Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions

F. Dias; A. I. Dyachenko; V. E. Zakharov

doi:10.1016/j.physleta.2007.09.027

Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions

F. Dias, A. I. Dyachenko, V. E. Zakharov

Research output: Contribution to journal › Article › peer-review

108 Scopus citations

Abstract

Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.

Original language	English (US)
Pages (from-to)	1297-1302
Number of pages	6
Journal	Physics Letters, Section A: General, Atomic and Solid State Physics
Volume	372
Issue number	8
DOIs	https://doi.org/10.1016/j.physleta.2007.09.027
State	Published - Feb 18 2008
Externally published	Yes

Keywords

Free surface
Navier-Stokes equations
Potential flow
Viscosity

ASJC Scopus subject areas

Physics and Astronomy(all)

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10.1016/j.physleta.2007.09.027

Cite this

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title = "Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions",

abstract = "Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schr{\"o}dinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.",

keywords = "Free surface, Navier-Stokes equations, Potential flow, Viscosity",

author = "F. Dias and Dyachenko, {A. I.} and Zakharov, {V. E.}",

note = "Funding Information: The authors are grateful to the referees for their valuable comments and suggestions. A.I. Dyachenko is grateful for support from Centre National de la Recherche Scientifique (CNRS). This work was also supported by the US Army Corps of Engineers Grant W912HZ-05-P-0351, by ONR Grant N00014-03-1-0648, NSF Grant DMS 0404577, RFBR Grant 06-01-00665, the Program “Fundamental Problems in Nonlinear Dynamics” from the RAS Presidium, and Grant “Leading Scientific Schools of Russia”. F. Dias thanks E.O. Tuck and L. Lazauskas for interesting discussions and references on the topic of viscous damping and water waves, as well as the staff of the library of the University of Adelaide, Australia, for providing access to the first edition of Lamb's book on hydrodynamics.",

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N1 - Funding Information: The authors are grateful to the referees for their valuable comments and suggestions. A.I. Dyachenko is grateful for support from Centre National de la Recherche Scientifique (CNRS). This work was also supported by the US Army Corps of Engineers Grant W912HZ-05-P-0351, by ONR Grant N00014-03-1-0648, NSF Grant DMS 0404577, RFBR Grant 06-01-00665, the Program “Fundamental Problems in Nonlinear Dynamics” from the RAS Presidium, and Grant “Leading Scientific Schools of Russia”. F. Dias thanks E.O. Tuck and L. Lazauskas for interesting discussions and references on the topic of viscous damping and water waves, as well as the staff of the library of the University of Adelaide, Australia, for providing access to the first edition of Lamb's book on hydrodynamics.

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N2 - Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.

AB - Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.

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Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions

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Keywords

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