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6D buoy theory: Lumped buoy added mass, damping and drag |
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6D buoy theory: Lumped buoy added mass, damping and drag |
The added mass and fluid inertia loads on a lumped buoy can be modelled in two different ways, either by supplying full 6x6 added mass and inertia matrices or by giving only the diagonal values of each. The complete matrices allow a fully general model, while to use only the diagonal data is a simplified view which does not allow for interaction between the different direction components.
The added mass load $f_\mathrm{A}$ for each translational component follows the inertial part of Morison's equation. Writing $\ab$ for the buoy acceleration relative to earth in the component direction, we have the general form \begin{equation} \label{fAdiagonal} f_\mathrm{A} = (\DM + \Ca^\mathrm{t}\DM)\af - \Ca^\mathrm{t}\DM\ \ab \end{equation} where
$\DM$ is the instantaneous reference mass, given by $\PW M_\textrm{ref}$
$\PW$ is the proportion wet of the buoy
$M_\textrm{ref}$ is the reference hydrodynamic mass for this component, given on the data form
$\Ca^\mathrm{t}$ is the translational added mass coefficient for this component, given on the data form
$\af$ is the component of fluid acceleration, relative to earth
$\ab=\af-\ar$, where $\ar$ is the component of fluid acceleration relative to the buoy.
The first term in expression (\ref{fAdiagonal}) can also be written as $(1{+}\Ca^\mathrm{t})\DM \af$, 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 the buoy is \begin{equation} \label{fADiagonal} f_\mathrm{A} = \Cm \DM\ \af - \Ca^\mathrm{t} \DM\ \ab \end{equation} where ~ for $\Cm$ is understood to mean the usual $(1{+}\Ca^\mathrm{t})$ form.
Thus far, we have only considered translational added mass and inertia. The rotational effects are simpler: since we deem the fluid to be irrotational, the rotational fluid acceleration is zero and there is no rotational analogue of the $\Cm\DM\af$ term. We are left with an added moment of inertia $m_\mathrm{A}$ due to the buoy's angular acceleration \begin{equation} \label{mADiagonal} m_\mathrm{A} = -\Ca^r\ \DI\ \vec{\omega}_\mathrm{b}' \end{equation} where
$\DI$ is the instantaneous reference inertia, given by $\PW I_\textrm{ref}$
$I_\textrm{ref}$ is the reference hydrodynamic inertia for this component, given on the data form (so $\PW I_\textrm{ref}$ is the instantaneous reference inertia)
$\Ca^r$ is the rotational added mass coefficient for this component, given on the data form
$\vec{\omega}_\mathrm{b}'$ is the buoy angular acceleration (relative to earth).
Writing out equations (\ref{fADiagonal}} and (\ref{mADiagonal}) in full in local axes directions then, the forces and moments due to added mass are \begin{equation} \begin{aligned} f_\mathrm{Ax} &= \C{mx} \DMx\ a_\mathrm{fx} - \C{ax}^t \DMx\ a_\mathrm{bx} \\ f_\mathrm{Ay} &= \C{my} \DMy\ a_\mathrm{fy} - \C{ay}^t \DMy\ a_\mathrm{by} \\ f_\mathrm{Az} &= \C{mz} \DMz\ a_\mathrm{fz} - \C{az}^t \DMz\ a_\mathrm{bz} \\ m_\mathrm{Ax} &= -\C{ax}^r \Delta_\mathrm{Ix} \vec{\omega}_\mathrm{bx}' \\ m_\mathrm{Ay} &= -\C{ay}^r \Delta_\mathrm{Iy} \vec{\omega}_\mathrm{by}' \\ m_\mathrm{Az} &= -\C{az}^r \Delta_\mathrm{Iz} \vec{\omega}_\mathrm{bz}' \end{aligned} \end{equation} where
$\DMx = \PW M_\textrm{ref,x}$ and similarly for $y$ and $z$
$\Delta_\mathrm{Ix} = \PW I_\textrm{ref,x}$ and similarly for $y$ and $z$.
When you specify the full 6x6 added mass and damping matrices, the given added mass matrix is simply scaled by the proportion wet and added into the buoy's inertia (also known as the virtual mass matrix). This inertia matrix is considered with respect to local axes and is applied at the centre of wetted volume of the buoy.
