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.

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.

Second Order Difference Frequency Wave-Current Loading Using Kelvin-Newman Approximation

2020 ◽  
Author(s):  
Farid P. Bakti ◽  
Moo-Hyun Kim
Keyword(s):  
Second Order ◽  
The Body ◽  
Difference Frequency ◽  
Computational Time ◽  
Forward Speed ◽  
Frequency Wave ◽  
Order Difference ◽  
First Order ◽  
Current Interaction ◽  
Current Loading

Abstract Kelvin & Newman introduced a linearization method to include the current (or forward speed) effect into the diffraction & radiation wave field for large-slender floating bodies. The K-N method assumes a steady far-field current while disregarding the steady potential field due to the presence of the body. The method is proven to be reliable when the Froude number is relatively small, the body shape is relatively slender (∂∂x≪∂∂y,∂∂z), and the sea condition is mild. This requirement is fulfilled for typical FPSOs and ship-shaped vessels in a typical current (or forward speed) condition. Several studies suggested that the presence of the current might change the first order hydrodynamic coefficients such as the first order diffraction force, added mass, and radiation damping. Currents also contributed to a change in the second-order slowly-varying drift force. However, the effect of current in the second-order difference-frequency force is yet to be investigated. By expanding the Kelvin-Newman approximation up to the second order, and solving the problem in the frequency domain, we can save computational time while expanding the accuracy of the scheme. The second order quadratic force is the main focus of this study, since it is the main contributor to the total second order difference frequency forces especially near the diagonal. By implementing the Kelvin-Newman wave current interaction approach up to the wave’s second order, we can assess the performance of the Kelvin-Newman wave current interaction formulation in various sea conditions.

A Second-Order Theory for the Potential Flow About Thin Hydrofoils

1984 ◽  
Vol 28 (01) ◽  
pp. 55-64
Author(s):  
Colen Kennell ◽  
Allen Plotkin
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Potential Flow ◽  
Order Theory ◽  
Wave Drag ◽  
Second Order ◽  
The Body ◽  
Surface Shape ◽  
Surface Boundary ◽  
Free Surface Boundary Condition

This research addresses the potential flow about a thin two-dimensional hydrofoil moving with constant velocity at a fixed depth beneath a free surface. The thickness-to-chord ratio of the hydrofoil and disturbances to the free stream are assumed to be small. These small perturbation assumptions are used to produce first-and second-order subproblems structured to provide consistent approximations to boundary conditions on the body and the free surface. Nonlinear corrections to the free-surface boundary condition are included at second order. Each subproblem is solved by a distribution of sources and vortices on the chord line and doublets on the free surface. After analytic determination of source and doublet strengths, a singular integral equation for the vortex strength is derived. This integral equation is reduced to a Fredholm integral equation which is solved numerically. Lift, wave drag, and free-surface shape are calculated for a flat plate and a Joukowski hydrofoil. The importance of free-surface effects relative to body effects is examined by a parametric variation of Froude number and depth of submergence.

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.

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.

Characteristics of Steep Second-Order Random Waves in Finite and Shallow Water

2011 ◽  
Author(s):  
Carl Trygve Stansberg
Keyword(s):  
Time Series ◽  
Shallow Water ◽  
Frequency Component ◽  
Second Order ◽  
Difference Frequency ◽  
Nonlinear Interactions ◽  
Random Waves ◽  
Theoretical Formulation ◽  
Sum Frequency ◽  
The Difference

The theoretical formulation of second-order random waves in deep and finite water is reviewed. In particular, the increased nonlinear interactions with decreasing depth are addressed, including both the sum-frequency as well as the slowly varying difference-frequency components. Depth-defined limitations in the valid range for random waves are suggested based on the Ursell number. Numerical time series realizations at various depths and for two sea states are obtained by an efficient bifrequency summation procedure. Resulting time series show moderate average second-order energy contents, except for the steep sea state Hs = 15m, Tp = 14s in depths of 30m and 20m which are outside the suggested valid second-order range. The two largest wave events from the simulations are studied in particular for the different depths. Nonlinear interactions increase significantly with decreasing depth. Still, within the valid range, extreme second-order crests and peak particle velocities are only moderately increased with decreasing depth, while the negative peaks increase significantly. This is because the difference-frequency component almost compensates for the sum-frequency part at crests, while it is opposite at troughs. Maximum slopes, however, are clearly increased in shallow water, eventually leading to increased breaking (which is beyond second order of course). Velocity profiles under the crests are also shown, confirming the findings from the elevation.

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.

The effect of non-linearity at the free surface on flow past a submerged cylinder

1965 ◽  
Vol 22 (2) ◽  
pp. 401-414 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Free Surface ◽  
Surface Condition ◽  
Second Order ◽  
Order Effects ◽  
Surface Boundary ◽  
First Order ◽  
Flow Past ◽  
Free Surface Boundary Condition ◽  
Non Linear ◽  
Wave Induced

Plane potential flow past a circular cylinder beneath a free surface under gravity is investigated in order to determine the importance or otherwise of non-linear effects from the free-surface boundary condition. It is shown that non-linear second-order corrections to the first-order linearized expressions for the wave-induced forces on the cylinder are considerably larger than second-order effects which are present even with a linear free-surface condition. Further evidence for the importance of non-linearity is presented in the form of streamline plots of the first-order solution showing strange behaviour at wave crests.

Analysis of the First Order and Slowly Varying Motions of an Axisymmetric Floating Body in Bichromatic Waves

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
João Pessoa ◽  
Nuno Fonseca ◽  
C. Guedes Soares
Keyword(s):  
Vertical Cylinder ◽  
Vertical Axis ◽  
Second Order ◽  
The Body ◽  
Floating Body ◽  
Drift Motion ◽  
First Order ◽  
Motion Responses ◽  
Simple Geometry ◽  
Good Agreement

The paper presents an experimental and numerical investigation on the motions of a floating body of simple geometry subjected to harmonic and biharmonic waves. The experiments were carried out in three different water depths representing shallow and deep water. The body is axisymmetric about the vertical axis, like a vertical cylinder with a rounded bottom, and it is kept in place with a soft mooring system. The experimental results include the first order motion responses, the steady drift motion offset in regular waves and the slowly varying motions due to second order interaction in biharmonic waves. The hydrodynamic problem is solved numerically with a second order boundary element method. The results show a good agreement of the numerical calculations with the experiments.

From First-Order Coherence to Higher-Order Coherence of Synchrotron Radiation

1998 ◽  
Vol 5 (3) ◽  
pp. 305-308 ◽  
Author(s):  
Tsuneaki Miyahara
Keyword(s):  
Synchrotron Radiation ◽  
Free Electron ◽  
Higher Order ◽  
Second Order ◽  
Free Electron Lasers ◽  
First Order ◽  
Noise Characteristics ◽  
The Difference

The difference between first-order and second-order coherence of synchrotron radiation is discussed in relation to how they can be measured and how they affect the noise characteristics of future free-electron lasers.

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