A Practical O(Δω) Approximation of Low Frequency Wave Loads

2013 ◽  
Author(s):  
Charles Monroy ◽  
Guillaume de Hauteclocque ◽  
Xiao-Bo Chen
Keyword(s):  
Free Surface ◽  
Order Term ◽  
Low Frequency ◽  
Radial Distance ◽  
Second Order ◽  
Radial Coordinate ◽  
Wave Loads ◽  
Surface Integral ◽  
Frequency Wave ◽  
The Difference

The low-frequency quadratic transfer function (QTF) is defined as the second-order wave loads occurring at the frequency equal to the difference frequency (ω1 − ω2) of two wave frequencies (ω1, ω2) in bi-chromatic waves of unit amplitude. The exact formulation of the QTF which is recalled here is difficult to implement due to numerical convergence problems mainly related to the evaluation of an arduous free surface integral. This is why several approximations have been used for practical engineering studies. They have been the subject of a detailed review in [5]. Following this work, two closely-related formulations are investigated in this paper. In [2], the classical formulations of QTF are examined by an analysis based on the Taylor development with respect to Δω for Δω ≪ 1 and an expansion of QTF in power of Δω is then obtained. It is shown that the zeroth-order term is a pure real function equal to the drift loads and that the term of order O(Δω) is a pure imaginary function. The second-order low-frequency wave loading of order O(Δω) contains a free-surface integral representing the second-order corrective forcing on the free surface. Since the integrand is of order O(1/R4) with R as the radial coordinate, the free-surface integral converges rapidly with the radial distance. Unlike what has been assumed in previous studies of particular cases, this free-surface contribution is, in general, not negligible for high Δω compared to other components and the complete QTF. Depending whether we use the O(Δω) approximation for the whole QTF or only for this free surface integral, it leads to two different approximations. The first one is called original O(Δω) approximation, because it is on this form that the O(Δω) approximation was first described in [2]. If we use the O(Δω) approximation only for the free surface integral, we call this approximation the practical O(Δω) approximation. It is shown in this paper that the original formulation fails to predict the behaviour of the QTF even for small O(Δω). Comparison for the O(Δω) approximation of the free surface integral is performed against the analytical solution and the exact numerical formulation. The results are improved compared to when we neglect this free surface integral for the range of Δω of interest, but still the agreement with the exact solution is not ideal. A path for further improvement is finally proposed.

Approximation of Second-Order Low-Frequency Wave Loading in Multi-Directional Waves

2010 ◽  
Author(s):  
Flavia C. Rezende ◽  
Xiao-bo Chen
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Wave Loads ◽  
Wave Loading ◽  
Frequency Wave ◽  
New Developments ◽  
Directional Waves ◽  
Novel Method ◽  
The Difference ◽  
Accurate Scheme

Further to the studies by Chen & Rezende (OMAE2009) on the quadratic transfer function (QTF) of low-frequency wave loading in which the QTF is developed by the series expansion associated with the difference-frequency up to the order-Δω2, new formulations have been developed in order to take into account the effect of interactions between waves of different headings. It provides a novel method to evaluate the low-frequency second-order wave loads in a more accurate than usual order-Δω approximation (often called Newman approximation) and more efficient way comparing to the computation of complete QTF in multi-directional waves. New developments including numerical results of different components of QTF are presented here. Furthermore, the time-series reconstruction of excitation loads by quadruple sums in the motion simulation of mooring systems is analyzed and a new efficient and accurate scheme using only a triple sum is demonstrated.

