Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-l8tfn Total loading time: 0.25 Render date: 2022-06-29T20:46:12.950Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

TRANSCRITICAL FLOW PAST AN OBSTACLE

Part of: Waves

Published online by Cambridge University Press:  20 May 2011

R. GRIMSHAW*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK (email: R.H.J.Grimshaw@lboro.ac.uk)
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that transcritical flow past an obstacle may generate undular bores propagating away from the obstacle. This flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical simulations and asymptotic analyses have shown that the unsteady undular bores are connected by a locally steady solution over the obstacle. In this paper we present an overview of the underlying theory, together with some recent work on the case where the obstacle has a large width.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Akylas, T. R., “On the excitation of long nonlinear water waves by moving pressure distribution”, J. Fluid Mech. 141 (1984) 455466.CrossRefGoogle Scholar
[2]Binder, B. J., Dias, F. and Vanden-Broeck, J.-M., “Steady free-surface flow past an uneven channel bottom”, Theor. Comput. Fluid Dyn. 20 (2006) 125144.CrossRefGoogle Scholar
[3]Binder, B. J., Vanden-Broeck, J.-M. and Dias, F., “Forced solitary waves and fronts past submerged obstacles”, Chaos 15 (2005) 037106.CrossRefGoogle ScholarPubMed
[4]Clarke, S. R. and Grimshaw, R. H. J., “Resonantly generated internal waves in a contraction”, J. Fluid Mech. 274 (1994) 139161.CrossRefGoogle Scholar
[5]Cole, S. L., “Transient waves produced by flow past a bump”, Wave Motion 7 (1985) 579587.CrossRefGoogle Scholar
[6]Dias, F. and Vanden-Broeck, J.-M., “Generalized critical free surface flows”, J. Engrg. Math. 42 (2002) 291301.CrossRefGoogle Scholar
[7]Dias, F. and Vanden-Broeck, J.-M., “Steady two-layer flows over an obstacle”, Philos. Trans. R. Soc. Ser. A 360 (2002) 21372154.CrossRefGoogle ScholarPubMed
[8]Dias, F. and Vanden-Broeck, J.-M., “Trapped waves between submerged obstacles”, J. Fluid Mech. 509 (2004) 93102.CrossRefGoogle Scholar
[9]Dias, F. and Vanden-Broeck, J.-M., “Two-layer hydraulic falls over an obstacle”, Eur. J. Mech. B Fluids 23 (2004) 879898.CrossRefGoogle Scholar
[10]Ee, B. K. and Clarke, S. R., “Weakly dispersive hydraulic flows in a contraction: parametric solutions and linear stability analysis”, Phys. Fluids 19 (2007) 056601.CrossRefGoogle Scholar
[11]Ee, B. K. and Clarke, S. R., “Weakly dispersive hydraulic flows in a contraction: nonlinear stability analysis”, Wave Motion 45 (2008) 927939.CrossRefGoogle Scholar
[12]Ee, B. K., Grimshaw, R. H. J., Zhang, D.-H. and Chow, K. W., “Steady transcritical flow over a hole: parametric map of solutions of the forced Korteweg–de Vries equation”, Phys. Fluids 22 (2010) 056602.CrossRefGoogle Scholar
[13]El, G. A., Grimshaw, R. H. J. and Smyth, N. F., “Transcritical shallow-water flow past topography: finite-amplitude theory”, J. Fluid Mech. 640 (2009) 187214.CrossRefGoogle Scholar
[14]Ertekin, R. C., Webster, W. C. and Wehausen, J. V., “Ship generated solitons”, Proc. 15th Symp. Naval Hydrodyn., Hamburg (National Academy Press, Washington, DC, 1984) 347–364.Google Scholar
[15]Ertekin, R. C., Webster, W. C and Wehausen, J. V., “Waves caused by a moving disturbance in a shallow channel of finite width”, J. Fluid Mech. 169 (1986) 275292.CrossRefGoogle Scholar
[16]Grimshaw, R. H.  J., Chan, K. H. and Chow, K. W., “Transcritical flow of a stratified fluid: the forced extended Korteweg–de Vries model”, Phys. Fluids 14 (2002) 755774.CrossRefGoogle Scholar
