Abstract
The finite element method (FEM) is employed to analyze the resonant oscillations of the liquid confined within multiple or an array of floating bodies with fully nonlinear boundary conditions on the free surface and the body surface in two dimensions. The velocity potentials at each time step are obtained through the FEM with 8-node quadratic shape functions. The finite element linear system is solved by the conjugate gradient (CG) method with a symmetric successive overelaxlation (SSOR) preconditioner. The waves at the open boundary are absorbed by the combination of the damping zone method and the Sommerfeld-Orlanski equation. Numerical examples are given by an array of floating wedge-shaped cylinders and rectangular cylinders. Results are provided for heave motions including wave elevations, profiles and hydrodynamic forces. Comparisons are made in several cases with the results obtained from the second order solution in the time domain. It is found that the wave amplitude in the middle region of the array is larger than those in other places, and the hydrodynamic force on a cylinder increases with the cylinder closing to the middle of the array.
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References
Cointe, R., Geyer, P., King, B., Molin, B. and Tramoni, M., 1990. Nonlinear and linear motions of a rectangular barge in a perfect fluid, Proc. 18th Symp. on Naval Hydro., Ann Arbor, Michigan, 85–98.
Evans, D. V. and Porter, R., 1997. Near-trapping of waves by circular arrays of vertical cylinders, Appl. Ocean Res., 19(2): 83–99.
Faltinsen, O. M. and Timokha, A. N., 2001. An adaptive multimodal approach to nonlinear sloshing in a rectangular tank, J. Fluid Mech., 432, 167–200.
Faltinsen, O. M., 1978. A numerical non-linear method of sloshing in tanks with two-dimensional flow, J. Ship Res., 22(3): 193–202.
He, W. Z., 1999. Water surface wave radiation generated by multiple cylinders oscillating with identical frequency, Applied Mathematics and Mechanics, 20(10): 1150–1159. (in Chinese)
Ma, Q. W., Wu, G. X. and Eatock Taylor, R., 2001a. Finite element simulation of fully nonlinear interaction between vertical cylinders and steep waves, Part 1: Methodology and numerical procedure, Int. J. Num. Methods Fluids, 36, 265–285.
Ma, Q. W., Wu, G. X. and Eatock Taylor, R., 2001b. Finite element simulation of fully nonlinear interaction between vertical cylinders and steep waves, Part 2: Numerical results and validation, Int. J. Num. Methods Fluids, 36, 287–308.
Malenica, Š., Eatock Taylor, R. and Huang, J. B., 1999. Second order water wave diffraction by an array of vertical cylinders, J. Fluid Mech., 390, 349–373.
Maniar, H. D. and Newman, J. N., 1997. Wave diffraction by a long array of cylinders, J. Fluid Mech., 339(01): 309–330.
Orlanski, I., 1976. A simple boundary condition for unbounded hyperbolic flows, J. Comput.Phys., 21, 251–269.
Tanizawa, K., 1996. Long time fully nonlinear simulation of floating body motions with artificial damping zone, Journal of the Society of Naval Architectures of Japan, 180, 311–319.
Ursell, F., 1951. Trapping modes in the theory of surface waves, Proceedings of the Cambridge Philosophical Society, 47(2): 347–358.
Wang, C. Z and Khoo, B. C., 2005. Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations, Ocean Eng., 32(2): 107–133.
Wang, C. Z, Wu, G. X. and Drake, K. R., 2007. Interactions between fully nonlinear water wave and non-wall-sided 3D structures, Ocean Eng., 34(8–9): 1182–1196.
Wang, C. Z. and Wu, G. X., 2006. An unstructured mesh based finite element simulation of wave interactions with non-wall-sided bodies, J. Fluids Struct., 22(4): 441–461.
Wang, C. Z. and Wu, G. X., 2007. Time domain analysis of second order wave diffraction by an array of vertical cylinders, J. Fluids Struct., 23(4): 605–631.
Wang, C. Z. and Wu, G. X., 2008. Analysis of second order resonance in wave interactions with floating bodies through a finite element method, Ocean Eng., 35(8–9): 717–726.
Wang, C. Z. and Wu, G. X., 2010. Interactions between fully nonlinear water waves and an array of cylinders in a wave tank, Ocean Eng., 37(4): 400–417.
Wang, C. Z., Wu, G. X. and Khoo, B. C., 2011. Fully nonlinear simulation of resonant motion of liquid confined between floating structures, Comput. Fluids, 44(1): 89–101.
Wu, G. X. and Eatock Taylor, R., 1994. Finite element analysis of two-dimensional non-linear transient water waves, Appl. Ocean Res., 16, 363–372.
Wu, G. X. and Eatock Taylor, R., 2003. The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies, Ocean Eng., 30(3): 387–400.
Wu, G. X., 1998. Hydrodynamic force on a rigid body during impact with liquid, J. Fluids Struct., 12(5): 549–559.
Wu, G. X., 2007. Second order resonance of sloshing in a tank, Ocean Eng., 34(17–18): 2345–2349.
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This work was financially supported by the Fundamental Research Funds for the Central Universities and NPRP 08-691-2-289 grant from Qatar National Research Fund (QNRF).
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Huang, Hc., Wang, Cz. & Leng, Jx. Fully nonlinear simulations of wave resonance by an array of cylinders in vertical motions. China Ocean Eng 27, 87–98 (2013). https://doi.org/10.1007/s13344-013-0008-x
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DOI: https://doi.org/10.1007/s13344-013-0008-x