Topology optimization method for conjugate natural convection with high solid-to-gas thermal conductivity ratios based on continuous adjoint method

Conjugate natural convection (CNC) is widely used in power generation energy storage, indoor heating, food processing, and electronic equipment cooling. Researchers optimize heat sink structures to meet CNC enhancement requirements under lightweight constraints. Topology optimization (TO) offers high design freedom and rapid structural optimization. Therefore, it has become an ideal method for the lightweight design of CNC heat sink. However, applying to CNC systems with high solid-to-gas thermal conductivity ratios (e.g., aluminum and air approaching 10⁴) poses significant numerical stability challenges. Meanwhile, the widely used discrete adjoint TO method suffers from excessive computational loads in three-dimensional problems. Meanwhile, reduced-order solving methods simplify governing equations but often sacrifice calculation accuracy. This study proposes a Continuous Adjoint-based Full order Topology Optimization for Natural Convection (CAFTO-NC) for three-dimensional CNC problems. The adjoint governing equations are derived for the CNC governing equations based on the finite volume method. This approach constructs a three-dimensional steady-state TO framework that balances computational efficiency, accuracy and robust convergence even under extreme thermal conductivity contrasts. We implemented the method on the open-source platform OpenFOAM. TO studies were conducted for heat dissipation structures with different base materials under two-dimensional and three-dimensional conditions to evaluate the method's capability handling high conductivity ratios. The results show that the thermal resistance of the three-dimensional topology structure is reduced by 64.31% compared to traditional aluminum straight fins of equal weight. This value decreases by 34.54% compared to a substrate with low thermal conductivity (2.57 W/m/K). CNC heat dissipation is constrained by both the conduction thermal resistance of the substrate and the convection thermal resistance of the surface. At low heat flux (100 W/m²), the optimization method identifies conduction thermal resistance as the dominant factor. Consequently, it prioritizes enhancing thermal conduction uniformity to reduce conduction thermal resistance. Conversely, at high heat flux (above 1000 W/m²), the method identifies CNC thermal resistance as dominant. The optimization direction thus shifts to increasing the heat dissipation area. These results demonstrate that the proposed method effectively improves the heat transfer performance of CNC heat sinks and handles extreme material property differences robustly. It exhibits significant potential for three-dimensional high-degree-of-freedom designs.