Viscous flow past a cylinder close to a free surface: Benchmarks with steady, periodic and metastable responses, solved by meshfree and mesh-based schemes

The question of whether it is possible to set relevant, robust and reliable benchmarks for viscous free- surface flows with complex free-surface dynamics is investigated in this work. The proposed method for finding an answer to this question consists of selecting three conditions leading to increasing flow com- plexity and to simulate them using three well established solvers based on diverse numerical techniques. In the three conditions, a submerged horizontal cylinder in an uniform current perpendicular to its axis is considered, the Reynolds number is fixed to 180, and the analysis is limited to a 2D framework. While the unbounded solution for such flow is well established, adding a free surface and setting the submergence ratio and the Froude number in certain ranges, challenging free-surface dynamics takes place. In the spe- cific conditions selected, phenomena of increasing complexity are identified and studied with: (i) ? + -SPH, an enhanced version of the Smoothed Particle Hydrodynamics method, (ii) a single-phase Finite Volume scheme with a Level Set function for tracking the free-surface (LS-FVM), and (iii) a two-phase Finite Vol- ume with a Volume-of-Fluid algorithm to treat the gas/liquid interface (VOF-FVM). It is shown that the test-cases, even being geometrically simple, present intricate complexities, such as alternate metastable states in the wake, linked to the strong non-linearities induced by the interactions between the wake's vorticity and the free surface. It is also shown that the solvers considered are able to depict a consistent representation of these complex flows, useful as benchmarks for other solvers and methods. An addi- tional research question, investigating whether the improvements of the ? + variant of the SPH method are necessary for simulating specific aspects of the flows treated in the paper, is also posed and discussed.