CHEN Wei-qi

(Key Lab of Science and Technology on Hydrodynamics,China Ship Scientific Research Center,Wuxi 214082,China)

A Theoretical Study for Three-dimensional Cavity Re-entrant Jets

CHEN Wei-qi

(Key Lab of Science and Technology on Hydrodynamics,China Ship Scientific Research Center,Wuxi 214082,China)

Abstract:An important reason for various patterns of cavity shedding is the cavity re-entrant jet impinging against the cavity wall in a certain direction.However,the direction of the cavity re-entrant jet is currently considered as an uncertainty,which needs to be determined artificially.This paper focused on the three-dimensional cavity in an infinite flow field with gravity.A system of algebraic equations with trigonometric functions,which can calculate the angle and the sectional area of the reentrant jet,was deducted based on the potential flow theory and integral equations.A theoretical analytic formula for the angle and the sectional area of the re-entrant jet on the condition of small angles was obtained.The study shows that the angle of a cavity re-entrant jet relates to the attack angle and resistance of the cavitator and the cavity buoyancy.

Key words:cavity;cavity re-entrant jet;cavity hydrodynamics

0 Introduction

It occurs the cavitating effect in the high-speed movement of a cavitator(an underwater vehicle).As the pressure at the tail of a cavity is much higher than that inside the cavity,a jet flow will be generated at the tail of cavity under the action of adverse pressure gradient,which is called the cavity re-entrant jet.Previous studies showed that the key reason for the integral or partial cavity shedding is that the cavity re-entrant jet strikes the front section of cavity interface[1].The cavity shedding will cause negative influences to the performance of an underwater vehicle,such as noise and the abrupt change of pressure distribution,so it is of great importance to study the cavity re-entrant jet.

One of the key reasons for various patterns of cavity shedding is the different positions of the cavity wall where the jet strikes at,so the position depends on the direction of re-entrant jets.As both of the attack angle and the buoyancy will cause the asymmetry of the cavity,which leads to the direction change of the re-entrant jet,it is important to study the influence of factors such as the attack angle,and the buoyancy to the direction of a re-entrant jet.Early studies on cavities with re-entrant jets had been carried out successfully by Knapp,Daily&Hammitt[2]and Wu[3-4]using conformal mapping,and so on.Modern CFD gave detail information in-side the cavity.But little work with analytical expression of the direction and the section of the re-entrant jet has been reported yet.

The author[7]studied the theoretical model of the thickness of an axisymmetric and a twodimensional symmetric cavity re-entrant jet,and provided the analytical expression.This paper tries to solve the quantitative relationship among the cavity re-entrant jets and the cavity movement direction,the attack angle and the buoyancy.

1 Mathematical model of a steady cavity re-entrant jet

When the cavitator is far away from the water surface,which means the water surface has no influence on the cavitator,it can be seen that the cavitator is moving in an infinite flow field with gravity.As seen in Fig.1,S0is the surface of the cavitator,Scis the cavity wall,S is the sectional area of the cavity re-entrant jet which can be considered as uniform,Sb=S+Sc+S0is the closed surfaces that consist of the above three surfaces.Then let S∞be a closed surface surrounding the cavity in the distance.As a result,a control volumewith the fluid volume value of τ is formed.

Fig.1 The sketch of cavity movement in an infinite flow field with gravity

For a steady flow,the flow velocity potential on coordinate systemcan be expressed as

According to the boundary conditionthe velocity potential of the relative flow is

Applying momentum theorem for the control volumeproduces

According to the Archimedean principle,it is easy to see that the cavity buoyancy B is compatible with the relation

The acting force on the closed surfaces Sb=S+Sc+S0is as follows:

It needs to be noted that the pressure pcin the cavity is considered as constant,so for an arbitrary closed surface,we have

Therefore,Eq.(5)can be written as

The pressure p on the cavity wall surface Scand the sectional area surface S is equal to the cavity pressure pc,so the last two items on the right side of the Eq.(7)vanish and then it can be simplified as follows:

where R is the acting force on the cavitator.

By substituting Eq.(4)and Eq.(8)into Eq.(3),we have

Now consider two items on the right side of Eq.(11).on the cavity surface Scandcavitator surface S0,whileon the section of the re-entrant jet,whereis the velocity magnitude of the cavity re-entrant jet.Therefore the first item on the right side can be written as

The physical meaning of the second item on the right side of Eq.(9)is the momentum through the surround surface S∞.According to Eq.(2),the flow velocity tends to be a constant as.Then

It can be noted that the flow satisfied the continuity theorem on the closed surface,as

As the existence of the re-entrant jet,the flux through Sbis not zero.Then

Substituting Eq.(13)into Eq.(12),and then substituting the formula obtained into Eq.(11)produces

Then substituting Eq.(14)and Eq.(10)into Eq.(9),we have

The vector equation(15)reflects the relationship among the re-entrant jet velocity u1,the acting force R on the caivtator and the cavity buoyancy B.It is obvious that equation(15)is true without any restriction on cavitator shape,directions ofin oxyz.Therefore,it is called the mathematical model of a 3D cavity with re-entrant jet.

