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TRANSCRITICAL FLOW PAST AN OBSTACLE

Part of: Waves

Published online by Cambridge University Press:  20 May 2011

R. GRIMSHAW
R. GRIMSHAW*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK (email: R.H.J.Grimshaw@lboro.ac.uk)
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Abstract

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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.

Keywords

solitary wavestranscritical flowundular bore

MSC classification

Secondary: 74J30: Nonlinear waves
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 1 , July 2010 , pp. 2 - 26
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) 455–466.Google 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) 125–144.Google 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 Scholar
[4]Clarke, S. R. and Grimshaw, R. H. J., “Resonantly generated internal waves in a contraction”, J. Fluid Mech. 274 (1994) 139–161.Google Scholar
[5]Cole, S. L., “Transient waves produced by flow past a bump”, Wave Motion 7 (1985) 579–587.Google Scholar
[6]Dias, F. and Vanden-Broeck, J.-M., “Generalized critical free surface flows”, J. Engrg. Math. 42 (2002) 291–301.Google Scholar
[7]Dias, F. and Vanden-Broeck, J.-M., “Steady two-layer flows over an obstacle”, Philos. Trans. R. Soc. Ser. A 360 (2002) 2137–2154.Google Scholar
[8]Dias, F. and Vanden-Broeck, J.-M., “Trapped waves between submerged obstacles”, J. Fluid Mech. 509 (2004) 93–102.Google Scholar
[9]Dias, F. and Vanden-Broeck, J.-M., “Two-layer hydraulic falls over an obstacle”, Eur. J. Mech. B Fluids 23 (2004) 879–898.Google 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.Google Scholar
[11]Ee, B. K. and Clarke, S. R., “Weakly dispersive hydraulic flows in a contraction: nonlinear stability analysis”, Wave Motion 45 (2008) 927–939.Google 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.Google 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) 187–214.Google 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) 275–292.Google 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) 755–774.Google Scholar
[17]Grimshaw, R., Pelinovsky, E., Talipova, T. and Kurkina, A., “Internal solitary waves: propagation, deformation and disintegration”, Nonlinear Proc. Geoph. 17 (2010) 633–649.CrossRefGoogle Scholar
[18]Grimshaw, R. H. J. and Smyth, N. F., “Resonant flow of a stratified fluid over topography”, J. Fluid Mech. 169 (1986) 429–464.Google 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) 235–254.Google Scholar
[20]Grimshaw, R. H. J., Zhang, D.-H. and Chow, K. W., “Transcritical flow over a hole”, Stud. Appl. Math. 122 (2009) 235–248.Google 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) 223–252.Google Scholar
[22]Gurevich, A. V. and Pitaevskii, L. P., “Nonstationary structure of a collisionless shock wave”, Sov. Phys. JETP 38 (1974) 291–297.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) 569–593.Google 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) 263–288.Google Scholar
[27]Mei, C. C., “Radiation of solitons by slender bodies advancing in a shallow channel”, J. Fluid Mech. 162 (1986) 53–67.Google Scholar
[28]Melville, W. K. and Helfrich, K. R, “Transcritical two-layer flow over topography”, J. Fluid Mech. 178 (1987) 31–52.Google Scholar
[29]Porter, A. and Smyth, N. F., “Modelling the morning glory of the Gulf of Carpentaria”, J. Fluid Mech. 454 (2002) 1–20.Google Scholar
[30]Smyth, N. F., “Modulation theory solution for resonant flow over topography”, Proc. R. Soc. A 409 (1987) 79–97.Google 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) 75–99.Google 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) 119–135.Google 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) 341–350.Google Scholar