- Email: [email protected]

STUDY ON CAVITATION FLOWS OF UNDERWATER VEHICLE MA Cheng Department of Naval Architecture and Ocean Engineering, Harbin University of Engineering, Harbin, 150001 & Institute of Naval Vessels, Naval Academy of Armament, Beijing 100073, China E-mail: [email protected] JIA Di Institute of Naval Vessels, Naval Academy of Armament, Beijing 100073, China QIAN Zheng-fang Institute of Naval Vessels, Naval Academy of Armament, Beijing 100073, China FENG Ding-hua College of Aerospace and Material Engineering, National Univ. of Defense Technology, Changsha 410073, China ABSTRACT: In this paper, numerical simulation was utilized to deal with the cavity generated at the tip and surface of vehicle. As for the water flow field of cavitation, the two-dimension axial symmetric Hess-Smith method, which was based on the potential theory and panel technique, was employed to deal with the problem. And surface pressure coefficient and drag character of cavity at axial symmetric body were analyzed. The tip drag of cavitators with different dimensions were also computed and discussed. KEY WORDS: underwater vehicle, cavitation, water flow field, drag coefficient

1. Introduction When high-speed vehicles move under water, cavitation generally occurs if the pressure in certain locations drops below the vapor pressure. Cavitation can be observed[1] in a wide variety of propulsion and power systems like pumps, marine propellers, POD propulsion[2], hydrofoils and other underwater bodies. Usually, undesirable cavitation causes structural damage, noise and power loss. It can distort water flow to rob pumps, turbines, hydrofoils, and propellers of operational efficiency. It can also lead to violent shock waves (from rapid bubble collapse), which cause pitting and erosion of metal surfaces. On the other hand, drag can be reduced on bodies surrounded fully or partially with a natural or gas-ventilated cavity. Generally, the ultimate speed of underwater vehicle is 40m/s～50m/s. If cavity surrounds fully the vehicle’s body, drag will be reduced obviously. It is called supercavitation. In this condition, the speed of vehicle can break 100m/s. If supercavitation is used for weapon, it will has great significance. Just like airplanes having realized sound barrier breakthrough, if underwater vehicles can move

fast in steady supercavity, underwater weapons will have revolutionary change. So many countries such as America, Russia, Britain, Germany, Japan etc. have done many researches.

Fig. 1 Supercavitation around high-speed fish torpedo

2. Types of Cavitation Different types of cavitation can be observed depending on the flow conditions and geometry. Each type of cavitation has distinct characteristics. İNANÇ ŞENOCAK[3] has done some researches. Five major types of cavitation will be described in the following pictures. Traveling cavitation A type of cavitation in which individual transient cavities or bubbles form in the liquid and move with it as they expand or shrink during their life cycles. It is typically observed on hydrofoils at small angle of attack. To the naked eye, traveling cavitation may appear as sheet cavitation. Individual transient spherical bubbles can be distinguished by snapshot photographs. Traveling cavitation on a hydrofoil is shown in Fig. 2(a). Cloud cavitation It is caused by vorticity shed into the flow field. It causes strong vibration, noise and erosion. The shedding of cloud cavitation is periodic and the re-

Biography: Ma Cheng (1963-), Male, Master, Doctor Candidate 373

entrant jet is the basic mechanism to generate cloudcavitation. The process is shown in Fig. 2(b). Sheet cavitation It is also known as fixed, attached, cavity or pocket cavitation. Sheet cavitation is stable in a quasi-steady sense. The phenomenon is shown in Fig. 2(c). The interface between the liquid and the vapor can be smooth and transparent or it can have the shape of a highly turbulent boiling surface. Downstream flow, which contains large scale eddies, is dominated by bubble clusters. Supercavitation If cavity grows in such a way to envelope the whole solid body then this type of cavitation is referred to as supercavitation. Supercavitation is desirable to achieve viscous drag reduction on underwater vehicles operating at high speeds. Other than viscous drag reduction, lift force acting on hydrofoils can be increased by creating a supercavity on the upper surface at a constant pressure. A typical supercavitating hydrofoil is shown in Fig. 2(d). Vortex cavitation It occurs on the tips of rotating blades (Fig. 2(e)). Cavities form in the cores of vortices in regions of high shear. This type of cavitation is not restricted to rotating blades. It can also occur in the separation zones of bluff bodies.

