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J. Marine Sci. Appl.DOI:

Numerical calculation of supercavitating flow over disk cavitator of under water subsonic projectile

Qingchang Meng*, Zhihong Zhang and Jubin Liu

College of Science, Naval University of Engineering, Wuhan 430033, China

Abstract: To deal with the effect of compressible fluid on the supercavitating flow over the subsonic disk cavitator of projectile, a finite volume method has been formulated based on the ideal compressible potential theory. By using the continuity, equation and Tait state equation as well as Riabouchinsky closure model, an "inverse problem" solution has been presented for the supercavitating flow. According to the impenetrable condition on the surface of supercavity, a new iterative method for the supercavity shape has been designed to deal with the effect of compressibility on supercavity shape, pressure drag coefficient and density field. By this method, the very low cavitation number can be computed. The calculated results agree well with the experimental data and empirical formula. At subsonic condition, the fluid compressibility will make supercavity length and radius increase. The supercavity expands, but remains spheroid. The effect on the first 1/3 part of supercavity is not obvious. The drag coefficient of projectile increases as the cavitation number or mach number increases. With mach number increasing, the compressibility is more and more significant. The compressibility must be considered as far as the accurate calculation of supercavitating flow is concerned.Keywords: projectile; supercavity; finite volume method; potential flow; fluid mechanics

Article ID:

1 Introduction1

Supercavity can be used for significantly reducing the viscous resistance of underwater body. There are two kinds of supercavitating weapons: a large-scale supercavitating torpedo and a small-scale supercavitating projectile. The large-scale supercavitating torpedo is similar to the "shkval" supercavitating torpedo equipped in Russian army with its speed at 100m/s order of magnitude. The small-scale supercavitating projectile is similar to the "Rapid Airborne Mine Clearance System" and "Adaptable High Speed Undersea Munitions" developed by the U.S. army, with its speed at 1000m/s order of magnitude (Kirschner et al., 2001; Savchenko, 2001; Semenenko, 2001a; Semenenko, 2001b; Kam, 2006). The high-speed projectile can be used to intercept torpedoes, destroy mines, break barriers and deal

with divers. At present, compared with the research on the supercavitating torpedo, the research on the supercavitating projectile is not enough, especially when the projectile speed is close to or exceeds the sonic speed (1450m/s) and fluid compressibility will have great effect on the supercavity shape. Only with its nose exposed to water, the projectile tends to “tail slap”, which will influence the trajectory and structural strength (Meng et al., 2009).

Usually a proper cavitator is set in the front of projectile to form a suitable supercavity. The cavitator is divided into two types: slender cone cavitator and disk cavitator. For the slender cone cavitator, Serebryakov (Serebryakov, 1997; Serebryakov, 2001; Serebryakov and Schnerr, 2003, Serebryakov, 2006) deduced asymptotic solution for supercavity shape from Laplace equation and energy equation, based on the slender body theory (SBT) and matched asymptotic expansion method (MAEM). The method is only used to calculate the supercavity shape and the drag coefficient of cavitator, instead of solving the entire flow field. Vasin (1987a; 1987b; 2001), using the SBT, computed the relation between the supercavity slenderness ratio and the cavitation number for subsonic and supersonic flow respectively. Zhang and Meng (2010) improved the accuracy of supercavity shape, and analyzed the effect of compressibility on supercavity under the conditions of the high speed impact of the projectile.

Vlasenko (2003) did experiment with the supercavitating projectile with the mach number from 0.54 to 0.77 and the experimental data and empirical formula were in good agreement. Ohtani et al., (2006) made a successful trial firing with the truncated cone projectile and clearly observed the straight projectile trajectory and supercavity. Hrubes (2001)chose an appropriate disk-cavitator projectile and did experiment after tradeoff between drag and cavity size for the expected speed range. A clear supercavity was observed and the stability mechanism of the projectile was verified. The experiment made by Gu et al. (2005) showed that although the drag of projectile with a slender cone cavitator is small, when the hydrodynamic force acting on the cavitator does not

1Received date: Foundation item: Supported by the National Natural Science Foundation of China (Grant No.51309230) and China Postdoctoral Science Foundation (2014T70992 and 2013M542531).*Corresponding author Email: [email protected]

© Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015

get through the gravity center of projectile, the pitch rate tends to vary easily. The "flap" effect will appear, which will have a great impact on the trajectory stability. In practice, a disk is usually used as a cavitator of projectile.

