Ye-jun GONG （龔也君）， Jie-min ZHAN （詹杰民）， Tian-zeng LI （李天贈(zèng)）

Department of Applied Mechanics and Engineering， Sun Yat-sen University， Guangzhou 510275， China，

E-mail: yejungong@126.com

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Numerical investigation of the effect of rotation on cavitating flows over axisymmetric cavitators*

Ye-jun GONG （龔也君）， Jie-min ZHAN （詹杰民）， Tian-zeng LI （李天贈(zèng)）

Department of Applied Mechanics and Engineering， Sun Yat-sen University， Guangzhou 510275， China，

E-mail: yejungong@126.com

The rotating axisymmetric cavitator is widely applied in underwater vehicles， and its rotational motion affects the cavitating flow over the cavitator. This study focuses on the effect of rotation on the flow structure in the cavity bubble. Unsteady 2-D/3-D numerical simulations of cavitating flows over axisymmetric cavitators are performed using the volume of fraction （VOF） method and the Sauer-Schnerr cavitation model. Firstly， the 2-D simulation of cavitating flow over a circular disk or a cone cavitator is carried out at various cavitation numbers （0.15， 0.175， 0.2， 0.225 and 0.25）. The simulated cavity lengths and drag coefficients are compared with the experimental data， the theoretical estimations and the published numerical results. Then the 3-D simulations of cavitating flows over the same axisymmetric cavitators with different rotating speeds are performed using the sliding mesh model（SMM）. The effect of rotation on the cavity shape and the internal flow structure is analyzed.

rotation cavitator， volume of fraction （VOF）， sliding mesh model

Introduction

The supercavitation technology is to use the cavitation effect for a high-speed underwater vehicle to create a gas bubble， which engulfs the whole body of the vehicle and greatly reduces its skin friction， and then the underwater vehicle can travel at a higher speed. The main applications of the supercavitation technology are in high-speed underwater vehicles， e.g.， the supercavitating propeller （Brave-class， Vosper， Portsmouth， England， 1958）， the supercavitating projectile（RAMICS， U.S. Navy Forces， 1994） and the supercavitating torpedo （Barracuda， Diehl BGT Defence，Berlin， German， 2004）. In recent years， the supercavitation technology attracts more and more interest［1，2］.

The supercavitation occurs when the cavitation numberis lower than 0.1. The cavitation number，， depends on the vapor pressure， the ambient pressure， the fluid densityand the free stream velocityThe supercavitation condition，， can be reached through: （1） increasing， （2） decreasingof a closed-circuit cavitation tunnel， or （3） increasingthrough ventilating noncondensable gases［1］. The third way is called the ventilated supercavitation as in contrast to the natural supercavitation.

In practical applications， the leading edge of the submerged body， i.e.， the “cavitator”， is designed to enhance the supercavitation. The shape of the cavitator is one of the main factors affecting the cavity shape［3］. Additionally， the rotation of the cavitator can also influence the cavity shape［4］. The rotating cavitator is commonly seen in industrial and military fields，and has attracted the attention of scientists for decades. The aim of this study is to investigate the effect of rotation on the cavitating flow structure inside the cavity bubble.

The supercavitation technology has been investigated by means of experimental measurements， semiempirical models and numerical simulations［5，6］. With the development of computational fluid dynamics（CFD） and computer technologies in recent years， the numerical method is used to visualize the internal flow structure inside the vapor cavity. The re-entrant jet in the cavity closure region forces the gas convection in the cavity bubble and induces the vortex flow， which is hard to be measured in laboratory. Obviously， when the cavitator is rotating with a high speed， the internal flow structure in the gas bubble is even more complex under the action of inertial forces. In this study， 3-D cavitating flows over a circular disk and a cone cavitator are numerically simulated to see how the rotational movement of the cavitator influences the flow structure of the cavitating flow.

In the early numerical investigations of the cavitating flow， empirical correlations are widely used［5］，and the flow was assumed to be inviscid［7］. However，the viscous effect on the shape of the cavity was observed experimentally. Furthermore， the linearized theory can not predict the vortex structure in the cavitating flow. In the Street’s research on the rotating cavitation，the vorticity is assumed to be constant， and the stream function of the rotational flow is resolved by solving a simplified Poission’s equation［4］.

