Optimal mixing of a passive scalar by supercritical 3D plane Poiseuille flow*

We consider a passive zero-mean scalar field organised into two layers of different concentration, in a 3D plane channel subjected to a constant along-stream pressure gradient. We employ a fully nonlinear adjoint-looping approach to identify the optimal initial perturbation of the velocity field with given initial energy which yields maximal mixing by a target time horizon, in the sense of minimisation of the spatially-integrated variance of the concentration field. Foures \emph{et.al.} (JFM, 2014) considered 2D plane Poiseuille flow at a sufficiently low (subcritical) $Re \sim 500$ to not be subject to flow instabilities, and demonstrated that the initial perturbation which maximizes the time-averaged energy gain of the flow leads to weak mixing, and is qualitatively different from the optimal initial ``mixing'' perturbation which exploits classical Taylor dispersion. We generalise this study to the optimisation of mixing three-dimensional flows at a range of significantly higher (supercritical) Reynolds numbers, showing how the potential triggering of ``strong'' flow instabilities modifies the structure of the optimal initial mixing perturbation qualitatively.