Journal of Korean Society of Coastal and Ocean Engineers (한국해안·해양공학회논문집)

Korean Society of Coastal and Ocean Engineers (한국해안·해양공학회)
권호 표지
DOI QR코드

DOI QR Code

Linear Spectral Method for Simulating the Generation of Regular Waves by a Moving Bottom in a 3-dimensional Space

3차원 공간에서 바닥의 움직임에 의한 규칙파의 생성을 모의할 수 있는 선형 스펙트럼법

  • Jae-Sang Jung (Hwa-an Project Office, Korea Rural Community Corporation) ;
  • Changhoon Lee (Dept. of Civil and Environmental Engineering, Sejong University)
  • 정재상 (한국농어촌공사 화안사업단) ;
  • 이창훈 (세종대학교 건설환경공학과)
  • Received : 2024.03.26
  • Accepted : 2024.04.17
  • Published : 2024.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we introduce a linear spectral method capable of simulating wave generation and transformation caused by a moving bottom in a 3-dimensional space. The governing equations are linear dynamic free-surface boundary conditions and linear kinematic free-surface boundary conditions, which are solved in Fourier space. Solved velocity potential and free-surface displacement should satisfy continuity equation and kinematic bottom boundary condition. For numerical analysis, a 4th order Runge-Kutta method was utilized to analyze the time integral. The results obtained in Fourier space can be converted into velocity potential and free-surface displacement in a real space using inverse Fourier transform. Regular waves generated by various types of moving bottoms were simulated with the linear spectral method. Additionally, obliquely generated regular waves using specified bottom movements were simulated. The results obtained from the spectral method were compared to analytical solutions, showing good agreement between the two.

본 연구에서는 3차원 공간에서 바닥의 움직임에 따른 선형파의 생성을 모의할 수 있는 스펙트럼 법을 소개한다. 지배방정식은 선형의 동역학적 및 운동학적 자유수면 경계조건이며, 두 식은 Fourier 공간에서 해석된다. 해석된 속도포텐셜 및 자유수면변위는 연속방정식과 운동학적 바닥경계조건을 항상 만족해야 한다. 수치해석에서 시간 적분은 4차 Runge-Kutta 법을 이용하여 해석하였다. Fourier 공간에서 해석한 결과는 Fourier 역변환을 통해 실제 공간에서의 속도포텐셜과 자유수면변위로 표현된다. 본 수치모델을 이용하여 다양한 형상의 바닥이 규칙적으로 움직이는 경우 생성되는 규칙파에 대해 모의하였다. 또한 바닥의 움직임을 이용하여 비스듬히 전파하는 규칙파의 생성도 모의하였다. 수치모델의 결과는 해석해와 비교하였으며, 거의 일치하는 결과를 보였다.

Keywords

Acknowledgement

이 논문은 2024년도 해양수산부 재원으로 해양수산과학기술진흥원의 지원을 받아 수행된 연구임(RS-2023-00239837, 유망기술 Scale-up사업).

