## 2 Numerical Methods and Target Model  
### Large Eddy Simulation  
Compressible LES techniques were used to accurately predict external flow and acoustic disturbances around a vehicle running at a speed of 110 km/h. The governing equations of LES were obtained by applying a low-pass spatial filter to the Navier-Stokes equations, thus computing only large eddies directly. The three-dimensional compressible unsteady filtered Navier-Stokes equations with the Smagorinski-Lilly model to predict subgrid-scale turbulent viscosity can be written in the following form:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho\overline{u}_{i})}{\partial x _{i}}=0, \tag{1}\]  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}=\frac{\partial\sigma_ {ij}}{\partial x_{j}}-\frac{\partial\overline{p}}{\partial x_{j}}-\frac{ \partial\tau_{ij}}{\partial x_{j}}, \tag{2}\]  
\[\frac{\partial\Big{(}\rho\overline{h}_{s}\Big{)}}{\partial t}+\frac{\partial \Big{(}\rho\overline{u}_{i}\overline{h}_{s}\Big{)}}{\partial x_{j}}-\frac{ \partial\overline{p}}{\partial t}-\overline{u}_{j}\frac{\partial\overline{p} }{\partial x_{i}}-\frac{\partial}{\partial x_{i}}\bigg{(}\lambda\frac{ \partial\overline{T}}{\partial x_{i}}\bigg{)}=-\frac{\partial}{\partial x_{ j}}\Big{[}\rho\Big{(}\overline{u_{i}h_{s}}-\overline{u}_{i}\overline{h}_{s}\Big{)} \Big{]}, \tag{3}\]  
where \(\sigma_{ij}\) is the stress tensor due to molecular viscosity, which is defined as  
\[\sigma_{ij}=\left[\mu\left(\frac{\partial\overline{u}_{i}}{\partial x_{j}}+ \frac{\partial\overline{u}_{j}}{\partial x_{i}}\right)\right]-\frac{2}{3}\mu \frac{\partial\overline{u}_{l}}{\partial x_{l}}\delta_{ij}, \tag{4}\]  
and \(\tau_{ij}\) is the subgrid-scale stress tensor, defined using the Boussinesq hypothesis in the form  
\[\tau_{ij}=-2\mu_{t}\bigg{(}\overline{S}_{ij}-\frac{1}{3}\overline{S}_{kl} \delta_{ij}\bigg{)}, \tag{5}\]  
were \(\mu_{t}\) is the subgrid-scale turbulent viscosity, and \(\overline{S}_{ij}\) is the strain-rate tensor for the resolved scale, which represents the length scale of turbulence components less than the grid size, in the form  
\[\overline{S}_{ij}=\frac{1}{2}\bigg{(}\frac{\partial\overline{u}_{i}}{\partial \overline{x}_{j}}+\frac{\partial\overline{u}_{j}}{\partial\overline{x}_{i}} \bigg{)}. \tag{6}\]  
For the LES Smagorinsky model, the subgrid-scale turbulent viscosity is defined by  
\[\mu_{t}=\rho L_{s}^{2}|\overline{S}|, \tag{7}\]  
where \(|\overline{S}|=\sqrt{\overline{S}_{ij}\overline{S}_{ij}}\), and \(L_{s}\) is the mixing length for the subgrid-scale defined as  
\[L_{s}=\min(\kappa d,C_{s}\Delta), \tag{8}\]  
were, \(\kappa\) 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 (3), \(h_{s}\) and \(\lambda\) are the sensible enthalpy and thermal conductivity, respectively. The subgrid enthalpy flux term on the right side of Equation (3) is approximated using the gradient hypothesis in the form  
\[\left[\rho\Big{(}\overline{u_{i}h_{s}}-\overline{u}_{i}\overline{h}_{s}\Big{)} \right]=-\frac{\mu_{t}}{Pr_{t}}C_{p}\frac{\partial\overline{T}}{\partial x_{ j}}, \tag{9}\]  
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 1. In the normal direction of the wall, the following equation is imposed as the wall boundary for the LES:  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial\big{(}\rho \overline{u}_{i}\overline{u}_{j}\big{)}}{\partial x_{j}}=\frac{\partial\sigma_ {ij}}{\partial x_{j}}-\frac{\partial\overline{p}}{\partial x_{j}}-\frac{ \partial\tau_{ij}}{\partial x_{j}}, \tag{2}\]0  
where \(u_{w}\) is the fluid velocity in the normal direction to the wall.