Impact of Roughness Spanwise Wavelength on Secondary Flow Rearrangement

Spanwise heterogeneity in surface roughness generates secondary mean flows in a rough-wall turbulent boundary layer. This study investigates the influence of roughness spanwise wavelength on the arrangement of these secondary flows using direct numerical simulation (DNS). We systematically vary the spanwise wavelength, $S/δ$⁠, from $π/16$ to $2π$⁠, while maintaining constant roughness height and surface coverage density. Here, S represents the roughness spanwise wavelength, and $δ$ denotes the outer length scale, which in this case is the half-channel height. Secondary flows are observed in all DNS cases, but their configurations depend on the spanwise wavelength. Specifically, small wavelengths result in low-momentum pathways (LMPs) above the roughness elements, whereas large wavelengths lead to high-momentum pathways (HMPs) at these locations. To elucidate the mechanisms behind this rearrangement, we analyze the mean streamwise vorticity transport equation. The findings indicate that shear stress anisotropy induces another pair of secondary vortices above the roughness elements as the spanwise wavelength increases. Given that secondary flows are features of the mean flow, we further examine the budgets of the dispersive kinetic energy (DKE). The analyses reveal that at small spanwise wavelengths, shear production primarily generates DKE, while at large wavelengths, wake production becomes the dominant energy source. Building on these insights, we propose a refined classification for surfaces with spanwise heterogeneity.