Numerical Simulation on Near-water-surface Skipping Motion of Trans-media Vehicle

When a trans-media vehicle impacts the water surface at a small angle, it will bounce and take off and then re-enter the water, which is called near-water-surface Skipping Motion. The study of this kind of movement is of great significance to the high maneuvering penetration of ships or aircraft near sea level. This paper adopted the method of numerical simulation, selected the SST k-ω turbulence model, combining the N-S equation and the six-degree-of-freedom (6DOF) algorithm, and used the overall dynamic grid technology to calculate the vehicle entry condition at speed of 50m/s, 100m/s, 150m/s &10°, 20°, 30° angle of attack, and the near-water-surface skipping motion characteristics of the vehicle were explored, which provided a reference for studying the high maneuvering motion of offshore plane trans-media vehicles.


Introduction
Near-water-surface skipping motion is a kind of special bounce movement after an object hits the water, "Stone Skipping" is one of the typical near-water-surface skipping motion.Bocquet [1] first proposed the numerical study of near-water-surface skipping motion.He established a mathematical model for the " Stone Skipping " of circular and square discs by dimensional analysis, and gave the conditions of disc bounce and the law of energy dissipation in the process of bounce.On the basis of Bocquet's research, Christophe Clanet [2] et al. studied the disc bounce experiment by using high-speed photography equipment in 2004, and gave the law between the water entry angle, water entry speed, speed angle and rotational speed of the disc, and concluded that the ideal water entry angle of the disc was about 20°.In recent years, some scholars believe that the near-water-surface skipping motion provides a strong reference for the study of high maneuvering motion of trans-media vehicles near sea level [3], and try to use its principles to realize the high-maneuverability of offshore-level vehicles, as well as new concepts trans-media vehicles [4][5][6][7].Those make the near-water-surface skipping motion become one of the research hotspots of trans-media vehicles.
The near-water-surface skipping motion of trans-media vehicles involves complex multiphase flow and the interaction between the fluid and the large-scale motion of the vehicle, especially the violent change of the vehicle force caused by the high-speed collision of the vehicle and the crushing and splash of the free surface, which has strong nonlinearity [8][9].In this paper, by studying the ship with high speed motion near the water surface, the characteristics of near-water-surface skipping motion of the vehicle were explored, which provided a reference for studying the high maneuvering motion of transmedia vehicles at near sea level.

Governing Equation
Near-water-surface skipping motion involves three-phase coupling of solid, liquid and gas.Incompressible fluid is selected in the simulation without considering heat exchange.VOF model is used to simulate the multiphase flow field structure of water, vapor and non-condensable gas.In the calculation process, VOF multiphase flow model treats the fluid medium in the whole flow field as a single fluid medium, which is homogeneous and variable in density and viscosity.Different fluid media regions are distinguished according to the volume fraction of each phase in the same grid.
The continuity equation describing the mixed flow field is: The momentum equation is: In the formula, i = 1,2,3, representing different fluid phases. represents the velocity component of the medium. represents turbulent viscosity coefficient;  and  are the density and dynamic viscosity of the mixed medium respectively, and their expressions are: ∑   ( 4 )

Calculation Model and Boundary Conditions
The calculation model used in this paper is a simplified vehicle, the length is 1.6m, the longitudinal height is 0.18m, the lateral maximum width is 0.2m, the total area of the bottom surface is 0.18m 2 , the total volume is 3.45e -2 m 3 , its density is 2.7 times of water, and the total mass is about 93.2kg.The shape of vehicle is shown in Figure 1.The centroid is 0.879 m away from the head of the vehicle, which is set as the origin of the coordinate system.The axis of the vehicle inclines at θ angle with the horizontal direction, and the gravity direction is vertical downward.The whole computational domain is a sphere with a radius of 5m. Figure 2 shows the computational domain in the xoz plane at the initial time.The upper blue area is the air, the lower red area is the water, and the middle boundary between the two areas is the free surface.
The calculation adopted the overall dynamic mesh technology, and the outer surface of the sphere was set as the pressure outlet as a whole.Different surface pressures and different phases were defined by loading udf.At the initial time, the height of the centroid of the vehicle from the water surface was 0.5 meters.The vehicle had initial velocity both in the horizontal direction and vertical direction.The motion law of the vehicle was solved by the unsteady Reynolds-averaged N~S equation coupled with the 6DOF equation.The object was assumed to be a rigid body, ignoring the surface tension and considering the effect of gravity.SST (Shear Stress Transport) k-ω turbulence model was selected.The pressure field and velocity field adopted SIMPLE algorithm.The implicit scheme was adopted in time, and the second-order upwind scheme was adopted in space.The time step was set to 10 -5 s, so that the residual error could reach 10 -4 after 20 iterations in each time step [10].

Dynamic mesh technology and mesh subdivision
Due to the large motion range of the vehicle, traditional dynamic mesh method will produce large mesh deformation in the mesh reconstruction, which will bring great errors to the calculation.Although the overset mesh method can improve the accuracy, the separation of background mesh and component mesh greatly increases the number of grids, and also increases the computational complexity.
Considering the calculation accuracy and efficiency comprehensively, the global dynamic mesh technology was adopted in this paper.The calculation domain included a spherical region with a radius of 5 meters around the vehicle, which ensured that the grid around the vehicle had good quality and reduced the total number of grids to facilitate calculation [11].The near-wall grids of the vehicle are shown in Figure 3.The final number of grids was 2583194.

