeds. Senocak and Shyy [27] have further extended the PISO algorithm to enhance the coupling of cavitation and turbulence models and to handle the large density ratio associated with cavitation.

Based on our previous study [28], which investigated the overall cavitating flow structure and assessed the computational capability, we further apply pressure-based algorithm, with the turbulence closure achieved by the ε−k turbulence equations, combined with the Senocak and Shyy [20] cavitation model to simulate turbulent cavitating flows in a hollow-jet valve. The pressure-based operator splitting method is used for time-dependent computations. Our simulations employ different turbulence models, which consider the non-equilibrium effects of the turbulence, and, different pressure-density corrections related to the definition of speed of sound in liquid-vapor mixture [20, 27]. The numerical solutions and discussions are qualitatively assessed with the experimental observations of Wang [15]. While paucity of sufficient experimental data restricts us from making direct numerical comparisons, we certainly make pointed observations on the changes in flow time scales and overall structure due to compressibility and turbulence modeling variations.

2 Theoretical formulation

2.1 Governing equations The set of governing equations comprises the

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

0)(

=∂

∂+

∂∂

j

jmm

xu

tρρ (1)

)]32)([(

)()(

ijk

k

i

j

j

it

ji

j

jimim

xu

xu

xu

xxp

xuu

tu

δµµ

ρρ

∂∂−

∂∂

+∂∂+

∂∂+

∂∂−=

∂∂

+∂

∂

(2)

)()( −+ +=

∂∂

+∂

∂ mmxu

t j

jll��

αα (3)

The mixture density and turbulent viscosity are:

ερ

µαραρρ µ2

);1(kCm

tlvllm =−+= (4)

where, ll µρ , are liquid density and viscosity, lα is the liquid volume fraction. 2.2 Cavitation modeling

Physically, the cavitation process is governed by the thermodynamics and the 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 are adopted from Senocak and Shyy [20]. Surface tension and buoyancy effects are neglected considering the typical situation that the Weber and Froude numbers are large. The source terms adopted here are:

∞

+

∞

−

−−−−

=

−−−

=

tVVppm

tVVppm

vlnInv

vl

vlnInvv

vll

)()(),0(1()()(

),0(

2,,

2,,

ρρα

ρρραρ

)Max

Min

�

�

(5)

where,

9.0,1

1

,

,,

,

−=−

−=

∇∇

=⋅=

fVf

V

nnuV

nv

v

l

v

l

nI

l

lnv

ρρρρ

αα���

(6)

∞U is a characteristic velocity scale; here it is chosen as the inlet velocity. The time scale ∞t is defined as the ratio of the characteristic length to the reference velocity (D/U). The nominal density ratio ( lv ρρ ) is the ratio between thermodynamic values of density of vapor and liquid phases at the corresponding flow condition. f is found satisfactorily by numerical experimentation. 2.3 Turbulence modeling

For the system closure, the original ε−k turbulence model with wall functions is presented as follows:

3

( )( )

[( ) ]

m jm

j

tt m

j k j

u kkt x

kx x

ρρ

µρ ε µσ

∂∂ +∂ ∂

∂ ∂= Ρ − + +∂ ∂

(7)

2

1 2

( )( )

P [( ) ]

m jm

j

tt m

j j

ut x

C Ck k x xε ε

ε

ρ ερ ε

µε ε ερ µσ

∂∂ +∂ ∂

∂ ∂= − + +∂ ∂

(8)

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 ux x x

δτ τ ρ ν

∂∂ ∂′ ′ ′ ′= = − = − +∂ ∂ ∂

(9)

2.3.1 Equilibrium model The empirical coefficients originally proposed by Launder and Spalding [29], assuming local equilibrium between production and dissipation of turbulent kinetic energy, for model are as follows: 44.11 =εC , 92.12 =εC , 3.1=εσ ,

0.1=kσ . These parameters are called Model 1. 2.3.2 Non-equilibrium modifications Turbulence modeling is a major concern in numerical simulation of high Reynolds number flows. The coefficients in the original model do not respond the non-equilibrium effects when a turbulent flow is not in equilibrium [6]. There are several ways to count the non-equilibrium effect into the ε equation: (a) Stationary non-equilibrium model (Model 2) The basic idea behind the model is to employ a functional form for the dissipation coefficient 1Cε , which will add a locally dependent dynamic time scale to the turbulence [6]. The model is especially well suited for flows with non-equilibrium between production and dissipation, and tends towards the classical model when Pt ε= .

