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Numerical Simulation of One-Dimensional Madsen-Sørensen Extended Boussinesq Equations Using Crowhurst-Zhenquan Scheme

Crowhurst-Zhenquan 방법을 이용한 1차원 Madsen-Sørensen 확장형 Boussinesq 방정식의 수치 시뮬레이션

  • Kang, Sangmuk (Department of Naval Architecture & Ocean Engineering, Pusan National University) ;
  • Park, Jinsoo (Department of Naval Architecture & Ocean Engineering, Pusan National University) ;
  • Jang, Taek Soo (Department of Naval Architecture & Ocean Engineering, Pusan National University)
  • 강상묵 (부산대학교 조선해양공학과) ;
  • 박진수 (부산대학교 조선해양공학과) ;
  • 장택수 (부산대학교 조선해양공학과)
  • Received : 2017.04.14
  • Accepted : 2017.10.19
  • Published : 2017.10.31

Abstract

The aim of this paper is to apply the Crowhurst-Zhenquan scheme to one-dimensional Madsen-Sørensen extended Boussinesq equations. In order to verify the application of the aforementioned scheme, the propagation of solitary waves was simulated for two different cases of submarine topography; e.g., a plane beach and submerged breakwater. The simulated results are compared to the results of recent studies and show favorable agreement. The behavior of progressive waves is also investigated.

Keywords

References

  1. Boussinesq, J., 1872. Theorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de Mathematiques Pures et Appliquees, 2(17), 55-108.
  2. Crowhurst, P., Zhenquan, L., 2013. Numerical Solutions of One-Dimensional Shallow Water Equations. UKSim 15th International Conference on IEEE, 55-60.
  3. Ghadimi, P., Rahimzadeh, A., Chekab, M., 2016. Numerical Investigation of Free Surface Elevation and Celerity of Solitary Waves Passing over Submerged Trapezoidal Breakwaters. The International Journal of Multiphysics, 9(1), 61-74. https://doi.org/10.1260/1750-9548.9.1.61
  4. Grilli, S.T., Losada, M.A., Martin, F., 1994. Characteristics of Solitary Wave Breaking Induced by Breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 120(1), 74-92. https://doi.org/10.1061/(ASCE)0733-950X(1994)120:1(74)
  5. Jang, T.S., 2017. A New Dispersion-Relation Preserving Method for Integrating the Classical Boussinesq Equation. Communications in Nonlinear Science Numerical Simulation, 43, 118-138. https://doi.org/10.1016/j.cnsns.2016.06.025
  6. Jang, T.S., 2018a. An improvement of convergence of a dispersionrelation preserving method for the classical Boussinesq equation. Communications in Nonlinear Science and Numerical Simulation, 56, 144-160. https://doi.org/10.1016/j.cnsns.2017.07.024
  7. Jang, T.S., 2018b. A new functional iterative algorithm for the regularized long-wave equation using an integral equation formalism. Journal of Scientific Computing, In press.
  8. Kang, S., Park, J., Jang, T.S., 2017. Numerical Simulation of 1D Shallow Water Waves Using Madsen-Sorensen Extended Boussinesq Equations on Slowly Varying Bottom. Proceedings of the Spring Conference of the Korea Ocean Engineering Society, Busan
  9. Klopman, G., 2010. Variational Boussinesq Modelling of Surface Gravity Waves over Bathymetry. Doctoral dissertation, Wohrmann Print Service.
  10. Lee, C., Cho, Y.S., 2000. Extendability of Extended Boussinesq Equations : 1. Linear Dispersion Relation. Journal of Korean Society of Civil Engineering, 20(4B), 545-552.
  11. Madsen, O.S., Mei, C.C., 1969. The Transformation of a Solitary Wave over Uneven Bottom. Journal of Fluid Mechanics, 39(04), 781-791. https://doi.org/10.1017/S0022112069002461
  12. Madsen, P.A., Sorensen, O.R., 1992. A New Form of the Boussinesq Equations with Improved Linear Dispersion Characteristics. Part 2. A Slowly Varing Bathymetry. Coastal Engineering, 18, 183-204. https://doi.org/10.1016/0378-3839(92)90019-Q
  13. Peregrine, D.H., 1967. Long Waves on a Beach. Journal of Fluid Mechanics, 27, 815-827. https://doi.org/10.1017/S0022112067002605
  14. Seabra-Santos, F.J., Renouard, D.P., Temperville, A.M., 1987. Numerical and Experimental Study of the Transformation of a Solitary Wave over a Shelf or Isolated Obstacle. Journal of Fluid Mechanics, 176, 117-137. https://doi.org/10.1017/S0022112087000594
  15. Wang, X., Liu, P.L.F., 2011. An Explicit Finite Difference Model for Simulating Weakly Nonlinear and Weakly Dispersive Waves over Slowly Varying Water Depth. Coastal Engineering, 58(2), 173-183. https://doi.org/10.1016/j.coastaleng.2010.09.008
  16. Wikipedia, 2016. Wave Shoaling. [Online] (Updated November 2016) Available at : https://en.wikipedia.org/wiki/Wave_shoaling/ [Accessed November 2016]