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Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

Published online by Cambridge University Press:  07 January 2020

Sunao Murashige Open the ORCID record for Sunao Murashige [Opens in a new window]  and
Wooyoung Choi Open the ORCID record for Wooyoung Choi [Opens in a new window]
Sunao Murashige*
Affiliation:
Department of Mathematics and Informatics, Ibaraki University, Mito, Ibaraki, 310-8512, Japan
Wooyoung Choi
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
*
†Email address for correspondence: sunao.murashige.sci@vc.ibaraki.ac.jp
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Abstract

This paper describes linear stability analysis of the two-dimensional steady motion of periodic deep-water waves with symmetric non-overhanging profiles propagating on a linear shear current, namely a vertically sheared current with constant vorticity. In order to investigate numerically with high accuracy the stability of large-amplitude waves, we adopt a formulation using conformal mapping, in which the time-varying water surface is always mapped onto the real axis of a complex plane. This formulation allows us to apply numerical methods developed for large-amplitude irrotational waves without a shear current directly to the present problem, and reduces the linear stability problem to a generalized eigenvalue problem. Numerical solutions describe both super- and sub-harmonic instabilities of the periodic waves for a wide range of wave amplitudes and clarify how the behaviours of dominant eigenvalues change with the strength of the shear current. In particular, it is shown that, even in the presence of a linear shear current, the steady periodic waves lose stability due to superharmonic disturbances at the wave amplitude where the wave energy attains an extremum, similarly to the case of irrotational waves without a shear current. It is also found that re-stabilization with an increase in wave amplitude characterizes subharmonic instability for weak shear currents, but the re-stabilization disappears for strong shear currents.

