Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T14:53:44.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Effect of vorticity on steady water waves

Published online by Cambridge University Press:  11 July 2008

JOY KO  and
WALTER STRAUSS
JOY KO
Affiliation:
Mathematics Department, Brown University, Box 1917, Providence, RI 02912, USA
WALTER STRAUSS
Affiliation:
Mathematics Department, Brown University, Box 1917, Providence, RI 02912, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Two-dimensional, finite-depth periodic steady water waves with variable vorticity ω=γ(ψ) and large amplitude a are computed for a large number of cases. In particular, the effect of a shear layer at the top, the middle or the bottom is considered. The maximum amplitude amax varies monotonically with the vorticity function γ(ċ). It is increasing if the stagnation point is at the crest, and is decreasing if the stagnation point is in the interior of the fluid or on the bottom. Relationships between the amplitude, hydraulic head, depth and mass flux are investigated.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 608 , 10 August 2008 , pp. 197 - 215
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Benjamin, T. B. 1962 The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97–116.CrossRefGoogle Scholar
Constantin, A. & Escher, J. 2004 Symmetry of steady deep-water waves with vorticity. Eur. J. Appl. Maths 15, 755–768.CrossRefGoogle Scholar
Constantin, A. & Strauss, W 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Maths 57, 481–527.CrossRefGoogle Scholar
Constantin, A. & Strauss, W. 2007 Rotational steady water waves near stagnation. Phil. Trans. R. Soc. Lond. 365, 2227–2239.Google ScholarPubMed
Dalrymple, R. A. 1977 A numerical model for periodic finite amplitude waves on a rotational fluid. J. Comput. Phys. 24, 29–42.CrossRefGoogle Scholar
Dubreil-Jacotin, M.-L. 1934 Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie. J. Math. Pures Appl. 13, 217–291.Google Scholar
Hur, V. M. 2007 Symmetry of steady periodic surface water waves with vorticity. Phil. Trans. R. Soc. Lond. 365, 2203–2214.Google Scholar
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.CrossRefGoogle Scholar
Ko, J. & Strauss, W. 2008 Large-amplitude steady rotational water waves. Eur. J. Mech. B Fluids 27, 96–109.CrossRefGoogle Scholar
Lighthill, J. 2001 Waves in Fluids. Cambridge University Press.Google Scholar
Milne-Thomson, L. M. 1996 Theoretical Hydrodynamics. Dover.Google Scholar
Miroshnikov, V. A. 2002 The Boussinesq–Rayleigh approximation for rotational solitary waves on shallow water with uniform vorticity. J. Fluid Mech. 456, 1–32.CrossRefGoogle Scholar
Okamoto, H. & Shóji, M. 2001 The Mathematical Theory of Permanent Progressive Water-Waves. World Scientific.CrossRefGoogle Scholar
Sha, H. & Vanden-Broeck, J.-M. 1995 Solitary waves on water of finite depth with a surface or bottom shear layer. Phys. Fluids 7, 1048–1055.CrossRefGoogle Scholar
Simmen, J. A. & Saffman, P. G. 1985 Steady deep-water waves on a linear shear current. Stud. Appl. Maths 73, 35–57.CrossRefGoogle Scholar
Swan, C., Cummins, I. P. & James, R. L. 2001 An experimental study of two-dimensional surface water waves propagating on depth-varying currents. Part 1. Regular waves. J. Fluid Mech. 428, 273–304.CrossRefGoogle Scholar
Teles da Silva, A. F. & Peregrine, D. H. 1988 Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302.CrossRefGoogle Scholar
Thomas, G. P. 1990 Wavecurrent interactions: an experimental and numerical study. Part 2. Nonlinear waves. J. Fluid Mech. 216, 505–536.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 1994 Steep solitary waves in water of finite depth with constant vorticity. J. Fluid Mech. 274, 339–348.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 1995 New families of steep solitary waves in water of finite depth with constant vorticity. Eur. J. Mech. B Fluids 14, 761–774.Google Scholar
Varvaruca, E. 2008 On some properties of travelling water waves with vorticity. SIAM J. Math. Anal. 39, 1686–1692.CrossRefGoogle Scholar