To further study the complex cavitating flow structure， it is necessary to develop new cavitation models. Based on the boundary iteration methodology，a boundary method was developed for the study of the steady sheet cavitation and it can also be used for the propeller cavitation［8］. The Rayleigh-Plesset equation describing the bubble dynamics was used to simulate the cavitation in a rotational supercavitating evaporator［9，10］. For the unsteady turbulent cavitating flows，several multiphase CFD methods were developed， and the homogeneous equilibrium model （HEM） was widely employed［11］. One type of the HEM method is based on the barotropic law of state［12］， which is widely used but can not capture the vorticity production in a closure region due to the cavity collapse. Another HEM method is the transport-equation based cavitation model （TEM）， which solves the transport equation for the vapor volume fraction with a source term describing the mass transfer due to the cavitation process. The source term is calculated using the mass transfer cavitation model［13，14］. In the past decade， the TEM was applied extensively in the numerical simulation of the cavitation phenomena［15］.

In this study， the volume of fluid （VOF， Hirt，1981） method is applied to capture the boundary of the cavity bubble under the TEM framework. The VOF technique can capture the small scale vortex flow structure in the gas bubble， therefore， it was applied widely in the simulation of the natural super-cavitation， the ventilated super-cavitation and the rotating cavitation［1，16］. The TEM+VOF method can capture the abundant vortex structures in the cavity bubble behind the rotating cavitator， where the rotational movement of the cavitator is accounted for by using the sliding mesh model （SMM） in the ANSYS Fluent. The present numerical method will be verified by the experimental measurements of cavitating flows over a circular disk and a cone cavitator. Then the validated method is applied to the 3-D simulations of cavitating flows over the same cavitators with different rotating speeds. The cavity bubble shape， the drag coefficient of the cavitator， and the flow structure in the cavity bubble will be compared and analyzed.

Fig.1 Computational domain and mesh in 2-D disk case

1. Mathematical modeling

1.1 Governing equation

The governing equations of the two-phase cavitating flow are as follows:

The governing Navier-Stokes equations are time averaged under the Reynolds-averaged Navier-Stokes（RANS） framework， and the generated fluctuation terms are modeled with the introduction of two new terms， the turbulent kinetic energyand the dissipation rate. The two new turbulent variables are closed by the RNG （renormalization-group）equations （Yakhot 1986）.

Fig.2 Computational mesh in 3-D disk case， the black region is the moving zone in （b）

1.2 Volume of fraction （VOF） method

The liquid-gas interface of the cavity bubble is tracked using the VOF method. The volume fraction of the liquid phase and the gas phase are denoted byand， which satisfy the following relations:

Then any material property （e.g.， the densityand the viscosity， of the water-vapor mixture can be rewritten as

The boundary of the cavity bubble is reconstructed based on the calculated liquid phase volume fraction taking the value of 0.5， i.e.，， using the piecewise linear interface construction （PLIC） method （Rider 1998） or the modified high resolution interface capturing （HRIC） method （Muzaferija 1998）. The modified HRIC method can not capture the interface as sharply as the PLIC method does， but it takes less computational resources.

Fig.3 Nondimensional cavity lengthsand drag coefficientsin 2-D disk case at various cavitation numbers0.15， 0.175， 0.2， 0.225 and 0.25

Fig.4 Cavity shape at circled peak point （Figs.3（a）， 3（b）） in 2-D disk case

1.3 Cavitation model

The cavitation process is resolved by the transport equation of

1.4 Surface tension model

The surface tension force in Eq.（2） is modeled by the Brackbill’s surface tension model （1992）

where the surface tension coefficient0.027 N/m. The curvatureof the liquid-gas interface is defined as

2. Numerical results

2.1 Circular disk

2.1.1 Computation setup

One circular disk with diameteris fixed vertically in the computation domain as shownin Fig.1（a）. The inlet boundary is located at a placefrom the disk， and the inflow velocity is fixed to be constant， i.e.，. The distance between the outlet boundary and the disk is set as， which is long enough to avoid the influence from the outflow boundary. The top side of the computation domain is a slippery wall， i.e.， the wall shear stress is zero. In the 2-D simulation， the bottom of the computation domain is configured as the rotation axis. The 2-D structured 51 019 cell mesh is shown in Fig.1（c）， and the grid is refined near the disk as shown in Fig.1（b）.

Fig.5 The instantaneous nondimensional cavity lengthin 3-D disk case

Fig.6 The cavity shapes in 3-D disk cases at 0.55 s with different angular speeds， colored by the turbulent kinetic energy

In the 3-D simulation， the 2-D computation domain rotates around the axis， to generate a cylindrical computation domain composed by 1 680 624 computation cells as shown in Fig.2， which is a little coarser than the 2-D mesh near the disk， as limited by the computation condition. Meanwhile， to save the computation time， the distance between the disk and the outlet is also shortened to30DAdditionally， in the transient sliding mesh method（SMM）， the computation mesh is divided into several zones as in Fig.2（b）， and the moving zone will move with its own rotating speed relative to the adjacent zone along the mesh interface［19，20］.

