Morison elements
$\newcommand{\dn}{d\urm{n}}$ $\newcommand{\da}{d\urm{a}}$ $\newcommand{\CDx}{\C{Dx}}$ $\newcommand{\CDy}{\C{Dy}}$ $\newcommand{\CDz}{\C{Dz}}$ $\newcommand{\fD}{\vec{f}\urm{D}}$ $\newcommand{\fDn}{\vec{f}\urm{Dn}}$ $\newcommand{\fDx}{f\urm{Dx}}$ $\newcommand{\fDy}{f\urm{Dy}}$ $\newcommand{\fDz}{f\urm{Dz}}$ $\newcommand{\vn}{\vec{v}\urm{n}}$ $\newcommand{\vr}{\vec{v}\urm{r}}$ $\newcommand{\vx}{v\urm{x}}$ $\newcommand{\vy}{v\urm{y}}$ $\newcommand{\vz}{v\urm{z}}$ $\newcommand{\Cax}{\C{ax}}$ $\newcommand{\Cay}{\C{ay}}$ $\newcommand{\Caz}{\C{az}}$ $\newcommand{\Cmx}{\C{mx}}$ $\newcommand{\Cmy}{\C{my}}$ $\newcommand{\Cmz}{\C{mz}}$ $\newcommand{\fA}{\vec{f}\urm{A}}$ $\newcommand{\fAx}{f\urm{Ax}}$ $\newcommand{\fAy}{f\urm{Ay}}$ $\newcommand{\fAz}{f\urm{Az}}$ $\newcommand{\af}{\vec{a}\urm{f}}$ $\newcommand{\afx}{a\urm{fx}}$ $\newcommand{\afy}{a\urm{fy}}$ $\newcommand{\afz}{a\urm{fz}}$ $\newcommand{\ae}{\vec{a}\urm{e}}$ $\newcommand{\aex}{a\urm{ex}}$ $\newcommand{\aey}{a\urm{ey}}$ $\newcommand{\aez}{a\urm{ez}}$

Morison elements are collections of cylinders which attract hydrodynamic forces. Morison elements can be rigidly attached to either a vessel or a 6D buoy, referred to as the owner of the elements. The functionality is identical for both types of owner.

Morison element data

The elements are defined on the Morison elements page of the owner's data form.

Element type

The element type holds the hydrodynamic forces data and the drawing data for the element.

Position and orientation

The position defines the location of end A of the element, relative to the owner's origin and with respect to the owner axes. The element's $z$-axis is defined by the azimuth and declination angles. The $z$-axis points from end A towards end B along the axis of the cylinder. The $x$-axis and $y$-axis are the normal directions of the cylinder, defined by the gamma angle.

Length, $L$

The length of the element.

Number of segments, $N$

The element is discretised into $N$ sub-elements of length $l=L/N$.

Morison element type data

The Morison element type data define the hydrodynamic properties of the Morison elements. Multiple element types can be defined, with data specified on the Morison element type form.

Drag diameters

The normal drag diameter, $\dn$, and the axial drag diameter, $\da$. If $\da$ is set to ~ then the value of $\dn$ is used.

Drag coefficients

The drag coefficients $\CDx$, $\CDy$ and $\CDz$, with respect to the element's local axes.

The axial coefficient, $\CDz$, is constant, while the normal coefficients, $\CDx$ and $\CDy$, may take, independently, the form of:

The normal coefficients often take the same value; this can be indicated conveniently by setting ~ for $\CDy$, to mean 'same as $\CDx$'.

Hydrodynamic diameters

The normal hydrodynamic diameter, $h_n$, and the axial hydrodynamic diameter, $h_a$. If $h_a$ is set to ~ then the value of $h_n$ is used.

Hydrodynamic diameters $h_n$ and $h_a$ are entirely independent from the drag diameters $\dn$ and $\da$.

The hydrodynamic diameters are used to define the volume of the Morison element. The element volume per unit length is an area, and OrcaFlex uses circle areas calculated from these hydrodynamic diameters to define volume in the element normal and axial directions.

If your input data for added mass or fluid inertia provides a reference area, this can be input directly in the cross-section area column of the data form. The normal diameter $h_n$ will automatically update with the corresponding circle diameter.

