Modal and nonmodal stability of the laminar flow in a channel with longitudinal riblets

Modal and nonmodal stability analysis of a channel flow with longitudinal riblets are investigated. To this extent, a method based on the coupling between a two-dimensional stability problem and a computational framework for $n$-periodic systems based on Bloch waves is proposed. Unstable modes from linear stability can be retrieved. The influence of the riblets on the most amplified flow structures is investigated through a transient growth analysis: a resonance is found when the riblet wavenumber is equal to the wavenumber of the optimal streaks in a smooth channel flow. For large riblet spacing, a wavenumber lock-in regime, in which the streaks dynamics and wavelength are totally controlled by the riblet spacing, is observed. Physically, a modulation of the streaks amplitude in the spanwise direction via a beating mechanism is seen. These phenomena are characteristic of dynamic systems spatially forced and may exhibit geometric frustration. Similar results were found in the study of secondary flows in Rayleigh-Bénard convection with wavy walls. A resolvent analysis is also performed: it is found that riblets lead to the development of oblique waves that may trigger an early transition.