Unsteady supercavitating flow past cones*

262

Journal of Hydrodynamics Ser.B, 2006,18(3): 262-272

sdlj.chinajournal.net.cn

UNSTEADY SUPERCAVITATING FLOW PAST CONES* HU Chao Institute of Aerospace Engineering and Applied mechanics, Tongji Universtity, Shanghai 200092, China Department of Aerospace Engineering and Mechanics, Harbin Institute of Technology, Harbin 150001, China, E-mail:[email protected] YANG Hong-lan, ZHAO Cun-bao, HUANG Wen-hu Department of Aerospace Engineering and Mechanics, Harbin Institute of Technology, Harbin 150001, China

(Received March 8, 2005) ABSTRACT: Based on integral equation method, the study of unsteady supercavitating flow past cones is presented. The shape and length of supercavity are calculated respectively using the finite difference time discretization method. The characteristics of the shape and length of supercavities, which vary with the cone’s angle and cavitation number, are investigated respectively, the varied features of some supercavity scales are analyzed when the flow field is perturbed periodically. The curves relationship between cavity length and cavitation number, which are based respectively on present method and other theories, are discussed and compared. It is obviously shown that the supercavity changes have two characteristics: retardance and wave. These results obtained would be useful in the case of design and analysis of cavitator under water. KEY WORDS: supercavitation, cavitator, unsteady flow, slender body theory, integral equation method

1. INTRODUCTION As a natural phenomenon, cavitations would occur when the flow pressure is close to vapor pressure. The early research on cavitation flow mainly focused on destruction and protection of cavitation phenomenon to power equipment. It was later discovered during research on drag reduction caused by cavitations that hydrodynamic drag of a body under water, especially the friction drag, is reduced greatly when cavitation occurs. A number of basic

theories and experiments [1-26] on formation, controlling and stability of cavity were developed. It is obvious that cavitation phenomenon, especially supercavitation is always an important research subject in the mechanics field. The problem of cavitating hydrofoils has been studied by many researchers. Super-and partiallycavitating 2D and 3D hydrofoils have been investigated based mainly on boundary integral equations and numerical methods. Rowe and Biottianux[1] analyzed 2D and 3D partially-cavitating hydrofoil and Semenenko [2] investigated the problem of flow through a cascade of slender oscillating hydrofoils in both partial cavitation stage and supercavitation stage. In the paper, the flow was considered as potential, and the hydrofoil oscillations were supposed to be small and co-phased. He focused main attention on calculation of a variable length, shape, and volume of the cavities and also on investigation of their influence on the cascade performance. Semenenko[3] investigated the problem on low-disturbed flow around two-dimensional supercavitation and ventilated oscillating hydrofoils under a free surface. A method to calculate the length and shape of 2D unsteady supercavitaties past slender wedges and around slender hydrofoils was developed by Semenenko. The problem on cavitating flow in

* Project supported by the National Nature Science Foundation of China (Grant No: 10572045) and Distinguished Young Scholar Science Foundation of Heilongjiang Province of China(Grant No: JC-9). Biography: HU Chao (1961-), Male, Ph.D., Professor

