Numerical investigation of the ventilated cavitating flow around an under-water vehicle based on a three-component cavitation model

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2010,22(6):753-759 DOI: 10.1016/S1001-6058(09)60113-X

NUMERICAL INVESTIGATION OF THE VENTILATED CAVITATING FLOW AROUND AN UNDER-WATER VEHICLE BASED ON A THREE-COMPONENT CAVITATION MODEL * JI Bin, LUO Xian-wu State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China, E-mail: [email protected] PENG Xiao-xing China Ship Scientific Research Center, Wuxi 214082, China ZHANG Yao, WU Yu-lin, XU Hong-yuan State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China

(Received June 4, 2010, Revised July 26, 2010)

Abstract: Based on the Reynolds-Averaged Navier-Stokes equations and mass transfer model, an approach, where a three-component cavitation model is proposed, is presented to simulate ventilated cavitating flow as well as natural cavitation. In the proposed cavitation model, the initial content of nucleus in the local flow field is updated instantaneously, and is coupled with the Rayleigh-Plesset equation to capture the cavity development. The proposed model is applied to simulate the cavitating flow around an under-water vehicle in different cavitation conditions. The results indicate that for the natural and ventilated cavitation simulation, the predicted cavitation characteristics including the cavity length, cavity diameter and cavity shape agree satisfactorily with the analytic and experimental results, for the ventilated cavitation, the proposed methods reproduce the special behavior that the axial line of the cavity bends and rises at the tail part. The study concludes that the ventilated flow rate of the non-condensate gas influences the development of natural cavitation as well as ventilated cavitation, and the vapor cavity is suppressed remarkably by the gas cavity with the increase of the gas ventilation. Key words: under-water vehicle, natural cavitation, ventilated cavitation, three-component model

1. Introduction  Engineering interests in natural and ventilated cavitation around under-water vehicles have attracted people for modeling the large scale cavity for decades. In this application, the proximity of local pressure to the vapor pressure of the liquid can bring about natural cavitation. At a cavitation number lower than 0.1, natural super-cavitation occurs, and the drag force acting on the vehicles is much smaller than that in the usual condition. There are different ways to achieve the

* Project supported by the National Natural Science Foundation of China (Grant Nos. 50976061, 50676044) the Natural Science Foundation of Beijing (Grant No. 3072008). Biogranphy: JI Bin (1982-), Male, Ph. D. Candidate

super-cavitation[1]: (1) by accelerating the incoming flow velocity vf , for example to make it higher than 45 m/s at sea level, (2) by reducing the ambient pressure pf , which is only feasible in closed-circuit water tunnels, or (3) by increasing the cavity pressure pc through the ventilation of a cavity with some non-condensable gas. The super-cavitation induced by the first and the second ways is referred as vaporous or natural super-cavitation, and that by the third way is ventilated super-cavitation. Thus, it is possible to obtain artificial cavities globally similar to the vapor cavities generated by natural cavitation by injecting gas into the low pressure regions of liquid flows. There are two main differences between the artificial and natural super-cavity[2]. Firstly, the non-condensable characteristics of the gas cause

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different behaviors in the aft part of the cavity. This furthermore depends upon the ambient pressure due to the effects of gas compressibility, secondly, gravitational effects can be expected, as large cavities are obtained even for small velocities. One key parameter in all of these is the gas flow-rate ventilated into the cavity in order to obtain a desirable cavity length. Besides the ventilated hydrofoils[3,4] or propellers, and super-cavitating vehicle[5-7], the ventilation is also used to alleviate violent pressure fluctuations in a draft tube of hydraulic turbines. To perform such applications successfully, it is essential that those cavitating flows are reasonably predicted by the numerical simulation. In traditional cavitation models, the assumption of homogenous equilibrium medium, where the possible slip between liquid and vapor interface is neglected and the liquid-vapor mixture is treated as a single fluid that satisfies Navier-Stokes equation, is applied. The key challenge for this kind of models is how to define the mixed density of the single fluid. One approach is based on the state equation[8,9], and another method is solving the additional transport equation by introducing the source term to express the mass transfer between vapor and liquid[10-13]. In order to explain the origin of cavitation, non-condensable gas (or cavitation nuclei) is assumed to be premixed with liquid, and the fraction of the gas remains uniform and constant in the whole calculation domain (such as Singhal et al.[12]). However, the assumption of a constant gas fraction is not suitable to model the ventilated cavitation due to the gas ventilation during the calculation. In this article, a three-component model is proposed with the consideration of gas ventilation for simulating natural and ventilated cavitation around an under-water vehicle. Based on those results presented in the literatures, the cavitating flows in different operation conditions are analyzed. 2. Governing equations and cavitation model For the proposed model, the fluid is assumed to be a mixture of liquid, vapor and non-condensable gas. The flow is assumed to be homogeneous, so that the multiphase fluid components are assumed to share the same velocity and pressure. The continuity and the momentum equations for the mixture flow are as follows

