Two water waves in subsurface interaction on a shear flow (original) (raw)

Two-dimensional Models for Nonlinear Vorticity Waves in Shear Flows

Studies in Applied Mathematics, 1998

The evolution of weakly nonlinear wave perturbations in shear flows of stratified fluid is investigated for large Reynolds numbers. The study is focused on the vorticity waves, i.e., the wave-like motions caused by the mean flow vorticity gradient. A situation typical of the upper ocean is considered. The shear flow is supposed to be localized near the surface and to have no inflection points. The vertical scale of stratification is much larger than that of the shear current. Description of the dynamics of essentially three-dimensional wave perturbations is reduced by a systematic asymptotic procedure to a single nonlinear evolution integrodifferential equation for Ž . 2q1 -dimensions. The small parameters are the ratio of the vertical scale of the shear to the typical wavelength of the perturbations and the amplitude parameter. The equation does not contain viscous terms, but the regime of evolution it describes occurs owing to small but finite viscosity. The viscosity inhibits generation of strongly nonlinear vortices in the critical layer. Possible existence of localized two-dimensional stationary solutions of the equation is investigated. Axially symmetric soliton solutions are found for a fluid of arbitrary depth in the limit of vanishing stratification. In stratified flows a linear resonant interaction between shear flow perturbations and internal waves is found to play the major role. The radiation damping of vorticity waves due to these resonances makes the existence of similar lump solitary structures in stratified fluid impossible.

Nonlinear interaction of shear flow with a free surface

Journal of Fluid Mechanics, 1994

In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches ...

Nonlinear hydroelastic waves on a linear shear current at finite depth

Journal of Fluid Mechanics, 2019

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find ...

Large-amplitude interfacial waves on a linear shear flow in the presence of a current

Journal of Fluid Mechanics, 1993

The properties of two-dimensional steady periodic interfacial gravity waves between two fluids in relative motion and of constant vorticities and finite depths are investigated analytically and numerically. Particular attention is given to the effect of uniform vorticity, in the presence of a current velocity, on the two factors (identified in the literature as dynamical and geometrical limits) which limit the existence of steady gravity wave solutions. The dynamical limit to the existence of steady solutions is found to be significantly influenced by the uniform vorticity of the lower fluid. In particular, the effect of non-zero vorticity is qualitatively different between a very shallow and a relatively deep lower fluid. Profiles and flow fields corresponding to very steep waves are calculated for a wide range of parameter values. The effect of uniform vorticity on the interfacial wave structure is demonstrated through a direct comparison with irrotational waves. For negative vorticity and high current velocity, a new flow structure is found, consisting of a closed eddy attached to the interface below the crest. Resemblance with shallow water waves breaking under strong air flow, (described in the experimental literature as roll waves) is noted.

On the interaction of deep water waves and exponential shear currents

Zeitschrift für angewandte Mathematik und Physik, 2009

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil-Jacotin transformation is used to transfer the original exponentially nonlinear boundaryvalue problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fluid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.

On internal wave–shear flow resonance in shallow water

Journal of Fluid Mechanics, 1998

The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as 'vorticity waves', which enables us to treat the wave-flow resonance as the resonant wave-wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the 'fast' solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; 'subcritical' localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking. † Present address:

Explicit wave action conservation for water waves on vertically sheared flows

Ocean Modelling

This paper addresses a major shortcoming of the current generation of wave models, namely their inability to describe wave propagation upon ambient currents with vertical shear. The wave action conservation equation (WAE) for linear waves propagating in horizontally inhomogeneous vertically-sheared currents is derived following Voronovich (1976). The resulting WAE specifies conservation of a certain depth-averaged quantity, the wave action, a product of the wave amplitude squared, eigenfunctions and functions of the eigenvalues of the boundary value problem for water waves upon a vertically sheared current. The formulation of the WAE is made explicit using known asymptotic solutions of the boundary value problem which exploit the smallness of the current magnitude compared to the wave phase velocity and/or its vertical shear and curvature; the adopted approximations are shown to be sufficient for most of the conceivable applications. In the limit of vanishing current shear, the new formulation reduces to that of Bretherton & Garrett (1968) without shear and the invariant is calculated with the current magnitude taken at the free surface. It is shown that in realistic oceanic conditions, the neglect of the vertical structure of the currents in wave modelling which is currently universal might lead to significant errors in wave amplitude. The new WAE which takes into account the vertical shear can be better coupled to modern circulation models which resolve the three-dimensional structure of the uppermost layer of the ocean.

Oscillating sources in a shear flow with a free surface

arXiv: Fluid Dynamics, 2016

We report on progress on the free surface flow in the presence of submerged oscillating line sources (2D) or point sources (3D) when a simple shear flow is present varying linearly with depth. Such sources are in routine use as Green functions in the realm of potential theory for calculating wave-body interactions, but no such theory exists in for rotational flow. We solve the linearized problem in 2D and 3D from first principles, based on the Euler equations, when the sources are at rest relative to the undisturbed surface. Both in 2D and 3D a new type of solution appears compared to irrotational case, a critical layer-like flow whose surface manifestation ("wave") drifts downstream from the source at the velocity of the flow at the source depth. We analyse the additional vorticity in light of the vorticity equation and provide a simple physical argument why a critical layer is a necessary consequence of Kelvin's circulation theorem. In 3D a related critical layer phe...

Surface waves on shear currents: solution of the boundary-value problem

Journal of Fluid Mechanics, 1993

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terns of an infinite series in powers of a certain parameter E , which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that E be less than unity.