Line theory: Drag chains

A drag chain is an attachment to a line that applies a force to the node to which it is attached. The force consists of the tension in the drag chain and so is in the direction in which the chain is hanging. This direction is determined by the relative velocity of the water past the chain – the faster the flow then the greater the angle of the drag chain to the vertical. The details of this calculation are described in more detail below.

Consider the drag chain shown above, and let

$d=$ drag chain effective diameter

$l=$ drag chain length

$\theta=$ drag chain declination from vertical

$\vec{v}=$ horizontal relative velocity = (horizontal fluid velocity at $\vec{a}$) – (horizontal node velocity at $\vec{a}$).

Then the incidence angle $\alpha$ between the horizontal relative velocity vector $\vec{v}$ and the drag chain is $\alpha=\theta{-}90\degree$ and the normal $(\fn)$ and axial $(\fa)$ drag forces are given by \begin{align} \fn = \tfrac12 \rho\ l d\ \C{Dn}\!(\alpha)\ \vn\lvert\vn\rvert \\ \fa = \tfrac12 \rho\ \pi l d\ \C{Da}\!(\alpha)\ \va\lvert\va\rvert \end{align} where

$\vn = \lvert\vec{v}\rvert\sin\alpha=$ normal component of relative velocity

$\va = \lvert\vec{v}\rvert\cos\alpha=$ axial component of relative velocity

$\rho=$ water density

$\C{Dn}\!(\alpha), \C{Da}\!(\alpha)=$ normal and axial drag coefficients for incidence angle $\alpha$.

The inertia of the drag chain is assumed to be small enough to be neglected, so we assume that the drag chain is always in equilibrium under the action of three forces:

• the chain's wet weight, $\vec{w}$
• the fluid drag on the chain
• the tension being applied by the line at the top of the chain.

Because the tensile force being applied by the line is axial to the chain, the components of wet weight and drag normal to the chain must balance. In other words, the direction in which the drag chain hangs is that in which the chain is in force balance in the direction normal to the chain. The remaining net force on the chain in the axial direction, $\fa$, is then applied to the line.

The chain is therefore deemed to hang in the same vertical plane as the relative velocity vector $\vec{v}$ and at angle $\theta$ to the vertical. Angle $\theta$ is calculated by iteration until the sum of the normal components of drag force and wet weight is zero.

Drag chain interaction with seabed

Drag chains, in OrcaFlex, interact with the seabed in a fairly simplistic way that is designed to achieve two primary effects:

• as the chain is lowered down onto the seabed, the wet weight of the chain is steadily reduced as more of the chain becomes supported by the seabed
• if the chain is dragged across the seabed, then an opposing friction force $\mu \vec{r}$ is generated, where $\mu$ is the friction coefficient and $\vec{r}$ the seabed reaction.

The seabed interaction model is as follows:

• At any given time, OrcaFlex first calculates how much of the drag chain would be supported by the seabed, assuming that the chain hangs vertically straight down from the line. The drag chain then consists of two parts: the supported part and the remaining hanging part.
• The hanging part of the chain is then analysed as described above.
• The supported part of the chain is modelled as if it is lying on the seabed directly beneath the node to which the chain is attached. As the node moves laterally, so does the supported chain (below the node), generating a friction force that is then applied to the node.

Note that the division of the drag chain into a hanging length and a supported length is done before the hanging length is analysed, and so is done with the chain vertical. This means that if current drag causes the chain to hang at an angle to the vertical then the supported length will generally have been overestimated and the hanging length correspondingly underestimated. This is an inaccuracy that cannot easily be avoided at the moment.