1P1.15 0.25 tCε ε

= + ⋅ , 90.12 =εC , 15.1=εσ , 8927.0=kσ .

(b) Non-stationary model (Model 3) The concept of the stationary model follows that of Younis [30]. The subtle difference between Model 2 and Model 3 lies in the fact that 1Cε and, as a result, the dynamic time scales in Model 3, unlike Model 2, are strongly coupled to the overall time scales in the flow.

)/||38.01(44.11 QtQkC

∂∂⋅⋅+×=

εε (10)

where, 2/)( 222 wvukQ +++= , 92.12 =εC , 3.1=εσ , 0.1=kσ .

The time dependent term in Eq. (10) is simply discretized using first order difference, with the current and previous time step values:

tQQ

tQ ttt

∆−=

∂∂ ∆−

(c) Non-stationary, non-equilibrium model (Model 4) Based on the concepts presented above, we also investigate the combined modeling approach.

1P(1.15 0.25 ) (1 0.38 | | / )t k QC Q

tε ε ε∂= + × + ⋅ ⋅∂

,

90.12 =εC , 15.1=εσ , 8927.0=kσ . 2.4 Non-dimensional parameters

Reynolds number, cavitation number and the pressure coefficient are defined as:

l

l DUµ

ρ ∞=Re ; 2

2

21 V

PP

l

v

ρσ −

= ; 2

2

21

∞

−=

U

PPClρ

p (11)

where, D is the diameter of the valve needle, V is the mean flow velocity at the valve section through the needle seal (tip), P2 is the outlet pressure, Pv is the saturated vapor pressure. The value of cavitation number σ is taken as that in Wang [15], which is 0.9.

3 Numerical method The present Navier-Stokes solver employs

pressure-based algorithms and the finite volume approach [6, 22]. The governing equations are solved on multi-block structured curvilinear grids. To represent the cavitation dynamics, a transport equation model is adopted along with the particular source terms suggested by Senocak and Shyy [20]. The pressure-based method described in Senocak and Shyy [19] is utilized for steady-state computations of turbulent cavitating flows. One of the key features of this method is to reformulate the pressure correction equation to exhibit a convective-diffusive nature.

For unsteady flow computations, PISO algorithm is used. In this algorithm the system of discretized equations are solved through predictor and corrector steps without the need for iteration. A detailed description of the algorithm for turbulent cavitating flow computations is documented in Senocak and Shyy [27]. 3.1 Speed of sound

Several recent studies [27, 31] have pointed out that the harmonic speed of sound in a two-phase mixture is significantly attenuated. As a result, multiphase flowfields are characterized by widely different flow regimes, such as incompressible in pure liquid phase, low Mach compressible in the pure vapor phase, and transonic or

4

supersonic in the mixture. Consequently, correct and accurate representation of the speed of sound is imperative.

In the present algorithm, the correction equation of density and pressure is as suggested by Senocak and Shyy [19]:

pppplp PCPC ′=′−=′ )1( αρ (12) where, C is an arbitrary constant which does not affect the final converged solution due to the pressure correction nature.

In the unsteady computations the relation between

ρC and the speed of sound is:

2

1)(cP

C s =∂∂= ρ

ρ (13)

In the current study, for the pressure-density coupling, there are two different ways [27] to define the speed of sound: The first definition, as seen below, is fairly an algebraic approximation where the value of C, which is merely chosen to yield faster convergence, has no physical significance.

SoS-1: )1(1)( 2 ls CcP

C αρρ −≈=

∂∂= (14)

The second 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 .