wU w ( U u j ) + =0 wt wx j

(1)

w  U ui w  U ui u j wp w + = U fi  + g wt wx j wxi wx j ª § wu wu j 2 wuk · º  G ij ¸ » « P + Pt ¨¨ i + ¸ «¬ © wx j wxi 3 wxk ¹ »¼

(2)

where ui and fi are the velocity and body force in the i direction, respectively, p is the mixture pressure, P and Pt are laminar viscosity and turbulent viscosity. The mixture density is defined by

U = D v Uv + D g U g + 1  D v  D g Ul

(3)

where D is the volume fraction of one component. The subscripts v , g and l refer to the components of the vapor, non-condensable gas and liquid respectively. A cavitation process is governed by the mass transfer equations. Equation (4) gives the conservation equation of vapor volume fraction, and Eq.(5) shows the conservation equation of gas volume fraction. Note that the source term m + and m  in Eq.(4) represent the effect for the evaporation and condensation during phase transition.

w ( UvD v ) w ( UvD v u j ) § ī ’U D · + = ’<¨ v v v ¸ + U wt wx j © ¹

m +  m 



w U gD g wt

(4)

+ w  U D u = ’<§ ī ’U D g

wx j

g

j

¨ ©

g

g

U

g

· ¸ ¹

(5)

where * is the diffusion coefficient. According to the Rayleigh-Plesset equation, the size change for a single vapor/gas bubble is assumed to be driven by the pressure difference between the local static pressure p and the vapor pressure pv . By ignoring the second-order derivative of bubble radius, which is dominant only for rapid bubble acceleration, the Rayleigh-Plesset equation can be written in the following form

dR 2 pv  p = dt 3 Ul

(6)

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where R denotes the radius of the spherical bubble wall. The number of bubbles per unit volume, i.e., Nb , is dependent on the direction of phase transition[14]. During the bubble growth, i.e., vaporization, Nb is given by

N b = 1  D v  D g

3D g

(7)

4ʌR 3

During the condensation, Nb can be calculated by

Nb =

3D v 4ʌR3

(8)

16 mm ( 1.6 Dn ), and the body length of 107 mm ( 10.7 Dn ) as specified by Travis[16]. In order to save the computational resource, a half cylindrical zone including the vehicle has been selected as the calculation domain. The domain is 100 Dn in length, and 10 Dn for the radius of the semi cylinder. The inlet of the domain is 30 Dn upstream from the test

The total mass transfer rate in a unit volume is

§ U 4ʌR · d¨ v ¸ 3 ¹ dR © m = N b = 4ʌN b U v R 2 dt dt

3. Computational domain and boundary conditions For model application, the cavitation over a super-cavitating vehicle has been simulated. Figure 1 shows the computational domain including an under-water vehicle marked as “test body”. The vehicle has the following geometrical parameters: head diameter Dn of 10 mm, the body diameter of

body, while the outlet of the domain is 70 Dn downstream from the test body.

