Practical Ship Hydrodynamics

Ship manoeuvring 183
the rudder. In addition a form factor has to be taken into account to yield the
factor 2.5.
For hydrofoils the Reynolds number is defined as:
V Ð c
Rn D
where V is the inflow velocity (for rudders usually VA), c the mean chord
length and ³ 1.35 Ð 10 6 m 2 /s the kinematic viscosity of water at 10°C.
Table 5.5 shows the good agreement of the approximate formulae with
model test measurements of Whicker and Fehlner (1958) (columns 1 to 6)
and Thieme (1958) (other columns). Thieme’s results suffer somewhat from
small Reynolds numbers. Rudder Reynolds numbers behind a large ship are in
the vicinity of Rn D 5 Ð 10 7 . Too small Reynolds numbers result in larger drag
coefficients, a backward shift of the centre of effort of the rudder force and
smaller stall angles ˛s (by up to a factor of 2) than in reality. The Reynolds
number of column 13 corresponds approximately to the conditions in model
manoeuvring experiments. However, the strong turbulence behind a ship model
and its propeller act similarly to a larger Reynolds number in these experiments.
Table 5.5 Measured (M) and computed (C) force and moment coefficients of different
profiles (Thieme (1958), Whicker and Fehlner (1958)); Y: independent from profile shape;
∗ : uncertain values, probably due to experimental technique
NACA NACA NACA
Profile – C 0015 – C 0015 – C 0015
^ 1 1 2 2 3 3
⊲t/c⊳max C 15 C 15 C 15
at x/c 30 30 30
Rn/10 6 2.7 2.7 2.7 2.7 2.7 2.7
CL at ˛ D 10° 0.27 0.27 0.44 0.44 0.55 0.55
CL at ˛ D 20° 0.59 0.60 0.92 0.93 1.14 1.10
CL at ˛ D ˛s 1.17 1.26 1.33 1.33 1.32 1.25
˛s[°] 38.5 28.7 23.0
CL/CD at ˛ D 10° 8.11 7.26 10.45 10.35 12.28 12.40
CL/CD at ˛ D 20° 4.62 4.25 5.70 5.79 6.63 7.05
CL/CD at ˛ D ˛s 2.28 2.20 3.88 4.00 5.76 6.00
cs/c at ˛ D 10° 0.17 0.16 0.18 0.19 0.19 0.18
cs/c at ˛ D ˛s 0.30 0.31 0.24 0.25 0.23 0.23
NACA NACA IFS62 IFS61 IFS58 Plate NACA
Profile 0015 0025 TR 25 TR 25 TR 15 t/c D 0.03 0015
^ 1 1 1 1 1 1 1
⊲t/c⊳max 15 25 25 25 15 3 15
at x/c 30 30 20 20 25 – 30
Rn/10 6 0.79 0.78 0.78 0.79 0.79 0.71 0.20
CL at ˛ D 10° 0.29 0.27 0.33 0.32 0.32 0.34 0.35
CL at ˛ D 20° 0.62 0.59 0.71 0.69 0.67 0.72 0.55
CL at ˛ D ˛s 1.06 1.34 1.48 1.34 1.18 1.14 0.72
˛s[°] 33.8 46.0 Ł 46.0 Ł 41.0 Ł 33.5 40 Ł 35.0
CL/CD at ˛ D 10° 7.20 5.40 4.70 4.00 6.40 3.80 2.80
CL/CD at ˛ D 20° 4.40 4.20 3.60 3.60 3.90 2.50 1.75
CL/CD at ˛ D ˛s 2.30 1.70 1.50 1.80 2.40 1.30 1.19
cs/c at ˛ D 10° 0.18 0.20 0.27 0.26 0.25 0.28 0.28
cs/c at ˛ D ˛s 0.35 0.35 0.36 0.25 0.33 0.41 0.43