We investigate the optimisation of mixing in stratified, 2D plane Poiseuille flow by means of a fully nonlinear direct-adjoint looping method, where the initial perturbation leading to maximal mixing efficiency over a given time horizon is determined. A recent study by Foures et al. [Foures et al., J. Fluid Mech., 729, 692 (2014)] has demonstrated that the widely-used approach consisting in maximizing the time-averaged kinetic energy results in largely suboptimal suppression of a passive scalar variance. We extend these results to stratified flows, where the mixing of the (active) scalar density has an energetic cost and thus has an observable dynamic effect. We identify and compare the initial perturbations which either maximize the kinetic energy transient growth or minimize a specific ‘mix- norm’ [Thiffeault, Nonlinearity 25, R1 (2012)] at a chosen target time. The suboptimality of the former strategy, as well as the robustness and computational efficiency of the mix-norm-based optimisation, are demonstrated for all degrees of stratification considered. By analysing the time evolution of the kinetic energy and potential energy reservoirs in the flows triggered by these optimal perturbations, we finally discuss the physical mechanisms involved in achieving efficient irreversible mixing.

This is joint work with C.P. Caulfield.