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6D buoys: Hydrodynamic properties of a rectangular box |
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6D buoys: Hydrodynamic properties of a rectangular box |
Lumped buoys are generalised six degree of freedom objects with indeterminate geometry: only their height is defined. It is therefore necessary to define their hydrodynamic properties such as inertia, drag area, and added mass explicitly as data items. This can be a difficult task, especially where the buoy is used to represent a complex shape such as a midwater arch of the sort used to support a flexible riser system.
We cannot give a universal step-by-step procedure for this, given the widely-varying geometry of different objects. Instead we provide, as an example, the derivation of the hydrodynamic properties in 6 degrees of freedom for a rectangular box. This gives a general indication of the way in which the problem may be approached.
The same analysis applies to 3D buoys, since they likewise have no defined geometry. In this case, however, the rotational properties are not required.
| Figure: | Rectangular box geometry |
The drag areas are given by \begin{aligned} A\urm{x} &= l\urm{y} l\urm{z}\ \text{ in the $x$ direction} \\ A\urm{y} &= l\urm{x} l\urm{z}\ \text{ in the $y$ direction} \\ A\urm{z} &= l\urm{x} l\urm{y}\ \text{ in the $z$ direction} \end{aligned}
These are obtained from ESDU 71016, figure 1, which gives data for the drag on isolated rectangular blocks with one face normal to the flow. The dimensions of the block are $a$ in the flow direction, $b$ and $c$ normal to the flow with $c{\gt}b$.
Their figure plots drag coefficient $C\urm{x}$ against $(a/b)$ for discrete values of (c/b) from 1 to $\infty$. $C\urm{x}$ is in the range 0.9 to 2.75 for blocks with square corners.
| Note: | ESDU 71016 uses $C_d$ for the force in the flow direction; $C_x$ for the force normal to the face. For present purposes the two are identical. |
There is no standard data source. As an approximation, we assume that the drag force contribution $\ud\!f$ from an elementary area $\ud\! A$ is given by \begin{equation} \ud\!f = \frac12 \rho\ |\vec{v}|^2 \C{d}\ \ud\! A \end{equation} where $\C{d}$ is assumed to be the same for all points on the surface.
| Note: | This assumption is not strictly valid. ESDU 71016 gives pressure distributions for sample blocks in uniform flow which show that the pressure is greatest at the centre and least at the edges, but we do not allow for this here. |
| Figure: | Integration for rotational drag properties |
Consider the box rotating about the $Bx$ axis. The areas $A\urm{y}$ and $A\urm{z}$ will attract drag forces which will result in moments about $Bx$. For the area $A\urm{y}$, consider an elementary strip as shown. For an angular velocity $\omega$ about $Bx$, the drag force on the strip is \begin{equation} \ud\!f = \frac12 \rho\ \omega z\ |\omega z| \C{d}\ x\ \ud\! z \end{equation} and the moment of this force about $Bx$ is \begin{align} \ud\!m &= \frac12 \rho\ \omega z\ |\omega z| \C{d}\ x\ \ud\! z\ z \\ &= \frac12 \rho\ \omega\ |\omega| \C{d}\ x\ z^3\ \ud\! z \end{align} Total moment $m$ is then obtained by integration. Due to the $v|v|$ form of the drag force, simple integration from -Z/2 to +Z/2 gives $m{=}0$, so we integrate from 0 to $z/2$ and double, resulting in \begin{equation} m = \frac12 \rho\ \omega |\omega| \C{d}\ \frac{xz^4}{32} \end{equation} OrcaFlex calculates the drag moment by \begin{equation} m = \frac12 \rho\ \omega |\omega| \C{d,m}\ A\urm{m} \end{equation} where $\C{d,m}$ and $A\urm{m}$ are the drag coefficient and moment of area respectively, so we set \begin{align} \C{d,m} &= \C{d} \\ A\urm{m} &= \frac{xz^4}{32} \end{align} This is the drag moment contribution about $Bx$ from the $A\urm{y}$ area; there is a similar corresponding contribution from the $A\urm{z}$ area. Since $\C{d}$ generally differs for the two, it is convenient to calculate the sum of the $(\C{d}\,A\urm{m})$ products for both and then simply set $A\urm{m}$ to this value and $\C{d}$ to 1.
