Effects of streamwise-grooved spanwise-periodic roughness arrays on boundary-layer instability

We investigate the impact of streamwise-grooved and spanwise-periodic surface roughness arrays on the lower-branch viscous Tollmien–Schlichting (TS) instability in the boundary layer over an otherwise flat plate. The streamwise length scale and spanwise spacing of the arrays are of $O(L)$ and $O(\textit{Re}^{-3/8}L)$ , respectively, with the latter being comparable to the characteristic wavelength of the TS modes, where $L$ is the distance from the leading edge of the plate to the peak location of the roughness arrays and $\textit{Re}$ denotes the Reynolds number based on $L$ , assumed to be large. The characteristic height of the roughness arrays is of $O(\textit{Re}^{-3/8}L)$ , which is greater than the boundary-layer thickness and is the required asymptotic threshold for generating $O(1)$ streaks. We show that this nonlinear streaky flow is governed by three-dimensional (3-D) boundary-layer equations supplemented by a Laplace equation in an inviscid upper deck. Prandtl’s transformation is applied to convert the curved boundary to a flat one, which not only reduces computational complexity by avoiding meshing the geometry, but also shows that the spanwise undulation of the roughness arrays enhances transverse diffusion. The Laplace equation is solved to provide the spanwise pressure gradient and velocity, which drive the streaks. The boundary-layer equations are solved efficiently using a streamwise marching scheme. The linear viscous instability of the resulting streaky flow is analysed; by exploiting the asymptotic structure, the bi-global eigenvalue problem is reduced to a one-dimensional one, where the stability is found to be controlled by the spanwise-dependent wall shear and the shape function of the roughness arrays. The results suggest that two-dimensional and weakly 3-D low-frequency modes are stabilised, while most other modes are destabilised. The present formulation provides a convenient tool for predicting streaky flows induced by riblet-like roughness of fairly large height and furthermore assessing their viscous instability properties.