## 2 Gas-Solid-Coupling Model of Spalling Blocks-Train-Tunnel-Air
### Basic Theory of Gas-Solid Coupling  
(1) Turbulence model  
The modified out-of-body vortex turbulence model (DDES) that is based on the two equations of SST \(k\)-\(\omega\) was selected to simulate an unsteady, incompressible transient flow field based on the basic equations of fluid mechanics. This turbulence model has been widely used to simulate high-speed-train flow field structures [29; 30; 31; 32]. According to [PERSON] [33], the equations governing SST \(k\)-\(\omega\) are as follows:  
\[\frac{\partial\rho k}{\partial t}+u_{i}\frac{\partial\rho k}{\partial x_{i}}= \frac{\partial}{\partial x_{j}}\Bigg{[}\Gamma_{k}\frac{\partial k}{\partial x _{j}}\Bigg{]}+G_{k}-Y_{k} \tag{1}\]  
Figure 1: Impact on driving safety of spalling blocks comprising a tunnel lining. **(a)** Flaky block; **(b)** blocky block; **(c)** The falling block collided with the train. \[\frac{\partial\rho\omega}{\partial t}+u_{i}\frac{\partial\rho k\omega}{\partial x_{i} }=\frac{\partial}{\partial x_{j}}\Bigg{[}\Gamma_{\omega}\frac{\partial\omega}{ \partial x_{j}}\Bigg{]}+G_{\omega}-Y_{\omega}+D_{\omega} \tag{2}\]  
where \(k\) represents turbulent kinetic energy, \(w\) represents the turbulent kinetic energy dissipation rate, \(G_{\omega}\) represents turbulent kinetic energy caused by a velocity gradient, \(\Gamma_{k}\) and \(\Gamma_{w}\) represent the convection terms of \(k\) and \(\omega\), \(Y_{k}\) and \(Y_{w}\) represent the effective diffusion terms of \(k\) and \(\omega\) caused by turbulence, and \(D_{w}\) represents the cross-convection term. Turbulent shear stress was considered a part of turbulent viscosity. 2 Gas-solid-coupling interaction  
The pressure on the lining surface under train wind is equal to the sum of the wind pressure generated by the wind and the aerodynamic pressure (pressure wave) formed by the compressed air of the train. This pressure is given as follows:  
\[p_{j}=\frac{n}{1}(p_{vj}+p_{wj})=\frac{n}{1}(\frac{1}{2}\rho v_{j}{}^{2}+p_{wj}) \tag{3}\]  
where \(j\) represents the number of blocks on the lining surface, 1, 2, \(\cdots\)\(n\); \(p_{j}\), \(p_{vj}\), and \(p_{wj}\) represent total pressure, wind pressure, and aerodynamic pressure, respectively, on the surface of block \(j\), newtons per square meter; and \(v_{vj}\) represents the wind speed on the lining surface, meters per second. The external force acting on a spalling block's center of mass and the external moment around this point were obtained by integrating the compressive strength of the block as follows:  
\[\left\{ \begin{array}{c}F_{x}=\sum_{i}^{n}p_{j}\cos(\alpha_{j})s_{j}\\ F_{y}=\sum_{i}^{n}p_{j}\cos(\beta_{j})s_{j}-m\rho g\\ F_{z}=\sum_{i}^{n}p_{j}\cos(\varphi_{i})s_{j}\end{array} \right. \tag{4}\]  
\[\left\{ \begin{array}{c}M_{cx}=\sum_{i}^{n}p_{j}\cos(\alpha_{j})s_{j}r_{cxj} \\ M_{cy}=\sum_{i}^{n}p_{j}\cos(\beta_{j})s_{j}r_{cyj}\\ M_{cz}=\sum_{i}^{n}p_{j}\cos(\varphi_{i})s_{j}r_{czj}\end{array} \right. \tag{5}\]  
where s\({}_{j}\) represents the surface area of block \(j\), square meters; \(\alpha_{j}\), \(\beta_{j}\), and \(\varphi_{j}\) represent the angle between the normal vector outside block \(j\) and the x, y, and z axes, respectively; \(r_{cij}\) represents the component of the distance from the surface center of block \(j\) to its center of mass along direction \(i\), \(i\) = \(x\), \(y\), and \(z\), meters; g represents the acceleration due to gravity, meters per second squared; \(F_{i}\) represents the external force along direction \(i\), \(i\) = \(x\), \(y\), and \(z\)  
Figure 2: Flow chart of gas-solid-coupling calculation simulation. newtons; and \(M_{i}\) represents the closing moment of the lining block around the centroid axis in direction \(i\), \(i=x,y\), and \(z\), newton-meters. The acceleration of the center of mass and the angular acceleration of rotation around the center of mass axis acted on by the train wind pressure and aerodynamic pressure on the spalling block were obtained as follows:  
\[\left\{