SoS-2: ||||

)(1)(11

112

−+

−+

−−

=∆∆≈=

∂∂=

ii

iis PPPcP

Cρρρρ

ξρ (15)

4 Results and discussions

As documented in Wang [15], Fig. 1 shows the geometry and the main configurations of the valve. A key component, the needle, is used to control the flow rate by moving through the X-axis. A cylindrical seal supports the needle. There are six struts called splitters supporting the cylinder. Fig. 2 illustrates the computational domain in selected planes according to the geometry. A multi-block structured curvilinear grid is adopted to facilitate the computation. Fig. 2(a) shows the configuration on the X-Y plane and Fig. 2(b) on the Y-Z plane. In this study, the splitter thickness is neglected and its shape is considered rectangular. The chosen Reynolds number is 5×105 and the cavitation number is 0.9, with the liquid-vapor density ratio

vl ρρ being 1000. The computations are conducted with a valve opening of 33% using non-dimensional equations. Firstly, the steady single-phase turbulent flow solution without cavitation effects is computed, which in turn is adopted as the initial condition for the time dependent computation.

Fig. 3 shows the variation in cavity shapes and location with time for Model 1. As observed from Fig. 3(a) and Fig. 3(b), the cavity oscillation for SoS-1 is distinguished by a significantly higher frequency and a moderately higher amplitude than for SoS-2. These observations, as also discussed later, are a precursor to the significant impact produced by the speed of sound definition on the flow time scales. The qualitative shape and location of the cavity for both SoS-1 and SoS-2 is, however, similar to the experimental observations (Fig. 3(c)). Senocak and Shyy [27] pointed out that SoS-2 performed better than SoS-1 in simulating the oscillatory behavior of the cavitating flow in the convergent-divergent nozzles over a wide range of cavitation numbers. They indicated that, unlike SoS-1, SoS-2, which closely mimics the speed of sound definition, successfully produced quasi-steady solution at high cavitation numbers and a periodic solution at low cavitation numbers. Lack of substantial experimental data prevents us from making direct numerical comparison at this stage. Various experimental studies have shown that besides oscillating, the cavities incept, grow, detach from the needle tip and get periodically transported downstream (Fig. 3(c)). The periodic detachment and inception of cavities is, however, difficult to capture through currently known turbulence and SoS models [27]. Consequently, fundamental understanding on the above modeling aspects, which is discussed in the following sections, is imperative to encounter the above limitation. The following discussions serve to probe the sensitivity of the various modeling concepts, guided by our qualitative, but incomplete, insight into the fluid physics.

Fig. 4 shows that the cavity shape and location, and oscillation frequency obtained by turbulence Model 2, 3 and 4 fairly resemble those yielded by Model 1. Fig. 5 further illustrates that Model 2 and Model 4 significantly reduce kinetic energy k and dissipation in the proximity of cavity as compared to Model 1. The reduction in k is higher than reduction of dissipation leading to a drop in local viscosity near the cavitation zone, as seen from Fig. 6. Despite these differences, it is interesting to note that the qualitative flow structure and time scales, as seen before, are not severally affected. A common observation from Fig. 5 and Fig. 6 is that effect of non-stationary term in the turbulence model (Model 3) is weaker than the effect of non-stationary term of Model 2. The qualitative similarities of between contour plots of Model 4, which is essentially a combination of Model 2 and 3, and Model 2 reinforce the above argument.

The implications of speed of sound (SoS) definition are quantified by a correlation study (Table 1 & Table 2) and spectral analysis (Fig. 8) on pressure-density time history at various nodes in cavitation vicinity. The Pearson’s correlation (r) between pressure and density is calculated as follows:

5

2 22 2

; , ( ) ( )

( )( )

X YXY

Nr X P YX Y

X YN N

ρ−

= = =

− −

∑ ∑∑

∑ ∑∑ ∑

(16)

Table 1 and Table 2 indicate that within similar variation limits of liquid volume fraction, SoS-1 has a broader range of pressure-density correlation coefficients than SoS-2. Furthermore, SoS-2 consistently exhibits much stronger pressure-density coefficients than SoS-1. The distinct effects produced by the speed-of-sound definition are further corroborated by the dramatic time scale differences observed from the time history and FFT plots of pressure and density in Fig. 7(b), Fig. 7(c), Fig. 8(a), Fig. 8(b) and Fig 8(c). SoS-1 plots clearly indicate dominance of a single high-frequency compressibility effect unlike SoS-2, which are characterized by a wider bandwidth for pressure and density, and a lower dominant frequency. It is worth noting here that though the plots in Fig. 7(b), 7(c) and 8(a) are plotted with a smaller time range, we have used a convincingly long time history for our analysis.