There are two ways, viscous flow theory and potential flow theory, to research the high-speed supercavitating flow over disk cavitator projectile. Kunz (Kunz et al. 2001, Lindau and Kunz 2003, Lindau et al. 2003) and Pellone et al. (2004), as representative, constructed a preconditioned, homogenous, multiphase, unsteady Reynolds Averaged Navier-Stokes scheme (RANS), and developed a Computational Fluid Dynamics (CFD) method to simulate the supercavitating flow. The method demands rigorous computational conditions. Zhang and Fu (2009) using the Fluent software, studied the compressible supercavitating flow generated by the projectile, and obtained the temperature distribution around the projectile. Yi et al. (2008) also using Fluent software, studied the natural supercavity over the disk cavitator, and simulated the changes of underwater supercavity and drag coefficient. Vasin and Semenenko represent the potential flow method. For the ideal irrotational fluid, Vasin (1996; 1997; 1998, 2001a, 2001b) solved subsonic and supersonic flow with finite difference method (FDM) according to the continuity equation. However, the practicability of FDM is limited especially for complex boundary conditions, and its conservation should be improved. Semenenko (1997; 2001a; 2001b) and Pellone et al. (2004) developed a program to simulate the supercavitating unsteady motion based on the principle of independence of the cavity sections expansions.

As for disk cavitator, the purpose of this paper is to present a finite volume method (FVM) for computing the supercavitating flow over the disk cavitator based on the ideal compressible potential theory. By means of continuity equation and Tait state equation, a solution has been proposed for the "inverse problem" that is, giving the supercavity length in advance, and then deducing the cavitation number and supercavity shape. According to the impenetrable condition on supercavity surface, a new iterative method has been designed for the supercavity shape. Based on solving the subsonic supercavitating flow over the disk cavitator, the supercavity shape and density characteristics have been obtained. It is possible to compute the supercavitating flow when cavitation number is very small and the range of cavitation number is expanded. The computational results have been compared with the experimental data and empirical formula, and the feasibility of the program and reliability of the results has been verified. The method can be used for forecasting the supercavity shape and flow characteristics of the subsonic projectile with disk cavitator, further providing a basis for computing dynamics and trajectory of the supercavitating projectile.

2 Supercavitating flow model 2.1 Computational region

An ideal steady irrotational flow over disk cavitator is

established for a computational model. By symmetry, only a profile is considered as shown in Fig. 1. Riabouchinsky closure model is adopted. x and r are axial and radial coordinates respectively, and V is the free stream velocity. AB represents the disk cavitator with radius as Rn and CD as the mirror disk. HE is the symmetry axis. HG and FE are 1/4 arc length with Point A and Point D as the center of the circle respectively. The arc radius is determined in proportion as the supercavity length Lc. EFGH is the outer boundary, where GF is parallel to the symmetry axis. BC is the supercavity surface to be solved and Rc is the maximum radius of supercavity.

Fig. 1 Computational region

2.2 Governing equations

In the computational region, the continuity equation is satisfied:

11\*MERGEFORMAT ()

Axial component u and radial component v are expressed by velocity potential :

， 22\*MERGEFORMAT ()

Pressure p is computed by the Tait state equation:

， 33\*MERGEFORMAT ()where, n=7.15 is the adiabatic index of water and subscript represents the infinite parameter.

Density is derived by Tait state equation and Bernoulli equation:

44\*MERGEFORMAT ()

According to Eq. 1, 2 and 4, velocity potential and density can be solved.

