A model for slip and drag in turbulent flows over superhydrophobic surfaces with surfactant

Superhydrophobic surfaces (SHSs) can reduce the friction drag in turbulent flows. In the laminar regime, it has been shown that trace amounts of surfactant can negate this drag reduction, at times rendering these surfaces no better than solid walls (Peaudecerf et al., Proc. Natl. Acad. Sci. USA 114(28), 7254-9, 2017). However, surfactant effects on the drag-reducing properties of SHSs have not yet been studied under turbulent flow conditions, where predicting the effects of surfactant in direct numerical simulations remains expensive by today's standards. We present a model for turbulent flow inclusive of surfactant, in either a channel or boundary-layer configuration, over long but finite-length streamwise ridges that are periodic in the spanwise direction, with period $P$ and gas fraction $\varphi$. We adopt a technique based on a shifted log law to acquire an expression for the drag reduction. The average streamwise and spanwise slip lengths are derived by introducing a local laminar model within the viscous sublayer, whereby the effect of surfactant is modelled by modifying the average streamwise and spanwise slip lengths. Our model agrees with available laboratory experimental data from the literature when conditions are clean (surfactant-free), or when there are low surfactant levels. However, we find an appreciable drag increase for larger background surfactant concentrations that are characteristic of turbulent flows over SHSs for marine applications.