6D buoy theory: Spar buoy and towed fish drag

The second order hydrodynamic and aerodynamic loads on a spar buoy or towed fish are calculated and applied separately on each cylinder.

These loads are calculated as follows.

Drag forces

The drag forces are calculated, as with lines, using the cross-flow principle. In the local $x$ and $y$ directions, i.e. normal to the cylinder axis, the drag forces are given by \begin{equation} \begin{aligned} \fDx &= -\tfrac12\ p\ \rho\ \CDn^\mathrm{f} \An \vx \lvert \vn \rvert \\ \fDy &= -\tfrac12\ p\ \rho\ \CDn^\mathrm{f} \An \vy \lvert \vn \rvert \end{aligned} \end{equation} where

$p$ is the instantaneous proportion wet, $\PW$, of the cylinder, i.e. the proportion of the cylinder volume which is submerged for hydrodynamics, or the instantaneous proportion dry, $1 - \PW$, for aerodynamics

$\rho$ is the fluid density

$\CDn^\mathrm{f}$ is the drag force coefficient for the normal direction

$\An$ is the drag area for the normal direction

$\vn$ is the component of buoy velocity, relative to the fluid, in direction normal to the cylinder axis

$\vx$ and $\vy$ are the $x$- and $y$-direction components of $\vn$.

In the $z$ direction, i.e. parallel to the cylinder axis, the drag force is given by \begin{equation} \fDz = -\tfrac12\ p\ \rho\ \CDa^\mathrm{f} \Aa \vz \lvert \vz \rvert \end{equation} where

$\CDa^f$ is the drag force coefficient for the axial direction

$\Aa$ is the drag area for the axial direction

$\vz$ is the component of buoy velocity, relative to the fluid, in the $z$-direction.

Drag moments

Drag moments are similarly calculated, again applying the cross-flow principle.

About the local $x$ and $y$ directions the drag moments are given by \begin{equation} \begin{aligned} \mDx &= -\tfrac12\ p\ \rho\ \CDn^\mathrm{m} \AMn \wx \lvert \wn \rvert \\ \mDy &= -\tfrac12\ p\ \rho\ \CDn^\mathrm{m} \AMn \wy \lvert \wn \rvert \end{aligned}\label{mDxy} \end{equation} where

$\CDn^\mathrm{m}$ is the drag moment coefficient for the normal direction

$\AMn$ is the drag area moment for the normal direction

$\wn$ is the component of buoy angular velocity, relative to earth, in the direction normal to the cylinder axis. The fluid is treated as irrotational, so its angular velocity is taken to be zero.

$\wx$ and $\wy$ are the $x$- and $y$-direction components of $\wn$.

About the local $z$-direction, the drag moment is given by \begin{equation} \label{mDz} \mDz = -\tfrac12\ p\ \rho\ \CDa^\mathrm{m} \AMa \wz \lvert \wz \rvert \end{equation} where

$\CDa^\mathrm{m}$ is the drag moment coefficient for the axial direction

$\AMa$ is the drag area moment for the axial direction

$\wz$ is the $z$-component of the angular velocity of the buoy relative to earth.

Drag area moments

The drag area moments in equations (\ref{mDxy}) and (\ref{mDz}) are the rectified third moments of drag area about the axis of rotation. So drag area moment $= \Sigma A\lvert r \rvert^3$, where $A$ is an element of drag area at an (absolute) distance $\lvert r \rvert$ from the axis of rotation. The modulus $\lvert r \rvert$ arises from the drag term in Morison's equation. The area moment should have dimensions $L^5$. Note that the axial area moment is about the cylinder axis, and the normal area moment is about the normal to that axis through the cylinder centre.

We may derive the following results for simple bodies:

• For a rectangle of length $l$ and width $w$, the third moment of area about the line in the plane of the rectangle and through its centre in the length direction is $lw^4\!/32$. The third moment of area about the line in the plane of the rectangle and through its centre in the width direction is $wl^4\!/32$.
• For a circular disc of diameter $d$, the third moment of area about a line in the plane of the disc and through its centre is $d^5\!/60$.
• We can use the two results above to calculate reasonable drag area moment values to use for a spar buoy cylinder of length $l$ and diameter $d$. Consider the area moment for the normal direction, i.e. about a line through the cylinder centre and normal to the cylinder axis. If the curved surface of the cylinder is exposed to drag then we can account for its contribution to the area moment by using its projection onto the plane defined by the cylinder axis and the normal direction. This projected area is a rectangle of length $l$ and width $d$, and the line crosses it in the width direction, so the normal area moment is $dl^4\!/32$. If either of the circular end discs of the cylinder is also exposed to drag, then we must account for their contribution to the area moment.

Munk moment

Slender bodies in near-axial flow experience a destabilising moment called the Munk moment. This arises from potential flow theory and is distinct from (and additional to) any moments associated with viscous drag. It is only well-defined for a fully-submerged body.

Newman (1977) (p 341) derives the term and points out that it "acts on a non-lifting body in steady translation". Thwaites (1960) (pp 399-401) gives an alternative derivation and provides numerical values for spheroids.

Note that for bluff bodies the flow tends to separate over the afterbody, which has the effect of reducing the Munk moment to a value less than the potential flow theory would suggest. See Mueller (1968).

The Munk moment effect can be modelled in OrcaFlex by specifying a non-zero Munk moment coefficient for a spar buoy or towed fish. OrcaFlex then applies a Munk moment given by \begin{equation} m_\mathrm{M} = C_\mathrm{MM}\ m\ \tfrac12 \sin(2\alpha)\ \lvert \vec{v} \rvert^2 \end{equation} where

$C_\mathrm{MM}$ is the Munk moment coefficient

$m$ is the mass of water instantaneously displaced. If the buoy is surface-piercing then this allows for the proportion of the buoy that is in the fluid. Note, however, that the value of $C_\mathrm{MM}$ is ill-defined for a partially submerged body.

$\vec{v}$ is the flow velocity relative to the buoy, at the point on the stack axis that is half way between the ends of the stack.

$\alpha$ is the angle between the relative flow velocity $\vec{v}$ and the buoy axis.

The moment $m_\mathrm{M}$ is applied about the direction normal to the plane of the buoy axis and the relative flow vector $\vec{v}$, in the direction that tries to increase the angle $\alpha$.