Numerical Research on Ventilated Supercavity Shape and Flow Structure in the Turning Motion

(School of Astronautics,Harbin Institute of Technology,Harbin 150001,China)

Numerical Research on Ventilated Supercavity Shape and Flow Structure in the Turning Motion

ZHANG Guang,YU Kai-ping,ZHOU Jing-jun

(School of Astronautics,Harbin Institute of Technology,Harbin 150001,China)

For supercavitating vehicle,the prediction of cavity shape in the process of maneuver is an urgent problem to be investigated.In this paper,in the framework of two-fluid multiphase flow model,a three dimensional numerical model was presented for solving ventilated cavitating flow at the turning motion.By solving the RANS equations and SST(Shear Stress Transport)turbulence equations,the ventilated supercavity of disc cavitator was predictd and compared with the results of Logvinovich independence principle of the cavity section expansion,the comparsion results verified the accuracy of numerical model.On this basis,the influence of different turning radiuses to the cavity shape was analyzed and the flow structure was also investigated under the condition of different Froude number.

turning motion;ventilated supercavity;two fluid multiphase flow model;numerical simulation

Biography：ZHANG Guang(1983-),male,Ph.D.student of Harbin Institute of Technology,E-mail:zhangguang925@163.com;YU Kai-ping(1968-),male,professor/tutor of Harbin Institute of Technology.

1 Introduction

Supercavitation has become a hot topic due to its potential to significantly enhance the speed of undersea weapons and vehicles.Unlike traditional motion mode of underwater vehicle motion mode,all or most of the supercavitating vehicle surface are surrounded by cavity,and only the cavitator and part of the tail in contact with water.The force state and stability of vehicle are closely related with deformation of supercavity shape[1].

Based on a large number of experimental researches,foreign scholars summarized a series of formula for calculating cavity shape and analyzed the main factors that affect the cavity shape in the early stage[2-3];recently,using particle image velocimetry test system(PIV)and acoustic test system,the wake details of ventilated cavity were decribed[4]and the properties of the cavity contents were measured[5].These studies have improved our understanding of the physics phenomena.In addition,Chinese researchers have conducted certain experiments and numerical simulations on several aspects of supercavities[6-8].From the above studies,most studies of the cavity shape and flow structure focused on rectilinear motion supercavitating flow.However,with the development of related technologies on supercavitation,more advanced propulsion and control technologes are gradually used in supercavitating weapons,which makes the supercavitating weapons more accurate and agile in underwater navigation.In order to maintain the stability of supercavitating vehicle duing the maneuvering navigation,accurate predictions of supercavity shape is very important.

In this paper,based on finite volume method using the two fluid multiphase flow model and SST(Shear Stress Transport)[6-7]with the consideration of gas-water interaction and gravity effect,a three dimensional numerical simulations approach was presented for solving ventilated cavitating flow at the turning motion.Main research is on the ventilated supercavity shapes and supercavitating flow structure at the turning motion.

2 Numerical methods

2.1 Basic governing equations

Ventilated cavitation flows is a complicated multiphase viscous flow problem,this study involves only the interaction between two phases of water and gas.The basic approach adopted to simulate consists of solving the continuity equation,standard 3-D Navier Stokes equations,the volume fraction equation and turbulence equations.

The continuity equation is:

The momentum equation is:

here,SMα=γαρg+γαρU2/R,is user defined source term.

The volume fraction equation is:

SST turbulence model[9]:

SST turbulence model was developed based on Baseline(BSL)k-ω model.That Baseline k-ω solves two transport equations,one is for the turbulent kinetic energy,k,and the other is for the turbulence frequency,ω.The two equations are as follows:

k equation：

where Pkis the production rate of turbulence,μtis the eddy viscosity.

The proper transport behavior could be got by a limiter to the formulation of the eddyviscosity.So the SST turbulence model could be obtained.

The SST model accounts for the transport of the turbulence shear stress and gives highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients.

