• Title/Summary/Keyword: modified mild slope equation

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Elliptic Numerical Wave Model Solving Modified Mild Slope Equation (수정완경사방정식의 타원형 수치모형)

  • YOON JONG-TAE
    • Journal of Ocean Engineering and Technology
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    • v.18 no.4 s.59
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    • pp.40-45
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    • 2004
  • An efficient numerical model of the modified mild slope equation, based on the robust iterative method is presented. The model developed is verified against other numerical experimental results, related to wave reflection from an arc-shaped bar and wave transformation over a circular shoal. The results show that the modified mild slope equation model is capable of producing accurate results for wave propagation in a region where water depth varies substantially, while the conventional mild slope equation model yeilds large errors, as the mild slope assumption is violated.

A Study on the Extension of Mild Slope Equation (완경사 방정식의 확장에 관한 연구)

  • 천제호;김재중;윤항묵
    • Journal of Ocean Engineering and Technology
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    • v.18 no.2
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    • pp.18-24
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    • 2004
  • In this study, the Mild slope equation is extended to both rapidly varying topography and nonlinear waves, using the Hamiltonian principle. It is shown that this equation is equivalent to the modified mild-slope equation (Kirby and Misra, 1998) for small amplitude wave, and it is the same form with the nonlinear mild-slope equation (Isobe, 1994) for slowly varying bottom topography. Comparing its numerical solutions with the results of some hydraulic experiments, there is good agreement between them.

Reassessment of the Mild Slope Equations (완경사 파랑식들의 재평가)

  • Seo, Seung-Nam
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.19 no.6
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    • pp.521-532
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    • 2007
  • In the derivation of mild slope equation, a Galerkin method is used to rigorously form the Sturm-Liouville problem of depth dependent functions. By use of the canonical transformation to the dependent variable of the equation a reduced Helmholtz equation is obtained which exclusively consists of terms proportional to wave number, bottom slope and bottom curvature. Through numerical studies the behavior of terms is shown to play an important role in wave transformations over variable depth and it is proved that their relative magnitudes limit applicability of the mild slope equation(MSE) against the modified mild slope equation(MMSE).

A Note on the Modified Mild-Slope Equation (修正 緩傾斜方程式에 대한 小考)

  • Kyung Doug Suh;Woo Sun Park;Chang Hoon Lee
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.10 no.2
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    • pp.55-63
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    • 1998
  • Recently the modified mild-slope equation has been developed by several researchers using different approaches, which, compared to the Berkhoff's mild-slope equation, includes additional terms proportional to the square of bottom slope and to the bottom curvature. By examining this equation, it is shown that both terms are equally important in intermediate-depth water, but in shallow water the influence of the bottom curvature term diminishes while that of the bottom slope square term remains significant. In order to examine the importance of these terms in more detail, the modified mild-slope equation and the Berkhoff's mild-slope equation are tested for the problems of wave reflection from a plane slope, a non-plane slope, and periodic ripples. It is shown that, when only the bottom slope is concerned, the mild-slope equation can give accurate results up to a slope of 1 in 1 rather than 1 in 3, which, until now, has been known as the limiting bottom slope for its proper application. It is also shown that the bottom curvature term plays an important role in modeling wave propagation over a bottom topography with relatively mild variation, but, where the bottom slope is not small, the bottom slope square term should also be included for more accurate results.

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A Parabolic Model to the Modified Mild Slope Equation (수정 완경사 파랑식에 대한 포물형 근사식 모형)

  • Seo, Seung-Nam;Lee, Jong-Chan
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.18 no.4
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    • pp.360-371
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    • 2006
  • In order to calculate waves propagating into the shallow water region, a generalized parabolic approximate model is presented. The model is derived from the modified mild slope equation and includes all the existing parabolic models presented in the paper. Numerical results are presented in comparison to laboratory data of Berkhoff et al.(1982). The existing parabolic model shows almost same accuracy against the modified parabolic model and both results of models stand in closer agreement to the laboratory data. Therefore the existing parabolic model based on mild slope equation is a useful tool to compute shallow water waves which turns out to be more fast and stable in computational aspect.

Efficient Iterative Solvers for Modified Mild Slope Equation (수정완경사방정식을 위한 반복기법의 효율성 비교)

  • Yoon, Jong-Tae;Park, Seung-Min
    • Journal of Ocean Engineering and Technology
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    • v.20 no.6 s.73
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    • pp.61-66
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    • 2006
  • Two iterative solvers are applied to solve the modified mild slope equation. The elliptic formulation of the governing equation is selected for numerical treatment because it is partly suited for complex wave fields, like those encountered inside harbors. The requirement that the computational model should be capable of dealing with a large problem domain is addressed by implementing and testing two iterative solvers, which are based on the Stabilized Bi-Conjugate Gradient Method (BiCGSTAB) and Generalized Conjugate Gradient Method (GCGM). The characteristics of the solvers are compared, using the results for Berkhoff's shoal test, used widely as a benchmark in coastal modeling. It is shown that the GCGM algorithm has a better convergence rate than BiCGSTAB, and preconditioning of these algorithms gives more than half a reduction of computational cost.

A Study on the Extension of Mild Slope Equation (완경사 방정식의 확장에 관한 연구)

  • Chun, Je-Ho;Kim, Jae-Joong
    • Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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    • 2003.05a
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    • pp.72-77
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    • 2003
  • In this study, Mild slope equation is extended to both of rapidly varying topography and nonlinear waves in a Hamiltonian formulation. It is shown that its linearzed form is the same as the modified mild-slope equation proposed by Kirby and Misra(1998) And assuming that the bottom slopes are very slowly, it is the equivalent with nonlinear mild-slope equation proposed by Isobe(]994) for the monochromatic wave. Using finite-difference method, it is solved numerically and verified, comparing with the results of some hydraulic experiments. A good agreement between them is shown.

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Solution Comparisons of Modified Mild Slope Equation and EFEM Plane-wave Approximation (수정 완경사파랑식과 EFEM 평면파 근사식의 해 비교)

  • Seo, Seung-Nam
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.21 no.2
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    • pp.117-126
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    • 2009
  • In order to test the accuracy between the modified mild slope equation (MMSE) without evanescent modes and the plane-wave approximation (PA) of eigenfunction expansion method, various numerical results from both models are presented. In this study, analytical solutions of two models are employed, one based on the MMSE derived by Porter (2003) and the other on the scatterer method of PA by Seo (2008a). Judging from direct comparisons against existing results of rapidly varying topography, the PA model gives better predictions of the wave propagation than the MMSE model.

Elliptic Numerical Wave Model Solving Modified Mild Slope Equation with Nonlinear Shoaling and Wave Breaking (비선형 천수와 쇄파를 고려한 수정완경사방정식의 타원형 수치모형)

  • Yoon, Jong-Tae
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.21 no.1
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    • pp.39-44
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    • 2009
  • To improve the accuracy of numerical simulation of wave trans- formation across the surf zone, nonlinear shoaling effect based on Shuto's empirical formula and breaking mechanism are induced in the elliptic modified mild slope equation. The variations of shoaling coefficient with relative depth and deep water wave steepness are successfully reproduced and show good agreements with Shuto's formula. Breaking experiments show larger wave height distributions than linear model due to nonlinear shoaling but breaking mechanism shows a little bit larger damping in 1/20 beach slope experiment.