Further, the expression of sonic speed a is obtained as follows:

55\*

2

Journal of Marine Science and Application

MERGEFORMAT ()In the process of calculation, length, velocity and density

are non-dimensionalized by cavitator radius Rn, free stream velocity V and free stream density respectively.

2.3 Boundary conditionsThe impenetrable condition is satisfied on the symmetry

axis HA, DE, the cavitator AB, its mirror disk CD and the supercavity surface BC:

66\* MERGEFORMAT()

The pressure in the supercavity is the saturated vapor pressure, and the following dynamic boundary condition is satisfied on the supercavity surface BC:

77\* MERGEFORMAT ()For the pressure and velocity on the supercavity surface

are constants, while the direction of velocity is tangent to the supercavity surface, therefore

88\* MERGEFORMAT()where s is the arc length calculated from Point B on the supercavity surface, and V is the velocity on the supercavity surface.

On the outer boundary HGFE, the velocity potential is:

99\* MERGEFORMAT()

3 Numerical method3.1 Grid generation

First, the supercavity length Lc is given and then supercavity shape is initialized. When M, the initial supercavity surface is taken as a line connecting Point B and Point C. ABCD and HGFE constitute a pair of edges of computational structured grid, and another pair of edges is composed of AH and DE. Furthermore, the grid nodes are distributed on the boundaries as shown in Table 1. In addition, the denser grids are generated near the cavitator, mirror disk and supercavity surface to improve efficiency and be convenient for observing pressure or density diversification as shown in Fig. 2. The corner coordinates and the control volume node coordinates are calculated with the bilinear interpolation method. Finally the structured grid for FVM is generated as shown in Fig. 3, where and are the local coordinates.

Table 1 Numbers of node and element on boundaries

Boundary AB, CD BC

HA, DE

HGFE

NodeElement

nr+1

nr

nc+1

nc

ny+1

ny

2nr+nc+1

2nr+nc

Fig. 2 Grid on physical plane

Fig. 3 Grid on Computational plane

3.2 Calculation of velocity potentialWhen M=0, the free stream potential is taken as the initial

velocity potential. When M>0, the computed result of lower Mach number is taken as the initial value for the next computational step. Considering the boundary conditions, and using the integrating governing equation at each element, we get linear equations about potential. Gauss-Seidel iterative method is used to solve the equations to obtain the potential distribution, which meet the boundary conditions in the whole flow field.

The gradient of velocity potential is computed with Gauss Theorem. The relationship between the control volume node P and adjacent nodes is shown in Fig. 4.

Fig. 4 Relationship between control volume nodes

For a single volume element

1010\*MERGEFORMAT ()

3

where e, w, n, and s at Point e, w, n and s are calculated with the linear interpolation. Taking e for example:

， 1111\*MERGEFORMAT ()

After the potentials on the inner nodes are solved, the potentials on boundaries should be computed specially. Taking the potential on the cavitator for example, as shown in Fig. 5, the numbered dots represent the control volume nodes. Similarly, the numbered diamonds represent the corner nodes. For nodes i=1nr, the dot product of potential gradients and vector d of inner adjacent nodes is as follows:

1212\*MERGEFORMAT ()Where n and are the normal unit vector and the tangential unit vector respectively, as shown in Fig. 6.

Fig. 5 Nodes distribution on cavitator

Fig. 6 Relation between nodes on boundary

For the solid boundary,n=0, and the equation is derived as follows:


When i =1 and i=nr, Eq. (13) is written as follows:



For a one-dimension boundary problem, the equation is discretized as tridiagonal equations. Then the Gauss-Seidel iterative method is used to solve the potentials on the boundaries including the closure disk and the symmetry axis. For the outer boundary EFGH, the open condition is satisfied and the potential is computed by the extrapolation from the inner potential.

