Research on the hydrodynamic characteristics of cavitating grid fins*

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Journal of Hydrodynamics Ser.B, 2006,18(5): 537-541

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RESEARCH ON THE HYDRODYNAMIC CHARACTERISTICS OF CAVITATING GRID FINS* WU Lei, LU Chuan-jing, HUANG Tao, LI Jie Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China E-mail:[email protected] (Received June 6, 2005)

ABSTRACT: A two-phase mixture model based on the solution for the Navier–Stokes equations has been utilized in calculating the hydrodynamic characteristics of cavitating honeycomb grid fins with different configurations. The calculation results of lift, drag, and hinge moment coefficients are presented in various cavitation numbers and angles of attack, and its hydrodynamic features are also analyzed. The calculation results indicate that cavitation will reduce the lift/drag ratio of grid fins. The increment of horizontal blades as lift surface cannot unendingly improve lift because of the disturbance between the blades. KEY WORDS: grid fin, cavitating flow, hydrodynamic characteristic, equivalence

1. INTRODUCTION Since the introduction of the grid fin in the mid1980s, it has been indicated that the use of grid fin as an innovative aerodynamic stabilizer and/or a control surface has some advantages when compared with the conventional planar fin [1,2]. The grid fin can be effectively used as a lift surface in subsonic, transonic, and supersonic flows, especially at high angles of attack. It also shows excellent properties with regard to structure, strength, weight, manufacture, and setting [3-6] . In 1993, Russian scholars substituted four grid fins for conventional rudders on middle-range air-to-air missiles. Chinese scholars have carried forward the researches in this field since 1990, and applied grid fins to the spaceship “SHEN-ZHOU” [7] . The research on the hydrodynamic characteristics of

grid fins has been considerably accounted for since 1990 [8]. Grid fins can be used to increase the stability of ships while encountering waves. These also work as stabilizers and/or control surfaces for submarine launched weapons, which travel underwater with high speed for a period of time before they reach the surface of water. Cavitating usually occurs when grid fins are applied on high- speed vehicles. It is very important to study the hydrodynamic characteristics of cavitating grid fins. In this article, three-dimensional steady cavitating flows around different kinds of grid fins have been numerically studied. The Navier–Stokes equations have been solved using the finite volume method. The two-equation k − ε turbulence model has been used to simulate the turbulent flow. A two-phase mixture model has been applied to account for the two-phase flow of water/vapor and the transition from one phase to the other.

2. NUMERICAL METHOD 2.1Configuration of grid fins A sketch of the different kinds of grid fins is illustrated in Fig.1. The grid fins are named according to the geometry as OPEN, SINGLE, DOUBLE, THREE, FOUR, CROSS, and DIAGONAL. For example, the SINGLE fin has one single horizontal blade, and the CROSS fin has two blades arranged like the cross symbol (+). All fins have the same outer frame, with the geometric dimensions, span length

* Project supported by the National Natural Science Foundation of China (Grant No: 10372061) and the National Defense Technology Key Laboratory on Hydrodynamics. Biography: WU Lei (1974-), Male, Ph.D., Lecturer

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L=152.4mm, height H=76.2mm, and chord length b=25.4mm. To reduce the profile drag, the upper leading edges on the outer frame have been designed as wedges, which take the level of 15o.

∂ ∂p ∂ ( ρu j u i ) = − + ∂x j ∂xi ∂x j

⎡ ⎛ 1 ⎞⎤ ⎢ μ ⎜ eij − 3 eii δ ij ⎟⎥ ⎠⎦ ⎣ ⎝ (4)

∂u i ∂u j , u denotes the velocity + ∂x j ∂xi vectors, and p is the pressure. To solve the equations, where eij =

the bubble two-phase flow hypothesis introduced by Kubota et al. in 1992 [9] is adopted. It is assumed that the number of bubbles per unit volume of liquid n is known in advance. The bubble radius R is controlled by the Rayleigh-Plesset equation

Fig.1 Sketch of the grid fins

2.2 Two-phase mixture model Cavitating flow is typically a two-phase flow, which includes water and vapor. To model the two fluids as a continuum using one set of partial differential equations, the density ρ should be continuous and differentiable over the entire flow domain. By assuming a small transitional area δ between the two fluids, the constitutive relations for the density and the dynamic viscosity are

ρ = αρ vapor + (1 − α ) ρ water μ = αμ vapor + (1 − α ) μ water

(1)

where the subscripts denote the different fluids. The indicator function α is defined as

α ( x, t ) = 0 for the point ( x, t ) inside water α ( x, t ) = 1 for the point ( x, t ) inside vapor α ( x, t ) = 0 < αδ < 1

(2a)

(2b)

P −P d 2 R 3 ⎛ dR ⎞ R 2 + ⎜ ⎟ = v ρ water dt 2 ⎝ dt ⎠ 2

where, Pv is the saturation pressure of water. Thus,

α can be calculated using the number of bubbles per unit volume n and the individual bubble volume, which is a function of R . The finite volume method and the SIMPLE algorithm have been used to solve the equations. The two-equation k − ε turbulence model provides the turbulence closure. The numerical procedure is according to Wu et al. [10]. Simulating the performance of the fins inside water tunnel, Fig.2 shows the sketch of the computational domain, which is discretized into 350000 to 550000 cells for different fins. The fins are fixed at the bottom of the tunnel, and the coordinate system is set as shown in Fig.2 (b). The collocated arrangement is chosen to store all variables in the same set of grid points and to use the same control volumes for all variables.

