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

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cam.issuedOnline2018-08-23
dc.contributor.authorCaulfield, CP
dc.contributor.authorMarcotte, Florence
dc.contributor.orcidCaulfield, Colm-cille [0000-0002-3170-9480]
dc.date.accessioned2018-11-01T14:02:25Z
dc.date.available2018-11-01T14:02:25Z
dc.date.issued2018-10-25
dc.description.abstractWe 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.
dc.identifier.doi10.17863/CAM.31866
dc.identifier.eissn1469-7645
dc.identifier.issn1469-7645
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/284490
dc.language.isoeng
dc.publisherCambridge University Press
dc.publisher.urlhttps://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/optimal-mixing-in-twodimensional-stratified-plane-poiseuille-flow-at-finite-peclet-and-richardson-numbers/12FF98FCFC088755D54917BE22E45D07
dc.subjectmixing enhancement
dc.subjectstratified flows
dc.subjectvariational methods
dc.titleOptimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers
dc.typeArticle
dcterms.dateAccepted2018-07-12
prism.endingPage385
prism.publicationDate2018
prism.publicationNameJournal of Fluid Mechanics
prism.startingPage359
prism.volume853
pubs.funder-project-idEngineering and Physical Sciences Research Council (EP/K034529/1)
rioxxterms.licenseref.startdate2018-10-25
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserved
rioxxterms.typeJournal Article/Review
rioxxterms.versionAM
rioxxterms.versionofrecord10.1017/jfm.2018.565

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