Fig.7 Vortex structure in 3-D disk cases with different angular speeds at 0.55 s， visualized by the iso-surface of swirling strength of 0.1 and colored by velocity magnitude

2.1.2 2-D results

The cavity length， denoted by， is one of the most important characteristics of the cavity shape. In the 2-D simulation， the boundary of the cavity bubble is captured by using two VOF schemes， the PLIC and the modified HRIC （simplified as HRIC in the following context）. Fig.3（a） and Fig.3（b） show the instantaneous dimensionless cavity lengthcurves calculated by using the PLIC and the HRIC， respectively. The cavity bubble keeps developing untilreaches the first peak point， labeled by a circular signin Figs.3（a）-3（b）. Thevalue fluctuates around the peak value for a short while. During thisperiod， the cavity bubble is relatively stable and maintains a spheroid shape in Fig.4. Thenstarts to decrease once the bubble shedding starts. At last， the cavity length curve begins oscillating， due to the unsteadiness of the cavitating flow.

Fig.9 Computational domain and mesh in 2-D cone case

The labeled peak values of thecurves in Figs.3（a）， 3（b） are plotted versus the cavitation numbers in Fig.3（c）， and the cavity shapes at the relative peak points are shown in Fig.4. Obviously， the cavity length decreases when the cavitation number increases. Furthermore， the present simulation results obtained by using the PLIC and the HRIC are both consistent with the experimental data. Fard et al.［16］simulated the cavitating flow over the disk by using the Rayleigh-Plesset transport equation based cavitation model and the shear stress transport （SST）turbulence model with a 3-D O-type structured mesh［16］. In the Fard’s study， the cavity length was statistically averaged in 30 s， and the result was closer to the Richardt’s analytical estimation［5］. Note that the Richardt’s theoretical relation is only valid for， and the relation is extended to the high cavitation numbers in Fig.3（c）. We cannot claim that our results are better overwhelmingly than the Fard’s results， because ourcavity length data are estimated by using a different approach. Nonetheless， there is little doubt that the present numerical simulated cavity length is reliable.

Fig.10 Computational mesh in 3-D cone case， the black region is the moving zone in （b）

Fig.11 Nondimensional cavity lengthsand drag coefficientsin 2-D cone case at various cavitation numbers0.15， 0.175， 0.2， 0.225 and 0.25

The simulated drag coefficientsare also compared with the experimental， theoretical and previously published results in Fig.3（d）. The non-dimensional drag coefficientis defined as

Fig.12 Cavity shape at circled peak point （Fig.11（a）-11（b）） in 2-D cone case

However， the vortex structure in the irrotational 3-D case is different from that in the rotational 3-D cases， as shown by the iso-surfaces of the swirling strength in Fig.7. The rotational movement of the disk reduces the small-scale eddies. In Fig.6， we may also observe that the small and slim water column attaching to the bottom of the disk in the irrotational case disappears in the rotational cases. To see more details of the vortex structure， Fig.8 shows the velocity vector plot is shown on several cross sections located2.5D，andfrom the disk bottom， where the cavity length is around. Obviously， the flow is dominated by the rotational movement of the cavitator at， and the rotation has less effect on the back end of the cavity bubble. We will revisit this pointin the discussion of the 3-D cone case.

2.2 Cone cavitator

2.2.1 Computation setup

The cone cavitator is composed of a cone with an angle ofand a circular disk， exactly the same as in the disk case， and the boundary conditions are also the same， as shown in Fig.9（a）. After the grid dependence study， a 2-D 50 364 cell mesh （Fig.9（c）） and a 3-D 1 148 702 cell mesh （Fig.10） are constructed. Similar to the disk cases， 2-D simulations of cavitating flows over the cone cavitator are performed at different cavitation numbersby using the PLIC and the HRIC. The 3-D simulation is carried out atwith rotation speeds，， by using the HRIC.

2.2.2 2-D results

Similar to the disk cases， all calculated instantaneous dimensionless cavity lengths have the peakpoints denoted by circular signs as in Figs.11（a）， 11（b）. Thevalue at the peak point is extracted as the dimensionless cavity length for each cavitation number，and is consistent with the experimental data， as shown in Fig.3（c）. Furthermore， thevalues calculated by the PLIC and the HRIC are almost the same.

