Direct numerical simulation of turbulent boundary layer over rough and superhydrophobic surfaces

Abstract

This study investigated the effects of the surface texture on the characteristics of spatially developing turbulent boundary layer (TBLs) and its influence on the turbulent structures. The surface textures employed in the present study are the two-dimensional (2D) rough surface and superhydrophobic surface. To achieve this, direct numerical simulations (DNSs) of spatially developing turbulent boundary layers over sparsely-spaced two-dimensional rod-roughened walls were performed. The rod elements were periodically arranged along the streamwise direction with pitches of $p_x/k$ = 8, 16, 32, 64, and 128, where $p_x$ is the streamwise spacing of the rods, and k is the roughness height. The Reynolds number based on the momentum thickness was varied from $Re_\theta$ = 300-1400, and the height of the roughness element was k = $1.5\theta_in$, where θin is the momentum thickness at the inlet. The characteristics of the TBLs, such as the friction velocity, mean velocity, and Reynolds stresses over the rod-roughened walls, were examined by varying the spacing of the roughness features ($8 \le p_x/k \le 128$). The outer-layer similarity between the rough and smooth walls was established for the sparsely-distributed rough walls ($p_x/k \ge 32$) based on the profiles of the Reynolds stresses, whereas those are not for $p_x/k$ = 8 and 16. Inspection of the interaction between outer-layer large-scale motions and near-wall small-scale motions using two-point amplitude modulation (AM) covariance showed that modulation effect of large-scale motions on near-wall small-scale motions was strongly disturbed over the rough wall for $p_x/k$ = 8 and 16. For $p_x/k \ge 32$, the flow that passed through the upstream roughness element transitioned to a smooth wall flow between the consecutive rods. The strong influence of the surface roughness in the outer layer for $p_x/k$ = 8 and 16 was attributed to large-scale erupting motions by the surface roughness, creating both upward shift of the near-wall turbulent energy and active energy production in the outer layer with little influence on the near-wall region. DNS of spatially developing TBL over a convergent and divergent superhydrophobic surface (SHS) was performed. The convergent and divergent SHS was aligned in the streamwise direction. The SHS was modeled as a pattern of slip and no-slip surfaces. The Reθ was varied from 800 to 1400. The gas fraction of the convergent and divergent SHS was the same as that of the straight SHS, keeping the slip area constant. The slip velocity in the convergent SHS and the longer pattern was higher than that of the straight SHS. An optimal streamwise length of the convergent and divergent SHS was obtained. The convergent and divergent SHS gave more drag reduction than the straight SHS. The mean velocity profile indicated the effect of the SHS extends to the $y/\delta$ = 0.45. The convergent and divergent SHS led to the modification of near wall-turbulent structures. The convergent and divergent SHS had a relatively larger damping effect on near-wall turbulence than the straight SHS. The pre-multiplied spanwise energy spectra showed the movement of near-wall energy peak and appearance of the new energy peak with larger spanwise wavelength. Two-point AM covariance analysis exhibited the modification of near-wall structures and the relative insensitivity of the outer-wall structures.