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```assumed to be always parallel to the lines for other Froude numbers, so that the
di¤erence between the total resistance coe¢cient and the line Fn ! 0 is independent of
the Reynolds number. This di¤erence is now called wave resistance.
In this way the total resistance is split up into:
- the frictional resistance (from the plate line),
- the form resistance and
- the wave resistance.
Form resistance and wave resistance together constitute Froude’s residual resistance.
Form Resistance.
Several hypotheses have been made about the relation between the form resistance and the
frictional resistance. As already mentioned, Froude assumed that the form resistance was
independent of the Reynolds number. He therefore included it as a part of the residual
resistance. Hughes in 1953 assumed that the form drag was proportional to the viscous
resistance and multiplied the viscous resistance coe¢cient, Cf , by a constant factor, k, as
shown in …gure 4.10.
The Reynolds-dependent component of the resistance thus becomes (1 + k)¢Cf . The factor
(1 + k) is called the form factor. Froude’s method is still used, but Hughes’ approach of
is the most widely adopted one and is the accepted standard by the ITTC.
4-22 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA
Figure 4.10: Hughes Extrapolation Method
The form factor is in‡uenced primarily by the shape of the after part of the ship but it is
often given as a function of the block coe¢cient, CB, of the ship as a whole. The block
coe¢cient is the ratio between the ship’s displacement volume and that of a rectangular
box in which the ship’s underwater volume ”just …t”.
Form factors associated with typical block coe¢cients are listed below.
CB 1 + k
<0 .7 1 .10 -1 .15
0 .7- 0 .8 1 .15 -1 .20
>0 .8 1 .20 -1 .30
Wave Resistance.
The wave resistance coe¢cient of the model, Cw, over the speed range of the model is now
found by subtracting the frictional and form resistance coe¢cients from the measured total
resistance coe¢cient:
Cw = Ct ¡ (1 + k) ¢ Cf (4.44)
4.7.3 Extrapolation of Resistance Tests
Given the components of the total resistance of the model, one must extrapolate this data
to full scale. The resistance of the model is generally measured from a low speed up to the
design speed. The model design speed is set by maintaining the full scale Froude number.
Equation 4.43 is used to express the total resistance in dimensionless form.
The wetted surface, S , is taken as the wetted length from keel to still water line of the
frames, integrated over the ship length and multiplied by two, to account for both sides
of the ship. Note that the ship length is measured directly along the center line and not
along the water line.
The Froude number is maintained during the model test. This means that the wave
resistance coe¢cient, Cw , at model scale and at full scale are the same. The total resistance
coe¢cient of the ship, Ct ship, can therefore be found from:
jCt ship = (1 + k) ¢ Cf ship +Cw+ Caj (4.45)
The form factor, k, and the wave resistance coe¢cient, Cw , are found directly from the
model test. The frictional resistance coe¢cient at full scale, Cf ship, can be read from the
plate line using the full scale Reynolds number.
The additional resistance coe¢cient, Ca, is a new element. This coe¢cient is a correlation
coe¢cient based on experience at full scale. It accounts both for extrapolation errors due to
the various assumptions made and for e¤ects at full scale which are not present at the model.
Such e¤ects include the relatively rough surface at full scale in relation to the boundary
layer thickness. Moreover, Ca contains a correction for the di¤erence in air resistance of
the above water part of the hull of model and prototype. [Holtrop and Mennen, 1982] give
a simple formula for Ca as a fraction of the length of the ship:
Ca = 0:006 ¢ (LWL + 100)¡0:16 ¡ 0:00205 (4.46)
in which LWL is the length of the water plane in meters.
The additional resistance coe¢cient decreases in this formula with increasing length and
this can be attributed, at least partly, to roughness e¤ects. The e¤ect of surface roughness
in general requires special attention.
4.7.4 Resistance Prediction Methods
Thus, the total resistance of the ship follows from:¯¯¯¯
Rt ship =
1
2
½V 2 ¢ Ct ship ¢ S
¯¯¯¯
(4.47)
and the resistance coe¢cient, Cts, has to be determined.
A number of methods to determine the still water resistance coe¢cients of ships, based
on (systematic series of) model test data, are given in the literature. A very well known
method, developed at MARIN, is described by [Holtrop, 1977], [Holtrop and Mennen, 1982]
and [Holtrop, 1984]. The method is based on the results of resistance tests carried out by
MARIN during a large number of years and is available in a computerized format. The
reader is referred to these reports for a detailed description of this method, often indicated
by the ”Holtrop and Mennen” method. An example for a tug of the correlation be-
tween the results of this ”Holtrop and Mennen” method and model test results is given in
…gure 4.11.
Like all environmental phenomena, wind has a stochastic nature which greatly depends on
time and location. It is usually characterized by fairly large ‡uctuations in velocity and
4-24 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA
Figure 4.11: Comparison of Resistance Prediction with Model Test Results
direction. It is common meteorological practice to give the wind velocity in terms of the
average over a certain interval of time, varying from 1 to 60 minutes or more.
Local winds are generally de…ned in terms of the average velocity and average direction
at a standard height of 10 meters above the still water level. A number of empirical and
theoretical formulas are available in the literature to determine the wind velocity at other
elevations. An adequate vertical distribution of the true wind speed z meters above sea
level is represented by:
Vtw(z)
Vtw(10)
=
³ z
10
´0:11
(at sea) (4.48)
in which:
Vtw(z) = true wind speed at z meters height above the water surface
Vtw(10) = true wind speed at 10 meters height above the water surface
Equation 4.48 is for sea conditions and results from the fact that the sea is surprisingly
smooth from an aerodynamic point of view - about like a well mowed soccer …eld.
On land, equation 4.48 has a di¤erent exponent:
Vtw(z)
Vtw(10)
=
³ z
10
´0:16
(on land) (4.49)
At sea, the variation in the mean wind velocity is small compared to the wave period.
The ‡uctuations around the mean wind speed will impose dynamic forces on an o¤shore
structure, but in general these aerodynamic forces may be neglected in comparison with
the hydrodynamic forces, when considering the structures dynamic behavior. The wind
will be considered as steady, both in magnitude and direction, resulting in constant forces
and a constant moment on a …xed ‡oating or a sailing body.
The wind plays two roles in the behavior of a ‡oating body: