J.-M. Vanden-Broeck

Dynamics of steep two-dimensional gravity-capillary solitary waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

Two-dimensional generalized solitary waves and periodic waves under an ice sheet.

Two-dimensional gravity waves travelling under an ice sheet are studied. The flow is assumed to be potential. Weakly nonlinear solutions are derived and fully nonlinear solutions are calculated numerically. Periodic waves and generalized solitary waves are studied.

Effect of an electric field on film flow down a corrugated wall at zero Reynolds number

The effect of an electric field on a liquid layer flowing down an inclined, corrugated wall at zero Reynolds number is investigated. The layer is taken to be either a perfect conductor or a perfect dielectric. The region above the layer is assumed to be a perfect dielectric. Steady flow down a wall with small-amplitude sinusoidal corrugations is considered, and it is shown how the electric field can be used to control the amplitude of the free-surface deflection and the phase shift between the free surface and the wall profile. Steady flow over walls with large amplitude sinusoidal corrugations or other-shaped indentations is studied by using the boundary-element method. Results for flow into a wide rectangular trench are compared to previous model predictions based on the lubrication approximation. For a perfect-conductor film, the results confirm that the height of the capillary ridge, which appears above a downward step, monotonically decreases as the electric field strength increases. Solutions for a ...

Numerical study of interfacial solitary waves propagating under an elastic sheet.

Steady solitary and generalized solitary waves of a two-fluid problem where the upper layer is under a flexible elastic sheet are considered as a model for internal waves under an ice-covered ocean. The fluid consists of two layers of constant densities, separated by an interface. The elastic sheet resists bending forces and is mathematically described by a fully nonlinear thin shell model. Fully localized solitary waves are computed via a boundary integral method. Progression along the various branches of solutions shows that barotropic (i.e. surface modes) wave-packet solitary wave branches end with the free surface approaching the interface. On the other hand, the limiting configurations of long baroclinic (i.e. internal) solitary waves are characterized by an infinite broadening in the horizontal direction. Baroclinic wave-packet modes also exist for a large range of amplitudes and generalized solitary waves are computed in a case of a long internal mode in resonance with surface modes. In contrast to the pure gravity case (i.e without an elastic cover), these generalized solitary waves exhibit new Wilton-ripple-like periodic trains in the far field.

Multilump symmetric and nonsymmetric gravity-capillary solitary waves in deep water

Multilump gravity-capillary solitary waves propagating in a fluid of infinite depth are computed numerically. The study is based on a weakly nonlinear and dispersive partial differential equation (PDE) with weak variations in the spanwise direction, a model derived by Akers and Milewski [Stud. Appl. Math., 122 (2009), pp. 249--274]. For a two-dimensional fluid, this model agrees qualitatively well with the full Euler equations for the bifurcation curves, wave profiles, and dynamics of solitary waves. Fully localized solitary waves are then computed for three-dimensional fluids. New symmetric lump solutions are computed by using a continuation method to follow the branch of elevation waves. It is then found that the branch of elevation waves has multiple turning points from which new solutions, consisting of multiple lumps separated by smaller oscillations, bifurcate. Nonsymmetric solitary waves, which also feature a multilump structure, are computed and found to appear via spontaneous symmetry-breaking bi...

Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves

Hydroelastic waves propagating at a constant velocity at the surface of a fluid are considered. The flow is assumed to be two-dimensional and potential. Gravity is included in the dynamic boundary condition. The fluid is bounded above by an elastic sheet which is described by the Plotnikov-Toland model. Fully nonlinear solutions are computed by a series truncation method. The findings generalised previous results where the sheet was described by a simplified model known as the Kirchhoff-Love model. Periodic and generalised solitary waves are computed. The results strongly suggest that there are no true solitary waves (i.e., solitary waves characterised by a flat free surface in the far field). Detailed comparisons with results obtained with the Kirchhoff-Love model and for the related problem of gravity capillary waves are also presented.

Steady dark solitary flexural gravity waves

The nonlinear Schrödinger (NLS) equation describes the modulational limit of many surface water wave problems. Dark solitary waves of the NLS equation asymptote to a constant in the far field and have a localized decrease to zero amplitude at the origin, corresponding to water wave solutions that asymptote to a uniform periodic Stokes wave in the far field and decreasing oscillations near the origin. It is natural to ask whether these dark solitary waves can be found in the irrotational Euler equations. In this paper, we find such solutions in the context of flexural-gravity waves, which are often used as a model for waves in ice-covered water. This is a situation in which the NLS equation predicts steadily travelling dark solitons. The solution branches of dark solitons are continued, and one branch leads to fully localized solutions at large amplitudes.

On satisfying the radiation condition in free-surface flows

Binder & Vanden-Broeck (2005) showed there are no subcritical or critical solutions satisfying the radiation condition for steady flows past a flat plate. By using a weakly nonlinear analysis, it is shown that such flows exist for a curved plate. Fully nonlinear solutions are computed by a boundary integral equation method, and new nonlinear solutions for supercritical and generalized critical flows past a curved plate are presented.

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

Unsteady propagating bubbles in an unbounded Hele-Shaw cell are considered numerically in the case of zero surface tension. The instability of elliptical bubbles and their evolution toward a stable circular boundary, with speed twice that of the fluid speed at infinity, is studied numerically and by stability analysis. Numerical simulations of bubbles demonstrate that the important role played by singularities of the Schwarz function of the bubble boundary in determining the evolution of the bubble. When the singularity lies close to the initial bubble, two types of topological change are observed: (i) bubble splitting into multiple bubbles and (ii) a finite fluid blob pinching off inside the bubble region.

On asymmetric generalized solitary gravity–capillary waves in finite depth

Generalized solitary waves propagating at the surface of a fluid of finite depth are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. Both the effects of gravity and surface tension are included. It is shown that in addition to the classical symmetric waves, there are new asymmetric solutions. These new branches of solutions bifurcate from the branches of symmetric waves. The detailed bifurcation diagrams as well as typical wave profiles are presented.