Optimal mixing in twodimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers
dc.contributor.author  Caulfield, Colmcille  
dc.contributor.author  Marcotte, Florence  
dc.date.accessioned  20181101T14:02:25Z  
dc.date.available  20181101T14:02:25Z  
dc.date.issued  20181025  
dc.identifier.issn  14697645  
dc.identifier.uri  https://www.repository.cam.ac.uk/handle/1810/284490  
dc.description.abstract  We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable densitystratified 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 twodimensional 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 directadjointlooping (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 timeaveraged 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 `mixnorm' 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 longtime variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the timeaveragedkineticenergymaximising perturbations are significantly suboptimal at mixing compared to the mixnormminimising perturbations, and also that minimising the mixnorm remains (for densitystratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mixnormminimising 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.publisher  Cambridge University Press  
dc.title  Optimal mixing in twodimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers  
dc.type  Article  
prism.endingPage  385  
prism.publicationDate  2018  
prism.publicationName  Journal of Fluid Mechanics  
prism.startingPage  359  
prism.volume  853  
dc.identifier.doi  10.17863/CAM.31866  
dcterms.dateAccepted  20180712  
rioxxterms.versionofrecord  10.1017/jfm.2018.565  
rioxxterms.licenseref.uri  http://www.rioxx.net/licenses/allrightsreserved  
rioxxterms.licenseref.startdate  20181025  
dc.contributor.orcid  Caulfield, Colmcille [0000000231709480]  
dc.identifier.eissn  14697645  
dc.publisher.url  https://www.cambridge.org/core/journals/journaloffluidmechanics/article/optimalmixingintwodimensionalstratifiedplanepoiseuilleflowatfinitepecletandrichardsonnumbers/12FF98FCFC088755D54917BE22E45D07  
rioxxterms.type  Journal Article/Review  
pubs.funderprojectid  Engineering and Physical Sciences Research Council (EP/K034529/1)  
cam.issuedOnline  20180823  
dc.identifier.url  https://www.cambridge.org/core/journals/journaloffluidmechanics/article/optimalmixingintwodimensionalstratifiedplanepoiseuilleflowatfinitepecletandrichardsonnumbers/12FF98FCFC088755D54917BE22E45D07  
rioxxterms.freetoread.startdate  20190223 
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