Turbulent drag reduction by hydrophobic surfaces with shear-dependent slip length

Abstract

The stabilisation of a parabolic equilibrium profile in a three-dimensional (3D) turbulent channel flow for an incompressible fluid is addressed with the objective of achieving drag reduction. The formulation of this problem stems from Balogh’s work [1] where Lyapunov stability analysis was used to devise prototype feedback laws and prove global stability of the solutions. This treatment only considers the controller as a mathematical artefact, but it can actually be linked to physical control strategies modelling hydrophobic surfaces and porous media. In the former, only linear slip velocity boundary conditions (BC) were considered [8]. However, experiments [2] have suggested that the slip length may be shear-dependent. Motivated by these, the effect on drag reduction of a shear-dependent slip length surface is examined in the present study using Direct Numerical Simulations (DNS) at Re τ0 = u τ0 δ/ν ≃ 180. δ is the channel half height, u τ0 the wall-shear velocity for regular no-slip walls channel and ν the kinematic viscosity. The theoretical analysis in [5], is extended to this new model. The proposed formulation shows that the skin-friction coefficient can be reduced by tuning the parameters in the shear-dependent slip length model. The results, which verified by DNS simulations, show that by taking a slip length value based on a constant slip model [8] and combining it within a shear-dependent model, up to 50% drag reduction can be obtained. The effect of control is further assessed by formulating the Fukagata identity [4] with general boundaries; the weighted Reynolds shear-stress for each quadrant shows an enhanced reduction in the sweep/ejection events compared to the constant slip model.