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6D buoy theory: Spar buoy and towed fish added mass and damping |
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6D buoy theory: Spar buoy and towed fish added mass and damping |
The first order hydrodynamic loads on a spar buoy or towed fish can be modelled in two different ways, by giving either values for each cylinder or wave load RAOs and matrices for the whole buoy.
When you choose values for each cylinder, the added mass and damping effects are calculated separately for each cylinder. The resulting effects (force, moment, increased mass and rotational inertia), for each cylinder, are applied at the wetted centroid of the volume of water displaced by that cylinder.
In its simplest form, the added mass load on each cylinder follows the inertia component of Morison's equation. Writing $\ac$ for the cylinder acceleration relative to earth, we have \begin{equation} \label{fAconst} \fa = (\Delta + \Ca\Delta)\af - \Ca\Delta\ \ac \end{equation} where
$\fa$ is the added mass load on the cylinder
$\Delta$ is the reference mass, the mass of fluid displaced instantaneously by the cylinder
$\Ca$ is the constant added mass coefficient for the cylinder
$\af$ is the fluid acceleration relative to earth
$\ac=\af-\ar$, where $\ar$ is the fluid acceleration relative to the cylinder.
The first term in expression (\ref{fAconst}) can also be written as $(1{+}\Ca)\Delta \af$, and this is the form which is used when the added inertia coefficient $\Cm$ is assigned the value '~'. The $1$ in $(1{+}\Ca)$ represents what is known as the Froude-Krylov force; the $\Ca$ term gives rise to the added mass force.
$\Cm$ may also take a numerical value, and so the more general form for the added mass load on a cylinder is \begin{equation} \fa = \Cm \Delta\ \af - \Ca \Delta\ \ac \end{equation} where ~ for $\Cm$ is understood to mean the usual $(1{+}\Ca)$ form.
Thus far, we have only considered translational added mass and inertia. The rotational effects are simpler: since we deem the sea to be irrotational, the rotational fluid acceleration is zero and there is no rotational analogue of the $\Cm\Delta\af$ term. We are left with an added moment of inertia $\ma$ due to the cylinder's angular acceleration \begin{equation} \ma = -\PW\ I_\mathrm{a}\ \vec{\omega}_\mathrm{c}' \end{equation} where
$\PW$ is the instantaneous proportion wet of the cylinder, i.e. the proportion of the cylinder volume which is submerged
$I_\mathrm{a}$ is the added moment of inertia for the cylinder
$\vec{\omega}_\mathrm{c}'$ is the cylinder angular acceleration (relative to earth).
In terms of components in buoy local axes directions then, consider a spar buoy, for which the cylinder axis direction is the buoy $z$-direction. The forces and moments in these directions due to added mass are \begin{equation} \begin{aligned} \fax &= \Cmn \Dn\ \afx - \Can \Dn\ \acx \\ \fay &= \Cmn \Dn\ \afy - \Can \Dn\ \acy \\ \faz &= \Cma \Da\ \afz - \Caa \Da\ \acz \\ \max &= -\PW \Ian\ \omega_\mathrm{cx}' \\ \may &= -\PW \Ian\ \omega_\mathrm{cy}' \\ \maz &= -\PW \Iaa\ \omega_\mathrm{cz}' \end{aligned} \end{equation} where the subscripts $\mathrm{n}$ and $\mathrm{a}$ denote the normal and axial components respectively. The components of the displaced reference mass, $\Dn$ and $\Da$, need particular care for hollow cylinders; these are described below.
For a towed fish the cylinder axis direction is instead the buoy $x$-direction. So for a towed fish the subscripts $x$ and $z$ in these equations are interchanged, so that the axial values of the coefficients and reference masses and inertias are used in the equations for $\fax$ and $\max$, and the normal direction values are used in the equations for $\faz$ and $\maz$.
