水下圆柱体带空泡轴向绕流研究

金大桥,王 聪,魏英杰,邢彦江,邹振祝

(哈尔滨工业大学航天学院,哈尔滨 150001)

Introduction

In liquid flow with a circular cylinder immersed in it,cavitation generally occurs where the p ressure drops below the vapor pressure,and the negative pressures are relieved by the formation of gas-filled or vapor-filled cavities.Cavitating flow around circular cylinder is always turbulent and has great effects on hydrodynamic characteristics.Thus,it is significant to studyaxial cavitating flow around underwater circular cylinder in basic research and engineeringapplications.

At present,a considerable amount of theoretical and experimental researches on cavity and supercavity have been conducted.The studies of low-speed underwater circular cylinder focuson the vortex kinemat-icsand kinetic analysis at the conditions of different Reynolds numbers[1].With the increasing of the speed,cavitation p lays a more and more important role in liquid flow around cylinder because of the unsteady characteristics.It causes unsteady hydrodynamics,severe noise,structural erosion,and vibration p roblems.Many scholars have investigated the cavitation flows around circular cylinder immersed in water.G◦V◦Logvinovich studied the relationship between the drag coefficient of the cavitator and the cavitation number,and the shapes of natural supercavitation based on the potential theory and experimentalmethod[2].Cao Wei studied the shapes characteristics of natural supercavitation around a seriesof high-speed projectiles through experiments[3].Applying the distributing of sources and sinks on the surfacesof the revolution body and the cavity,the partial cavitating flow was numerical studied by Jia Caijuan[4].Characteristics of cavitating flow around revolution body were researched through commercial CFD code Fluent6.0 by Fu Hui-ping who focused on geometrical characteristics of partial cavity around distinct head forms by 2D axisymmetric solver[5].Using the homogeneous equilibrium cavity model with the solution of the transport of liquid mass fraction,Chen Ying made some numerical simulations of cavitation flows around an underwater vehicle and the results showed high accuracy in predication of cavity shapes[6].To sum up,most of the above literatures focused on the cavity shapesof cavitatorsorsome other revolution geometric bodies.

The numerical simulation of natural axial cavitation flow around underwater circular cylinder is investigated in the presentwork,which isbased on homogeneousequilibrium theory.The relationship of cavity shape and drag coefficient at different slenderness ratio of cylinder bodies is analyzed.The projectile experiments are conducted in the currentwork.The numerical simulation results and the experiment results are in good agreementwith each other.

1 Governing equations and numerical solutions

1.1 Governing equations

Based on the homogeneous equilibrium theory, using the full cavitationmodel[7],the governing equations are as follows：

The continuity equation ofmixture given by：

Whereρmis the density ofmixture,vmis themass-averaged velocity.

The momentum equationsofm ixture given by：

Whereμmis the viscosity of themixture of liquid and vapor,F is the body force,vdr,kis the drift velocity for secondary phase,p is the pressure of flow field,g is gravity acceleration,n is the number of phases.

The transport equation for themass faction of the vapor phase given by：

Where f is themass fraction of vapor,vvis the velocity vector of vapor,γis effective exchange coefficient,Reand Rcare the vapor generation and condensation rate terms respectively.

The equations of rate of phase change of them ixture：

Where pvis the liquid saturation vapor pressure at the given temperature,ρlis the densityof liquid,pvis the density of vapor,Vchis the characteristic velocity,σ is the cavitation number,Ceand Ccareempirical constants,the default values are Ce=0.02 and Cc= 0.01.

1.2 Numerical solutions

The flow field inlet and upper boundary is set to velocity-inlet zone;the walls of the circular cylinder are set to wall zones;the outflow of the flow field is set to pressure-outlet zone(Figure 1).The pressure-velocity coupling algorithm is set to SIMPLEC.The Second-order accuracy discremination schemes are chosen for the pressure,themomentum,and the volume fraction of the vapour.

Figure 1 Schematic diagram ofmesh and boundary conditions

2 Projectile experiment

The projectile experiment system is composed of launch system,trial tank,measurement and control system,protection system(Figure 2).The launch system is composed of light gas gun and gas storage device.The measurement and control system is composed of two sets of net target tomeasure the velocity of the circular cylinder and turn on the high speed camera.The circular cylinder p rojectile is launched into the trial tank by the light gas gun,and at the same time,the high speed camera is turned on to shoot themovementof the circu lar cylinder in thewater.The shape of cavity is obtained from the photograph shot by the high speed camera.

Figure 2 Schematic diagram of experim ent equipm ent

3 Results analysis and discussion

3.1 Effects of slenderness ratio variation on the cavity shape

Researches on cavitation showed that the scale effects can beneglected.The definition of slenderness ratio of circular cylinder is as follow：

Where L is the length of cylinder body,D is the diameter of cylinder body.

The cavitation numberwhich is one of themost important parameters describing the occurrence of cavitation and the characteristics of cavitating flow,it is defined as：

Where p∞and V are reference pressure and velocity, respectively,ρis the density of the liquid.

A number of numerical simulations have been conducted in the range of the slenderness ratioλ= 0.25～20.In general case,the necessary condition to form supercavitation is that the cavitation number is less than 0.1[8].For the cavitation numberσ= 0.220 andσ=0.040,the cavity shapes in the range of the slenderness ratioλ=0.25～20 are shown in Figure 3 respectively.At a larger cavitation number, the supercavitation can be formed when the slenderness ratio of the cylinder is small.However,the cavity closes on the body of the cylinder and the cavity can't envelop the entire cylinder,as the slenderness ratio increases.

