Optimal stabilization of a flow past a partially hydrophobic circular cylinder

Hydrophobic surfaces, enabling flow slip past a solid boundary, can be effective for passive flow control applications. In the present study, a computational investigation of flow past a circular cylinder with slip conditions is performed, at low values of Reynolds number, Re. Slip is modeled based on the Navier model. When slip conditions are applied on the entire cylinder surface, the present results demonstrate the stabilizing effect of increasing the non-dimensional slip length, b∗ = b/D, b being the slip length and D the cylinder diameter, in agreement with recent studies. In particular, the Kármán vortex street is supressed at a critical value of b∗, which is an increasing function of Re. Further, it is shown that, for the same levels of b∗, the wake can be stabilized by implementing slip conditions only on a part of the cylinder surface. Guided by this observation, the problem of fully or partially suppressing the Kármán vortex street by means of a partially hydrophobic cylinder is addressed by formulating a multi-objective optimization problem, in which the product of slip length and hydrophobic area quantifies the control effort; a second objective function, characterizing flow unsteadiness, is thereby introduced. The optimization results demonstrate that, both for full and partial suppression of the Kármán vortex street, a proper use of partial hydrophobicity can lead to a significant reduction in passive control effort, in comparison to the case of the fully hydrophobic cylinder. Computed optimal solutions of the Pareto front are characterized by means of local stability calculations based on an Orr-Sommerfeld solver. It is shown that flow stabilization is attained when a global intensity of absolute instability, involving local absolute growth rates and the streamwise extent of absolute instability, is sufficiently reduced.