Variational Nonlinear Optimization in Fluid Dynamics: The Case of a Channel Flow with Superhydrophobic Walls

Abstract

Variational optimization has been recently applied to nonlinear systems with many degrees of freedom such as shear flows undergoing transition to turbulence. This technique has unveiled powerful energy growth mechanisms able to produce typical coherent structures currently observed in transition and turbulence. However, it is still not clear the extent to which these nonlinear optimal energy growth mechanisms are robust with respect to external disturbances or wall imperfections. Within this framework, this work aims at investigating how nano-roughnesses such as those of superhydrophobic surfaces affect optimal energy growth mechanisms relying on nonlinearity. Nonlinear optimizations have been carried out in a channel flow with no-slip and slippery boundaries, mimicking the presence of superhydrophobic surfaces. For increasing slip length, the energy threshold for obtaining hairpin-like nonlinear optimal perturbations slightly rises, scaling approximately with Re−2.36 no matter the slip length. The corresponding energy gain increases with Re with a slope that reduces with the slip length, being almost halved for the largest slip and Reynolds number considered. This suggests a strong effect of boundary slip on the energy growth of these perturbations. While energy is considerably decreased, the shape of the optimal perturbation barely changes, indicating the robustness of optimal perturbations with respect to wall slip.

Flow stability and sequence of bifurcations in a cubic cavity driven by a constant shear stress

“…Kalaev focused on the temporal behaviour of the flow and roughly quantified the transition to turbulence in terms of a shear stress Reynolds number. The same configuration, albeit with an outer liquid flow driving a gas flow in a cavity, is also a viable approach to modelling geometry-induced hydrophobicity of surfaces (Ybert et al 2007;Cherubini, Picella & Robinet 2021), although it is often modelled using Navier's slip condition (see e.g. Lauga, Brenner & Stone 2007;Schönecker & Hardt 2013) to avoid a discretisation of the cavity.…”

Turbulent transition in a channel with superhydrophobic walls: anisotropic slip and shear misalignment effects

“…The validity of linear stability results on laminar–turbulent transition itself, which is an intrinsically nonlinear phenomenon, has been recently assessed by Picella, Robinet & Cherubini (2019, 2020), who have confirmed that SH surfaces strongly influence transition induced by wall-close disturbances, such as TS waves, even at subcritical Reynolds number, but have a weak effect on the subcritical growth of coherent structures lying farther from the wall, such as streaks and streamwise vortices. Cherubini, Picella & Robinet (2021) reported a strong effect of boundary slip on the transient growth of nonlinear optimal perturbations: in particular, while the maximal energy growth is considerably decreased, the shape of the optimal perturbation barely changes, indicating the robustness of optimal perturbations with respect to wall slip.…”