On the roughness instability of growing boundary layers

The streamwise vortex instability of boundary layers caused by wall roughness in the form of surface undulations is investigated. The instability is characterised by a roughness parameter depending on the geometry and fluid properties. At $O(1)$ values of disturbances develop on the same length scale as the basic boundary layer flow. The instability is driven by a boundary condition relating the disturbance wall shears in the streamwise and normal directions. The undulations have a wavelength comparable with the boundary layer depth and the amplitude is asymptotically small compared with the depth. If the roughness parameter is large then, apart from a narrow window of vortex wavenumbers, the instability responds in a quasi-parallel manner. Falkner–Skan boundary layers are considered in detail and the dependence on the angle of the wedge associated with the flows investigated. A particular susceptibility to roughness instabilities of flows past 90∘ wedges is uncovered. The limits of small and large wavenumbers are considered and universal results given for the critical roughness height h and wavelength b needed for instability.