# A Study on the Movement Characteristics of Fuel in the Fuel Tank during the Maneuvering Process
## 2 Research Object and Numerical Method
### Numerical Calculation Method and Validation  
The motion of fuel can be described using the Navier-Stokes (N-S) equations and the incompressibility condition. The governing equations of a free surface incompressible flow in Lagrangian form are provided by the following PDEs. \[\frac{DV}{Dt}=-\frac{1}{\rho}\
abla p+\
u\
abla^{2}V \tag{1}\]  
\[\
abla\cdot V=0 \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline X(m) & 12.6 & 13.3 & 12.8 & 12.5 & 12.8 & 12.5 & 13.1 & 13.2 \\ Y(m) & 3.7 & 4.8 & 4.8 & 4.3 & 4.0 & 3.8 & 3.6 & 3.9 \\ Z(m) & 0.1 & 0.08 & 0.08 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Virtual Sensor Number and Location. Figure 1: Physical model of the fuel tank. (**a**) The shape of the fuel tank; (**b**) Internal structure; (**c**) Flow holes. where \(\rho\) is the fluid density, \(p\) is the pressure, \(\
u\) is the kinematic viscosity. As FPM does not require the support of grid and therefore fluid values are known at discrete \"interpolation\" points, the Based on Kernel Function Interpolation, a correspondence between discrete particles and continuous physical fields, can be established, achieving the discretization of the fluid-fuel-flow physical field. This article uses the meshless method with a finite set of points. The FPM generates discrete particle points using a nodal distribution approach and solves algebraic equations discretely. This method determines the computed fluid domain based on the characteristics of the set model and generates computational nodes within the fluid domain. As the computational fluid domain may change over time, the computational fluid domain at a certain moment is defined as \(\Omega(t)\), with its boundary denoted as \(\partial\Omega(t)\). A series of computational particle points \(X=\{x_{1},\cdots,x_{n}\}\subset\overline{\Omega}\) is generated within the computational domain, including interior points \(X_{i}=\{x_{1},\cdots,x_{n_{i}}\}\subset\Omega\) and boundary points \(X_{b}=\{x_{n_{i+1}},\cdots,x_{n}\}\subset\partial\Omega\). Starting from the boundaries and advancing simultaneously into the computational domain, the direction of advancement is along the normal direction of the boundary points. When generating particle points, the inner and outer boundaries of the computational domain are numbered and their advancing directions are specified. During the advancement process, particle points continuously search for points generated by different numbered boundaries. If the distance between two points is less than the influence domain radius h, the generation of points in this direction is considered to have satisfied the calculation requirements. This method has good fitting effects for various boundaries and can reduce the number of particle points and improve computational efficiency by using ratio-based generation methods. In numerical simulations, it is common to approximate the values of a function and include its derivatives at a specific point. At a nearby point \(X_{i}=(x_{i},y_{i},z_{i})\), we can perform a Taylor series expansion and retain the first-order derivative, resulting in a smooth function for point \(X=(x,y,z)\), which can be described by the following expression:  
\[f(X)=f(X_{i})+\Big{(}X^{d}-X_{i}^{d}\Big{)}f_{i,d}+r(X-X_{i})^{2} \tag{3}\]  
here, \(d\) is the dimension index from x to z, and \(r(X-X_{i})^{2}\) represents the remainder of the Taylor expansion.