## 2 Computational Model of External Flow Field
### Governing Equations  
Compressibility is a basic property of a fluid. Any fluid is compressible, but when the influence of the change of fluid density on the flow can be ignored, the incompressible flow assumption can be adopted, that is, the density is constant. Usually, the train speed does not exceed 350 km/h (the corresponding Mach number is 0.286), and the influence of the change of air density on the flow is negligible and incompressible flow can be used. However, in the present paper, the speed of the high-speed maglev train is 600 km/h, and the corresponding Mach number is 0.49, which is significantly higher than 0.3. The compressibility of the air should be considered. In addition, the low vacuum tube is a closed structure, and the compressibility of the air should also be considered. Therefore, the flow field of the high-speed maglev train in the low vacuum tube can be described by the three-dimensional compressible Navier-Stokes equation.  
The flow field around the high-speed maglev train is a very complex turbulent flow field. It is necessary to simulate the turbulent flow to obtain the flow field of the high-speed maglev train. In order to study the characteristics of the aerodynamic noise source of the high-speed maglev train, the fluctuating characteristics of the surface pressure of the high-speed maglev train should be accurately computed; therefore, the large eddy simulation (LES) method is used to simulate the turbulent flow field of the high-speed maglev train [21; 22].  
The basic idea of the LES is to decompose the instantaneous turbulent flow including fluctuating motion into two parts, namely, large-scale eddy motion and small-scale eddy motion, through a certain filtering method. The large-scale vortex motion can be obtained by directly solving the Navier-Stokes equation, and the influence of the small-scale vortex motion on the large-scale vortex motion is expressed in the motion equation as a stress term similar to the Reynolds stress, which is called sub-grid Reynolds stress. Sub-grid Reynolds stress can be simulated by establishing the model. After filtering the Navier-Stokes equation, the governing equations of the LES can be obtained as:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\big{(}\rho\overline{u}_{j}\big{)} }{\partial x_{j}}=0 \tag{4}\]  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}=-\frac{\partial \overline{p}}{\partial x_{i}}+\frac{\partial\overline{\tau}_{ij}}{\partial x_ {j}}-\frac{\partial\tau_{ij}^{l}}{\partial x_{j}} \tag{5}\]  
where \(\overline{u}_{i}\) is the velocity component after spatial filtering, \(\overline{\tau}_{ij}\) is the viscous stress tensor after spatial filtering, and \(\tau_{ij}^{l}\) is the sub-grid Reynolds stress.  
In order to close the equations, it is necessary to model the sub-grid Reynolds stress. The commonly used model is the eddy viscosity model:  
\[\tau_{ij}^{l}-\frac{1}{3}\overline{\tau}_{kk}\delta_{ij}=-2\mu_{i}\overline{S} _{ij} \tag{6}\]  
where \(\mu_{i}\) is the sub-grid viscosity coefficient and \(\overline{S}_{ij}\) is the sub-grid tensor rotation rate, which can be expressed as:  
\[\overline{S}_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial\overline{u}_{i}}{\partial x _{j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\Bigg{)} \tag{7}\]  
The Smagorinsky-Lilly model is commonly used to compute the sub-grid viscosity coefficient. In this model, the sub-grid viscosity coefficient can be expressed as:  
\[\mu_{t}=\rho L_{s}\sqrt{2\overline{S}_{ij}\overline{S}_{ij}} \tag{8}\]  
where \(L_{s}\) is the sub-grid mixing length, and can be computed by  
\[L_{s}=\min\Big{(}\kappa d,C_{s}V^{1/3}\Big{)} \tag{9}\]  
where \(C_{s}\) is the Smagorinsky constant, \(\kappa\) is the von Karman constant, \(d\) is the distance to the nearest wall, and \(V\) is the volume of the computation cell. \(C_{s}\) is 0.1 in the Smagorinsky-Lilly model.