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
Similar content being viewed by others
Availability of data and materials
The authors declare that all of the material is owned by them.
References
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)
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)
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)
Chakrabarti, S.K.: Wave interaction with multiple horizontal cylinders. Appl Ocean Res 1(4), 213–216 (1979)
Chian, C., Ertekin, R.C.: Diffraction of solitary waves by submerged horizontal cylinders. Wave Motion 15, 121–142 (1992)
Cruz, J.: Ocean wave energy: current status and future perspectives. Springer-Verlag, Berlin (2008)
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)
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)
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)
Greenhow, M., Ahn, S.I.: Added mass and damping of horizontal circular cylinder sections. Ocean Engng 15(5), 495–504 (1988)
Higdon, R.L.: Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math Comput 47, 437–459 (1986)
Higdon, R.L.: Numerical absorbing boundary conditions for the wave equation. Math Comput 49, 65–90 (1987)
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)
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)
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)
Li, Y., Lin, M.: Hydrodynamic coefficients induced by waves and currents for submerged circular cylinder. Proced Engng 4, 253–261 (2010)
Liao, Z.P.: Extrapolation non-reflecting boundary conditions. Wave Motion 24, 117–138 (1996)
Makarenko, N.I.: Nonlinear interaction of submerged cylinder with free surface. J Offshore Mech Arct 125, 72–75 (2003)
McCauley, G., Wolgamot, H., Orszaghova, J., Draper, S.: Linear hydrodynamic modelling of arrays of submerged oscillating cylinders. Appl Ocean Res 81, 1–14 (2018)
Mehlum, E.: A circular cylinder in water waves. Appl Ocean Res 2(4), 171–177 (1980)
Ning, D., Zhou, Y., Zhang, C.: Hydrodynamic modeling of a novel dual-chamber owc wave energy converter. Appl Ocean Res 78, 180–191 (2018)
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)
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)
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
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
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
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
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
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
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)
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)
Thomas, G.P., Evans, D.V.: A hydrodynamic model of a submerged lenticular wave energy device. Appl Ocean Res 5(2), 69–79 (1983)
Tyvand, P.A.: Wave radiation and diffraction from a small submerged circular cylinder. Fluid Dyn Res 9, 279–288 (1992)
Venugopal, V., Varyani, K.S., Barltrop, N.D.: Wave force coefficients for horizontally submerged rectangular cylinders. Ocean Engng 33, 1669–1704 (2006)
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)
Wu, G.X.: Second-order wave radiation by a submerged horizontal circular cylinder. Appl Ocean Res 15, 293–303 (1993)
Wu, T.M.: Hydrodynamic loadings acting on submerged bodies advancing in waves-an assessment of accuracy. Engng Anal Bound Elem 13, 21–33 (1994)
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)
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
Contributions
Un-Ryong Rim contributed to methodology and supervision. Pil-Sung Dong was involved in calculation. Chol-Guk Jang contributed to writing.
Corresponding author
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.
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-023-00682-x