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Article: Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

TitleAsymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration
Authors
KeywordsAsymptotic solutions
Asymptotic theories
Ekman numbers
Fluid motions
Geophysical and geological flows
Issue Date2012
PublisherCambridge University Press. The Journal's web site is located at http://journals.cambridge.org/action/displayJournal?jid=FLM
Citation
Journal of Fluid Mechanics, 2012, v. 692, p. 420-445 How to Cite?
AbstractAbstract We consider a homogeneous fluid of viscosity u confined within an oblate spheroidal cavity, x 2/a 2 + y 2/a 2 + z 2/(a 2 (1-E 2))= 1, with eccentricity 0 1 and Po > 1 in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial-temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude O(Po), and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When |ω-2/(2-E 2)| > O(E 1/2), the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes O(Po/E 1/2) at E > 1 and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for E > 1 describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at E > 1. © 2012 Cambridge University Press.
Persistent Identifierhttp://hdl.handle.net/10722/156282
ISSN
2015 Impact Factor: 2.514
2015 SCImago Journal Rankings: 1.450
ISI Accession Number ID
Funding AgencyGrant Number
UK NERC
STFC
Leverhulme Trust
Institute of Mathematical Sciences
Chinese University of Hong Kong
Hong Kong RGC700310
NSFC10633030
CAS
Shanghai Supercomputer Center
Swiss National Supercomputing Center
Funding Information:

K.Z. would like to thank J. M. Aurnou, F. H. Busse, J. L. Margot and J. Noir for helpful discussions. In particular, J. Noir pointed out that the term. Po/(omega) over cap) cos(Omega<INF>0</INF>(omega) over capt)(y) over cap x u in (2.3) is physically required because the rotation vector Omega<INF>0</INF> in planetary latitudinal libration is fixed in the inertial frame. K.Z. is supported by UK NERC, STFC and Leverhulme Trust grants. Part of this work (K.Z.) was carried out at, and supported by, the Institute of Mathematical Sciences, the Chinese University of Hong Kong. K. H. C. is supported by Hong Kong RGC grant/700310 and X. L. is supported by NSFC/10633030 and CAS grants. The parallel computation is supported by Shanghai Supercomputer Center and Swiss National Supercomputing Center.

References

 

DC FieldValueLanguage
dc.contributor.authorZhang, Ken_US
dc.contributor.authorChan, KHen_US
dc.contributor.authorLiao, Xen_US
dc.date.accessioned2012-08-08T08:41:10Z-
dc.date.available2012-08-08T08:41:10Z-
dc.date.issued2012en_US
dc.identifier.citationJournal of Fluid Mechanics, 2012, v. 692, p. 420-445en_US
dc.identifier.issn0022-1120en_US
dc.identifier.urihttp://hdl.handle.net/10722/156282-
dc.description.abstractAbstract We consider a homogeneous fluid of viscosity u confined within an oblate spheroidal cavity, x 2/a 2 + y 2/a 2 + z 2/(a 2 (1-E 2))= 1, with eccentricity 0 <E < 1. The spheroidal container rotates rapidly with an angular velocity Ω 0, which is fixed in an inertial frame and defines a small Ekman number E= u/(a 2Ω 0), and undergoes weak latitudinal libration with frequency ω |Ω 0| and amplitude Po|Ω 0|, where Po is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When |ω-2/(2-E 2)| < O(E 1/2), an asymptotic solution for E > 1 and Po > 1 in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial-temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude O(Po), and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When |ω-2/(2-E 2)| > O(E 1/2), the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes O(Po/E 1/2) at E > 1 and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for E > 1 describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at E > 1. © 2012 Cambridge University Press.en_US
dc.languageengen_US
dc.publisherCambridge University Press. The Journal's web site is located at http://journals.cambridge.org/action/displayJournal?jid=FLMen_US
dc.relation.ispartofJournal of Fluid Mechanicsen_US
dc.rightsJournal of Fluid Mechanics. Copyright © Cambridge University Press.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectAsymptotic solutionsen_US
dc.subjectAsymptotic theoriesen_US
dc.subjectEkman numbers-
dc.subjectFluid motions-
dc.subjectGeophysical and geological flows-
dc.titleAsymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal librationen_US
dc.typeArticleen_US
dc.identifier.emailChan, KH: mkhchan@hku.hken_US
dc.identifier.authorityChan, KH=rp00664en_US
dc.description.naturepublished_or_final_versionen_US
dc.identifier.doi10.1017/jfm.2011.521en_US
dc.identifier.scopuseid_2-s2.0-84857344390en_US
dc.identifier.hkuros202756-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-84857344390&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume692en_US
dc.identifier.spage420en_US
dc.identifier.epage445en_US
dc.identifier.eissn1469-7645-
dc.identifier.isiWOS:000299883400019-
dc.publisher.placeUnited Kingdomen_US
dc.identifier.scopusauthoridLiao, X=7202134147en_US
dc.identifier.scopusauthoridChan, KH=7406033542en_US
dc.identifier.scopusauthoridZhang, K=7404451892en_US

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