Numerical study on characteristics of supercavitating flow around the variable-lateral-force cavitator

A control scheme named the variable-lateral-force cavitator, which is focused on the control of lift force, drag force and lateral forces for underwater supercavity vehicles was proposed, and the supercavitating flow around the cavitator was investigated numerically using the mixture multiphase flow model. It is verified that the forces of pitching, yawing, drag and lift, as well as the supercavity size of the underwater vehicle can be effectively regulated through the movements of the control element of the variable-lateral-force cavitator in the radial and circumferential directions. In addition, if the control element on either side protrudes to a height of 5% of the diameter of the front cavitator, an amount of forces of pitching and yawing equivalent to 30% of the drag force will be produced, and the supercavity section appears concave inwards simultaneously. It is also found that both the drag force and lift force of the variable-lateral-force cavitator decline as the angle of attack increases.


Introduction
The control on the profiles of unsteady supercavity is important for underwater supercavity vehicles, generally speaking, the dimensions of supercavity can be regulated by altering the cavitation number and drag coefficient of cavitator. Taking example for the underwater vehicles with relatively low velocity, gas-ventilation into the cavity can effectively alter the cavitation number and the shape of supercavity. While for vehicles approaching to the regime of vapor supercavity flow with a higher velocity, the size of supercavity can be adjusted through the variation of the drag coefficients of cavitator.
The traditional variable-drag cavitator designed by Savchenko (2001), for example, is capable of regulating the size of supercavity through the variation of drag coefficients of the cavitator without changing its diameter. While for high-speed projectiles, a configuration which includes a cylindrical telescoping cavitator design capable of providing projectile nose shape change was proposed (Gieseke, 2011). The researchers in the Hydrodynamics Research In-stitute of the National Academy of Sciences (1997) presented a kind of device called polygon cavitator, whose characteristics are the shallow grooves paved with small carved facets, thus to produce a stable moment forcing the vehicle system in disturbed return to its initial state. In the Navy Undersea Warfare Center of USA (2001), the shape and function of cavitators were changed using mechanical way to meet the installation requirements of control surface and sensors.
A method of using cavitator as the control surface and the depth being controlled by changing its rudder angle was proposed (Chen and Luo, 2009), and the simulation results demonstrate that this control method makes the deviation of depth smaller. In order to retain the profiles of supercavity and keep the balance of underwater supercavity vehicles, and the excessive angle of attack for the cavitator is inadvisable, researchers (Luo et al., 2007;Chen et al., 2012Chen et al., , 2013Quan et al., 2008aQuan et al., , 2008bXiang et al., 2011) showed that too large angle of attack would cause the underwater vehicles to be destabilized and deviated from the original states of equilibrium. Firstly, the obvious difference of cavity profiles caused by the excessive angle would lead to a large scale of diversity of pressure distributions between the upstream face and the downstream face, thus it brings severe impacts to the force state and manipulation condition for underwater vehicles. Secondly, the angle of attack for vehicle will obviously change the axial symmetry of supercavity, and meanwhile the length of supercavity would decrease gradually with the increase of the angle of attack. This means that more gas is needed to keep the dimension for the ventilated supercavity (Yi et al., 2008;Chen and Lu, 2008).
Aimed to solve the above problems existing in the mode of cavitator deflection, a control scheme based on the theory of traditional variable-drag cavitator is proposed in this paper. This variable-lateral-force cavitator scheme is focused on the control of lift force, drag force and lateral forces (including pitching force and yawing force) for underwater vehicles through the variation of the effective wetted area and working diameter of the front cavitator, whose advantage is that the control scheme can effectively evade from deflecting the cavitator when the coefficients of lift forces and lateral forces are changed. Therefore, the shape of supercavity would not be minished, in other words, the wetted area of underwater supercavity vehicles would not be extended, and thus it can avoid the risk of losing stability caused by the anterior position of the pressure center from the center of gravity. On the other hand, since the control of vehicle is realized through the variation of protruding heights of the variable-lateral-force cavitator in lateral directions, this implies considerably less demand of motive forces under the same condition, compared with that of the deflection mode of traditional cavitators.
Numerical investigations on the three dimensional supercavitating flow around the variable-lateral-force cavitator were implemented to verify the ability of cavitator to the adjustment of lift force, drag force and lateral forces. It is confirmed that the variable-lateral-force cavitator can effectively adjust the supercavity shape, lift force, drag force, pitching force and yawing force for underwater supercavity vehicles.

