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```can therefore be computed
by: ¯¯¯¯
Xw =
1
2
½airV 2rw ¢ CXw(®rw) ¢ AT
¯¯¯¯
¯¯¯¯
Yw =
1
2
½airV 2rw ¢ CYw(®rw) ¢ AL
¯¯¯¯
¯¯¯¯
Nw =
1
2
½airV 2rw ¢ CNw(®rw) ¢ AL ¢ L
¯¯¯¯
(4.50)
in which:
Xw = steady longitudinal wind force (N)
Yw = steady lateral wind force (N)
Nw = steady horizontal wind moment (Nm)
½air ¼ ½water=800 = density of air (kg/m3)
Vrw = relative wind velocity (m/s)
®rw = relative wind direction (-), from astern is zero
AT = transverse projected wind area (m2)
AL = lateral projected wind area (m2)
L = length of the ship (m)
C¤w(®rw) = ®rw-dependent wind load coe¢cient (-)
Note that it is a ”normal” convention to refer to the true wind direction as the direction
from which the wind comes, while waves and currents are usually referred to in terms of
where they are going. A North-West wind will cause South-East waves, therefore!
4.8.1 Wind Loads on Moored Ships
For moored ships, only the true wind speed and direction determine the longitudinal and
lateral forces and the yaw moment on the ship, as given in …gure 4.12. Because of the
absence of a steady velocity of the structure, the relative wind is similar to the true wind:
jVrw = Vtwj and j®rw = ®twj (4.51)
The total force and moment experienced by an object exposed to the wind is partly of
viscous origin (pressure drag) and partly due to potential e¤ects (lift force). For blunt
4-26 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA
Figure 4.12: De…nitions Used here for Forces and Moments
bodies, the wind force is regarded as independent of the Reynolds number and proportional
to the square of the wind velocity.
[Remery and van Oortmerssen, 1973] collected the wind data on 11 various tanker hulls.
Their wind force and moment coe¢cients were expanded in Fourier series as a function
of the angle of incidence. From the harmonic analysis, it was found that a …fth order
representation of the wind data is su¢ciently accurate, at least for preliminary design
purposes:
CXw = a0 +
5X
n=1
an sin(n ¢ ®rw)
CYw =
5X
n=1
bn sin(n ¢ ®rw)
CNw =
5X
n=1
cn sin(n ¢ ®rw) (4.52)
with wind coe¢cients as listed below.
Tanke r No . 1 2 3 4 5 6 7 8 9 10 11
Len gth Lpp 225 m 225 m 225 m 225 m 172 m 150 m 150 m
B rid ge Lo cat ion at 12L at
1
2L a ft a ft a t
1
2L at
1
2L a ft a ft a ft a ft a ft
a0 -0 .131 -0 .079 -0 .028 0 .014 -0 .074 -0 .055 -0 .038 -0 .039 -0 .042 - 0.075 -0 .051
a1 0 .738 0 .615 0 .799 0 .732 1 .050 0 .748 0 .830 0 .646 0 .487 0.711 0 .577
a2 -0 .058 -0 .104 -0 .077 -0 .055 0 .017 0 .018 0 .031 0 .034 -0 .072 - 0.082 -0 .058
a3 0 .059 0 .085 -0 .054 -0 .017 -0 .062 -0 .012 0 .012 0 .024 0 .109 0.043 0 .051
a4 0 .108 0 .076 0 .018 -0 .018 0 .080 0 .015 0 .021 -0 .031 0 .075 0.064 0 .062
a5 -0 .001 0 .025 -0 .018 -0 .058 -0 .110 -0 .151 -0 .072 -0 .090 -0 .047 - 0.038 0 .006
b1 0 .786 0 .880 0 .697 0 .785 0 .707 0 .731 0 .718 0 .735 0 .764 0.819 0 .879
b2 0 .039 0 .004 0 .036 0 .014 -0 .013 -0 .014 0 .032 0 .003 0 .037 0.051 0 .026
b3 0 .003 0 .003 0 .018 0 .014 0 .028 0 .016 0 .010 0 .004 0 .052 0.023 0 .014
b4 0 .034 -0 .004 0 .028 0 .015 0 .007 0 .001 -0 .001 -0 .005 0 .016 0.032 0 .031
b5 -0 .019 -0 .003 -0 .023 -0 .020 -0 .044 -0 .025 -0 .040 -0 .017 -0 .003 - 0.032 -0 .029
10 ¢ c1 -0 .451 -0 .338 -0 .765 -0 .524 -0 .216 -0 .059 -0 .526 -0 .335 -1 .025 - 0.881 -0 .644
10 ¢ c2 -0 .617 -0 .800 -0 .571 -0 .738 -0 .531 -0 .730 -0 .596 -0 .722 -0 .721 - 0.681 -0 .726
10 ¢ c3 -0 .110 -0 .080 -0 .166 -0 .175 -0 .063 -0 .035 -0 .111 -0 .090 -0 .345 - 0.202 -0 .244
10 ¢ c4 -0 .110 -0 .096 -0 .146 -0 .089 -0 .073 -0 .017 -0 .113 -0 .047 -0 .127 - 0.145 -0 .076
10 ¢ c5 -0 .010 -0 .013 0 .021 -0 .021 0 .024 -0 .013 0 .099 0 .067 -0 .022 0.039 0 .024
Figure 4.13 shows, as an example, the measured wind forces and moment together with
their Fourier approximation, for one of the tankers.
