OrcaFlex Manual - Orcina

w
163
Theory, Vessel Theory
ω is the vessel yaw rate, in radians per second, due to any low-frequency primary motion of the vessel
ρ is the water density.
These yaw rate drag loads are then applied at the current load origin.
Warning: Danger of double-counting: If manoeuvring load is also included then it may include potential
theory surge and sway force contributions that are quadratic in ω. To avoid counting these
contributions twice, if manoeuvring load and manoeuvring load are both included then you must
ensure that the yaw rate drag factors only include the contributions due to viscous effects.
Estimating the Yaw Rate Drag Factors
The above formulae (1) are based on a simple strip theory estimate of the drag loads on a yawing vessel, as given by
Wichers (1979). Consider the simplest situation where the vessel centre is stationary but the vessel is yawing at rate
ω about that centre. Let us also assume that the area exposed to sway drag is a simple rectangle of height D (the
draught) and length L (the length between perpendiculars), and that for simplicity we choose to put the load origin
at the centre of that area.
We now divide the drag area into vertical strips of width δx and consider the sway drag load on the strip at distance
x forward of the centre. The strip's area is Dδx and its sway velocity due to the yaw rate is ωx, so we can estimate
the sway drag load on it by ½ρCdDδx(ωx)|ωx| where Cd is the drag coefficient, which we assume to be the same for
all the strips.
These sway drag loads from each strip, and their moments about the centre, are then integrated to give the total
sway force and a contribution to yaw moment. When we do this integral the sway forces from corresponding strips
forward and aft of centre have the same magnitude but opposite direction, so they cancel and the total sway drag
force is therefore zero. However the yaw moment terms from forward and aft of centre have the same magnitude
and same direction, so they reinforce, giving a significant yaw drag moment. In fact the integral gives the following
yaw moment.
Yaw Moment = (½ρω 2 )(CdDL 4 /32) (2)
The same argument can be applied to the drag forces in the surge direction, with length L being replaced by width
W. But for a slender vessel W is much less than L, so the surge force contribution to yaw moment is generally
negligible.
Comparing equations (1) and (2) we see that the Kyaw corresponds to the bracketed term (CdDL 4 /32) in equation (2).
This is in fact a combination of a drag coefficient and the 3rd moment of drag area about the centre.
This strip theory argument therefore concludes that for a slender ship, with the hydrodynamic drag load origin at
the centre, then we can estimate the yaw rate drag factors by:
Ksurge = Ksway = 0
Kyaw = (CdDL 4 /32)
where Cd is some appropriate drag coefficient. However, there are a lot of questionable assumptions in the strip
theory argument. Indeed Wichers (1979) found that the strip theory results significantly underestimated the actual
yaw drag measured in model tests, unless the Cd value was increased to about 5, which is rather high for a drag
coefficient. So if more specific data is available, e.g. from model test, then we recommend setting the yaw rate drag
factors to values that best fit your data.
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 vω. This is an interaction between sway velocity and yaw
rate and its effect is to significantly increase the yaw moment.
OrcaFlex does not yet include this interaction effect. The reason for this is that it is difficult to model. Wichers
(1979) included them 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. However the method has some theoretical difficulties,
since the formulae break down when ω is zero.
Orcina is studying this, with a view to implementing a more accurate yaw rate drag model in a future release of
OrcaFlex. In the meantime we recommend that you specify yaw rate drag factors that are appropriate to the
conditions prevailing in the case being modelled. See the papers by Wichers for further information.