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Waveless subcritical flow past symmetric bottom topography

Published online by Cambridge University Press:  09 November 2012

R. J. HOLMES ,
G. C. HOCKING ,
L. K. FORBES  and
N. Y. BAILLARD
R. J. HOLMES
Affiliation:
Department of Mathematics and Statistics, Murdoch University, Perth, Western Australia emails: [email protected], [email protected]
G. C. HOCKING
Affiliation:
Department of Mathematics and Statistics, Murdoch University, Perth, Western Australia emails: [email protected], [email protected]
L. K. FORBES
Affiliation:
School of Mathematics and Physics, University of Tasmania, Hobart, Australia email: [email protected]
N. Y. BAILLARD
Affiliation:
Bureau of Meteorology, Perth, Western Australia email: [email protected]
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Abstract

The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

Keywords

Stream flowFree-surface flowWater wavesPotential flow
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 2 , April 2013 , pp. 213 - 230
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Baines, P. G. (1987) Upstream blocking and airflow over mountains. Ann. Rev. Fluid Mech. 19, 75–97.Google Scholar
[2]Belward, S. R. (1999) Fully nonlinear flow over successive obstacles: Hydraulic fall and supercritical flows. J. Aust. Math. Soc. Ser. B 40, 447–458.Google Scholar
[3]Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. (2005) Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106-1–13.Google Scholar
[4]Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. (2008) Influence of rapid changes on free surface flows. IMA J. Appl. Math. 73, 254–273.Google Scholar
[5]Choi, J. W. (2002) Free surface waves over a depression. Bull. Aust. Math. Soc. 65 329–335.Google Scholar
[6]Dias, F. & Vanden-Broeck, J.-M. (2002) Generalised critical free surface flows. J. Eng. Math. 42, 291–301.Google Scholar
[7]Dias, F. & Vanden-Broeck, J.-M. (2004) Trapped waves between submerged obstacles. J. Fluid. Mech. 509, 93–102.Google Scholar
[8]Forbes, L. K. (1981) On the wave resistance of a submerged semi-elliptical body. J. Eng. Math. 15, 287–298.Google Scholar
[9]Forbes, L. K. (1982) Non-linear, drag-free flow over a submerged semi-elliptical body. J. Eng. Math. 16, 171–180.Google Scholar
[10]Forbes, L. K. (1985) On the effects of non-linearity in free-surface flow about a submerged point vortex. J. Eng. Math. 19, 139–155.Google Scholar
[11]Forbes, L. K. (1988) Critical free-surface flow over a semi-circular obstruction. J. Eng. Math. 22, 3–13.Google Scholar
[12]Forbes, L. K. & Schwartz, L. W. (1982) Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314.Google Scholar
[13]Higgins, P. J., Read, W. W. & Belward, S. R. (2006) A series-solution method for free-boundary problems arising from flow over topography. J. Eng. Math. 54, 345–358.Google Scholar
[14]Hocking, G. C. (2006) Steady Prandtl-Batchelor flows past a circular cylinder. ANZIAM J. 48, 165–177.Google Scholar
[15]Hocking, G. C. & Forbes, L. K. (2001) Supercritical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth. J. Fluid Mech. 428, 333–348.Google Scholar
[16]Hocking, G. C., Holmes, R. J. & Forbes, L. K. (2012) A note on waveless subcritical flow past a submerged semi-ellipse. J. Eng. Math. (In press).Google Scholar
[17]Lamb, H. (1932) Hydrodynamics, 6th ed., Cambridge University Press, Cambridge, UK.Google Scholar
[18]Long, R. R. (1972) Finite amplitude disturbances in the flow of inviscid rotating and stratified fluids over obstacles. Ann. Rev. Fluid Mech. 4, 69–92.Google Scholar
[19]Pratt, L. J. (1984) On nonlinear flow with multiple obstructions. J. Atmos. Sci. 41, 1214–1225.Google Scholar
[20]Tuck, E. O. (1989) A submerged body with zero wave resistance. J. Ship Res. 33, 81–83.Google Scholar
[21]Vanden-Broeck, J.-M. (1987) Free-surface flow over an obstruction in a channel. Phys. Fluids 30, 2315–2317.Google Scholar
[22]Zhang, Y. & Zhu, S. (1996) Open channel flow past a bottom obstruction. J. Eng. Math. 30, 487–499.Google Scholar
[23]Zhang, Y. & Zhu, S. (1996) A comparison study of nonlinear waves generated behind a semicircular trench. Proc. R. Soc. Lond. 452, 1563–1584.Google Scholar