Motion response prediction by hybrid panel-stick models for a semi-submersible with bracings

An Erratum to this article is available

Abstract

A diffraction-radiation analysis is usually required when the hydrodynamic interactions between structural members occur in short waves. For bracings or small cylindrical members, which play important roles in the vicinity of the natural frequency of a floating platform, special care should be taken into account for the effect of viscous damping. Two hybrid panel-stick models are, therefore, developed, through the combination of the standard diffraction-radiation method and the Morison’s formulae, considering the effect of small members differently. The fluid velocity is obtained directly by the panel model. The viscous fluid force is calculated for individual members by the stick model. A semi-submersible type platform with a number of fine cylindrical structures, which is designed as a floating foundation for multiple wind turbines, is analyzed as a numerical example. The results show that viscous force has significant influence on the hydrodynamic behavior of the floating body and can successfully be considered by the proposed hybrid models.

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References

  1. 1.

    Leblanc L, Petitjean F, Roy FL, Chen XB (1993) A mixed panel-stick hydrodynamic model applied to fatigue life assessment of semi-submersibles. In: Proceedings of 12th International Conference on Ocean, Offshore and Arctic Engineering, Glasgow, Scotland

  2. 2.

    Hooft JP (1972) Hydrodynamic aspects of semi-submersible platforms. PhD thesis, Delft University of Technology

  3. 3.

    Malenica S, Sireta FX, Bigot F, Derbanne Q, Chen XB (2010). An efficient hydro structure interface for mixed panel-stick hydrodynamic model. In: Proceedings of 25th International Workshop on Water Waves and Floating Bodies, Harbin, China

  4. 4.

    Veer RV (2008) Application of linearized Morison load in pipe lay stinger design. In: Proceedings of 27th International Conference on Ocean, Offshore and Arctic Engineering, Estoril, Portugal

  5. 5.

    Faltinsen OM (1990) Sea loads on ships and offshore structures. Cambridge University Press

  6. 6.

    Newman JN (1999) Heave response of a semi-submersible near resonance. In: Proceedings of 14th International Workshop on Water Waves and Floating Bodies, Port Huron, USA

  7. 7.

    Hu CH, Sueyoshi M, Kyozuka Y, Yoshida S, Ohya Y (2014) Development of new floating platform for multiple ocean renewable energy. In: Proceedings of the Grand Renewable Energy International Conference 2014, Tokyo

  8. 8.

    Newman JN (1986) Distributions of sources and normal dipoles over a quadrilateral panel. J Eng Math 20:113–126

    Article  Google Scholar 

  9. 9.

    Webster WC (1975) The flow about arbitrary three-dimensional smooth bodies. J Ship Res 19(4):206–218

    Google Scholar 

  10. 10.

    Kashiwagi M, Takagi K, Yoshida H, Murai M, Higo Y (2003) Fluid dynamics of floating bodies in practice: Part 1 numerical computation method of the motion response problems. Seizando Press (in Japanese)

  11. 11.

    Newman JN (1985) Algorithms for free-surface Green function. J Eng Math 19:57–67

    Article  MATH  Google Scholar 

  12. 12.

    Liu YY, Iwashita H, Hu CH (2014) A calculation method for finite depth free-surface Green function. Int J Naval Archit Ocean Eng 7(2) (To be appear)

  13. 13.

    Saad Y, Schultz MH (1986) GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Zhao Y, Graham JMR (1996) An iterative method for boundary element solution of large offshore structures using the GMRES solver. Ocean Eng 23(6):483–495

    Article  Google Scholar 

  15. 15.

    Hu CH, Sueyoshi M, Liu C, Liu YY (2014) Hydrodynamic analysis of a semi-submersible type floating wind turbine. J Ocean Wind Energy 1(4):202–208

    Google Scholar 

Download references

Acknowledgments

This research is supported in part by Grants-in-Aid for Scientific Research (B), MEXT (no. 15H04215). We would like to thank ClassNK (Nippon Kaiji Kyoukai), Oshima Shipbuilding Co. Ltd., Shin Kurushima Dockyard Co. Ltd., and Tsuneishi Holdings Corp. for funding this study and for permission to publish this paper. The first author gratefully acknowledges the financial support provided by the MEXT Scholarship (Grant no. 123471) from Japanese Government during the three-year Ph.D. research.

