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Free surface in two-dimensional potential flow: singularities, invariants and virtual fluid

Published online by Cambridge University Press:  28 November 2022

A.I. Dyachenko Open the ORCID record for A.I. Dyachenko [Opens in a new window] ,
S.A. Dyachenko Open the ORCID record for S.A. Dyachenko [Opens in a new window]  and
V.E. Zakharov
A.I. Dyachenko
Affiliation:
Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia Skolkovo Institute of Science and Technology, Moscow, Russia National Research University Higher School of Economics, Myasnitskaya 20, Moscow 101000, Russia
S.A. Dyachenko*
Affiliation:
Department of Mathematics, University at Buffalo, SUNY Buffalo, NY 14260, USA
V.E. Zakharov
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
*
†Email address for correspondence: [email protected]
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Abstract

We study a two-dimensional (2-D) potential flow of an ideal fluid with a free surface with decaying conditions at infinity. By using the conformal variables approach, we study a particular solution of the Euler equations having a pair of square-root branch points in the conformal plane, and find that the analytic continuation of the fluid complex potential and conformal map define a flow in the entire complex plane, excluding a vertical cut between the branch points. The expanded domain is called the ‘virtual’ fluid, and it contains a vortex sheet whose dynamics is equivalent to the equations of motion posed at the free surface. The equations of fluid motion are analytically continued to both sides of the vertical branch cut (the vortex sheet), and additional time invariants associated with the topology of the conformal plane and Kelvin's theorem for a virtual fluid are explored. We called them ‘winding’ and virtual circulation. This result can be generalized to a system of many cuts connecting many branch points, resulting in a pair of invariants for each pair of branch points. We develop an asymptotic theory that shows how a solution originating from a single vertical cut forms a singularity at the free surface in infinite time, the rate of singularity approach is double exponential and supersedes the previous result of the short branch cut theory with finite time singularity formation. The present work offers a new look at fluid dynamics with a free surface by unifying the problem of motion of vortex sheets, and the problem of 2-D water waves. A particularly interesting question that arises in this context is whether instabilities of the virtual vortex sheet are related to breaking of steep ocean waves when gravity effects are included.

