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Analytical methods to predict the surf-riding threshold and the wave-blocking threshold in astern seas

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Abstract

For the safe design and operation of high-speed craft, it is important to predict the behavior of these craft in following and astern quartering seas as it is well known that serious problems can occur when a vessel is forced by the waves to travel at wave speed. The surf-riding threshold is the lower speed limit above which the vessel will be forced to travel at wave speed (usually on the face of the wave) which is generally accepted to be a prerequisite for broaching. While a vessel travelling below this speed will experience significant changes in its longitudinal speed in the wave, it will not be forced to travel at wave speed. For high-speed craft, the wave-blocking threshold also becomes important. This is the upper speed, below which the vessel will also be forced to travel at wave speed (usually on the back of the wave) and is related to the possibility of bow diving. By the application of a polynomial approximation to the wave-induced surge force, including the nonlinear surge equation, an analytical formula to predict both the surf-riding and the wave-blocking thresholds is proposed. Comparative results of the surf-riding threshold and wave-blocking threshold predicted utilizing the proposed formula and the thresholds predicted using numerical bifurcation analysis indicate fairly good agreement. In addition, previously proposed analytical formulae are examined. It is concluded that predictions of these thresholds obtained using the analytical formulae based on a continuous piecewise linear approximation and Melnikov’s method agree well when used to predict these thresholds with predictions obtained from numerical bifurcation analysis and those obtained experimentally using free-running models. As a result, it is considered that these two calculation methods could be recommended for the early design stage tool for avoiding broaching and bow diving.

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Acknowledgments

The work was supported by the US Office of Naval Research Contract No. 0014-06-1-0646 under the administration of Dr. Patrick Purtell. This research was partly supported by a Grant-in-Aid for Scientific Research of the Japan Society for Promotion of Science (No. 24360355). The authors express their sincere gratitude to the above organization. Further, the authors are grateful to John Kecsmar from Ad Hoc Marine Designs Ltd. for his comprehensive review for preparing his paper as an expert in small craft technology and native English speaker.

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  1. Atsuo Maki

    Present address: Naval System Research Centre, Technical Research and Development Institute, Ministry of Defence, 2-2-1 Nakameguro, Meguro, Tokyo, 153-8630, Japan

Authors and Affiliations

  1. Graduate School of Maritime Sciences, Kobe University, 5-1-1 Fukae-Minami, Higashinada, Kobe, 658-0022, Japan

    Atsuo Maki

  2. Department of Naval Architecture and Ocean Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, 565-0971, Japan

    Naoya Umeda

  3. Higher Colleges of Technology, Abu Dhabi, UAE

    Martin Renilson

  4. Centre for Advanced Information Technology, Tokushima University, Minami-Josanjima, Tokushima, 770-8506, Japan

    Tetsushi Ueta

Authors
  1. Atsuo Maki
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  2. Naoya Umeda
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  3. Martin Renilson
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  4. Tetsushi Ueta
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Corresponding author

Correspondence to Atsuo Maki.

Appendix. Proof that the phase of the sinusoidal function can be neglected in the surge equation

Appendix. Proof that the phase of the sinusoidal function can be neglected in the surge equation

The simplified surge equation is as follows:

$$\left( {m + m_{x} } \right)\ddot{\xi }_{G} + \beta \left( n \right)\dot{\xi }_{G} + f\sin k\left( {\xi_{G} - \xi_{P} } \right) = T\left( {c_{w} ,{\kern 1pt} n} \right) - R\left( {c_{w} } \right)$$
(A1)

where ξ P represents the phase of wave-induced surge force. When

$$\xi^{\prime}_{G} = \xi_{G} - \xi_{P}$$
(A2)

the following can be obtained:

$$\left( {m + m_{x} } \right)\ddot{\xi^{\prime}}_{G} + \beta \left( n \right)\dot{\xi^{\prime}}_{G} + f\sin k\xi^{\prime}_{G} = T\left( {c_{w} ,{\kern 1pt} n} \right) - R\left( {c_{w} } \right)$$
(A3)

Comparison of A1 and A3 demonstrates that the phase, ξ P , does not affect the surf-riding threshold.

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Maki, A., Umeda, N., Renilson, M. et al. Analytical methods to predict the surf-riding threshold and the wave-blocking threshold in astern seas. J Mar Sci Technol 19, 415–424 (2014). https://doi.org/10.1007/s00773-014-0257-2

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  • DOI: https://doi.org/10.1007/s00773-014-0257-2

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