Arbitrary Distribution Of Vorticity Research Articles

On Loops in Water Wave Branches and Monotonicity of Water Waves

We give a quite simple approach how to prove the absence of loops in bifurcation branches of water waves in rotational case with arbitrary vorticity distribution. The supporting flow may contain stagnation points and critical layers and water surface is allowed to be overhanging. Monotonicity properties of the free surface are presented. Especially simple criterium of absence of loops is given for bifurcation branches when the bifurcation parameter is the water wave period. We show that there are no loops if you start from a water wave with a positive/negative vertical component of velocity on the positive half period.

Nonexistence of Subcritical Solitary Waves

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.

Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed.

This paper considers the pressure-streamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. The application of such work is to both near shore and offshore environments, and in particular, for linear waves we provide a description of the role which the pressure function on the seabed plays in determining the free-surface profile elevation. Our approach is shown to provide a good approximation for a range of current conditions.This article is part of the theme issue 'Nonlinear water waves'.

Solitary Water Waves of Large Amplitude Generated by Surface Pressure

We consider exact nonlinear solitary water waves on a shear flow with an arbitrary distribution of vorticity. Ignoring surface tension, we impose a non-constant pressure on the free surface. Starting from a uniform shear flow with a flat free surface and a supercritical wave speed, we vary the surface pressure and use a continuation argument to construct a global connected set of symmetric solitary waves. This set includes waves of depression whose profiles increase monotonically from a central trough where the surface pressure is at its lowest, as well as waves of elevation whose profiles decrease monotonically from a central crest where the surface pressure is at its highest. There may also be two waves in this connected set with identical surface pressure, only one of which is a wave of depression.

The Froude number for solitary water waves with vorticity

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

Large-Amplitude Solitary Water Waves with Vorticity

We provide the first construction of exact solitary waves of large amplitude with an arbitrary distribution of vorticity. We use continuation to construct a global connected set of symmetric solitary waves of elevation, whose profiles decrease monotonically on either side of a central crest. This generalizes the classical result of Amick and Toland.

A PROBLEM IN THE CLASSICAL THEORY OF WATER WAVES: WEAKLY NONLINEAR WAVES IN THE PRESENCE OF VORTICITY

The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type ("soliton" equations), but with vorticity. The familiar problem of propagation governed by the Korteweg–de Vries (KdV) equation is introduced, but allowing for an arbitrary distribution of vorticity. The effect of the constant vorticity on the solitary wave is described. The corresponding problem for the Nonlinear Schrödinger (NLS) equation is briefly mentioned but not explored here. The problem of two-way propagation (admitting head-on collisions), as described by the Boussinesq equation, is examined next. This leads to a new equation: the Boussinesq-type equation valid for constant vorticity. However, this cannot be transformed into an integrable Boussinesq equation (as is possible for the corresponding KdV and NLS equations). The solitary-wave solution for this new equation is presented. A description of the Camassa–Holm equation for water waves, with constant vorticity, with its solitary-wave solution, is described. Finally, we outline the problem of propagation of small-amplitude, large-radius ring waves over a flow with vorticity (representing a background flow in one direction). Some properties of this flow, for constant vorticity, are described.

Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity

This paper presents an existence theory for small-amplitude Stokes and solitary-wave solutions to the classical water-wave problem in the absence of surface tension and with an arbitrary distribution of vorticity. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial coordinate is the time-like variable. A centre-manifold technique is used to reduce the system to a locally equivalent Hamiltonian system with one degree of freedom for values of a dimensionless parameter α near its critical value α ⋆ . The phase portrait of the reduced system contains a homoclinic orbit for α < α ⋆ and a family of periodic orbits for α > α ⋆ ; the corresponding solutions of the water-wave problem are respectively a solitary wave of elevation and a family of Stokes waves.

Spatial Dynamics Methods for Solitary Gravity-Capillary Water Waves with an Arbitrary Distribution of Vorticity

This paper presents existence theories for several families of small-amplitude solitary-wave solutions to the classical two-dimensional water-wave problem in the presence of surface tension and with an arbitrary distribution of vorticity. Moreover, the established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which the horizontal spatial direction is the timelike variable. A center-manifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.

