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.

翻译：湍流对边界附近网格的精细度要求极高以确保计算精度。在拓扑优化（TO）背景下，如此精细的网格变得不切实际，而常用方法受限于精度不足与边界层厚度的高估。壁面函数是降低计算需求的自然途径，但由于基于密度的拓扑优化采用弥散式设计参数化，壁面函数无法自然施加。本文针对雷诺平均纳维-斯托克斯（RANS）标准k-epsilon模型提出一种隐式壁面函数公式，该公式直接从设计变量的梯度提取壁面法向信息，并建立基于罚函数的壁面函数施加方式，无需依赖贴体网格。该方法为高雷诺数湍流拓扑优化提供了可靠路径，在保持基于密度拓扑优化灵活性的同时，实现了与显式壁面贴体分析相当的边界层精度。此外，由于壁面效应通过壁面函数建模，可在显著粗化的网格上获得精确解，从而大幅降低计算成本。该方法在雷诺数高达Re = 2e5的三个经典基准案例中得到验证：弯管、U型弯道和特斯拉阀。在所有案例中，所提方法均能准确还原近壁速度剖面，与采用显式壁面函数的贴体网格验证模拟高度吻合。相比之下，未采用所提壁面函数处理的传统湍流拓扑优化公式会错误预测边界层发展，并产生次优结果。