References

  1. Craig, W. and Sulem, C. (1993). Numerical simulation of gravity waves. Journal of Computational Physics, 108, 73-83.  https://doi.org/10.1006/jcph.1993.1164
  2. Dommermuth, D.G. and Yue, D.K.P. (1987). A high-order spectral method for the study of nonlinear gravity waves. Journal of Fluid Mechanics, 184, 267-288.  https://doi.org/10.1017/S002211208700288X
  3. Dytykh, D., Dias, F. and Kervella, Y. (2006). Linear theory of wave generation by a moving bottom. Comptes Rendus Mathematique, 343(7), 499-504.  https://doi.org/10.1016/j.crma.2006.09.016
  4. Dutykh, D. and Dias, F. (2009). Energy of tsunami waves generated by bottom motion. Proceedings of the Royal Society A, 465(2103), 725-744. 
  5. Guyenne, P. and Nicholls, D.P. (2007). A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM Journal of Scientific Computation, 30(1), 81-101.  https://doi.org/10.1137/060666214
  6. Hammack, J.L. (1973). A note on tsunamis: their generation and propagation in an ocean of uniform depth. Journal of Fluid Mechanics, 60(4), 769-799.  https://doi.org/10.1017/S0022112073000479
  7. Jung, J.-S. (2023). A simplified numerical method for simulating the generation of linear waves by a moving bottom. Journal of Korean Society of Coastal and Ocean Engineers, 35(2), 41-48 (in Korean).  https://doi.org/10.9765/KSCOE.2023.35.2.41
  8. Jung, J.-S. and Lee, C. (2022a). An analytical study of regular waves generated by bottom wave makers in a 3-dimensional wave basin. Journal of Korean Society of Coastal and Ocean Engineers, 34(4), 93-99 (in Korean).  https://doi.org/10.9765/KSCOE.2022.34.4.93
  9. Jung, J.-S. and Lee, C. (2022b). Development of analytical solutions on velocities of regular waves generated by bottom wave makers in a flume. Journal of Korean Society of Coastal and Ocean Engineers, 34(3), 58-71 (in Korean).  https://doi.org/10.9765/KSCOE.2022.34.3.58
  10. Jung, J.-S., Lee, C. and Park, Y.S. (2021). Variation of wave forces along a semi-infinite breakwater due to wave diffraction. Journal of Waterway Port Coastal and Ocean Engineering, 147(5), 04021028. 
  11. Jung, J.-S., Lee, C., Tran, M.T. and Park, Y.S. (2023). Generation of linear waves with bottom wave makers using analytical solution and extended mild-slope equations. Journal of Waterway Port Coastal and Ocean Engineering, 149(5), 04023013. 
  12. Jung, J.-S., Pham, V.K. and Lee, C. (2018). A study of performance of bottom moving wave maker: comparison of analytical solution and numerical analysis. Proceeding of the Korean Association of Ocean Science and Technology, Jeju ICC, Rep. of Korea, 54-57 (in Korean). 
  13. Jung, T. and Son, S. (2021). Active tsunami generation by tectonic seafloor deformations of arbitrary geometry considering rupture kinematics. Wave Motion, 100, 102683. 
  14. Klahn, M., Madsen, P.A. and Fuhrman, D.R. (2020). On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves. Proceedings of the Royal Society A, 476(2243), 20200436. 
  15. Lee, S.-J., Yates, G.T. and Wu, T.Y. (1989). Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. Journal of Fluid Mechanics, 199, 569-593.  https://doi.org/10.1017/S0022112089000492
  16. Lu, H., Park, Y.S. and Cho, Y.-S. (2017a). Modelling of long waves generated by bottom-tilting wave maker. Coastal Engineering, 122, 1-9.  https://doi.org/10.1016/j.coastaleng.2017.01.007
  17. Lu, H., Park, Y.S. and Cho, Y.-S. (2017b). Investigation of long waves generated by bottom-tilting wave maker. Coastal Engineering Journal, 59(04), 1750018. 
  18. Madsen, P.A. and Hansen, A.B. (2012). Transient waves generated by a moving bottom obstacle: a new near-field solution. Journal of Fluid Mechanics, 697, 237-272.  https://doi.org/10.1017/jfm.2012.54
  19. Mahjouri, S., Shabani, R., Badiei, P. and Rezazadeh, G. (2021). A bottom mounted wavemaker in water wave flumes. Journal of Hydraulic Research, 59(4), 663-669.  https://doi.org/10.1080/00221686.2020.1818314
  20. Tanioka, Y. and Satake, K. (1996). Tsunami generation by horizontal displacement of ocean bottom. Geophysical Research Letters, 23(8), 861-864.  https://doi.org/10.1029/96GL00736
  21. West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M. and Milton, R.L. (1987). A new numerical method for surface hydrodynamics. Journal of Geophysical Research: Oceans, 92(C11), 11803-11824.  https://doi.org/10.1029/JC092iC11p11803
  22. Whittaker, C.N., Nokes, R.I., Lo, H.Y., Liu, P.L.F. and Davidson, M.J. (2017). Physical and numerical modelling of tsunami generation by a moving obstacle at the bottom boundary. Environmental Fluid Mechanics, 17, 929-958.  https://doi.org/10.1007/s10652-017-9526-z
  23. Wu, T.Y.T. (1987). Generation of upstream advancing solitons by moving disturbances. Journal of Fluid Mechanics, 184, 75-99.  https://doi.org/10.1017/S0022112087002817
  24. Zakharov, V.E. (1968). Stability of periodic waves on finite amplitude on the surface of a deep fluid. Zhurnal Prikladnoi Mekhaniki I Technicheskoi Fiziki, 9(2), 86-94.