Perturbation growth over self-sustaining process in wall turbulence beyond the Lyapunov time

The evolution of small perturbations applied to turbulent Couette flow is examined over a long time horizon, until the perturbed flow field becomes completely decorrelated from the original state. To elucidate the fundamental physical processes involved, we focus on the minimal flow unit, where the dynamics of coherent structures is well understood in terms of the self-sustaining process (Hamilton et al. J. Fluid Mech. vol. 287, 1995, pp. 317–348). As expected, in the short term, perturbations exhibit exponential growth governed by the leading Lyapunov exponent. This mechanism is driven by the streamwise-dependent flow, which is known to involve intense turbulent dissipation events within the self-sustaining process, consistent with previous findings. Beyond the initial exponential phase, we observe a slow, sustained growth of perturbations over a long period – spanning tens of integral time scales – before eventual saturation. During this stage, the perturbation energy increases approximately linearly with time. While this behaviour resembles observations and predictions in other turbulent flows, the underlying physical process here is fundamentally different. Specifically, the perturbation field during this period is dominated by streaky structures and the growth mechanism is linked to the saturation of the wall-normal streak length scale at the largest dimension permitted by the flow geometry (i.e. the channel height). Finally, an evaluation of the dominant production term components reveals that the well-known lift-up effect is primarily responsible for the growth of these streaky perturbations.