Transient Growth in Streaky Unbounded Shear Flow: A symbiosis of Orr and Push-over mechanisms

Transient growth mechanisms operating on streaky shear flows are believed important for sustaining near-wall turbulence. Of the three individual mechanisms present - Orr, lift-up and 'push over' - Lozano-Duran et. al. (J. Fluid Mech. 914, A8, 2021) have recently observed that both Orr and push over need to be present to sustain turbulent fluctuations given streaky (streamwise-independent) base fields whereas lift-up does not. We show here, using Kelvin's model of unbounded constant shear augmented by spanwise-periodic streaks, that this is because the push-over mechanism can act in concert with a Orr mechanism based upon the streaks to produce much-enhanced transient growth. The model clarifies the transient growth mechanism originally found by Schoppa & Hussain, (J. Fluid Mech. 453, 57-108, 2002) and finds that this is one half of a linear instability mechanism centred at the spanwise inflexion points observed originally by Swearingen & Blackwelder (J. Fluid Mech. 182, 255-290, 1987). The instability and even transient growth acting on its own are found to have the correct nonlinear feedback to generate streamwise rolls which can then re-energise the assumed streaks through lift-up indicating a sustaining cycle. Our results therefore support the view that while lift-up is believed central for the roll-to-streak regenerative process, it is Orr and push-over mechanisms that are both key for the streak-to-roll regenerative process in near-wall turbulence.