Fig.13 The instantaneous nondimensional cavity lengthin 3-D cone case

Fig14 The cavity shapes in 3-D cone cases at 0.4 s with different rotating speeds， colored by the turbulent kinetic energy

With respect to the drag coefficient， the present PLIC simulatedvalues fit very well with the experimental data， while the HRICvalues are closer to the Richard’s theoretical relation［5］. Both are better than the Fard’s results［16］. Obviously， in the cone case，the predictions of the drag coefficients are better than those in the disk case.

Figure 12 shows the cavity shape at the peak point mentioned above for each cavitation number by using the PLIC and the HRIC. Again， the stagnation point is observed， and the PLIC captures more sharply the cavity boundary than the HRIC. Unlike the disk case， the size of the vapor bubble in the cone case is smaller， and the HRIC fails to capture the small eddies observed in the closure region of the cavity bubble behind the disk. In the cone case， the advantage of the PLIC over the HRIC is not as obvious as in the disk case.

2.2.3 3-D results

Fig.15 Vortex structure in 3-D cone cases with different rotating speeds at 0.4 s， visualized by the iso-surface of swirling strength of 0.1 and colored by velocity magnitude

Unlike the 3-D disk case， the difference between the three 3-D cases with different rotating speeds are much greater. The cavity length decreases when the rotating speed increases， and the difference of thecavity size is obvious in Fig.14. Additionally， the drag coefficients are 0.32，0.32 and 0.28 in the 3-D cases with angular speeds，and， respectively. When the rotation speed increases， the cavity length decreases， and the reentrant jet may produce a thrust on the cone cavitator.

Fig.16 Cross-sectional vector field plot of 3-D cone cases at 0.4 s. From left to right， the cross section is located at places1.5D，andfrom the cone bottom. Gray part indicates the water， while white part is the vapor

Compared with the disk case， the rotational movement of the cavitator has a greater influence on the cavity shape， as shown in Fig.14 and on the vortex structure， as shown in Fig.15. The small water drop attached behind the cone bottom in the irrotational case vanishes in the rotational cases. Meanwhile， the small-scale eddies are also reduced with the increase of the angular speed. Figure 16 confirms the above observations. The flow keeps rotating to the end of the bubble in the rotational cases， and the reentrant jet travels nearer to the cone bottom than the irrotational case.

3. Conclusions

The 2-D cavitating flows over a circular disk and a cone cavitator are simulated by using the Sauer-Schnerr cavitation model and the VOF method with the PLIC or HRIC scheme. The simulated cavity lengths of all cases are consistent with the experimental measurements. The simulated drag coefficients of the cone cases also fit well with the experimental data， while thevalues of the disk cases are underestimated by around 10% for the PLIC and 20% for the HRIC， as acceptable for the 2-D simulations.

Limited by the computation condition， the VOF+ HRIC method is used for the 3-D simulation of the cavitating flows over axisymmetric cavitators with angular speedsandWhen the cavitator is irrotational， the gas flow behind the cavitator is disturbed by the reentrant jet to generate small-scale eddies. However， the fluid force is not strong enough to remove the tiny water drops attached to the cavitator bottom. When the cavitator is rotating，the gas flow near the cavitator is governed by the rotational movement and a large-scale vortex， which repel the small-scale eddies， so that the small water drops and vortex structures are reduced in the rotational cases.

Furthermore， the rotational movement of the cone cavitator has a greater influence on the cavitating flow compared with that of the circular disk. With the increase of the angular speed， the cavity bubble behind the cone cavitator shrinks faster with less small-scale eddies， and the reentrant jet travels nearer to the cavi-tator bottom. This may generates a thrust and reduces the drag coefficient as observed in the 3-D case with. There are two possible reasons. One is because the surface area of the upwind side of the cone cavitator is greater compared with the circular disk. Another one is that the cavity bubble in the cone case is short enough such that the rotation movement of the cavitator can influence the whole bubble.

References

［1］ RASHIDI I.， PASANDIDEH-FARD M. and PASSANDIDEH-FARD M. et al. Numerical and experimental study of a ventilated supercavitating vehicle［J］. Journal of Fluids Engineering， 2014， 136（10）: 216-223.

［2］ WANG Yi-wei， HUANG Chen-guang and FANG Xin et al. Characteristics of the re-entry jet in the cloud cavitating flow over a submerged axisymmetric projectile［J］. Chinese Journal of Hydrodynamics， 2013， 28（1）: 23-29（in Chinese）.