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case of arbitrary time-dependence was investigated based on solving integral equations using the finite difference time discrimination method [4]. The mathematical model of the plane unsteady supercavity that’s based on the 2-nd linearized Tulin’s scheme with infinite wake and finite pressure at infinity was constructed by Semennko[5]. Stability of the ventilated supercavity in both the infinite stream and the free jet were also investigated. The obtained results were compared with both the experimental data on the plane ventilated cavity pulsation and previous solution based on Tulin’s linearized theory. The research on axisymmetric cavitation flows, especially unsteady supercavitating flows, had been done little. Nesteruk[6] investigated the solution to slender axisymmetric cavity in a heavy liquid. The influence of a supercavitating model design parameters on model’s range and its perturbed motion features were investigated by Putilin[7] and some important results were obtained. The axisymmetric cavitator is classified into several types, such as cone, disc etc. The drag reduction using supercavitation technology is a key method, in body under water movement at high speed. Therefore it is necessary to investigate first the formation, movement, and stability of supercavities. Semenenko[8] constructed a solution of the problem on 2D unsteady cavities in an unbounded flow based on the linearized scheme of Tulin and investigated a global stability of the stationary gas-filled 2D cavities. Experimental and computer-simulation results of the phenomena accompanied unsteady supercavitation movement bodies under water that were presented by Semenenko and Savchenko[9]. Serebryakov[10] analyzed the problem of high speed movement on slender axisymmetric bodies under water with supercavitation. Based on the singular integral equations method, Semenenko[11] investigated 2D unsteady cavities past a wedge at aperiodic time dependence and the influence of elasticity of gas filling the ventilated supercavity Serebryakov [12] investigated the dynamical behavior of slender movement body during high speed in supercavitating flow while at the same time he analyzed the problem on formation and reduction of cavitation drag. Nesteruk [13] reviewed theoretical and numerical results based on the integral-differential equations for the form of an unsteady axisymmetrical supercavity and obtained with the use of slender body theory. At the present, numerous researches studying the dynamical behavior of cavitating flow are being developed in many domestic scientific research units. Xie et al. [14]calculated the developed cavity shape and the hydrodynamics characteristics of cavitating axisymmetric bodies by a numerical method. Cheng investigate 2D partial-cavitating and Lu [15]

hydrofoils based on the boundary integral equation method[15]. Fu and Li [16] analyzed steady partiallycavitation flow around axisymmetric body based on panel method, which employed sources which distribute on the axisymmetric body and cavity surfaces. Fu et al. [17] investigated the characteristics of flow around cavitating bodies of revolution of a single-fluid with variable density approach. The drag reduction mechanism of cavitating flow was also probed. The following are 2 types of methods applied to analyze the cavitating flow problem. First, determine the cavity shape and size when the cavitation number is pre-determined (direct problem). Second, determine the cavity shape, cavitation number, et.al when the cavity size is pre-determined (inverse problem). The relations between cavity shape or size and cavitation number as well as the characteristics of these time-dependence must first be studied in order to determine the formation and development regularity of the cavity. The macroscopic dynamical characteristics of partially and super-cavitating flow had been successfully investigated based on potential flow theory before. The relations between physical parameters of supercavitating flow and the characteristics of the time-dependence can be simply determined based on the methods above. However these methods ignore some minor details of cavitating flow such as two-phase flow and turbulent flow. At the same time, these methods can be interpreted as ideal gas-phase and no fluid in cavitation region due to the fluid being non-viscous and incompressible, while the flow is irrotational outside. Experimental investigation on ventilated cavitation for slender axisymmetric body were done by Xie and He[18] . In this experiment the shape of cavity, the base pressure and the steady hydrodynamic forces and moments are examined in detail in the experiment. Liu et al. [19] conducted an experimental study in a cavitation tunnel for four axisymmetric bodies at different degrees of attack angle. The frequency characteristics of hydrodynamic loads on axisymmetric slender bodies were obtained. Gu et al. [20] investigated the features of four stages of the pellet water-entry: collision, formation of the flow, cavity opening, cavity closure. The behavior of supercavitating and cavitating flow around a conical body of revolution with and without ventilation at several angles of attack was studied experimentally by Feng et al.[21]. conducted experiments to Yuan et al. [22] investigate asymmetry of ventilated supercavity configuration. The relations between supercavity asymmetry and the related factors were found qualitatively and quantitatively. Huang et al.[23] outlined the current research situation of reducing drag by microbubbles and discussed the mechanism of

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reducing drag by use of micro-bubbles in turbulent environment. In addition Gu et al. [24] carried out experiments in high speed water tunnel for investigating the cavities characters of headforms and their influence on the drag of slender body. Zhang et al. [25] carried out experimental investigations on ventilated cavity flow pattern of slender bodies and their control methods in the high speed water tunnel. Natural and ventilated cavitations generated on a smooth - nosed axisymmetric body were studied experimentally by Feng et al.[26]. It can be shown from above literatures that little theoretical research on unsteady supercavitating flow past slender body has been undertaken thus far thus making important to extend the investigation into the characteristics of cavitating flow for the movement body under water. This paper investigates supercavitating flow past cones using integral equation method. The shape and length of unsteady supercavity have been calculated. Some numerical results based on the finite difference time discretization method, such as the variation history of the supercavity length at variation of the conical shape and cavitation number and at small perturbation of cross section, are presented.

