Author

# M. M. Rienecker

Bio: M. M. Rienecker is an academic researcher from University of New South Wales. The author has contributed to research in topics: Conservative vector field & Fourier series. The author has an hindex of 3, co-authored 3 publications receiving 597 citations.

##### Papers

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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.

424 citations

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TL;DR: In this paper, a numerical method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface and variable bed topography, which is applied to the unsteady motion of non-breaking water waves of arbitrary magnitude over a horizontal bed.

Abstract: A numerical method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface and variable bed topography. It is applied to the unsteady motion of non-breaking water waves of arbitrary magnitude over a horizontal bed. All horizontal variation is approximated by truncated Fourier series. This and finite-difference representation of the time variation are the only necessary approximations. Although the method loses accuracy if the waves become sharp-crested at any stage, when applied to non-breaking waves the method is capable of high accuracy.The interaction of one solitary wave overtaking another was studied using the Fourier method. Results support experimental evidence for the applicability of the Korteweg-de Vries equation to this problem since the waves during interaction are long and low. However, some deviations from the theoretical predictions were observed - the overtaking high wave grew significantly at the expense of the low wave, and the predicted phase shift was found to be only roughly described by theory. A mechanism is suggested for all such solitary-wave interactions during which the high and fast rear wave passes fluid forward to the front wave, exchanging identities while the two waves have only partly coalesced; this explains the observed forward phase shift of the high wave.For solitary waves travelling in opposite directions, the interaction is quite different in that the amplitude of motion during interaction is large. A number of such interactions were studied using the Fourier method, and the waves after interaction were also found to be significantly modified - they were not steady waves of translation. There was a change of wave height and propagation speed, shown by the present results to be proportional to the cube of the initial wave height but not contained in third-order theoretical results. When the interaction is interpreted as a solitary wave being reflected by a wall, third-order theory is shown to provide excellent results for the maximum run-up at the wall, but to be in error in the phase change of the wave after reflection. In fact, it is shown that the spatial phase change depends strongly on the place at which it is measured because the reflected wave travels with a different speed. In view of this, it is suggested that the apparent time phase shift at the wall is the least-ambiguous measure of the change.

172 citations

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29 Jan 1980TL;DR: In this article, the authors describe numerical methods for the accurate solution of the nonlinear equations for water waves propagating on irrotational flow over a horizontal bed and compare results with previous theoretical and experimental results.

Abstract: This paper describes numerical methods for the accurate solution of the nonlinear equations for water waves propagating on irrotational flow over a horizontal bed. Fourier approximation is used throughout. Firstly, the problem of waves propagating without change is considered, giving a set of nonlinear equations which may be conveniently solved by Newton's method. It is emphasized that the usual specification of water depth, wave height and wave period is not enough to solve the problem - an assumption as to wave speed or,mean current or mass flux must be included. Comparing results with previous theoretical and experimental results, good .agreement was obtained. In the second part un^ steady wave motion is examined, and a numerical method proposed for studying the evolution of unsteady disturbances. This is applied to the case of a solitary wave being reflected by a vertical wall. Close agreement with experimental results is obtained. In addition, design criteria for force and moment on the wall are suggested.

19 citations

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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

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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

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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

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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