Control equations
The theory of homogeneous multiphase flow, which regards the density of fluids inside and outside the supercavity as variable and that there is no interface existing between the fluids, is applied to the model establishment of supercavitating flow, and the phases of the entire mixture are allowed to be interpenetrated (Fu et al., 2004;Chen and Lu, 2008).
The equations of continuity (Eq. (1)) and momentum (Eq. (2)) for the mixture, as well as the volume (or mass) fraction equation for the secondary phases (vapor) (Eq. (3)) are solved respectively. (2) where u i is velocity; ρ m represents the mixture density; μ m indicates the mixture viscosity; ρ l , ρ v and ρ g stand for the density of water liquid, vapor, and non-condensable gas, respectively; while α l , α v and α g are the corresponding volume fractions.

Cavitation model
The vapor mass fraction f v is governed by the following transport equation: The source terms vapor generation R e and condensation rates R c can be the functions of flow parameters and fluid properties (such as the liquid and vapor phase densities, saturation pressure, and liquid-vapor surface tension) (Singhal et al., 2002): where C e and C c are the empirical constants, C e =0.02, C c =0.01; σ is the surface tension coefficient; p is local pressure; the phase-change threshold pressure p v , the mass fractions of vapor f v , and the mass fractions of non-condensable gas f g can be expressed as: where p sat is the saturation vapor pressure.

Turbulence model
The standard k-ε two-equation model is chosen to simulate the effect of turbulence, and the turbulence kinetic energy k and its rate of dissipation ε are obtained from the following transport equations: where σ k and σ ε are the turbulent Prandtl numbers for k and ε, respectively; G k and G b are the turbulence kinetic energy generation due to the mean velocity gradients and buoyancy, respectively; Y M is the fluctuating dilatation contribution in compressible turbulence to the overall dissipation rate.
The turbulent viscosity μ t is computed: where C μ is a constant. Fig. 1 shows the operating scheme for the traditional variable-drag cavitator, which alters the drag coefficient and profiles of supercavity by changing the position of the outside body 1 relative to the center element 2 without changing the diameter of cavitator (Savchenko, 1997(Savchenko, , 2001Savchenko et al., 1998).

Traditional variable-drag cavitator
Numerical calculation of unsteady natural supercavitating flow around the traditional variable-drag cavitator has been implemented and the analysis in detail can be found in reference (Hu et al., 2011), and part of the conclusions was cited for comparison in this paper.
The variation curves of dimensionless length and drag coefficients of supercavity with working stroke are presented in Fig. 2. Here the cavity length L c is non-dimensionalized with the diameter of cavitator D n and the drag coefficient c x is defined as: where F x demonstrates the drag force acting on the cavitator in the direction of the x axis, U ∞ is the velocity of the ambient flow, and A is the reference area of cavitator. It can be seen that the length of supercavity represents a consistent varying pattern with the maximum diameter, while the drag coefficient shows an unsynchronized but similar varying trend with the supercavity size, and the changes of both the supercavity length and maximum diameter lag behind the variation of the drag coefficient.

Operating principle
Though the traditional variable-drag cavitator can rapidly change the drag coefficient and supercavity shape, it is difficult to implement precisely due to the considerable pressure concentrated in front of the cavitator.
Since the pressure of the cavitator in lateral directions is relatively low, and the variation of the cavitator diameter can also make alterations to the size of supercavity, a mechanism named the variable-lateral-force cavitator in Fig. 3 was proposed, of which the working diameter and effective wetted area can be altered.   HU Xiao et al. China Ocean Eng., 2017, Vol. 31, No. 1, P. 123-129 125 The operating principle can be described as follows combined with Fig. 3a: the movable element 3, which can be moved in the axial direction, is constituted of a conical movable section and a cylindrical operating lever is placed inside the front cavitator 1, and the control element 2, which can be moved in the radial direction is placed in the tail part. Both elements 2 and 3 are designed into four pieces of valves, and there is a pair of curved steel piece installed on each side of the valve piece (seen in Fig. 3b), through the movements of element 3 in the axial direction, the curved steel pieces will slip relatively to each other in circumferential direction, thus to alter the working diameter of the front cavitator. Therefore, the forces acting on the front cavitator and the size of supercavity will be changed. If the relative displacements of the curved steel pieces on either side of the control element are inconsistent, the protruding heights of the front cavitator on the longitudinal (or lateral) plane would be different. Non-symmetrical forces, i.e., the lift force and lateral forces will be generated on the front cone cavitator. Consequently, the supercavity shape would be changed, and the posture of the underwater supercavity vehicle could be effectively regulated and controlled (Hu et al., 2011).

