Practical Ship Hydrodynamics

182 Practical Ship Hydrodynamics
Figure 5.14 illustrates the CL formula. The first term in the CL formula follows
from potential thin-foil theory for the limiting aspect ratios 3 D 0and3 D1.
For other aspect ratios it is an approximation to theoretical and experimental
results. The first term in the CD formula is the induced resistance due to the
generation of trailing vortices behind the foil. The equation includes a 10%
increase of the minimum induced drag which occurs for an elliptical load
distribution over the rudder height. The first term in the CQN formula would
be a good approximation of the theoretical moment in ideal fluid if 0.5 were
used instead of the empirical value 0.47. The second terms in the formulae
for CL and CD follow from the assumption of an additional resistance-like
force depending quadratically on the velocity component V Ð sin ˛ which is
perpendicular to the rudder plane. A resistance coefficient CQ ³ 1maybe
used for rudders with a sharp upper and lower edge. Rounding the edges
(which is difficult in practice) would lead to much smaller CQ values. The
second term in the CQN formula follows from the assumption that this force
component acts at 75% chord length from the leading edge. CD0 in the formula
for CD approximates the surface friction. We may approximate:
C L
CD0 D 2.5 Ð
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
0.075
⊲log Rn 2⊳ 2
Range of maximum
lift coefficient
2.5
2.0
L 3.0
1.5
1.2
1.0
0.8
Usual maximum
rudder angle d
0° 10° 20° 30° 40° 50°
a
Figure 5.14 Lift coefficient CL versus angle of attack ˛ with the aspect ratio as parameter
This is 2.5 times the frictional resistance coefficient following the ITTC 1957
formula. CD0 refers to the rudder area which is about half the wetted area of
0.6