# 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  
A schematic representation of the physical configuration is reported in Figure 10, where the mutual exchange of the boundary conditions at the interface is depicted. Note that the fluid region is described by a squared cavity of dimension \(l\times l\). In this work, two geometric configurations are considered to take into account the solid region with thicknesses of \(t_{1}=0.2l\) and \(t_{2}=0.8l\), where \(l\) is the length side of the cavity. The physical and geometrical configuration were implemented following the work of [PERSON] et al. [32], with the idea of reproducing numerical results described in the literature. Several physical and geometrical configurations were analyzed in [32], changing the \(Pr\) number, the \(Ra\) number, the conductivity ratio \(K\), the solid wall thickness \(t\), and its geometrical position (hot side or cold side). Regarding \(K\), this parameter is defined as the ratio between solid and fluid thermal conductivity as  
\[[H]K=\frac{k_{s}}{k_{f}}\,, \tag{5}\]  
where the subscripts \(s\) and \(f\) refer to solid and fluid regions, respectively. We present the numerical simulation of these tests considering only a few configurations. In particular, the \(Pr\) number was considered fixed and equal to 0.015, while two \(Ra\) numbers were considered (\(10^{3}\) and \(10^{5}\)). Three values of \(K\) were investigated, equal to \(K_{1}=0.1\), \(K_{2}=1\), and \(K_{3}=10\). Concerning the solid wall position, only one configuration was taken into account, where the solid represents the cold side of the physical domain. Regarding the initial condition for the temperature field, it is worth noting that two different thermal conductivities are present in the whole simulated region (fluid + solid), for which we have that \(\Theta\in[0,1]\). Therefore, the linear behavior of the temperature distribution between cold and hot walls, which is the initial condition, has to take into account two different \(k_{i}\) with two different region widths. In our simulation, we fix the initial temperature as  
\[\Theta(x)= \begin{cases}\dfrac{x}{s_{1}+s_{2}K}&x\in\ solid\\ \dfrac{x}{s_{1}/K+s_{2}}+\Theta_{b}&x\in\ fluid\,,\end{cases}  \tag{6}\]  
where \(\Theta_{b}\) represents the initial temperature on the interface and is equal to \(s_{1}/(s_{1}+s_{2}K)\), with \(s_{1}\) and \(s_{2}\) as the widths of the two regions. Naturally, (6) arises from the well-known solution of the temperature distribution inside a wall with two different regions with no convective motion. Considering the boundary conditions on \(\Theta\), which reads \(\Theta=0\) on the cold wall (solid) and \(\Theta=1\) on the hot wall (fluid), this initial condition ensures that the temperature flux is always with the right sign, i.e., the thermal flux passes through the common boundary from the fluid to the solid region. The convergence criteria that are adopted are based on the residual of the linear system generated in the discretization of the problem. In particular, in OpenFOAM, all variables are solved to a precision of \(1\times 10^{-6}\), meaning that the relative residual of each equation reaches that value at each time step. Each iteration also includes a single nonorthogonal correction step. Given the uniform discretization of the grid, no further correction steps are required. The pressure equation is solved up to \(1\times 10^{-8}\) since its solution influences the mass conservation. In FEMuS, the convergence criteria are also based on the linear algebra residual and kept at the same value as the ones in OpenFOAM when running in split or monolithic configuration. In Figures 11-13, the contour of the nondimensional temperature \(\Theta\) and the velocity stream function \(\Psi\) are reported for the simulated cases. Concerning the temperature contour \(\theta\), we can observe a squeeze of the lines toward the solid region (gray-colored domain) with the decrease in the conductivity ratio \(K\). Regarding the changing \(Ra\) numbers, its increase results in a stretching of the temperature contour.