JFM classification

Instability: Instability Wakes/Jets: Shear layers Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A41
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Banner, M. L. & Phillips, O. M. 1974 On the incipient breaking of small scale waves. J. Fluid Mech. 65, 647–656.CrossRefGoogle Scholar
Banner, M. L. & Song, J. B. 2002 On determining the onset and strength of breaking for deep water waves. Part II: influence of wind forcing and surface shear. J. Phys. Oceanogr. 32, 2559–2570.CrossRefGoogle Scholar
Benjamin, T. B. 1962 The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97–116.CrossRefGoogle Scholar
Branger, H., Ramamonjiarisoa, A. & Kharif, C. 1986 On the subharmonic instability of very steep deep water gravity wave. J. Méc. Théor. Appl. 5, 515–550.Google Scholar
Chalikov, D. & Sheinin, D. 2005 Modeling extreme waves based on equations of potential flow with a free surface. J. Comput. Phys. 210, 247–273.CrossRefGoogle Scholar
Chen, B. & Saffman, P. G. 1985 Three-dimensional stability and bifurcation of capillary and gravity waves on deep water. Stud. Appl. Maths 72, 125–147.CrossRefGoogle Scholar
Choi, W. 2003 Strongly nonlinear long gravity waves in uniform shear flows. Phy. Rev. E 68, 026305.CrossRefGoogle ScholarPubMed
Choi, W. 2009 Nonlinear surface waves interacting with a linear shear current. Maths Comput. Simul. 80, 29–36.CrossRefGoogle Scholar
Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. 125, 756–760.CrossRefGoogle Scholar
Dosaev, A. S., Troitskaya, Y. I. & Shishina, M. I. 2017 Simulation of surface gravity waves in the Dyachenko variables on the free boundary of flow with constant vorticity. Fluid Dyn. 52, 58–70.CrossRefGoogle Scholar
Dyachenko, A. L. & Hur, V. M. 2019a Stokes waves with constant vorticity: folds, gaps and fluid bubbles. J. Fluid Mech. 878, 502–521.CrossRefGoogle Scholar
Dyachenko, A. L. & Hur, V. M. 2019b Stokes waves with constant vorticity: I. Numerical computation. Stud. Appl. Maths 142, 162–189.CrossRefGoogle Scholar
Dyachenko, A. L., Lushnikov, P. M. & Korotkevich, A. O. 2016 Branch cuts of Stokes wave on deep water. Part I: numerical solution and Padé approximation. Stud. Appl. Maths 137, 419–472.CrossRefGoogle Scholar
Dyachenko, A. L., Zakharov, V. E. & Kuznetsov, E. A. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 73–79.CrossRefGoogle Scholar
Francius, M. & Kharif, C. 2017 Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity. J. Fluid Mech. 830, 631–659.CrossRefGoogle Scholar
Freeman, N. C. & Johnson, R. S. 1970 Shallow water waves on shear flows. J. Fluid Mech. 42, 401–409.CrossRefGoogle Scholar
Hsu, H.-C., Francius, M., Montalvo, P. & Kharif, C. 2016 Gravity-capillary waves in finite depth on flows of constant vorticity. Proc. R. Soc. Lond. A 472, 20160363.CrossRefGoogle ScholarPubMed
King, F. W. 2009 Hilbert Transforms. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1978a The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics. Proc. R. Soc. Lond. A 360, 471–488.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1978b The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A 360, 489–505.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1986 Bifurcation and instability in gravity waves. Proc. R. Soc. Lond. A 403, 167–187.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching and analytic continuation. J. Fluid Mech. 85, 769–786.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Tanaka, M. 1997 On the crest instabilities of steep surface waves. J. Fluid Mech. 336, 51–68.CrossRefGoogle Scholar
Lushnikov, P. M., Dyachenko, A. L. & Silantyev, D. A. 2017 New conformal mapping for adaptive resolving of the complex singularities of Stokes wave. Proc. R. Soc. Lond. A 473, 20170198.CrossRefGoogle ScholarPubMed
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406, 115–125.CrossRefGoogle Scholar
McLean, J. W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.CrossRefGoogle Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Dover.CrossRefGoogle 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
Moreira, R. M. & Chacaltana, J. T. A. 2015 Vorticity effects on nonlinear wave-current interactions in deep water. J. Fluid Mech. 778, 314–334.CrossRefGoogle Scholar
Murashige, S. & Choi, W. 2017 A numerical study on parasitic capillary waves using unsteady conformal mapping. J. Comput. Phys. 328, 234–257.CrossRefGoogle Scholar
Murashige, S. & Tanaka, K. 2015 A new method of convergence acceleration of series expansion for analytic functions in the complex domain. Japan J. Indust. Appl. Maths 32, 95–117.CrossRefGoogle Scholar
Okamura, M. & Oikawa, M. 1989 The linear stability of finite amplitude surface waves on a linear shearing flow. J. Phys. Soc. Japan 58, 2386–2396.CrossRefGoogle Scholar
Ovshannikov, L. 1974 To the shallow water theory foundation. Arch. Mech. 26, 407–422.Google Scholar
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.CrossRefGoogle Scholar
Ruban, V. P. 2008 Explicit equations for two-dimensional water waves with constant vorticity. Phys. Rev. E 77, 037302.Google ScholarPubMed
Saffman, P. G. 1985 The superharmonic instability of finite-amplitude water waves. J. Fluid Mech. 159, 169–174.CrossRefGoogle Scholar
Sato, N. & Yamada, M. 2019 Superharmonic instability of nonlinear traveling wave solutions in Hamiltonian systems. J. Fluid Mech. 876, 896–911.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
Tanaka, M. 1983 The stability of steep gravity waves. J. Phys. Soc. Japan 52, 3047–3055.CrossRefGoogle Scholar
Tanaka, M. 1985 The stability of steep gravity waves. Part 2. J. Fluid Mech. 156, 281–289.CrossRefGoogle Scholar
Tanaka, M. 1986 The stability of solitary waves. Phys. Fluids 29, 650–655.CrossRefGoogle Scholar
Teles da Silva, A. F. & Peregrine, D. H. 1988 Steep, steady waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302.CrossRefGoogle Scholar
Thomas, G. P. 1981 Wave-current interactions: an experimental and numerical study. Part 1. Linear waves. J. Fluid Mech. 110, 457–474.CrossRefGoogle Scholar
Thomas, G. P. 1990 Wave-current interactions: an experimental and numerical study. Part 2. Nonlinear waves. J. Fluid Mech. 216, 505–536.CrossRefGoogle Scholar
Thomas, R., Kharif, C. & Manna, M. 2012 A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity. Phys. Fluids 24, 127102.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, 207–217.CrossRefGoogle Scholar
Wahlén, E. 2007 A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315.CrossRefGoogle Scholar
Yao, A. & Wu, C. H. 2005 Incipient breaking of unsteady waves on sheared currents. Phys. Fluids 17, 082104.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.Google Scholar
Zhang, J. & Melville, W. K. 1987 Three-dimensional instabilities of nonlinear gravity–capillary waves. J. Fluid Mech. 174, 187–208.CrossRefGoogle Scholar