Similarity and structure of wall turbulence with lateral wall shear stress variations

Wall-bounded turbulence, where it occurs in engineering or nature, is commonly subjected to spatial variations in wall shear stress. A prime example is spatially varying roughness. Here, we investigate the configuration where the wall shear stress varies only in the lateral direction. The investigation is idealised in order to focus on one aspect, namely, the similarity and structure of turbulent inertial motion over an imposed scale of stress variation. To this end, we analyse data from direct numerical simulation (DNS) of pressure-driven turbulent flow through a channel bounded by walls of laterally alternating patches of high and low wall shear stress. The wall shear stress is imposed as a Neumann boundary condition such that the wall shear stress ratio is fixed at 3 while the lateral spacing s of the uniform-stress patches is varied from 0.39 to 6.28 of the half-channel height delta. We find that global outer-layer similarity is maintained when s is less than approximately 0.39 delta while local outer-layer similarity is recovered when s is greater than approximately 6.28 delta. However, the transition between the two regimes through s approximate to delta is not monotonic owing to the presence of secondary roll motions that extend across the whole cross-section of the flow. Importantly, these secondary roll motions are associated with an amplified skin-friction coefficient relative to both the small- and large-s/delta limits. It is found that the relationship between the secondary roll motions and the mean isovels is reversed through this transition from low longitudinal velocity over low stress at small s/delta to high longitudinal velocity over low stress at large s/delta.