2.2 The Logvinovich principle of independence

The principle of independence is stated as follows:Each cross section of a cavity expands relatively to the trajectory of the center of a cavitator which happens almost independently from the following or the previous body motion.The expansion occurs according to the definite law which is dependent only upon the difference between the pressure at infinity and the pressure within the cavity,the speed,the size,and the drag of a body at the moment when a body passes the plane of the considered section.Relevant equations see Ref.[10].

3 Physics model and boundary conditions

The disk cavitator model is used with diameter of Dn=10mm.Hexahedral elements are employed as shown in Fig.1.The velocity inlet boundary and pressure outlet boundary are defined at inlet and outlet of computational region,pressure distribution is specified at pressure outlet boundary(see Fig.2).The mass inlet boundary is defined at the blowhole.At external boundary,the free-slip wall boundary condition is specified.Using the relative motion method to investigate turning motion of ventilated cavitating flow,attentions need to be paid on a few points.First,to create computational domain as the turning radius of cavitator;second,the inlet velocity distribution of computational domain is initialed according to turning motion characteristics.Finally,a source term is added to the right side of the momentum equation in order to simulate the actual pressure environment.

Fig.1 Computation model

Fig.2 Computational region and boundary conditions

Fig.3 shows the pressure distributions of the middle horizontal cross section(depth H=10m)in calculation domain with and without an additional momentum source term.The pressure curves on the two lines along radial direction in Fig.3 are shown in Fig.4.It is seen that the pressure of the model without source item increases along radial direction,which is not corresponding to the actual flow field.While,the model equipped with an source item has pressure distribution that approaches to the actual status.

Fig.3 Pressure distribution added before and after the source term

Fig.4 Radial pressure distribution comparison

Ventilated cavitation number σc,Froude number Fr,and ventilation coefficientare defined as follows:

4 The simulation results and discussions

where Q is the volume flow rate at the pressure of cavity inside.

4.1 The influence of turning radiuses to the ventilated cavity shape

First based on CFD method and the principle of independence,supercavities produced from cavitator under the rectilinear motion and turning motion(Turn radius are respectively R1=2m,R2=1m,R3=0.5m)are simulated.In this simulation,ventilation coefficient=0.094 5,Froude number Fr=95.83.Fig.5(top view)shows a comparison of simulation results between the two methods.The upper is CFD method simulation results and the other is the principle of independence simulation results.The auxiliary line is cavitator trajectory,and the contour surface whose air volume fraction is 0.5 is taken as cavity surface in CFD method.In Fig.5,from the comparison of ventilated cavity shapes under the same conditions,the simulation results of both methods are in good agreement.Different from the symmetry cavity produced under rectilinear motion,the cavity produced from turning motion has a bending deformation which increases with the increasing of turning radius.

Fig.5 Comparison of the simulation results based on two method

In addition,as the definition of principles of independent,the cavity shape calculated by this method has a typical characteristic that the cavity axis and cavitator motion trajectory coincide with each other.CFD method used in the paper can correctly describe the cavity feature as shown in Fig.5.It also proves that CFD method can effectively predict ventilated cavity shapes under the turning motion.Unlike the the principle of independence,CFD method based on the viscosity multiphase flow model may subtly capture ventilated cavity flow details.It can be seen clearly reentrant jet regime at the tail of the cavity under four motion condition.

Meanwhile,the influence of turning motion on the cavity shape is investigated under the condition of low Froude number.In this simulation,ventilation coefficient=0.094 5,Froude number Fr=12.76.Under such conditions,significant gravity effect can be seen from Fig.6,the process of turning motion cavity addition to bending deformation.The cavitiy tail was obvious upward drift and showing twin-vortex tubes mode.Fig.7(Top view)shows comparison of cavity shapes,as can be seen bending axis of cavities with strictly consistent motion trajectory of cavitator.

