# Numerical Coupling between a FEM Code and the FVM Code OpenFOAM Using the MED Library
## 4 Numerical Results
### Buoyant-Driven Cavity
#### 4.1.2 Simulations Results  
The numerical fields resulting from the computation have been nondimensionalized with the following:  
\[\mathbf{x}^{*}=\frac{\mathbf{x}}{L}\,,\quad\mathbf{u}^{*}=\frac{\mathbf{u}\,L}{\alpha}\,,\quad \Theta=\frac{T-T_{C}}{\Delta T}\,, \tag{2}\]  
where \(T_{C}\) represents the Dirichlet boundary condition for the temperature on the cold wall. Naturally, by considering these new variables, such as the nondimensional temperature \(\Theta\), the nonhomogeneous Dirichlet boundary conditions change their specific values: on the cold wall, we now have \(\Theta=0\), while on the hot one, we have \(\Theta=1\). Therefore, the numerical results are described by using only the nondimensionalized variables. In Figures 4-7, isolines of the contour of the velocity magnitude (\(|\mathbf{u}|\)), the nondimensional velocity components (\(u^{*},v^{*}\)), and the nondimensional temperature (\(\Theta\)) are reported for four \(Ra\) numbers (from \(10^{3}\) up to \(10^{6}\)) for the two coupling algorithms (\(c_{1}\) and \(c_{2}\)). Reference results of the physical field contours can be widely found in the literature for this problem. For this reason, the interested reader can refer to [27] and references therein. For every case of \(Ra\) number, the contour isolines are in agreement with the data found in the literature. In Table 1, a grid convergence analysis is reported for the maximum value of the \(v^{*}\) component evaluated at \(y^{*}=0.5\) for the case of \(Ra=10^{5}\). In particular, three types of grid size were investigated, corresponding, respectively, to 400, 1600, and 6400 elements (\(n_{el}\)), for each of the four simulation setups. The four simulations show the same convergence behavior, with a similar value for the finest grid solution. A comparison with [12] for the same simulations was reported since we used the same grid refinement. It can be noticed that our results are consistent with the ones already published. Figure 4: From **left to right**, the contour of velocity magnitude, nondimensional velocity components, and nondimensional temperature. Coupling algorithm \(c_{1}\) (**top**) and \(c_{2}\) (**bottom**) for the case with \(Ra=10^{3}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n_{el}\) & \(F\) & \(OF\) & \(c_{1}\) & \(c_{2}\) & [12] \\ \hline \(20\times 20\) & 73.639 & 65.186 & 66.789 & 71.908 & 73.241 \\ \(40\times 40\) & 73.615 & 72.470 & 73.140 & 73.244 & 73.189 \\ \(80\times 80\) & 73.617 & 73.337 & 73.515 & 73.681 & 73.168 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grid convergence of the \(v^{*}_{max}\) value at \(y^{*}=0.5\) for the case with \(Ra=10^{5}\), and comparison with the same data of [12], for three different level of discretization. Figure 5: From **left to right**, the contour of velocity magnitude, nondimensional velocity components, and nondimensional temperature. Coupling algorithm \(c_{1}\) (**top**) and \(c_{2}\) (**bottom**) for the case with \(Ra=10^{4}\). Figure 6: From **left to right**, the contour of velocity magnitude, nondimensional velocity components, and nondimensional temperature. Coupling algorithm \(c_{1}\) (**top**) and \(c_{2}\) (**bottom**) for the case with \(Ra=10^{5}\). In Tables 2 and 3, we report the maximum value of the nondimensional velocity components, \(u^{*}\) and \(v^{*}\), evaluated, respectively, at the planes \(x^{*}=0.5\) and \(v^{*}=0.5\) with different \(Ra\) numbers and compare them with the same data taken from the literature. In general, a good agreement can be noticed for the maximum value of the nondimensional velocity components. For the case of \(Ra=10^{6}\), however, a slight difference is present concerning other literature data, although the four simulations presented in this work exhibit values close to each other. In Figures 8 and 9, the nondimensional velocity components and the nondimensional temperature are reported for every type of the four simulations (\(F\), \(OF\), \(c_{1}\), \(c_{2}\)) and for every \(Ra\) number.