scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A Fourier approximation method for steady water waves

01 Mar 1981-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 104, Iss: -1, pp 119-137
TL;DR: In this article, a finite Fourier series is used to give a set of nonlinear equations which can be solved using Newton's method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed.
Abstract: A method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed is presented. No analytical approximations are made. A finite Fourier series, similar to Dean's stream function series, is used to give a set of nonlinear equations which can be solved using Newton's method. Application to laboratory and field situations is emphasized throughout. When compared with known results for wave speed, results from the method agree closely. Results for fluid velocities are compared with experiment and agreement found to be good, unlike results from analytical theories for high waves.The problem of shoaling waves can conveniently be studied using the present method because of its validity for all wavelengths except the solitary wave limit, using the conventional first-order approximation that on a sloping bottom the waves at any depth act as if the bed were horizontal. Wave period, energy flux and mass flux are conserved. Comparisons with experimental results show good agreement.
Citations
More filters
Journal ArticleDOI
TL;DR: A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of a two-phase flow where the interface can merge/break and the flow can have a high Reynolds number.

825 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed a robust numerical method for modeling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness.
Abstract: We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number ( N = O (1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M , and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness ( ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

616 citations

Journal ArticleDOI
TL;DR: The algorithm is based upon Fick's law of diffusion and shifts particles in a manner that prevents highly anisotropic distributions and the onset of numerical instability, and is validated against analytical solutions for an internal flow at higher Reynolds numbers than previously.

513 citations


Cites background or methods from "A Fourier approximation method for ..."

  • ...(34) is related to the wavenumber by the dispersion equation, x(2) 1⁄4 gk tanhðkdÞ: ð36Þ Even with small wave heights, free-surface waves show slightly non-sinusoidal form and comparisons are made with the highly accurate stream function theory of Rienecker and Fenton [24] based on Fourier approximations....

    [...]

  • ...(34) is related to the wavenumber by the dispersion equation, x2 ¼ gk tanhðkdÞ: ð36Þ Even with small wave heights, free-surface waves show slightly non-sinusoidal form and comparisons are made with the highly accurate stream function theory of Rienecker and Fenton [24] based on Fourier approximations....

    [...]

  • ...Regular waves are generated by a piston-type paddle for several periods and wave heights; velocities and pressures are then compared against accurate stream-function theory [24]....

    [...]

Journal ArticleDOI
TL;DR: In this article, an alternative Stokes theory for steady waves in water of constant depth is presented where the expansion parameter is the wave steepness itself, and the first step in application requires the solution of one nonlinear equation, rather than two or three simultaneously as has been previously necessary.
Abstract: An alternative Stokes theory for steady waves in water of constant depth is presented where the expansion parameter is the wave steepness itself. The first step in application requires the solution of one nonlinear equation, rather than two or three simultaneously as has been previously necessary. In addition to the usually specified design parameters of wave height, period and water depth, it is also necessary to specify the current or mass flux to apply any steady wave theory. The reason being that the waves almost always travel on some finite current and the apparent wave period is actually a Dopplershifted period. Most previous theories have ignored this, and their application has been indefinite, if not wrong, at first order. A numerical method for testing theoretical results is proposed, which shows that two existing theories are wrong at fifth order, while the present theory and that of Chappelear are correct. Comparisons with experiments and accurate numerical results show that the present theory ...

488 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation, and is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness δ of the boundary layer.
Abstract: It was shown by Stokes that in a water wave the particles of fluid possess, apart from their orbital motion, a steady second-order drift velocity (usually called the mass-transport velocity). Recent experiments, however, have indicated that the mass-transport velocity can be very different from that predicted by Stokes on the assumption of a perfect, non-viscous fluid. In this paper a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation. Part I deals especially with the interior of the fluid. It is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness $\delta $ of the boundary layer: when a$^{2}$/$\delta ^{2}$ is small the diffusion of vorticity takes place by viscous 'conduction'; when a$^{2}$/$\delta ^{2}$ is large, by convection with the mass-transport velocity. Appropriate field equations for the stream function of the mass transport are derived. The boundary layers, however, require separate consideration. In part II special attention is given to the boundary layers, and a general theory is developed for two types of oscillating boundary: when the velocities are prescribed at the boundary, and when the stresses are prescribed. Whenever the motion is simple-harmonic the equations of motion can be integrated exactly. A general method is described for determining the mass transport throughout the fluid in the presence of an oscillating body, or with an oscillating stress at the boundary. In part III, the general method of solution described in parts I and II is applied to the cases of a progressive and a standing wave in water of uniform depth. The solutions are markedly different from the perfect-fluid solutions with irrotational motion. The chief characteristic of the progressive-wave solution is a strong forward velocity near the bottom. The predicted maximum velocity near the bottom agrees well with that observed by Bagnold.

1,186 citations

Journal ArticleDOI
TL;DR: In this paper, Stokes' infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic, which is valid for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth.
Abstract: Stokes' infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes' expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Pade approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth.

362 citations

Journal ArticleDOI
TL;DR: In this article, an analytical stream function expression representing a nonlinear gravity water wave is applied both to the representation of measured wave forms and also to nonlinear theoretical waves, where the stream function form is chosen so that it is a solution to the Laplace equation and the bottom boundary condition; the parameters are chosen by a numerical perturbation procedure that provides a best fit to the kinematic and dynamic free surface boundary conditions.
Abstract: An analytical stream function expression representing a nonlinear gravity water wave is applied both to the representation of measured wave forms and also to nonlinear theoretical waves. The stream function form is chosen so that it is a solution to the Laplace equation and the bottom boundary condition; the parameters in the stream function expression are chosen by a numerical perturbation procedure that provides a best fit to the kinematic and dynamic free surface boundary conditions. The boundary condition errors associated with the nonlinear stream function representation of four measured wave profiles are compared with estimates of the corresponding errors associated with a linear representation. The stream function method is judged more accurate than a linear method if the wave height is greater than 50% of the breaking height. The stream function method is also applied to represent theoretical waves for which only the wave height and period are available to characterize the wave profile. It is demonstrated that the method represents an improvement over other available nonlinear procedures. The method can be employed to represent wave conditions which include a prescribed uniform steady current and a specified pressure distribution on the free surface.

348 citations

Journal ArticleDOI
TL;DR: In this paper, the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible.
Abstract: Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Pade approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Pade-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

322 citations