［3］ CAO Wei， WEI Ying-jie and WANG Cong. Current status，problems and applications of supercavitation technology［J］. Advances in Mechanics， 2006， 36（4）: 571-579..

［4］ STREET R. L. A linearized theory for rotational supercavitating flow［J］. Journal of Fluid Mechanics， 1963， 17: 513-545.

［5］ FRANC J. P.， MICHEL J. M. Fundamentals of cavitation［M］. Dordrecht The Netherlands: Kluwer Academic Publishers， 2004.

［6］ PARK S.， RHEE S. H. Computational analysis of turbulent super-cavitating flow around a two-dimensional wedge-shaped cavitator geometry［J］. Computers and Fluids， 2012， 70: 73-85，

［7］ ALVAREZA A.， BERTRAMB V. and GUALDESIC L. Hull hydrodynamic optimization of autonomous underwater vehicles operating at snorkeling depth［J］. Ocean Engineering， 2009， 36（1）: 105-112.

［8］ PEREIRA F.， SALVATORE F. and FELICE F. D. Measurement and modeling of propeller cavitation in uniform inflow［J］. Journal of Fluids Engineering， 2004， 126（4）: 671-679.

［9］ LIKHACHEV D. S.， LI F. Modeling of rotational supercavitating evaporator and the geometrical characteristics of supercavity within［J］. Physics， Mechanics and Astronomy， 2014， 57（3）: 541-554.

［10］ WEI Qun， Chen Hong-xun and ZHANG Rui. Numerical research on the performances of slot hydrofoil［J］. Journal of Hydrodynamics， 2015， 27（5）: 105-111.

［11］ ROOHI E.， ZAHIRI A. P. and PASSANDIDEH-FARD M. Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model［J］. Applied Mathematical Modelling， 2013， 37（9）: 6469-6488.

［12］ BARRE S.， ROLLAND J. and BOITEL G. et al. Experiments and modelling of cavitating flows in venturi: Attached sheet cavitation［J］. European Journal of Mechanics B/Fluids， 2009， 28（3）: 444-464.

［13］ FRIKHA S.， COUTIER-DELGOSHA O. and ASTOLFI J. A. Influence of the cavitation model on the simulation of cloud cavitation on 2-D foil section［J］. International Journal of Rotating Machinery， 2008， 2008: ID 146234.

［14］ MORGUT M.， NOBILE E. and BILU? I. Comparison of mass transfer models for the numerical prediction of sheet cavitation around a hydrofoil［J］. International Journal of Multiphase Flow， 2011， 37（6）: 620-626，

［15］ SRINIVASANA V.， SALAZARB A. J. and SAITOC K. Modeling the disintegration of cavitating turbulent liquid jets using a novel VOF-CIMD approach［J］. Chemical Engineering Science， 2010， 65（9）: 2782-2796.

［16］ FARD M. B.， NIKSERESHT A. H. Numerical simulation of unsteady 3-D cavitating flows over axisymmetric cavitators［J］. Scientia Iranica， 2012， 19（5）: 1258-1264.

［17］ SAUER J.， SCHNERR G. H. Unsteady cavitating flow: A new cavitation model based on amodified front capturing method and bubble dynamics［C］. Proc. of FEDSM， 4th Fluids Engineering Summer Conference. Boston， MA，USA， 2000.

［18］ SCHNERR G. H.， SAUER J. Physical and numerical modeling of unsteady cavitation dynamics［C］. Fourth International Conference on Multiphase Flow. New Orleans， USA， 2001.

［19］ MO?TěK M.， KUKUKOVá A. and JAHODA M. et al.Comparison of different techniques for modelling of flow field and homogenization in stirred vessels［J］. Chemical Papers， 2005， 59（6a）: 380-385.

［20］ DU Te-zhuan， HUANG Chen-guang and WANG Yi-wei et al. Investigation of dynamic mesh technique and unsteady cavitation flows［J］. Chinese Journal of Hydrodynamics， 2010， 25（2）: 190-198（in Chinese）.

February 5， 2015， Revised May 21， 2015）

* Project Supported by the Sepcial Research Program of Public Welfare and Capacity Building in Guangdong Province（Grant No. 2015A020216008）.

Biography: Ye-jun GONG （1980-）， Female， Ph. D.， Research Associate Professor

Jie-min ZHAN，

E-mail: cejmzhan@vip.163.com