Skip to main content
SpringerLink
Log in

Free surface wave interaction with a submerged body using a DtN boundary condition

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

Recently, Rim (Ocean Engng 239:711, 2021; J Ocean Engng Mar Energy 9:41-51, 2023 ) suggested an exact DtN artificial boundary condition to study the three-dimensional wave diffraction by stationary bodies. This paper is concerned with three-dimensional linear interaction between a submerged oscillating body with arbitrary shape and the regular water wave with finite depth. An exact Dirichlet-to-Neumann (DtN) boundary condition on a virtual cylindrical surface is derived, where the virtual surface is chosen so as to enclose the body and extract an interior subdomain with finite volume from the horizontally unbounded water domain. The DtN boundary condition is then applied to solve the interaction between the body and the linear wave in the interior subdomain by using boundary integral equation. Based on verification of the present model for a submerged vertical cylinder, the model is extended to the case of a submerged chamfer box with fillet radius in order to study 6-DoF oscillatory motion of the body under the free surface wave.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Availability of data and materials

The authors declare that all of the material is owned by them.

References

  1. Aranha, J.A.P., Pinto, M.O.: Wave radiation by a deep submerged cylinder with application to ocean structure. Appl Ocean Res 15, 327–335 (1994)

    Article  Google Scholar 

  2. Bai, J., Ma, N., Gu, X.: Study of interaction between wave-current and the horizontal cylinder located near the free surface. Appl Ocean Res 67, 44–58 (2017)

    Article  Google Scholar 

  3. Berggren, L., Johansson, M.: Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate. Appl Ocean Res 14, 51–58 (1992)

    Article  Google Scholar 

  4. Chakrabarti, S.K.: Wave interaction with multiple horizontal cylinders. Appl Ocean Res 1(4), 213–216 (1979)

    Article  Google Scholar 

  5. Chian, C., Ertekin, R.C.: Diffraction of solitary waves by submerged horizontal cylinders. Wave Motion 15, 121–142 (1992)

    Article  MathSciNet  Google Scholar 

  6. Cruz, J.: Ocean wave energy: current status and future perspectives. Springer-Verlag, Berlin (2008)

    Book  Google Scholar 

  7. Ding, B., Sergiienko, N., Meng, F., Cazzolato, B., Hardy, P., Arjomandi, M.: The application of modal analysis to the design of multi-mode point absorber wave energy converters. Ocean Engng 171, 603–618 (2019)

    Article  Google Scholar 

  8. Dong, M.S., Miao, G.P., Zhu, R.C., Fan, J.: Analytic solution for wave diffraction of a submerged hollow sphere with an opening hole. J Hydrodyn 22(3), 295–304 (2010)

    Article  Google Scholar 

  9. Eatock Taylor, R., Hung, S.M., Chau, F.P.: On the distribution of second order pressure on a vertical circular cylinder. Appl Ocean Res 11(4), 183–193 (1989)

    Article  Google Scholar 

  10. Greenhow, M., Ahn, S.I.: Added mass and damping of horizontal circular cylinder sections. Ocean Engng 15(5), 495–504 (1988)

    Article  Google Scholar 

  11. Higdon, R.L.: Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math Comput 47, 437–459 (1986)

    Google Scholar 

  12. Higdon, R.L.: Numerical absorbing boundary conditions for the wave equation. Math Comput 49, 65–90 (1987)

    Article  MathSciNet  Google Scholar 

  13. Hill, J., Laycock, S., Chai, S., Balash, C., Morand, H.: Hydrodynamic loads and response of a mid water arch structure. Ocean Engng 83, 76–86 (2014)

    Article  Google Scholar 

  14. Iskandarani, M., Liu, P.L.F.: Multiple scattering of surface water-waves and wave forces on cylinder arrays. Appl Ocean Res 10(4), 170–180 (1988)

    Article  Google Scholar 

  15. Jiang, S.C., Gou, Y., Teng, B., Ning, D.Z.: Analytical solution of a wave diffraction problem on a submerged cylinder. J Engng Mech 140(1), 225–232 (2014)

    Google Scholar 

  16. Li, Y., Lin, M.: Hydrodynamic coefficients induced by waves and currents for submerged circular cylinder. Proced Engng 4, 253–261 (2010)

    Article  Google Scholar 

  17. Liao, Z.P.: Extrapolation non-reflecting boundary conditions. Wave Motion 24, 117–138 (1996)

    Article  MathSciNet  Google Scholar 

  18. Makarenko, N.I.: Nonlinear interaction of submerged cylinder with free surface. J Offshore Mech Arct 125, 72–75 (2003)

    Article  Google Scholar 

  19. McCauley, G., Wolgamot, H., Orszaghova, J., Draper, S.: Linear hydrodynamic modelling of arrays of submerged oscillating cylinders. Appl Ocean Res 81, 1–14 (2018)

    Article  Google Scholar 

  20. Mehlum, E.: A circular cylinder in water waves. Appl Ocean Res 2(4), 171–177 (1980)

    Article  Google Scholar 

  21. Ning, D., Zhou, Y., Zhang, C.: Hydrodynamic modeling of a novel dual-chamber owc wave energy converter. Appl Ocean Res 78, 180–191 (2018)

    Article  Google Scholar 

  22. Pagani, C.D., Pierotti, D.: Exact solution of the wave-resistance problem for a submerged cylinder. i. linearized theory. Arch Ration Mech Anal 149, 271–288 (1999)

