Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers

Abstract

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\nu$ and density diffusivity $\kappa$. The initial diffusive error function density distribution varies continuously so that $\rho \in [\bar{\rho} - (1/2)\rho_0, \bar{\rho} + (1/2) \rho_0]$. A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$. We consider flows with a finite P\'eclet number $Pe= U_m h /\kappa=500$ and Prandtl number $Pr=\nu/\kappa=1$, and a range of bulk Richardson numbers $Ri_b= g \rho_0 h /(\bar{\rho} U^2) \in [0,1]$ where $U_m$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimization problems, extending the optimal mixing results of Foures, Caulfield \& Schmid (2014) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximize the time-averaged perturbation kinetic energy developed by the perturbations over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$) of a `mix-norm' as first introduced by Mathew, Mezic \& Petzold (2005), further discussed by Thi eault (2012) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged-kinetic-energy-maximising perturbations are significantly suboptimal at mixing compared to the mix-norm-minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm-minimising optimal perturbations can still trigger substantial mixing for $Ri_b \lesssim 0.3$. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into `available potential energy', which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic energy reservoirs.