2.

THE EQUATIONS OF UNSTEADY SUPERCAVITY AND ITS SOLUTIONS We will consider the problem on unsteady supercavitating flow past a slender cone, as indicated in Fig.1. The direction of inflow is assumed to be parallel with the symmetric axis of the cone and in the case of neglecting fluid’s gravity an axisymmetric supercavity can be formed behind the cone. We will select cylindrical coordinate system (r ,θ , x) , therefore this problem can be translated into a two-dimension problem because all parameters concerned with flow are independent of θ . The radiuses of meridian plane of slender cone and supercavity are given separately as r = r ( x, t ) and r = R ( x , t ) . The flow is assumed to be potential. The flow domain of the study represents a plane with a slit along the interval 0 ≤ x ≤ l (t ) .

The velocity potential of corresponding to inflow is given as

incident wave ϕ , perturbation (i )

ϕ ( b ) , and total velocity (i ) (b ) potential of wave field is ϕ = ϕ + ϕ . For an

velocity potential is given as

incompressible fluid, the unsteady Bernoulli’s equation, namely Cauchy-Lagrange integral equation, is given as following

∂ϕ 1 2 p c 1 2 p∞ + Vc + = V∞ + ∂t 2 ρ 2 ρ

(1)

According to Eq.(1), we can derive the following formula,

∂ϕ σ v (V ∞2 ) −1 + c = ∂t V∞ 2

(2)

where V∞ , Vc represent the velocity of inflow along the x-axis and velocity of cavity surface, given as Vc = V∞ + vc . σ being the cavitation number is given as σ = ( p∞ − pc ) /

1 ρV∞2 . 2

Characteristic length and characteristic velocity are respectively the height of cone a and inflow velocity V∞ ,so the characteristic time is a / V∞ . For this reason, other non-dimensional variables have following forms (The superscript “ ～ ” designates nondimensional variables): the reduced frequency ω~ = ωa / V∞ , the velocity v c = v c / V ∞ , the cavity

~ l = l / a , the velocity potential ~ ϕ = ϕ /(aV∞ ) . Nondimensional coordinates are x = a~ x and t = a t .

length

V∞

According to dynamical boundary condition and Eq.(1), we can obtain these following nondimensional equations about perturbation velocity potential,

∂ ϕ (b ) ∂ ϕ ( b ) σ + = ∂t ∂x 2

(3)

We suppose that the expression of cone’s angle and cavitation number at an arbitrary time will have the following forms,

α = α 0 + δ1α 0 sin(ω t ) = α 0 + δ1α 0 sin(ω t ) Fig.1 Schematics of unsteady supercavitating flow pasta cone

(4a)

σ = σ 0 − δ 2σ 0 sin(ω t ) = σ 0 − δ 2σ 0 sin(ω t ) (4b)

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where δ1 , δ 2 are respective proportion factors of fluctuation range determining the cone’s angle and cavitation number, ( δ1 = 1/ 3, δ 2 = 1/ 2 while calculating). After regularizing those kinematical and dynamic boundary conditions’ expressions by introducing these non-dimensional variables and leaving out the superscript “ ～ ”, we obtained the following relationship,

vr = ϕ r = Nr ( x, t ), 0 < x < 1, r = 0

(5a)

vr = ϕ r = NR ( x, t ),1 < x < l (t ), r = 0 1 2

ψ = Nϕ = σ (t ),1 < x < l (t ), r = 0

Laplace transforms of equality (7b) has following form,

S x ( x, p) + pS = q ( x, p) So

boundary condition turns into S (0, p ) = 0 corresponding to S (0, t ) = 0 . The solution of above Eq.(8) concerning S ( x, p ) can be obtained, as following, x

S ( x, p ) = ∫ e − p ( x − s ) q ( s, p )ds

∫

l (t )

0

q ( s, t ) 1 ds = x−s 4π

∫

l (t )

0

(5c)

q ( s, t ) i x−s

si gn( x − s)ds

We can obtain the relationship between cross-sectional area of cavity and source strength based on the inverse transforms of above equation with time delay, x

