Vessel theory: Current and wind loads

These loads are an important source of damping when modelling vessel slow drift. For a discussion of the various damping sources see damping effects on vessel slow drift.

The current and wind drag loads on a vessel are, if included, calculated using the data on the current and wind load pages of the vessel type data form. The loads conveniently split into those due to translational relative velocity and due to yaw rate.

Drag loads due to translational relative velocity

The drag loads due to translational velocity of the sea and air past the vessel are calculated using the standard OCIMF method outlined below. For further details see Oil Companies International Marine Forum, 1994.

The OCIMF method calculates the surge, sway and yaw drag loads on a stationary vessel. OrcaFlex extends this to a moving vessel by replacing the current (or wind) velocity used in the OCIMF method with the relative translational velocity of the current (or wind) past the vessel.

More precisely, the relative velocity used is that of the current (or wind), relative to the low frequency primary motion of the vessel, at the current (or wind) load origin. In other words the relative velocity includes the current but not the waves, and it includes the low frequency primary motion of the vessel, but not the wave frequency motion or the superimposed motion.

In addition, OrcaFlex extends the OCIMF method to provide heave, roll and pitch loads.

Note that the OCIMF method does not include any drag due to yaw angular velocity of the vessel, since there is none for a stationary vessel. Nor does the OrcaFlex extension of OCIMF add these yaw rate drag terms, since the OCIMF method has no framework for them. They are therefore calculated separately in OrcaFlex – see drag loads due to yaw rate below.

Notes: The OCIMF method is intended for tankers, but could be applied to other vessel types providing suitable data are obtained.
The wind load is only included if the include wind loads on vessels option is enabled in the environment data.
Forces arising from current and wind acceleration are outside the scope of this model and are therefore ignored.
Warning: The current and wind loads are based on theory for surface vessels and are not suitable for submerged vessels.

The drag loads due to surge and sway relative velocity are given by \begin{equation} \begin{aligned} \fx &= \tfrac12\ C\urm{surge}\ \rho\ \lvert\vec{v}\rvert^2 A\urm{surge} \\ \fy &= \tfrac12\ C\urm{sway}\ \rho\ \lvert\vec{v}\rvert^2 A\urm{sway} \\ \fz &= \tfrac12\ C\urm{heave}\ \rho\ \lvert\vec{v}\rvert^2 A\urm{heave} \\ \mx &= \tfrac12\ C\urm{roll}\ \rho\ \lvert\vec{v}\rvert^2 A\urm{roll} \\ \my &= \tfrac12\ C\urm{pitch}\ \rho\ \lvert\vec{v}\rvert^2 A\urm{pitch} \\ \mz &= \tfrac12\ C\urm{yaw}\ \rho\ \lvert\vec{v}\rvert^2 A\urm{yaw} \end{aligned} \end{equation} where

$\fx$, $\fy$ and $\fz$ are the drag force in the x-, y- and z-directions.

$\mx$, $\my$ and $\mz$ are the drag moment about the x-, y- and z-directions.

$C\urm{surge}$, $C\urm{sway}$, $C\urm{heave}$, $C\urm{roll}$, $C\urm{pitch}$ and $C\urm{yaw}$ are the surge, sway, heave, roll, pitch and yaw coefficients for the current or wind direction relative to the low frequency vessel heading.

$\rho$ is the water density (for current) or air density (wind).

$\vec{v}$ is the relative velocity of the sea or air past the vessel. For wind loads, $\vec{v}$ is based on the wind velocity specified in the data (i.e. the wind velocity at 10m above mean water level). For current, $\vec{v}$ is based on the current velocity at the low frequency instantaneous position of the load origin (if this is above the water surface, then the current velocity at the surface is used). In both cases $\vec{v}$ is taken relative to the vessel: it includes allowance for the translational velocity of the load origin due to any low frequency primary motion of the vessel, but does not include any wave frequency or superimposed motion.

$A\urm{surge}$, $A\urm{sway}$, $A\urm{heave}$, $A\urm{roll}$, $A\urm{pitch}$ and $A\urm{yaw}$ are the surge, sway and heave areas and the roll, pitch and yaw area moments. For current these correspond to the exposed areas below the waterline, and for wind to the exposed areas above the waterline.

The force and moment act at the current or wind load origin. The axes in use are those of the primary low frequency heading frame.

Note: The OCIMF standard method uses $A\urm{yaw}{=}L^2D$ for current loads, and $A\urm{yaw}{=}LA\urm{sway}\ $ for wind loads, where $L$ is the length between perpendiculars and $D$ is the draught. For current loads (not wind loads) the OCIMF standard method uses the same value, $LD$, for both the surge and sway areas $A\urm{surge}$ and $A\urm{sway}$.

Drag loads due to yaw rate

A vessel rotating in yaw will generate a drag moment resisting the yaw rate, but – since it is designed for stationary vessels – the OCIMF method described above does not include this drag load.

For wind drag these yaw rate terms are insignificant and so are omitted by OrcaFlex. They may, however, be important for current drag, so we model them with the formulae \begin{equation} \begin{aligned} \fx &= \tfrac12\ \rho\ \lvert\omega\rvert\ \omega\ K\!\urm{surge} \\ \fy &= \tfrac12\ \rho\ \lvert\omega\rvert\ \omega\ K\!\urm{sway} \\ \mz &= \tfrac12\ \rho\ \lvert\omega\rvert\ \omega\ K\!\urm{yaw} \end{aligned} \label{YawRateDrag} %split or aligned must have \label here for \ref to work \end{equation} where

$\rho$ is the water density

$\omega$ is the vessel yaw rate, in radians per second, due to the low frequency primary motion of the vessel

$K\!\urm{surge}, K\!\urm{sway}, K\!\urm{yaw}$ are the yaw rate drag factors on the current load page of the vessel type data form

These yaw rate drag loads are applied at the current load origin in the axes of the primary low frequency heading frame. We recommend setting the yaw rate drag factors to values that best fit your available data, e.g. from model tests.

Warning: Danger of double-counting. If manoeuvring load is also included, it may include potential theory surge and sway force contributions that are quadratic in yaw rate $\omega$. To avoid counting these contributions twice, if both manoeuvring load and drag loads due to yaw rate are included you must ensure that the yaw rate drag factors only include the contributions due to viscous effects.

Interaction with sway rate

Further complications arise if the vessel is swaying as well as yawing. In this case the integral in the above strip theory argument turns out to give an extra term involving $\lvert\vec{v}\rvert\ \omega$. This is an interaction between sway velocity and yaw rate and its effect is to significantly increase the yaw moment.

OrcaFlex does not include this difficult interaction effect. Wichers (1979) included it in his strip theory model, but as described above the model's results did not match experimental results particularly well. He returned to the problem in his PhD thesis (Wichers, 1988) and developed a more accurate empirical approach based on model test data, but the method has some theoretical difficulties since the formulae break down when $\omega$ is zero.

Since there is no consensus on the best way to model this interaction, we recommend that you use yaw rate drag factors that are appropriate to the conditions prevailing in the case being modelled. See the papers by Wichers for further information.