Sensitivity analysis of SST k-ω model for fluid-thermal coupled topology optimization

This study presents a sensitivity analysis of the SST k-ω model for fluid-thermal coupled topology optimization (TopO) problems. The SST k-ω model inherently contains discontinuous functions, such as min and max operators, which preclude its direct differentiation and thereby hinder its incorporation into the TopO framework. To overcome this difficulty, a Heaviside-type step function H(x) is introduced to distinguish the active branches of the piecewise-defined functions. The non-differentiable components of the SST k-ω model are then analysed on a case-by-case basis, and the equation-solving sequence is carefully arranged to enable the formulation of a continuous adjoint-based sensitivity analysis. To assess the effectiveness of the derived sensitivities, the obtained sensitivity fields are employed to design a cooling-channel structure. Guided by the sensitivities, multi-branch structures with smooth fluid-solid interfaces and continuous flow channels are obtained. Compared with the “frozen turbulence” assumption, incorporating the full sensitivity of the turbulence model leads to improved optimization performance. In addition, the mechanisms by which turbulence affects the sensitivity analysis are systematically analysed, thereby clarifying the physical interpretation of the resulting sensitivities.