2 Sectional area and the direction of a steady cavity re-entrant jet

For a special case of Eq.(15),whileare all in oxy plane,the simplified expression of the sectional area and the direction of a cavity re-entrant jet can be obtained from Eq.(15).

According to the projections of the vector equation(15)in directions of cavitator resistance and lift,the following two algebraic equations can be obtained:

Comparing the two equations,an equation of trigonometric function in the direction of reentrant jets can be obtained,as

Based on the steady Bernoulli equation

where hsis the depth of the re-entrant jet section S below the water surface.The magnitude of the re-entrant jet velocity is

where σ is the cavitation number at the position of the re-entrant jet section S,which can be expressed as

?關(guān)于“特蘭托公會議對造型藝術(shù)的激進改變”，可參見：émile Male,L'Art religieux de la fin du XVIe siècle,du XVIIe siècle et du XVIIIe siècle ：étude sur l'iconographie après le Concile de Trente.Italie-France-Espagne-Flanders，Paris，1951.

Substituting Eq.(19)into Eq.(17)produces

Substituting Eq.(19)into the first equation in Eq.(16),the sectional area of the re-entrant jet can be written as

In Eq.(21),as all parameters are known except θ,the value of θ can be solved in principle.However,the simplified analytic expression can not be obtained as it is an equation of trigonometric function.It is found from experiments that θ is small when α andare both small,and hereBased on the above analysis,an simplified analytic expression of θ can be obtained,as

3 Solutions of several special cases

(A)For an axisymmetric cavity in an infinite flow field without gravity,B=0,θ=0,α=0.

From Eq.(16),there is

which is as the same as the classic results of Sedov[9].

The sectional area of the re-entrant jet can be expressed as

(B)For a horizontal cavity in an infinite flow field with gravity,β=0,α≠0.

From Eqs.(17)and(19),there is

It can be seen that the ratio of buoyancy to resistancehas an important influence to the direction of the cavity re-entrant jet.

(C)For a vertical cavity in an infinite flow field with gravity,is small.

Substituting them into Eqs.(22)and(23),the angle and the sectional area of the re-entrant jet can be expressed as

4 Conclusions

This paper focuses on a three-dimensional cavity with re-entrant jet in an infinite flow field with gravity.In the potential flow field,an equation is proposed which describes the relationship of the acting force of cavitator,the flow velocity,the cavity buoyancy,the direction of the re-entrant jet and the sectional area of the cavity re-entrant jet using integral momentum equations and continuum equations.Then an analytical formula for the direction of cavity re-entrant jet and sectional area is deduced using above equations.The formula can be used to calculate the sectional area and the direction of a three-dimensional cavity re-entrant jet,and the method can be easily applied two-dimensional problems.

The study in this paper shows that the sectional area and the angle of a three-dimensional cavity re-entrant jet can be quantitatively determined analytical expressions.

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[7]Chen Weiqi,Yan Kai,Wang Baoshou.Modeling of the thickness of the cavity reentrance jet in bounded flow field[C]//The 8th National Conference on Fluid Mechanics.Lanzhou,China,2014.(in Chinese)

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三維空泡回射流的理論研究

陳瑋琪
（中國船舶科學研究中心 水動力學重點實驗室，江蘇 無錫214082）

空泡回射流以某個方向沖擊空泡壁面是導致空泡發(fā)生不同脫落方式的重要原因，但是空泡回射流的方向目前認為是一個不確定量，需要人為指定。該文針對無界重力流場中的三維空泡，基于勢流假設和積分方程推導了一個計算回射流角度和回射流截面面積的三角函數(shù)代數(shù)方程組，并且在小角度條件下給出了回射流角度和截面面積的理論解析式。研究表明，空泡回射流角度與空化器的攻角、阻力和空泡浮力有關(guān)。

空泡；空泡回射流；空泡水動力

0302 0352

A

水動力學重點實驗室基金（6142203030702）

陳瑋琪（1971-），男，博士，中國船舶科學研究中心研究員。

10.3969/j.issn.1007-7294.2017.09.001

Article ID： 1007-7294（2017）09-1055-07

Received date：2017-06-19

Foundation item:Supported by Foundation of Science and Technology on Hydrodynamics Laboratory(No.6142203030702)

Biography：CHEN Wei-qi(1971-),male,Ph.D.,researcher,E-mail:tiger_cwq@aliyun.com.