3.3 Computing velocity on supercavity surfaceAfter solving the potentials on the boundaries, velocity on

the supercavity surface Vc is computed to judge whether FVM is convergent. Eq. 8 provides the following:

1616\*MERGEFORMAT ()Where C

(k) and B(k) are the potentials on Point C and Point

B, and the superscript k is the iterative times of the supercavity shape. sc is the arc length from Point C to Point B.

3.4 Method for upgrading supercavity shape/n on the supercavity surface is computed to judge if

the supercavity shape meets the impenetrable condition. From the dot product of the gradient expression of potential and vector d, the expression /n is derived as follows:


According to Eq. 17, /n on the supercavity surface is computed by the backward difference scheme. Condition 6 is satisfied on the supercavity surface. Otherwise, the supercavity shape needs to be modified. Tangent velocity Vc

on the supercavity surface has been computed with the iterative method, corresponding to the normal velocity Vc = /n. Then the velocity vector of each node on the supercavity surface can be calculated by the following expression:

1818\*MERGEFORMAT ()Where i, j are the unit vectors of x and r direction respectively on the physical plane.

On the supercavity surface:


Keeping coordinate x, Eq. 19 is numerically solved and the new coordinate r is obtained, which is used to update the supercavity shape. The new supercavity shape regenerates the computational grid. Recalculating the velocity potential and repeating the above processes result in an accurate supercavity shape. For M>0,the calculated supercavity shape and other parameters at lower Mach number are used as the initial values for the next step. Flowchart is shown in Fig. 7.

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Fig. 7 Flowchart

3.5 Calculation of density gradientFor the compressible fluid, the density gradient needs to

be computed after the continuity equation is discretized. Gauss Theorem computes the density gradients on the inner nodes as follows:


The density gradients on the boundaries are computed with the least square method. For each node as follows:

2121\*MERGEFORMAT ()Where l12 is the vector from Point 2 to Point 1 as shown in Fig. 6. The expression is derived as follows:


Similarly, for Node 3 and Node 4,

2323\*

MERGEFORMAT ()


The square sum of the above expressions should be minimal. Finally, the expression /x and /y are deduced.

4 Results and analysis4.1 Supercavity shape

Given the supercavity length in advance, and then the cavitation number is computed, where =(p-pc)/(0.5V

2). Next, the cavity length is amended according to the cavitation number required, until the correct cavity shape is computed. The computed supercavity length and radius are in good agreement with the empirical formula and experimental data (Vlasenko, 2003) as shown in Fig. 8 and Fig. 9, which verifies that the algorithm can be used for incompressible supercavitating flow. Supercavity length and radius are decreased with cavitation number increasing.

Fig. 8 Relation of supercavity radius and cavitation number

Fig. 9 Relation of cavity length and cavitation number

For incompressible flow (M=0), Fig. 10 shows that the supercavity length decreases with cavitation number increasing. The algorithm can be applied for the case of small cavitation number. Taking two cases =1.010-4 and =1.010-3 (M=0.8) for example, Fig. 11 shows the two supercavity shapes. So the range of cavitation number is extended and the supercavitating flow at very small cavitation number can be calculated.

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Fig. 10 Cavity shapes at different cavitation number

Fig. 11 Cavity shapes at small cavitation number

For high-speed supercavitating flow, the fluid compressibility should be considered able to more accurately describe the supercavity shape. For =0.0235, the incompressible supercavity shape (solid line), compressible supercavity shape (dashed line) and the result of Vasin (1996) (dots) are compared together in Fig. 12. The dashed line and dotted line are in good agreement, which verifies the algorithm for the compressible flow (M=0.8). The solid line does not take into account the compressibility and the result is significantly different from dashed line.

Furthermore, to illustrate the compressible effect on the supercavity of small cavitation number, taking =0.0025 for example, the supercavity shapes at mach numbers from 0.1 to 0.9 are compared in Fig. 13. The result shows that the compressibility makes supercavity length and radius increase in subsonic flow. The supercavity expands, but remains spheroid. The effect on the first 1/3 part of supercavity is not obvious as shown in Fig. 13.