Flow

for the point ( x, t ) inside the transitional area

wall

(2c) Therefore, the continuity equation and the momentum equations, which govern the flow field according to the two-phase mixture model, can be written as:

∂ ( ρu i ) = 0 ∂xi

(5)

(a)

z

(3) y

x

x y

z

(b)

Fig.2 Sketch of the computational domain

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3.RESULTS AND DISCUSSION The fin chord length b, the fin wetted area S=L i H, and the inflow velocity U ∞ are considered as the reference length, area, and velocity, respectively. The cavitation number σ , the lift coefficient CL, the drag coefficient CD, and the pitching-moment coefficient CM are defined as follows:

0.15

P∞ − Pv FY , CL = , 2 0.5 ρ waterU ∞ 0.5 ρ waterU ∞2 b

CL =

FX

0.5 ρ waterU b 2 ∞

, CM =

0.1 0.075

MZ

0.05

0.5 ρ waterU S 2 ∞

In the simulations described in this study, the cavitation inception number σ 0 ranges between 0.75 and 1.0. Figures 3 and 4 illustrate the lift and drag coefficients for the cavitation number σ = 0.4 as a function of the incidence angle β on the OPEN fin, respectively, by comparing with the noncavitating cases. It is obvious that the lifts decrease and the drags increase because of cavitating. The former is caused by the reduction of the lift surfaces, which are covered with cavity. The latter is caused by the increment of the profile drag. 0.6

Non-cavitating flow Cavitating flow

CL

0.4

Non-cavitating flow Cavitating flow

0.125 CD

σ=

fins coincide with that of the OPEN fin. Cavitating causes a decrease of the lift/drag ratio, and consequently reduces the efficiency of grid fins.

Fig.4

0

5

15

10

β

20

Cavitating ( σ =0.4) and non-cavitating drag coefficients as a function of incidence angle β ( o) on OPEN fin

By comparing the five lift curves of OPEN, SINGLE, DOUBLE, THREE, and FOUR fins in Fig.5 and Fig.8, it can be found that the lift curve goes up and tends to a limit line with the increment of the number of horizontal blades. The more the horizontal blades, the more are the lifting surfaces that are provided. However, the lift does not magnify endlessly because the cavity covers the blades, which blocks the spaces and causes higher disturbances between the blades. The five drag curves, shown in Fig.6 and Fig.9, indicate that the drags increase with the increment of the number of horizontal blades. The spaces between every two curves are almost equal, which indicates that the effect of a single horizontal blade to drag is almost equivalent in these cases.

0.2

1

0 0

5

10

β

15

20

0.8

CL

0.6

Fig.3

Cavitating (σ=0.4) and noncavitating lift coefficients as a function of the incidence angle β ( o) on OPEN fin

0.4 0.2 0

-0.2

Figs.5-10 illustrate the hydrodynamic forces, which changes with the cavitation number or the angle of attack on the different fins. For all fins, the lift curves move down with the development of cavitation. The drag curves first move up as inception of cavitation, and then tend to drop, but are still greater than that in full-wetted conditions when the cavitation number is less than 0.4. The results on the different

OPEN SINGLE DOUBLE THREE FOUR CROSS DIAGONAL

Fig.5

0

0.5

1

1.5

σ

2

2.5

Lift coefficients as a function of the cavitation number

σ( β =10o)

Previous researches [7,11] indicate that the aerodynamic characteristics of gird fins are equivalent in a large range of the Mach number and angle of

540

0.3

0.2

OPEN SINGLE DOUBLE THREE FOUR CROSS DIAGONAL

0.2

0.15 CD

0.25

OPEN SINGLE DOUBLE THREE FOUR CROSS DIAGONAL

CD

0.1

0.15 0.05 0.1 0.05 0

Fig.6

0 0.5

1

2

1.5

σ

2.5

Drag coefficients as a function of the cavitation number

σ ( β =10o)

OPEN SINGLE DOUBLE THREE FOUR CROSS DIAGONAL

0.35 0.3

5

15

20

OPEN SINGLE DOUBLE THREE FOUR CROSS DIAGONAL

0.2 0.15

0.2

10

β

Fig.9 Drag coefficients as a function of the incidence angle β ( o ) (σ=0.4)

