Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

Nonlinear Air Gap Analysis of a Semi-submersible Unit in Time Domain
Wei Wei, Chengming Liu, Jinghui Duan, Wei Sun, Yuanhua Liang
Offshore Engineering Technology Center, China Classification Society
Tianjin, China

Nevertheless, it is difficult to simulate the six degree of freedom
motion and complicated mooring system of the platform. Moreover, itis time-consuming.

ABSTRACT
Based on the second-order Stokes wave theory and introducing a wave
spectrum transformation method, the wave spectrum of nonlinear
waves can be obtained from the wave spectrum of linear waves. Then,the nonlinear wave elevation time history can be described as thefunction of the wave spectrum of nonlinear waves and linear wave
elevation responses obtained from frequency domain. The nonlinear air
gap of a semi-submersible unit, including the effect of nonlinear wave
elevations and nonlinear wave excitation forces, is investigated in timedomain. Comparisons between the linear and nonlinear air gap are
made to investigate the nonlinear effect. The numerical results indicatethat the nonlinearity has a significant effect on the air gap.

In this paper, applying the second-order Stokes wave theory (Stokes,1847) and introducing a wave spectrum transformation method, thewave spectrum of nonlinear waves is introduced. Then, the nonlinear
wave elevation time history can be described as the function of thewave spectrum of nonlinear waves and linear wave elevation responses
obtained from frequency domain. Newman’s approximation, where thequadratic transform functions (QTFs) off diagonal terms are estimatedfrom the steady drift force transfer function, is adopted to obtain thesecond-order difference-frequency wave forces. Moreover, byintroducing convolution theorem, the frequency-domain first-orderwave excitation forces and hydrodynamic coefficients are transformed
into time-domain motion equation. Finally, the nonlinear responses ofthe floating structure are obtained. To reveal the nonlinear effect on,comparisons between the linear and nonlinear air gap are made.

KEY WORDS: second-order Stokes wave theory; wave spectrum
transformation method; air gap; nonlinear effect

INTRODUCTION
Floating platform is widely used in the field of offshore oil and gas
exploration. For floating submersible platform, the air gap problem isrelatively complicated owing to the coupling effect of its own motions
and waves. Not only the radiation, diffraction and second-ordernonlinear motions, but also the nonlinear wave itself, should beconsidered in the air gap analysis (DNVGL-OTG-13, 2019).

THEORETICAL BACKGROUND

Due to the complicated nature of the air gap, model test is the mostused methodology in the past, which is expensive to cost (Kriebel andWallendorf, 2001; Alexandre, 2006). With the development ofcomputers and the linear method based on the first-order potential
theory can offer more accurate of platform motion (Pessoa and Moe,2017). However, the wave elevation of any position may beunderestimated (Stansberg and Baarholm, 2005). For nonlinear
problems, focuses are on the non-linear diffraction, nonlinear platform
motions and nonlinear waves (Lu and Zhao, 2020; Sweetman andWinterstein, 2002; Lin and Treakle, 2000; Huo and Zhang, 2015). Less
attention has been given to the combination of all these nonlineareffects. Some researchers adopted CFD method (Wu and Danmeier,2014; Liang and Yang, 2010), which can capture the characteristics offlow field and wave elevation nearby the column accurately.

( ) =Nonlinear Waves
For regular waves, the first-order velocity potential in finite water depth
can be written as,1

iag cosh k ( z + d )cos(t − kx) cosh(kd )(1)where a is the wave amplitude; g is the acceleration of gravity;  is
the wave frequency; d is the water depth; x and z are location
dependent variables; t is time variable; k is the wave number, k = 2  , is wave length.
Then the first-order wave elevation for regular waves can be expressedas, (1) (t ) = a cos(t − kx )2049(2)