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Transcritical flow over obstacles and holes: forced Korteweg–de Vries framework

Published online by Cambridge University Press:  25 October 2019

Roger H. J. Grimshaw Open the ORCID record for Roger H. J. Grimshaw [Opens in a new window]  and
Montri Maleewong Open the ORCID record for Montri Maleewong [Opens in a new window]
Roger H. J. Grimshaw
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
Montri Maleewong*
Affiliation:
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok, 10900, Thailand
*
†Email address for correspondence: montri.m@ku.th
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Abstract

This paper extends a previous study of free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation, to an analogous study of flow over two localised holes, or a combination of an obstacle and a hole. Importantly the terminology obstacle or hole can be reversed for a stratified fluid and refers more precisely to the relative polarity of the forcing and the solitary wave solution of the unforced Korteweg–de Vries equation. As in the previous study, our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. In the transcritical regime at early times, undular bores are produced upstream and downstream of each forcing site. We then describe the interaction of these undular bores between the forcing sites, and the outcome at very large times.

JFM classification

Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 660 - 678
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Copyright
© 2019 Cambridge University Press 

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References

Akylas, T. R. 1984 On the excitation of long nonlinear water waves by moving pressure distribution. J. Fluid Mech. 141, 455–466.Google Scholar
Baines, P. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Binder, B., Dias, F. & Vanden-Broeck, J.-M. 2006 Steady free-surface flow past an uneven channel bottom. Theor. Comput. Fluid Dyn. 20, 125–144.Google Scholar
Binder, B. J., Vanden-Broeck, J.-M. & Dias, F. 2005 Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106.Google Scholar
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579–587.Google Scholar
El, G., Grimshaw, R. & Smyth, N. 2006 Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027214.Google Scholar
El, G., Grimshaw, R. & Smyth, N. 2008 Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory. Physica D 237, 2423–2435.Google Scholar
El, G., Grimshaw, R. & Smyth, N. 2009 Transcritical shallow-water flow past topography: finite-amplitude theory. J. Fluid Mech. 640, 187–214.Google Scholar
El, G. & Hoefer, M. 2016 Dispersive shock waves and modulation theory. Physica D 333, 11–65.Google Scholar
Grimshaw, R. 2010 Transcritical flow past an obstacle. ANZIAM J. 52, 1–25.Google Scholar
Grimshaw, R. & Helfrich, K. R. 2018 Internal solitary wave generation by tidal flow over topography. J. Fluid Mech. 839, 387–407.Google Scholar
Grimshaw, R. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Grimshaw, R., Zhang, D. & Chow, K. 2007 Generation of solitary waves by trancritical flow over a step. J. Fluid Mech. 587, 235–354.Google Scholar
Grimshaw, R. H. J. & Maleewong, M. 2015 Critical controls in transcritical shallow-water flow over obstacles. J. Fluid Mech. 780, 480–502.Google Scholar
Grimshaw, R. H. J. & Maleewong, M. 2016 Transcritical flow over two obstacles: forced Korteweg-de Vries framework. J. Fluid Mech. 809, 918–940.Google Scholar
Kim, H. & Choi, H. 2018 A study of wave trapping between two obstacles in the forced Korteweg-de Vries equation. J. Engng Maths 108, 197–208.Google Scholar
Lee, S.-J., Yates, G. & Wu, T.-Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569–593.Google Scholar