The paper is concerned with an adjoint complement to the Volume-of-Fluid (VoF) method for immiscible two-phase flows, e.g. air and water, which is widely used in marine engineering due to its computational efficiency. The particular challenge of the primal and the corresponding adjoint VoF-approach refers to the sharp interface treatment featuring discontinuous physical properties. Both the continuous adjoint two-phase system (integration-by-parts) and the corresponding dual compressive convection schemes (summation-by-parts) are derived for two prominent compressive convection schemes, namely the High Resolution Interface Capturing Scheme (HRIC) and Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM). The dual scheme rigorously mirrors the primal Normalized-Variable-Diagram (NVD) stencils. Attention is restricted to steady state applications. Thus both the primal and the dual procedures are performed in pseudo-time and the backward integration of the dual approach is performed around the (pseudo-temporal) converged primal field. Therefore, the adjoint system experiences the same time step size restrictions as the primal system, is independent of the primal time horizon and forms a robust as well as an a priori stable adjoint solution process.

The paper analyses the primal and adjoint equations for an engineering model problem. An analytical solution to the model problem is initially presented, which displays that the adjoint part does not offer a unique, non-trivial solution. As a remedy, an additional diffusive concentration term is introduced to the adjoint concentration equation. The imposed heuristic modification violates the dual consistency but strongly regularizes the solution of the adjoint system. The modification can be justified by reference to phase-separating diffuse-interface models and inheres a free mobility-parameter. Numerical results obtained from the modified approach are benchmarked against the analytical solution for the model problem. Supplementary, the influence of the modification on the sensitivities obtained from simulations for the two-dimensional flow around a submerged hydrofoil at Froude and Reynolds numbers of practical interest are discussed for a range of mobility-parameters. The final application refers to a shape-optimization of a generic 3D underwater vehicle and underlines a negligible influence of the free mobility parameter, even for an objective functional that directly depends on the manipulated (dual) field quantity.

本文涉及对不相溶的两相流（例如空气和水）的流体体积（VoF）方法的辅助补充，该方法由于其计算效率而广泛用于海洋工程。原始的和相应的伴随VoF方法的特殊挑战是指具有不连续物理特性的尖锐界面处理。对于两个突出的压缩对流方案（即高分辨率接口捕获方案（HRIC）和压缩式），导出了连续伴随的两相系统（逐部分积分）和相应的双重压缩对流方案（逐部分求和）。任意网格的接口捕获方案（CICSAM）。对偶方案严格反映了原始规范化可变图（NVD）模具。注意仅限于稳态应用。因此，原始过程和对偶过程都在伪时间内执行，并且对偶方法的后向合并是在（伪时态）会聚原始域上执行的。因此，伴随系统经历与原始系统相同的时间步长限制，独立于原始时间范围并形成健壮以及先验稳定的伴随解决方案过程。

本文分析了工程模型问题的原始方程和伴随方程。最初提出了模型问题的解析解决方案，该解决方案表明，伴随部分没有提供唯一的，非平凡的解决方案。作为补救措施，将附加的扩散浓度项引入到伴随浓度方程式中。施加的启发式修改违反了双重一致性，但强烈规范了伴随系统的解决方案。可以通过参考相分离的扩散界面模型来证明这种修改是正确的，并且这里包含了自由迁移率参数。从改进方法获得的数值结果以模型问题的解析解为基准。补充，讨论了在一定的迁移率参数范围内，修改对从实际感兴趣的Froude和Reynolds数围绕淹没水翼的二维流动模拟获得的灵敏度的影响。最终应用涉及通用3D水下航行器的形状优化，并强调了自由移动性参数的可忽略不计的影响，即使对于直接依赖于操纵（双重）场量的目标功能也是如此。