The influence of turbulence modeling on flow time scales can be quantitatively confirmed from Fig 7(d) and the FFT plots in Fig 9. These figures strengthen our previous argument that the effect of various turbulence models on the flow time scales is moderate as compared to that of SoS. Model 3 qualitatively exhibits a distinct spectral distribution as compared to other turbulence models. The FFT plots of turbulent kinetic energy (Fig. 10) also indicate a discernible spectrum for Model 3. However, the spectral plots in Fig. 9 and Fig. 10 for Model 4 are qualitatively similar to those of Model 2. This observation conforms to our earlier belief that non-stationary terms in the Model 4 have a stronger impact on the flow than the time-dependent terms. Nonetheless, it is worth noting that the influence of time dependent term, though insignificant compared to the non-stationary term, produces a conspicuous difference in the flow time scales merely by itself.

5 Summary and Conclusions

In the current study, unified pressure-based algorithm has been used to simulate the turbulent cavitating flow in a hollow-jet valve. The standard ε−k model has been modified to account for the non-equilibrium and non-stationary turbulent effects. Furthermore, two different Speed of Sound (compressibility) models have been investigated. The numerical results obtained by these aforesaid models have been compared with the experimental results and with each other to yield insights on their impact on the flow. Correlation study on pressure-density time history and FFT analysis on various flow parameters has been employed to quantify the differences caused by the numerical models.

The time dependent solutions are characterized by auto-oscillations of the cavity formed at the tip of the needle. The qualitative shape and location of the cavity yielded by SoS-1 and SoS-2 are similar to the experimental observations however; significant differences are observed in the flow time scales for the two cases. Weak pressure-density correlations, a relatively high dominative frequency and a low bandwidth are the key attributes to the pressure and density variations in SoS-1 case. The SoS-2 case demonstrates an exactly opposite behavior for pressure- density in the cavitation vicinity. This radical disparity reveals the severe impact of SoS definition on the flow scales.

The effects of the various turbulence models are, however, comparatively less striking. Though the various turbulence models induce noticeable differences in the kinetic energy (k) and turbulent viscosity distribution in the flow domain, the cumulative influence of these parameters on the overall flow structure and time scales is weak. It is worth mentioning that the non-equilibrium terms weigh more than the non-stationary terms, which is evident from the similarities in the computations for Model 2 and Model 4. The relatively insignificant non-stationary modification (Model 3) to the turbulence model, nevertheless, causes distinct alterations to the flow spectrum merely by its presence.

It is envisioned that comparing the order of magnitude of the non-equilibrium and non-stationary terms during the computations over a wide range of physical parameters can provide a rationale for their currently observed behavior. But more than the above, there is a need to seek alternative strategies beyond the two-equation turbulence closures, so that one can accurately predict the innate time scales in the cavitation zone with greater resolution. A detailed investigation on the behavior of local speed of sound with Mach number for various computational cases, and in turn its comparison with the known data can precisely appraise the applicability of SoS models. These investigations would advance us towards the ultimate art of simulating the periodic cavity detachment and are strongly recommended as a part of future research.

6 Acknowledgments

The present study has been supported in part by NASA and ONR.

References 1. Athavale, M.M., Li, H.Y., Jiang, Y. and Singhal, A.K.

(2000) “Application of the Full Cavitation Model to Pumps and Inducers,” paper of the 8th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISEOMAC-8).

2. Athavale, M.M and Singhal, A.K (2001) “Numerical Analysis of Cavitating Flows in Rocket Turbopump Elements,” AIAA-2001-3400.

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3. Medvitz, R.B., Kunz, R.F., Boger, D.A., Lindau, J.W. and Yocum, A.M. (2002) “Performance Analysis of cavitating Flow in Centrifugal Pumps Using Multiphase CFD,” J. Fluids Eng., vol.124, pp. 377-383.