[17]Grimshaw, R., Pelinovsky, E., Talipova, T. and Kurkina, A., “Internal solitary waves: propagation, deformation and disintegration”, Nonlinear Proc. Geoph. 17 (2010) 633649.CrossRefGoogle Scholar
[18]Grimshaw, R. H. J. and Smyth, N. F., “Resonant flow of a stratified fluid over topography”, J. Fluid Mech. 169 (1986) 429464.CrossRefGoogle Scholar
[19]Grimshaw, R. H. J., Zhang, D.-H. and Chow, K. W., “Generation of solitary waves by transcritical flow over a step”, J. Fluid Mech. 587 (2007) 235254.CrossRefGoogle Scholar
[20]Grimshaw, R. H. J., Zhang, D.-H. and Chow, K. W., “Transcritical flow over a hole”, Stud. Appl. Math. 122 (2009) 235248.CrossRefGoogle Scholar
[21]Grue, J., Friis, H. A., Palm, E. and Rusas, P.-O., “A method for computing unsteady fully nonlinear interfacial waves”, J. Fluid Mech. 351 (1997) 223252.CrossRefGoogle Scholar
[22]Gurevich, A. V. and Pitaevskii, L. P., “Nonstationary structure of a collisionless shock wave”, Sov. Phys. JETP 38 (1974) 291297.Google Scholar
[23]Huang, D. B., Sibul, O. J., Webster, W. C., Wehausen, J. V., Wu, D. M. and Wu, T. Y, “Ships moving in the transcritical range”, Proc. Conf. on Behaviour of Ships in Restricted Waters, Varna, Bulgaria (1982) 26-1–26-10.Google Scholar
[24]Lee, S. J., Yates, G. T. and Wu, T. Y., “Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances”, J. Fluid Mech. 199 (1989) 569593.CrossRefGoogle Scholar
[25]Lighthill, J., Waves in fluids (Cambridge University Press, Cambridge, 1978).Google Scholar
[26]Marchant, T. R. and Smyth, N. F., “The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography”, J. Fluid Mech. 221 (1990) 263288.CrossRefGoogle Scholar
[27]Mei, C. C., “Radiation of solitons by slender bodies advancing in a shallow channel”, J. Fluid Mech. 162 (1986) 5367.CrossRefGoogle Scholar
[28]Melville, W. K. and Helfrich, K. R, “Transcritical two-layer flow over topography”, J. Fluid Mech. 178 (1987) 3152.CrossRefGoogle Scholar
[29]Porter, A. and Smyth, N. F., “Modelling the morning glory of the Gulf of Carpentaria”, J. Fluid Mech. 454 (2002) 120.CrossRefGoogle Scholar
[30]Smyth, N. F., “Modulation theory solution for resonant flow over topography”, Proc. R. Soc. A 409 (1987) 7997.CrossRefGoogle Scholar
[31]Thews, J. G. and Landweber, L., “The influence of shallow water on the resistance of a cruiser model”, US Experimental Model Basin, Washington, DC, 1934, Report 408.Google Scholar
[32]Whitham, G. B., Linear and nonlinear waves (Wiley, New York, 1974).Google Scholar
[33]Wu, T. Y., “Generation of upstream advancing solitons by moving disturbances”, J. Fluid Mech. 184 (1987) 7599.CrossRefGoogle Scholar
[34]Wu, D. M. and Wu, T. Y., “Three-dimensional nonlinear long waves due to moving surface pressure”, Proc. 14th Symp. Naval Hydrodyn. (National Academy Press, Washington, DC, 1982) 103–129.Google Scholar
[35]Zhang, D.-H. and Chwang, A. T., “On solitary waves forced by underwater moving objects”, J. Fluid Mech. 389 (1999) 119135.CrossRefGoogle Scholar
[36]Zhang, D.-H. and Chwang, A. T., “Generation of solitary waves by forward- and backward-step bottom forcing”, J. Fluid Mech. 432 (2001) 341350.Google Scholar
You have Access
13
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

TRANSCRITICAL FLOW PAST AN OBSTACLE
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

TRANSCRITICAL FLOW PAST AN OBSTACLE
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

TRANSCRITICAL FLOW PAST AN OBSTACLE
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *