# Research on the Water Entry of the Fuselage Cylindrical Structure Based on the Improved SPH Model
## 2 Improved SPH Model
### Riemann-Based Contact Algorithm for Fluids and Solids  
This section is divided into subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.  
When implementing boundary conditions using the virtual particle method in SPH, virtual particles need to be arranged in the initial stage, with their positions either fixed or determined by the fluid particles. This method ensures conservation and facilitates the implementation of slip and non-slip boundary conditions [52]. However, this method is complicated for moving and complex boundaries. Therefore, a modified virtual particle method based on the Riemann model was proposed [53], which can retain the advantages of the virtual particle method and apply to moving and complex boundaries.  
This paper implements the contact algorithm based on the following conditions. At a fluid-solid interface, solid particles provide velocity boundary conditions to fluid particles, and fluid particles provide pressure boundary conditions to the solid particles. The solid particles providing velocity boundary conditions to the fluid particles achieve this by including the contribution of the velocities of neighbor solid particles in the discretized continuity equation of the fluid. The fluid particles providing pressure boundary conditions to the solid particles are achieved by defining a virtual pressure, \(p^{\prime}\), for the solid particles. Equation (12) shows that the virtual pressure is obtained by interpolating the fluid pressure.  
The first term represents the pressure interpolation of the fluid particles, and the second term is the correction pressure considering hydrostatic pressure [54].  
\[p^{\prime}=\frac{\sum\limits_{f}p_{f}W_{sf}}{\sum\limits_{f}W_{sf}}+\frac{\sum \limits_{f}\rho_{f}\big{(}\mathbf{g}-\mathbf{a}_{s}\big{)}\cdot\Big{(}\mathbf{r}_{j}-\mathbf{r} _{f}\Big{)}W_{sf}}{\sum\limits_{f}W_{sf}} \tag{12}\]  
where \(f\) represents the fluid particles within the supported domain of the solid particle, and \(\mathbf{a}_{s}\) is the acceleration of the solid particle.  
Then, as shown in Figure 3, the solid particles can be considered pre-arranged virtual particles whose positions are not fixed and are updated based on the solid control equations. The virtual pressure, \(p^{\prime}\), is assigned to the solid particles. Once the virtual pressure, \(p^{\prime}\), is obtained, the interaction force between the fluid and solid can be realized through the gradients of fluid pressure and solid virtual pressure.  
To improve the stability of the fluid-solid interface, this paper also treats the fluid-solid interface as a discontinuous Riemann model [53]. The contact force between the fluid and solid is modified by solving the intermediate state pressure, \(p^{\star}\). Finally, according to the momentum equation, the interaction force between particles \(i\) and \(j\) can be expressed through Equation (13):  
\[\mathbf{f}_{ij}=2m_{i}p^{\star}V_{j}^{2}\
abla W_{ij} \tag{13}\]  
Moreover, to achieve the no-slip boundary condition between the fluid and solid, the velocity, \(\mathbf{v}\), of the solid particle can be modified as shown in Equation (14) [54], where \(j\) represents the fluid particles within the supported domain of solid particle \(i\), and \(\mathbf{v}_{s}\). is the velocity of the solid itself.  
\[\mathbf{v}=2\mathbf{v}_{s}-\sum\limits_{j}\frac{\mathbf{v}_{j}W_{ij}}{W_{ij}} \tag{14}\]