Adjoint complement to the volume-of-fluid method for immiscible flows
Adjoint complement to the volume-of-fluid method for immiscible flows
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DOI：
10.1016/j.jcp.2021.110411
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发表时间：
2020-09
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影响因子：
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通讯作者：
Niklas Kühl;J. Kröger;Martin Siebenborn;M. Hinze;T. Rung
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文献类型：
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作者：
Niklas Kühl;J. Kröger;Martin Siebenborn;M. Hinze;T. Rung
本文关注的是不混溶两相流的流体体积 (VoF) 方法的伴随补充，例如空气和水，由于其计算效率而广泛应用于海洋工程。原始和相应的伴随 VoF 方法的特殊挑战是指具有不连续物理属性的尖锐界面处理。连续伴随两相系统（分部分积分）和相应的双压缩对流方案（分部分求和）都是针对两种著名的压缩对流方案，即高分辨率界面捕获方案（HRIC）和任意网格压缩界面捕获方案（CICSAM）推导的。双方案严格反映了原始标准化变量图 (NVD) 模板。注意力仅限于稳态应用。因此，原始过程和对偶过程都是在伪时间中执行的，并且对偶方法的后向积分是围绕（伪时间）收敛的原始场执行的。因此，伴随系统与原始系统经历相同的时间步长限制，独立于原始时间范围，形成鲁棒且先验稳定的伴随求解过程。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，这表明伴随部分没有提供唯一的、不平凡的解。作为补救措施，在伴随浓度方程中引入了附加的扩散浓度项。强加的启发式修改违反了对偶一致性，但强烈地规范了伴随系统的解决方案。该修改可以通过参考相分离扩散界面模型来证明，并且具有自由迁移率参数。从修改方法获得的数值结果以模型问题的解析解为基准。作为补充，针对一系列流动性参数，讨论了修改对实际感兴趣的弗劳德数和雷诺数下水下水翼周围二维流动的模拟所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，并强调自由移动参数的影响可以忽略不计，即使对于直接依赖于操纵（双）场量的目标函数也是如此。
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.