Theory: Morison elements
$\newcommand{\vn}{\vec{v}\urm{n}}$ $\newcommand{\vx}{v\urm{x}}$ $\newcommand{\vy}{v\urm{y}}$ $\newcommand{\vz}{v\urm{z}}$ $\newcommand{\dn}{d\urm{n}}$ $\newcommand{\da}{d\urm{a}}$ $\newcommand{\CD}[1]{\C{D #1}}$ $\newcommand{\fD}{\vec{f}\urm{D}}$ $\newcommand{\hn}{h\urm{n}}$ $\newcommand{\ha}{h\urm{a}}$ $\newcommand{\Vn}{V\urm{n}}$ $\newcommand{\Va}{V\urm{a}}$ $\newcommand{\pw}{p\urm{w}}$ $\newcommand{\fA}{\vec{f}\urm{A}}$ $\newcommand{\f}[1]{f\urm{#1}}$ $\newcommand{\Cm}[1]{\C{m #1}}$ $\newcommand{\Ca}[1]{\C{a #1}}$ $\newcommand{\aobject}{\vec{a}_{\textrm{object}}}$ $\newcommand{\afluid}{\vec{a}_{\textrm{fluid}}}$

Morison elements are collections of cylinders which attract hydrodynamic forces. They can be used to model loads which would otherwise not be captured in your diffraction analysis. For example, loads on parts of a structure which are not included in your panel mesh, or hydrodynamic drag loads.

Note: Morison elements can be used to improve the displacement RAOs and results which depend on them, such as QTFs and sea state RAOs. Morison elements do not affect results for load RAOs, added mass or damping.

Morison elements are rigidly attached to a body, referred to as the owner of the elements. Each element is discretised into $N$ sub-elements. Forces are calculated separately for each sub-element, and applied at the centroid of the sub-element.

Submergence

For sub-elements that pierce the sea surface, the proportion wet is found using the method for calculating submergence of line segments in OrcaFlex. For this calculation, the geometry is defined by the length, $l$, and the normal hydrodynamic diameter, $\hn$, of the sub-element. We denote proportion wet as $\pw$ when it appears in the equations below.

Drag force

Drag is determined by $\vec{v}$, the relative velocity between the fluid and a sub-element. Drag is quadratic in $\vec{v}$ and therefore stochastic linearisation must be performed, for a specific sea state, to obtain a linearised drag model which can contribute to the first-order equation of motion in OrcaWave.

The drag force is calculated using the cross-flow principle. That is, the relative fluid velocity $\vec{v}$ is split into its components $\vn$ and $\vz$ normal and parallel to the cylinder axis. The normal drag force is then determined by $\vn$ and its $x$- and $y$-components $\vx, \vy$; the parallel drag force is determined by $\vz$. The drag force vector, $\fD$, for a sub-element is given by \begin{equation} \begin{aligned}%implicit alignment is {rlrlrl...} \f{Dx} &= \tfrac12\ \pw \rho\ \dn l\ \CD{x} \vx \lvert\vn\rvert \\ \f{Dy} &= \tfrac12\ \pw \rho\ \dn l\ \CD{y} \vy \lvert\vn\rvert \\ \f{Dz} &= \tfrac12\ \pw \rho\ \pi\da l\ \CD{z} \vz \lvert\vz\rvert \end{aligned} \end{equation} where

$\fD = (\f{Dx},\f{Dy},\f{Dz})$ in element local coordinates

$\pw=$ proportion wet, as noted above

$\rho=$ water density

$\dn$, $\da$ are the element's drag diameters

$l=$ length of the sub-element

$\CD{x},$, $\CD{y}$, $\CD{z}$ are the element's drag coefficients

Inertia force

The inertia force is determined by the accelerations of the fluid and the sub-element. In contrast to drag, this force is linear and therefore there is no requirement for a specific sea state and stochastic linearisation. The force vector, $\fA$, for a sub-element is given by \begin{equation} \left( \begin{array}{c} \f{Ax} \\ \f{Ay} \\ \f{Az} \end{array} \right) = \pw \rho \left( \begin{array}{ccc} \Vn\Cm{x} & 0 & 0 \\ 0 & \Vn\Cm{y} & 0 \\ 0 & 0 & \Va\Cm{z} \end{array} \right) \afluid \,-\, \pw \rho \left( \begin{array}{ccc} \Vn\Ca{x} & 0 & 0 \\ 0 & \Vn\Ca{y} & 0 \\ 0 & 0 & \Va\Ca{z} \end{array} \right) \aobject \end{equation} where

$\fA = (\f{Ax},\f{Ay},\f{Az})$ in element local coordinates

$\Vn = \pi (\hn/2)^2 l$ is the element's volume based on the hydrodynamic diameter $\hn$; similarly $\Va = \pi (\ha/2)^2 l$

$\Ca{x},$, $\Ca{y}$, $\Ca{z}$ are the element's added mass coefficients

$\Cm{x},$, $\Cm{y}$, $\Cm{z}$ are the element's fluid inertia coefficients

$\aobject$ is the acceleration of the sub-element relative to earth

$\afluid$ is the acceleration of the fluid relative to earth