Fig.6 Cavity shapes at turning motion Fr=12.76

Fig.7 Comparison of cavitiy shapes Fr=12.76

From calculated results it can be found that maintaining the same ventilation coefficient,the length and maximum diameter of cavity remain unchanged under the same Froude number.In order to analyze the reasons,10 pressure monitoring points are arranged on the cavity axis to investigate the changes of the pressure in the cavity under different turning motion.Fig.8 shows comparison of pressure distribution within the cavity when Fr=95.83 and Fr=12.76.It can be found that the pressure distribution is not uniform within cavity.The pressure gradually increases from head to tail of cavity.In the case of Fr=95.83,pressure gradient within cavity is larger.Comparing with the rectilinear motion,the turning motion leads to lower pressure within cavity.The maximum relative difference the average pressure is about 0.16%;When Fr=12.76,pressure gradient within cavity is relatively small,and the turning motion make the pressure increase.The maximum relative difference between the average pressure is about 0.01%.

Fig.8 Comparison of internal pressure distribution of the ventilated cavity

From the above analysis,it shows that turning motion has little effect on pressure distribution within the cavity.According the definition of ventilated cavitation number,the venti-lated cavitation number does not change when maintaining the ventilation coefficient under different turning motion conditions.

4.2 The influence of turning radiuses to flow structure

Based on the above results,the flow structure of ventilated cavitaty is analyzed under the condition of different froude number.Fig.5 shows the streamline distribution of flow field under four motion conditions when Fr=95.83.At cavity tail streamlines collection and come into the cavity forming reentrant jet.Because of the interaction between the outflow and reentrant jet,the internal of cavitaty form gas vortex.The gas vortex is symmetrical in linear motion,while in turning motion the gas vortex is not.

Fig.9 The streamline of velocity(Fr=79.83)

Fig.10 The streamline at the tail of cavity(Fr=12.76)

When Fr=12.76,the lower surface of cavity tail raises up because of the gravity effect.The velocity of gas within the region is reduced.With the further development of raising,the rear cavity is split into two vortex tube,gas-water mixture discharges to the downstream with high speed in the form of twin-vortex tubes.From the streamline distribution within the vortex tube it can also be found that under the linear motion the streamlines distributions in the two vortex tubes are the same,while the streamline distributions are different significantly under the turning motion.

Quantitative analysis for effects of turning motion on gas leakage of twin-vortex tubes is implemented by monitoring the gas flow within two vortex tube.The introduced parameters fmand fvare defined as follows:

Fig.11 Top view of cavity shape

Tab.1 Comparison parameters fv,fmof the four motion conditions

Values of fvand fmunder four motion conditions are listed in Tab.1.From the variation of fm,the gas leakage quantities of the two vortex tubes at the cavitiy tail are equal under linear motion(≈).Under turning motions,the gas leakage of the two vortex tubes at the cavity tail comes up to be m˙1＞m˙2,and the difference becomes larger with the decreasing of the turning radius.Under turning motions,the surface velocity of cavities is different that v1＞v2shown in Fig.11,which causes the flow at v1side more gas away from the cavity.To decrease turning radius,the difference between v1and v2becomes larger(that means fvincreases),which leads to the increasing of fmthat represents the gas leakage difference between the two vortex tubes at the tails of cavities.Since fvhas no linear relation with fm,the surface velocity of cavity is only one of main factors of gas leakage.

5 Conclusions

In this paper,a three-dimensional numerical model was presented for solving ventilated cavitating flow at the turning motion.The influences of different turning radius on ventilated cavitaty and flow structure were investigated.The validity of the model was verified by the principle of independence.The main conclusions are as follows:

(1)Under the turning motion,ventilated supercavity has bending deformation which depends on the turning radius,and the cavity axis coincides with the cavitator motion trajectory.

(2)The effect of the turning on internal pressure of ventilated supercavity is small.Under the same Froude number,the same ventilated cavitation number is obtained by maintaining the same ventilation coefficient,and the size of the cavity is not changed.

(3)The model in the paper visually simulates the flow structures of the inner eddy and the ventilated cavitating.The different surface velocity of cavities in turning motion leads to obvious difference of gas leakage between the two vortex tubes at the tail of the ventilated cavities.

