Density-based topology optimization for Navier-Stokes flow with free-slip boundary conditions

Implementing free-slip boundary conditions in density-based topology optimization presents several unique challenges. Firstly, the method lacks an explicit representation of solid-fluid interfaces, making the enforcement of boundary-specific conditions non-trivial. Secondly, defining the normal vectors required for applying free-slip conditions is not straightforward in a diffuse domain description. In this study, we propose an approach to incorporate free-slip boundary conditions into density-based topology optimization by enforcing zero tangential viscous stresses along the walls. This effectively eliminates wall shear stresses, diminishing the influence of viscous terms on pressure loss and enhancing the importance of inertial effects. The effectiveness of the proposed method is demonstrated through a series of benchmark problems, including short and long double-pipe configurations, where no connection forms between the pipes, and a bending pipe case, where curvature naturally emerges in the optimized design even for a low Reynolds number. Additionally, a Tesla valve is investigated, revealing a significantly different geometry under free-slip conditions compared to the traditional no-slip scenario. The results highlight the dual significance of the free-slip boundary conditions: not only are they physically relevant for applications such as microfluidics, biomedical, and super-hydrophobic surfaces, but they also yield designs that resemble those expected in inertia-dominated, high-Reynolds-number flows. These findings suggest that incorporating slip boundary conditions in topology optimization enables a broader and more realistic topology optimized design space.