4. Duttweiler, M.E. and Brennen, C.E. (2002) “Surge Instability on a Cavitating Propeller,” J. Fluid Mech., vol. 458, pp. 133-152.

5. Wang, G., Senocak, I., Shyy, W., Ikohagi, T. and Cao, S. (2001) “Dynamics of Attached Turbulent Cavitating Flows,” Progress in Aerospace Sciences, vol. 37, pp. 551-581.

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

7. Shyy, W., Liu, J. and Ouyang, H. (1997) “Multi-Resolution Computations for Fluid Flow and Heat/Mass Transfer,” in Advances in Numerical Heat Transfer, Minkowycz, W. J., and Sparrow, E. M. (editors), vol. 1, pp. 137-169, Taylor & Francis, Washington, DC.

8. Martin, A.R., Saltiel, C. and Shyy, W. (1998) “Frictional Losses and Convection Heat Transfer in Sparse, Periodic Cylinder Arrays in Cross Flow,” International Journal of Heat and Mass Transfer, vol. 41, pp. 2383-2397.

9. Shyy, W. (2002) “Multi-Scale Computational Heat Transfer with Moving Solidification Boundaries,” International Journal for Heat and Fluid Flow, vol. 23, pp. 278-287.

10. Garikipati, K.(2002) “A Variational Multiscale Method to Embed Micromechanical Surface Laws in the Macromechanical Continuum Formulation,” Computer Modeling in Engineering and Sciences, vol. 3, no. 2, pp. 175-184.

11. Wang, Y., Wang, Q. J., and Lin, C. (2002) “Mixed Lubrication of Coupled Journal-Thrust Bearing Systems,” Computer Modeling in Engineering and Sciences, vol. 3, no. 4, pp. 517-530.

12. Oba, R., Ito, Y. and Miyakura, H. (1985) “Cavitation Observation in a Jet-Flow Gate-Valve,” JSME Int. J., Series B, vol. 28, pp. 1036-1043.

13. Tani, K., Ito, Y., Oba, R., Yamada, M. and Onishi, Y. (1991) “High-Speed Stereo-Flow Observations around a Butterfly Valve,” Mem. Inst. Fluid Sci., Tohoku Univ., vol. 2, pp. 101-112.

14. Tani, K., Ito, Y., Oba, R. and Iwasaki, M. (1991) “Flow Visualization in the High Shear Flow on Cavitation Erosion around a Butterfly Valve,” Trans. JSME, vol. 57, pp. 1525-1529.

15. Wang, G. (1999) “A Study on Safety Assessment of a Hollow-Jet Valve Accompanied with Cavitation,” PhD Dissertation, Tohoku University, Sendai, Japan.

16. Singhal, A.K., Vaidya, N. and Leonard, A.D. (1997) “Multi-Dimensional Simulation of Cavitating Flows

Using a PDF Model for Phase Change,” ASME paper FEDSM97-3272, Proc. of 1997 ASME fluids Engineering Division Summer Meeting.

17. 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 Pre-conditioned Navier-Stokes Method for Two-Phase Flows with Application to Cavitation Prediction,” Computers & Fluids, vol. 29, pp. 849-875.

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

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

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

21. Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C.

22. Shyy, W. (1994) Computational Modeling for Fluid Flow and Interfacial Transport, Elsevier, Amsterdam, The Netherlands (Revised print 1997)

23. Issa, R.I. (1985) “Solution of the Implicitly Discretised Fluid Flow Equations by Operator-Splitting,” J. Comput. Phys., vol. 62, pp. 40-65.

24. Bressloff, N.W. (2001) “A Parallel Pressure Implicit Splitting of Operator Algorithm Applied to Flows at All Speed,” Int. J. Numer.. Meth. FL., vol. 36 (5), pp. 497-518.

25. Oliveira, P.J. and Issa, R.I. (2001) “An Improved PISO Algorithm for the Computation of Buoyancy-Driven Flows,” Numerical Heat Transfer, Part B, vol. 40, pp. 473-493.