velocity, respectively, pc the vapor pressure at the bulk temperature of the liquid, and ρ the density of the liquid. The particular importance in the study of cavitation is the minimum pressure coefficient, C pmin given by

C p min

⎛V ⎞ p − p∞ = min = 1 − ⎜⎜ ⎟⎟ 1 ⎝ V∞ ⎠ ρV∞ 2 2

2

where pmin is the minimum pressure in the liquid. The “classical theory for scaling vaporous limited cavitation” states that, σ = −C pmin and C pmin = const . 3.2 Control formulation With the scope of potential flow, fluid motions are governed by the Laplace equation: ∇ 2Φ = 0 (3) where Φ = Φ ∞ + φ is the velocity potential of the

flow field; Φ ∞ is the far field velocity potential ; φ K is the perturbation potential ; the velocity is V = ∇Φ . It is assumed that the pressure within the cavity is constant. The shape of the cavity is unknown and is a function of its length and inside pressure. In order to solve Eq. (3) uniquely, several boundary conditions[2] have been identified for 2-D cavitating vehicle. (1) Far field Far away from the vehicle, the flow is nit disturbed, and a uniform distribution is assumed: ∇Φ → V∞ (4) (2) Kinematic condition On the surface of the hydrofoil and the cavity, the flow is tangent to the surface:

G G G ∂ϕ ∂Φ ∞ =− = −∇Φ ∞ ⋅ n = −V∞ ⋅ n ∂n ∂n where n is the unit normal to the surface.

Fig. 2 Visualization of different types of cavitation (a) Traveling cavitation, (b) Cloud cavitation, (c) Sheet cavitation, (d) Supercavitation, (e) Vortex cavitation

3. Partial Cavitation 3.1 Basic definitions Cavitating flows are commonly described by the cavitaion number σ , expressed as

p − pc σ= ∞ 1 ρV∞ 2 2

where 374

p∞ and

(1)

V are the reference pressure and

(2)

(5)

(3) Dynamic condition The pressure inside the cavity is constant. That is on the cavity surface: p = pc (6) where pc is the vapor pressure. And the cavitation number:

σ=

p∞ − pc q = ( c )2 −1 1 V∞ ρV∞ 2

(7)

where qc is the tangential velocity on the cavity surface. (4) Kutta condition ∇ϕ < +∞ (8) At the edge of the vehicle, velocities are finite.

(5) Closure condition

afterbody is 0.2m long.

(9) h( s L ) = 0 The thickness at the cavity trailing edge is equal to zero. 3.3 Integral formulation Considering an arbitrary 3-D body, we use the panel-element method[4] to deal with the problem. Continuous sources and dipoles are distributed on the body’s surface. The perturbation potential can be computed: ϕ ( x, y , z ) = − ∫∫ S

σ (Q ) rPQ

dS

σ (ξ ,η , ζ )dS

= − ∫∫

(10) ( x − ξ )2 + ( y − η )2 + ( z − ζ )2 where σ (Q ) is the source and dipole strength, and ( x, y, z ) (ξ ,η , ζ ) are coordinate dimensions. The boundary condition of the surface states that the normal velocity of the surface is zero: G G ∂ϕ K (11) |S = −V∞ ⋅ n |S ∇ϕ ⋅ n | S = ∂n Combining Eqs. (10) and (11), the following equation can be deduced: S

G G ∂ 1 2πσ ( P) − ∫∫ ( )σ (Q)dS = −V∞ ⋅ n ( P) ∂n rPQ S

(12)

3.4 Numerical discretization The surface boundary S is divided into N straight-line panels and the integral equation (12) is performed for each panel i : N K K ∂ ⎛⎜ 1 ⎞⎟ (13) ds = −V∞i ⋅ ni 2πσ i − ∑ σ j ∫∫ ∂ni ⎜⎝ rij ⎟⎠ j =1 Equation (13) can be expressed: N

