# Dynamic Pressure Analysis of Hemispherical Shell Vibrating in Unbounded Compressible Fluid
## 3 Numerical Validation  
### CFD Method  
It is well-known that the second-order method is the most stable discretization scheme, which is derived by [PERSON]'s series:  
\[\frac{\partial^{2}u}{\partial x^{2}}=\frac{\partial}{\partial}[\frac{\partial u }{\partial x}]=\underset{\Delta x->0}{lim}\frac{\left(\partial u/\partial x \right)_{i+1/2}-\left(\partial u/\partial x\right)_{i-1/2}}{\Delta x}\approx \frac{u_{i+1}-2u_{i}+u_{i-1}}{\left(\Delta x\right)^{2}}+O(\Delta x^{2}) \tag{36}\]  
\[\frac{\partial^{2}u}{\partial x\partial y}\approx\frac{u_{i+1,j+1}-u_{i+1,j-1} -u_{i-1,j+1}+u_{i-1,j-1}}{4\Delta x\Delta y}+O(\Delta x^{2},\Delta y^{2}) \tag{37}\]  
Equations (36) and (37) are one and two-dimensional schemes, respectively, in which \(u\) is the value of physical quantities such as velocity and pressure, and \(x,y\) are the coordinates. Usually, the grid will not be the rectangular shape. Let the transformation of domain be as Equation (36), shown in Figure 3a. \[\xi=\xi(x,y),\eta=\eta(x,y) \tag{38}\]  
In that case, the differential can be obtained by the chain rule, and the second-order differential cab be rewritten as Equations (39) and (40):  
\[\frac{\partial^{2}u}{\partial x^{2}}=\frac{\partial u}{\partial\xi}\frac{ \partial^{2}\xi}{\partial x^{2}}+\frac{\partial u}{\partial\eta}\frac{ \partial^{2}\eta}{\partial x^{2}}+2\frac{\partial^{2}u}{\partial\xi\partial \eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}+\frac{ \partial^{2}u}{\partial\xi^{2}}{\left(\frac{\partial\xi}{\partial x}\right)^{2 }}+\frac{\partial^{2}u}{\partial\eta^{2}}{\left(\frac{\partial\eta}{\partial x }\right)^{2}} \tag{39}\]  
\[\frac{\partial^{2}u}{\partial y^{2}}=\frac{\partial u}{\partial\xi}\frac{ \partial^{2}\xi}{\partial y^{2}}+\frac{\partial u}{\partial\eta}\frac{ \partial^{2}\eta}{\partial y^{2}}+2\frac{\partial^{2}u}{\partial\xi\partial \eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}+\frac{ \partial^{2}u}{\partial\xi^{2}}{\left(\frac{\partial\xi}{\partial y}\right)^{2 }}+\frac{\partial^{2}u}{\partial\eta^{2}}{\left(\frac{\partial\eta}{\partial y }\right)^{2}} \tag{40}\]  
On the basis of the preceding analysis, in order to find and check the validity of the theory, a hemispherical shell submerged in unbounded fluid was examined using a CFD analysis using a commercial computer code, ANSYS (release 17.0, Ansys Company, Canonsburg, PA, USA) [27]. The CFD model has the identical shell geometry, boundary conditions, and material properties that will be used in the theoretical calculation. Similar applications can be found in some papers [28; 29]. The radius of the hemispherical shell is set to 10 m, and the unbounded fluid is simulated as a far field boundary whose radius is 500 m (see Figure 3b). The physical properties of the fluid are as follows: density = 1.225 kg/m\({}^{3}\), sound velocity c = 340 m/s. The viscosity of the fluid is neglected in both the theoretical analysis and the CFD calculation. The purpose of the present paper is to verify the principle of pressure under the vibration of the shell. The movement of the hemispherical shell was set as uniform vibration in the radial direction, which was also the boundary of the fluid. As for the numerical simulation parameters, the far field boundary is set to be the pressure far field, reference pressure to zero, movement boundary to \(U(t)=sin(10t)\), calculating time to 1.0 s, and the delta-time to 0.001 s. A simple scheme second-order method was accepted for the CFD simulation, and the initial condition of fluid was set to be stationary. For comparison, both the compressible fluid case and the incompressible fluid case were carried out. To verify the solution, the line from \(r_{s}\) to \(R\), which data was obtained from, is shown as LINE-X in Figure 3b. In order to demonstrate the influence of the discretization, the numerical result with different meshing sizes is displayed. Table 1 shows the comparison of the total energy of all over the fluid at time = 0.2 s with different density of shell mesh.