# Numerical Simulation of Intrachamber Processes in the Power Plant
## 2 Materials and Methods
### Numerical Schemes and Algorithms  
To approximate the equations of conservation (6-22) in space, the method of control volumes [1] was used. The partitioning of the calculated area in space was performed using prismatic elements (hexagons) with a total number of 1,480,000 elements [30]. Basic equations were discretized using a monotonic central difference scheme with the application of the PISO pressure correction algorithm and the Van-Lyre minmod limiter. As for orthogonal sections of the grid, the normal velocity gradients on the cell surface, required for the diffusion terms in calculating by Gauss theorem, are obtained from the velocity values at the centers of neighboring cells using a second-order scheme. An iterative procedure for error correction is used. An implicit Euler method is applied to discretize the time derivative. To solve the system of linear algebraic equations obtained during the discretization of equations, the iterative method of conjugate gradients with preconditioning for symmetric matrices is used, while for asymmetric matrices, it is the method of biconjugate gradients with preconditioning. For symmetric matrices, a procedure based on the simplified partial factorization scheme by [PERSON] was chosen as a preconditioner, while for asymmetric matrices, it was the preconditioner based on the simplified partial LU factorization scheme. Modeling was performed by means of computer simulation of gas dynamics and heat exchange processes in the open integrated platform for numerical simulation of continuum mechanics problems (openFoam) [31; 32; 33]. The visualization of the computation results was performed using the free software ParaView.  
To verify the proposed numerical schemes and algorithms, the problem of gas flow in the flow paths of the four-nozzle model power plant [12; 34; 35; 36; 37; 38; 39; 40; 41; 42] with a radial slot channel of the mass supply was solved. The calculated area is shown in Figure 1; the temperature, pressure, and gas velocity were assigned on the mass supply surfaces, non-reflective boundary conditions were set at the nozzle outlet, and adhesion and impermeability conditions were set on solid surfaces. The numerical experiment is described in detail in [20].  
As a result of the calculated data analysis, pressure distribution over the nozzle bottom and the end surface is obtained (Figure 2). The arrangement of the experimental pressure sensors at the end face of the charge and the cover is schematically shown in the upper part of the figure, with arrows indicating which curve belongs to which experimental section.  
A comparison of the calculated and experimental distributions of the relative pressure over the surface of the nozzle bottom and the end face of the slot charge (Figure 2) showed good qualitative and quantitative agreement between experimental and calculated data (except for the first and last points (relative error of 1.9% and 0.95%, respectively), the maximum relative error of the calculated data does not exceed 0.87%).  
\begin{table}
\begin{tabular}{c c} \hline
**Boundary** & **Condition** \\ \hline Mass supply surface Inlet & \(T=2500K\), \(G=15\) m/s, \(\frac{du}{dT}=f(p)\). \\ \hline Nozzle outlet & \(\frac{dp}{\partial\mathbf{m}}=0\). \\ \hline Nozzle bottom and end surface (walls) & \(U=V=W=0\), \(T_{wall}\) (\(x,y,z\), \(t\)) \(\mid_{wall}=T_{gas}(x,y,z,t)\mid_{gas}\), \(\lambda_{wall}\Big{(}\frac{\partial T_{wall}}{\partial\mathbf{m}}\Big{)}_{S _{wall}}=\lambda_{gas}\Big{(}\frac{\partial T_{gas}}{\partial\mathbf{m}}\Big{)} _{S_{wall}}\) \\ \hline \end{tabular}
\end{table}
Table 1: Boundary conditions.