Fifth order gravity wave theory
29 Jan 2011-Vol. 1, Iss: 7, pp 10-10
TL;DR: In this article, the results of the fifth order theory and values of the various coefficients as a function of the parameter d/L were presented for gravity waves of greater steepness.
Abstract: In dealing with problems connected with gravity waves, scientists and engineers frequently find it necessary to make lengthy theoretical calculations involving such wave characteristics as wave height, wave length, period, and water depth. Several approximate theoretical expressions have been derived relating the above parameters. Airy, for instance, contributed a very valuable and complete theory for waves traveling over a horizontal bottom in any depth of water. Due to the simplicity of the Airy theory, it is frequently used by engineers. This theory, however, was developed for waves of very small heights and is inaccurate for waves of finite height. Stokes presented a similar solution for waves of finite height by use of trigonometric series. Using five terms in the series, this solution will extend the range covered by the Airy theory to waves of greater steepness. No attempt has been made in this paper to specify the range where the theory is applicable. The coefficients in these series are very complicated and for a numerical problem, the calculations become very tedious. Because of this difficulty, this theory would be very little used by engineers unless the value of the coefficient is presented in tabular form. The purpose of this paper is to present the results of the fifth order theory and values of the various coefficients as a function of the parameter d/L.
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TL;DR: In this paper, the authors considered some aspects of water impact and green water loading by numerically investigating a dambreak problem and water entry problems, based on the Navier-Stokes equations that describe the flow of a viscous fluid.
Abstract: In this paper, some aspects of water impact and green water loading are considered by numerically investigating a dambreak problem and water entry problems. The numerical method is based on the Navier-Stokes equations that describe the flow of an incompressible viscous fluid. The equations are discretised on a fixed Cartesian grid using the finite volume method. Even though very small cut cells can appear when moving an object through the fixed grid, the method is stable. The free surface is displaced using the Volume-of-Fluid method together with a local height function, resulting in a strictly mass conserving method. The choice of boundary conditions at the free surface appears to be crucial for the accuracy and robustness of the method. For validation, results of a dambreak simulation are shown that can be compared with measurements. A box has been placed in the flow, as a model for a container on the deck of an offshore floater on which forces are calculated. The water entry problem has been investigated by dropping wedges with different dead-rise angles, a cylinder and a cone into calm water with a prescribed velocity. The resulting free surface dynamics, with the sideways jets, has been compared with photographs of experiments. Also a comparison of slamming coefficients with theory and experimental results has been made. Finally, a drop test with a free falling wedge has been simulated.
618 citations
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
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482 citations
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424 citations
TL;DR: In this article, landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity, and four wave types were determined: weakly nonlinear oscillatory wave, nonlinear transition wave, solitary-like wave and dissipative transient bore.
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298 citations