• Journal of Ocean Engineering and Technology
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• v.36 no.2
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• pp.101-107
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• 2022

Dean (1965) proposed the use of the root mean square error (RMSE) in the dynamic free surface boundary condition (DFSBC) and kinematic free-surface boundary condition (KFSBC) as an error evaluation criterion for wave theories. There are well known wave theories with RMSE more than 1%, such as Airy theory, Stokes theory, Dean's stream function theory, Fenton's theory, and trochodial theory for deep-water waves. However, none of them can be applied for deep-water breaking waves. The purpose of this study is to provide a closed-form solution for deep-water waves with RMSE less than 1% even for breaking waves. This study is based on a previous study (Shin, 2016), and all flow fields were simplified for deep-water waves. For a closed-form solution, all Fourier series coefficients and all related parameters are presented with Newton's polynomials, which were determined by curve fitting data (Shin, 2016). For verification, a wave in Miche's limit was calculated, and, the profiles, velocities, and the accelerations were compared with those of 5^th-order Stokes theory. The results give greater velocities and acceleration than 5^th-order Stokes theory, and the wavelength depends on the wave height. The results satisfy the Laplace equation, bottom boundary condition (BBC), and KFSBC, while Stokes theory satisfies only the Laplace equation and BBC. RMSE in DFSBC less than 7.25×10^-2% was obtained. The series order of the proposed method is three, but the series order of 5^th-order Stokes theory is five. Nevertheless, this study provides less RMSE than 5^th-order Stokes theory. As a result, the method is suitable for offshore structural design.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.4 no.4
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• pp.187-200
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• 1992

A general scheme is developed to determine the Langrangian motions of water particles by the Eulerian velocity at their mean positions by using Taylor's theorem. Utilizing the Stokes finite-amplitude wave theory, the orbital motions and the mass transport velocity including the effects of higher-order wave components are determined. The fifth-order approximation of orbital motion gives very good predictions of actual water particle motion in Stokes fifth-order wave theory except near the free-surface. The fifth-order theory predicts the mass transport velocity less than that given by the existing second-order theory over the whole water depth.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.7 no.4
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• pp.329-339
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• 1995

To clarify the surface drift velocity in gravity waves. experimental data measured in a two-dimensional wave flume were compared with theoretical values predicted by the Stokes 2nd- and 5th- order theories as well as by the conduction solution or Longuet-Hinggins (1953). Relative water depth and wave height ranged 0.040.13. For a closed flume condition, Stokes 2nd-order theory gives lower values than the experimental data, and the differences increase as both relative water depth and wave height increase. Based on the observed data of the surface drift velocities, a modified Parabolic model of the return current velocity Profile has been suggested, which is Proved to fit better to the existing experimental data of mass transport velocity profiles in a closed wave flume than the models of Longuet-Hinggins (1953) and Stokes wave theories do.

• Proceedings of the Korea Water Resources Association Conference
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• 1988.07a
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• pp.29-36
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• 1988

A general scheme is developed which determines the Lagrangian motions of water particles by the Eulerian velocity at their mean positions by use of Taylor's theorem. Utilizing the Stokes finite-amplitude wave theory, the mass transport velocity which includes the effects of higher-order wave components is determined. The fifth-order theory predicts the mass transport velocity less than that given by the existing second-order theory over the whole depth. Limited experimental data for changes in wave celerity in closed wave flumes are compared with the theoretical predictions.

• Journal of Advanced Research in Ocean Engineering
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• v.2 no.1
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• pp.1-11
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• 2016

The objective of this paper is to present an analytical solution in deep water waves and verify the validity of the theory (Shin, 2015). Hence this is a follow-up to Shin (2015). Instead of a variational approach, another approach was considered for a more accurate assessment in this study. The products of two coefficients were not neglected in this study. The two wave profiles from the KFSBC and DFSBC were evaluated at N discrete points on the free-surface, and the combination coefficients were determined for when the two curves pass the discrete points. Thus, the solution satisfies the differential equation (DE), bottom boundary condition (BBC), and the kinematic free surface boundary condition (KFSBC) exactly. The error in the dynamic free surface boundary condition (DFSBC) is less than 0.003%. The wave theory was simplified based on the assumption tanh $D{\approx}1$ in this paper. Unlike the perturbation method, the results are possible for steep waves and can be calculated without iteration. The result is very simple compared to the 5th Stokes' theory. Stokes' breaking-wave criterion has been checked in this study.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.23 no.6
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• pp.401-406
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• 2011