Fig. 12 Compressible effect on the supercavity

Fig. 13 Supercavity shapes at different Mach numbers

The supercavity slenderness ratio (=Lc/Rk, where Rk is the maximum of supercavity radius) considering fluid compressibility is compared well with literature (Serebryakov 1997) as shown in Fig. 14, which also verifies the algorithm. As the cavitation number increases, the slenderness ratio decreases. To facilitate the analysis of compressibility effects on the supercavity, Fig. 15 shows the relationship of slenderness ratio and cavitation number at different mach number. At the same cavitation number, the slenderness ratio of the supercavity increases as the mach number increases.

Fig. 14 Relation of slenderness ratio and cavitation number（M=0.8）

Fig. 15 Relation of slenderness ratio and cavitation number at different mach numbers

4.2 Density

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The density distribution around the supercavity is computed for Lc=200. At low mach number (M=0.2), the maximum of dimensionless density is 1.019, which is equal to the stagnation density of the adiabatic isentropic fluid. The maximum change of density does not exceed 1.9%, and the compressible effect is not obvious. At high mach number (M=0.8), the maximum change of density is 19.35% when the compressible effect is great as shown in Fig.16. Since the mach number increases, the effect on the density field is more and more prominent.

(a) Global contour0.9927

0.9927

1.0 2

1.02

1.02

1.02

1.02

1.02

1.04

1.04

1.04

1.0 4

1.04

1.04

1.06

1.0 6

1.08

1.0 8

1.08

1.08

1.1

1.12

1.14

1.14

1.16

1.16

1.18

1.18

-8 -6 -4 -2 0 2 4 6 8-3

-2

-1

0

1

2

3

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

(b) Details near the cavitator

Fig. 16 Density contour（M=0.8）4.3 Pressure drag coefficient

When =0.02, the present pressure drag coefficient compares well with empirical formula as shown in Fig. 17. With the increase of mach number, the drag coefficient increases slightly. Fig. 18 shows the relation of drag coefficient and cavitation number at different mach numbers. At the same mach number, the drag coefficient has linear growth with cavitation number, which agree with the empirical formula (Vasin, 1996).

Fig. 17 Relation of drag coefficient and mach number

Fig. 18 Relation of drag coefficient and cavitation number at different mach number

5 ConclusionsThis paper has presented a finite volume method based on

the ideal compressible potential theory to deal with the effect of compressible fluid on supercavitating flow over subsonic disk cavitator of projectile. In the paper, continuity equation is solved using Riabouchinsky closure model and combining it with Tait equation. An “inverse problem” solution is proposed for the supercavitating flow, that is, the supercavity length is given first, when the cavitation number, the supercavity shape and the density field are solved. A new iterative method for the supercavity shape is designed according to the impenetrable condition on the supercavity surface. By this method, the very low cavitation number can be computed and the range of cavitation number is expanded to 10-410-2. The comparison of the computed result with the experimental data and the empirical formula verifies the program.

From the computation of subsonic supercavitating flow over disk cavitator, the relationship has been described about supercavity length, maximum radius and cavitation number, and the effect of fluid compressibility on the supercavity shape and density has been analyzed. At subsonic condition, the fluid compressibility will make supercavity length and radius increase. The supercavity expands, but remains spheroid. The effect on the first 1/3 part of supercavity is not obvious. With mach number increasing, the effect on the density field is more and more prominent. The drag coefficient of projectile increases with cavitation number or mach number increasing. With Mach number increasing, the compressibility is more and more significant. The compressibility must be considered as far as the accurate calculation and the application of supercavitating flow are concerned.

More accurate discretization scheme should be developed to improve the algorithm, and further studies on the supersonic supercavitating flow will be summarized in our next study.

AcknowledgementWe would like to thank Dr. Tao Miao for closely

following our work and making several useful suggestions.

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水下亚声速圆盘空化器射弹超空泡流动数值计算方法研究孟庆昌，张志宏，刘巨斌

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