Added mass coefficients

The added mass coefficients $\Cax$, $\Cay$ and $\Caz$, with respect to the element's local axes.

The axial coefficient, $\Caz$, is constant, while the normal coefficients, $\Cax$ and $\Cay$, may take, independently, the form of:

The normal coefficients often take the same value; this can be indicated conveniently by setting ~ for $\Cay$, to mean 'same as $\Cax$'.

Fluid inertia coefficients

The fluid inertia coefficients $\Cmx$, $\Cmy$ and $\Cmz$, with respect to the element's local axes.

These coefficients can be given an explicit constant value, or can be set to '~'. Each direction is treated independently. The use of ~ for a fluid inertia coefficient instructs OrcaFlex to calculate the coefficient using the equation $\Cm = 1 + \Ca$. If the referenced added mass coefficient is variable, then the fluid inertia coefficient will also vary.

Modelling

Each element is discretised into $N$ sub-elements. The hydrodynamic forces are then calculated separately for each sub-element, and applied at the centre of the sub-element.

Submergence

For elements that pierce the sea surface, the proportion wet of the sub-elements is calculated using the same method as used for lines, with the sub-element circumference determined by $h_n$. We denote proportion wet as $p\urm{w}$ when it appears in the equations below.

Drag calculation

The drag force is calculated using the cross-flow principle. That is, the fluid velocity $\vr$ relative to the sub-element is split into its components $\vn$ and $\vz$ normal and parallel to the element $z$-axis. The drag force normal to the element $z$-axis is then determined by $\vn$ and its $x$- and $y$-components $\vx, \vy$; the drag force parallel to the element $z$-axis is determined by $\vz$.

The drag force vector, $\fD$, for a sub-element is given by \begin{align} \fDx &= \tfrac12\ p\urm{w} \rho\ \dn l\ \CDx \vx \lvert\vn\rvert \\ \fDy &= \tfrac12\ p\urm{w} \rho\ \dn l\ \CDy \vy \lvert\vn\rvert \\ \fDz &= \tfrac12\ p\urm{w} \rho\ \pi\da l\ \CDz \vz \lvert\vz\rvert \end{align} where

$\rho=$ fluid density

$p\urm{w}=$ proportion wet, as noted above

$\fD = (\fDx,\fDy,\fDz)$ in element local coordinates

Added mass and fluid inertia calculations

For the constant coefficient case the added mass force is \begin{align} \fAx &= \Cmx\Delta\ \afx - \Cax\Delta\ \aex \\ \fAy &= \Cmy\Delta\ \afy - \Cay\Delta\ \aey \\ \fAz &= \Cmz\Delta\ \afz - \Caz\Delta\ \aez \end{align} where

$\fA$ is the added mass force on the sub-element

$\Delta$ is the mass of fluid displaced by the sub-element. The displaced volume is calculated using diameters $h_n$ and $h_a$, as described above.

$\af$ is the fluid acceleration relative to earth

$\ae$ is the sub-element acceleration relative to earth

There are further considerations that apply if coefficients are variable, and when $\Cm$ is calculated based on $\Ca$.

For any contribution to a component of $\fA$ which has a constant coefficient, $\Delta$ is the instantaneous displaced mass, which accounts for surface piercing using the same approach as for line segments, using sub-element proportion wet. Note that when fluid inertia coefficient is calculated as $\Cm = 1 + \Ca$, then the contribution from the '1' here has constant coefficient, and therefore uses instantaneous displaced mass.

Variable $\Ca$ may be used explicitly for components of the force from $\ae$, or implicitly via application of $\Cm = 1 + \Ca$ for the force from $\af$. When $\Ca$ is variable, the displaced mass $\Delta$ is taken from the total reference volume for the sub-element - proportion wet is not used. We assume that force variation due to proportion wet is accounted for as part of the variable coefficient data source.

The fluid kinematics and surface elevation used in the hydrodynamic forces calculations take account of the wave calculation method and disturbance vessel (if any) specified by the owner. For elements attached to a vessel, which do not support a choice of wave calculation method or disturbance vessel, the wave kinematics are always calculated at the element's instantaneous position (exact) in the undisturbed sea state.