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VIM simulation on a cylindrical floating structure

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Abstract

Vortex-induced motion (VIM) on a cylindrical floating structure can be easily found in ocean strong current. Since the VIM on the structure with long-term low-frequency motion causes fatigue damage of the structure’s mooring lines and risers, precise VIM assessment is needed for the safe evaluation of them. In the standard of the International Organization for Standardization ISO19901-7, ‘Specific requirements on stationkeeping systems for floating offshore structures and mobile offshore units’, for instance, a concrete method of assessing VIM displacement is not represented in the standard document, though the requirement on the VIM demands for an assessment on the basis of a proper way. In this paper, a VIM time-domain simulation method on a floating structure with circular cylinder form, that is, for example a Spar, an MPSO (mono-column type floating production storage and off-loading) and so on, is shown using the wake oscillator model. Transverse and in-line VIM are treated for the floater’s motion. The assessment quality of the simulation on the VIM amplitudes of floaters in current is confirmed by model test data.

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Abbreviations

\(b\) :

Oscillation amplitude of wake tail length (m)

\(C_{\text{c}}\) :

Damping coefficient (kg/s)

\(C_{\text{D}}\) :

Drag coefficient of a cylindrical floater

\(C_{\text{D0}}\) :

Drag coefficient without oscillation of a cylindrical floater

\(C_{\text{Dir}}\) :

Increased ratio of the drag coefficient \(C_{\text{D}}\)

\(C_{\text{L}}\) :

Lift coefficient

\(C_{\text{L0}}\) :

Lift coefficient on a fixed floater

\(d\) :

Draft of a cylindrical floater (m)

\(D\) :

Diameter of a cylindrical floater (m)

\(F_{{{\text{D}}x}} ,\;F_{{{\text{D}}y}}\) :

x- and y-directional drag working on the cylindrical floater surface (kg m/s2)

\(f\) :

Lift slope coefficient (=1.16)

\(f_{\text{v}}\) :

Oscillation frequency of the wake (1/s)

\(F_{\text{x}} ,\;F_{\text{y}}\) :

External force for in-line and transverse directions (kg m/s2)

\(h\) :

Breadth of the wake oscillator (m)

\(I\) :

Moment of inertia on the wake oscillator (kg m2)

\(K\) :

Restoring moment acting on the oscillator (kg m2/s2)

\(K_{\text{c}}\) :

Restoring coefficient of mooring stiffness (kg/s2)

\(\bar{l}\) :

Averaged half wake length of the wake oscillator (m)

\(\bar{l}^{*}\) :

1/2 non-dimensional average length of the wake oscillator (=1.10)

\(F_{{{\text{L}}x}} ,\;F_{{{\text{L}}y}}\) :

x-directional induced drag caused by the wake oscillator lift and y-directional wake oscillator lift (kg m/s2)

\(M\) :

Mass of a cylindrical floater (kg)

\(M_{\text{a}}\) :

Added mass of a cylindrical floater (kg)

\(n\) :

Mass ratio of a cylindrical floater (=\(\rho D^{2} d/(2(M + M_{\text{a}} ))\))

\(R_{e}\) :

Reynolds number

\(\,S\) :

Strouhal number on a stationary cylinder

\(S_{\text{v}}\) :

Reverse parameter of specified reduced velocity (=\(1/V_{\text{rin}}\))

\(t\) :

Progress time (s)

\(t^{\prime}\) :

Non-dimensional progress time

\(T_{\text{n}}\) :

Natural period of mooring condition of a cylindrical floater in no towing (=1/f 0)

\(V_{\text{rin}}\) :

Reduced velocity in case with relatively large VIM amplitude (=9.0)

\(V\) :

Current velocity (m/s)

\(V_{\text{r}}\) :

Reduced velocity

\(x\) :

In-line position of a cylindrical floater (m)

\(X\) :

Non-dimensional in-line position (=\(x/D\))

\(y\) :

