## 2 Materials and Methods
### Turbulence Model  
The stream of the air and methane mixture flowing through the longwall and longwall headings, as well as through the initial section of the goaves with caving, is of a turbulent nature. The analyses related to the flow of gas mixtures, including air, use the Reynolds number as the criterion specifying the type of flow, whose critical value makes it possible to determine the critical state of the flow separating the area of static laminar flow from the turbulent flow. Therefore, by taking a time average of the Navier-Stokes equations, the Reynolds-averaged Navier-Stokes (RANS) equations have the following form [46]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{i})}{\partial x_{i}}=0 \tag{12}\]\[\frac{\partial(\rho v_{i})}{\partial t}+\frac{\partial\Big{(}\rho v_{i}v_{j}\Big{)} }{\partial x_{j}}=-\frac{\partial}{\partial x_{i}}+\frac{\partial}{\partial x_{ j}}\bigg{[}\mu\Big{(}\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{ \partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial v_{l}}{\partial u_{l}} \Big{)}\bigg{]}+\frac{\partial}{\partial x_{j}}\bigg{(}-\overline{\rho v_{i}^ {\prime}v_{j}^{\prime}}\bigg{)} \tag{13}\]  
As can be seen from Equation (11), a new variable, the Reynolds stress \(\overline{\rho v_{i}^{\prime}v_{j}^{\prime}}\) is introduced to the equations and it must be solved to achieve the closure of the equations. Two approaches are adopted to calculate the Reynolds stress, i.e., the Reynolds stress models (RSM) and the Boussinesq hypothesis. The Reynolds stresses are related to the mean velocity gradients in the Boussinesq hypothesis [46]:  
\[-\ \overline{\rho v_{i}^{\prime}v_{j}^{\prime}}=\ \mu_{l}\left(\frac{ \partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right) -\frac{2}{3}\bigg{(}\rho k+\mu_{l}\frac{\partial v}{\partial x_{k}}\bigg{)} \delta_{ij} \tag{14}\]  
In the turbulence model \(k-\varepsilon\), in the standard variation, the basic Navier-Stokes equation has been transformed into the Reynolds averaged equation. This equation includes an additional term in the form of the Reynolds stress tensor. Due to this term, the set of equations is not closed. To close the set of equations, it is necessary to introduce additional differential equations, which include the equation of kinetic turbulent energy and the equation of kinetic turbulent energy dissipation in the following form [46]:  
\[\rho\frac{\partial k}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho kv_{i })=\ \frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}}{\sigma_{k}})\frac{ \partial k}{\partial x_{j}})]+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k} \tag{15}\]  
\[\rho\frac{\partial\varepsilon}{\partial t}+\frac{\partial}{\partial xi}(\rho \varepsilon v_{i})=\ \frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}}{\sigma_{ \varepsilon}})\frac{\partial\varepsilon}{\partial x_{j}})]+C_{1\varepsilon} \frac{\varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b})-C_{2\varepsilon\rho}\frac{ \varepsilon^{2}}{k}+S_{\varepsilon} \tag{16}\]  
where: \(C_{1\varepsilon}\), \(C_{2\varepsilon\rho}\), \(C_{3\varepsilon}\) are constans, \(\sigma_{k}\), \(\sigma_{\varepsilon}\) are turbulent Prandtl numbers for \(k\) and \(\varepsilon\), \(G_{b}\) is the generation of turbulence kinetic energy due to buoyancy, \(G_{k}\) is the generation of turbulence kinetic energy due to the mean velocity gradients, \(Y_{M}\) is contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, \(S_{k}\), \(S_{\varepsilon}\) are user-defined source terms. In the Ansys Fluent software, the porous medium (the goaves with caving) is represented as a fluid characterised by two additional parameters. These parameters include porosity and the permeability coefficient. In order for the tests to take into account the porous medium through which the flow occurs, it is required to consider the source term \(S_{i}\) in the equation of momentum preservation.