Scaling of drag reduction and logarithmic profile in the turbulent boundary layer over micro-grated superhydrophobic surfaces

We developed a dedicated, high-resolution skin-friction balance in a water tunnel to measure turbulent drag reduction over micro-grate-patterned superhydrophobic (SHPO) surfaces at the Reynolds number Re_L ranging from 4.1 × 10⁵ to 6.9 × 10⁵ and achieved a significant drag reduction of up to 46%. The correlation between drag reduction and surface topology was investigated. By considering air fraction, micro-grate gap, and meniscus curvature, an empirical scaling for drag reduction was proposed, which reconciles the widely scattered drag reduction data in the literature. This scaling law could provide a valuable guidance on future design of effective SHPO surfaces for real-world applications. The scaling of the logarithmic layer was also analyzed under the condition that the outer layer has not fully adapted to the SHPO wall manipulation, a common occurrence in experiments due to the limited length of fabricated SHPO surfaces. The slope of the logarithmic layer was found to increase with the drag reduction. Moreover, a theoretical expression describing the slope and up-shifting level of the logarithmic profile was proposed. These results are insightful, providing a new perspective for researchers to examine their velocity profile and drag reduction data in turbulent boundary layers.