Qingchang Meng, Zhihong Zhang and Jubin Liu

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

1 Introduction1

Supercavitation can be used for significantly reducing the viscous resistance of an underwater body. There are two kinds of supercavitating weapons: the large-scale supercavitating torpedo and the small-scale supercavitating projectile. The large-scale supercavitating torpedo is similar to the “Shkval”supercavitating torpedo equipped in Russian army with its speed at 100 m/s order of magnitude. The second one 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 1 000 m/s order of magnitude(Kirschneret al., 2001; Savchenko, 2001; Semenenko,2001a; Semenenko, 2001b; Ng, 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 (1 450 m/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 (Menget 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: the slender cone cavitator and the disk cavitator. For the slender cone cavitator, Serebryakov (1997; 2001; 2006)Serebryakov and Schnerr (2003) 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; 2001a), using the SBT,computed the relation between the supercavity slenderness ratio and the cavitation numbers for subsonic and supersonic flows. Zhanget al. (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. Ohtaniet 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 Guet 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 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, the viscous flow theory and potential flow theory, to research the high-speed supercavitating flow over disk cavitator projectile. Kunzet al. (2001), Lindau and Kunz (2003), Lindauet al. (2003) and Pelloneet al. (2004),as representatives, 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. Yiet al. (2008) also using Fluent software, studied the natural supercavity over the disk cavitator, and simulated the changes of underwater supercavitation and drag coefficient. Vasin and Semenenko represented 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 Pelloneet 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 the 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 have 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.xandrare axial and radial coordinates, respectively, andis the free stream velocity.ABrepresents the disk cavitator with radius asRnandCDas the mirror disk.HEis the symmetry axis.HGandFEare 1/4 arc length with pointAand pointDas the center of the circle respectively. The arc radius is determined in proportion as the supercavity lengthLc.EFGHis the outer boundary, whereGFis parallel to the symmetry axis.BCis the supercavity surface to be solved andRcis the maximum radius of supercavity.

Fig. 1 Computational region

2.2 Governing equations

In the computational region, the continuity equation is satisfied:

Axial componentuand radial componentvare expressed by velocity potentialf:

Pressurepis computed by the Tait state equation:

wheren=7.15 is the adiabatic index of water and subscript￥ represents the infinite parameter.

Densityris derived by Tait state equation and Bernoulli equation:

According to Eqs. (1), (2) and (4), velocity potentialfand densityrcan be solved.

Further, the expression of sonic speedais obtained as follows:

In the process of calculation, length, velocity and density are non-dimensionalized by cavitator radiusRn, free stream velocityand free stream densityr￥, respectively.

2.3 Boundary conditions

The impenetrable condition is satisfied on the symmetry axisHA,DE, the cavitatorAB, its mirror diskCDand the supercavity surfaceBC:

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

For the pressure and velocity on the supercavity surface are constants, while the direction of velocity is tangent to the supercavity surface, therefore

wheresis the arc length calculated from pointBon the supercavity surface, andV￥the velocity on the supercavity surface.

On the outer boundaryHGFE, the velocity potentialfis

3 Numerical method

3.1 Grid generation

First, the supercavity lengthLcis given and then supercavity shape is initialized. WhenM￥＝0, the initial supercavity surface is taken as a line connecting pointBand pointC.ABCDandHGFEconstitute a pair of edges of computational structured grid, and another pair of edges is composed ofAHandDE. 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 computational efficiency. It is 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, wherexandhare the local coordinates.

Fig. 2 Grid on physical plane

Fig. 3 Grid on computational plane

Table 1 Numbers of node and element on boundaries

3.2 Calculation of velocity potential

WhenM=0, the free stream potential is taken as the initial velocity potential. WhenM>gt;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’s theorem. The relationship between the control volume nodePand adjacent nodes is shown in Fig. 4.

Fig. 4 Relationship between control volume nodes

For a single volume element,

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 nodesi=1–nr, the dot product of potential gradients and vectordof inner adjacent nodes is as follows:

wherenandtare 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

Wheni=1 andi=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 boundaryEFGH, the open condition is satisfied and the potential is computed by the extrapolation from the inner potential.

3.3 Computing velocity on supercavity surface

After solving the potentials on the boundaries, velocity on the supercavity surfaceVcis computed to judge whether the FVM is convergent. Eq. (8) provides the following:.

where?C(k)and?B(k)are the potentials on pointsCandB, the superscriptkis the iterative times of the supercavity shape,andscis the arc length from pointCto pointB.

3.4 Method for upgrading supercavity shape

wherei,jare the unit vectors of directionsxandron the physical plane, respectively.

On the supercavity surface:

Keeping coordinatex, Eq. (19) is numerically solved and the new coordinateris 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. ForM>gt;0, the calculated supercavity shape and other parameters at lower Mach numbers are used as the initial values for the next step. Flowchart of supercavity profile is shown in Fig. 7.

Fig. 7 Flowchart of supercavity profile

3.5 Calculation of density gradient

For the compressible fluid, the density gradient needs to be computed after the continuity equation is discretized.Gauss’s 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:

wherel12is the vector from node 2 to node 1 as shown in Fig. 6. The expression is derived as follows:

Similarly, for Node 3 and Node 4,

The square sum of the above expressions should be minimal. Finally, the expression ?ρ/?xand ?ρ/?yare deduced.

4 Results and analysis

4.1 Supercavity shape

Given the supercavity length in advance, and then the cavitation numbersis computed, whereNext, 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 Figs. 8 and 9, which verifies that the algorithm can be used for the incompressible supercavitating flow. Supercavity length and radius decrease with the cavitation number increasing.

For incompressible flow (M=0), Fig. 10 shows that the supercavity length decreases with the cavitation number increasing. The algorithm can be applied for the case of small cavitation number. Taking two casesandfor 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.

Fig. 8 Relation of supercavity radius and cavitation number

Fig. 9 Relation of cavity length and cavitation number

Fig. 10 Cavity shapes at different cavitation numbers

Fig. 11 Cavity shapes at a small cavitation number

Fig. 12 Compressible effect on the supercavity

Fig. 13 Supercavity shapes at different Mach numbers

For the high-speed supercavitating flow, the fluid compressibility should be considered able to more accurately describe the supercavity shape. Fors=0.023 5,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 the dashed line.

Furthermore, to illustrate the compressible effect on the supercavity of small cavitation number, takings=0.002 5 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.

The supercavity slenderness ratiois 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 the slenderness ratio and cavitation number at different Mach numbers. 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

The density distribution around the supercavity is computed forLc=200. At a 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 a high Mach number (M=0.8), the maximum change of density is 19.35%when the compressible effect is great. Since the Mach number increases, the effect on the density field is more and more prominent.

4.3 Pressure drag coefficient

Whens=0.02, the present pressure drag coefficient compares well with empirical formula as shown in Fig. 16.With the increase of Mach number, the drag coefficient increases slightly. Fig. 17 shows the relation of drag coefficient and cavitation number at different Mach numbers. At the same Mach number, the drag coefficient increase linearly with cavitation number increasing, which agrees with the empirical formula (Vasin, 1996).

Fig. 16 Relation of drag coefficient and Mach number

Fig. 17 Relation of drag coefficient and cavitation number at different Mach number

5 Conclusions

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, i.e., 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?4–10?2. The comparison of the computed result with the experimental data and the empirical formula verifies the program.

At the 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.

Acknowledgement

We would like to thank Dr. Tao Miao for closely following our work and making several useful suggestions.

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