JFM classification

Vortex Flows: Contour dynamics Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 952 , 10 December 2022 , A30
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Ablowitz, M.J., Kaup, D.J., Newell, A.C. & Segur, H. 1974 The inverse scattering transform-fourier analysis for nonlinear problems. Stud. Appl. Maths 53 (4), 249–315.CrossRefGoogle Scholar
Alpert, B., Greengard, L. & Hagström, T. 2000 Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal. 37 (4), 1138–1164.CrossRefGoogle Scholar
Ambrose, D.M., Camassa, R., Marzuola, J.L., McLaughlin, R.M., Robinson, Q. & Wilkening, J. 2022 Numerical algorithms for water waves with background flow over obstacles and topography. Adv. Comput. Math. 48 (4), 1–62.CrossRefGoogle Scholar
Arsénio, D., Dormy, E. & Lacave, C. 2020 The vortex method for two-dimensional ideal flows in exterior domains. SIAM J. Math. Anal. 52 (4), 3881–3961.CrossRefGoogle Scholar
Baker, G.R. & Xie, C. 2011 Singularities in the complex physical plane for deep water waves. J. Fluid Mech. 685, 83–116.CrossRefGoogle Scholar
Castro, A., Córboda, D., Fefferman, C., Gancedo, F. & Gómez-Serrano, J. 2013 Finite time singularities for the free boundary incompressible euler equations. Ann. Math. 178 (3), 1061–1134.CrossRefGoogle Scholar
Crapper, G.D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540.CrossRefGoogle Scholar
Crowdy, D.G. 2000 A new approach to free surface euler flows with capillarity. Stud. Appl. Maths 105 (1), 35–58.CrossRefGoogle Scholar
Dyachenko, A.I. 2001 On the dynamics of an ideal fluid with a free surface. Dokl. Math. 63, 115–117.Google Scholar
Dyachenko, A.I., Dyachenko, S.A., Lushnikov, P.M. & Zakharov, V.E. 2019 Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion. J. Fluid Mech. 874, 891–925.CrossRefGoogle Scholar
Dyachenko, A.I., Dyachenko, S.A., Lushnikov, P.M. & Zakharov, V.E. 2021 Short branch cut approximation in two-dimensional hydrodynamics with free surface. Proc. R. Soc. A 477 (2249), 20200811.CrossRefGoogle Scholar
Dyachenko, A.I., Kuznetsov, E.A., Spector, M.D. & Zakharov, V.E. 1996 Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1), 73–79.CrossRefGoogle Scholar
Dyachenko, A.I., Lvov, Y.V. & Zakharov, V.E. 1995 Five-wave interaction on the surface of deep fluid. Physica D 87 (1–4), 233–261.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 (10), 829–840.Google Scholar
Dyachenko, S.A., Lushnikov, P.M. & Korotkevich, A.O. 2014 Complex singularity of a stokes wave. JETP Lett. 98 (11), 675–679.CrossRefGoogle Scholar
Dyachenko, S.A., Lushnikov, P.M. & Korotkevich, A.O. 2016 Branch cuts of stokes wave on deep water. Part 1. Numerical solution and Padé approximation. Stud. Appl. Maths 137 (4), 419–472.CrossRefGoogle Scholar
Dyachenko, S.A. & Mikyoung Hur, V. 2019 Stokes waves with constant vorticity: folds, gaps and fluid bubbles. J. Fluid Mech. 878, 502–521.CrossRefGoogle Scholar
Dyachenko, S.A. & Newell, A.C. 2016 Whitecapping. Stud. Appl. Maths 137 (2), 199–213.CrossRefGoogle Scholar
Galin, L.A. 1945 Unsteady filtration with a free surface. C. R. L'Acad. Sci. l'URSS 47, 246–249.Google Scholar
Gardner, C.S., Greene, J.M., Kruskal, M.D. & Miura, R.M. 1967 Method for solving the Korteweg-Devries equation. Phys. Rev. Lett. 19 (19), 1095.CrossRefGoogle Scholar
Grant, M.A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59 (2), 257–262.CrossRefGoogle Scholar
Haziot, S.V., Mikyoung Hur, V., Strauss, W., Toland, J.F., Wahlén, E., Walsh, S. & Wheeler, M.H. 2022 Traveling water waves – the ebb and flow of two centuries. Quart. Appl. Math. 80 (2), 317–401.CrossRefGoogle Scholar
Karabut, E.A. & Zhuravleva, E.N. 2014 Unsteady flows with a zero acceleration on the free boundary. J. Fluid Mech. 754, 308–331.CrossRefGoogle Scholar
Karabut, E.A., Zhuravleva, E.N. & Zubarev, N.M. 2020 Application of transport equations for constructing exact solutions for the problem of motion of a fluid with a free boundary. J. Fluid Mech. 890, A13.CrossRefGoogle Scholar
Kolmogorov, A.N. 1954 On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. In Dokl. Akad. Nauk SSSR 98 (527), 2–3.Google Scholar
Kuznetsov, E.A., Spector, M.D. & Zakharov, V.E. 1993 Surface singularities of ideal fluid. Phys. Lett. A 182 (4–6), 387–393.CrossRefGoogle Scholar
Liu, J.-G. & Pego, R.L. 2021 In search of local singularities in ideal potential flows with free surface. arXiv:2108.00445.Google Scholar
Longuet-Higgins, M.S. 1972 A class of exact, time-dependent, free-surface flows. J. Fluid Mech. 55 (3), 529–543.CrossRefGoogle Scholar
Lushnikov, P.M. 2016 Structure and location of branch point singularities for stokes waves on deep water. J. Fluid Mech. 800, 557–594.CrossRefGoogle Scholar
Lushnikov, P.M. & Zakharov, V.E. 2021 Poles and branch cuts in free surface hydrodynamics. Water Waves 3, 251–266.CrossRefGoogle Scholar
Mikyoung Hur, V. & Wheeler, M.H. 2020 Exact free surfaces in constant vorticity flows. J. Fluid Mech. 896, R1.Google Scholar
Ovsyannikov, L.V. 1973 Dynamika sploshnoi sredy, lavrentiev institute of hydrodynamics. Sib. Branch Acad. Sci. USSR 15, 104.Google Scholar
Polubarinova-Kochina, P.Y. 1945 On the displacement of the oil-bearing contour. C. R. L'Acad. Sci. l'URSS 47, 250–254.Google Scholar
Shabat, A. & Zakharov, V. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34 (1), 62.Google Scholar
Stokes, G.G. 1880 Mathematical and Physical Papers, vol. 1. Cambridge University Press.Google Scholar
Tanveer, S. 1991 Singularities in water waves and Rayleigh–Taylor instability. Proc. R. Soc. Lond. A 435 (1893), 137–158.Google Scholar
Tanveer, S. 1993 Singularities in the classical Rayleigh–Taylor flow: formation and subsequent motion. Proc. R. Soc. Lond. A 441 (1913), 501–525.Google Scholar
Wilkening, J. 2021 Traveling-standing water waves. Fluids 6 (5), 187.CrossRefGoogle Scholar
Wilkening, J. & Vasan, V. 2015 Comparison of five methods of computing the dirichlet-neumann operator for the water wave problem. Contemp. Maths 635, 175–210.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. 9 (2), 190–194.CrossRefGoogle Scholar
Zakharov, V.E. 2020 Integration of a deep fluid equation with a free surface. Theor. Math. Phys. 202 (3), 285–294.CrossRefGoogle Scholar
Zakharov, V.E. & Dyachenko, A.I. 1996 High-jacobian approximation in the free surface dynamics of an ideal fluid. Physica D 98 (2), 652–664.CrossRefGoogle Scholar
Zakharov, V.E., Dyachenko, A.I. & Vasilyev, O.A. 2002 New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface. Eur. J. Mech. B/Fluids 21 (3), 283–291.CrossRefGoogle Scholar
Zakharov, V.E. & Faddeev, L.D. 1971 Korteweg–de vries equation: a completely integrable hamiltonian system. Funktsional'nyi Analiz i ego Prilozheniya 5 (4), 18–27.Google Scholar
Zakharov, V.E. & Shabat, A.B. 1974 A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funktsional'nyi Analiz i ego Prilozheniya 8 (3), 43–53.Google Scholar
Zakharov, V.E. & Shabat, A.B. 1979 Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Applics. 13 (3), 166–174.CrossRefGoogle Scholar