Steady Gravity Waves Research Articles

Steady Radiating Gravity waves: An Exponential Asymptotics Approach

AbstractThe radiation of steady surface gravity waves by a uniform stream $$U_{0}$$ U 0 over locally confined (width $$L$$ L ) smooth topography is analyzed based on potential flow theory. The linear solution to this classical problem is readily found by Fourier transforms, and the nonlinear response has been studied extensively by numerical methods. Here, an asymptotic analysis is made for subcritical flow $$D/\lambda > 1$$ D / λ > 1 in the low-Froude-number ($$F^{2} \equiv \lambda /L \ll 1$$ F 2 ≡ λ / L ≪ 1 ) limit, where $$\lambda = U_{0}^{2} /g$$ λ = U 0 2 / g is the lengthscale of radiating gravity waves and $$D$$ D is the uniform water depth. In this regime, the downstream wave amplitude, although formally exponentially small with respect to $$F$$ F , is determined by a fully nonlinear mechanism even for small topography amplitude. It is argued that this mechanism controls the wave response for a broad range of flow conditions, in contrast to linear theory which has very limited validity.

Recovery of steady rotational wave profiles from pressure measurements at the bed

We derive equations relating the pressure at a flat seabed and the free-surface profile for steady gravity waves with constant vorticity. The resulting set of nonlinear equations enables the recovery of the free surface from pressure measurements at the bed. Furthermore, the flow vorticity (unknown a priori) is determined solely from the bottom pressure as part of the recovery method. This approach is applicable even in the presence of stagnation points and its efficiency is illustrated via numerical examples.

Solitary waves on constant vorticity flows with an interior stagnation point

Abstract

Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth: Derivation and Numerical Solution

The nonlinear two-dimensional problem describing periodic steady gravity waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko’s equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the $$2 \pi $$ -periodic Hilbert transform $${\mathcal {C}}$$ used in the equation for deep water, the equation obtained here contains a certain operator $${\mathcal {B}}_r$$ , which is the sum of $${\mathcal {C}}$$ and a compact operator depending on a parameter related to the depth of water. Numerical computations are based on an equivalent form of Babenko’s equation derived by virtue of the spectral decomposition of the operator $${\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t$$ . Bifurcation curves and wave profiles of the extreme form are obtained numerically.

An Approximation of the Steady Gravity Wave Induced by Mountainous Topography* *Supported by the Natural Science Foundation of China under Grant Nos. 91437215, 41575047, and The Third Scientific Experiment of Tibetan Plateau under Grant No. GYHY201406001

An approximation of the orography gravity wave, which is induced by mountainous topography, is considered in this study. By assuming that the horizontal wind is a linear function with respect to the height, the approximating equation for the orography gravity waves is obtained. Four topography functions are considered in this study and the orography gravity wave are obtained. The dynamics of the orography gravity wave is then discussed by considering the effect of the surface topography and background horizontal wind.

New exact relations for steady irrotational two-dimensional gravity and capillary surface waves.

Steady two-dimensional surface capillary-gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained.This article is part of the theme issue 'Nonlinear water waves'.

Bounds for Solutions to the Problem of Steady Water Waves with Vorticity

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the ...

Traveling Gravity Water Waves with Critical Layers

We establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions.

On bounds and non-existence in the problem of steady waves with vorticity

AbstractFor the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

Steady water waves with vorticity: an analysis of the dispersion equation

AbstractTwo-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm–Liouville problem considered by Constantin & Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481–527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133–175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm–Liouville problem mentioned in (i) has an eigenvalue is obtained.

Steady water waves with vorticity: spatial Hamiltonian structure

AbstractSpatial dynamical systems are obtained for two-dimensional steady gravity waves with vorticity on water of finite depth. These systems have Hamiltonian structure and Hamiltonian is essentially the flow–force invariant.

Steady gravity waves due to a submerged source

AbstractIn the low-Froude-number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Steady linearized flow past a submerged source is considered, and exponential asymptotic methods are applied to determine the behaviour of the free-surface gravity waves. The free surface is found to contain longitudinal and transverse waves that switch on rapidly across curves known as Stokes lines on the free surface. The longitudinal waves are present everywhere downstream of the singularity, while the transverse waves are restricted to two downstream wedges. As the depth of the source approaches the surface, the familiar Kelvin-wedge wave behaviour is recovered.

