1 Introduction

Ventilated cavitation consists of creating gas cavities in a liquid flow by injection of non-condensable gas such as air(Stinebring and Holl 1979).Contrary to natural cavitation(Coutier-Delgosha et al.2007),this type of cavitation is usually not harm ful.Some parameters such as the air flow rate,the injection frequency,and the morphology and size of the injected bubblesenable to control and even reduce the damage and performance loss due to natural cavitation,and thus enable to increase the performance of hydraulic machines(Kopriva et al.2005;Am romin et al.2011).

Although it can be controlled,ventilated cavitation is complex and is characterized by various gas regimes:portion gasleakage,twin vortex gas-leakage,and gas-leakage from pulsating surpercavities.Previous studies(Stinebring and Holl 1979;Swanson and O’Neill 1951)have focused on the transition between the reentrant jet(portion gas-leakage)and the twin vortex regimes.For low air flow rates(short air cavity),a reentrant jet regime is usually observed,while for high ones(long air cavity),a twin vortex regime is obtained.These authorsprovide a physical explanation for the transition between these two regimes and emphasize its dependence on the ventilated cavitation number σc and the Froude number Fr.

步骤1 计算估计值Δt0,将Delta机器人抓取到达G的位置时刻Δt0作为迭代初始值。如果在水平方向上没有匀速运动,则最大速度需要满足那么:

In the present work,a 2D code initially devoted to natural cavitation was extended to the transport of dissolved noncondensable gas,in order to take into account the air injection.The flow over a conical head body previously tested by Stinebring and Holl(1979)has been calculated and a detailed comparison with the experimental data has been performed to demonstrate the capability of the code to reproduce the different flow patterns associated with ventilated cavitation(Adama Maiga et al.2013).Further analysis has been focused on the transition between the reentrant jet and twin vortex flow regimes.

The present paper is divided into two parts.The first one is devoted to the theoretical formulation,i.e.,the governing equations and the cavitation and turbulence models.The second part is dedicated to the results and discussion.

2 Theoretical Formulation

2.1 Governing Equations

The numerical model is based on a single-fluid approach,where the two phases(liquid/vapor)are treated as a single homogeneous fluid,characterized by an average density and velocity.The balance equations for the mixture are the follow ing:

The source terms˙m-.and˙m+.in Eq.7 govern the evaporation and condensation rates.Among the numerous available models(Reboud et al.1998;Kunz et al.1999;Singhal et al.2002),it is the simple version from that proposed by Singhal et al.(2002)that is used:

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2.2 Cavitation Model

By considering a control volume V we have

The volume fractions of the vapor and non-condensable gas are governed by the two following transport equations:

Results show that for natural sigma higher than 1,where the phenomenon of the natural cavitation is negligible,there is almost no difference between the curves.For values between 1 and 0.5,where cavitation is obtained at high Reynolds number,there is a slight decrease in the transition coefficients CQt for these highest Reynolds numbers.However,the major effects are obtained for sigma σ=0.5,where the effects of cavitation are important:a large decrease of coefficients CQt is obtained.Consequently,the range of flow conditions leading to the reentrant jet case is almost divided by two.By further reducing the natural cavitation number,the area of the occurrence of the reentrant jet will be completely removed.This is due to the coupling with natural cavitation,which greatly influences the development of the cavity.Indeed,more natural cavitation is associated with a lower local pressure,i.e.,a higher expansion of the air and therefore the grow of the cavity.

The influence of the initial single-phase flow without air injection(subcavitant flow)on the transition from the reentrant jet to the twin vortex regime is investigated hereafter.

with αv the volume fraction of the vapor phase and αng the volume fraction of the non-condensable gas.The mixture density ρm is given by

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with Ce=0.02 and Cc=0.01 the empirical constants,V∞the free-stream velocity,and pv the saturated vapor pressure which is 2000 for water.

Fig.1 a Experimental setup.b Mesh for CFD.c Zoom on conical head body mesh and location of air injection

Fig.2 Experimental and numerical photos of the air evolution in the reentrant jet regime

2.3 Turbulence Model

Natural and ventilated cavitation are turbulent phenomena.The k-epsilon RNG turbulence model is used to calculate the turbulent viscosity:

with n=10;for more details about the numerical model,it can be found in Coutier-Delgosha et al.(2007).

3 Results and Discussion

3.1 Two Dimensional Wedge and Formation of Reentrant Jet and Twin Vortex Regimes

Figure 1a,b shows a scheme of the experimental model and the mesh used in the numerical approach,respectively.The model is a conical head body(45°apex angle,1.0 in.diameter conical head)without after body.Figure 1c shows the location of air injection,which is consistent with the experimental setup.

The main similarity parameters in this configuration are the natural cavitation numberthe sigma is based on the outlet pressure,the ventilated cavitation number the Froude numberand the ventilation air flow coefficientw iththe volume flow rate of air,D the model diameter,and pc the cavity pressure(in the numerical calculations,it is an average pressure of the cavity which is estimated).

Experimental studies show that for low air flow rates(short air cavity),a reentrant jet regime is usually observed,while for high ones(long air cavity),a twin vortex regime is obtained.Figures 2 and 3 compare numerical photos with experimental ones(Stinebring et al.2001)in a reentrant jet regime and that twin vortex one,respectively.In both regime cases,the comparison shows a very good agreement between the photos.

Fig.3 Experimental and numerical photos of the air evolution in the twin vortex regime

Figures 4 and 5 illustrate the results:(i)an air pocket cycle in the reentrant jet regime(CQ=0.014)and(ii)the different phases in the air pocket formation in the twin vortex regime(CQ=0.051),respectively.

At low flow rate(CQ=0.014),a water jet flowing upstream at mid-height inside the cavity can be noticed in Fig.4a.When the water jet reaches a certain position,it cuts the cavity which is then convected downstream by the flow.At the higher flow rate(CQ=0.051),Fig.5a,b shows the formation of cavity and vortex tubes.In the case of 2D simulations,the second tube is not observed;it is symmetric with the first.The evolutions of the cavity and the vortex,for low and high flow rates,show that the two regimes observed experimentally,reentrant jet and twin vortex regimes,are qualitatively correctly obtained by the CFD.For more details about the validation of the results,see Adama Maiga et al.(2013).

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3.2 Transition Condition Between Reentrant Jet and Twin Vortex Flow Regimes

The condition of transition from the reentrant jet to the twin vortex regime has been reported previously to be a function of σc Fr the product of the ventilated cavitation number with the Froude number.Figure 6 presents σc Fr for various flow conditions.The hollow bars correspond to the reentrant jet regime and the full ones the twin vortex regime.It confirms that the twin vortex regime is obtained forσc Fr＜ 1,while the reentrant jet regime is obtained for σc Fr＞ 1.

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where ρm is the mixture density is the mixture velocity,and πm is the tensor for the external forces.These equations are coupled with a cavitation model to determ ine the mixture density.

For that purpose,a transition ventilation air flow coefficient CQt is defined.It is a sort of bisection method that is used to calculate the transition ventilation air flow coefficient CQt.For one Reynolds number,as soon as one has a CQ of a jet reentrant regime and one of a twin vortex regime,a new calculation is initiated with a CQm1 equal to the average of the CQ of the regimes jet reentrant and twin vortex.If this new calculation provides a jet reentrant regime,so in a new calculation is initiated with a CQm2 equal to the average of the twin vortex CQ and CQm1.Otherwise,CQm2 equal to the average of the jet reentrant CQ and CQm1.This process is repeated until the difference between the twin vortex CQ and the jet reentrant one is less than 0.01 and the transition ventilation air flow coefficient CQt equal to the average of this CQ.

The numerical results and experimental measurements(Stinebring and Holl 1979;Adama Maiga et al.2013;Swanson and O’Neill 1951)show that the more the freestream velocity V∞is decreased,the lowest is the critical air flow rate where the transition to the twin vortex regimeoccurs.This can be explained by the fact that at low free-stream velocity V∞,the instabilities behind the body are less intense so the cavity can grow without being convected,even with low air flow rates.Conversely,with a high free-stream velocity V∞,the instabilities are more intense and will likely destabilize the cavity development.

