Model for turbulent drag reduction of superhydrophobic surfaces in large Reynolds number flows*

Superhydrophobic surfaces (SHS) can reduce drag for flows over surfaces owing to a gas layer trapped in the SHS texture. Drag reduction (DR) by SHS could significantly reduce energy consumption and carbon emissions in industrial and naval applications such as maritime shipping, which accounts globally for 2% CO2 and 13% NOx and SOx emissions. Current models predict that DR increases relatively rapidly with friction Reynolds number (Re_τ) with DR -> 100% as Re_τ -> ∞. This has been questioned in light of recent numerical and experimental data showing a departure from this trend, with possible saturation in DR, and a regime transition at large Re_τ when the SHS texture size is of the order of P⁺ ~ 10 (in wall units). For P⁺ >> 10, we model the flow over parallel SHS ridges in a plane channel of height  using a turbulent heterogeneous layer near the SHS, together with a turbulent homogeneous layer with a shifted log law in the bulk. Using conservation of mass, momentum and continuity of velocity, we predict DR as a function of all input parameters. In the limit parallel SHS ridges in a plane channel of height 2H using a turbulent heterogeneous layer near the SHS and a turbulent homogeneous layer as a shifted log law in the bulk. Using conservation of mass, momentum and continuity of velocity, we predict DR as a function of all input parameters. In the limit P/H<<1, which corresponds to most applications, our model predicts a DR that asymptotically approaches the gas fraction, with a logarithmic dependence on P/H and Re_τ.  Our prediction agrees with all available numerical data, where the DR shows a logarithmic increase towards the gas fraction as P⁺, Re_τ->∞.  Our model provides the physical mechanisms for this distinct regime of SHS DR at large Reynolds numbers, and testable predictions for the design and optimisation of SHS.