Nonlinear optimisation of scalar mixing in plane Poiseuille flow with finite diffusivity

We consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in 2D plane Poiseuille flow at finite Reynolds number and P\'eclet number, $Re \sim Pe \sim O(10^3)$. We use a nonlinear-adjoint-looping approach to minimise the variance of the scalar concentration $\theta$ at various target times $T$, subject either to a finite kinetic energy initial disturbance, or wall velocity perturbation. We show that both optimal initial perturbations and optimal wall excitation strategies which minimise the variance of $\theta$ are distinct from the equivalent perturbations which maximise the time-averaged energy gain of disturbance at $t=T$, and that these ``gain'' perturbations can often be poor at scalar mixing. We also identify perturbations and excitation strategies which minimise the distribution of $\theta$ at the target time relative to a particular Sobolev norm of negative index, a ``mix-norm'' as used in flows with no diffusion to measure ``mixing'' in the sense of ergodic theory (G. Mathew, I. Mezic, \& L. Petzold 2005 {\it Physica D}, {\bf 211}, 23-46). We show the close connection between these mix-norm perturbations and the optimal variance perturbations, all of which initially increase gradients to ensure good mixing at later times.