    Article  MathSciNet  Google Scholar 

  23. Pagani, C.D., Pierotti, D.: Exact solution of the wave-resistance problem for a submerged cylinder. ii. the non-linear problem. Arch Ration Mech Anal 149, 289–327 (1999)

    Article  MathSciNet  Google Scholar 

  24. Rim, U.R.: Wave diffraction by floating bodies in water of finite depth using an exact DtN boundary condition. Ocean Engng 239(109), 711 (2021). https://doi.org/10.1016/j.oceaneng.2021.109711

    Article  Google Scholar 

  25. Rim UR (2023a) Numerical simulation for water wave motion over undulated seabed in front of a vertical wall by using an exact DtN boundary condition. In: 6th International conference on mathematics and statistics, Leipzig, Germany, https://doi.org/10.1145/3613347.3613367,available online 30 August 2023

  26. Rim, U.R.: Wave interaction with multiple bodies bottom-mounted on an undulated seabed using an exact DtN artificial boundary condition. J Ocean Engng Mar Energy 9(2), 41–51 (2023). https://doi.org/10.1007/s40722-022-00275-6

    Article  Google Scholar 

  27. Rim, U.R.: Wave radiation by a floating body in water of finite depth using an exact DtN boundary condition. J Ocean Univ China 22, 1497–1504 (2023). https://doi.org/10.1007/s11802-023-5358-2

    Article  Google Scholar 

  28. Rim, U.R., Choe, G.H., Ri, N.H., Jon, M.H., Pae, W.S., Han, U.H.: An exact DtN artificial boundary condition for motion analysis of water wave with undulated seabed. Wave Motion 116(103), 063 (2023). https://doi.org/10.1016/j.wavemoti.2022.103063

    Article  MathSciNet  Google Scholar 

  29. Rim, U.R., Ri, Y.G., Do, W.C., Dong, P.S., Kim, C.W., Kim, J.S.: Wave diffraction and radiation by a fully-submerged body in front of a vertical wall by using an exact DtN artificial boundary condition. Engng Anal Bound Elem 151, 19–29 (2023). https://doi.org/10.1016/j.enganabound.2023.02.053

    Article  MathSciNet  Google Scholar 

  30. Shao, Y.L., Faltinsen, O.M.: Use of body-fixed coordinate system in analysis of weakly nonlinear wave-body problems. Appl Ocean Res 32, 20–33 (2010)

    Article  Google Scholar 

  31. Silva, A.F.T., Peregrine, D.H.: Nonlinear perturbations on a free surface induced by a submerged body: a boundary integral approach. Engng Anal Bound Elem 7(4), 214–222 (1990)

    Article  Google Scholar 

  32. Thomas, G.P., Evans, D.V.: A hydrodynamic model of a submerged lenticular wave energy device. Appl Ocean Res 5(2), 69–79 (1983)

    Article  Google Scholar 

  33. Tyvand, P.A.: Wave radiation and diffraction from a small submerged circular cylinder. Fluid Dyn Res 9, 279–288 (1992)

    Article  Google Scholar 

  34. Venugopal, V., Varyani, K.S., Barltrop, N.D.: Wave force coefficients for horizontally submerged rectangular cylinders. Ocean Engng 33, 1669–1704 (2006)

    Article  Google Scholar 

  35. Wu, B.J., Zheng, Y.H., You, Y.G., Jie, D.S., Chen, Y.: On diffraction and radiation problem for two cylinders in water of finite depth. Ocean Engng 33, 679–704 (2006)

    Article  Google Scholar 

  36. Wu, G.X.: Second-order wave radiation by a submerged horizontal circular cylinder. Appl Ocean Res 15, 293–303 (1993)

    Article  Google Scholar 

  37. Wu, T.M.: Hydrodynamic loadings acting on submerged bodies advancing in waves-an assessment of accuracy. Engng Anal Bound Elem 13, 21–33 (1994)

    Article  Google Scholar 

  38. Xu, G., Duan, W.Y.: Time-domain simulation of wave-structure interaction based on multi-transmitting formula coupled with damping zone method for radiation boundary condition. Appl Ocean Res 42, 136–143 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the anonymous reviewers of this paper for their constructive comments.

Funding

The authors declare no funding with this paper.

Author information

Authors and Affiliations

  1. Institute of Ocean Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

    Un-Ryong Rim

  2. Faculty of Naval Architecture and Ocean Engineering, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea

    Pil-Sung Dong & Chol-Guk Jang

Authors
  1. Un-Ryong Rim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pil-Sung Dong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chol-Guk Jang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Un-Ryong Rim contributed to methodology and supervision. Pil-Sung Dong was involved in calculation. Chol-Guk Jang contributed to writing.

Corresponding author

Correspondence to Un-Ryong Rim.

Ethics declarations

Conflict of interest

The authors declare no competing interests with this paper.

Ethical Approval

The authors demonstrate that they adhered to the accepted ethical standards of the present paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rim, UR., Dong, PS. & Jang, CG. Free surface wave interaction with a submerged body using a DtN boundary condition. Theor. Comput. Fluid Dyn. 38, 75–87 (2024). https://doi.org/10.1007/s00162-023-00682-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-023-00682-x

Keywords

Search

Navigation