Vessel theory: Sea state disturbance

Sea state disturbance modelling is based on the potential theory of inviscid fluids, which is the approach used in vessel response diffraction programs. In this approach, the fluid elevation, motion and pressure distributions for a wave are all based on a complex-valued potential function, $\Phi(\vec{x}, t)$, which is a function of position, $\vec{x} = (x,y,Z)$, and time, $t$. The position coordinates $x$, $y$ and $Z$ are relative to a horizontal frame of reference with axes parallel to the vessel's primary low frequency heading axes. The $xy$ origin is at the low frequency $xy$ position of the vessel origin; the $Z$ origin at the mean sea surface level, so $-Z$ is the depth of the point below the mean sea surface level.

The sea surface elevation, $\eta(\vec{x}, t)$, for this wave is computed in terms of the real part of the time-derivative of the complex velocity potential at the sea surface \begin{equation} \eta(\vec{x}, t) = \Re \left\{ - \frac{1}{g} \left. \frac{\partial \Phi}{\partial t} \right|_{Z=0} \right\} \end{equation} where $g$ is the acceleration due to gravity. The velocity field within the fluid is given by the real part of the gradient of $\Phi(\vec{x}, t)$ \begin{equation} \vec{v}(\vec{x}, t) = \Re \{ \nabla \Phi(\vec{x}, t) \} \end{equation} where $\nabla$ is the gradient operator. All the desired properties (surface elevation, velocity and acceleration) of the wave can be derived from its velocity potential function $\Phi(\vec{x}, t)$ and gradient $\nabla \Phi(\vec{x}, t)$. Disturbance effects can therefore be represented by data which specify how the velocity potential and its gradient are affected by the disturbance.

Velocity potential disturbance

Consider a single Airy wave. If there were no vessel present, then the velocity potential would be that of the undisturbed wave, $\Phi_\urm(\vec{x}, t)$, given by \begin{equation} \label{eq:undisturbed} \Phi_\urm(\vec{x}, t) = \i A_\urm(\vec{x}) \exp\left( \i [\omega t - \varphi_\urm(\vec{x})] \right) \end{equation} where

$A_\urm(\vec{x}) = \frac{A g \cosh[k(Z+h)]}{\omega \cosh(kh)} =$ the amplitude of the undisturbed velocity potential at $\vec{x}$

$A =$ the undisturbed wave elevation amplitude

$k=\frac{2 \pi}{\lambda} =$ the wave number, where $\lambda$ is the wavelength

$h =$ water depth

$\varphi_\urm(\vec{x})=$ phase lag (in radians) of the undisturbed velocity potential at position $\vec{x}$, at simulation time zero

$\omega = \frac{2 \pi}{T} =$ wave angular frequency, where $T$ is the wave period.

The presence of (and response of) the vessel (or multibody group of vessels) disturbs the wave, and the resulting disturbed velocity potential function is $\Phi_\drm(\vec{x}, t)$, given by \begin{equation} \Phi_\drm(\vec{x}, t) = \i A_\drm(\vec{x}) \exp\left( \i [\omega t - \varphi_\drm(\vec{x})] \right) \end{equation} where subscript $\drm$ denotes the values for the disturbed wave. $A_\drm(\vec{x})$ and $\varphi_\drm(\vec{x})$ are the amplitude and phase lag (at simulation time zero) of the disturbed velocity potential at position $\vec{x}$. The angular frequency, $\omega$, is not affected by the disturbance.

The complex-valued disturbance RAO, $\mathscr{P}(\vec{x})$, is defined as the ratio of the disturbed potential to the undisturbed potential, given by \begin{equation} \mathscr{P}(\vec{x}) = \frac{\Phi_\drm(\vec{x}, t)}{\Phi_\urm(\vec{x}, t)} = \frac{A_\drm(\vec{x})}{A_\urm(\vec{x})} \exp\left( \i [\varphi_\urm(\vec{x}) - \varphi_\drm(\vec{x})] \right) \end{equation} Note that this disturbance RAO is independent of time, since the time variation, $\mathrm{e}^{\i \omega t}$, has cancelled out in this ratio.

OrcaFlex can calculate the disturbed velocity potential of each wave from $\mathscr{P}(\vec{x})$, by multiplying the known undisturbed potential, given by (\ref{eq:undisturbed}), by $\mathscr{P}(\vec{x})$ for that wave \begin{equation} \Phi_\drm(\vec{x}, t) = \mathscr{P}(\vec{x}) \Phi_\urm(\vec{x}, t) \end{equation} OrcaFlex can also calculate the disturbed potential gradient from the spatial derivative of that equation \begin{equation} \nabla \Phi_\drm(\vec{x}, t) = \mathscr{P}(\vec{x}) \nabla \Phi_\urm(\vec{x}, t) + \Phi_\urm(\vec{x}, t) \nabla \mathscr{P}(\vec{x}) \end{equation}

Disturbed wave field

The user disturbance RAO data (after allowing for whether the vessel type phase convention is leads or lags) specify the amplitudes and phases, $\lvert\mathscr{P}(\vec{x})\rvert$, $\arg (\mathscr{P}(\vec{x}) )$ and $\lvert\nabla_i \mathscr{P}(\vec{x})\rvert$, $\arg (\nabla_i \mathscr{P}(\vec{x}))$ of this complex-valued disturbance RAO and its gradient, as a function of position relative to the vessel, $\vec{x}$, and the direction and period/frequency of the wave component. $\nabla_i$ is the $i$-th component of the gradient operator, $\nabla$

OrcaFlex can calculate $\mathscr{P}(\vec{x})$ from these data, and hence use the above equations to calculate the disturbed velocity potential, $\Phi_\drm$, and its gradient, $\nabla \Phi_\drm$, for the wave component from the corresponding undisturbed potential $\Phi_\urm$ and gradient $\nabla \Phi_\urm$. All the desired properties of the disturbed wave component can then be derived from this disturbed velocity potential and its gradient. The properties of the whole wave field then follow from summing over all wave components in the sea state.