S ( x, t ) = N −1q ( x, t ) = ∫ q ( s, t − x + s ) ds 0

1 ≤ x ≤ l (t )

boundary

condition

(10)

So the radius of cross-section of cavity is R( x, t )= S ( x, t ) π . We will obtain the following singular integral equation through substituting formula (6) into Eq. (5c)

∫

∂ q ( s, t ) si gn( x − s )ds − i 2 ∂t ( x − s) l (t ) q ( s, t ) ∫0 x − s s i gn( x − s)ds − 2πσ (t ) = 0, 0 < x < l (t )

l (t )

(11)

(6)

where sign(⋅) is Sign function. According to slender body theory, we can obtain these source strengths located respectively on symmetric axis of cone and cavity, q ( x, t ) = 2π rvr ,

q( x, t ) = 2π RvR ,

(9)

(5b)

0

1 4π

its

0

where N = D Dt = ∂ ∂ t + ∂ ∂ x is regularized linear differential operator, ϕ ,ψ are respectively perturbation velocity potential and perturbation acceleration potential. In the unsteady cavitating flow case, cavitation number σ (t ) , cavity length l (t ) and source strength q ( x, t ) all depend on the time variable. Based on slender body theory, perturbation velocity potential in flow field can be described by distributing point source on x-axis, and have the following form,

ϕ=

(8)

moreover, is

its

kinematical

vr = Nr ( x, t )

or

vr = Nr ( x, t ) . So we have the following relations,

q ( x, t ) = Ns ( x, t )

(7a)

q ( x, t ) = NS ( x, t )

(7b)

where s ( x, t ) , S ( x, t ) are cross-sectional area of cone and cavity.

respectively

It is easy to calculate this integral on interval [0, 1] because source strength q ( x, t ) on interval [0, 1] is known. Thus, in the general case this kind of problem can be come down to initial value and boundary value problem. The initial conditions is following form,

ϕ ( x , r , t ) t = 0 = ϕ 0 ( x, r ) ， l (t ) t =0 = l0 , σ (t ) t =0 = σ 0

(12)

The solution to this problem can be obtained by means of solving Eq. (11). A restrictive condition, namely the condition of Newmann boundary, must be added to Eq. (11) in order to obtain the solution, as following

266

∫

l (t )

0

(n)

q ( s , t ) ds = 0

(13)

For each iterative step, the functions B1 ( x) ,

B3( n ) ( x) is known if the cone shape is given, at the

We note that application of this restrictive condition can ensure boundedness of the pressure when approaching infinite in flow field. In addition, based on equality (7b), in the steady flow case the cross-section area of cavity has following form,

same time the value l (n ) is considered to be known. Then the linear algebraic equations with respect to q ( n ) ( s ) and σ ( n ) can be obtained.

x

S ( x , t ) = S ( x ) = ∫ q ( s ) ds 0

(14)

Therefore, the cross-section area of cavity at trailing-edge is zero, l

0 = ∫ q ( s ) ds 0

(15) It can be known that Eq. (13) is also a condition of cavity closure in the steady flow case by comparing Eq. (15) with Eq. (13). In the case of unsteady flow, Eq. (13) is still right, but the cavity is unclosed.

3. NUMERICAL ALGORITHM The set of the Eqs. (11) and (13) is non–linear because the integrand and the cavity length l(t) is all the unknown time–function. Its solution is sought numerically in sequential moments t ( n ) = t ( n −1) + Δt ( n = 2,3,...) with the initial conditions

t (1) = 0 , q (1) ( x ) = q0 ( x ) and l (1) = l0 . Then, for the n-th time step the Eqs. (11) and (13) may be written in the following form l( n)

∫

1

q ( n ) ( s) 1 si gn( x − s ) ds − i 2 Δt ( x − s)

q( s) (n) ∫1 x − s s i gn( x − s)ds − 2πσ = 1 ( n−1) B1( n ) ( x) − B ( x), Δt 2 l( n)

∫

l(n)

1

Fig.2 History of the supercavity length at variation of the cone’s angle

1 < x < l (n)

(16)

q ( n ) ( s ) ds = B3( n )

(17)

( n −1)

where B2

=∫

l ( n−1)

