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Stability of waves on fluid of infinite depth with constant vorticity

Published online by Cambridge University Press:  17 February 2022

M.G. Blyth Open the ORCID record for M.G. Blyth [Opens in a new window]  and
E.I. Părău Open the ORCID record for E.I. Părău [Opens in a new window]
M.G. Blyth*
Affiliation:
School of Mathematics, University of East Anglia, NorwichNR4 7TJ, UK
E.I. Părău
Affiliation:
School of Mathematics, University of East Anglia, NorwichNR4 7TJ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., vol. 896, 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method. Both superharmonic and subharmonic perturbations are considered. Instability is shown to occur for any non-zero amplitude wave.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 936 , 10 April 2022 , A46
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Akers, B. & Nicholls, D.P. 2012 Spectral stability of deep two-dimensional gravity water waves: repeated eigenvalues. SIAM J. Appl. Maths 72 (2), 689–711.CrossRefGoogle Scholar
Benjamin, T.B. 1967 Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A 299 (1456), 59–76.Google Scholar
Benjamin, T.B & Feir, J.E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27 (03), 417–430.CrossRefGoogle Scholar
Blyth, M.G. & Părău, E.I. 2016 The stability of capillary waves on fluid sheets. J. Fluid Mech. 806, 5–34.CrossRefGoogle Scholar
Chen, B. & Saffman, P.G. 1985 Three-dimensional stability and bifurcation of capillary and gravity waves on deep water. Stud. Appl. Maths 77 (2), 125–147.CrossRefGoogle Scholar
Choi, W. 2009 Nonlinear surface waves interacting with a linear shear current. Math. Comput. Simul. 80 (1), 29–36.CrossRefGoogle Scholar
Crapper, G.D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (06), 532–540.CrossRefGoogle Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.CrossRefGoogle Scholar
Deconinck, B. & Trichtchenko, O. 2014 Stability of periodic gravity waves in the presence of surface tension. Eur. J. Mech. (B/Fluids) 46, 97–108.CrossRefGoogle Scholar
Dyachenko, A.I., Zakharov, V.E. & Kuznetsov, E.A. 1996 Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys. Rep. 22, 829–840.Google Scholar
Dyachenko, S.A. & Hur, V.M. 2019 a Stokes waves with constant vorticity: folds, gaps and fluid bubbles. J. Fluid Mech. 878, 502–521.CrossRefGoogle Scholar
Dyachenko, S.A. & Hur, V.M. 2019 b Stokes waves with constant vorticity: I. Numerical computation. Stud. Appl. Maths 142 (2), 162–189.CrossRefGoogle Scholar
Hogan, S.J. 1988 The superharmonic normal mode instabilities of nonlinear deep-water capillary waves. J. Fluid Mech. 190, 165–177.CrossRefGoogle Scholar
Hur, V.M. & Vanden-Broeck, J.-M. 2020 A new application of Crapper's exact solution to waves in constant vorticity flows. Eur. J. Mech. (B/Fluids) 83, 190–194.CrossRefGoogle Scholar
Hur, V.M. & Wheeler, M.H. 2020 Exact free surfaces in constant vorticity flows. J. Fluid Mech. 896, R1.CrossRefGoogle Scholar
Kapitula, T. & Promislow, K. 2013 Spectral and Dynamical Stability of Nonlinear Waves. Springer.CrossRefGoogle Scholar
Lighthill, M.J. 1965 Contributions to the theory of waves in non-linear dispersive systems. IMA J. Appl. Maths 1 (3), 269–306.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1978 The instabilities of gravity waves of finite amplitude in deep water. I. superharmonics. Proc. R. Soc. Lond. A 360 (1703), 471–488.Google Scholar
McLean, J.W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.CrossRefGoogle Scholar
Murashige, S. & Choi, W. 2020 Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping. J. Fluid Mech. 885, A41.CrossRefGoogle Scholar
Saffman, P.G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.CrossRefGoogle Scholar
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52 (9), 3047–3055.CrossRefGoogle Scholar
Tiron, R. & Choi, W. 2012 Linear stability of finite-amplitude capillary waves on water of infinite depth. J. Fluid Mech. 696, 402–422.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 1996 Periodic waves with constant vorticity in water of infinite depth. IMA J. Appl. Maths 56 (3), 207–217.CrossRefGoogle Scholar
Whitham, G.B. 1967 Non-linear dispersion of water waves. J. Fluid Mech. 27 (2), 399–412.CrossRefGoogle Scholar