Density-based topology optimization for turbulent fluid flow using the standard k-epsilon RANS model with wall-functions imposed through an implicit wall penalty formulation

Abstract: Turbulent flows have high requirements for very fine meshes near the boundary to ensure accuracy. In the context of topology optimization (TO), such fine meshes become unrealistic and common approaches are hampered by low accuracy and overestimation of boundary layer thickness. Wall-functions are a natural way to ease the computational requirements, but they are not naturally imposed in density-based TO due to the diffuse design parametrization. We propose an implicit wall-function formulation for the Reynolds-Averaged Navier-Stokes (RANS), standard k-epsilon model that extracts wall-normal information directly from the gradient of the design variable and enables a penalty-based formulation for imposing wall-functions to the RANS equations, without the need for body-fitted meshes. The method provides a reliable route to high Reynolds number turbulent topology optimization, delivering boundary layer accuracy comparable to explicit-wall body-fitted analyses, while retaining the flexibility of density-based TO. Furthermore, because wall effects are modeled using wall-functions, accurate solutions are obtained on substantially coarser meshes, leading to significant reductions in computational cost. The approach is validated on three canonical benchmarks over Reynolds numbers up to Re = 2e5: a pipe-bend; a U-bend; and a Tesla-valve. Across all cases, the proposed method accurately recovers near-wall velocity profiles, closely matching verification simulations on body-fitted meshes with explicit wall-functions. In contrast, a conventional turbulent TO formulation, without the proposed wall-function treatment, mispredicts boundary-layer development and yields sub-optimal results.