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Transcritical shallow-water flow past topography: finite-amplitude theory

Published online by Cambridge University Press:  04 November 2009

G. A. EL ,
R. H. J. GRIMSHAW  and
N. F. SMYTH
G. A. EL*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
R. H. J. GRIMSHAW
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
N. F. SMYTH
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh, Scotland EH9 3JZ, UK
*
†Email address for correspondence: G.El@lboro.ac.uk
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Abstract

We consider shallow-water flow past a broad bottom ridge, localized in the flow direction, using the framework of the forced Su–Gardner (SG) system of equations, with a primary focus on the transcritical regime when the Froude number of the oncoming flow is close to unity. These equations are an asymptotic long-wave approximation of the full Euler system, obtained without a simultaneous expansion in the wave amplitude, and hence are expected to be superior to the usual weakly nonlinear Boussinesq-type models in reproducing the quantitative features of fully nonlinear shallow-water flows. A combination of the local transcritical hydraulic solution over the localized topography, which produces upstream and downstream hydraulic jumps, and unsteady undular bore solutions describing the resolution of these hydraulic jumps, is used to describe various flow regimes depending on the combination of the topography height and the Froude number. We take advantage of the recently developed modulation theory of SG undular bores to derive the main parameters of transcritical fully nonlinear shallow-water flow, such as the leading solitary wave amplitudes for the upstream and downstream undular bores, the speeds of the undular bores edges and the drag force. Our results confirm that most of the features of the previously developed description in the framework of the unidirectional forced Korteweg–de Vries (KdV) model hold up qualitatively for finite amplitude waves, while the quantitative description can be obtained in the framework of the bidirectional forced SG system. Our analytic solutions agree with numerical simulations of the forced SG equations within the range of applicability of these equations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 187 - 214
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Camassa, R., Holm, D. D. & Levermore, C. D. 1997 Long-time shallow-water equations with a varying bottom. J. Fluid Mech. 349, 173–189.CrossRefGoogle Scholar
Dellar, P. J. 2003 Dispersive shallow water magnetohydrodynamics. Phys. Plasmas 10, 581–590.CrossRefGoogle Scholar
El, G. A. 2005 Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15, 037103.CrossRefGoogle ScholarPubMed
El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2006 Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104.CrossRefGoogle Scholar
El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2008 Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory. Physica D 237, 2423–2435.CrossRefGoogle Scholar
El, G. A., Khodorovskii, V. V. & Tyurina, A. V. 2005 Undular bore transition in bi-directional conservative wave dynamics. Physica D 206, 232–251.CrossRefGoogle Scholar
Ertekin, R. C., Webster, W. C. & Wehausen, J. V. 1986 Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech. 169, 275–292.CrossRefGoogle Scholar
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. 289, 373–404.Google Scholar
Gavrilyuk, S. L. 1994 Large amplitude oscillations and their “thermodynamics” for continua with “memory”. Eur. J. Mech. B/Fluids 13, 753–764.Google Scholar
Gobbi, M. F., Kirby, J. T. & Wei, G. 2000 A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to o(kh)4. J. Fluid Mech. 405, 181–210.CrossRefGoogle Scholar
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.CrossRefGoogle Scholar
Grimshaw, R. H. J. & Smyth, N. F. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.CrossRefGoogle Scholar
Grimshaw, R. H. J., Zhang, D. H. & Chow, K. W. 2007 Generation of solitary waves by transcritical flow over a step. J. Fluid Mech. 587, 235–354.CrossRefGoogle Scholar
Gurevich, A. V. & Pitaevskii, L. P. 1974 Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38, 291–297.Google Scholar
Johnson, R. S. 2002 Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech. 455, 63–82.CrossRefGoogle Scholar
Kim, J. W., Bai, K. J., Ertekin, R. C. & Webster, W. C. 2001 A derivation of the Green–Naghdi equations for irrotational flow. J. Engng Math. 40, 17–24.CrossRefGoogle Scholar
Lamb, K. G. & Yan, L. 1996 The evolution of internal wave undular bores: comparison of a fully nonlinear numerical model with weakly nonlinear theory. J. Phys. Oceanogr. 26, 2712–2734.2.0.CO;2>CrossRefGoogle Scholar
Madsen, P. A., Bingham, H. B. & Schäffer, H. A. 2003 Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104.CrossRefGoogle Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley and Sons.Google Scholar
Miles, J. W. & Salmon, R. 1985 Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519–531.CrossRefGoogle Scholar
Nadiga, T., Margolin, L. G. & Smolarkiewicz, P. K. 1996 Different approximations of shallow fluid flow over an obstacle. Phys. Fluids 8, 2066–2077.CrossRefGoogle Scholar
Smyth, N. F. 1987 Modulation theory for resonant flow over topography. Proc. R. Soc. Lond. A 409, 79–97.Google Scholar
Su, C. H. & Gardner, C. S. 1969 Korteweg–de Vries equation and generalisations III. Derivation of the Kortwewg–de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539.CrossRefGoogle Scholar
Tyurina, A. V. & El, G. A. 1999 Hydrodynamics of modulated finite-amplitude waves in dispersive media. J. Exp. Theor. Phys. 88, 615–625.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley and Sons.Google Scholar