# Shape Optimization of the Streamlined Train Head for Reducing Aerodynamic Resistance and Noise
## 3 Train Aerodynamic Model
### Computational Model  
The three-dimensional parametric model described in Section 3.1 is used to set up a train model with three cars. The computational domain as well as the boundary conditions is shown in Figure 3. The front of the computational domain is set as velocity inlet boundary, and the rear is set as pressure outlet boundary. The two sides and top of the domain are set as symmetry boundaries. The bottom of the domain is set as sliding wall boundary, and the train is set as the stationary wall. The train speed in the present paper is 300 km/h. Thus, the flow around the train is considered as in-compressible steady flow. The commercial software Fluent is used to simulate the flow field of the train, the standard \(k\)-\(\varepsilon\) turbulence model is used to resolve the turbulent flow, and the standard wall function is used to resolve the boundary layer.  
Figure 2: Left half of streamlined head.  
The aerodynamic resistance of a train is directly obtained by aerodynamic calculation, and the aerodynamic noise can be described by dipole noise source on the train surface. [PERSON] [19] analyzed the noise generated by gas flowing through solid surface, which can be expressed as  
\[p^{\prime}(\mathbf{z},t)=\frac{1}{4\pi a_{0}}\int_{S}\frac{(z_{i}-y_{i})n_{i}}{r ^{2}}\frac{\partial p(\mathbf{y},\tau)}{\partial t}dS(\mathbf{y}) \tag{8}\]  
where \(\mathbf{y}\) and \(\mathbf{z}\) are the positions of near-field points and far-field points, respectively; \(y_{i}\) and \(z_{i}\) are components of \(\mathbf{y}\) and \(\mathbf{z}\); \(t\) is time; \(\tau\) is delay time, \(\tau=t-r/a_{0}\); \(p^{\prime}\) is sound pressure; \(a_{0}\) is sound velocity; \(S\) is solid surface; \(r\) is the distance between \(\mathbf{y}\) and \(\mathbf{z}\); \(n_{i}\) is the unit normal vector component of \(S\).  
Based on Formula (8), the total acoustic power \(P_{A}\) emitted from the entire body surface can be computed from  
\[P_{A}=\int_{S}\frac{A_{c}(\mathbf{y})}{12\rho_{0}\pi a_{0}^{3}}\left[\frac{ \partial\overline{p(\mathbf{y},\tau)}}{\partial t}\right]^{2}dS(\mathbf{y}) \equiv\int_{S}I(\mathbf{y})dS(\mathbf{y}) \tag{9}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline \(x_{1}\), \([-400,400]\)/(mm) & \(x_{2}\), \([-100,200]\)/(mm) & \(x_{3}\), \([-200,200]\)/(mm) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Range of design variables and corresponding deformation.  
Figure 3: Computational domain and boundary condition.  
where \(A_{c}\) is the correlation region and \(I(\mathbf{y})\) is the dipole noise source on the solid surface, which can be interpreted as the contribution of the noise per unit area of solid surface to the total sound pressure.  
The sound power level \(L_{p}\) is defined as  
\[L_{p}=10\log(P_{A}/P_{r}) \tag{10}\]  
where \(P_{r}\) is reference sound power, \(P_{r}=10^{-12}\text{W}/\text{m}^{3}\).  
After the flow filed of the train is computed, the broadband noise source module in the commercial software Fluent is activated. Then, the dipole noise source on the streamlined head can be obtained.  
Grid-independence test is conducted aiming at eliminating the influence of grid density on flow field simulation. Three sets of mesh with different grid density are generated, and the information of grid setting and calculation results are shown in Table 2. The three meshes have the same settings of boundary layers. The height of first layer is 2 mm, the growth ratio is 1.1, and there are 8 layers in total. As presented in Table 1, when the maximum size of the surface grid is densified from 250 mm (mesh1) to 200 mm (mesh2), the aerodynamic resistance decreases by 4.55%, and the dipole noise source increases by 0.5 dB. When the maximum size of the surface grid is densified from from 200 mm (mesh2) to 150 mm (mesh3), the aerodynamic resistance increases by 0.86%, and the dipole noise source increases by 0.1 dB. Therefore, mesh2 is chosen for the subsequent aerodynamic calculations.