3

(9)

Combining Eqs.(6)-(9), the source terms in Eq.(4) for the vaporization and condensation can be expressed as

m + = Ce

3U v (1  D v  D g ) max(D g , D nuc ) Rb

Fig.1 Computational domain

g



2 max(pv  p,0) 3 Ul m  = Cc

3U vD v Rb

2 max(p  pv ,0) 3 Ul

(10)

(11)

where Ce and Cc are empirical coefficients for different phase transition process, and their values may be 50 and 0.01, as recommended by Zwart et al.[15]. D nuc is the fractions of non-condensable gas in liquid, and its value is around 5×10í4 for most practical cases. In usual cases, the typical bubble size Rb is 1×10í6 m in water. When the ventilated cavitating flow is treated, the volume fractions of the vapor and gas are calculated by solving the mass transfer equations such as Eqs.(4) and (5) combined with Eqs.(10) and (11). The gas ventilation in the computation domain is used for the boundary condition during the equation solution. In a word, the content of the nucleus in the local flow field given by Eq.(10) is updated instantaneously.

For ensuring computation accuracy, the mesh generation has been conducted carefully. In total domain, the structural grids having the node number of 455 034 are formed. Figure 2 shows the mesh near the cavitator and gas deflector. Since the interaction between the near-wall flow and cavity should be taken into consideration, the mesh near the wall of test body is well refined so as to ensure the non-dimensional normal distance from the wall i.e., y + . From Fig.3, it is noted that the value of y + at the wall surface of test body is smaller than 100.

Fig.2 Structural grids generated near the cavitator and gas deflector

The boundary conditions are as follows: (1) At the domain inlet, the uniform velocity vf

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is set. (2) At the domain outlet, an averaged static pressure pf is set. (3) The middle plane of the cylinder is set as the symmetry boundary. (4) The non-slip condition is applied to the surface of test body, while the free-slip wall condition is adopted to the out wall of the domain. Three-dimensional turbulent cavitating flow in the domain has been calculated under natural cavitation and ventilated cavitation conditions. The k  Z SST turbulence model is used. For convenience, the solver of a commercial CFD code CFX coupled with the proposed cavitation model inserted by a user defined file has been applied for the calculation.

In Fig.4, the natural cavities around the vehicle at six kinds of the cavitation number from 0.1 to 0.5 are displayed. Since the location of cavity closure is difficult to measure due to the unsteadiness of cavitating flow, the cavity length Lc is usually defined as twice of the distance from the leading edge of cavity to the location with the maximum cavity diameter of Dc , which is illustrated in Fig.5. In this study, Dc and Lc were determined by examining the contour of the vapor volume fraction D v of 0.1.

Fig.5 Schematic of axisymmetric super-cavity over a disk cavitator

Fig.3 Distribution of y + at the vehicle wall surface

4. Results and discussions 4.1 Natural cavity In order to specify the cavitation condition of the flow, a natural cavitation number, V v , is defined as

Vv =

pf  pv U vf 2 2

Fig.6 Comparisons of numerical results with analytic solution of cavity shape

Based on the asymptotic solution for main dimensions of the super cavity past a disk cavitator, the empirical formulae for the dimensionless cavity length Lc / Dn and maximum cavity diameter

Dc / Dn at a small cavitation number are derived and given by the following equations[17]

Fig.4 Numerical results of natural cavity at different cavitation number

Lc 1+ V v ln 1 = 0.91 Vv Vv Dn

(12)

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Dc 1+ V v = 0.91 Vv Dn

(13)

qg = Qg /(vf Dn2 ) , where

Qg

is the volumetric

flow-rate of ventilated gas.

The validity of Eqs.(12) and (13) for under-water bodies with a disc cavitator has been confirmed extensively[17,18]. From the results in Fig.4, it is seen that the cavity grows gradually from the cavitation inception at V v = 0.5 , to partial cavitation at

V v = 0.3, 0.2 and 0.15, and finally to supe- cavitation at V v = 0.1 . Figure 6 shows the comparison of dimensionless values of Lc / Dn and Dc / Dn between the numerical simulation and the solution from Eqs.(12) and (13). A good agreement is achieved in both cases.