Efficient Computations of Second-Order Low-Frequency Wave Load

2009 ◽  
Author(s):  
Xiao-Bo Chen ◽  
Fla´via Rezende
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Wave Loads ◽  
Wave Loading ◽  
Wave Load ◽  
Frequency Wave ◽  
New Developments ◽  
Novel Method ◽  
The Difference ◽  
New Formulation

As the main source of resonant excitations to most offshore moored systems like floating LNG terminals, the low-frequency wave loading is the critical input to motion simulations which are important for the design. Further to the analysis presented by Chen & Duan (2007) and Chen & Rezende (2008) on the quadratic transfer function (QTF) of low-frequency wave loading, the new formulation of QTF is developed by the series expansion of the second-order wave loading with respect to the difference-frequency upto the order-2. It provides a novel method to evaluate the low-frequency second-order wave loads in a more accurate than usual order-0 approximation (often called Newman approximation) and more efficient way comparing to the computation of complete QTF. New developments including numerical results of different components of QTF are presented here. Furthermore, the time-series reconstruction of excitation loads in the motion simulation of mooring systems is analyzed and a new efficient and accurate scheme is demonstrated.

Accuracy Effects of Low Frequency Wave Loads on Ships Offshore to LNG Marine Terminal Design

2004 ◽  
Author(s):  
Monica J. Holboke ◽  
Robert G. Grant
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
Surface Boundary ◽  
Wave Loads ◽  
Wave Load ◽  
Frequency Wave ◽  
First Order ◽  
Terminal Design

This paper presents the results of a two-body analysis for a moored ship sheltered by a breakwater in shallow water with and without free surface forcing in the low frequency wave load calculation. The low frequency wave loads are determined by second order interactions from the first order. The free surface forcing term arises from the free surface boundary condition, which is trivial to first order but is not at second order. We demonstrate in the frequency domain the importance of this term in a two-body analysis. Additionally, we show how inaccurate calculations of the off-diagonal terms of the Quadratic Transfer Function can translate to over or under prediction of low frequency wave loads on moored ships sheltered by breakwaters in shallow water. Low frequency wave load accuracy has direct consequence for LNG marine terminal design. Generally, LNG marine terminals are sited in sheltered harbors, however increasingly they are being proposed in offshore locations where they will require protection from persistent waves and swells. Since breakwaters typically cost twice as much as the rest of the marine facilities, it is important to optimize their size, orientation and location. In a previous paper we described this optimization process [1], which identified a key step to be the transforming of waves just offshore the breakwater into wave loads on the moored ships. The ability to do this step accurately is of critical importance because if the loads are too large, the breakwater will be larger and more expensive than necessary and if the loads are too small, the terminal will experience excessive downtime and loss of revenue.

Review of Approximations to Evaluate Second-Order Low-Frequency Load

2012 ◽  
Author(s):  
Guillaume de Hauteclocque ◽  
Flávia Rezende ◽  
Olaf Waals ◽  
Xiao-Bo Chen
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
The Body ◽  
Difference Frequency ◽  
Surface Boundary ◽  
First Order ◽  
Potential Part ◽  
The Difference

The second order low-frequency loads are one of main sources of excitation for moored systems. These loads are usually decomposed into the quadratic part, contributed only by first order quantities and potential part contributed by the second order potentials. In shallow water the second order incoming and diffracted potentials give a significant contribution to the low frequency forces. Therefore, the accuracy on the determination of this parcel of the low-frequency loads is a key issue for the assessment of mooring lines and operability of systems moored in shallow water area, as for example LNG terminals. Due to the complexity in computing the second order diffraction potential, which would involve a non-homogeneous free surface boundary condition, the so-called Pinkster approximation has been proposed. This approximation is based on the assumption that the major contribution to the potential part of low-frequency loads is given by the second order potential of the undisturbed incoming waves. The methods to compute the wave forces related to the second order potentials are based on scaling of the first order wave induced forces. Another approximation recently formulated in Chen and Rezende consists of developing the second-order bi-frequency load into a series of different orders of the difference frequency. The potential contribution to the term proportional to the difference-frequency can be evaluated efficiently by involving an integral over a small zone on the free surface around the body. In the present paper, the existing approximations are revisited and compared to analytical solution of exact second-order load on a vertical cylinder and for the case of floating body (LNG) in shallow water. Some guidelines in the practical use of different approximations will be derived.

