Several practical applications in fluid mechanics have the interest to reduce energy dissipation by reducing the drag or pressure drop. An example of a solution is to turn the wetted surface into a super-hydrophobic surface (SHS). However, the whole surface coated may adversely affect some regions, depreciating the fluid flow. Furthermore, an SHS may not be cheap, being important to decide which regions to prioritize. Thus, in this work, the topology optimization method is applied to obtain an optimized design of SHS distribution. Derivation of the discrete adjoint problem applied to super-hydrophobic modeling is shown. The numerical implementation is done by using a computational fluid dynamics (CFD) code based on the finite volume method (FVM) as the primal and the adjoint solver, and the internal point optimization algorithm (IPOPT) as the optimizer. The SHS behavior is simplified by adopting the slip length model. Two test cases are presented: internal and external flows. For internal flow case, a stretched-S pipe is used, aiming to reduce the pressure drop; for external flow case, the foil NACA0015 is analyzed, aiming to reduce the drag. Both cases are assumed to be 2D, steady-state flow, using the properties of water, in a fully turbulent flow, and constrained by a maximum material distribution. Optimized topologies for several surface constraints (maximum amount of SH material used) are obtained, showing the regions to be prioritized based on the surface constraint limit. Additionally, the effects of slip length (the level of hydrophobicity), Reynolds number, and angle of attack are analyzed. The obtained results show that regions to be prioritized in order to reduce the dissipated energy are not always intuitive. Furthermore, depending on the operating condition, a fully SHS case may not be the best option.

流体力学中的一些实际应用具有通过减小阻力或压降来减少能量耗散的兴趣。解决方案的一个示例是将润湿的表面变成超疏水表面（SHS）。但是，整个涂层表面可能会对某些区域产生不利影响，从而降低流体流量。此外，SHS可能并不便宜，这对于确定优先区域至关重要。因此，在这项工作中，采用拓扑优化方法来获得SHS分布的优化设计。显示了应用于超疏水性建模的离散伴随问题的推导。通过使用基于有限体积法（FVM）作为原始和伴随解算器的计算流体力学（CFD）代码来完成数值实现，以及内部点优化算法（IPOPT）作为优化器。通过采用滑移长度模型可以简化SHS行为。给出了两个测试用例：内部和外部流。对于内部流动情况，使用拉伸S型管以减少压降。对于外部流动情况，分析了铝箔NACA0015，旨在减少阻力。两种情况都假设为二维稳态流，它利用水的特性在完全湍流中受最大材料分布的约束。获得了针对多个表面约束（使用的SH材料的最大量）的优化拓扑，显示了基于表面约束限制进行优先排序的区域。此外，分析了滑移长度（疏水性水平），雷诺数和攻角的影响。获得的结果表明，为了减少耗散能量而需要优先处理的区域并不总是直观的。此外，根据操作条件，完全SHS外壳可能不是最佳选择。