Steady Periodic Capillary‐Gravity Waves with Vorticity

In this paper we prove the existence of steady periodic two‐dimensional capillary‐gravity waves on flows with an arbitrary vorticity distribution. The original free‐surface problem is first transformed to a second‐order quasi‐linear elliptic equation with a second‐order quasi‐linear boundary condition in a fixed domain by a change of variables. We then use local bifurcation theory combined with the Schauder theory of elliptic equations with Venttsel boundary conditions and spectral theory in Pontryagin spaces to construct the solutions. We show that some bifurcation points are simple while others are double, a situation already known to occur in the case of irrotational capillary‐gravity waves.

On steady gravity waves with vorticity

Using bifurcation theory and methods from the Schauder theory of elliptic partial differential equations, we prove the existence of small-amplitude steady water waves with an arbitrary vorticity distribution.

Quelques remarques sur la dynamique Hamiltonienne des tourbillons

This work deals with the determination of a general formulation of the Hamiltonian approach applicable to an arbitrary distribution of vorticity. The numerical results display the importance of the respect of the laws of conservation. A model which guarantees the respect of them is presented.

Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet &amp; Widmayer (1974) and Weigand &amp; Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

Non–existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

Unsteady flow near a moving cylinder

General representations are derived for both the velocity potential and the surface pressure fluctuations induced by an arbitrary distribution of vorticity near a manoeuvring cylinder. The cylinder is inextensible and in unsteady motion. Its axis may be slightly curved, with radius of curvature large in comparison with the cylinder radius.Two model problems are considered in detail to investigate the effect of lateral displacements of a cylinder with an established boundary layer. The boundary layer on the flexible cylinder is found to be shed once the lateral displacement of the cylinder axis exceeds the boundary-layer thickness. The unsteady pressures generated by this vortex shedding are investigated.

Wave–current interactions: an experimental and numerical study. Part 2. Nonlinear waves

The interaction between a regular wavetrain and a current possessing an arbitrary distribution of vorticity, in two dimensions, is considered for waves of finite amplitude. A numerical model is constructed, primarily for use in the finite depth regime, extending the work of Dalrymple (1973, 1977) and this is used to predict the wavelength and the particle velocities under the waves. These predictions agree very well with experimentally obtained data and the importance of the vorticity in the wave–current interaction is clarified. Amplitude and wavelength modulations are considered for finite amplitude waves on a slowly varying irrotational current; moderate agreement is found between theory and experiment.

Extension of the Circle Theorems by Surface Source Distribution

The paper presents a closed-form analytical solution for the source strength distribution along the circumference of a two-dimensional circular cylinder that is required for producing an arbitrary distribution of normal velocity. Being suitable to be used with flows having arbitrary vorticity distribution, the present formulation can be considered as an alternative and extensive form of the circle theorems. Using the conformal transformation technique, the formulation also serves as a closed-form solution of Laplace’s equation in any two-dimensional flow domain that is reducible to the outer or inner region of a circular cylinder having arbitrary prescribed normal velocity over its boundary.

Wave-current interactions: an experimental and numerical study. Part 1. Linear waves

The interaction between a regular wavetrain and an adverse current containing an arbitrary distribution of vorticity, in two dimensions, is studied using a linear theory. The model is used to predict the wavelength and the particle velocities under the waves and these are found to agree well with experimentally obtained data for a number of current profiles. Surprisingly accurate predictions, for the profiles considered, were also obtained from an irrotational wave–current model in which the constant current has a value equal to the depth-averaged mean of the measured current profile. The changes in the wave amplitude as the current magnitude increases are predicted using an irrotational slowly varying model with good agreement being found between theory and experiment.

Effect of continuous variation of the density of a liquid on the waves generated by moving surface pressures

This study investigates the plane linear problem of steady-state internal waves in an ideal incompressible liquid with nonuniform density. The waves are generated by surface pressures applied in a bounded region which moves at constant velocity. It is assumed that the density in the unperturbed state varies continuously with depth, remaining constant in the upper and lower layers and varying according to an exponential law in the middle layer. The problem may be regarded, in particular, as a hydrodynamic model for the study of internal waves produced by a cyclone moving over the surface of the ocean. Analogous investigations for a homogeneous liquid were carried out in [1–3]; internal waves for a liquid with the above-mentioned law of density variation but with stationary pressure changes which are periodic with respect to time were studied in [4]. Problems analogous to the one considered here, both for exponential variation of density in the entire layer and for the case of a nonuniform layer near the surface, were investigated in [5, 6]. An analysis of non-linear waves of the steady-state type with arbitrary distribution of vorticity and density with respect to depth was carried out in [7, 8].

The stability of a helical vortex filament

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss &amp; Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.