Figure 4.13: Example of Wind Load Coe¢cients
4.8.2 Wind Loads on Other Moored Structures
The wind forces on other types of structures, as for instance semi-submersible platforms,
can be approximated by dividing the structure into a number of components, all with
a more or less elementary geometry, and estimating the wind force on each element.
Drag coe¢cients are given in the literature for a lot of simple geometrical forms, such
as spheres, ‡at plates and cylinders of various cross sectional shapes. [Hoerner, 1965] and
[Delany and Sorensen, 1970] are good sources of this information. The total wind load on
the structure is found by adding the contributions of all the individual component parts.
The fact that one element may in‡uence the wind …eld of another element is neglected in
this analysis.
4.8.3 Wind Loads on Sailing Ships
For sailing merchant ships and tankers, only the longitudinal wind resistance, Xw = Rw,
is of importance for determining a sustained sea speed.
The relative wind speed, Vrw, and the relative wind direction, ®rw, have to be determined
from this true wind speed, Vtw, and the true wind direction, ®tw, together with the forward
ship speed, Vs, and the heading of the ship, see …gure 4.14.
As opposed to the hydromechanical notation, for seamen head wind has a direction equal
to zero. The e¤ect of the lateral force and the yaw moment on the ship will be corrected
via a small course correction. Then the relative wind speed and the relative wind direction
(so head wind has a direction equal to zero) follows from:¯¯¯¯
Vrw =
q
V 2s + V
2
tw+ 2 ¢ Vs ¢ Vtw
¯¯¯¯
¯¯¯¯
®rw = arctan
µ
Vtw ¢ sin®tw
Vs+ Vtw ¢ cos®tw
¶¯¯¯¯
(4.53)
4-28 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA
Figure 4.14: Relative Wind
[Isherwood, 1973] published a reliable method for estimating the wind resistance. He ana-
lyzed the results of wind resistance experiments carried out at di¤erent laboratories with
models covering a wide range of merchant ships. He determined empirical formulas for the
two horizontal components of the wind force and the wind-induced horizontal moment on
any merchant ship form for a wind coming from any direction.
The longitudinal wind resistance is de…ned by:
¯¯¯¯
Xw =
1
2
½airV 2rw ¢ CXw ¢ AT
¯¯¯¯
(4.54)
with for the longitudinal wind resistance coe¢cient CXw:
CXw = A0 + A1 ¢
2AL
L2oa
+A2 ¢ 2AT
B2
+ A3 ¢ Loa
B
+A4 ¢ S
Loa
+ A5 ¢ C
Loa
+ A6 ¢M (4.55)
in which:
Loa = length over all (m)
B = beam (m)
S = length of perimeter of lateral projection excluding water line
and slender bodies such as masts and ventilators (m)
C = distance from bow of centroid of lateral projected area (m)
AL = lateral projected wind area (m2)
AT = transverse projected wind area (m2)
M = number of distinct groups of masts or kingposts seen in lateral
projection (-); kingposts close against the bridge are not included
The coe¢cients for equation 4.55 are listed below.
®rw A0 A1=10 A2 A3 A4 A5=10 A6
0± 2 .152 - 0.500 0 .243 - 0 .164 0 .000 0.000 0 .000
10± 1 .714 - 0.333 0 .145 - 0 .121 0 .000 0.000 0 .000
20± 1 .818 - 0.397 0 .211 - 0 .143 0 .000 0.000 0 .033
30± 1 .965 - 0.481 0 .243 - 0 .154 0 .000 0.000 0 .041
40± 2 .333 - 0.599 0 .247 - 0 .190 0 .000 0.000 0 .042
50± 1 .726 - 0.654 0 .189 - 0 .173 0 .348 0.000 0 .048
60± 0 .913 - 0.468 0 .000 - 0 .104 0 .482 0.000 0 .052
70± 0 .457 - 0.288 0 .000 - 0 .068 0 .346 0.000 0 .043
80± 0 .341 - 0.091 0 .000 - 0 .031 0 .000 0.000 0 .052
90± 0 .355 0.000 0 .000 0 .000 -0 .247 0.000 0 .018
100± 0 .601 0.000 0 .000 0 .000 -0 .347 0.000 -0 .020
110± 0 .651 0.129 0 .000 0 .000 -0 .582 0.000 -0 .031
120± 0 .564 0.254 0 .000 0 .000 -0 .748 0.000 -0 .024
130± -0 .142 0.358 0 .000 0 .047 -0 .700 0.000 -0 .028
140± -0 .677 0.364 0 .000 0 .069 -0 .529 0.000 -0 .032
150± -0 .723 0.314 0 .000 0```