# Conjugate Heat Transfer Advancements and Applications in Aerospace Engine Technology
## 2 Numerical Method of Conjugate Heat Transfer  
### Coupled Technique for Conjugate Heat Transfer  
In the realm of coupled field analysis, there exist two distinct methodological paradigms: the monolithic approach [10; 11; 12; 13] and the partitioned approach [6; 14; 15]. The monolithic approach amalgamates the fluid and solid domains into a unified system of equations that are addressed concurrently, sharing a singular computational infrastructure. Conversely, partitioned methods decompose the problem into separate fluid and solid regions that are calculated independently, with the domains connected via an exchange of physical quantities at the coupled boundaries. Within the partitioned coupling framework, a variety of computational techniques can be selectively implemented, including finite difference, finite volume, boundary element, and finite element methods, affording greater computational versatility. The partitioned approach has garnered significant attention from a myriad of investigators [16; 17; 18]. It partitions the problem into two subdomains, each governed by a separate solver, simulating the fluid and solid regions separately. This strategy allows for the seamless integration of existing fluid flow or heat conduction solvers into a comprehensive conjugate heat transfer solver without imposing significant alterations upon the original computational program. In this partitioning strategy, the exchange of information between the fluid and solid subdomains becomes pivotal. The conventional serial staggered (CSS) method, often employed in the partitioned approach, serves as a typical example. Here, physical quantities are exchanged post-convergence of flow and heat transfer in each subdomain, a technique colloquially referred to as explicit coupling technology. As depicted in Figure 1, [PERSON] et al. [19] utilized the CSS algorithm in their aero-thermal coupling study. The CSS scheme is straightforward to implement, and there are no modifications at the solver level for flow and heat transfer, which each iterate independently. The schematics in the upper portion of Figure 2 depict the procedural flowcharts for two distinct methods: Flux Forward Temperature Back (FFTB) and Temperature Forward Flux Back (TFFB). Both methodologies commence with an estimated coupled wall temperature, which is adopted as the preliminary boundary condition. Although this strategy expedites the rate of iterative convergence, it introduces marked fluctuations in the fluid's physical parameters with each iteration, potentially compromising the coupled system's stability. Within the heat transfer coefficient Forward Temperature Back (hFTB) methodology, the solid side's revised boundary conditions precipitate a corresponding adjustment in the resultant wall temperature, which is then relayed to the fluid domain. Empirical evidence has indicated that the stability of this triad of methods hinges on the attainment of a consistent and unique solution at the onset of the computational loop. Moreover, it has been observed that incorporating the wall temperature as the initial boundary condition during the iterative process can undermine the coupling's stability, particularly as the frequency of fluid iterations diminishes. Conversely, implementing heat flux as the boundary condition enhances stability. In light of these findings, [PERSON] et al. [15] introduced the heat transfer coefficient Forward Flux Back (hFFB) method as an innovative alternative. This methodology imposes heat  
Figure 1: Conventional serial staggered (CSS) aerothermal coupling scheme [19]. Figure 2: Overview of different coupling methods [15]. flow boundary conditions on the fluid domain, leading to a demonstrably stable and more rapid convergence in comparison to its predecessors. In recent years, the integration of CHT into these unsteady simulations presents a significant challenge. This is attributable to the necessity for exceedingly small-time steps to accurately capture the gas flow dynamics, particularly in scenarios involving turbine blades.