Optimal mixing in three-dimensional plane Poiseuille flow at high Peclet number
Journal of Fluid Mechanics
Cambridge University Press
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Caulfield, C., & Vermach, L. (2018). Optimal mixing in three-dimensional plane Poiseuille flow at high Peclet number. Journal of Fluid Mechanics, 850 875-923. https://doi.org/10.1017/jfm.2018.388
We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields `maximal' mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially-integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241-277) that the initial perturbation which maximizes the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial `mixing' perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ($Re=U_m h/\nu= 500$, and $Re = 3000$, where $U_m$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\nu$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the `mix-norm' of the concentration eld, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, pp. 23-46, 2005). We show that the `true' variance-based mixing strategy can be successfully and practically approximated by the mix-norm minimisation since we f ind that the mix-norm optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.
Is supplemented by: https://doi.org/10.17863/CAM.22239
This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. The research activity of C.P.C. is supported by EPSRC Programme Grant EP/K034529/1 entitled Mathematical Underpinnings of Strati ed Turbulence. This re- search was also supported in part by the National Science Foundation under Grant No. NSF PHY17-48958.
External DOI: https://doi.org/10.1017/jfm.2018.388
This record's URL: https://www.repository.cam.ac.uk/handle/1810/279455