OrcaFlex requires the added mass and inertia contributions to the mass matrix, plus the hydrodynamic mass and inertia values to be used for computation of wave forces. For each of the six degrees of freedom, three data items are required: hydrodynamic mass $H\!M$ (or inertia $H\!I$) and coefficients $\Ca$ and $\Cm$. Added mass is then calculated as $A\!M = H\!M\,\Ca$ and wave force as $H\!M\,\Cm\,a\urm{f}$ for water particle acceleration $a\urm{f}$.
On the usual assumptions intrinsic in the use of Morison's equation (that the body is small by comparison with the wavelength), the wave force is given by $(\Delta+A\!M) a\urm{f}$, where $\Delta$ is body displacement. Equating the two expressions for wave force, we get \begin{equation} H\!M\,\Cm\,a\urm{f} = (\Delta+A\!M) a\urm{f} \end{equation} For translational degrees of freedom then, set $H\!M = \Delta$, and it follows that $\Ca = A\!M / \Delta$ and $\Cm = 1{+}\Ca$.For rotational degrees of freedom, set $H\!I = \Delta\!I$, the moment of inertia of the displaced mass, then $\Ca = A\!I / \Delta\!I$ and, again, $\Cm = 1{+}\Ca$. $A\!I$ is the added inertia, the rotational analogue of added mass.
DNV-RP-C205, Table 6.2, gives added mass data for a square section prism accelerating along its axis. The square section sides are of length $a$, prism length is $b$, and data are given for discrete values of $b/a = 1.0$ and above. The reference volume is the volume of the body, which corresponds to our own definition in OrcaFlex. We can therefore use the calculated $\Ca$ without further adjustment.
Consider flow in the $x$ direction:
The area normal to the flow $= A\urm{x}$.
For a square of the same area, $a = \sqrt{A\urm{x}}$
Length in flow direction $= l\urm{x}$.
Hence $b/a = l\urm{x}/\sqrt{A\urm{x}}$.
$\Ca$ can thus be obtained from DNV-RP-C205 by interpolation, and then $\Cm = 1{+}\Ca$.
If $b/a \lt 1$ this approach fails and we instead use the data given in DNV-RP-C205 for rectangular flat plates. If $l\urm{y} \gt z$, the aspect ratio of the plate is $l\urm{y}/l\urm{z}$, and hence $C\!A$ from DNV-RP-C205 by interpolation. The reference volume in this case is that of a cylinder of diameter $l\urm{z}$, length $l\urm{y}$, and so \begin{equation} \text{added mass} = C\!A\ \rho\ \frac{\pi}{4} y\ z^2 = A\!M\urm{x} \text{ say} \end{equation} and then \begin{align} \Ca &= \frac{A\!M\urm{x}}{\Delta} \\ \Cm &= 1+\Ca \end{align}
| Note: | If $y{\lt}z$, then aspect ratio $= z/y$ and reference volume $= C\!A\,\rho \frac{\pi}{4} z\ y^2$. |
DNV-RP-C205 gives no data for hydrodynamic inertia of rotating bodies. The only data for 3D solids we are aware of are for spheroids: figure 4.8 of Newman 1977 gives the added inertia for coefficient for spheroids of varying aspect ratio, referred to the moment of inertia of the displaced mass. We assume that the same coefficient applies to the moment of inertia of the displaced mass of the rectangular block.
\begin{equation} \Delta\!I = \Delta (Y^2 + Z^2) / 12 \end{equation}
Using data for spheroids from Newman 1977,
Length in flow direction $= 2a = l\urm{x}$, so $a = l\urm{x}/2$.
Equivalent radius normal to flow, $b$, is given by $\pi b^2 = l\urm{y}l\urm{z}$, so $b = \sqrt{l\urm{y}l\urm{z}/\pi}$.
Hence Ca from Newman 1977:
For $b/a \leq 1.6$, $\Ca$ can be read from the upper figure where the value is referred to the moment of inertia of the displaced mass. In this case no further adjustment is required.
For $b/a \gt 1.6$, the coefficient $C\!A$ is read from the lower graph in which the reference volume is the sphere of radius $b$. In this case \begin{equation} \Ca = C\!A\ \frac{2b^3}{a(a^2+b^2)} \end{equation} In either case, $\Cm = 1+\Ca$.