Steady Periodic Water Waves with Constant Vorticity: Regularity and Local Bifurcation (original) (raw)

0 Steady Periodic Water Waves with Constant Vorticity: Regularity and Local Bifurcation

2016

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.

An alternative approach to study irrotational periodic gravity water waves

Zeitschrift für angewandte Mathematik und Physik

We are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

Steady periodic water waves with unbounded vorticity: equivalent formulations and existence results

arXiv (Cornell University), 2013

In this paper we consider the steady water wave problem for waves that possess a merely Lr−integrable vorticity, with r ∈ (1, ∞) being arbitrary. We first establish the equivalence of the three formulations-the velocity formulation, the stream function formulation, and the height function formulation-in the setting of strong solutions, regardless of the value of r. Based upon this result and using a suitable notion of weak solution for the height function formulation, we then establish, by means of local bifurcation theory, the existence of small amplitude capillary and capillary-gravity water waves with a Lr−integrable vorticity.

Existence of capillary-gravity water waves with piecewise constant vorticity

arXiv (Cornell University), 2013

In this paper we construct periodic capillarity-gravity water waves with a piecewise constant vorticity distribution. They describe water waves traveling on superposed linearly sheared currents that have different vorticities. This is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump. The solutions of the diffraction problem, found by using local bifurcation theory, are the desired solutions of the hydrodynamical problem.

Global bifurcation theory for periodic traveling interfacial gravity–capillary waves

Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2016

We consider the global bifurcation problem for spatially periodic traveling waves for two-dimensional gravity-capillary vortex sheets. The two fluids have arbitrary constant, non-negative densities (not both zero), the gravity parameter can be positive, negative, or zero, and the surface tension parameter is positive. Thus, included in the parameter set are the cases of pure capillary water waves and gravity-capillary water waves. Our choice of coordinates allows for the possibility that the fluid interface is not a graph over the horizontal. We use a technical reformulation which converts the traveling wave equations into a system of the form "identity plus compact." Rabinowitz' global bifurcation theorem is applied and the final conclusion is the existence of either a closed loop of solutions, or an unbounded set of nontrivial traveling wave solutions which contains waves which may move arbitrarily fast, become arbitrarily long, form singularities in the vorticity or curvature, or whose interfaces self-intersect.

On the Construction of Traveling Water Waves with Constant Vorticity and Infinite Boundary

International Journal of Mathematics and Mathematical Sciences

The issue of whether there is a closed orbit in the water waves in an infinite boundary condition is an outstanding open problem. In this work, we first discuss the various developments on the structure of water waves in the context of finite bottom conditions. We then focus on the behavior of water for the kinematic boundary for the infinite depth. We present some findings to address this issue by creating a water wave profile for the zero and constant vorticity conditions through the application of the Crandall–Rabinowitz theorem.

Oscillatory spatially periodic weakly nonlinear gravity waves on deep water

Journal of Fluid Mechanics, 1988

A weakly nonlinear Hamiltonian model is derived from the exact water wave equations to study the time evolution of spatially periodic wavetrains. The model assumes that the spatial spectrum of the wavetrain is formed by only three free waves, i.e. a carrier and two side bands. The model has the same symmetries and invariances as the exact equations. As a result, it is found that not only the permanent form travelling waves and their stability are important in describing the time evolution of the waves, but also a new kind of family of solutions which has two basic frequencies plays a crucial role in the dynamics of the waves. It is also shown that three is the minimum number of free waves which is necessary to have chaotic behaviour of water waves.

Gravity water flows with discontinuous vorticity and stagnation points

arXiv (Cornell University), 2015

We construct small-amplitude steady periodic gravity water waves arising as the free surface of water flows that contain stagnation points and possess a discontinuous distribution of vorticity in the sense that the flows consist of two layers of constant but different vorticities. We also describe the streamline pattern in the moving frame for the constructed flows.

N-Modal Steady Water Waves with Vorticity

Journal of Mathematical Fluid Mechanics, 2017

Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.

A new limiting form for steady periodic gravity waves with surface tension on deep water

Physics of Fluids, 1996

The method developed by Longuet-Higgins [J. Inst. Math. Appl. 22, 261 (1978)] for the computation of pure gravity waves is extended to capillary-gravity waves in deep water. Surface tension provides an additional term in the identities between the Fourier coefficients in Stokes' expansion. This term is then reduced to a simple function of the slope of the local tangent to the profile of the free surface. A set of nonlinear algebraic equations is derived and solved by using the Newton's method. A new family of limiting profiles of steady gravity waves with surface tension is found.