Geometric Aspects of Spatially Periodic Interfacial Waves

Dias, Frédéric; Bridges, Thomas J.

doi:10.1002/sapm199493293

Cited by 14 publications

(21 citation statements)

References 37 publications

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“…As pointed out by Dias and Bridges,37 there are no mathematical rigorous published works on the stability of finite amplitude interfacial standing wave solutions with surface tension, even though it is for the case of single-phase flow of ͉A͉ = 1, as far as we know.…”

Section: B Stable and Unstable Finite Amplitude Standing Wave Solutimentioning

confidence: 99%

Vortex sheet motion in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities with surface tension

Matsuoka

2009

Physics of Fluids

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Effect of compressibility on the Rayleigh-Taylor and Richtmyer-Meshkov instability induced nonlinear structure at two fluid interface Phys. Plasmas 16, 032303 (2009); 10.1063/1.3074789 Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified cylindrical shellsMotion of a planar interface in incompressible Richtmyer-Meshkov ͑RM͒ and Rayleigh-Taylor ͑RT͒ instabilities with surface tension is investigated numerically by using the boundary integral method. It is shown that when the Atwood number is small, an interface rolls up without regularization of the interfacial velocity. A phenomenon known as "pinching" in the physics of drops is observed in the final stage of calculations at various Atwood numbers and surface tension coefficients, and it is shown that this phenomenon is caused by a vortex dipole induced on the interface. It is also shown that when the surface tension coefficient is large, finite amplitude standing wave solutions exist for the RM instability. This standing wave solution is investigated in detail by nonlinear stability analysis. When gravity is taken into account ͑RT instability͒, linearly stable but nonlinearly unstable motion can occur under a critical condition that the frequency of the linear dispersion relation in the system is equal to zero. Further, it is shown that the growth rate of bubbles and spikes under this critical motion is neither of the exponential type nor of the power law type at both the linear stage and the asymptotic stage.

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Section: B Stable and Unstable Finite Amplitude Standing Wave Solutimentioning

confidence: 99%

Vortex sheet motion in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities with surface tension

Matsuoka

2009

Physics of Fluids

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“…In complete correspondence with (II.10), the envelope equation (II.17) can be recasted in the following variational problem: 19) where…”

Section: Basic Equationsmentioning

confidence: 99%

Deep-water internal solitary waves near critical density ratio

Агафонцев

Dias

Кузнецов

2007

Physica D: Nonlinear Phenomena

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Bifurcations of solitary waves propagating along the interface between two ideal fluids are considered. The study is based on a Hamiltonian approach. It concentrates on values of the density ratio close to a critical one, where the supercritical bifurcation changes to the subcritical one. As the solitary wave velocity approaches the minimum phase velocity of linear interfacial waves (the bifurcation point), the solitary wave solutions transform into envelope solitons. In order to describe their behavior and bifurcations, a generalized nonlinear Schrödinger equation describing the behavior of solitons and their bifurcations is derived. In comparison with the classical NLS equation this equation takes into account three additional nonlinear terms: the so-called Lifshitz term responsible for pulse steepening, a nonlocal term analogous to that first found by Dysthe for gravity waves and the six-wave interaction term. We study both analytically and numerically two solitary wave families of this equation for values of the density ratio ρ that are both above and below the critical density ratio ρcr. For ρ > ρcr, the soliton solution can be found explicitly at the bifurcation point. The maximum amplitude of such a soliton is proportional to √ ρ − ρcr, and at large distances the soliton amplitude decays algebraically. A stability analysis shows that solitons below the critical ratio are stable in the Lyapunov sense in the wide range of soliton parameters. Above the critical density ratio solitons are shown to be unstable with respect to finite perturbations.

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“…Moreover, all canonical transformations are exact and genuinely infinite dimensional, so that in principle no information is lost. This contrasts with the work of Dias & Bridges [7], who also use a Hamiltonian formulation to study standing and travelling waves in fluid interfaces, but their analysis is based upon a truncation to a finite number of Fourier modes. Our approximations to Euler flow are within the weakly nonlinear regime, and are effectively the approximation of a Hamiltonian by Taylor polynomials about a stationary point.…”

Section: Introductionmentioning

confidence: 93%

“…The action-frequency map is nondegenerate; writing I = (I 1 , I 9 ), one finds that its derivative is which is nonzero for all 0 ≤ ρ 1 < ρ ≤ 1. The corresponding fluid interface is shown in Figure 2; the solution is excited in twelve modes (modes 1,2,3,7,8,9,10,11,17,18,19,27) in physical coordinates.…”

Section: Theorem 8 the Hamiltonian Flow Of (39) Restricted To A Resomentioning

confidence: 99%

“…Miles [6] sets the Kelvin Helmholtz problem in a Lagrangian framework, recovering Thorpe's expression for the amplitude/frequency dependence of one-mode solutions. In more recent work, Dias & Bridges [7] examined standing and travelling waves of the Kelvin Helmholtz problem in a Hamiltonian setting, and discussed approximations to free interface motions which involve a low number of spatial modes. The present paper examines the class of standing wave solutions in much greater detail, admitting arbitrary density differences and having no a priori truncation of spatial modes.…”

Section: Introductionmentioning

confidence: 99%

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Normal forms for wave motion in fluid interfaces

Craig

Groves

2000

Wave Motion

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The subject of this paper is the dynamics of wave motion in the two-dimensional Kelvin-Helmholtz problem for an interface between two immiscible fluids of different densities. The difference of the mean flow between the two fluid bodies is taken to be zero, and the effects of surface tension are neglected. We transform the problem to Birkhoff normal form, in which a precise analysis can be made of classes of resonant solutions. This paper studies standing-wave solutions of the fourth-order normal form in particular detail. We find that that there are families of invariant resonant subsystems, which are nevertheless integrable. Within these families we describe the periodic and the time quasi-periodic standing waves, and determine their stability or instability. In particular we show that for a certain range of densities, a basic time-periodic standing wave with principal wave number k is unstable to modes with principal wave numbers k/4 and 9k/4, and we calculate the Lyapunov exponent of the instability. We furthermore show that the stable and unstable manifolds to these periodic solutions of the Birkhoff normal form are connected by a homoclinic orbit. This instability mechanism, as well as others that we describe, appears to be new, and its description is possible because of the precision afforded by the normal form. These results contrast with the case of the water wave problem described by Dyachenko & Zakharov [1] and Craig & Worfolk [2], where the fourth-order Birkhoff normal form is an integrable system, with all orbits undergoing stable almost-periodic motion, and instabilities arise only in normal forms to higher order.

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Geometric Aspects of Spatially Periodic Interfacial Waves

Cited by 14 publications

References 37 publications

Vortex sheet motion in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities with surface tension

Vortex sheet motion in incompressible Richtmyer–Meshkov and Rayleigh–Taylor instabilities with surface tension

Deep-water internal solitary waves near critical density ratio

Normal forms for wave motion in fluid interfaces

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