Fig.4 a Evolution of the volume fraction of air for CQ=0.014 and V∞=30 ft/s.b Evolution of the vorticity for CQ=0.014 and V∞=30 ft/s

Fig.5 a Evolution of the volume fraction of air for CQ=0.051 and V∞=30 ft/s.b Evolution and formation of the twin vortex for CQ=0.051 and V∞=30 ft/s

Fig.6 Occurrence of the two regimes according to σcFr

These instabilities are mostly dependent on the Reynolds number Re.Thus,at sigma σ=25(where the natural cavitation phenomenon is negligible),Fig.7 presents the transition ventilation air flow coefficient CQt function of the Reynolds number Re.The numerical calculations show that for Reynolds numbers lower than 1.5 106,as the instabilities behind body are less intense,only developed cavity appears(twin vortex regime).The curve shows a first phase,up to Re=8×106,where the coefficient CQt experiences great variations according to the Reynolds number.For higher Re numbers,CQt converges towards an asymptote.This result shows that for a Reynolds number greater than 8×106,one needs almost the same flow rate to obtain a cavity developed.On the other hand,it notes that the area under the curve represents the reentrant jet zone and one above the twin vortex zone.

Fig.7 Evolution of the transition ventilation air flow coefficient according to the Reynolds number for natural sigma=25

3.3 Natural Cavitation Influence

Concerning the influence of natural cavitation,as its appearance particularly depends on the natural cavitation number(natural sigma),Fig.8 presents the transition ventilation air flow coefficient CQt as a function of the Reynolds number Re,for different values of sigma.

with ρl the liquid density, ρv the vapor density, ρng the density of the non-condensable gas,and αl the volume fraction of the liquid.

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4 Conclusions

In the present work,a 2D numerical model devoted to the simulation of unsteady cavitating flows developed in Adama Maiga et al.(2013)is used to investigate the influence of natural cavitation on the reentrant jet and the twin vortex regimes and the transition between these two regimes.

As observed in previous works,the results show that the product of ventilated cavitation number and Froude number of the twin vortex regime is obtained when σc Fr＜ 1,while the reentrant jet regime corresponds to σc Fr＞ 1.

Fig.8 Evolution of the transition ventilation air flow coefficient according to the Reynolds number for different natural sigma

Neglecting the cavitation effects,the numerical calculations show that for Reynolds numbers lower than 1.5 106,as the instabilities behind body are less intense,only developed cavity appears(twin vortex regime).For values between 1.5 106＜Re＜8 106,the transition ventilation air flow coefficient CQt varies drastically with the Reynolds number.For Reynolds number higher than 8 106,the coefficient CQt converges towards an asymptote,so at high Reynolds numbers,the transition between reentrant jet and twin vortices does not depend on the Reynolds number.

These results are observed for natural sigma higher than 1,where the phenomenon of the natural cavitation is negligible.For values between 1 and 0.5,where some cavitation is obtained for high Reynolds number Re＞2.5 106,a slight decrease in the transitions coefficients CQt is observed.However,at sigma 0.5,where the effects of cavitation are important,a large decrease in the transitions coefficients CQt is noticed,and especially the range of flow conditions leading to the reentrant jet zone is almost divided by two.

These results confirm that the natural cavitation number significantly influences the development of ventilated cavitation.

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Acknowledgements This work was performed in the scope of project ANR-12-ASTR-0017-03 “F-DRAINH”in collaboration with the IRENav Laboratory(French Naval Academy,Brest,France)and the IMFT laboratory(Toulouse,France).

Nomenclature Cc,empirical constant for condensation term source;Ce,empirical constant for evaporation term source;Cμ,empirical constant for k-epsilon model;CQ,ventilation air flow coefficient;CQt,transition ventilation air flow coefficient;D,model diameter;Fr,Froude number;k,turbulent kinetic energy;˙m-.,condensation term source of the vapor volume fraction;˙m+.,evaporation term source of the vapor volume fraction;pv,saturated vapor pressure;pc,cavity pressure;,volume flow rate of air;Um,mixture velocity;V,control volume;V∞,free-stream velocity;αl,liquid volume fraction;αng,non-condensable gas volume fraction;αv,vapor volume fraction;ε,turbulent dissipation;μt,eddy viscosity;πε,tensor for the turbulent dissipation;πk,tensor for the turbulent kinetic energy;πm,tensor for the external forces;ρl,liquid density;ρm,mixture density;ρng,non-condensable gas density;ρv,vapor density;σ,natural cavitation number;σc,ventilated cavitation number

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