Motivated by extensive discussion in the literature, by experimental evidence and by recent direct numerical simulations, we study flows over hydrophobic surfaces with shear-dependent slip lengths and we report their drag-reduction properties. The laminar channel-flow and pipe-flow solutions are derived and the effects of hydrophobicity are quantified by the decrease of the streamwise pressure gradient for constant mass flow rate and by the increase of the mass flow rate for constant streamwise pressure gradient. The nonlinear Lyapunov stability analysis, first applied to a two-dimensional channel flow by Balogh et al. [“Stability enhancement by boundary control in 2-D channel flow,” IEEE Trans. Autom. Control 46, 1696-1711 (2001)], is employed on the three-dimensional channel flow with walls featuring shear-dependent slip lengths. The feedback law extracted through the stability analysis is recognized for the first time to coincide with the slip-length model used to represent the hydrophobic surfaces, thereby providing a precise physical interpretation for the feedback law advanced by Balogh et al. The theoretical framework by Fukagata et al. [“A theoretical prediction of friction drag reduction in turbulent flow by superhydrophobic surfaces,” Phys. Fluids 18, 051703 (2006)] is employed to model the drag-reduction effect engendered by the shear-dependent slip-length surfaces and the theoretical drag-reduction values are in very good agreement with our direct numerical simulation data. The turbulent drag reduction is measured as a function of the hydrophobic-surface parameters and is found to be a function of the time- and space-averaged slip length, irrespective of the local and instantaneous slip behaviour at the wall. For slip parameters and flow conditions that could be realized in the laboratory, the maximum computed turbulent drag reduction is 50% and the drag reduction effect degrades when slip along the spanwise direction is considered. The power spent by the turbulent flow on the hydrophobic walls is computed for the first time and is found to be a non-negligible portion of the power saved through drag reduction, thereby recognizing the hydrophobic surfaces as a passive-absorbing drag-reduction method. The turbulent flow is further investigated through flow visualizations and statistics of the relevant quantities, such as vorticity and strain rates. When rescaled in drag-reduction viscous units, the streamwise vortices over the hydrophobic surface are strongly altered, while the low-speed streaks maintain their characteristic spanwise spacing. We finally show that the reduction of vortex stretching and enstrophy production is primarily caused by the eigenvectors of the strain rate tensor orienting perpendicularly to the vorticity vector.
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