1

q ( n −1) ( s ) s i gn( x − s ) ds . x−s

3.1 Unsteady natural vapor supercavity In the case of natural vapor supercavity, the pressure in cavity is constant ( i.e. pc is const) and cavitation number depend only on the pressure p∞ and inflow velocity V∞ . Figure 2 gives examples of calculation of history of the natural vapor supercavity length with changing the cone’s angle at all variations

267

of reduced frequency. Figure 3 gives examples of history of the supercavity length changing with sinusoidal variation of cavitation number when cone angle is fixed. Continuous line and broken curve denote respectively the corresponding unsteady and quasistationary dependencies of cavity length l (t ) . In the quasistationary case of flow, they are calculated by omitting time part of the differential operator N = ∂ ∂ t + ∂ ∂ x in both the Eq. (11) and the relation (13).

asymmetrically when the initial value of supercavity length is longer and the variation frequency of cone’s angle is higher. (2) The higher the reduced frequency ω , the less the variation of cavity length. (3) The cavity length l (t ) change becomes non-monotone at sufficiently high variation frequency ω of cone’s angle. In Fig. 2(c), cavity length can reach new equilibrium value l1 = 4.6 from initial value l0 = 8.0 when the cone’s angle decreases from initial value α 0 to 2 / 3α 0 . It can be shown that the when variation frequency ω is higher, the cavity reach the balanced length with more time lagging. In Fig.3, the lower part of the figure represent the change of cavitation number σ (t ) / σ 0 with time t at given cone’s angle, the top curves represent the variation of cavity length l (t ) / l 0 with time t under the different frequency perturbation of cavitation number σ (t ) / σ 0 . Continuous line and broken curve denote respectively the corresponding unsteady and quasistationary dependencies of cavity length l (t ) . It can be shown that the results obtained from Fig.3 are similar with Fig.2.

Fig.3

Fig.4(a)

History of the supercavity shape with time at variation of cone’s angle( ω = 0.5)

Fig.4(b)

History of the supercavity shape with time

History of the supercavity length with time at variation of cavitation number

In Fig.2, (a) is similar process with (b), but initial value of supercavity length in (b) is bigger. The cone’s angle α increases in 4/3 times for different time intervals and then decreases to its initial value α 0 . Compared with quasistationary behavior of l (t ) , the unsteady supercavity behavior has the following characteristics, (1) The change of supercavity length occurs more

268

at variation of cone’s angle( ω

= 2)

According to the formula (10), Fig.4 shows the change curves of cavity length vs. time corresponding to Fig.2(b). Every curved face is composed of the cavity’s meridian line at different time. We can see that the cavity shape will go on changing for some times to go back to the initial steady shape when the perturbations of cone angle stop. Corresponding to Fig.3, the change of cavity shape with time under three kinds of frequency perturbation of unsteady cavitation number are depicted in Fig.5. If we assume the change of cavitation number will result from the variation of system pressure, as Fig.5 shown, the change of cavity shape can’t stop immediately and go on for some times and the cavity shape finally comes back to initial steady cavity shape when the perturbations of pressure stop.

s( x, t ) = πα 02 x 2 + A Re[ s* ( x)e−iω t ] q( x, t ) = πα 02 q0 ( x,l ) + A Re[q* ( x,l )i e − iω t ]

(18b)

σ (t ) = πα 02 σ 0 (l ) + A Re[σ * (l )e-iω t ]

(18c)

S ( x, t ) = S0 ( x) + A Re[ S * ( x, l )e−iω t ]

(18d)

We assume the order of several mechanical quantities: πα 02 ~ A ~ O (ε ) ， s∗ ~ q∗ ~ σ ∗ ~ S ∗ ~ q0 .