[1]Yang Li,Zhang Qingming.Current application and perspectives on supercavitation technology research[J].Tactical Missile Technology,2006(5):6-10.

[2]Semenenko V N.Artificial supercavitation physics and calculation[C]//VKI Special Course on Supercavitating Flows.Brussels:RTO-AVT and VKI,2001:RTO-EN-010(11).

[3]Savchenko Y N.Control of supercavitation flow and stability of supercavitating motion of bodies[C]//VKI Special Course on Supercavitating Flows.Brussels,2001:RTO-EN-010(14).

[4]Wosnik M,Arndt Roger E A.Measurements in high void-fraction bubbly wakes created by ventilated supercavitation[C]//Sixth International Symposium on Cavitation.Wageningen the Netherlands,2006.

[5]Xiong junwu,Chahine G L.Characterization of the content of the cavity behind a high-speed supercavitating body[J].ASME,Journal of Fluids Engineering,2007,129:136-145.

[6]Yu Kaiping,Zhou Jingjun,et al.A contribution to study on the lift of ventilated supercavitating vehicle with low Froude number[J].Journal of Fluids Engineering,2010,132.

[7]Zhou Jingjun,Yu Kaiping,Zhang Guang.Research on the process of supercavity development and the evaluation on two methods of studying the planing state[J].Journal of Ship Mechanics,2011,15(3):200-206.

[8]Hu Shiliang,Lu Chuanjing,Pan Zhancheng.Research on the gravity effect of ventilated cavitating flows[J].Chinese Journal of Hydrodynamics,2009,24(6):786-792.

[9]Menter F R,Kuntz M,Langtry R.Ten years of industrial experience with the SST turbulence model[C]//Proc of the 4th International Symposium on Turbulence,Heat and Mass Transfer,2003,4:625-632.

[10]Vasin A D.The principle of independence of the cavity sections expansion(Logvinovich’s principle)as the basis for investigation on cavitation flows[C]//VKI Special Course on Supercavitating Flows.Brussels:RTO2AVT and VKI,2001:RTO2EN2010(8):105-131.

轉(zhuǎn)彎運(yùn)動(dòng)通氣超空泡形態(tài)及流場(chǎng)結(jié)構(gòu)數(shù)值研究

張 廣，于開(kāi)平，周景軍
（哈爾濱工業(yè)大學(xué)航天學(xué)院，哈爾濱 150001）

超空泡航行體機(jī)動(dòng)過(guò)程中空泡形態(tài)的預(yù)測(cè)是目前該領(lǐng)域亟待研究的問(wèn)題。文章在兩流體多相流模型的框架內(nèi)建立了用于求解通氣超空泡流轉(zhuǎn)彎?rùn)C(jī)動(dòng)的三維數(shù)值模型。通過(guò)求解RANS方程和SST（Shear Stress Transport）湍流方程，預(yù)測(cè)了圓盤(pán)空化器轉(zhuǎn)彎?rùn)C(jī)動(dòng)條件下生成的通氣空泡形態(tài)，并同Logvinovich獨(dú)立膨脹原理的計(jì)算結(jié)果進(jìn)行對(duì)比，驗(yàn)證了文中數(shù)值模型的有效性。在此基礎(chǔ)上，分析了轉(zhuǎn)彎半徑對(duì)通氣空泡的形態(tài)尺度的影響，對(duì)比研究了不同弗魯?shù)聰?shù)條件下通氣空泡的流場(chǎng)結(jié)構(gòu)。

轉(zhuǎn)彎運(yùn)動(dòng)；通氣空泡；兩流體多相流模型；數(shù)值模擬

TV131.3+2

A

張 廣（1983-），男，哈爾濱工業(yè)大學(xué)博士研究生；

周景軍（1981-），男，哈爾濱工業(yè)大學(xué)博士研究生。

TV131.3+2

A

1007-7294（2011）12-1335-09

date：2011-08-11

Support by the major National Natural Science Foundation of China(Grant No.10832007)

于開(kāi)平（1968-），男，哈爾濱工業(yè)大學(xué)教授，博士生導(dǎo)師；