Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-10T06:16:07.485Z Has data issue: false hasContentIssue false
  • English
  • Français

Transcritical flow over two obstacles: forced Korteweg–de Vries framework

Published online by Cambridge University Press:  21 November 2016

Roger H. J. Grimshaw  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: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation in a suite of numerical simulations. Our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. The flow behaviour can be characterised by the Froude number and the maximum heights of the obstacles. In the transcritical regime at early times, undular bores are produced upstream and downstream of each obstacle. Our main aim is to describe the interaction of these undular bores between the obstacles, and to find the outcome at very large times. We find that the flow development can be defined in three stages. The first stage is described by the well-known development of undular bores upstream and downstream of each obstacle. The second stage is the interaction between the undular bore moving downstream from the first obstacle and the undular bore moving upstream from the second obstacle. The third stage is the very large time evolution of this interaction, when one of the obstacles controls criticality. For equal obstacle heights, our analytical and numerical results indicate that either one of the obstacles can control flow criticality, that being the first obstacle when the flow is slightly subcritical and the second obstacle otherwise. For unequal obstacle heights the larger obstacle controls criticality. The results obtained here complement a recent numerical study using the fully nonlinear, but non-dispersive, shallow water equations.

JFM classification

Waves/Free-surface Flows: Hydraulics Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 918 - 940
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akylas, T. R. 1984 On the excitation of long nonlinear water waves by moving pressure distribution. J. Fluid Mech. 141, 455–466.CrossRefGoogle 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
Chardard, F., Dias, F., Nguyen, H. & Vanden-Broeck, J.-M. 2011 Stability of some stationary solutions to the forced kdv equation with one or two bumps. J. Engng Maths 70, 175–189.CrossRefGoogle Scholar
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579–587.Google Scholar
Dias, F. & Vanden-Broeck, J. M. 2004 Trapped waves between submerged obstacles. J. Fluid Mech. 509, 93–102.Google Scholar
Donahue, A. S. & Shen, S. S. P. 2010 Stability of hydraulic fall and sub-critical cnoidal waves in water flows over a bump. J. Engng Maths 68, 197–205.Google Scholar
Ee, B. K. & Clarke, S. R. 2007 Weakly dispersive hydraulic flows in a contraction: parametric solutions and linear stability. Phys. Fluids 19, 056601.Google Scholar
Ee, B. K., Grimshaw, R. H. J., Chow, K. W. & Zhang, D.-H. 2011 Steady transcritical flow over a hole: Parametric map of solutions of the forced extended Korteweg-de Vries equation. Phys. Fluids 23, 04662.Google Scholar
Ee, B. K., Grimshaw, R. H. J., Zhang, D.-H. & Chow, K. W. 2010 Steady transcritical flow over an obstacle: Parametric map of solutions of the forced Korteweg-de Vries equation. Phys. Fluids 22, 056602.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
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A 289, 373–404.Google Scholar
Grimshaw, R. 2010 Transcritical flow past an obstacle. ANZIAM J. 52, 1–25.CrossRefGoogle Scholar
Grimshaw, R. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.CrossRefGoogle 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., Zhang, D.-H. & Chow, K. W. 2009 Transcritical flow over a hole. Stud. Appl. Maths. 122, 235–248.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
Gurevich, A. V. & Pitaevskii, L. P. 1974 Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38, 291–297.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
Pratt, L. J. 1984 On nonlinear flow with multiple obstructions. J. Atmos. Sci. 41, 1214–1225.Google Scholar
Whitham, G. B. 1965 Non-linear dispersive waves. Proc. R. Soc. Lond. A 283, 238–261.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar