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6D buoy theory: Slam force |
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6D buoy theory: Slam force |
The calculation of slam force on lumped buoys is a special case of the general cylinder-based model. For constant slam data, OrcaFlex uses a fixed user-defined value for waterplane area, while the variable data form has to make assumptions about the nature and orientation of the cylinder representing the lumped buoy.
Lumped buoys do not have a specific geometry. Consequently, it is not possible to calculate their waterplane area, and so you must define it (as a constant value, slam area) in order to use the constant slam coefficient model. This does cause some difficulty, however, since OrcaFlex cannot simply apply the resultant slam force the instant that the buoy makes contact with the surface. This would give a discontinuous step change in the load applied to the buoy, which cannot be permitted in a time domain simulation because it may preclude the existence of a dynamic equilibrium position. For the same reason, OrcaFlex cannot immediately stop applying the slam force the instant the buoy becomes fully-submerged.
To avoid this problem, the value of slam area is ramped up to its full value over the first 10% of the buoy's passage through the surface, and ramped down again to zero over the last 10% of the passage through the surface. This ensures that the slam force varies continuously at the start and end of penetration. In order to compensate for the slam force being reduced over this first and last 10% of the penetration, the slam area used is slightly increased during the middle 80% of the penetration through the surface. The ramping function that is applied to do this is shown in the following figure. It is a smooth ramping function that has been calculated to have the correct integral (one), so that the total slam impulse over a complete submergence or emergence is correct.
| Figure: | Lumped buoy slam force ramping |
In order to apply the variable data slamming model to lumped buoys, we must impute to them a circular cylindrical geometry. We therefore follow our buoyancy model assumptions and take each buoy for this purpose to be cylindrical in shape, with radius consistent with its user-specified volume and height, and its axis oriented in the instantaneous surface normal direction. This means that, for a lumped buoy, the half-span is constant and equal to half the user-specified height.
Spar buoys and towed fish, consisting of a stack of coaxial circular cylinders, follow the general slamming models precisely in both the constant slam coefficient and variable slam data cases: the slam load is evaluated independently for each cylinder of the stack, and these combine to give the net instantaneous slam load on the buoy. Note that hollow stacked buoys require special care.
For spar buoys and towed fish using constant slam coefficients, the discontinuity problem seen with lumped buoys does not arise in practice since the water plane area naturally rises from, and returns to, zero continuously as the buoy's passage through the surface begins and ends. No ramping is therefore applied to the slam force on spar buoys and towed fish. The waterplane area could in theory change discontinuously if the buoy end cylinder face hit the water exactly parallel to the surface, but in practice this situation is so unlikely to occur that we have not found it to be a problem. For the variable slam coefficient model, neither lumped nor stacked buoys have a discontinuity problem since the slam coefficients are required to take the value zero for $\small{h/s}\leq-1$ and to be continuous at $\small{h/s}=+1$.
For hollow buoys (stacked buoys only; lumped buoys may not be hollow), the slam area $A_\mathrm{w}$ in the constant coefficient calculation is given by \begin{equation} \label{Aw} A_\mathrm{w} = A_\mathrm{o} - \lvert\cos\theta\rvert A_\mathrm{i} \end{equation} where
$A_\mathrm{o}=$ area of water surface intersection of the whole cylinder, i.e. based on the cylinder outer diameter and including the intersection area of the hollow void
$\theta=$ angle between the buoy cylinder axis and the sea surface outward normal, $\vec{n}$
$A_\mathrm{i}=$ area of water surface intersection of the hollow void within the cylinder, i.e. based on the buoy cylinder inner diameter.
Formula (\ref{Aw}) for $A_\mathrm{w}$ for a hollow buoy is not theoretically based – it is simply an approximation that varies continuously between sensible limiting values:
For intermediate piercing angles $\theta$, the formula excludes a proportion $\cos\theta$ of the intersection area of the hollow void.
User-defined variable slam data, when used with a hollow spar buoy or towed fish, must include any effect due to the cylinders being hollow. The total volume $V_\mathrm{T}$ in the formulae for the variable slam data case is calculated by OrcaFlex from the outside diameter of the cylinder and does not allow for it being hollow. If the hollow buoy is free-flooding, then the rate of change of added mass coefficient with normalised submergence will depend on the inclination of the cylinder axis to the surface. The variable slam data specified should therefore be for the buoy orientation expected when the slam or water exit forces apply.
In all cases (all permutations of lumped/spar buoys and constant/variable coefficients), OrcaFlex is required to calculate $v_\mathrm{n}$, the component in the surface normal direction of the cylinder velocity relative to the fluid velocity. Both of these velocities are measured at the instantaneous centre of wetted volume of the corresponding cylinder.
For lumped buoys the slam force is applied at the instantaneous centre of wetted volume of the buoy, as with other hydrodynamic loads. This applies for both constant and variable slam data.
For spar buoys and towed fish, the slam force is calculated separately for each cylinder of the buoy. With constant slam data this force is then applied at the instantaneous centre of waterplane area of the corresponding cylinder. With variable slam data the force is applied at the instantaneous centre of wetted volume of the cylinder. In both cases, the slam force is correctly distributed along the length of a spar buoy or towed fish; for example if it penetrates the surface with its axis close to being parallel to the surface, so that multiple cylinders are piercing the surface simultaneously.