Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T20:04:28.232Z Has data issue: false hasContentIssue false
  • English
  • Français

Gravity–capillary waves in reduced models for wave–structure interactions

Published online by Cambridge University Press:  13 March 2020

Sean Jamshidi Open the ORCID record for Sean Jamshidi [Opens in a new window]  and
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]
Sean Jamshidi
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Philippe H. Trinh*
Affiliation:
Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with steady-state subcritical gravity–capillary waves that are produced by potential flow past a wave-making body. Such flows are characterised by two non-dimensional parameters: the Froude number, $F$, and the inverse Bond number, $T$. When the size of the wave-making body is formally small, there are two qualitatively different flow regimes and thus a single bifurcation curve in the $(F,T)$ plane. If, however, the size of the obstruction is of order one, then, in the limit $F,T\rightarrow 0$, Trinh & Chapman (J. Fluid Mech., vol. 724, 2013, pp. 392–424) have shown that the bifurcation curve widens into a band, within which there are four new flow regimes. Here, we use results from exponential asymptotics to show how, in this low-speed limit, the water-wave equations can be asymptotically reduced to a single differential equation, which we solve numerically to confirm one of the new classes of waves. The issue of numerically solving the full set of gravity–capillary equations for potential flow is also discussed.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Capillary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A18
Copyright
© The Author(s), 2020. Published by 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

Binder, B. J., Blyth, M. G. & McCue, S. W. 2013 Free-surface flow past arbitrary topography and an inverse approach for wave-free solutions. IMA J. Appl. Maths 78 (4), 685–696.CrossRefGoogle Scholar
Binder, B. J. & Vanden-Broeck, J. M. 2007 The effect of disturbances on the flows under a sluice gate and past an inclined plate. J. Fluid Mech. 576, 475–490.CrossRefGoogle Scholar
Chapman, S. J. & Vanden-Broeck, J. M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
Forbes, L. K. 1983 Free-surface flow over a semicircular obstruction, including the influence of gravity and surface tension. J. Fluid Mech. 127 (1), 283–297.CrossRefGoogle Scholar
Forbes, L. K. & Schwartz, L. W. 1982 Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114, 299–314.CrossRefGoogle Scholar
Grandison, S. & Vanden-Broeck, J. M. 2006 Truncation approximations for gravity–capillary free-surface flows. J. Engng Maths 54 (1), 89–97.CrossRefGoogle Scholar
King, A. C. & Bloor, M. G. 1987 Free-surface flow over a step. J. Fluid Mech. 182, 193–208.CrossRefGoogle Scholar
King, A. C. & Bloor, M. G. 1990 Free-surface flow of a stream obstructed by an arbitrary bed topography. Q. J. Mech. Appl. Maths 43 (1), 87–106.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Dover Publications.Google Scholar
Moreira, R. M. & Peregrine, D. H. 2010 Nonlinear interactions between a free-surface flow with surface tension and a submerged cylinder. J. Fluid Mech. 648, 485–507.CrossRefGoogle Scholar
Ogilvie, T. F.1968 Wave resistance: the low speed limit. Tech. Rep. Michigan University, Ann Arbor.Google Scholar
Părău, E. I. & Vanden-Broeck, J. M. 2002 Nonlinear two- and three-dimensional free surface flows due to moving disturbances. Eur. J. Mech. (B/Fluids) 21 (6), 643–656.CrossRefGoogle Scholar
Părău, E. I., Vanden-Broeck, J. M. & Cooker, M. J. 2007 Three-dimensional capillary–gravity waves generated by a moving disturbance. Phys. Fluids 19 (8), 082102.CrossRefGoogle Scholar
Părău, E. I., Vanden-Broeck, J. M. & Cooker, M. J. 2010 Time evolution of three-dimensional nonlinear gravity–capillary free-surface flows. J. Engng Maths 68 (3), 291–300.CrossRefGoogle Scholar
Rayleigh, Lord 1883 The form of standing waves on the surface of running water. Proc. Lond. Math. Soc. 1 (1), 69–78.CrossRefGoogle Scholar
Russell, J. S. 1845 Report on waves. In 14th Meeting of the British Association for the Advancement of Science, pp. 311–390. John Murray.Google Scholar
Scullen, D. C.1998 Accurate computation of steady nonlinear free-surface flows. PhD thesis, University of Adelaide.Google Scholar
Stoker, J. J. 1957 Water Waves. Interscience.Google Scholar
Trinh, P. H. 2016 A topological study of gravity free-surface waves generated by bluff bodies using the method of steepest descents. Proc. R. Soc. Lond. A 472, 20150833.CrossRefGoogle ScholarPubMed
Trinh, P. H. 2017 On reduced models for gravity waves generated by moving bodies. J. Fluid Mech. 813, 824–859.CrossRefGoogle Scholar
Trinh, P. H. & Chapman, S. J. 2013a New gravity–capillary waves at low speeds. Part 1: Linear theory. J. Fluid Mech. 724, 367–391.CrossRefGoogle Scholar
Trinh, P. H. & Chapman, S. J. 2013b New gravity–capillary waves at low speeds. Part 2: Nonlinear geometries. J. Fluid Mech. 724, 392–424.CrossRefGoogle Scholar
Vanden-Broeck, J. M. 2010 Gravity–Capillary Free-Surface Flows. Cambridge University Press.CrossRefGoogle Scholar
Vanden-Broeck, J. M. & Tuck, E. O. 1977 Computation of near-bow or stern flows using series expansion in the Froude number. In 2nd International Conference on Numerical Ship Hydrodynamics. University of California.Google Scholar