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Livro do Journee - Offshore Hydromechanics

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waves by the hull, which means the appearance of a
speed-dependent second type of resistance; wave making resistance. For an extrapolation
from ship model data to prototype data, this part mainly depends on the Froude number.
In the second part of the 19th century William Froude, 1810 - 1879, (as distinct from his
son Robert Edmund Froude from the Froude number) proposed the British Admiralty
to build what would become the world’s …rst towing tank at Torquay in England. He
had recently developed scaling laws for predicting the resistance of ships from tests on
ship models and he intended to use this tank for the required scale model experiments.
The British Admiralty accepted Froude’s proposal on the condition that he also used the
tank to investigate the ways of reducing the rolling motion of ships. That was when sails
were being replaced by steam driven propulsion systems and roll was becoming more of
a problem. On March 3 1872, the …rst indoor professional model test in the world was
carried out in a basin of 85 x 11 x 3 meter, as shown in …gure 4.8. About half a century
later, Froude’s towing tank was pulled down.
Figure 4.8: The World’s First Towing Tanks
In 1883, the world’s …rst privately owned towing tank, the Denny tank as shown in …gure
4.8 too, was built at Dumbarton in Scotland, using the original design of Froude’s tank
in Torquay. The Denny tank - with dimensions 93.00 x 6.80 x 2.75 meter - was closed in
1983, but it was re-opened in 1986 under the umbrella of the Scottish Maritime Museum.
The tank testing and museum parts are separate. Since 1989, the tank testing facility
operates under the management of Strathclyde University’s Department of Ship and Marine
Technology.
It has been the merit of William Froude to distinguish the components of the total hull
resistance, Rt, and to relate them to scaling laws; he distinguished between:
- a frictional resistance component, Rf, and
- a residual resistance component, Rr .
Then he made a very drastic simpli…cation, which has worked out remarkably well: Froude’s
…rst hypothesis was that these two components of the resistance are independent of each
other. The determination of these two components was a second problem, but he found a
simple way out of this problem. Froude’s second hypothesis was that the frictional part
4.7. SHIP STILL WATER RESISTANCE 4-19
of the resistance can be estimated by the drag of a ‡at plate with the same wetted area
and length as the ship or model. In principle, a ‡at plate (towed edgewise) has no wave re-
sistance and can therefore be investigated over a range of Reynolds numbers (Rn = VL=º)
without in‡uence of the wave-related Froude number (Fn = V=
p
gL).
4.7.1 Frictional Resistance
To determine the frictional resistance coe¢cient, William Froude’s approach yielded that
the frictional resistance coe¢cient was related to the resistance coe¢cient of a ‡at plate
with the same length and wetted surface as the ship or model hull:¯¯¯¯
Cf =
Rf
1
2½V
2S
¯¯¯¯
or:
¯¯¯¯
Rf =
1
2
½V 2 ¢ Cf ¢ S
¯¯¯¯
(4.39)
in which:
Cf = frictional resistance coe¢cient (-)
Rf = frictional resistance (N)
½ = density of water (kg/m3)
V = ship or model speed (m/s)
S = wetted surface of ship or model hull (m2)
He did numerous experiments to determine the resistance coe¢cients of a ‡at plate as
a function of the Reynolds number. He himself did not …nd a single relationship as a
function of the Reynolds number due to laminar ‡ow and edge e¤ects in his measurements.
His results not only depended on the Reynolds number but also on the length of the
plate. Several friction lines based only on the Reynolds number were developed later, both
theoretically using boundary layer theory and experimentally.
For laminar ‡ows, the resistance coe¢cient was formulated from boundary layer theory
by Blasius:
Blasius: Cf = 1:328 ¢
p
Rn (4.40)
So-called plate lines were developed for turbulent boundary layer ‡ows from the leading
edge. These plate lines were extended to include full scale Reynolds numbers. They have
relatively simple formulations, such as the Schoenherr Mean Line or the ITTC-1957
Line, which are de…ned as:
Schoenherr:
0:242p
Cf
= log10 (Rn ¢ Cf) (4.41)
ITTC-1957: Cf =
0:075
(log10 (Rn) ¡ 2)2
(4.42)
The latter one is accepted as a standard by the International Towing Tank Conference
(ITTC).
As a matter of fact it is not too important that a ‡at plate with a certain length and
wetted surface has a resistance coe¢cient exactly according to one of the mentioned lines.
The Froude hypothesis is crude and correlation factors are required afterwards to arrive
at correct extrapolations to full scale values. These correlation factors will depend on the
plate line which is used.
4-20 CHAPTER 4. CONSTANT REAL FLOW PHENOMENA
4.7.2 Residual Resistance
The determination of the resistance components of a ship’s hull can be illustrated with the
results of resistance tests with a series of models at various scales, the ”Simon Bolivar”
family of models. Resistance tests were carried out over a certain speed range for each
of the models. Each model had a di¤erent scale factor, ®. The total resistance (in non-
dimensional form) is shown in …gure 4.9.
Figure 4.9: Resistance Coe¢cients of the ”Simon Bolivar” Model Family
The total resistance force on a ship is made non-dimensional in the same way as the
frictional resistance above:¯¯¯¯
Ct =
Rt
1
2½V
2S
¯¯¯¯
or:
¯¯¯¯
Rt =
1
2
½V 2 ¢ Ct ¢ S
¯¯¯¯
(4.43)
in which:
Ct = total resistance coe¢cient (-)
Rt = total resistance (N)
½ = density of water (kg/m3)
V = ship or model speed (m/s)
S = wetted surface of ship or model hull (m2)
4.7. SHIP STILL WATER RESISTANCE 4-21
Ct is plotted as a function of the logarithm of the Reynolds number, Rn = V ¢ LWL =º, in
which LWL is the length at the water plane. Each curve of data points has been made
with a model of the given scale. The Schoenherr Mean Line follows from equation 4.41;
the other two curves connect the points for Fn is zero and for Fn is another constant,
respectively.
Similar lines can be drawn for other Froude numbers. Such a line of constant Froude
number is approximately parallel to this plate line. If all scaling laws could be satis…ed,
the resistance curves at all model scales would coincide. Because the Reynolds number is
not maintained, this is not the case and each model has a separate curve.
The residual resistance coe¢cient, Cr, at a certain Froude number is now the vertical
distance between the plate line and the line for that Froude number. When the plate
line and the line of constant Froude number are parallel, this means that the residual
resistance component is indeed independent of the Reynolds number. This is assumed to
be always the case in Froude’s method and the residual resistance at each Froude number
is determined by subtracting the calculated frictional resistance coe¢cient of a ‡at plate
according to equations 4.41 or 4.42 from the measured total resistance.
The assumption that the residual resistance is approximately independent of the Reynolds
number is not accurate in many cases. This means that the lines of constant Froude
number in …gure 4.9 are not exactly parallel. The residual resistance in that case contains
a component which depends on the Reynolds number.
To determine this component, the line for Fn ! 0 has been determined in …gure 4.9. When
the Froude number becomes su¢ciently small the generated waves will disappear and the
di¤erence between the total resistance coe¢cient and the plate line can only be associated
with the longer distance of water particles to travel along the (not ‡at) hull form and some
small viscous e¤ects. Thus the di¤erence between the plate line and the line Fn ! 0 has
to do with the form of the hull and is therefore called form resistance. The line Fn ! 0
is now