Numerical Simulation of the water-exit process of the missile based on Moving Particle Semi-Implicit method

The phenomenon of a water mound can show up in the process of water-exit of the missile [1], and the a water mound will break and splash in a very short time, which will exert an extra force on the missile and therefore will affects the movement of the missile as well as the gesture when the missile comes out the water. Therefore, investigation into the water-exit process of the missile is of great significance to the launch of the underwater missiles. The traditional grid methods always fail to simulate the whole the water-exit process because of the large deformation of the mesh. The paper adopts a meshless method, called Moving Particle Semi-Implicit method (MPS), to study the water-exit process of the missile. Based on verifying it by the dam break simulation, MPS is applied to simulate the water mound during the water-exit process, which is aiming at providing some reference for the relevant theoretical research.


Introduction
Moving particle semi-implicit method (MPS) is relatively new meshless method based on Lagrange particles [2], which replaces the mesh in the traditional numerical methods, applies some numerical operators according to the relationship of the neighboring particles to take the place of traditional difference terms. As such, the numerical diffusion resulted from solving the convection term can be avoided [3]. Meanwhile, the particle moves freely and separately under the external force and internal force from its neighboring particles, so MPS has absolute advantages in solving problems with large deformation, even with the splashing of free water surface.
During the water-exit process of the underwater launching missile, when the missile approaches the water free surface, some water will be carried and raised because of the high speed of the missile, which is called the phenomenon of a water mound. The water mound will grow larger and higher with the movement of the missile, and then because of the gravity force itself the bump will break away from the surface of the missile and splash. For such problems, the traditional mesh methods always lose efficacy because of the large deformation of the mesh.
Subsequently, the paper focuses on the application, and develops the relative numerical model of the MPS. The classical dam break model [4][5] is adopted to verify the program. On this base, the water-exit process of the missile is simulated numerically, and the mechanism and variation law of the water mound during this process is analyzed and summarized, which is aiming at providing some useful reference for the relative engineering practice.

Basis of MPS
For impressible viscous flows, the continuity equation (Law of mass conservation) and Navier-Stokes equation (Law of momentum conservation) are given by: Where  is fluid density, v is fluid velocity, f is the external force, p is pressure and  is the kinematic viscosity coefficient.

Numerical Model of Original MPS
Where r is the distance between particle, e r is the effective radius of particle, the value of which directly affects the number of neighboring particles. The number density of particle can be obtained by: The Gradient model is described by: , which is effective to avoid instability resulting from minus pressure gradient.
The Laplacian model of  at point i is expressed as Where  can be calculated by the following equation [8]: where  is a parameter below 1.0 and 0.97   is selected. The dynamic free surface condition is satisfied by taking the atmospheric pressure 0 atm p p   for the free surface particles.
In order to avoid misjudging the fluid particles near the boundary as free surface particles, the solid boundary is discretized into three layers of particles. Moreover, a repulsive force [11] is employed to prevent fluid particles from penetrating the solid boundary.

Verification of the numerical model
Dam break [12] is a classic example to simulate the fluid flow with free surface. Therefore, the semiimplicit method for moving particles is verified by the dam break simulation. The model adopted is shown in Fig.1, and it is discretized into 1122 particles, of which 648 are fluid particles, and 474 are solid wall particles and fluid boundary particles, as shown in Fig.2.        [13][14]. It can be seen from Fig. 4 that the results agree well.

simulation of water-exit of the missile
In this paper, the model of water-exit adopted is shown in Fig.5, the corresponding numerical model is shown in Fig.6, which contains 3712 particles, including 2860 fluid particles, 286 fluid boundary particles and 566 solid wall particles. The missile moves upward at a constant velocity of 1m/s from 2.2L below the free surface. Other calculation parameters are consistent with the dam break model.  Figure 6. Numerical model of water-exit of the missile. As can be seen from Fig.7, due to the shape of the missile and the viscosity of water, when it starts to move, bubble areas are gradually formed near the shoulder and tail of the missile [15][16]. Meanwhile, a water mound appears at the free surface. Then, with the movement of the missile, the water mound becomes more and more obvious.   The bubble area near the tail grows first and then begins to decrease. At t=0.072s, water impacts the tail of the missile, where jet impact occurs. At the same time, the bubble area around the shoulder becomes larger and moves backward, and it begins to break around t=0.078s. In addition, due to the reflection of the boundary of the solid wall, one layer of floating particles appears on the free surface within t=0.060s to t=0.120s. Finally, the missile gets out of the water with severe water splash. In this process, the bubble area near the shoulder of the missile gradually disappears. At the same time, the water mound will fall and break in a very short time and the liquid surface shows very large deformations.

Conclusion
In this paper, MPS is verified by investigating the dam break model that it is capable of dealing with fluid flows with large defamations. Based on these, MPS is applied to simulate the whole water-exit process of the missile, including the initial movement, steady movement in the water as well as piercing the water surface. This has great significance to investigate the forming, growing and breaking mechanism of the bubble areas near the should and the tail of the missile. Moreover, the results in the paper can provide some reference to the forming and breaking mechanism of the water bump during the water-exit process of the missile.
Due to the limitation of time and equipment, the number of particles used in the simulation is relatively small, which leads to the unsatisfactory effect of the simulation of the broken water mound. Therefore, the next research objective is to optimize the algorithm, expand the two-dimensional to threedimensional, to better simulate the water-exit process and then reveal the mechanism of the water mound phenomenon.