ε

~ σ 0 ~ S0 ~ O(1) ,

is small parameter,

ω is

exterior perturbation frequency, the functions q0 ( x, l ) , q ∗ ( x, l ) depend on the unsteady cavity length l (t ) like on a parameter. Substituting the expression(18) in the Eqs.(11),(13), we can obtain respectively the equations of steady part and perturbation part of flow, as the steady part has following singular integral equations,

q0 ( s ) si gn( x − s )ds − 2πσ 0 (l ) = 1 ( x − s)2 1 1 −2[ln(1 − ) + ] x (1 − x)

∫ Fig.5(a) History of the supercavity shape with timeat variation of cavitation number ( ω = 0.5)

(18a)

l

(19)

l

∫ q ( s)ds = −1 1

(20)

0

So the singular integral equations of perturbation part is l

iω ]si gn( x − ξ )dξ − (x − ξ ) 1 1 iω 2πσ * = − ∫ ϕ r* (ξ )[ + ]i 2 0 (x − ξ ) (x − ξ ) 1

∫ q (ξ )[ ( x − ξ ) 1

Simple harmonic perturbations of several mechanical quantities Now, we will investigate the flow in the case of cross-section perturbations of cone. The following exponential part in formula (18a)-(18d) represents perturbation part, s (x ) and S (x ) is respectively the cross-section of slender cone and supercavity

2

+

s i gn( x − ξ )dξ

Fig.5(b) History of the supercavity shape with time at variation of cavitation number ( ω = 2)

3.2

*

∫

l

1

1

q* (ξ )dξ = − ∫ ϕ r* (ξ )dξ

where ϕ r* (ξ ) = (

(21)

0

(22)

d − iω ) s* (ξ ) , s* (ξ ) = iξ 2 . dξ

The right parts of Eqs. (21) and (22) are known. After separating a real part and an imaginary part of

269

the Eqs. (21) and (22), we can obtain four equations. Then we can solve a set of seven equations, which include above four equations, the Eqs. (19),(20) and a equation obtained from formula(18c). Substituting formulae (18a) and (18b) in the formula (7b), we can obtain following equation through equalizing the right and left perturbation part of equation and omitting time factor,

∂ S * ( x, l ) − iω S * ( x , l ) = q * ( x , l ) ∂x

(23)

By solving this first order linear equation, the perturbation amplitude of cavity cross-section area can be obtained, as the following shown, x

S * ( x, l ) = ei ω x ∫ q* (ξ , l )e -iωξ dξ 0

sinusoidal vibration

are

increasing

when

the

frequency ω and the oscillating amplitude A of perturbation are increasing. The function of cavity length l (t ) is discontinuous when the frequency ω and the amplitude A exceed some value. Figure 7 describes change history graphs of cavity shape vs. time at three different cross-section perturbation frequencies of cone. As can be shown in Fig.7, the lower the perturbation frequency, the bigger the amplitude of cavity shape. The Riabouchinsky’s cavity close model is selected in the case of steady flow. S 0 ( x) represents the cross-section of steady supercavity. We can obtain the expression of cavity cross-section under small perturbation by * substituting S ( x, l ) in the formula of S ( x, t ) .

(24)

The following formulas can be obtained from the formula (18d) and (24),

S * ( x, l )ei kx e-i ωt = ei kx e-i ωt i

∫

x

0

q* (ξ , l )e-iωξ dξ = C ( x, k )ei ω ( x -t )

(25)

where, wave number k = ω V∞ = ω . We can see that the bigger the wave number corresponding to cavity shape, the higher the perturbation frequency, wave-motion on cavity surface travel at velocity V∞ .

Fig.6

Fig.7

Amplitude of cross-section of cavity at different frequency ( A =0.1)

Figures 8(a), 8(b) describes a change history graphs pertaining to cavity shape vs. time at several different perturbation frequencies. In fig.8, it can be seen that the higher the perturbation frequency, the more wave number of cavity. The oscillation of cavity boundary point travel at velocity V∞ along cavity surface.

Effect of cross-section perturbation on cavity length ( l0 =6.0 , ω

= 0.5)

In Fig.6 we have plotted a change history graph of l (t ) during a periods at three different relative amplitudes A= A/α 0 . These graphs show that the differences between cavity length’s fluctuation and

Fig.8(a) History of cavity shape with time

270

(ω

= 1， A =0.1)

known that when

Froude

Number’s

value is very

much bigger, the effect of fluid’s gravity can be neglect. Figure 9 gives the comparison of the relationship curves between cavity length and cavitation number, which continuous curve represents the present numerical results, broken curve represents the results of the literature [6].

