## 2 Materials and Methods
### Numerical Studies in ANSYS Fluent  
CFD numerical calculations enabling the evaluation of flow parameters were developed based on the FVM method [34; 35]. Model discretization, i.e., the extraction of the flow volume from the 3D model and its division into smaller cells of various shapes, the size of which is defined during the preparation of the numerical model mesh, makes it possible to search for the values of the working medium parameters, such as the flow velocity, pressure or temperature, in any defined cross-section of the model without the need to search for a general solution. Due to the extensive mesh generation capabilities of the Ansys Fluent environment, a tetrahedral mesh with polyhedral domains was used to search for a solution to the described problem, which in effect enabled the creation of a finite volume mesh with a polyhedral shape, highly reflecting the volume of compressed air inside the valve chambers, as well as optimizing the computation time. Polyhedral meshes are widely used in numerical simulations, constituting an optimal solution in terms of the quality of the obtained results, with a significantly shorter time necessary to obtain a solution to a given problem. The flow parameters of the stream in numerical methods are determined by numerically solving partial differential equations describing the behaviour of the velocity and pressure fields of a gas or liquid based on the principles of the conservation of momentum and mass, called the Navier-Stokes equations [36]. \[\frac{\partial p}{\partial t}+\
abla(pv)=0 \tag{7}\]  
\[p\frac{\partial v}{\partial t}=-\
abla p+\rho g+\mu\
abla^{2}v \tag{8}\]  
In fluid mechanics, two basic types of flow can be distinguished, namely laminar and turbulent, whereby in the considered case, the focus was on turbulent flow and the \(k\)-\(\varepsilon\) model was used with an additionally defined value of turbulence intensification \(I\) = 2% and a length \(L\) = 7 \(\times\) 10\({}^{-5}\) m. Such selected parameters in the case of a pneumatic brake valve and solving the problem using the RANS model in the authors' earlier works enabled the attainment of highly accurate turbulent flow simulation in near-wall areas and local flow constrictions resulting from the valve geometry. The RANS model allows one to take into account the influence of local vortices caused by turbulent flow on the flow parameters, considering the turbulent viscosity coefficient \(\mu_{t}\) (Equation (9)) describing the actual dynamic viscosity and considering the turbulent mixing of gas particles during movement depending on the turbulence kinetic energy \(k\) and dispersion energy \(\varepsilon\). \[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{9}\]  
\[C_{\mu}=\left(\frac{3C_{k}}{2}\right)^{-3}\cong 0.1 \tag{10}\]  
where: \(\mu_{t}\)--eddy viscosity, \(\rho\)--gas density, \(v_{t}\)--turbulent viscosity coefficient, \(\varepsilon\)--rate of dispersion, \(k\)--kinetic energy of vortices, \(C_{\mu}\)--constant -0.1, \(C_{k}\) = 1.4-1.5--Kolmogorov constant. Finally, after taking into account the turbulent viscosity coefficient, the equations of the turbulence kinetic energy \(k\) and dispersion energy \(\varepsilon\) take the form presented in Equations (11) and (12).