You may, if you wish, specify the normal added mass coefficient $\Can$ in a more precise form, using tabular variable data. Using a constant value, as above, means that the variation with depth of the added mass load is calculated by OrcaFlex through the displaced mass (for force) and proportion wet (for moment); choosing instead a variable form for $\Can$ means that the data become the source of the variation with depth. Specifically, you define the values of $\Can$ corresponding to a range of values of normalised submergence, $\hovers$, in a generalisation of DNV RP-H103 3.2.13. This allows a more precise definition of the variation of added mass with submergence than the constant coefficient. It also lets you define the nature of any variation when the cylinder is fully submerged, which is not possible for the constant coefficient case (since, when fully submerged, the displaced mass is constant).
As all the variation is in the coefficient values, the reference mass is constant in this case; we take it to be the mass of fluid displaced by the whole of the cylinder volume were it to be fully submerged, regardless of its actual instantaneous submergence and (for a hollow cylinder) as if its ends were closed (see reference fluid mass below). We denote this fully-submerged reference mass by $\Mn$.
Since the axial added mass coefficient $\Caa$ and the added inertia $\Cm$ and added moment of inertia $I_\mathrm{a}$ are here always of the constant form and not allowed by OrcaFlex to be variable, only the normal translational components differ under variable added mass. For a spar buoy, these components are $\fax$ and $\fay$, and writing $\Canhovers$ to signify the variable data form for $\Can$, these components become for $\Cm\neq\ \sim$ \begin{equation} \begin{aligned} \fax &= \Cmn\Dn\ \afx - \Canhovers\Mn\ \acx \\ \fay &= \Cmn\Dn\ \afy - \Canhovers\Mn\ \acy \end{aligned} \end{equation} and for $\Cm =\ \sim$ \begin{equation} \begin{aligned} \fax &= (\Dn+\Canhovers\Mn)\ \afx - \Canhovers\Mn\ \acx \\ \fay &= (\Dn+\Canhovers\Mn)\ \afy - \Canhovers\Mn\ \acy \end{aligned} \end{equation} Again, for a towed fish we simply interchange the $x$- and $z$-directions, so the axial terms are used in the equations for $\fax$ and $\max$, and the normal terms for $\faz$ and $\maz$.
| Notes: | If you choose to use variable added mass, you must provide data for at least two values of normalised submergence $\hovers$. The smallest value of $\hovers$ must be -1 (corresponding to the cylinder just about to enter the water), and on physical grounds the corresponding $\Ca$ value must be zero. There is no upper limit on the range of $\hovers$: the data will be truncated at both ends of the range. |
| Added mass is frequency-dependent, so you should take care to use data which correspond to the response frequency for your particular model. DNV, RP-H103, 3.2.13, for instance, provides a graph of $\Ca$ against $\hovers$, but this represents the high frequency limit so may well be inappropriate for your own model. |
For motion normal to the cylinder axis, the value used for the displaced reference fluid mass $\Dn$ is the mass of the fluid displaced by the submerged part of the whole of the cylinder cross section. So, if the cylinder is hollow (i.e. inner diameter is non-zero), $\Dn$ includes the fluid trapped inside the part of the cylinder that is below the surface. The fully-submerged reference mass $\Mn$ for the normal direction follows the same rule: if the cylinder is hollow, then $\Mn$ includes the fluid trapped inside the whole cylinder. In effect, a hollow cylinder is treated as if it were solid when determining the normal reference fluid mass $\Dn$ or $\Mn$.
For motion parallel to the cylinder axis, the value used for the axial reference fluid mass, $\Da$, does depend on whether the cylinder is hollow. If it is not hollow, then $\Da$ is the same as the value used for motion normal to the cylinder axis, $\Dn$, i.e. equal to the mass of the fluid displaced by the submerged part of the cylinder; if the cylinder is hollow, then $\Da$ is the mass of the fluid displaced by the submerged part of just the cylinder annulus, excluding the fluid trapped inside the part of the cylinder that is below the surface. (There is no variable data form of the coefficient $\Caa$, so no corresponding fully-submerged $M_\mathrm{a}$)
These values for the reference fluid mass and inertia are based on the assumption that for a hollow cylinder the trapped fluid contents are free to translate and rotate axially relative to the cylinder, but not free to move normal to the cylinder axis.