∑σ j =1

j

[2πδ ij − ∫∫ Sj

∂ ⎛⎜ 1 ∂ni ⎜⎝ rij

K K ∑ Aijσ j = −V∞i ⋅ ni N

⎞ K ⎟ds] = −V∞i ⋅ nKi ⎟ ⎠

(i = 1,2,3,⋅ ⋅ ⋅, N )

(14) (15)

j =1

r

l lD

D

K V∞

PC

T

λl L

o

z Fig. 3 Partially cavitating model

A termination model must be applied at the end of the cavity. A pressure recovery termination model[5] is employed, by which the velocity in a transition zone (of length λl as shown in Fig. 3) departs from its constant value on the cavity D and T according to a prescribed algebraic law in the transition zone between T and L .

∂ϕ ∂Φ ∞ = q c [1 − f ( s f )] + ∂s c ∂s c where f ( s f ) is defined as follows:

⎧0 ⎪ f (s f ) = ⎨ s f − sT ν ⎪A[ s − s ] ⎩ L T where s f

(17)

s f < sT sT ≤ s f ≤ sL

(18)

is the arc-length of the foil measured

beneath the cavity, measured from the cavity leading edge, and A(0 < A < 1) and ν (ν > 0) are arbitrary constants, and they are determined by experiment. The cavity detachment point D , as shown in Fig. 3, is the position of the lowest pressure. The computation condition is 20 m/s velocity of incoming flow. The number of panels is 150. In Eq. (16), arbitrary constants are A = 0.2 , λ = 0.1 , ν = 1.0 . By solving Eqs. (1) to (16), the strength value of source and dipole on every panel can be known. The accurate position of the cavity surface and the new cavity arc length, can then be determined and by enough iterations.

Aij is the normal velocity infection coefficient. The total velocity of panel i is VTi : N K K VTi = ∑ Bijσ j + V∞i ⋅ ti

(16)

j =1

BTi is tangent velocity infection coefficient.

3.5 Numerical results and discussion 3.5.1 A termination model The body model is axis symmetrical. Its head is sphericity with the diameter of 0.08m. The middle part is cylinder with the length of 0.76m, and the cone

3.5.2 Results The detachment point is the end point of the head sphericity by computing. In Fig. 4, curve 1 is the shape of partial cavity; curve 2 is the body surface of symmetrical vehicle; curve 3 is the distribution of surface pressure coefficient. Figure 6 shows the pressure coefficient distribution on body and cavity surface. By comparing two conditions 1 and 2, we can find that on the front part of cavity, the pressure nearly has no change. But in the transition zone, the pressure change intensely. 375

1954) have shown that the net tip force acts approximately along the axis of the body with zero net applied moment.

0.9 0.8 0.7 0.6 0.5 3

cp

0.4 0.3

1

0.6

0.2 0.1 0

0.5

-0.1

2

-0.2 -0.3 0.5

0.75

Cd

0.4 0.25

z

Fig. 5 Pressure coefficient distribution ( A = 0.2 , λ = 0.1 , ν = 1.0 , l = 0.5 , N = 150 )

0.3

A = 0.2

λ = 0.1 ν = 1.0

0.2 0.8 0.6

0.3

0.4

0.5

0.6

0.7

0.8

L 2

0.4

Fig. 7 Drag coefficients of different length cavities

1

cp

0.2 0

-0.2 -0.4

0.45 -0.6 -0.8

0.4 0.25

0.5

A = 0.2

0.75

z

Fig. 6 Pressure coefficient distribution in two conditions 1：without partial cavity; 2：with partial cavity

When cavity appears on the surface of underwater vehicle, the shape of surface is changed. This will have great effect on vehicle dynamic character. Considering that the flow is uncompress and inviscid, the drag character of cavities with different lengths. The diameter of reference section is 0.08m, and cavity lengths are from 0.3m to 0.8m ( 0.3 ≤ L ≤ 0.8 ). Figure 7 shows different length cavities’ effect on drag coefficient. With the increasing length, the drag is becoming less. Figure 8 shows the relation between cavitation number and cavity length. We can find that the longer cavity is, the smaller cavitation number is. 4. Supercavitation Underwater vehicles such as torpedoes and submarines are limited in maximum speed by the considerable drag produced by the flow friction on the hull skin. Speeds of 40 m/s (75 knots) are considered very high; most practical systems are limited to less than half this figure. Flows exhibiting cavities enveloping a moving body entirely are called ‘supercavitating’, and, since the liquid phase does not contact the moving body through most of its length, skin drag is almost negligible. In the case of pure supercavitating flight, forces produced by the flow of water vapor may be a significant stabilizing effect at very high speeds. In the case that the body touches the cavity walls, these contacts may be of long-duration (planing), or intermittent (impacts). In initial study, we assume that the only force on the body is due to the fluid force at the tip. Laboratory experiments (May, 1975), (Kiceniuk, 376

