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We introduce a novel method for the implementation of shape optimization for non-parameterized shapes in fluid dynamics applications, where we propose to use the shape derivative to determine deformation fields with the help of the p− Laplacian for p > 2 . This approach is closely related to the computation of steepest descent directions of the shape functional in the W1,∞ − topology and refers to the recent publication Deckelnick et al. (A novel W1,∞ approach to shape optimisation with Lipschitz domains, 2021), where this idea is proposed. Our approach is demonstrated for shape optimization related to drag-minimal free floating bodies. The method is validated against existing approaches with respect to convergence of the optimization algorithm, the obtained shape, and regarding the quality of the computational grid after large deformations. Our numerical results strongly indicate that shape optimization related to the W1,∞-topology—though numerically more demanding—seems to be superior over the classical approaches invoking Hilbert space methods, concerning the convergence, the obtained shapes and the mesh quality after large deformations, in particular when the optimal shape features sharp corners.
The paper is devoted to the simulation of maritime two-phase flows of air and water. Emphasis is put on an extension of the classical Volume-of-Fluid (VoF) method by a diffusive contribution derived from a Cahn-Hilliard (CH) model and its benefits for simulating immiscible, incompressible two-phase flows. Such flows are predominantly simulated with implicit VoF schemes, which mostly employ heuristic downwind-biased approximations for the concentration transport to mimic a sharp interface. This strategy introduces a severe time step restriction and requires pseudo time-stepping of steady flows. Our overall goal is a sound description of the free-surface region that alleviates artificial time-step restrictions, supports an efficient and robust upwind-based approximation framework, and inherently includes surface tension effects when needed. The Cahn-Hilliard Navier-Stokes (CH-NS) system is verified for an analytical Couette-flow example and the bubble formation under the influence of surface tension forces. 2D validation examples are concerned with laminar standing waves reaching from gravity to capillary scale as well as a submerged hydrofoil flow. The final application refers to the 3D flow around an experimentally investigated container vessel at fixed floatation for Re = 1.4 × 10 7 and Fn = 0.26. Results are compared with data obtained from VoF approaches, supplemented by analytical solutions and measurements. The study indicates the superior efficiency, resharpening capability, and wider predictive realm of the CH-based extension for free-surface flows with a confined spatial range of interface Courant numbers.