Wave Drift Forces and Low Frequency Damping on the Exwave Semi-Submersible

2017 ◽  
Author(s):  
Nuno Fonseca ◽  
Carl Trygve Stansberg
Keyword(s):  
Potential Flow ◽  
Low Frequency ◽  
Second Order ◽  
Difference Frequency ◽  
Frequency Wave ◽  
Wave Drift ◽  
The Difference ◽  
Wave Drift Forces ◽  
The Relationship ◽  
Drift Forces

The paper presents realistic horizontal wave drift force coefficients and low frequency damping coefficients for the Exwave semi-submersible under severe seastates. The analysis includes conditions with collinear waves and current. Model test data is used to identify the difference frequency wave exciting force coefficients based on a second order signal analysis technique. First, the slowly varying excitation is estimated from the relationship between the incoming wave and the low frequency motion using a linear oscillator. Then, the full quadratic transfer function (QTF) of the difference frequency wave exciting forces is defined from the relationship between the incoming waves and the second order force response. The process identifies also the linear low frequency damping. The paper presents results from cases selected from the EXWAVE JIP test matrix. The empirical wave drift coefficients are compared to potential flow predictions and to coefficients from a semi-empirical formula. The results show that the potential flow predictions largely underestimate the wave drift forces, especially at the low frequency range where severe seastates have most of the energy.

Wave Drift Forces and Low Frequency Damping on the Exwave FPSO

2017 ◽  
Author(s):  
Nuno Fonseca ◽  
Carl Trygve Stansberg
Keyword(s):  
Low Frequency ◽  
Second Order ◽  
Difference Frequency ◽  
Frequency Wave ◽  
Force Coefficients ◽  
Wave Drift ◽  
The Difference ◽  
Wave Drift Forces ◽  
The Relationship ◽  
Drift Forces

A method is followed in the present analysis to estimate realistic surge and sway wave drift force coefficients for the Exwave FPSO. Model test data is used to identify the difference frequency wave exciting force coefficients based on a second order signal analysis technique. First, the slowly varying excitation is estimated from the relationship between the incoming wave and the low frequency motion using a linear oscillator. Then, the full QTF of the difference frequency wave exciting forces is defined from the relationship between the incoming waves and the second order force response. The process identifies also the linearized low frequency damping. The paper presents results from a few cases selected from the Exwave JIP test matrix. Empirical mean wave drift coefficients are compared to potential flow predictions. It is shown that the latter underestimate the wave drift forces, especially at the lower frequency range where severe seastates have most of the energy. The sources for the discrepancies are discussed.

Hysteresis and nonlinear detuning in a spatial-resonance phenomenon

1973 ◽  
Vol 61 (2) ◽  
pp. 401-413
Author(s):  
Ian Huntley ◽  
Ronald Smith
Keyword(s):  
Free Surface ◽  
Experimental Work ◽  
Alternative Model ◽  
Pressure Field ◽  
Low Frequency ◽  
Theoretical Description ◽  
Resonance Phenomenon ◽  
Theoretical Problem ◽  
Frequency Wave ◽  
Spatial Resonance

The experimental work of Franklin, Price & Williams (1973) shows that for moderately large driving amplitudes there are features of spatial resonance that are not predicted by the model representation of Mahony & Smith (1972). We here derive an alternative model, which remains valid for moderately large driving amplitudes, and we are able to obtain a theoretical description of both hysteresis and nonlinear detuning of the low frequency wave response. An experiment in which surface waves were generated by a sinusoidal pressure field at the free surface (and which corresponds almost exactly to the theoretical problem) was conducted in order to test these predictions.

The Effect of Wave Directionality on Low Frequency Motions and Mooring Forces

2009 ◽  
Author(s):  
Olaf J. Waals
Keyword(s):  
Water Depth ◽  
Transfer Functions ◽  
Low Frequency ◽  
Sensitivity Study ◽  
Wave Loads ◽  
Offshore Structure ◽  
Frequency Wave ◽  
Drift Theory ◽  
Mooring Forces ◽  
Drift Forces

