Wave Diffraction Approximation

When the wavelength of waves impacting on an offshore structure becomes comparable with the dimensions of the structure (e.g. the diameter of a monopile), diffraction effects begin to occur whereas, waves that have wavelengths much smaller than the diameter of the structural member do not contribute substantially to the net force. Bladed provides two different approaches for including these effects: an approximation based on MacCamy-Fuchs theory, and a simple frequency cut-off.

MacCamy-Fuchs approximation

Explicit models of the diffraction phenomenon are highly specialised and computationally expensive, and it is therefore not generally feasible to incorporate them in time domain design tools. An alternative approach is to account for the wave diffraction effect by altering the hydrodynamic loads experienced by the structure via revised values of the inertia and drag coefficients \(C_m\) and \(C_d\), calculated as a function of the wave frequency. This approach, described by MacCamy and Fuchs in (MacCamy 1954), is only applicable to the frequency domain as it is not possible to alter the \(C_m\) and \(C_d\) coefficients in time domain analysis. Bladed includes a variation of this technique in the time domain, MacCamy Fuchs approximation, achieved by altering the wave energy spectrum rather than the \(C_m\) and \(C_d\) terms, to give the same result of hydrodynamic load on the structure over the full range of incident wave frequencies. In this procedure, it is assumed that the \(C_m\) term dominates the hydrodynamic loading, and so the wave energy spectrum is altered based solely on how the \(C_m\) coefficient varies as a function of wave frequency.

The procedure followed by Bladed is explained below:

Hydrodynamic force, \(F_{H}\), is given by Hydrodynamic Mass Force and Froude-Krylov Force

\[ \begin{equation} \begin{aligned} F_{H} &= \rho A \ddot{X} C_a + \rho A \ddot{X}\\ &= \rho A \ddot{X} (C_a \ + \ 1)\\ &= \rho A \ddot{X} C_{m} \end{aligned} \end{equation} \]

with water density \(\rho\), cross-sectional area of the the cylinder \(A\), the water particle horizontal acceleration \(\ddot{X}\), the hydrodynamic added mass coefficient \(C_a\), and the overall hydrodynamic inertia coefficient \(C_{m} = C_{a}\ + 1\).

In the case of a vertical cylinder with no diffraction effects, \(C_{a} = 1\). Therefore in the non-diffraction case, \(C_{m} = 2\).

In the MacCamy Fuchs approximation, the hydrodynamic inertia coefficient \(C_{m}\) has the following form

\[ \begin{equation} C_{m} = \frac{4 B (kr_{0})} {\pi (kr_{0})^2} \end{equation} \]

where \(k\) is the wave number, \(r_0\) is the radius of the cylinder and we define

\[ \begin{equation} B (kr_{0}) = \left (J_{1}^{’2}(kr_{0}) + Y_{1}^{’2}(kr_{0}) \right)^{-1/2} \end{equation} \]

This reduces \(C_{m}\) as \(k\) increases. As \(k\) tends to zero, \(C_{m}\) tends to a value of \(2\), equal to the non-diffraction case.

Since it is not possible to modify \(C_{m}\) in the time domain within Bladed, \(C_{m}\) is held fixed and the change in hydrodynamic forcing predicted by the MacCamy Fuchs approximation is reproduced by applying an identical functional form to the entire Hydrodynamic force equation, considering it to be a modification of \(\ddot{X}\) rather than \(C_{m}\).

This modification to \(\ddot{X}\) is produced via modifying the wave energy spectrum. The standard wave energy spectrum is the Pierson Moskowitz distribution, in which \(S(k) \propto \ H^{2}\). In sinusoidal wave theory, \(\ddot{X}\propto H\) (wave height) giving \(S(k)\propto \ddot{X}^{2}\).

Thus the wave energy spectrum is multiplied by a correction factor (\(f_c\)):

\[ \begin{equation} S(k)_\text{modified} = f_c \cdot S(k)_\text{unmodified} \end{equation} \]

The correction factor is the square of the normalised MacCamy Fuchs \(C_{m}\) function. Normalising \(C_{m}\) against the non-diffraction value of 2 we have:

\[ \begin{equation} {\overline{C_{m}}} = \frac {4 B (kr_{0})} {2 \pi (kr_{0})^{2}} \end{equation} \]

Therefore the correction factor is \(f_c= \left(\frac {2 B (kr_{0})} { \pi (kr_{0})^{2}}\right )^{2}\).

Thus:

\[ \begin{equation} \begin{aligned} S(k)_\text{modified} &= (\overline{C_{m}})^{2} \times S(k)_\text{unmodified}\\ ~\\ S(k)_\text{modified} &= \left(\frac {2 B (kr_{0})} { \pi (kr_{0})^{2}}\right )^{2} \times S(k)_\text{unmodified} \end{aligned} \end{equation} \]

Simple cut-off frequency

Applied forces are calculated from the wave particle kinematics at the member centreline, using a spectrum in which the high frequencies have been cut off. The frequency cut-off is based on experimental work by (Hogben and Standing 1975) which shows that the applied force on a cylinder falls off rapidly when the wave number (\(k\)) exceeds \(1 / r_0\) where \(r_0\) is the radius of the cylinder. Therefore:

\[ \begin{equation} S_{\zeta} (f)=0 \qquad \text{for} \qquad k \gt \frac{1}{r_0} \end{equation} \]

In Bladed, the radius \(r_0\) is taken as the minimum tower radius between the seabed and a height of \(3\) standard deviations of the wave elevation above the mean water level. At any instant, the wave elevation has a probability of \(99.85%\) of being within this range. Alternatively the cut-off frequency may be specified by the user.

Last updated 06-09-2024