Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers
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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 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.
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1469-7645