Operability of offshore moored ships can be affected by low frequency wave loads. The low frequency motions of a moored ship may limit the uptime of an offshore structure such as an LNG offloading terminal. The wave loads that cause the main excitation of these low frequency motions are usually computed using second order wave drift theory for long crested waves, which assumes that the low frequency components are only related to waves coming from the same direction. In this method short crested seas are dealt with as a summation of long crested seas, but no interaction between the wave components traveling in different directions is usually taken into account. This paper describes the results of a study to the effect of 2nd order low frequency wave loads in directional seas. For this study the drift forces related to the interaction between waves coming from different directions is also included. This is done by computing the quadratic transfer functions (QTF) for all possible combinations of wave components (frequencies and directions). Time traces of drift forces are generated and compared to the results without wave directional interaction after which the motions of an LNG carrier are simulated. A sensitivity study is carried out towards the number of direction steps and the water depth. Finally the motions of an LNG carrier in shallow water (15m water depth) are simulated and mooring forces are compared for various amounts of wave spreading.

Low Frequency Wave Forces and Wave Induced Motions of a FPSO in Shallow Water

2007 ◽  
Author(s):  
Longfei Xiao ◽  
Jianmin Yang ◽  
Zhiqiang Hu
Keyword(s):  
Shallow Water ◽  
Wave Spectrum ◽  
Low Frequency ◽  
Wave Force ◽  
Wave Forces ◽  
Frequency Wave ◽  
Wave Basin ◽  
The Difference ◽  
The One ◽  
Wave Induced

The low frequency (LF) response of a soft yoke moored 160kDWT FPSO in shallow water is investigated by conducting frequency domain computations and wave basin model tests. An incident wave with Hs = 4.1m and Tp = 8.9s is applied. An obvious LF part appears in the measured wave spectrum at water depth of 16.7m. As a result, the 1st order LF wave force exists and is much larger than the 2nd one. The difference of the spectrums is about one hundred times. The LF wave drift force increases enormously. Consequently, much larger resonant surge response is induced. The LF surge amplitude at h = 16.7m is about 7 times the one at h = 29.0m and 9 times the one in deep water, although the 2nd order response changes a little. Therefore, in very shallow water, LF part of incident waves should be taken into account carefully and LF wave forces and wave induced motions will be very serious.

Influence of Second-Order Difference-Frequency Wave Loads on the Floating Wind-Wave Hybrid Platform

2017 ◽  
Author(s):  
Hyebin Lee ◽  
Yoon Hyeok Bae ◽  
Kyong-Hwan Kim ◽  
Sewan Park ◽  
Keyyong Hong
Keyword(s):  
Wind Wave ◽  
Computational Effort ◽  
Second Order ◽  
Difference Frequency ◽  
Linear Wave ◽  
Wave Loads ◽  
Wave Load ◽  
Frequency Wave ◽  
Order Difference ◽  
Hybrid Platform

A wind-wave hybrid power generation system is a floating offshore energy platform which is equipped with a number of wind turbines and wave energy converters (WECs) to harvest energy from various resources. This wind-wave hybrid platform is moored by eight catenary lines to keep its position against wind-wave-current environment. In most cases, the resonant frequency of horizontal motion of moored platform is very low, so a resonance is hardly seen by numerical simulation with linear wave assumptions. However, the incident waves with different frequency components are accompanied by sum and difference frequency loads due to the nonlinearity of the waves. Typically, the magnitude of the second-order wave loads are small and negligible, but once the second-order wave loads excite the platform at its natural frequency, the resonance can take place, which results in adverse effects on the platform. In this paper, the second-order difference frequency wave load on the wind-wave hybrid platform is numerically assessed and time domain simulation by coupled platform-mooring dynamic analysis is carried out. As a result, the horizontal motions of the platform was highly excited and the increased motions led higher top tension of the mooring lines compared with the case of linear wave environment. Especially, the combination of the wind and wave loads excited the horizontal motions more and made the mooring top tension far higher than wave load was only applied. With regards to the second-order difference frequency wave load, the result with the Quadratic Transfer Function (QTF) is compared to the one with Newman’s approximation. As the simulation results between them was insignificant, the Newman’s approximation can be used instead of the complete QTF to reduce the computational effort.

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