## 2 Computational Methods and Model  
### Large Eddy Simulation  
The compressible LES techniques are used for the accurate, simultaneous computation of external flow disturbance and sound waves around a high-speed train running at 300 km/h in the open field and the tunnel. The 3-D compressible unsteady filtered Navier-Stokes equations with the Smagorinski-Lilly sub-grid model can be written in the form below:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho\overline{u}_{i})}{ \partial x_{i}}=0 \tag{1a}\] \[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial \left(\rho\overline{u}_{i}\overline{u}_{j}\right)}{\partial x_{j}}=\frac{ \partial\sigma_{ij}}{\partial x_{j}}-\frac{\partial\overline{p}}{\partial x_ {j}}-\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{1b}\]\[\frac{\partial\Big{(}\rho\overline{h_{s}}\Big{)}}{\partial t}+\frac{\partial\Big{(} \rho\overline{u_{i}}\overline{h_{s}}\Big{)}}{\partial x_{j}}-\frac{\partial \overline{p}}{\partial t}-\overline{u_{j}}\frac{\partial\overline{p}}{\partial x _{i}}-\frac{\partial}{\partial x_{i}}\Big{(}\lambda\frac{\partial\overline{T}} {\partial x_{i}}\Big{)}=-\frac{\partial}{\partial x_{j}}\Big{[}\rho\Big{(} \overline{u_{i}h_{s}}-\overline{u}_{i}\overline{h}_{s}\Big{)}\Big{]} \tag{1c}\]  
where \(\sigma_{ij}\) is the stress tensor defined as  
\[\sigma_{ij}=\Bigg{[}\mu\Bigg{(}\frac{\partial\overline{u}_{i}}{\partial x_{j} }+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\Bigg{)}\Bigg{]}-\frac{2}{3} \mu\frac{\partial\overline{u}_{l}}{\partial x_{l}}\delta_{ij}, \tag{2}\]  
and \(\tau_{ij}\) is the subgrid-scale stress tensor, defined using the Boussinesq hypothesis in the form,  
\[\tau_{ij}-\frac{1}{3}\tau_{kl}\delta_{ij}=-2\mu_{t}\bigg{(}\overline{S}_{ij}- \frac{1}{3}\overline{S}_{kl}\delta_{ij}\bigg{)} \tag{3}\]  
Here, \(\mu_{t}\) is the subgrid-scale turbulent viscosity and \(\overline{S}_{ij}\) is the strain-rate tensor for the resolved scale in the form,  
\[\overline{S}_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial\overline{u}_{i}}{\partial \overline{x}_{j}}+\frac{\partial\overline{u}_{j}}{\partial\overline{x}_{i}} \Bigg{)} \tag{4}\]  
For the LES Smagorinsky model, the subgrid-scale turbulent viscosity is defined by  
\[\mu_{t}=\rho L_{s}^{2}|\overline{S}| \tag{5}\]  
where \(\big{|}\overline{S}\big{|}=\sqrt{2\overline{S}_{ij}\overline{S}_{ij}}\) and \(L_{s}\) is the mixing length for the subgrid-scale defined as  
\[L_{s}=\min(\kappa d,C_{s}\Delta) \tag{6}\]  
Here, \(\kappa\) is the von Karman constant, \(d\) is the distance to the nearest wall, \(C_{s}\) is the Smagorinsky constant, and \(\Delta\) is the local grid size defined as \(\Delta=V^{1/3}\). In Equation (1c), \(h_{s}\) and \(\lambda\) are the sensible enthalpy and thermal conductivity, respectively. The subgrid enthalpy flux term on the right side of Equation (1c) is approximated using the gradient hypothesis  
\[\Big{[}\rho\Big{(}\overline{u_{i}h_{s}}-\overline{u}_{i}\overline{h}_{s} \Big{)}\Big{]}=-\frac{\mu_{t}}{Pr_{t}}C_{p}\frac{\partial\overline{T}}{ \partial x_{j}} \tag{7}\]  
where \(C_{p}\) is the specific heat for the fluid and \(Pr_{t}\) is the subgrid Prandtl number. The numerical discretization methods are listed in Table 2. The LES is numerically realized using the commercial software ANSYS Fluent (version: 18.0) [18]. The validity of current numerical methods was confirmed in the previous study [19], where the predicted surface pressure spectrum, which is one of the most critical quantities in the current study, was shown in excellent agreement with measured ones.