Figure 3 Cavity shapes vs slenderness ratios at different cavitation numbers

The results indicate that supercavitation can be easily formed with smaller slenderness ratio at a given cavitation number.With the increasing of theslenderness ratio,the cavitymay close on the body of cylinder.To get the supercavitation,the cavitation number must be decreased further.The cavity shapes in the range ofλ=0.25～20 have little differences atσ= 0.220 andσ=0.040 respectively.Hence,the cavity shape has little relationship with the slenderness ratio of cylinder at thismoment.

3.2 Comparison between numerical sim u lation and projectile experim ent

The results of numerical simulation and experiment both indicate that the cavity is incipient at forepart of cylinder first.When the cavitation number is larger,the partial cavity is formed and the head of the cylinder is enveloped.As the cavitation number decreases,the length and diameter of the cavity increase,and the supercavitationwill occur eventually. The comparison of cavity shapes between numerical results and experimental results at different cavitation number are shown in Figure 4(The slenderness ratio of circular cylinder is 10).The cavity shapes of numerical simulation and experimentare in great agreementwith each other.

Figure 4 Cavity shapes by numerical simulation and experiment

The ratio of the maximum length of cavity to the cylinder diameter is defined as dimensionless lengthand the ratio of themaximum diameter of cavity to cylinder diameter is defined as dimensionless diameter.Some resultsaccording to the resultsof the numerical simu lation and experiments are shown in Figure 5.

Figure 5 Dimension less supercavitation length and diam eter vs cavitations num ber

The dimensionless length and diameter in projectile experimentsare slightly larger than the numerical simulation data.The difference may be explained by the following reasons：firstly,as the velocity of the projectile body decreases,the hysteresis effect of the cavitation affects the cavity shapes;secondly,the contour of cavity is not very clear in experimental results,and some errors are brought into measurements.The cavity shape has a consistent trend both in the experiment and the numerical simulation,it changes exponentially w ith cavitation number.Fitting equations of the curvesare as follows：

Based on the experimental resuis expressed respectively as：

Based on the numerical simulation results,andis expressed respectively as：

3.3 Effects of supercavitation on d rag characteristics of circular cylinder

The total drag is defined as：

Where Fpis pressure drag,and Ffis friction drag.

The p ressure drag coefficient Cp,friction drag coefficientCf,total drag coefficient is Cddefined as：

Where F is the drag of circular cylinder.S is the characteristic area,which is the cross sectional area of circular cylinder.

As the cavitation forms,the type of flow around the circular cylinder changes to cavitating flow.Because the wet surface decreases,the drag of the circular cylinder decreases compared with all wet condition.Figure 6 shows that the drag coefficient varies with the cavitation number.When the cavitation number decreases from 0.22 to 0.124,the cavity forms on the head of the circular cylinder,and then a long near ellipsoidal cavity forms around the total circular cylinder.The long ellipsoidal cavity has a good streamline shape,which reduces the pressure drag coefficient of the circular cylinder.At the same time,the friction drag of the circular cylinder decreases,because thewet surface of the circular cylinder decreases.When the cavitation number is less than 0.124, the supercavitation forms,and the pressure drag coefficient still decreases slowly.The friction drag coefficient almost remains zero,because the wet surface is only the front end surface.The total drag coefficient almost is same as the pressure drag coefficient.

Figure 6 Drag coefficients vs cavitation num ber

When the cavitation number is 0.01,the total drag coefficient is 0.869.The relative error is 4.9% compared with the experimental data 0.828[9].The relative error is small(Figure 7).

Figure 7 D rag coefficient vs cavitation number

Figure 8 shows that the drag coefficient varies with the slenderness ratio of the circu lar cylinder under different cavitation number.Under the condition of big cavitation number,the d rag coefficient decreases,and then increases,as the slenderness ratio increases.When the slenderness ratio is2,the drag coefficient has a minimum.Figure 9 shows that the pressure drag coefficient,friction drag coefficient and total the pressure drag coefficient,friction drag coefficient and total coefficient vary with the slenderness ratio of the circular cylinder under that the cavitation number is 1.979.The pressure drag coefficient decreases,and then keeps invariant as the slenderness ratio is bigger than 2.On the other hand,the wet surface of the circular cylinder increases as the slenderness ratio increases,and so,the friction drag coefficient increases.Therefore,the total drag coefficient has am inimum,as the slenderness ratio is 2.Under the condition of small cavitation number,the friction drag coefficient almost is zero,and the total drag coefficient is equal to the pressure drag coefficient,because the circular cylinder isenveloped by the supercavitation except the frontend surface of it.Under the condition of same cavitation number,the variation of the slenderness ratio of the circular cylinder has no effecton the cavity shape,and so,the total drag coefficient is equal to the p ressure drag coefficient, which almost keeps invariant.

4 Conclusions

Based on numerical simulations and projectile experiments,the cavity shape and the drag characteristics of the cavitating flow around the circu lar cylinder are investigation under the conditions of different cavitation numbers and slenderness ratios.The main conclusions as follows：

The cavity shape and drag coefficients ofnumerical simulation are in good agreement with the experimental ones.The numerical simulations results were validated.

At a given cavitation number,the cavity shape has no apparentdifference atdifferentslenderness ratios of circular cylinder,and the cavity shape depends on the cavitation number.The cavity is closed on the cylinder when the slenderness ratio and cavitation number is both big;the supercavitation forms when the slenderness ratio or the cavitation number is small.Cavity shapes and their dimension less fitting equations are obtained by numerical simulation and experiments respectively.

Under the condition of big cavitation number, the total drags coefficient decreases and then increases,as the slenderness ratio increases.The total drag coefficienthas aminimum valuewhen the slenderness ratio is 2.Under the condition of small cavitation number,the slenderness ratio has no effects on the drag coefficients.For every given slenderness ratio at the small cavitation number,the cavitation number decreases,the drag coefficient decreases.

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