The fluid inertia load, considered with respect to local axes and applied at the centre of wetted volume, is given by the matrix equation \begin{equation} \begin{bmatrix}\vec{f}_\mathrm{A}\\ \vec{m}_\mathrm{A}\end{bmatrix} = \PW\ \mat{I} \begin{bmatrix}\vec{\af}\\ \vec{0}\end{bmatrix} \end{equation} where
$\vec{f}_\mathrm{A} = [f_\mathrm{Ax},f_\mathrm{Ay},f_\mathrm{Az}]^T$, $\vec{m}_\mathrm{A} = [m_\mathrm{Ax},m_\mathrm{Ay},m_\mathrm{Az}]^T=$ the resulting fluid inertia force and moment, in local buoy axes directions
$\mat{I}=$ the full $6{\times}6$ fluid inertia matrix
$\vec{a}_\mathrm{f} = [a_\mathrm{fx},a_\mathrm{fy},a_\mathrm{fz}]^T=$ components, in local buoy axes directions, of the water particle acceleration relative to the earth
$\vec{0}$ represents the fluid angular acceleration components, which are taken to be zero since the fluid is treated as irrotational.
The damping force applied in each local axis direction is given by \begin{equation} \begin{aligned} f_\mathrm{Dx} &= -\PW\ U\!D\!F_\mathrm{x}\ v_\mathrm{x} \\ f_\mathrm{Dy} &= -\PW\ U\!D\!F_\mathrm{y}\ v_\mathrm{y} \\ f_\mathrm{Dz} &= -\PW\ U\!D\!F_\mathrm{z}\ v_\mathrm{z} \\ \end{aligned} \end{equation} where
$U\!D\!F=$ specified unit damping force for each component direction
$\vec{v}=$ buoy velocity, either relative to the earth or relative to the fluid velocity as given in the buoy data
The damping moment applied in each local axis direction is given by \begin{equation} \begin{aligned} m_\mathrm{Dx} &= -\PW\ U\!D\!M_\mathrm{x}\ \omega_\mathrm{x} \\ m_\mathrm{Dy} &= -\PW\ U\!D\!M_\mathrm{y}\ \omega_\mathrm{y} \\ m_\mathrm{Dz} &= -\PW\ U\!D\!M_\mathrm{z}\ \omega_\mathrm{z} \end{aligned} \end{equation} where
$U\!D\!M=$ specified unit damping moment for each component direction
$\vec{\omega}=$ angular velocity of the buoy. Note that $\vec{\omega}$ does not include any contribution from fluid angular velocity: the fluid is treated as irrotational so has zero angular velocity.
Hydrodynamic and aerodynamic drag forces, represented by the drag term in Morison's equation, are applied to the buoy. The same drag formulation is used for hydrodynamic and aerodynamic drag forces.
| Note: | Aerodynamic drag is only included if the include wind loads on 6D buoys option is enabled in the environment data. |
The drag force applied in each local axis direction is given by \begin{equation} \begin{aligned} f_\mathrm{Rx} &= -\tfrac12\ p\ \rho\ \C{Dx}^\mathrm{t} A_\mathrm{x} v_\mathrm{x} \lvert \vec{v} \rvert \\ f_\mathrm{Ry} &= -\tfrac12\ p\ \rho\ \C{Dy}^\mathrm{t} A_\mathrm{y} v_\mathrm{y} \lvert \vec{v} \rvert \\ f_\mathrm{Rz} &= -\tfrac12\ p\ \rho\ \C{Dz}^\mathrm{t} A_\mathrm{z} v_\mathrm{z} \lvert \vec{v} \rvert \\ \end{aligned} \end{equation} where
$p=$ proportion wet $\PW$ for hydrodynamics, or proportion dry ($= 1 - \PW$) for aerodynamics, as appropriate
$\rho=$ fluid density
$\CD^\mathrm{t}=$ the given translational drag coefficient for each component
$A=$ the given drag area for each component
$\vec{v}=$ buoy velocity relative to the fluid velocity.
The drag moment applied about each local axis direction is given by \begin{equation} \begin{aligned} m_\mathrm{Rx} &= -\tfrac12\ p\ \rho\ \C{Dx}^\mathrm{r} A\!M_\mathrm{x}\ \omega_\mathrm{x} \lvert \vec{\omega} \rvert \\ m_\mathrm{Ry} &= -\tfrac12\ p\ \rho\ \C{Dy}^\mathrm{r} A\!M_\mathrm{y}\ \omega_\mathrm{y} \lvert \vec{\omega} \rvert \\ m_\mathrm{Rz} &= -\tfrac12\ p\ \rho\ \C{Dz}^\mathrm{r} A\!M_\mathrm{z}\ \omega_\mathrm{z} \lvert \vec{\omega} \rvert \\ \end{aligned} \end{equation} where
$\CD^\mathrm{r}=$ the given rotational drag coefficient for each component
$A\!M=$ the given drag area moment for each component
$\vec{\omega}=$ buoy angular velocity. Note that $\vec{\omega}$ does not include any contribution from fluid angular velocity: the fluid is treated as irrotational so has zero angular velocity.