Simulation results of the axis symmetrical supercavitating flow
The three-dimensional supercavitating flow around the variable-lateral-force cavitator under different control st-atuses was investigated using ANSYS FLUENT, here the x axis is defined as the direction of the axis of the front cavitator, of which the positive direction is towards right horizontally, and the y axis is in the radial direction orthogonal to the x axis, the xoy plane is considered the longitudinal plane, and the positive direction of the z axis is defined vertically downward.
Calculated results of the supercavitating flow under control states of Cases 1, 2 and 3 are shown in Figs. 4 and 5. To verify the accuracy of the simulation results, the drag coefficients and the maximum diameter of the supercavity for variable-lateral-force cavitator are compared with the results of the theoretical flow (Semenenko, 2001).
For disk cavitators, the drag coefficient c x can be calculated as (Semenenko, 2001): where c x0 is the cavitator drag coefficient when , and the value was established experimentally for the disk cavitator, empirical constants k = 0.9-1.0, A = 2.0. The mid-section diameter D c and the cavity length L c were ob-  tained by the semi-empirical relations of the same structure. While for the cone cavitator with a central angle of 2β, the drag coefficient c x and the main dimensions of supercavity D c and L c are calculated below (Semenenko, 2001): c x = c x 0 + (0:524 + 0:672 ) v ; 0 6 v 6 0:25; 1 12 6 6 1 2 : The contours of the supercavity density in Fig. 4 indicate that the size of supercavity can be significantly changed through the movement of the control element in the radial direction, and the dimension of the supercavity formed at the same cavitation number increases as the displacements of the control element in the radial direction rise.
The drag coefficients and the maximum diameters of supercavity under the control statuses of Cases 1, 2 and 3 compared with the potential flow (Semenenko, 2001;Kuklinski et al., 2001) are given in Figs. 5a and 5b, respectively, in which the dash dot line represents the theoretical results for disk cavitators, and the dash line stands for cone cavitators.
It can be seen that when the displacement of the control element in the radial direction is equal to 0, the variation law of both the drag coefficient and the maximum diameter of supercavity with the cavitation number are well consistent with the results of the potential flow for cone cavitators with the central angle of 60°, represents a smaller magnitude than the theoretical results, and this may be caused by the reason that the theory of potential flow does not take into account the influence of viscosity of fluid.
¾ v 6 0:150 For the control states of both Case 2 and Case 3, the control element protrudes from the tail part of the front cavitator, the law of variation of drag coefficients and the maximum dimensionless diameter of supercavity for the variable-lateral-force cavitator approaches the theoretical results of disk cavitators, which indicates the accuracy of simulation results. While for the control state of Case 3, the simulated drag coefficients are in perfect accordance with the theoretical values calculated by Eq. (14) for the disk when , and then the theoretical value becomes slightly larger than the simulated drag coefficients as the cavitation number increases.

Simulation results of the asymmetrical supercavitating flow
The results of supercavitating flow calculated under control states 4 and 5 when σ v =0.124 are shown in Figs. 6, 7 and 8, respectively.
The distributions of radius of supercavity on the longitudinal plane are presented in Fig. 6. It can be seen from the curves that the profile of supercavity has been obviously altered compared with that of Cases 1, 2 and 3 due to the asymmetry of cavitator, either the upper or lower portion of supercavity is symmetrical, the axis of supercavity deviates upward, and the radius of the lower supercavity is obviously smaller than that of the upside. Fig. 7a presents the contours of the density for the supercavity section corresponding to Case 5. It reveals that the lower section of supercavity appears concave inwards at the distance of about 25% of the supercavity length from the cavitator. The distributions of the pressure coefficient are given in Fig. 7b. The contours indicate that the pressure distribution on cavitator is not symmetrical, and the lines of pressure contour on the upper side of the cavitator is more concentrated than that on the down side, which demonstrates a relatively larger pressure gradient, thus leading to the generation of downward lift force. Here the lift force is defined as the force in the positive direction of the y axis acting on cavitator, and the coefficient of lift c y is expressed as follows:

F y
F y where is the force of lift acting on the front cavitator, U ∞ is the velocity of the ambient flow, and A demonstrates the reference area of cavitator. Fig. 8 shows the curves of coefficients of both lift and drag forces for cavitator at states 4 and 5. It can be seen that  the amount of the lift force equivalent to 30% of the drag force in magnitude is produced on cavitator due to the protuberance of the control element on the upper side. The more the height of protruding part is, the larger the lift force will be. It can be concluded that when the control element protrudes on any of the three sides, a considerable amount of pitching and yawing forces will be produced in the same way, therefore the gesture of vehicles can be well adjusted.
The transverse force arising on the cavitator non-symmetric about the free-stream results in the supercavity axis deformation (Savchenko et al., 1998). According to the theorem of momentum, the impulse of the transverse force on cavitator must correspond to the change of the momentum in the wake that is the same by magnitude and opposite by direction. This means that if the force on the cavitator is directed down, then the supercavity axis must be deflected up and vise versa.
The distribution of the supercavity axis deflection for Case 4 at σ v =0.124 is described in Fig. 9. It can be seen that the supercavity axis floats up on the longitudinal plane and the deflection becomes more serious when approaching the tail part of supercavity.
The same lift force will also arise on cavitator when the flow is not parallel with the x axis, thus causing the deformation of the supercavity axis. The supercavity characterist-ics of Case 4 at different angles of attack when σ v =0.194 are exhibited in Fig. 9. The relevant angles include 0°, 5°, 10°, 15°, 20°, 25° and 30°, where the angle of attack is defined as the inclined angle of the direction of inflow velocity to the x axis, and the positive value of the angle indicates the clockwise direction, while a negative value means anticlockwise direction.
The deformation of supercavity axis for the variable-lateral-force cavitator is presented in Fig. 10a. The curves reveal that the existence of attack angle causes the upward deformation decline to a certain degree, which is mainly reflected in the second part of supercavity, and the deflection diminishes with the increase of the attack angle.   The variation curves of the lift and drag coefficients for cavitator are shown in Figs. 10b and 10c, respectively. It can be found that when the angle of attack increases from 0°t o 10°, the drag force and lift force of variable-lateral-force cavitator have dropped nearly by 10% and 19.4%, respectively, and then the coefficient of lift rises about 4.6% as the angle of attack varies from 10° to 25°, which may be induced by the oscillation of unstable supercavity. Finally the change becomes gentle with the rise of the attack angle when the value of angle reaches 30°.

Conclusions
The varying laws of supercavity size with working strokes for the traditional variable-drag cavitator were analyzed. A new scheme named the variable-lateral-force cavitator was proposed on the basis of the theory of the traditional variable-drag cavitator, and the three-dimensional supercavitating flow around the cavitator was calculated. It is confirmed that the lift force, lateral forces and supercavity size can be effectively adjusted and controlled by the front cavitator. The main conclusions can be drawn as follows.
(1) It was found that the dimensions of supercavity under the same cavitation number grow larger as the displacement of the control element in the radial direction raises, which reflects the excellent advantages of the variable-lateral-force cavitator in changing the supercavity size compared with the cavitators with the fixed diameters and profiles.
(2) The amount of pitching and yawing forces equivalent to 30% of the drag in magnitude will be produced when the height of protruding part on either side is 5% of the diameter of cavitator, and the axis of supercavity will be deviated simultaneously.
(3) The lower section of supercavity appears concave inwards when the control element protrudes on the upper side.
(4) It was observed that the coefficients of lift and drag forces for cavitator decline with the increase of the attack angle, and the deflection of axis in the second part of supercavity decreases simultaneously.
Since the coefficient of drag for the variable-lateralforce cavitator will be raised to a small extent as the lift force and lateral force increase, the impetus should be promoted simultaneously to keep the velocity of underwater vehicles. Hence the research work in the next step should be focused on the solution to enhance the ratio of the lift force and drag force of the variable-lateral-force cavitator. It is hoped that the application value of the cavitator in engineering will be promoted through the verification and modification experiments on the control scheme.