Movement characteristics analysis
Fig. 4 The attitude change of near-water-surface skipping motion Figure 4 shows the attitude changes of a single skidding movement at different times when the horizontal vehicle's velocity was 100 m/s and the angle between the vehicle and the water surface was 20°.In order to facilitate observation, the main view and side view are selected for display.The interval time of each image is 0.05 s, so that it can show the whole process of a single near-water-surface skipping motion of the vehicle.
As can be seen from Figure 4, the lowest point of the vehicle first contacted with the water surface, and then part of the vehicle entered the water and disturbed the surrounding water, thus producing some depressions.In the whole process, according to the conclusion drawn by Wang Chao et al. [12], the vehicle could approximately slide and skip in the water with a constant incidence angle.After the vehicle left the water surface, its trajectory in the air was similar to a parabola.The trajectory and motion law of the vehicle were in good agreement with the experiment of Sun Shiming et al. [13], which showed that the numerical simulation method could accurately simulate the whole near-water-surface skipping motion process.

Analysis of near-water-surface skipping motion law of vehicle
In order to explore the influence of initial inclination angle θ on the process of skipping motion, numerical simulation was carried out for the vehicle with horizontal muzzle velocity of 100m/s and vertical muzzle velocity of 10m/s at three inclination angles of 10°, 20° and 30°.Since the lateral force (y direction force) of the vehicle was approximately balanced, the velocity change and trajectory deviation were not obvious, the near-water-surface skipping motion characteristics of the vehicle were only analyzed from the vertical and horizontal directions in the xoz plane.

Horizontal direction (x direction)
. Figure 5 and Figure 6 show the variation law of the horizontal velocity 'Vx' of the centroid and the resultant force 'Fx' of the horizontal direction with time when the vehicle moved under three attitudes.It can be seen from the figure that from the initial moment, after the vehicle touched the water surface, its velocity in the horizontal direction began to decline, and then the change gradually slowed down.Combined with the change of its trajectory and resultant force, it could be seen that the resistance of the vehicle in the horizontal direction suddenly increased after the first contact with the water surface, so that the speed began to decrease.Compared with the first skipping motion, the change of the resistance of the vehicle after the second contact was smaller, and the change range of the velocity was also reduced accordingly.After the vehicle bounced off the water surface, it moved parabolically in the air, and the horizontal velocity remained basically unchanged.The numerical simulation results were consistent with the actual physical laws.

Fig. 5 Horizontal velocity variation of vehicle
Fig. 6 Horizontal force variation of vehicle

Vertical direction (z direction)
. Figure 8 and Figure 9 show the variation law of the velocity 'Uz' of the vehicle in the vertical direction and the resultant force 'Fz' of the vehicle in the vertical direction with time.It can be seen from the figure that after the vehicle contacted the water surface at the initial time, it continued to decline under the inertial effect, and after reaching the lowest point of its trajectory, the vehicle began to rise gradually.Combined with the trajectory of the vehicle and the resultant force in the z direction, the vehicle kept downward motion under the action of gravity and inertia at the initial moment.With the volume of the vehicle immersed in water increases gradually, the lift force began to increase until the velocity in the vertical direction decreased to zero.At this time, the lift force of the vehicle was greater than the force of gravity.Under the combined action of the lift and gravity, the vehicle rebounded until it left the water surface.Since the force of air on the vehicle was very small, after a parabolic movement in the air, the vehicle re-entered the water, and then performed the next skipping motion until its energy was insufficient so that it could not rebound again.

Trajectory analysis of xoz plane.
Figure 9 shows the trajectory projection of three kinds of attitude vehicles in the xoz plane.From the comparison of motion trajectory and velocity change, it can be seen that when θ = 10°, the z-direction velocity of the vehicle rapidly decreased after the first skipping, and then the skipping process tended to be stable.When θ = 30°, the vehicle could maintain velocity in the vertical direction, but the horizontal velocity decreased quickly.Based on the velocity change and force condition, it could be concluded that when the angle between the vehicle and the horizontal direction was 20°, the slip movement process was relatively smooth, and the velocity and force change were relatively stable, which was consistent with the facts that Clanet et al. found in the experiment [2].Therefore, the attitude of θ = 20° was selected as the benchmark to further explore the near-water-surface skipping motion of the vehicle.

Analysis of control moment
On the premise that the inclination angle θ was 20°, the numerical simulation was carried out for three kinds of motion conditions with the horizontal motion velocity of 50 m/s, 100 m/s and 150 m/s respectively.Figure 10 shows the skipping trajectory of the vehicle at different speeds in the same time.Figure 11 and Figure 12 show the pitching moment and yaw moment required to maintain the attitude of the vehicle at different speeds.It can be seen from Figure 10 that the vehicle could not skip when entering the water at a low horizontal velocity.Comparing the force of the vehicle with the speed of 100m/s and 150m/s, it can be seen that with the increase of the speed of the vehicle, the control moment required to maintain its attitude also increased.When the speed of the vehicle was 150m/s, the order of the pitching moment required to maintain the attitude of the vehicle was up to 10 7 , and the order of the yaw moment required to maintain the attitude of the vehicle was up to 10 5 , these data were obtained at the lowest point of each skipping movement.The force and moment in the rolling direction were less than those in the pitching and yaw directions, so they are not given here.With the attenuation of the speed of the vehicle, the moment needed to maintaining its attitude gradually increased.From the change of the vehicle moment, it can be seen that pitching moment and yaw moment are both important to control the vehicle to maintain a stable attitude for continuous skipping motion.

Fig. 7
Fig. 7 Velocity variation in vertical direction Fig. 8 Force change in vertical direction

Fig. 9
Fig. 9 Comparison of xoz plane trajectory projection of vehicle