Transverse position of a cylindrical floater (m)

\(Y\) :

Non-dimensional transverse position (=\(y/D\))

\(\alpha\) :

Rotation angle of the wake oscillator (rad)

\(\alpha_{0}\) :

Wake rotation amplitude (rad)

\(\varGamma\) :

Wake circle strength

\(\eta\) :

Non-dimensional damping coefficient of a cylindrical floater (=\(C_{\text{c}} /(2\omega_{0} (M + M_{\text{a}} ))\))

\(\theta\) :

Relative angle for current velocity (rad)

\(\varLambda\) :

Constant parameter means the aspect ratio of a cylindrical floater (=\(2\pi d/D\))

\(\nu\) :

Non-dimensional circular frequency (=\(\omega_{\text{v}} /\omega_{0}\))

\(\nu_{\text{e}}\) :

Kinematic coefficient of water (m2/s)

\(\pi^{*}\) :

Value relating with \(\,S\) (=\(2\pi S\))

\(\rho\) :

Water density (kg/m3)

\(\omega_{0}\) :

Circular frequency of mooring lines keeping a cylindrical floater (=\(2\pi \,f_{0}\)) (1/s)

\(\omega_{\text{v}}\) :

Circular frequency of the wake (=\(2\pi f_{\text{v}}\)) (1/s)

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Acknowledgements

The VIM model tests were conducted by many researchers, Mr M. Saito, Mr H. Sato and Mr K. Ishida, Ocean engineering department, at our institute. The author thanks them. This research was supported by the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 26289344). This paper includes some reference model test data based on the consignment study of Maritime Bureau, Ministry of Land, Infrastructure, Transport and Tourism (MLIT), Japan, for making the safety guideline on floating wind turbine facilities. The author expresses the deepest appreciation to them.

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  1. National Maritime Research Institute (NMRI), 6-38-1 Shinkawa, Mitaka, Tokyo, 181-0004, Japan

    Toshifumi Fujiwara

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Correspondence to Toshifumi Fujiwara.

Appendix

Appendix

1.1 Relation between the \(P^{\prime}\) parameter and the lift coefficient on a fixed floater \(C_{{{\text{L}}0}}\) [9, 26]

Motion equation of the wake oscillator is replaced as follows by using the lift coefficient \(C_{\text{L}}\),

$$\ddot{C}_{\text{L}} - \varepsilon \omega_{\text{v}} \left( {1 - \frac{P'}{{f^{2} }}C_{\text{L}}^{2} } \right)\dot{C}_{\text{L}} + \omega_{\text{v}}^{2} C_{L} = 0.$$
(45)

Using \(t' = \omega_{\text{v}} t\), Eq. 45 is transformed to the non-dimensional form:

$$\ddot{C}_{\text{L}} - \varepsilon \left( {1 - \frac{P'}{{f^{2} }}C_{\text{L}}^{2} } \right)\dot{C}_{\text{L}} + C_{\text{L}} = 0.$$
(46)

Steady periodical solutions are obtained as follows:

$$C_{\text{L}} = \frac{2f}{{\sqrt {P^{\prime}} }}\cos t_{1} - \varepsilon \frac{f}{{4\sqrt {P^{\prime}} }}\sin 3t_{1}$$
(47)

and

$$C_{\text{L}} = - \varepsilon \frac{f}{{4\sqrt {P^{\prime}} }}\sin 3t_{1} .$$
(48)

The second term in Eq. 47 is able to recognize negligible value and the second solution, Eq. 48, is an unstable one. Then, as a result, the lift coefficient on a fixed floater \(C_{{{\text{L}}0}}\) is defined as follows:

$$C_{{{\text{L}}0}} = \frac{2f}{{\sqrt {P^{\prime}} }}.$$
(49)

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Fujiwara, T. VIM simulation on a cylindrical floating structure. J Mar Sci Technol 23, 288–301 (2018). https://doi.org/10.1007/s00773-017-0470-x

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