Laminar–turbulent patterns in plane Couette flow: influence of three-dimensional roughness

This direct numerical simulation study analyses the transition-to-turbulence in plane Couette flow (PCF) with three-dimensional (3-D) roughness. Square ribs of height $k=0.2h$ (where $h$ is the half-channel height) and a streamwise pitch separation of $\lambda =10k$ , classified as k -type roughness, are mounted on the stationary wall. This configuration features alternating rough and smooth zones in the spanwise direction and is referred to as 3-D k -type roughness. We compare the behaviour of 3-D k -type roughness in the transitional regime with that of two-dimensional (2-D) k -type roughness (Gokul & Narasimhamurthy 2024 J. Fluid Mech. vol. 1000, p. A40). The route-to-transition in 3-D k -type roughness confirms the formation of laminar–turbulent patterns, which exist in the transitional Reynolds number range $\textit{Re} \in [325, 350]$ . This range interestingly overlaps with those for smooth PCF ( $\textit{Re} \in [325, 400]$ ) and 2-D k -type roughness ( $\textit{Re} \in [300, 325]$ ), thereby indicating that 3-D k -type roughness amalgamates the characteristics of both the rough and smooth zones. In striking contrast to the 2-D k -type roughness, the laminar–turbulent bands in the 3-D k -type configuration are of non-uniform bandwidth. The 3-D k -type roughness surprisingly introduces continuous competition among bands of opposite orientations, a characteristic unique in this case. Due to this competition, the large-scale flow associated with the oblique bands is never fully aligned with the diagonal direction. The smooth zones in the 3-D k -type roughness exhibit complex interactions, evident from oscillatory velocity signals and multiple high-energy peaks in the frequency spectra, which likely contribute to the competing patterns with opposite orientations.