# 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
A comparison with literature data from [27], symbolized with circular markers, is also highlighted. Specifically, these plots refer to the variables' behavior at specific points in the domain: the line \(x^{*}=0.5\) for the \(u^{*}\) component and the line \(y^{*}=0.5\) for the \(v^{*}\) component and the temperature \(\Theta\). Regarding the latter variable, the plotted domain is restricted to \(x^{*}\in[0,0.2]\) (apart from \(Ra=10^{5}\)) since the literature data can be found only in this interval. Moreover, for \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(Ra\) & \(F\) & \(OF\) & \(c_{1}\) & \(c_{2}\) & [23] & [24] & [26] & [27] \\ \hline \(10^{3}\) & 3.66 & 3.59 & 3.64 & 3.70 & 3.63 & 3.68 & 3.65 & 3.49 \\ \(10^{4}\) & 16.24 & 16.22 & 16.19 & 16.33 & 16.18 & 16.10 & 16.18 & 16.12 \\ \(10^{5}\) & 35.70 & 35.71 & 35.75 & 35.80 & 34.81 & 34.00 & 34.77 & 33.39 \\ \(10^{6}\) & 80.79 & 81.03 & 83.16 & 78.47 & 65.33 & 65.40 & 64.69 & 65.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum value of \(u^{*}\) component at \(x^{*}=0.5\) for different \(Ra\) numbers and comparison with literature data. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(Ra\) & \(F\) & \(OF\) & \(c_{1}\) & \(c_{2}\) & [23] & [24] & [25] & [26] & [27] \\ \hline \(10^{3}\) & 3.69 & 3.60 & 3.68 & 3.73 & 3.68 & 3.73 & 3.69 & 3.70 & 3.69 \\ \(10^{4}\) & 19.80 & 19.76 & 19.72 & 19.88 & 19.51 & 19.90 & 19.63 & 19.62 & 19.76 \\ \(10^{5}\) & 73.62 & 73.34 & 73.52 & 73.68 & 68.22 & 70.00 & 68.85 & 68.69 & 70.63 \\ \(10^{6}\) & 234.80 & 234.66 & 227.41 & 229.06 & 216.75 & 228.00 & 221.60 & 220.83 & 227.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum value of \(v^{*}\) component at \(y^{*}=0.5\) for different \(Ra\) numbers and comparison with literature data. Figure 7: 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 \(Ra=10^{6}\). the same \(\Theta\), a good agreement with reference data published in [27] is present for every case and every type of simulation, including both coupled algorithms. For this reason, we do not provide a zoom on specific regions of the nondimensional temperature plot since the lines of the four simulations are almost overlapping. The same trend can also be seen for the \(v^{*}\) component, while some discrepancy can be noticed for the \(u^{*}\) component in the case of \(Ra=10^{6}\). On the other hand, each of our simulations seems to produce the same numerical solution, confirming the goodness of the simulations and coupling procedure. We provide a zoom of the plot in the region close to the maximum/minimum of the velocity components to better highlight the slight differences between the four simulations and the literature results. Figure 8: Nondimensional temperature \(\Theta\) (at \(y^{*}=0.5\), **top**) and nondimensional components \(u^{*}\) (at \(x^{*}=0.5\), **middle**) and \(v^{*}\) (at \(y^{*}=0.5\), **bottom**) for the four types of simulations (\(F\), \(OF\), \(c_{1}\), and \(c_{2}\)), with a comparison with literature data from Wan2001 [27] (circular markers). Case with \(Ra=10^{3}\) on the left and \(Ra=10^{4}\) on the right.