Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers
cam.issuedOnline | 2018-08-23 | |
dc.contributor.author | Caulfield, CP | |
dc.contributor.author | Marcotte, Florence | |
dc.contributor.orcid | Caulfield, Colm-cille [0000-0002-3170-9480] | |
dc.date.accessioned | 2018-11-01T14:02:25Z | |
dc.date.available | 2018-11-01T14:02:25Z | |
dc.date.issued | 2018-10-25 | |
dc.description.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. | |
dc.identifier.doi | 10.17863/CAM.31866 | |
dc.identifier.eissn | 1469-7645 | |
dc.identifier.issn | 1469-7645 | |
dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/284490 | |
dc.language.iso | eng | |
dc.publisher | Cambridge University Press | |
dc.publisher.url | https://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.subject | mixing enhancement | |
dc.subject | stratified flows | |
dc.subject | variational methods | |
dc.title | Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Peclet and Richardson numbers | |
dc.type | Article | |
dcterms.dateAccepted | 2018-07-12 | |
prism.endingPage | 385 | |
prism.publicationDate | 2018 | |
prism.publicationName | Journal of Fluid Mechanics | |
prism.startingPage | 359 | |
prism.volume | 853 | |
pubs.funder-project-id | Engineering and Physical Sciences Research Council (EP/K034529/1) | |
rioxxterms.licenseref.startdate | 2018-10-25 | |
rioxxterms.licenseref.uri | http://www.rioxx.net/licenses/all-rights-reserved | |
rioxxterms.type | Journal Article/Review | |
rioxxterms.version | AM | |
rioxxterms.versionofrecord | 10.1017/jfm.2018.565 |
Files
Original bundle
1 - 1 of 1
Loading...
- Name:
- manuscript_final.pdf
- Size:
- 6.89 MB
- Format:
- Adobe Portable Document Format
- Description:
- Accepted version
- Licence
- http://www.rioxx.net/licenses/all-rights-reserved
License bundle
1 - 1 of 1
No Thumbnail Available
- Name:
- DepositLicenceAgreement.pdf
- Size:
- 417.78 KB
- Format:
- Adobe Portable Document Format