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Correspondence to Changhong Hu.

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An erratum to this article is available at http://dx.doi.org/10.1007/s00773-016-0427-5.

Appendix

Appendix

A procedure for calculating natural periods of a semi-submersible is given here. Since in the neighborhood region of the natural periods, the wave damping is negligible. From Eq. 6 we know the free motion equation of a semi-submersible without damping is

$$\left\{ { - \omega^{2} \left( {\left[ M \right] + \left[ a \right]} \right) + \left( {\left[ K \right] + \left[ C \right]} \right)} \right\}\left\{ \xi \right\} = 0.$$
(22)

In these matrices, many of the elements are zeroes, only leaves the diagonal terms and some coupling terms are nonzero. We can thus further deduce characteristic equations for each mode from Eq. 22 as the following: for heave motion, we have

$$- \omega^{2} \left( {M_{33} + a_{33} } \right) + K_{33} + C_{33} = 0.$$
(23)

For yaw motion, we have

$$- \omega^{2} \left( {M_{66} + a_{66} } \right) + K_{66} = 0.$$
(24)

For surge and pitch motion, we have

$$A_{1} \omega^{4} + B_{1} \omega^{2} + C_{1} = 0,$$
(25)

where

$$A_{1} = \left( {M_{11} + a_{11} } \right)\left( {M_{55} + a_{55} } \right) - \left( {M_{15} + a_{15} } \right)\left( {M_{51} + a_{51} } \right),$$
(26)
$$B_{1} = K_{15} \left( {M_{51} + a_{51} } \right) + K_{51} \left( {M_{15} + a_{15} } \right) - \left( {K_{55} + C_{55} } \right)\left( {M_{11} + a_{11} } \right) - K_{11} \left( {M_{55} + a_{55} } \right),$$
(27)

and

$$C_{1} = \left( {K_{55} + C_{55} } \right)K_{11} - K_{15} K_{51} .$$
(28)

For sway and roll motion, we have

$$A_{2} \omega^{4} + B_{2} \omega^{2} + C_{2} = 0,$$
(29)

where

$$A_{2} = \left( {M_{22} + a_{22} } \right)\left( {M_{44} + a_{44} } \right) - \left( {M_{24} + a_{24} } \right)\left( {M_{42} + a_{42} } \right),$$
(30)
$$B_{2} = K_{24} \left( {M_{42} + a_{42} } \right) + K_{42} \left( {M_{24} + a_{24} } \right) - \left( {K_{44} + C_{44} } \right)\left( {M_{22} + a_{22} } \right) - K_{22} \left( {M_{44} + a_{44} } \right),$$
(31)

and

$$C_{2} = \left( {K_{44} + C_{44} } \right)K_{22} - K_{24} K_{42} .$$
(32)

It should be noted that, for Eqs. 25 or 29, the two modes are coupled and thus should be solved simultaneously. The two positive solutions of Eq. 25 correspond to the surge (the smaller) and the pitch (the larger) natural angular frequencies, respectively; similarly, the two positive solutions of Eq. 29 correspond to the sway (the smaller) and the roll (the larger) natural angular frequencies, respectively. In addition, since the added mass coefficients which depend on the wave frequency are contained, all the above equations need to be solved through iteration processes to find the exact values.

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Liu, Y., Hu, C., Sueyoshi, M. et al. Motion response prediction by hybrid panel-stick models for a semi-submersible with bracings. J Mar Sci Technol 21, 742–757 (2016). https://doi.org/10.1007/s00773-016-0390-1

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Keywords

  • Floating offshore wind turbine
  • Semi-submersible
  • Potential theory