Fig.8 Comparison of experimental and numerical results for ventilated cavity contour

Fig.7 Cavity comparison between the calculation and experiment at different gas ventilations ( V v = 1.0 )

4.2 Ventilated cavity simulation In the case of gas ventilated cavitation, the cavity changes with the amount of gas ventilation quantified by the dimensionless air entrainment coefficient

Figures 7 and 8 show the cavity with the comparison between the calculation and experiment for different gas ventilations at V v = 1.0 . In Fig.7, the left pictures are experimental photos of Travis[16], and the right ones are calculated distributions of gas volume fraction over the test body. Note that the legend of volume fraction for the calculation is the same as that in Fig.4, even in case that the cavity contents are different. In order to evaluate the comparison quantitatively, Fig.8 gives the comparison of cavity radius ( R ) in several profiles. Here the cavity shape by calculation is illustrated by the contour of gas volume fraction D g of 0.1, while for the experimental case, the shape is obtained from the interface between gas and liquid by image analysis. It

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is noted that the rear part of cavity was rather blurry and it is very difficult to distinguish this interface. Because the present natural cavitation number is much larger than the critical value of nature cavitation inception, the minimum static pressure in the flow field is larger than vapor pressure. Thus, there is no vapor in the cavity. Based on Figs.7 and 8, the following conclusions can be reached. (1) From the general characteristics of the cavitating flow such as cavity radius in several profiles, etc., the calculation results agree satisfactorily with the experimental results, though the supporting strut to fix the test body has not been included in the calculation to simplify the problem. (2) The thickness and length of the cavity over the vehicle increase with the increase of q g . The test

the test body tends to shift upward, and wedge into the cavity along the vehicle wall. This phenomenon is believed to be resulted from the gravity effect, and related to the gas leakage of the ventilated cavitation[2].

body has been wrapped totally by the surrounding cavity at qg = 0.32 . (3) The numerical simulation excellently reproduces the special behavior that the axial line of the cavity bends and rises in the tail part.

Fig.10 Vapor volume fraction (marked by “vapor”) and gas volume fraction (marked by “gas”) at different gas ventilated rate ( V v = 0.3 , the same legend as Fig.4)

Fig.9 Velocity vector around the vehicle ( qg = 0.32 )

The velocity vectors near the vehicle are shown at Fig.9, where the cavity shape is illustrated by the gas volume fraction D g of 0.1. Even though there are circulating flow and back flow downstream the disk cavitator and deflector, the main flow becomes more uniform due to the gas ventilation. The velocity field illustrates that the flow is not symmetric along the vehicle. In the rear part of the vehicle, the flow under

4.3 Mixed cavity simulation Figure 10 illustrates the cavitation around the vehicle associated with a gas ventilated rate at partial natural cavitation of V v = 0.3 . Note that the pictures marked by vapor show the vapor distributions, and those marked by gas show the distributions of the gas cavity. Without the ventilated gas, i.e., qg = 0 , there is a natural cavitation originated from the cavitator disk, and no gas cavity is found as shown in Fig.10(a). At q g of 0.04, the vapor bubble is suppressed to some extent, and there occurs obvious gas cavity just

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downstream of the disc as shown in Fig.10(b). With the increase of the gas ventilated rate from qg = 0.08 to qg = 0.16 , the vapor cavity area becomes much smaller, and disappears at last, while the gas volume fraction grows quickly in size, and covers the whole length of the vehicle body as shown in Fig.10(d). The numerical results indicate that the proposed cavitation model can predict the mixed cavitation reasonably, and the ventilated gas may have the potential to suppress the natural cavitation. 5. Conclusions A three-component model based on mass transfer equation has been proposed to simulate both the natural and ventilated cavitation. In the proposed cavitation model, the initial content of nucleus in the local flow field is updated instantaneously, and is coupled with the Rayleigh-Plesset equation so as to capture the cavity development. The proposed model has been applied to simulate the cavitating flow around an under-water vehicle in different cavitation condition. Based on the results, the following conclusions can be drawn: (1) For the natural and ventilated cavitation simulation, the predicted cavitation characteristics including the cavity length, cavity diameter and cavity shape agrees satisfactorily with the analytic and experimental results. (2) For the ventilated cavitation, the proposed methods reproduce reasonably the special behavior that the axial line of the cavity bends and rises at the tail part. (3) The ventilated flow rate of the non-condensate gas influences the development of natural cavitation as well as ventilated cavitation. With the increase of the gas ventilation, the vapor cavity is suppressed by the gas cavity remarkably. References [1] WOSNIK M., ARNDT R. E. A. Measurements in high void-fraction bubbly wakes created by ventilated supercavitation[C]. Proceedings of 6th International Symposium on Cavitation. Wageningen, The Netherlands, 2006. [2] FRANC J., MICHEL J. Fundamentals of cavitation[M]. Dordrecht, The Netherlands: Springer, 2005, 193-221. [3] KOPRIVA J., ARNDT R. E. A. and AMROMIN E. Improvement of hydrofoil performance by partial ventilated cavitation in steady flow and periodic gusts[J]. Journal of Fluids Engineering, 2008, 130(3): 31301.