Fig.8(b)

History of cavity shape with time

(ω

= 3， A =0.1)

4. ANALYSIS AND DISCUSSION In this paper, a fluid’s gravity isn’t considered. Nesteruk[6] investigated the shape of slender axisymmetric cavity in a heavy liquid. It is known that when Froude Number is approaches infinity, this case can be seen as omitting effect of fluid’s gravity. In the literature[6], the following formula represents the case of flying off of water for slender body,

ε=

σ x*2 x*3 + + 2 R0 R0′ x* + R02 2 2 ln ε 3Fr ln ε

x* = −

σ Fr 2 2

+ (

σ Fr 2 2

(26a)

Fig.9 Comparison of the relationship curves between cavity length and cavitation number ( Fr = 1000 ) (Continuous curve represents the present results, broken curve represents the results of the literature[6])

It can be seen in Fig.9 that two kinds of relationship curves between cavity length and cavitation number obtained, respectively based on the present method and the results of the literature [6], are basically coincident. But a little difference exits when cavity length is greatly shorten or greatly lengthened.

) 2 − 2 R0 R0′ Fr 2 ln ε (26b)

σ (1 − l1 ) 2 (1 − l1 )3 + + 2 R0 R0′ (1 − l1 ) + R02 = 0 2 ln ε 3Fr 2 ln ε (26c)

l1 = R0 (l1 / R0 ) = R0 [tan α ]−1

(26d)

where σ and Fr denote respectively cavitation number and Froude Number as known parameters. ε is maximal radius of cavity, x* is the location coordinate of maximal radius, l1 is length of slender cone, R0 is bottom radius, cavity length is l = 1 . By solving the Eqs.(26a),(26b),(26c),(26d), we can obtain the solution to four unknown parameters. We can solve this set of equations when cavitation number, Froude Number and the tangent of cone’s half- angle tan α = R0 / l1 are given. It is

Fig.10 Comparison of the relationship curves between cavity length and cavitation number ( Fr = 1000 )

In addition, we obtain respectively the relationship curves of 2D wedge and axisymmetric cone between cavity length and cavitation number based on the present method. Figure 10 gives the comparison of two kinds of calculative results. It can be seen in Fig.10 that the cavity length of 2D wedge is much longer than cone cavity length when the angle of 2D wedge, cone, and cavitation number are the same. Moreover, the perturbation of revolution bodies

271

is less than 2D movement body in the case of same inflow conditions.

5. CONCLUSIONS The change of slender shape can give rise to the change in supercavity length with the more time lagging. Thus meaning that the change of cavity shape can’t stop immediately and go on in order to reach new balance when the perturbations of slender cone’s shape stop. Furthermore, the less cavitation number, the more obvious the time-lagging phenomenon. Similarly, the change of cavitation number can give rise to the change of supercavity length and cavity shape with the more time lagging. Small sinusoidal perturbation of cross-section belonging to a slender body can give rise to the corresponding change of cavity length. When the perturbation amplitude increases, the cavity length oscillation differs more and more from the sinusoidal one. Small perturbation of cross-section of slender body can travel on cavity’s surface in wave form and forms the perturbation wave shape. It can be shown that when 2D wedge angle and cone’s angle of slender axisymmetric cone and cavitation number are the same, the cavity length of 2D wedge are longer than the cone. Giving rise that in the case of same inflow conditions, the perturbation of revolution bodies is less than 2D movement body. This paper gives some numerical results to explain the method. By comparing with other theory and numerical results, it has been proven effective to investigate the cavitation flow problem using the present method. In fact, we investigate the cavitation flow problem not only to find the relationships between cavitation number and cavity size or cavity shape but also to determine the drag of movement body under water. We can determine the pressure drag of the total drag based on potential theory, but other part drag results from the viscidity of water. After obtaining the initiative pressure distribution of axisymmetric body based on potential theory in inviscid fluid, we can recalculate the pressure distribution by means of viscous correction method and at last obtain the right results coinciding with real experiment data, (viscous correction is viscous boundary layer correction). This paper has investigated the former problem and has studied the later problem. The results obtained based on the present method can be effectively applied to investigate the supercavitation flow past movement body under water and to instruct cavitator’s design.

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