For each of the buoy local axes directions, each cylinder is subject to a damping force and damping moment given by \begin{align} f_\mathrm{D} &= -\PW\ U\!D\!F\ v \\ m_\mathrm{D} &= -\PW\ U\!D\!M\ \omega \end{align} where
$\PW=$ proportion wet of this cylinder
$U\!D\!F, U\!D\!M=$ specified unit damping force and unit damping moment for this direction of motion (normal or parallel to the cylinder axis)
$v=$ component, in this direction, of the translational velocity of the buoy at the instantaneous position of the centroid of the submerged part of this cylinder, either relative to the earth or relative to the fluid velocity (as given by the buoy data).
$\omega=$ component, in this direction, of the angular velocity of the buoy relative to the earth. (The fluid is treated as irrotational so its angular velocity is taken to be zero).
| Note: | If damping relative to fluid is chosen in the buoy data, then the velocity $v$ used in the damping force formula is the buoy velocity minus the fluid velocity (including current and waves) at the instantaneous position of the centroid of the submerged part of the cylinder. The angular velocity $\omega$ used in the damping moment formula is not affected by this – it is always the buoy angular velocity relative to the earth, since the fluid is treated as irrotational so its angular velocity is taken to be zero. |
| Note: | This option is only available for spar buoys. |
When you choose RAOs and matrices for buoy, the added mass and damping effects are calculated from the specified wave force and moment RAOs and added mass and damping matrices. In addition, the buoyancy force is calculated using the mean water level, rather than the instantaneous water surface. This excludes the wave-related buoyancy effects, since these are assumed to be accounted for in the loads specified by the RAOs.
| Warning: | The drag force is also calculated using the mean water level, although (nonlinear) drag cannot be completely accounted for in (linear) RAOs. |
The RAOs are used to calculate force and moment vectors that are proportional to the amplitude of the wave component. They are applied at the instantaneous position of the RAO, added mass and damping origin, but are specified with respect to a frame of reference that has $x$ horizontal in the wave direction, $y$ horizontal and normal to the wave direction, and $z$ vertically upwards. The surge RAO therefore specifies a force that acts at the RAO, added mass and damping origin in the wave direction, the heave RAO specifies a vertical force, and the pitch RAO specifies a moment acting about the horizontal line normal to the wave direction.
The added mass and damping matrices are also applied at the RAO, added mass and damping origin, and they are applied in the buoy axes directions, i.e. the surge added mass is applied in the buoy local $x$-direction, etc.
The added mass matrix is simply added into the buoy's inertia (also known as the virtual mass matrix).
The damping load is calculated using the matrix equation \begin{equation} \begin{bmatrix}\vec{f}_\mathrm{D}\\ \vec{m}_\mathrm{D}\end{bmatrix} = -\mat{D} \begin{bmatrix}\vec{v}\\ \vec{\omega}\end{bmatrix} \end{equation} where
$\vec{f}_\mathrm{D} = [f_\mathrm{Dx},f_\mathrm{Dy},f_\mathrm{Dz}]^T$, $\vec{m}_\mathrm{D} = [m_\mathrm{Dx},m_\mathrm{Dy},m_\mathrm{Dz}]^T=$ the resulting damping force and moment, in buoy local axes directions
$\mat{D}=$ the given six DOF damping matrix
$\vec{v} = [v_\mathrm{x},v_\mathrm{y},v_\mathrm{z}]^T=$ components, in buoy axes directions, of the buoy velocity at the RAO, added mass and damping origin, relative to the earth or relative to the fluid velocity (as specified in the buoy data)
$\vec{\omega} = [\omega_\mathrm{x},\omega_\mathrm{y},\omega_\mathrm{z}]^T=$ components, in buoy axes directions, of the buoy angular velocity.
| Note: | If damping relative to fluid is specified in the buoy data, then the velocity $\vec{v}$ used here excludes the fluid velocity due to waves, because the damping effects due to waves are assumed to be included in the RAOs. $\vec{v}$ is therefore taken to equal the buoy velocity minus the current velocity at the instantaneous position of the centroid of the submerged part of the cylinder. And $\vec{\omega}$ is simply equal to the buoy angular velocity relative to the earth, since the current has no angular velocity. Added mass effects are always calculated using acceleration relative to earth, even if the current acceleration is non-zero. |