[4] AMROMIN E., KOPRIVA J. and ARNDT R. E. A. et al. Hydrofoil drag reduction by partial cavitation[J]. Journal of Fluids Engineering, 2006, 128(5): 931-936. [5] CHEN Xin, LU Chuan-jing and LI Jie et al. The wall effect on ventilated cavitating flows in closed cavitation tunnels[J]. Journal of Hydrodynamics, 2008, 20(5): 561-566. [6] LEE Qi-tao, XUE Lei-ping and HE You-sheng. Experimental study of ventilated supercavities with a dynamics pitching model[J]. Journal of Hydrodynamics, 2008, 20(4): 456-460. [7] LI Jie, LU Chuan-jing and HUANG Xuan. Calculation of added mass of a vehicle running with cavity[J]. Journal of Hydrodynamics, 2010, 22(3): 312-318. [8] COUTIER-DELGOSHA O., REBOUD J. L. and DELANNOY Y. Numerical simulation of the unsteady behaviour of cavitating flows[J]. International Journal for Numerical Methods in Fluids, 2003, 42(5): 527-548. [9] WANG G., OSTOJA S. M. Large eddy simulation of sheet/cloud cavitation on a NACA0015 hydrofoil[J]. Applied Mathematical Modeling, 2007, 31(3): 417-447. [10] LUO Xian-wu, ZHANG Yao and PENG Jun-qi et al. Impeller inlet geometry effect on performance improvement for centrifugal pumps[J]. Journal of Mechanical Science and Technology, 2008, 22(10): 1971-1976. [11] SENOCAK I., SHYY W. Interfacial dynamics-based modelling of turbulent cavitating flows, Part-1: Model development and steady-state computations[J]. International Journal for Numerical Methods in Fluids, 2004, 44(9): 975-995. [12] SINGHAL A. K., ATHAVALE M. and LI H. et al. Mathematical basis and validation of the full cavitation model[J]. Journal of Fluids Engineering, 2002, 124(3): 617-624. [13] KUNZ R. F., BOGER D. A. and STINEBRING D. R. et al. A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction[J]. Computers and Fluids, 2000, 29(8): 849-875. [14] MEJRI I., BAKIR F. and REY R. et al. Comparison of computational results obtained from a homogeneous cavitation model with experimental investigations of three inducers[J]. Journal of Fluids Engineering, 2006, 128(6): 1308-1323. [15] ZWART P. J., GERBER A. G. and BELAMRI T. A Two-phase flow model for predicting cavitation dynamics[C]. Proceedings of International Conference on Multiphase Flow. Yokohama, Japan, 2004. [16] TRAVIS J. An experimental study of a ventilated supercavitating vehicle[D]. Master Thesis, Minnesota, USA: University of Minnesota, 2003. [17] CHEN Ying, LU Chuan-jing. A homogenousequilibrium-model based numerical code for cavitation flows and evaluation by computation cases[J]. Journal of Hydrodynamics, 2008, 20(2): 186-194. [18] JIA Li-ping, WANG Cong and WEI Ying-jie et al. Numerical simulation of artificial ventilated cavity[J]. Journal of Hydrodynamics, Ser. B, 2006, 18(3): 273-279.