The amplitude dependence of the mean velocity distortion for turbulence stabilisation in channel flow*

Flattening the mean velocity profile has recently been shown both experimentally and computationaly as an effective means to stabilise turbulence in pipe flow (Kühnen et al, 2018, Nat. Phys.). Yet, a subsequent recent study optmising this distortion with an adjoint-based method has revealed that decreasing the mean shear near the wall may be more effective than flattenning the mean velocity subject to a sufficiently high forcing amplitude used for the mean distortion (Marensi et al., 2020, J. Fluid Mech.). In this study, we systematically investigate the amplitude dependence for the optimal mean distortion using an ensemble variation technique, which enables us to study the regime of a relatively small-amplitude mean distortion that was not able to be studied previously. The mean distortion mechanism is implemented by applying a streamwise localised linear damping force, and the optimisation of its wall-normal profile is subsequently performed. At small amplitudes of the forcing, the most effective approach is to apply the damping in the vicinity of the half-channel height, similar to the original study by Kühnen et al (2018). However, on increasing the amplitude, the optimal damping coefficient profile moves progressively closer to the near-wall region, resulting in a distinctive 'M' shape, consistent with the findings of Marensi et al (2020). These observations suggest that there exist some non-trivial physical mechanisms in turbulence stabilisation, and this will be dicussed in detail in the final presentation.