## Morison's equation |

OrcaFlex calculates hydrodynamic loads on lines, 3D buoys and 6D buoys by an extended form of Morison's equation. This equation was originally formulated by Morison *et al* (1950), for calculating the wave loads on fixed vertical cylinders, as having two force components: one related to water particle acceleration, the *fluid inertia* force, and one related to water particle velocity, the *drag* force.

Their original equation can be written as \begin{equation} \vec{f} = {\underbrace{\vphantom{\frac12}\Cm \Delta\ \af}_{inertia}} + {\underbrace{\frac12\ \rho\ \C{d} A\ \vert\vf\vert \vf}_{drag}} \end{equation} where

$\vec{f}$ is the fluid force (per unit length) on the body

$\Cm$ is the inertia coefficient for the body

$\Delta$ is the mass of fluid displaced by the body

$\af$ is the fluid acceleration relative to earth

$\rho$ is the density of water

$\C{d}$ is the drag coefficient for the body

$A$ is the drag area

$\vf$ is the fluid velocity relative to earth.

The same principles can also be applied to a moving body, in which case the inertia term is reduced by the amount $\Ca \Delta \ab$ and the drag term uses the body-relative velocity, resulting in the extended form of Morison's equation we use in OrcaFlex \begin{equation} \vec{f} = \left( \Cm \Delta\ \af - \Ca \Delta\ \ab \right) + \frac12\ \rho\ \C{d} A\ \vert\vr\vert \vr \end{equation} where, in addition,

$\Ca$ is the added mass coefficient for the body

$\ab$ is the body acceleration relative to earth

$\vr$ is the fluid velocity relative to the body.

This is the most general form of the equation used in OrcaFlex. Commonly, however, the value of $\Cm$ is taken to be $1{+}\Ca$ (which we indicate in OrcaFlex by using the special value ~ for $\Cm$); in this case, the extended form of Morison's equation simplifies to this, possibly more familiar, form \begin{equation} \vec{f} = \left( \Delta\ \af + \Ca \Delta\ \ar \right) + \frac12\ \rho\ \C{d} A\ \vert\vr\vert \vr \end{equation} where we write $\ar = \af{-}\ab$ for the fluid acceleration relative to the body.

The term in parentheses is the inertia force, the other term is the drag force. The drag force will be familiar to most engineers, but the inertia force can cause confusion. The inertia force consists of two parts, one proportional to fluid acceleration relative to earth $\af$ (the *Froude-Krylov* component), and one proportional to fluid acceleration relative to the body $\ar$ (the *added mass* component).

To understand the *Froude-Krylov* component, imagine removing the body and replacing it with an equivalent volume of water. This water would have mass $\Delta$ and be undergoing an acceleration $\af$. It must therefore be experiencing a force $\Delta\af$. Now remove the water and put the body back: the same force must now act on the body. This is equivalent to saying that the Froude-Krylov force is the integral over the surface of the body of the pressure in the incident wave, undisturbed by the presence of the body. (Note the parallel with Archimedes' principle: in still water, the integral of the fluid pressure over the wetted surface must exactly balance the weight of the water displaced by the body.)

The *added mass* component is due to the distortion of the fluid flow by the presence of the body. A simple way to understand it is to consider a body accelerating through a stationary fluid. It can be shown that the force required to sustain the acceleration is proportional to the body acceleration and can be written
\begin{equation}
\vec{f}\urm{b} = (m + \Ca\Delta)\ \ab
\end{equation}
where

$\vec{f}\urm{b}$ is the total force on the body

$m$ is the mass of the body

$\Ca\Delta$ is a constant related to the shape of the body and its displacement

Another way of looking at the problem is in terms of energy. The total energy required to accelerate a body in a stationary fluid is the sum of the kinetic energy of the body itself, and the kinetic energy of the flow field about the body. These energies correspond to the terms $(ma)$ and $(\Ca\Delta a)$ respectively.

The term $\Ca\Delta$ has the dimensions of mass and has become known as the *added mass*. This is an unfortunate name which has caused much confusion over the years. It should **not** be viewed as a body of fluid trapped by and moving with the body. Some bodies are so shaped that this does occur, but this trapped water is a completely different matter. Trapped water occurs when the body contains a closed flooded space, or where a space is sufficiently closely surrounded to prevent free flow in and out. Trapped water should be treated as part of the body: the mass of the trapped water should be included in the body mass, and its volume should be included in the body volume.

For a more complete description of Morison's equation and a detailed derivation of the added mass component see Barltrop and Adams, 1991 and Faltinsen, 1990.