Stability of steady gravity waves generated by a moving localised pressure disturbance in water of finite depth

We study the stability of the steady waves forced by a moving localised pressure disturbance in water of finite depth. The steady waves take the form of a downstream wavetrain for subcritical flow, but for supercritical flow there is a solution branch for each specified forcing term, which has two localised solitary-like solutions for each Froude number, a small-amplitude and a large-amplitude solution. Our main purpose is to numerically investigate the stability of these steady waves by simulating the unsteady development from appropriate initial conditions. We use a forced Korteweg-de Vries model valid in the transcritical regime for weak forcing, and a fully nonlinear boundary integral simulation. The simulations using the forced Korteweg-de Vries model are in good agreement with the fully nonlinear simulations when the Froude number is near unity for small pressure forcing, as expected. We find that the steady subcritical downstream wavetrain is stable. For supercritical flow, the small-amplitude solitary-like wave which has bifurcated from a uniform flow is stable, whereas the large-amplitude solitary-like wave which has bifurcated from a free solitary wave is unstable. Furthermore, here a difference between the forced Korteweg-de Vries model and the fully nonlinear simulations is revealed. For moderate and large pressure forcing, the forced Korteweg-de Vries model predicts a stable solitary wave moving away from the pressure forcing, while the nonlinear simulation shows that this wave evolves to a breaking wave.

Efficient computation of steady solitary gravity waves

An efficient numerical method to compute solitary wave solutions to the free surface Euler equations is reported. It is based on the conformal mapping technique combined with an efficient Fourier pseudo-spectral method. The resulting nonlinear equation is solved via the Petviashvili iterative scheme. The computational results are compared to some existing approaches, such as Tanaka’s method and Fenton’s high-order asymptotic expansion. Several important integral quantities are computed for a large range of amplitudes. The integral representation of the velocity and acceleration fields in the bulk of the fluid is also provided.

Fast accurate computation of the fully nonlinear solitary surface gravity waves

In this short note, we present an easy to implement and fast algorithm for the computation of the steady solitary gravity wave solution of the free surface Euler equations in irrotational motion. First, the problem is reformulated in a fixed domain using the conformal mapping technique. Second, the problem is reduced to a single equation for the free surface. Third, this equation is solved using Petviashvili’s iterations together with pseudo-spectral discretisation. This method has a super-linear complexity, since the most demanding operations can be performed using a FFT algorithm. Moreover, when this algorithm is combined with the multi-precision floating point computations, the results can be obtained to any arbitrary accuracy.

No steady water waves of small amplitude are supported by a shear flow with a still free surface

AbstractThe two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

Reconstruction of the pressure in long-wave models with constant vorticity

The effect of constant background vorticity on the pressure beneath steady long gravity waves at the surface of a fluid is investigated. Using an asymptotic expansion for the streamfunction, we derive a model equation and a formula for the pressure in a flow with constant vorticity. The model equation was previously found by Benjamin (1962), [3], and is given in terms of the vorticity ω0, and three parameters Q,R and S representing the volume flux, total head and momentum flux, respectively.The focus of this work is on the reconstruction of the pressure from solutions of the model equation and the behavior of the surface wave profiles and the pressure distribution as the strength of the vorticity changes. In particular, it is shown that for strong enough vorticity, the maximum pressure is no longer located under the wave crest, and the fluid pressure near the surface can be below atmospheric pressure.

Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves

The velocity and other fields of steady two-dimensional surface gravity waves in irrotational motion are investigated numerically. Only symmetric waves with one crest per wavelength are considered, i.e. Stokes waves of finite amplitude, but not the highest waves, nor subharmonic and superharmonic bifurcations of Stokes waves. The numerical results are analysed, and several conjectures are made about the velocity and acceleration fields.

Bounds for steady water waves with vorticity

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution.

The Benjamin–Lighthill Conjecture for Steady Water Waves (revisited)

The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin–Lighthill conjecture is proved for these waves provided Bernoulli’s constant attains near-critical values. In fact this is a consequence of the following more general results. If Bernoulli’s constant is near-critical, then all corresponding waves have sufficiently small heights and slopes. Moreover, for every near-critical value of Bernoulli’s constant, there exist only the following waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave; this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch Rational Mech Anal 197, 433–488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope of this wave is also small.