# Research on the Water Entry of the Fuselage Cylindrical Structure Based on the Improved SPH Model
## 2 Improved SPH Model  
### Low-Dissipation Godunov SPH (GSPH) Method for Fluid  
For fluids, when a particle is in a compressible state, the repulsive force between particles increases and then decreases as particle spacing decreases. It is easy to cause the particle to oscillate near the equilibrium position, which causes the particle pressure to oscillate, thereby leading to instability [24]. Moreover, in the weakly compressible SPH model, which is often used to simulate the fluid, small-density variations may lead to pressure oscillation, especially when phenomena such as rolling and splashing occur, which can result in computational instability [48]. GSPH [42] (Godunov SPH) is an effective approach to alleviating tensile instability. The GSPH method is based on solving the Riemann problem between particles, as shown in Figure 1. Assuming the existence of a discontinuity interface between particles \(i\) and \(j\) located at the center point \((r_{i}+r_{j})/2\), the physical quantities on both sides of the discontinuity surface are denoted as \((\rho_{L},U_{L},p_{L},c_{L})=(\rho_{i},\
u_{i};e_{ij},p_{i},c_{j})\) and \((\rho_{R},U_{R},p_{R},c_{R})=(\rho_{j},\
u_{j};e_{ij},p_{j},c_{j})\), where \(\rho\), \(\
u\), \(p\) and \(c\) represent the density, velocity, pressure, and sound speed of the particles, respectively. Here, \(e_{ij}=-r_{ij}/\left.1\ r_{ij}\right|\) is the unit vector pointing from particle \(i\) to particle \(j\). Assuming that particles \(i\) and \(j\) have the same sound speed, \(c_{0}\), an approximate linear Riemann solution is introduced at the interface to obtain the intermediate velocity, \(U^{*}\), and pressure, \(p^{*}\). The form of the approximate Riemann solution is given by [42]  
\[\left\{ \begin{array}{l}U^{*}=\frac{U_{L}\rho_{L}+U_{R}\rho_{R}+1/c_{0}(p_{L }-p_{R})}{\rho_{L}+\rho_{R}}\\ p^{*}=\frac{P_{R}\rho_{L}+p_{L}\rho_{R}+\rho_{R}c_{0}(U_{L}-U_{R})}{\rho_{L}+ \rho_{R}}\end{array} \right. \tag{1}\]  
By replacing the average pressure and average velocity of particles \(i\) and \(j\) in the traditional SPH governing equations with the velocity, \(U^{*}\), and pressure, \(p^{*}\), the following GSPH governing equations can be obtained:  
\[\left\{ \begin{array}{l}\frac{d\rho_{i}}{dt}=2\rho_{i}\frac{\gamma}{\rho_{i}} (\upsilon_{i}-\upsilon^{*})\
abla W_{ij}\\ \frac{d\upsilon_{i}}{dt}=-2\frac{\gamma}{\rho_{i}\rho_{j}}(\frac{p^{*}}{\rho _{i}\rho_{j}})\
abla W_{ij}\end{array} \right. \tag{2}\]  
where \(\rho_{i}\) and \(\upsilon_{i}\) represent the density and velocity of particle \(i\). \(m_{j}\) and \(\rho_{j}\) represent the mass and density of particle \(j\), and \(\
abla W_{ij}\) represents the gradient of the kernel function. \[\left\{ \begin{array}{l}\upsilon^{*}=U^{*}e_{ij}+(\overline{\upsilon}_{ij}- \overline{U}e_{ij})\\ \overline{U}=(U_{L}+U_{R})/2\end{array} \right. \tag{3}\]  
where \(\overline{\upsilon}_{ij}\) represents the average velocity of particles \(i\) and \(j\). In the weakly compressible SPH model, defining the state equation that describes fluid pressure and density is necessary, as shown in Equation (4) [49]:  
\[P=\frac{\rho_{0}c_{0}}{\gamma}[(\frac{\rho}{\rho_{0}})^{\gamma}-1] \tag{4}\]  
where \(\rho_{0}\) is the initial density of the fluid, \(\gamma\) is the specific heat ratio (typically seven for water), and \(c_{0}\) is the sound speed. When using the approximate Riemann solver to solve for the intermediate state pressure, \(p^{*}\), it is found that the dissipation term \(1/2\rho c_{0}(U_{L}-U_{R})\) is introduced in the momentum equation. If \(c_{0}\) is taken as the real sound speed of the  
Figure 1: Riemann model between SPH particles. fluid, it will result in excessive dissipation, deviating from the actual results. To limit this dissipation, \(c_{0}\) in the calculation of \(p^{\star}\) is restricted as follows [50]:  
\[c_{0}= \begin{cases}\min(\frac{1}{2}\omega(|U_{L}|+|U_{R}|),c_{0}),&U_{L}\geq U_ {R}\\ 0,&U_{L}<U_{R}\end{cases}  \tag{5}\]  
where \(\omega\) is a constant.