Gas diffusion in and out of super-hydrophobic surface in transitional and turbulent boundary layers.*

The rate of gas diffusion in and out of a super-hydrophobic surface (SHS) located in boundary layers is investigated at varying Reynolds numbers and ambient pressures. The hierarchical SHS consists of nano-textured, $\approx $100 $\mu $m wide spanwise grooves. The boundary layers over the SHS under the Cassie-Baxter and Wenzel states as well as a smooth wall at same conditions are characterized by particle image velocimetry. The Reynolds number based on momentum thickness of the smooth wall, \textit{Re}$_{\Theta 0}$, ranges from 518 to 2088, covering transitional and turbulent boundary layer regimes. The mass diffusion rate is estimated by using microscopy to measure the time-evolution of plastron shape and volume. The data is used for calculating the Sherwood number based on smooth wall momentum thickness, \textit{Sh}$_{\Theta 0}$. As expected, the diffusion rate increases linearly with the under- or super-saturation level, i.e., \textit{Sh}$_{\Theta 0}$ is independent of ambient pressure. For the turbulent boundary layers, the data collapses onto \textit{Sh}$_{\Theta 0}=$0.47\textit{Re}$_{\Theta 0}^{\mathrm{0.77}}$. For the transitional boundary layer, \textit{Sh}$_{\Theta 0}$ is lower than the turbulent power law. When \textit{Sh}$_{\Theta 0}$ is plotted against the friction Reynolds number (\textit{Re}$_{\tau 0})$, both the transitional and turbulent boundary layer data collapse onto a single power law, \textit{Sh}$_{\Theta 0}=$0.34\textit{Re}$_{\tau 0}^{\mathrm{0.913}}$. Results scaled based on Wenzel state momentum thickness show very similar trends.