CM

CM

0.25

0

0.1

0.15 0.1

0.05

0.05 0

Fig.7

0

0.5

1

1.5

2

σ

2.5

Hinge moments as a function of the cavitation number σ( β =10o)

OPEN SINGLE DOUBLE THREE FOUR CROSS DIAGONAL

1

CL

0.8 0.6 0.4 0.2 0

Fig.8

0

5

10

β

15

20

Lift coefficients as a function of the incidence angle β ( o ) (σ=0.4)

attack. When the grid fins acquire the same outer frame and spacing t between blades, the aerodynamic lifts and moments are nearly equal. By considering the force curves of CROSS and DUPLICATE fins, it can be found that the curves are considerably closer to those of SINGLE and/or DOUBLE fins. As SINGLE

0

0

5

β

10

15

20

Fig.10 Hinge moments as a function of the incidence angle β ( o ) (σ=0.4)

and CROSS fins have the same outer frame and spacing, the lift curves of the two fins exactly coincide with each other and also with the hinge moment curves. The lift curves and the moment curves of DULPICATE fins are close to those of SINGLE or DOUBLE fins because of the similar configurations. Since the blades of DUPLICATE fins are oblique, these are different from the other fins, and hence increase considerably experiments to obtain the exact equivalent condition of hydrodynamics according to the various grid fin configurations. The drag curves of CROSS and DUPLICATE fins are closer to those of DOUBLE fins than those of SINGLE fins, as shown in Figs.6 and 9. It is obvious that these offer more similar drag surfaces. 4. CONCLUSIONS As a result of the mathematical modeling and the analysis, the following conclusions are drawn: (1) Grid fins can also perform as hydrodynamic stabilizers and/or control surfaces as in aerodynamics.

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Cavitating will reduce the lift/drag ratio of grid fins, and thus affect the efficiency. (2) Under the same flow conditions, the lift of the grid fins will not continuously increase with the increase in the number of blades. It will tend to a limit for the disturbance between the blades introduced by cavity. Hence, it is reasonable to specify the number of the blades according to the comprehensive influences of lift/drag, structure, strength, and manufacture. (3) The hydrodynamic characteristics of grid fins also satisfy the equivalence postulate as that of aerodynamics. When the grid fins acquire the same outer frame and spacing t between blades, the hydrodynamic lifts and moments obtain the equivalent relations.

REFERENCES [1]

[2]

[3]

WASHINGTON W. D., MILLER M. S. Grid fins a new concept for missile stability and control[R]. AIAA -93-0035, 1993. CHEN Shao-song, ZHAO Run-xiang, DING Ze-sheng et al. Study and comment on the aerodynamic characteristic of grid fins [J]. Journal of Ballistics, 1999, 11(2): 89-95. (in Chinese) WU Pin, MA Yong-gang, CHEN Chun. The research analysis of aerodynamic numerical simulation of grid fin [J]. Journal of Zhejiang University Science, 2005, 6(7): 741-746.

[4]

YANG Xiao-hui, WANG Chen-yao. Aerodynamic numerical simulation of three-dimensional complex flow on grid fin [J]. Journal of National University of Defense Technology, 1996, 18(3): 1-4. (in Chinese) [5] BURKHALTER J. E., FRANK H. M. Grid fin aerodynamics for missile applications in subsonic flow [J]. Journal of Spacecraft and Rocket, 1996, 33(1): 38-44. [6] SHEN Xia-ling, LU Zhong-rong, TONG Zi-li et al. Calculation of grid-body-engine configuration [J]. Journal of Beijing University of Aeronautics and Astronautics, 1996, 22(6): 727-732. (in Chinese) [7] LU Zhong-rong, YANG Ji. Calculation and analysis of grid fin-missile aerodynamic characteristics [J]. Missiles and Space Vehicles, 2001, 15(3): 7-12. (in Chinese) [8] FU Hui-ping, LU Chuan-jing, LI Jie et al. Research on hydrodynamic forces of cavitating grid fins [J]. Journal of Hydrodynamics, Ser. B, 2005, 17(2): 148-153. [9] KUBOTA A., KUTO H., YAMAGUCHI H. A new modeling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil section [J]. Journal of Fluid Mechanics, 1992, 240: 59-96. [10] WU Lei, LU Chuan-jing, XUE Lei-ping. An approach in modeling two-dimensional partially cavitating flow [J]. Journal of Hydrodynamics, Ser. B, 2002, 14(1): 45-51. [11] LU Zhong-rong, SHEN Xia-ling, TONG Zi-li et al. Calculation and analysis of grid fin configurations [J]. Journal of Beijing University of Aeronautics and Astronautics, 1996, 22(5): 575-580. (in Chinese)