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On bounds and non-existence in the problem of steady waves with vorticity
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Russian Academic Science, Russia.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2015 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 765, no R1Article in journal (Refereed) Published
Abstract [en]

For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoullis constant. (ii) If only one parallel shear flow exists for a given value of Bernoullis constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

Place, publisher, year, edition, pages
Cambridge University Press (CUP): STM Journals , 2015. Vol. 765, no R1
Keyword [en]
surface gravity waves; waves/free-surface flows
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-114417DOI: 10.1017/jfm.2014.747ISI: 000348130700001OAI: oai:DiVA.org:liu-114417DiVA: diva2:791981
Note

Funding Agencies|Swedish Research Council; G.S. Magnusons Foundation of the Royal Swedish Academy of Sciences; Linkoping University

Available from: 2015-03-02 Created: 2015-02-20 Last updated: 2017-12-04
In thesis
1. Small-amplitude steady water waves with vorticity
Open this publication in new window or tab >>Small-amplitude steady water waves with vorticity
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The problem of describing two-dimensional traveling water waves is considered. The water region is of finite depth and the interface between the region and the air is given by the graph of a function. We assume the flow to be incompressible and neglect the effects of surface tension. However we assume the flow to be rotational so that the vorticity distribution is a given function depending on the values of the stream function of the flow. The presence of vorticity increases the complexity of the problem and also leads to a wider class of solutions.

First we study unidirectional waves with vorticity and verify the Benjamin-Lighthill conjecture for flows whose Bernoulli constant is close to the critical one. For this purpose it is shown that every wave, whose slope is bounded by a fixed constant, is either a Stokes or a solitary wave. It is proved that the whole set of these waves is uniquely parametrised (up to translation) by the flow force which varies between its values for the supercritical and subcritical shear flows of constant depth. We also study large-amplitude unidirectional waves for which we prove bounds for the free-surface profile and for Bernoulli’s constant.

Second, we consider small-amplitude waves over flows with counter currents. Such flows admit layers, where the fluid flows in different directions. In this case we prove that the initial nonlinear free-boundary problem can be reduced to a finite-dimensional Hamiltonian system with a stable equilibrium point corresponding to a uniform stream. As an application of this result, we prove the existence of non-symmetric wave profiles. Furthermore, using a different method, we prove the existence of periodic waves with an arbitrary number of crests per period.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2017. 16 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1830
National Category
Fluid Mechanics and Acoustics Computational Mathematics Mathematical Analysis Ocean and River Engineering Applied Mechanics
Identifiers
urn:nbn:se:liu:diva-134243 (URN)10.3384/diss.diva-134243 (DOI)9789176855874 (ISBN)
Public defence
2017-02-24, Nobel BL32, B-huset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2017-01-30 Created: 2017-01-30 Last updated: 2017-02-09Bibliographically approved

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Kozlov, VladimirLokharu, Evgeniy

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