Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping)

“…It is based on the equations obtained in [4,5]. To our opinion this approach combines the advantages of the existing approaches, being free of these disadvantages.…”

Section: Introductionmentioning

confidence: 99%

New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface

Захаров

Dyachenko

Vasilyev

2002

European Journal of Mechanics - B/Fluids

162

151

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“…This formulation is based on the conformal mapping technique [22,23,40] that has been successfully used to compute numerically periodic gravity waves in deep water [15] and in finite depth [28,30]. More recently, this approach was adapted also to periodic capillary-gravity waves in deep water [35].…”

Section: Introductionmentioning

confidence: 99%

A plethora of generalised solitary gravity–capillary water waves

Clamond¹,

Dutykh

Durán

2015

J. Fluid Mech.

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The present study describes, first, an efficient algorithm for computing solutions in terms of capillary–gravity solitary waves of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.

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“…The derivation of the equations used here has its origins in Zakharov's Hamiltonian theory of water waves [7,16]; see also [13], but since we do not use variational methods for their analysis it is not appropriate to review that material here. Instead we rely on the description of the problem for a heavy membrane in [9,11] and for a weightless membrane in [4,12].…”

Section: Physical Problemmentioning

confidence: 99%

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

Baldi¹,

Toland²

2010

Interfaces Free Bound.

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The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic extensible rod, and the fluid beneath is supposed to be in steady two-dimensional irrotational motion under gravity. When the wavelength has been normalized to be 2π , and when gravity and the density of the undeformed membrane are prescribed, there are two free parameters in the problem: the speed of the wave and the drift velocity of the membrane.It is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. When the linearized problem has only one normalized solution, it is shown that the full nonlinear problem has a sheet of solutions consisting of a family of curves bifurcating from simple eigenvalues. Here one of the problem's parameters is used to index a family of bifurcation problems in which the other is the bifurcation parameter.When the linearized problem has two solutions, with wave numbers k and l such that max{k, l}/min{k, l} / ∈ Z, it is shown that there are three two-dimensional sheets of bifurcating solutions. One consists of "special" solutions with minimal period 2π/k; another consists of "special" solutions with minimal period 2π/ l; and the third, apart from those on the curves where it intersects the "special" sheets, consists of "general" solutions with minimal period 2π .The two sheets of "special" solutions are rather similar to those that occur when the linearized problem has only one solution. However, points where the first sheet or the second sheet intersects the third sheet are period-multiplying (or symmetry-breaking) secondary bifurcation points on primary branches of "special" solutions. This phenomenon is analogous to that of Wilton ripples, which arises in the classical water-wave problem when the surface tension has special values. In the case of Wilton ripples, the coefficient of surface tension and the wave speed are the problem's two parameters. In the present context, there are two speed parameters, meaning that the membrane elasticity does not need to be highly specified for this symmetry-breaking phenomenon to occur.

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Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping)

Cited by 192 publications

References 8 publications

New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface

New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface

A plethora of generalised solitary gravity–capillary water waves

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

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