# Numerical Coupling between a FEM Code and the FVM Code OpenFOAM Using the MED Library
## 4 Numerical Results
### Conjugate Heat Transfer (CHT)  
In this section, we present the second application implemented to test the data transfer between an FVM code and an FEM code in the domain defined by Figure 10. This test aims to investigate the data transfer through a boundary that connects these two domains. In this context, we analyze a conjugate heat transfer problem that describes a thermal exchange between two regions of different materials. In particular, we consider the heat exchange through a physical boundary between a solid and a fluid region. This kind of application finds significant attention in several scientific and engineering applications, such as solar heating [28], heat exchange [29], and nuclear energy production [30]. The solid is modeled as a two-dimensional isotropic material with constant material properties. Two domains are considered where different equations are solved. The first region represents a solid domain in which only the temperature equation has been solved, while in the second region, the momentum and temperature equations are solved for a buoyant fluid, employing the same system of equations described in (1). The only parameter which describes the temperature distribution in the solid is the thermal diffusivity \(\alpha\), which is defined as  
\[\alpha=\frac{k}{\rho\,c_{p}}\,, \tag{3}\]  
Figure 9: 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 [PERSON] [27] (circular markers). Case with \(Ra=10^{5}\) on the left and \(Ra=10^{6}\) on the right. where \(k\) is the thermal conductivity and \(c_{p}\) is the thermal capacity of the solid domain. Therefore, the heat equation in the solid reads as  
\[\frac{\partial T}{\partial t}=\alpha\Delta T\,. \tag{4}\]  
We recall that in the following discussion, the physical field has been nondimensionalized; thus, the temperature field \(T\) is transformed into the corresponding \(\Theta\) by using (2). The peculiarity of this kind of physical setup is the mutual exchange of the boundary condition values at the interface between the two regions. At the interface, the fluid problem is defined by a nonhomogeneous Dirichlet boundary condition, while the solid problem is defined by a nonhomogeneous Neumann boundary condition. Specifically, the temperature field in the solid at the boundary is used as the boundary condition for the fluid region, while the heat flux computed at the same interface for the fluid region is the boundary condition for the temperature equation of the solid. This setup is commonly adopted for conjugate heat transfer simulations, as detailed in [31]. For the solid subdomain, the boundary conditions include two homogeneous Neumann boundary conditions that are imposed at the top and bottom boundaries, while a fixed temperature is imposed on one lateral side (the cold wall). The opposite wall is the exchange interface with the fluid subdomain, where a nonhomogeneous flux condition with \(\mathbf{q}_{w}\) computed from the temperature field near the wall in the fluid region is imposed at every time step. In the fluid subdomain, the boundary conditions for the velocity field are the same as the ones that we described for the cavity in the previous section. For the temperature field, the top and bottom walls are adiabatic, i.e., there is no flux at those walls. There is a Dirichlet condition on the other two boundaries. The nondimensional temperature \(\Theta\) is fixed at the hot wall with value \(1\), while the other wall is the exchange interface with the solid subdomain. Here, the temperature profile is dictated by the solution of the temperature equation in the solid region.