Modelling turbulent drag reduction for superhydrophobic surfaces with surfactant

Superhydrophobic surfaces (SHS) can reduce the friction experienced at a boundary in turbulent fluid flows. In the laminar regime, it has been demonstrated that naturally-occurring surfactants can negate drag reduction, at times even rendering SHS no better than solid walls. However, extending these findings to turbulent flow remains challenging, as the full numerical solution to the equations which govern the fluid and surfactant are expensive. To address this challenge, we present a theory for both internal or external turbulent flows, over a periodic array of longitudinal or transverse ridges, in the presence of a small concentration of soluble surfactant. To deduce an expression for the turbulent drag reduction, we adopt a technique based upon the shifted log-law, whereas we obtain the slip lengths from local solutions due to laminar theory in the presence of surfactant. We are thereby able to examine how the slip and drag depend on the parameters that characterise surfactant transport and the SHS geometry. This allows one to predict the optimal configurations for turbulent drag reduction in the presence of surfactant. Finally, we summarise several key numerical and experimental works within the literature, which we use to evaluate the effectiveness of our model.