Linear and nonlinear interactions between stationary cross-flow instabilities and a smooth surface hump

Smooth surface features were recently found to stabilise stationary cross-flow instability (CFI) of swept-wing boundary layers, thus holding potential for passive laminar flow control. Notably, the effect of surface features on the transition location exhibited a significant dependence on the CFI amplitude. In this work, numerical solutions of the harmonic Navier–Stokes (HNS) equations are used to explore the impact of a smooth surface hump on the linear and nonlinear development of stationary CFI under various perturbation amplitudes. Linear simulations identify regions of successive inhibited and enhanced perturbation growth. Despite the recovery of the base flow and perturbation kinetic energy to the reference (i.e. no-hump) state, significantly reduced perturbation growth is observed. The distorted perturbation profile due to the interaction with the hump is postulated to be responsible for this. Increasing the perturbation amplitude results in a response of the flow that is qualitatively similar to the linear case, albeit with increasing local destabilisation of new fundamental (i.e. primary wavelength) structures and higher-order harmonics near the wall. An energy budget analysis reveals that the growth of the fundamental incoming CFI is inhibited through the reduced effectiveness of the lift-up mechanism downstream of the hump. This is preceded by a spatial perturbation shape deformation, governed by (spanwise) transport terms. The results suggest that stabilisation of incoming stationary CFI via smooth surface humps is most effective at low incoming perturbation amplitudes. At higher perturbation amplitudes, newly formed near-wall structures, pre-conditioned by the incoming CFI, overtake the incoming CFI and could anticipate the transition process.