Three-dimensional instabilities for two circular lid-driven cavities are investigated by a linear stability analysis and direct numerical simulations using high order spectral techniques. Two circular geometries have been analysed and compared: a circular cavity with a horizontal top boundary and a circular cavity with a circular lid. Compared to more classic results for squared and rectangular lid driven cavities, the corners of these rounded geometries have been partially or totally removed. Critical Reynolds numbers, neutral curves, and three dimensional structures associated with the least stable modes have been identified by the linear stability analysis and then confirmed by spectral direct numerical simulations. We show that the geometries that present fewer sharp corners have enhanced stability: the circular cavity with a flat lid presents the first bifurcation at $\left(R{e}_c,k_c\right)\approx \left(1362,25\right)$ whilst the circular lid bifurcates at $\left(R{e}_c,k_c\right)\approx \left(1438,18\right)$ , where Re_c is the critical Reynolds number based on the cavity diameter and lid tangential velocity, and k_c is the spanwise wavenumber. Neutral curves and properties of the leading three-dimensional flow structures are documented, and analogies to instabilities in other lid-driven cavities are discussed. Additionally, we include results for the adjoint problem and the structural sensitivity 3D iso-maps (i.e., wavemaker regions), to show that the cavity corners play a relevant role in the generation of 3D instabilities.

In this work, the flow normal to a rectangular flat plate with sharp angles in the presence of the free surface and gravity has been studied numerically, using a three dimensional (3D) large Eddy simulation methodology. Most of the previous studies on this geometry consider either periodic assumptions or laminar regime Re ∼ O(10³). The numerical study described here, with a fully 3D simulation of the flat plate in turbulent regime Re ∼ O(10⁵), had not been reported earlier. Important differences have been found with respect to the laminar case or when periodic boundary conditions are assumed. The influence of the gravity force and the free surface interface on the hydrodynamics of the problem is extremely relevant for industrial applications. The simulations are initially validated using a reference case with a plate with sharp corners and a single phase laminar 3D-periodic configuration. The time averaged drag force, the velocity, and pressure fields are compared as well against the case of the plate with smooth corners. Finally, in the case of the two-phase 3D numerical simulations, the drag force and the vorticity fields in the near wake of the plate are compared to recent experimental work [S. Satheesh and F. J. Huera-Huarte, “Effect of the free surface on a flat plate translating normal to the flow,” Ocean Eng. 171, 458–468 (2019)]. These experiments were used as a guideline for the computational setup; consequently, the values of most of the dimensionless parameters are the same. A second scenario where the free surface is replaced by a solid wall is considered. Two important observations show up after this research: first, the characteristic unsteady frequencies that are found associated with two periodic regimes disappear when 3D cases are evaluated. In such a scenario, no dominant frequencies are found. Second, a critical submergence depth (distance) between the upper part of the plate and the free surface (or solid wall) has been found, where the drag force shows a maximum value. These observations are discussed in relation to the pressure distribution, the vortex structures formed at the wake, and the gap flow formed between the plate and the free surface.