# Conjugate Heat Transfer Advancements and Applications in Aerospace Engine Technology
## 2 Numerical Method of Conjugate Heat Transfer
### Mesh Adaptability in Conjugate Heat Transfer  
Within the realm of partitioned techniques for conveying conjugate heat transfer in air-cooled turbines, a plethora of investigations [32; 33; 34; 35] have adopted a fully conformal meshing strategy, ensuring an exact correspondence between the fluid and solid nodes at the fluid-solid interface, as depicted in Figure 5. Nevertheless, in tackling the complexities inherent to large-scale engineering challenges, it is commonplace to encounter considerable discrepancies in the topology and resolution of grids across different media domains, as exemplified in Figure 6. One of the intrinsic challenges of conjugate heat transfer simulations lies in the necessity for a meticulous alignment of boundary points along the solid-fluid interface [36], which imposes stringent constraints on the preservation of boundary heat flux in engineering applications that necessitate such simulations. Consequently, the development of a data interpolation methodology that upholds conservation principles across non-conforming grids emerges as a pivotal consideration for the successful implementation of conjugate heat transfer simulations within extensive engineering frameworks.  
Figure 4: Scheme of the iterative unsteady CHT model (uFFTB) [29].  
Figure 5: Commonly used fluid–solid grids for conjugate heat transfer [32].  
The transfer of heat flux at the interface necessitates that conservation be guaranteed, and for the entire interface, the overall conservation is satisfied:  
\[\int_{S}F_{s}ds=\int_{S}F_{f}ds \tag{1}\]  
where \(S\) refers to the entire interface; \(F_{s}\) and \(F_{f}\) refer to the flux on the solid and fluid sides, respectively.  
Regarding fluid-fluid and fluid-solid interfaces, congruent grids facilitate robust flux transfer and uphold the principle of flux conservation. For mismatched grids, the patched grid algorithm of the conservation class is used to implement data interpolation [38]. [PERSON] et al. [39] pioneered the \"local supermesh\" approach, grounded in the concept of element intersection, and effectively integrated a sophisticated 3D element intersection algorithm [40] into an unstructured mesh framework. Subsequent research endeavors [41; 42] have further expanded on this technique, advancing both two-dimensional and three-dimensional hybrid grid cutting capabilities. These advancements have been successfully applied to the intricate calculations involving overlapping grids within the flow field, yielding promising outcomes.  
Building upon the concept of \"local supermesh\", this paper extends its application to conjugate heat transfer calculations. By utilizing \"local supermesh\", heat flux conservation is maintained across relatively flat interfaces. For coupling regions with significant curvature, the heat flux is decomposed along both the normal and tangential directions of the interface. The superposition of the heat flux vectors from multiple subdomains ensures the conservation of heat inflow and outflow. Illustrating this with the fluid-solid interface, Figure 7 depicts the discretized unpaired interface. Here, the continuous lines represent the fluid mesh, whereas the dashed lines denote the solid mesh. The conservation relationship of heat flux is shown as follows:  
\[F_{s}(s_{0})=\frac{\sum_{i=0}^{4}A_{i}F_{i}(f_{i})}{\sum_{n=0}^{4}A_{n}}=\sum _{i=0}^{4}c_{i}F_{f}(f_{i}),\ c_{i}=\frac{A_{i}}{\sum_{n=0}^{4}A_{n}} \tag{2}\]  
where \(f_{i}\) designates a fluid element; \(s_{0}\) denotes a solid element; and \(A_{i}\) symbolizes the area of intersection between the fluid element \(f_{i}\) and the solid element \(s_{0}\). The interpolation of the solid element's flux is derived from that of the adjacent fluid element, with \(c_{i}\) signifying the interpolation coefficient.  
Figure 6: Non-matching meshes in 2D [37].