In this study, generation methods of oblique incident wave are newly proposed and examined using the fully non-linear numerical model with non-reflected wave generation system(LES-WASS-3D). In order to verify, free surface elevation and horizontal velocities are compared with $3^{rd}$ -order Stokes wave theory in 3-D oblique incident wave field. As a results, it is revealed that the numerical results by newly proposed technique are in good agreement with the theory.

• Journal of Ocean Engineering and Technology
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• v.17 no.6
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• pp.7-15
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• 2003

A Digital Wave Tank simulation technique, based on a finite-difference method and a modified marker-and-cell (MAC) algorithm, is applied in order to investigate the characteristics of nonlinear Tsunami propagations and their interactions with a 2D sloping beach, Ohkushiri Island, and to predict maximum wove run-up around the island. The Navier-Stokes (NS) and continuity equation are governed in the computational domain, and the boundary values are updated at each time step, by a finite-difference time-marching scheme in the frame of the rectangular coordinate system. The fully nonlinear, kinematic, free-surface condition is satisfied by the modified marker-density function technique. The near shore Tsunami is assumed to be a solitary wave, and is generated from the numerical wave-maker in the developed Digital Wave Tank. The simulation results are compared with the experiments and other numerical methods, based on the shallow-water wave theory.

• Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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• 2003.05a
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• pp.231-239
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• 2003

A Digital Wave Tank simulation technique based on a finite-difference method and a modified marker-and-cell (MAC) algorithm is applied to investigate the characteristics of nonlinear Tsunami propagations and their interactions with a 2D sloping beach and Ohkushiri island, and to predict maximum wave run-up around the island. The Navier-Stokes (NS) and continuity equation are governed in the computational domain and the boundary values updated at each time step by a finite-difference time-marching scheme in the frame of rectangular coordinate system. The fully nonlinear kinematic free-surface condition is satisfied by the modified marker-density function technique. The Nearshore Tsunami is assumed to be a solitary wave and generated from the numerical wavemaker in the developed Digital Wave Tank. The simulation results are compared with the experiments and other numerical methods based on the shallow-water wave theory.

• Water Engineering Research
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• v.1 no.4
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• pp.293-298
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• 2000

Optimal values of $\alpha$ characterizing the linear dispersion property in the modified Boussinesq equations are determined by minimizing the combined relative errors of the phase and group velocities. The value of $\alpha$ is fixed in previous studies, whereas it is varying in the present study. The phase and group velocities are calculated by using variable $\alpha$ and compared to those of the linear Stokes wave theory and previous studies. It is found that the present study produces the best match to the linear Stokes theory.

• 한국전산유체공학회:학술대회논문집
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• 2004.03a
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• pp.119-125
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• 2004

Unsteady nonlinear wave motions on the free surface over a plane beach of constant slope are numerically simulated using a finite difference method in rectangular grid system. Two-dimensional Navier-Stokes equations and the continuity equation are used for the computations. Irregular leg lengths and stars are employed near the boundaries of body and free surface to satisfy the boundary conditions. Also, the free surface which consists of markers or segments is determined every time step with the satisfaction of kinematic and dynamic free surface conditions. Moreover, marker-density method is also adopted to allow plunging jets impinging on the free surface. The second-order Stokes wave theory and solitary wave theory are employed for the generation of waves on the inflow boundary. For the simulation of wave breaking phenomena, the computations are carried out with the plane beach of constant slope in surf zone. The results are compared with each other. The marker-density method is better then the hybrid method. Also they are compared with other existing experimental results. The Agreement between the experimental data and the computation results is good.