Offshore Technology

Nonlinear Superposition Methods Applied to Continuous Ocean Wave Spectra

[+] Author and Article Information
Thomas B. Johannessen

Centre for Ships and Offshore Structures (CeSOS), Norwegian University of Science and Technology, Trondheim N-7491, Norwaythomas.b.johannessen@ntnu.no

J. Offshore Mech. Arct. Eng 134(1), 011302 (Oct 13, 2011) (14 pages) doi:10.1115/1.4003518 History: Received July 22, 2009; Revised December 07, 2010; Published October 13, 2011; Online October 13, 2011

The present paper addresses the challenges associated with applying weakly nonlinear mode-coupled solutions for wave interaction problems to irregular waves with continuous spectra. Unlike the linear solution, the nonlinear solutions will be strongly dependent on cut-off frequency for problems such as the wave elevation itself or loads on a slender cylinder used together with typical ocean wave spectra. It is found that the divergence of the solutions with respect to the cut-off frequency is related to the nonlinear interaction between waves with very different frequencies. This is, in turn, linked to a long standing discussion about the ability of mode-coupled methods to describe the modulation of a short wave due to the presence of a long wave. In cases where nonlinear properties associated with a measured or assumed history of the surface elevation is sought, it is not necessary to calculate accurately the nonlinear evolution of the wave field in space and time. For such cases it is shown that results which are independent of frequency cut-off may be obtained by introducing a maximum bandwidth in frequency between waves which are allowed to interact. It is shown that a suitable bandwidth can be found by applying this method to the problem of back-calculating a linear wave profile from a measured wave profile. In order to verify that this choice of bandwidth is suitable for second and third order terms, nonlinear loads on a slender vertical cylinder are calculated using the FNV method of Faltinsen, Newman, and Vinje (1995, “Nonlinear Wave Loads on a Slender, Vertical Cylinder,” J. Fluid Mech., 289, pp. 179–198). The method is used to compare loads calculated based on measured surface elevations with measurements of loads on two cylinders with different diameters. This comparison indicates that the bandwidth formulation is suitable and that the FNV solution gives a reasonable estimate of loading on slender cylinders. There are, however, loading mechanisms that the FNV solution does not describe, notably the secondary loading cycle first observed by Grue (1993, Higher Harmonic Wave Exciting Forces on a Vertical Cylinder, Institute of Mathematics, University of Oslo, Preprint No. 2). Finally, the method is employed to calculate the ringing response on a large concrete gravity base platform. The base moment response is calculated using the FNV loading on the shafts and linear loads from a standard diffraction code, together with a structural finite element beam model. Comparison with results from a recent model testing campaign shows a remarkable agreement between the present method and the measured response.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 7

Linearized surface elevation, effect of varying dω

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Figure 9

Comparison between the FNV solution and measurements, largest crest

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Figure 10

Comparison between the FNV solution and measurements, largest force

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Figure 11

The troll A GBS (reproduced with the kind permission of Statoil Hydro)

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Figure 1

Log-linear plot of spectral density of some linear wave properties

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Figure 2

Relation between frequency and wavenumber for regular waves

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Figure 3

Relation between frequency and wavenumber for nonlinear broad banded waves

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Figure 8

Comparison between the regular and irregular wave FNV solutions for three events

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Figure 4

Illustration of maximum frequency bandwidth approach with ω1≤ω2

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Figure 12

Model testing (reproduced with the kind permission of Statoil Hydro)

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Figure 13

Model test seastates

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Figure 14

Model test seastate linearization properties, effect of varying dω

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Figure 5

Analysis scheme for irregular waves

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Figure 6

Properties of linearized seastate, effect of varying dω

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Figure 15

Power spectra of overturning moment excitation (Tn1 is the first eigenperiod of the GBS)

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Figure 16

Ringing response of the troll A GBS (Tn1 is the first eigenperiod of the GBS)



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