λ = 0.1 ν = 1.0

0.35

0.3

0.25

0.2

0.4

0.5

0.6

0.7

L

Fig. 8 Cavitation numbers of different length cavities

Fig. 9 Disk cavitator

Fig. 10 Conical cavitator

The supercavity usually is generated by some instruments. Photographs[6] of cavities produced by disk and cone cavitators are shown in Fig. 9 and 10. Here, the drag character of cavitator will be discussed. Carrying out F = ma , we have:

mU = ∑ F D

(19)

60 55

The magnitude FD of the tip force is: 1 (20) FD = ρAcU 2 C D 2 where ρ is density of water, Ac the cross-sectional

50

dc=0.01

U,m/s

45 40 35 30

are of the tip, U the forward velocity, C D = k cos 2 α ,

25

k = cx 0 a nondimensional constant, α attack angle.

20 15

In the computation process, we chose the flowing conditions: The vehicle’s mass m is 0.15kg; the constant k is 0.8; the initialized velocity U of vehicle is 60m/s. And there are three different diameters of disk cavitators are inspected. They are d c = 0.01 m, d c =

0

0.1

0.2

t,second

(a) 60

50

0.025 m, d c = 0.05 m,. In 0.25 second movement duration, the velocity of vehicle decreases, and the trend becomes intense with diameter increasing. This is shown in Fig. 11.

dc=0.025

U,m/s

40

30

20

5. CONCLUSIONS In this paper, partially cavitation and supercavitation were studied. By assuming some proper conditions, we found several simple and useful conclusions: (1) On the front partial cavity surface, the pressure nearly had no change. But in the transition zone of cavity, the pressure changes intensely. (2) Considering that the flow is uncompress and inviscid, the drag of vehicle became less when the cavity was longer. Supercavity had the least drag. (3) The tip cavitator’s drag was a great part of the whole vehicle’s, and the bigger cavitator’s dimension was, the more obvious drag was.

10

0

0.1

0.2

t,second

(b) 60

50 dc=0.05

U,m/s

40

30

20

10

References 0

0.1

0.2

t,second

[1]

[2] [3] [4]

Zhang Dong, Liu Zhi-ping, Xiang Ya-ping. Cavitation Inception Witnessed by Sound Pressure Level in Model Test and Prototype Observation Journal of Hydrodynamics (Ser. B), 2004, 16(6): 227-232. Ma Cheng Improvement of trailing vortex model for single-screw POD propulsion Journal of Hydro- dynamics (Ser. B), 2004, 16(4)，492-500 İNANÇ ŞENOCAK. Computational Methodology for the Simulation of Turbulent Cavitating Llows. Ph. D. Dissertation. University of Florida. 2002 Jiahn-Horng Chen and Ying-Chieh Weng. A Semi-Direct Method for Analysis of Flow Past aTwo-Dimensional Supercavitating Hydrofoil. Proc Natl. Sci. Counc. (A) Vol. 23. No. 5, 1999. pp. 591-589

(c) Fig.11 effect of cavitator on velocity

(a) d c = 0.01 m

(b) d c = 0.025 m (c) d c = 0.05 m [5]

[6]

Spyros A.Kinnas and Neale. Fine. A numerical nonlinear analysis of the flow around two-and three-dimension partially cavitating. J. Fluid Mech.(1993), vol. 254, pp. 151-181 . DAVID R. STINGEBRING, JOH N E. DZIELSKI, JULES W. LINDAU, and ROBERT F. KUNZ. Supercavitation Research and Development. Applied Research Laboratory / Penn State University. 2002

377