## 2 Numerical Calculation Method  
### Governing Equation  
The research methods for the aerodynamic effect of train tunnels mainly include theoretical analysis, actual vehicle testing, model experiments, and numerical simulation. Considering the advantages of low cost and convenient simulation of different working conditions in numerical simulations, this paper adopts the method of numerical simulation to analyze the MPW characteristics generated by trains entering tunnels with different hoods. The numerical simulation of the aerodynamic effects of high-speed railway tunnels is based on the fluid equation governed by the laws of physical conservation, including the conservation of mass, momentum, and energy, which can be expressed as:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{u})=0 \tag{1}\]  
\[\frac{\partial(\rho\mathbf{u})}{\partial t}+\
abla\cdot(\rho\mathbf{u})=-\
abla p+ \
abla\cdot(\mathbf{\tau})+\rho g \tag{2}\]  
\[\frac{\partial(\rho E)}{\partial t}+\
abla\cdot(\mathbf{u}(\rho E+p))=\
abla\cdot [k_{\text{eff}}\
abla\mathbf{T}+(\mathbf{\tau}\cdot\mathbf{u})] \tag{3}\]  
where \(t\) is time; \(\rho\) represents fluid density; \(\mathbf{u}\) denotes fluid velocity vector; \(p\) represents fluid pressure; \(g\) denotes the gravity; \(\mathbf{\tau}\) is the viscous stress tensor; \(E\) represents the sum of kinetic energy and internal energy; and \(k_{\text{eff}}\) represents the effective thermal conductivity coefficient. For Newtonian fluid, the viscous stress is proportional to the deformation rate of the fluid, which can be described as:  
\[\mathbf{\tau}=\mu\left[\left(\
abla\mathbf{u}+\
abla\mathbf{u}^{T}\right)-\frac{2}{3} \
abla\cdot\mathbf{u}\mathbf{l}\right]=\mu\left[\left(\frac{\partial u_{i}}{\partial x _{\text{j}}}+\frac{\partial u_{j}}{\partial x_{\text{i}}}\right)-\frac{2}{3} \frac{\partial u_{k}}{\partial x_{\text{k}}}\sigma_{\text{ij}}\right] \tag{4}\]  
where \(\mu\) is the dynamic viscosity of air, and \(\mathbf{l}\) represents unit tensor. Air based on the _ideal gas_ model is adopted in this study, and the density for compressible flows can be obtained by \(\rho=(p_{0}+p)/RT\), where \(p_{0}\) is the operating pressure, and \(R\) denotes the specific gas constant. For air at near-standard conditions, \(R=287\) J/(kg\(\cdot\)K). The dynamic viscosity is derived by _[PERSON]'s law_ with three coefficients, which can be expressed as:  
\[\mu=\mu_{0}\left(\frac{T}{T_{0}}\right)^{3/2}\frac{T_{0}+S}{T+S} \tag{5}\]where \(\mu_{0}\) and \(T_{0}\) denote the reference viscosity and reference temperature, respectively; and \(S\) is the Sutherland constant. For air at moderate temperature and pressure, \(\mu_{0}=1.716\times 10^{-5}\) kg/(m\(\cdot\)s), \(T_{0}=273\) K, and \(S=111\) K. The effective thermal conductivity coefficient is calculated according to the turbulence model, and the \(RNG\)\(k\)-\(e\) turbulence model is used [33]. The model constants are given as \(C_{\mu}=0.0845\), \(C_{1\epsilon}=1.42\), and \(C_{2\epsilon}=1.68\), and the turbulent Prandtl number at the wall is fixed as \(Pr_{\text{t}}=0.85\). Since the source terms and species diffusion are not involved in this study, the values of \(S_{\text{m}}\), \(S_{\text{h}}\), and \(J_{\text{j}}\) are zero. Although the variables of the three governing equations are different, they all represent the conservation of the physical quantities in the fluid. If the general variable in the equation is represented by \(\phi\), the governing equation can be written in the following general form:  
\[\frac{\partial(\rho\phi)}{\partial t}+\
abla\cdot(\rho u\phi)=\
abla\cdot( \Gamma\
abla\phi)+S_{\phi} \tag{6}\]  
The left two terms represent the transient and convective terms, respectively, while the right two terms represent the diffusion and source terms, respectively. For a specific equation, \(\phi\), \(\Gamma\), and \(S_{\phi}\) have a specific form.