Abstract
Waves in deep water with constant vorticity in the region bounded by the free surface and the infinitely deep plane bottom are considered. Using the conformal variables and the conformal transform technique, a system of exact integro-differential equations solved relative to the derivatives with respect to time is derived and the equivalent system of equations is obtained in the Dyachenko variables. The efficiency of using the obtained system in the Dyachenko variables for investigating surface wave dynamics on the current of infinite depth with constant vorticity is demonstrated with reference to numerical experiments.
Similar content being viewed by others
References
T.B. Benjamin, “The Solitary Wave on a Stream with an Arbitrary Distribution of Vorticity,” J. Fluid Mech. 12, 97–116 (1962).
N.C. Freeman and R.S. Johnson, “Shallow Water Waves on Shear Flows,” J. Fluid Mech. 42, 401–409 (1970).
C.S. Yih, “SurfaceWaves in Flowing Water,” J. Fluid Mech. 51, 209–220 (1972).
R.S. Johnson, “Ring Waves on the Surface of Shear Flows: a Linear and Nonlinear Theory,” J. Fluid Mech. 215, 145–160 (1990).
K.A. Belibassakis, “A Coupled-Mode Model for the Scattering of Water Waves by Shearing Currents in Variable Bathymetry,” J. Fluid Mech. 578, 413–434 (2007).
V.A. Miroshnikov, “The Boussinesq–Rayleigh Approximation for Rotational Solitary Waves on Shallow Water with Uniform Vorticity,” J. Fluid Mech. 456, 1–32 (2002).
W. Choi, “Strongly Nonlinear Long Gravity Waves in Uniform Shear Flows,” Phys. Rev. E. 68, 1–7 (2003).
A.F. Teles da Silva and D.H. Peregrine, “Steep, Steady Surface Waves on Water of Finite Depth with Constant Vorticity,” J. Fluid Mech. 195, 281–302 (1988).
D.I. Pullin and R.H.J. Grimshaw, “Finite-Amplitude Solitary Waves at the Interface Between Two Homogeneous Fluids,” Phys. Fluids 31, No. 12, 3550–3559 (1988).
J.-M. Vanden-Broeck, “Steep Solitary Waves in Water of Finite Depth with Constant Vorticity,” J. Fluid Mech. 274, 339–348 (1994).
R.S. Johnson, “On the Modulation of Water Waves on Shear Flows,” Proc. R. Soc. Lond. A. 347, 537–546 (1976).
J.C. Li, W.H. Hui, and M.A. Donelan, “Effects of Velocity Shear on the Stability of Surface Deep Water Wave Trains,” in: K. Horikawa and H. Maruo (Eds.) Nonlinear Water Waves (Springer, 1987), 213–220.
M. Oikawa, K. Chow, and D.J. Benney, “The Propagation of NonlinearWave Packets in a Shear Flow with a Free Surface,” Stud. Appl. Math. 76, 69–92 (1987).
R. Thomas, C. Kharif, and M. Manna, “A Nonlinear Schrödinger Equation forWater Waves on Finite Depth with Constant Vorticity,” Phys. Fluids 24, 127102 (2012).
A.I. Dyachenko, V.E. Zakharov, and E.A. Kuznetsov, “Nonlinear Dynamics of the Free Surface of an Ideal Fluid,” Fizika Plazmy 22, No. 10, 916–928 (1996).
D. Chalikov and D. Sheinin, “Numerical Modeling of Surface Waves Based on Principal Equations of Potential Wave Dynamics,” Technical Note (NOAA/NCEP/OMB, 1996). 54 p.
D. Chalikov and D. Sheinin, “Direct Modeling of One-Dimensional Nonlinear Potential Waves,” Advances in Fluid Mechanics 17, 207–258 (1998).
V.E. Zakharov, A.I. Dyachenko, and O.A. Vasilyev, “New Method for Numerical Simulation of a Nonstationary Potential Flow of Incompressible Fluid with a Free Surface,” Eur. J. Mech. B-Fluids 21, No. 3, 283–291 (2002).
A.I. Dyachenko and V.E. Zakharov, “Modulation Instability of StokesWave→FreakWave,” Pis’ma v ZhETF 81, No. 6, 318–322 (2005).
V.E. Zakharov, A.I. Dyachenko, and A.O. Prokofiev, “FreakWaves as Nonlinear Stage of StokesWaveModulation Instability,” Eur. J. Mech. B-Fluids 25, No. 5, 677–692 (2006).
A.I. Dyachenko, E.A. Kuznetsov, M.D. Spector, and V.E. Zakharov, “Analytical Description of the Free Surface Dynamics of an Ideal Fluid (Canonical Formalism and Conformal Mapping),” Phys. Lett. A. 221, Nos. 1–2, 73–79 (1996).
L.V. Ovsyannikov, “Plane Problem of the Time-Dependent Motion of Fluid with Free Boundaries,” Continuum Dynamics, 8, 22–26 (1971).
V.P. Ruban, “Explicit Equations for Two-Dimensional Water Waves with Constant Vorticity,” Phys. Rev. E. 77, 037302 (2008).
J.A. Simmen and P.G. Saffman, “Steady Deep-Water Waves on a Linear Shear Current,” Stud. Appl. Maths. 73, 35–57 (1985).
V.I. Shrira, “NonlinearWaves on the Surface of a Fluid Layer with Constant Vorticity,” Dokl. Akad. Nauk USSR, 286, No. 6, 1332–1336 (1986).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.S. Dosaev, Yu.I. Troitskaya, M.I. Shishina, 2017, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2017, Vol. 52, No. 1, pp. 62–73.
Rights and permissions
About this article
Cite this article
Dosaev, A.S., Troitskaya, Y.I. & Shishina, M.I. Simulation of surface gravity waves in the Dyachenko variables on the free boundary of flow with constant vorticity. Fluid Dyn 52, 58–70 (2017). https://doi.org/10.1134/S0015462817010069
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0015462817010069