26. Thakur, S.S. and Wright, J.F (2002) “An Operator-Splitting Algorithm for Unsteady Flows at All Speed in Complex Geometries,” to be published.

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

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

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

30. Personal communication: Bassam Younis. 31. Venkateswaran, S., Lindau, J.W., Kunz, R.F. and

Merkle C.L. (2002) “Computation of Multiphase Mixture Flows with Compressibility Effects,” Journal of Computational Physics, 180, 54-77.

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Table 1 Time averaged liquid volume fraction v/s pressure-density correlation at multiple points in vicinity of cavity (SoS-1, Model 1)

Time averaged liquid volume fraction (αl)

Pearson’s correlation (r) between pressure and density (SoS-1)

0.862 -0.674 0.882 -0.023 0.936 0.093 0.970 -0.045 0.978 -0.077 0.983 -0.096 0.986 -0.103 0.988 -0.116

Mean(r) = -0.13; SD(r) = 0.23

Table 2 Time averaged liquid volume fraction v/s pressure-density correlation at multiple points in vicinity of cavity (SoS-2, Model 1)

Time averaged liquid volume fraction (αl)

Pearson’s correlation (r) between pressure and density (SoS-2)

0.858 -0.733 0.868 -0.246 0.934 -0.246 0.968 -0.420 0.977 -0.421 0.982 -0.426 0.985 -0.412 0.987 -0.426

Mean(r) = -0.42; SD(r) = 0.15

(1) Splitter; (2) Cylinder; (3) Plunger; (4) Needle; (5) Needle seal overlay;

(6) Seal seat inlay; (7) Stroke; (8) Ventilation duct; (9) Gear

Figure 1 Geometry of the valve [15]

8

Inlet

Pipe centerValve

Upper boundary

outlet

Lower boundary

Splitter

X

Y

Z

Splittersection

Middle section

600

(a) Computational domain in X-Y plane (b) Computational domain in Y-Z plane

Figure 2 Computational domain & boundary conditions

3

4

5

5 0.994 0.983 0.962 0.91 0.8

34

5

5 0.994 0.983 0.962 0.91 0.8

(a) t*=0.32 (left) and t*= 0.77 (right), SoS-2, Model 1

3

54

5 0.994 0.983 0.962 0.91 0.8

5

4

5 0.994 0.983 0.962 0.91 0.8

(b) t*=5.92 (left) and t*=5.86 (right), SoS-1, Model 1

(c) Experiment observations: cavity at needle tip (left) & cavitation aspects around the needle (right) [15]

Figure 3 Middle section density contours at different time instances (the mixture of vapor and liquid inside the outer line forms the cavity)

Needle Tip

9

43

5

5 0.994 0.983 0.962 0.91 0.8

5

43

5 0.994 0.983 0.962 0.91 0.8

(a) t*=0.30 (left) and t*=0.72 (right), Model 2

3

4

5

5 0.994 0.983 0.962 0.91 0.8

4

5

3

5 0.994 0.983 0.962 0.91 0.8

(b) t*=0.28 (left) and t*=0.72 (right), Model 3

3

4

5

5 0.994 0.983 0.962 0.91 0.8

45

5 0.994 0.983 0.962 0.91 0.8

(c) t*=0.28 (left) and t*=0.7 (right), Model 4

Figure 4 Density distributions for turbulence Model 2, 3 and 4; SoS-2

10

0.40060.34720.29380.24040.18700.13350.08010.0267

32.027727.757423.487019.216614.946310.67596.40562.1352

(a) Kinetic energy (left) and dissipation (right) at t*=0. 32, Model 1 (original model)

0.28360.24570.20790.17010.13230.09450.05670.0189

18.789516.284213.778911.27378.76846.26323.75791.2527

(b) Kinetic energy (left) and dissipation (right) at t*=0.3, Model 2 (stationary non-equilibrium)

0.28050.24310.20570.16830.13090.09350.05610.0187

28.454324.660420.866517.072613.27879.48485.69091.8970

(c) Kinetic energy (left) and dissipation (right) at t*=0.28, Model 3 (non-stationary)

