Effects of roughness spacing on boundary-layer transition due to distributed surface roughness

The influence of roughness distribution with different streamwise and spanwise spacings on laminar-turbulent transition in wall-bounded flows is investigated using direct numerical simulation and global linear stability analysis. The results are compared to an isolated roughness element with aspect ratio \eta=1. The small spanwise spacing (\lambda_z=2.5h) inhibits the formation of hairpin vortices and prevents transition to occur. For \lambda_z=5h, the hairpin vortices are induced by the first-row roughness, perturbing the downstream shear layer and cause transition. The periodicity of the primary hairpin vortices is independent on the streamwise spacing and the distributed surface roughness leads to lower critical Re_h for instability to occur. When the streamwise spacing is comparable to the region of flow separation (\lambda_x=5h), the high-momentum fluid hardly moves downward into the cavities and the wake flow has little impact on the following roughness elements. The leading unstable varicose mode is associated with the central low-speed streaks along the aligned roughness elements. For larger streamwise spacing ($\lambda_x=10h$), two distinct modes are obtained from global stability analysis. The first one presents varicose symmetry, corresponding to the primary hairpin vortex shedding induced by the first-row roughness. The high-speed streaks form in the longitudinal grooves are also found to be unstable and modulated with the varicose mode. The second one is a sinuous mode with lower frequency, induced as the wake flow of the first-row roughness runs onto the second row, and extracts energy from the spanwise shear between the high- and low-speed streaks.