Modelling Pushover Transient Growth

The presence of wall-tangent shear, such as that set up by a streak field, gives rise to an additional source of transient growth in wall-bounded shear flows beyond the better-known `lift-up' and Orr processes. This additional, poorly studied mechanism, dubbed `pushover’ by Lozano-Duran et. al. (2021), has been found to play an important role in sustaining turbulence at least at low Reynolds numbers Re. To examine this, we consider a simple model of unbounded wall-normal constant shear to which a (spatially) sinusoidally-varying spanwise shear is added. Along with investigating the relative importance of each mechanism, of particular interest are the possible interactions between the pushover and Orr mechanisms, and the timescales that the growth processes act on. We hope to discuss whether the pushover mechanism acts on an O(Re) time scale (perhaps expected in analogy with lift-up) and if the lift-up timescale remains O(Re) or does some interaction with pushover speed that growth process up.