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6D buoy theory: Overview |
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6D buoy theory: Overview |
6D buoys are treated as rigid bodies with six degrees of freedom: three translational and three rotational. We describe here the various contributions to their equation of motion; there is also some more detailed documentation available at www.orcina.com/wp-content/uploads/Spar-Buoy-Equation-of-Motion.pdf for spar buoys in particular.
The weight force is \begin{equation} \vec{f}_\mathrm{w} = -m\ g\ \vec{u}_\mathrm{Z} \end{equation} where
$m=$ buoy mass
$g=$ acceleration due to gravity
$\vec{u}_\mathrm{Z}$ is a unit vector vertically upwards.
This force is applied (vertically downwards, due to the minus sign) at the buoy centre of mass.
The instantaneous proportion wet, $\PW$, is an important property of a surface-piercing lumped buoy and is used in a large part of the theoretical calculations. It is defined as \begin{equation} \PW = h\wet / h \end{equation} where
$h\wet=$ wetted height $= \frac12 h + (\text{depth of the specified centre of volume below the surface})$, taking depth as negative above the surface, so $0 \leq h\wet \leq h$
$h=$ buoy height, as given on the buoy data form.
Further notation:
$\rho=$ water density
$V\wet=$ wetted volume $= \PW V$
$V=$ volume, as given on the buoy data form
With this notation, the buoyancy force on a lumped buoy is given by \begin{equation} \vec{f}_\mathrm{b} = \rho\ g\ V\wet \vec{u}_\mathrm{Z} \end{equation} and is applied (vertically upwards) at the centre of wetted volume of the buoy, which is taken to be \begin{equation} (1-\PW) \frac{h}{2} \end{equation} below the specified centre of volume.
Under this model the proportion wet $\PW$, buoyancy force and centre of wetted volume all vary linearly with vertical position of the centre of volume relative to the surface. So, for instance,
This model is equivalent to considering the buoy, for buoyancy purposes, as a cylinder of the given volume and height, which pivots about its centre of volume such that its axis is always normal to the instantaneous water surface tangent plane.
| Warning: | This lumped buoy surface-piercing model is not realistic or accurate – OrcaFlex does not have enough geometry information for lumped buoys to model the surface-piercing accurately. It is just a simple way to give a buoyancy force, wetted volume and centre of wetted volume that vary in the right sort of way as the buoy passes through the surface. The model gives no contribution to roll and pitch stiffness from free surface effects when the buoy pierces the surface, so static stability of a floating lumped buoy is therefore not correctly represented. To model surface-piercing accurately you will need to use a spar buoy or towed fish, or multiple such buoys connected together to represent the actual buoy geometry. |
For spar buoys and towed fish, separate buoyancy forces are calculated and applied to each cylinder. The buoyancy force on an individual cylinder is given by \begin{equation} \vec{f}_\mathrm{b} = \rho\ g\ V\wet \vec{u}_\mathrm{Z} \end{equation} where in this case $V\wet$ is the volume of that part of the cylinder that is below the water surface. This force is applied (vertically upwards) at the centroid of that wetted volume.
| Note: | The precise form of the water surface used to calculate the cylinder $V\wet$ depends on the choice of added mass and damping data. If the data are specified by values for each cylinder, then the instantaneous water surface is used, while if the specification is by RAOs and matrices for buoy, then the (horizontal) mean water surface is used; this latter excludes buoyancy effects due to the waves, which are assumed to be included in the RAOs. |
In general, the hydrodynamic loads are calculated using Morison's equation, with additional components described below. The exception to this is if spar buoy added mass and damping effects are specified by wave load RAOs and added mass and damping matrices.
For a lumped buoy, the hydrodynamic loads are calculated using the fluid kinematics at the buoy centre of wetted volume. These loads are scaled by the buoy proportion wet and applied at the centre of wetted volume.
For a spar buoy or towed fish, the calculation of hydrodynamic loads depends on the form of added mass and damping chosen.
For precise details of the calculations, see added mass and damping and drag.
You can apply additional damping forces and moments that are directly proportional to the fluid velocity and angular velocity relative to the buoy. See lumped buoy damping and added mass and damping for details of these calculations.
The main use of these terms is to represent wave radiation damping for surface-piercing buoys. Data may be obtained theoretically from a 3D diffraction model of the buoy or, more commonly, from empirical results such as a roll decay test.
Rotation of the body relative to the fluid generates hydrodynamic moments which are analogous to the hydrodynamic forces given by Morison's equation. These fluid moments are calculated based on the angular velocity and acceleration of the buoy, without any contribution due to angular velocity and acceleration of the fluid. This is because the sea is treated as being irrotational, so there is no unique definition of fluid angular velocity and angular acceleration at a point. The sea angular velocity and acceleration are therefore taken to be zero.
If you are in any doubt as to the correctness of the model, then we recommend setting the moment terms to zero. Hydrodynamic moments will then be omitted completely for a lumped buoy. For a spar buoy represented by several cylindrical sections, moments will arise naturally as a result of the distribution of hydrodynamic forces along the buoy axis.
See the technical note hydrodynamic properties of a rectangular box for an example of the way in which hydrodynamic data may be derived.