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6D buoys: Spar buoy and towed fish added mass and damping |
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6D buoys: Spar buoy and towed fish added mass and damping |
There are two choices to be made on the added mass and damping page of the spar buoy and towed fish data form, regarding the way in which these first order effects are modelled:
The added mass and damping effects are calculated separately for each cylinder using Morison's equation.
Separate added mass coefficients $\Ca$ are given for flow normal to ($x$ and $y$ directions) and axially along ($z$-direction) the cylinder. The coefficients for the normal direction may be variable with submergence; axial coefficients must be constant. Translational added mass effects are calculated using the displaced mass as the reference mass for each cylinder: specifically, instantaneous displacement for constant coefficients and fully-submerged displacement for variable coefficients.
Translational inertia coefficients $\Cm$ are given in the same way for normal and axial directions, though only constant values are allowed here. A value of ~ tells OrcaFlex to use the usual "Froude-Krylov plus added mass" formulation for inertia.
| Note: | The Froude-Krylov force will always be applied to the buoy if Cm is set to '~', even if $\Ca=0$. To suppress all fluid inertia effects you need to set both $\Ca$ and $\Cm$ to zero. |
Rotational added inertia is specified directly: no reference inertia is invoked. Separate values are given for rotation about the cylinder axis and normal to that axis.
The damping forces and moments are the hydrodynamic loads that are proportional to cylinder velocity and angular velocity, respectively. You specify whether the velocity used is the cylinder velocity relative to the earth or to the fluid. This only affects the translational velocity: since the fluid is considered irrotational, the angular velocity is the same in either case.
These damping terms are primarily intended to represent radiation damping on 6D buoys, in which case damping relative to earth should normally be chosen. This generally applies to surface-piercing buoys, where the waves generated by the buoy motion effectively extract energy from the buoy motion.
For a fully-submerged buoy, however, wave radiation damping will not normally arise (unless it is close to the surface). The damping data can then instead be used to represent linear skin friction, in which case damping relative to fluid should be chosen. Note, however, that such linear damping is usually not significant compared to the quadratic drag.
The damping forces and moments are given by the unit damping force and unit damping moment for the normal and axial directions. These are the force and moment that the cylinder will experience, in that direction, when the cylinder velocity (relative to earth or fluid, for damping forces) in that direction is 1 length unit per unit time.
The linear hydrodynamic effects are specified by giving wave force and moment RAOs, and added mass and damping matrices. Normally these data would come from an external source, such as a diffraction program.
This option was developed primarily for CALM buoys with diameter in the range 5m to 15m, and is only appropriate for circularly-symmetric spar buoys whose axis undergoes only small oscillations about vertical. It is therefore not available for towed fish.
| Warning: | The RAOs and added mass and damping matrices apply to the mean position of the buoy. They are not modified to account for any variations in buoy attitude or immersion, so this option is only suitable for buoys that undergo small oscillations about their mean position. |
The coordinates, with respect to buoy axes, of the point on the buoy at which the RAOs and added mass and damping matrices are applied. At this point,
Because the buoy is assumed to be axisymmetric, the $x$ and $y$ coordinates of the RAO origin should coincide with those of the stack base origin position: OrcaFlex will warn you if this is not the case.
This table of RAOs gives the amplitudes and phases of the surge force, heave force and pitch moment RAOs, as a function of wave angular frequency. The RAOs must follow the conventions that the force and moment amplitudes are those applied at the RAO, added mass and damping origin due to a unit amplitude wave, and the phases are lags, in degrees, from the time the wave crest passes that origin until the maximum positive force or moment occurs.
In reality, added mass and damping matrices are frequency-dependent (as for OrcaFlex vessels), but for buoys you can only enter the matrices for one frequency. You should give the values for the principal frequency of excitation expected.
The added mass and damping matrices are $6{\times}6$ symmetric matrices, so only the cells on and above the leading diagonal are editable. The cells below the leading diagonal are non-editable and always reflect the corresponding cell values above the diagonal.
The units of the $3{\times}3$ blocks of the added mass matrix are \begin{equation} \nonumber \left[ \begin{matrix} M & ML \\ ML & ML^2 \end{matrix} \right] \end{equation} where $M$ and $L$ denote the units of mass and length, respectively.
The units of the $3{\times}3$ blocks of the damping matrix are \begin{equation} \nonumber \left[ \begin{matrix} \dfrac{F}{L/T} & \dfrac{F}{\rad/T} \\ \dfrac{FL}{L/T} & \dfrac{FL}{\rad/T} \end{matrix} \right] \end{equation} where $F$, $L$ and $T$ denote the units of force, length and time, respectively.
Because the buoy is assumed to be axisymmetric, these matrices should both have the axisymmetric form \begin{equation} \nonumber \left[ \begin{matrix} a & 0 & 0 & 0 & b & 0 \\ 0 & a & 0 & -b & 0 & 0 \\ 0 & 0 & c & 0 & 0 & 0 \\ 0 & -b & 0 & d & 0 & 0 \\ b & 0 & 0 & 0 & d & 0 \\ 0 & 0 & 0 & 0 & 0 & e \end{matrix} \right] \end{equation}