0.16920.14670.12410.10150.07900.05640.03380.0113

18.845716.332913.820211.30748.79476.28193.76921.2564

(d) Kinetic energy (left) and dissipation (right) at t*=0.28, Model 4 (non-stationary non-equilibrium)

Figure 5 Turbulent kinetic energy and dissipation contours at the biggest cavity size instance, SoS-2

11

4

1

63

2 4

5

X

Y

3 3.5

0.25

0.5

0.75

6 0.01365 0.01074 0.00763 0.00292 0.00191 0.0010

1 23

4 5

X

Y

2.8 3 3.2 3.4 3.6 3.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

6 0.01365 0.01074 0.00763 0.00292 0.00191 0.0010

(a) Model 1, t*=0.32 (b) Model 2, t*=0.3

3

1 2 34

1 1

3

X

Y

2.8 3 3.2 3.4 3.6 3.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

6 0.01365 0.01074 0.00763 0.00292 0.00191 0.0010

4

1 23

X

Y

2.8 3 3.2 3.4 3.6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

6 0.01365 0.01074 0.00763 0.00292 0.00191 0.0010

(c) Model 3, t*=0.28 (d) Model 4, t*=0.28

Figure 6 Viscosity contours for different turbulence models at the biggest cavity size instance, SoS-2

12

A B

C D

(a) Samples locations

t*

Cp

0 1 2

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

ABCD

t*

Cp

0 1 2

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

ABCD

(b) Original ε−k model (Model 1), SoS-1 (c) Original ε−k model (Model 1), SoS-2

t*

Cp

0 0.5 1 1.5-3

-2.5

-2

-1.5

Model 1Model 2Model 3Model 4

(d) Comparisons of different models at location c, SoS-2

Figure 7 Pressure time evolutions at selected locations in the flow domain

13

t/T

Cp

0 0.5 1 1.5 2 2.5-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

Sos-2

Sos-1

t/T

ρ/ρ 0

0 0.5 1 1.5 2 2.50.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

Sos-2

Sos-1

(a) Pressure and density time history at a point in the cavitation vicinity, Model 1

0 5 10 15 20 250

1

2

3

4

5

Frequency

Pres

sure

/Hz

0 5 10 15 20 25

0

0.05

0.1

0.15

Frequency

Den

sity

/Hz

(b) Spectral analysis on pressure and density at a point in the cavitation vicinity, SoS-1, Model-1

0 5 10 15 20 250

0.5

1

1.5

Frequency

Pres

sure

/Hz

0 5 10 15 20 25

0

0.01

0.02

0.03

Frequency

Den

sity

/Hz

(c) Spectral analysis on pressure and density at a point in the cavitation vicinity, SoS-2, Model-1

Figure 8 Figures showing time evolution and spectrum of pressure and density for the two SoS definitions

14

0 10 200

0.5

1

Frequency

Pres

sure

/Hz

0 10 200

0.05

Frequency

Den

sity/

Hz

(a) Spectral analysis on pressure and density, SoS-2, Model-1

0 10 200

0.5

1

Frequency

Pres

sure

/Hz

0 10 20

0

0.05

FrequencyD

ensi

ty/H

z

(b) Spectral analysis on pressure and density, SoS-2, Model-2

0 10 200

0.5

1

Frequency

Pres

sure

/Hz

0 10 200

0.05

Frequency

Den

sity

/Hz

(c) Spectral analysis on pressure and density, SoS-2, Model-3

0 10 200

0.5

1

Frequency

Pres

sure

/Hz

0 10 200

0.05

Frequency

Den

sity

/Hz

(d) Spectral analysis on pressure and density, SoS-2, Model-4

Figure 9 Figures showing spectral strength of pressure and density for various turbulence models at a point in the cavitation

vicinity

15

0 10 200

0.01

0.02

Frequency

Turb

. K.E

./Hz

0 10 200

0.01

0.02

Frequency

Turb

. K.E

./Hz

(a) Model 2 (b) Model 3

0 10 200

0.01

0.02

Frequency

Turb

. K.E

./Hz

(c) Model 4

Figure 10 Spectral strength of turbulent K.E at a point in the cavitation vicinity, SoS-2