# Numerical Coupling between a FEM Code and the FVM Code OpenFOAM Using the MED Library
## 4 Numerical Results
### Conjugate Heat Transfer (CHT)
#### 4.2.2 Simulations Results
As the Rayleigh number increases, the configuration of the solution moves from temperature stratification towards a recirculation cell. The streamlines contour \(\psi\), instead, tends to have a more circular shape for higher Rayleigh numbers. In both cases, the increase in the conductivity ratios (\(K\)) has the slight effect of enlarging the velocity zone of interest away from the middle of the domain. Considering the nondimensional temperature, we can notice a different behavior for the isolines with different conductivity ratios \(K\). When \(K<1\), the largest part of the temperature gradient is located in the solid region, while for values of \(K>1\), the same consideration can be drawn for the fluid region. Naturally, increasing the \(Ra\) number, we reobtain the classical temperature isolines of a buoyant cavity, where the isolines distribution still follows the previous discussion of the conductivity ratio \(K\). These considerations can be applied also in the case of a solid wall thickness equal to \(t_{2}\). Regarding the velocity stream function \(\Psi\), the major difference can be noticed between the simulations with a different \(Ra\) number: for \(Ra=10^{3}\), the order of magnitude is lower than 1, while for \(Ra=10^{5}\), we reach values up to 8. Furthermore, the conductivity ratio \(K\) influences this value, for which an increase in \(K\) produces an increase in the stream function magnitude. In Figure 14, the local Nusselt number on the interface is reported for the case of the solid wall thickness equal to \(t_{1}\). The Nusselt number is a global index that effectively summarizes the heat exchange between a solid and a fluid, and it is relevant in many engineering applications of interest in the nuclear field and, in general, when there are complex heat exchange configurations at play. This parameter was computed as the normal gradient of the nondimensional temperature on the interface and represents the total ratio between the convective and the conductive heat transfer over the boundary interface. In particular, we have  
\[Nu_{l}=\frac{\partial\Theta}{\partial n}\,. \tag{7}\]Figure 11: Simulations with solid thickness \(t_{1}\) and \(Ra=10^{3}\). From left to right, contour of nondimensional temperature \(\Theta\) (**top**) and velocity stream function \(\Psi\) (**bottom**) for \(K=0.1\), \(1\), \(10\). Figure 12: Simulations with solid thickness \(t_{1}\) and \(Ra=10^{5}\). From left to right, contour of nondimensional temperature \(\Theta\) (**top**) and velocity stream function \(\Psi\) (**bottom**) for \(K=0.1\), \(1\), \(10\). A comparison with the reference literature data is reported with the white circular markers (data from [32]), and the gray markers were obtained with the boundary data algorithm presented in this work. In general, the estimation of the Nusselt number is in good agreement with the literature data. In the case of \(Ra=10^{5}\) and \(K=10\), there is a slight overestimation for \(y^{*}\) between 0.4 and 0.8. This is mostly due to the mesh resolution near the wall since the discretization is uniform on the wall domain, while the computation of the normal derivative near the wall would require a larger set of points. A slight overestimation concerning the literature data is obtained for the case with \(Ra=10^{5}\) and \(K=10\). In Table 4, the average Nusselt number \(\overline{Nu_{l}}\) on the shared boundary between the two regions is reported, with a comparison of the same parameter presented in [32]. A good agreement with the literature data is achieved for every simulation. Figure 14: Local boundary Nusselt number for the case with solid thickness \(t1\): gray markers are the simulations with \(Ra=10^{3}\) on top (\(K=0.1\) on the (**left**) and \(K=10\) on the (**right**)) and \(Ra=10^{5}\) on the bottom (\(K=0.1\) on the left and \(K=10\) on the right). A comparison with data from [32] is reported (white circular markers). Figure 13: Simulations with solid thickness \(t_{2}\) and \(Ra=10^{5}\).