Effect of Interface Curvature on Turbulent Skin-Friction Drag Reduction with Super-Hydrophobic Micro-Grooves

Effect of interface curvature on Drag Reduction (DR) with Super-Hydrophobic (SH) Micro-Grooves (MGs) was investigated by DNS with lattice Boltzmann methods. The liquid/gas interfaces in the SH MGs were modeled as curved, stationary, shear-free boundaries, with the interface shape determined from the Young-Laplace equation. The full range of interface protrusion angles, ranging from $0^o$ to $-90^o$, were investigated. DRs of 35\% to 63\% were realized in DNS, in turbulent channel flows at a $Re_{bulk}=7200$ ($Re_{\tau_0} \approx 222$) with longitudinal MGs of size $14 \le g^{+0} \le 56$ \& $g^{+0}/w^{+0} = 7$ on both walls, where $g^{+0}$ and $w^{+0}$ denote the widths and spacings of the MGs, in wall units of the base flow, respectively. The presence of interface curvature led to increases of 2.3\% to 4.5\% in the magnitude of DR, and drops of -3.5\% to -13.5\% in the slip velocity, at low protrusion angles, and drops of -2.2\% to -12.5\% in the magnitude of DR, and either drops of up to -16.5\% or increases of up to 6\% in the slip velocity, at high protrusion angles, compared to flat interfaces. In addition, the instantaneous pressure fluctuations on curved SH interfaces at low protrusion angles were significantly lower (by a factor of $\sim 2$) than those on flat interfaces.