2. 5.1 Introduction
When a ship moves forward through the water at a constant
velocity, V. Its forward motion is going to generate:
a) dynamic pressure on the hull, producing a resultant force in
the longitudinal direction and opposite to the advancing
direction; and
b) tangential stresses on the immersed (or wetted) surface due to
the viscosity; their resultant force is also opposite to the ship’s
moving direction.
The total force opposite to the motion is called the resistance of
the ship or “drag.” The resistance components most concerned
arise from one of the two forces; namely normal dynamic
pressures or tangential stresses on the ship surface.
3. The ship actually moves at the same time through two fluids,
water and air, with widely different density. While the lower part
of the hull is moving through water, the upper part is moving
through air. Like moving in the water, the upper part of the ship
moving in the air is also subject to the same types of forces
(dynamic pressures and tangential stresses).
Because , the air resistance is usually much smaller
than the water resistance, except for those aerostatic support of
hydrodynamic support crafts.
Summary: Water resistance (submerged part of a hull)
Air resistance (upper part of hull &
superstructure)
a w
4. 5.2 Types of Water Resistances
1. Wave-Making Resistance: belongs to the category of
normal dynamic pressures. Due to these dynamic pressures
waves are generated on the surface of water and spread away
from a ship. Waves possess energy. Thus a ship making
waves means a loss of its energy. Wave-making resistance
is important to surface ships, especially those of high speeds,
but may be negligible to submarines.
2. Frictional Resistance: arising due to the viscosity of water,
i.e. tangential stresses. Because of viscosity & velocity
gradient in the direction normal to the ship hull, there is a
mass of fluid being dragged along with a ship. Energy
necessary to drag the mass of fluid is the work done by the
ship against the frictional resistance.
5. 3. Eddy-making Resistance: contributed from normal pressure
applied on a hull. Due to the viscosity of the fluid, the flow
separates from the surface of a hull and eddies (vortices) are
formed. These eddies induce the changes in the velocity
field and thus change the normal pressures on a hull. The
changes in the pressure field around a ship result in the
eddy-making resistance.
4. Air resistance (mainly resulting from wind resistance).
5. Appendage resistances: are caused by the appendages of a
ship, such as propellers, rudders and bilge keels.
6. 5.3 Dimensional Analysis of Ship Resistance
• The purpose of studying “Dimensional Analysis” (D.A)
D. A is helpful to classify and compute various types of
resistances, by examining the basic laws governing the
resistances of a body moving through a fluid.
Although CFD has made considerable progresses, the present
practice still depends on ship model test to determine the
resistances of the ship. D.A is especially useful in data
analysis of ship model test, which may deduce the resistances
of the corresponding prototype ship.
7. • The foundation of dimensional analysis (review)
D. A is based on the principle that an equation which expresses a
physical relationship must be dimensionally homogenous.
In other words, the physical units of all terms at both sides of an
equation must be the same, e.g.
2
, F
A R P
A
In general, all physical units can be expressed by 3 fundamental
units, such as mass-length-time or force-length-time.
Buckingham theory: if there are n dimensional variables in a
physical equation, described by m fundamental dimensions, they
may be grouped into n – m dimensionless variables.
8. • Dimensional Analysis of model test of resistance
1 2 3
1 2 3
( , , , , , , , , , )
For a comparison between geometrically similar bodies
(a ship and its model), their nondimensional ratios
, , , (ratioes) etc are the same. Thus,
( )
Ba
a b c d e f
R f L etc V g p
R f L V g p
2 3 2 2
3 2 2
sed on the dimensional homogeneity
( - mass, - length, - time)
b c d e f
a
a b c d e f b d f c d e f
M L T
ML M L M L M
f L
T L T TL T T L
f L M T
9. 2
3 1 (1)
1 (2)
2 2 2 (3)
We have 6 parameters and 3 eq.s relating
them, thus 1 ,
2 2 2 , 2 .
d e
a b c d e f
b d f
c d e f
b d f
c d e f a d e
R f L
1 2 2 2
2 2
2 2
,
1
2
/ , is the dynamic viscosity
& the kinematic viscosity
d f d e f d e f
d f
e
p V g p
R Lg p
f
LV V V
V L
10. 2 2
2 2
2
, , & are dimensionless coefficients. They
1
2
are related to the similarity laws between the model and prototype.
The (total) . ,
1
2
where the wetted surface o
T
R LV V V
Lg p
V L
R
C
V S
S
resistance coeff
2
2
f a hull has is proportional to ;
Reynolds number, Re , related to the friction resistance;
Froude Number, Fr , related to the wave-making resistance
Euler Number, Eu , not significant to
1
2
L
LV
V
Lg
p
V
the resistance.
11.
2
1
2
, Re,Fr
T
R VL V
C f f
SV Lg
When a model and its prototype are geometrically similar and
their two dimensionless coefficients (Re, Fr) are the same,
their resistance coefficients (CT) should be the same.
Dimensional analysis reduces the number of the related
parameters involved in model tests. However, it can take the
problem no further than the above conclusion.
12. 5.4 Model Tests of Ship Resistance
• Model tests are widely used in the design and study of large
engineering constructions, such as harbor, breakwater, bridge
constructions, and ship buildings.
• A ship model is geometrically similar to its prototype. The
size of the model is usually much smaller than that of the
ship.
• Ship model tests are employed to predict the resistance, the
interaction between the hull and the propeller, seakeeping
properties of a ship, etc. Therefore, model tests are very
important in ship design and ship research. Here we focus on
model resistance tests.
13. • Ship Resistance and Model Test
Model resistance tests are usually carried out in a towing tank. A
towing tank is a long and narrow basin. Small towing tanks are
about 200-300’ long, 15-30’ wide, 5-9’ deep. Large ones, e.g.
U.S. Navy, the David Taylor Model Basin has a length of
2775’, a width of 51’ and a depth of 22’.
A ship model (at a fixed displacement and a naked hull (no
appendage, 4-7’ for small towing tank, 12-30’ for large one) is
towed at a constant velocity by a mechanically propelled towing
carriage (see website below). The resistance of the model at the
constant velocity is recorded by the instruments on the carriage.
Usually the test is carried at a number of constant velocities, and
a resistance curve is thus obtained.
http://www.dt.navy.mil/hyd/fac/tow-bas/hig-spe-bas/index.html
18. • Determining the Resistance of a ship based on its
model test
When a ship and its model are geometrically (all characteristics
& dimensions are in the same ratio) and dynamically similar, we
may use Eq (5.1) to determine the resistance of a ship based on
the measured data from its model test. Namely,
2 2
1 1
2 2
2 2
1
2
2 2
1
2
when Re Re ,
or (5.1)
m s m s
T T
m s
s m
s s s
s s s
m m m m m m
Fr Fr and
R R
C C
SV SV
S V
R S V
R S V S V
19. Geometrical similarity indicates the main characteristics of a
model & its prototype are in the same ratio.
or , for a model and its prototype
having the same Fr & Re, then we require
1
, & ,
if both are run in water at the similar density &
temperature, .
Since 1
s
m
s s s m s
m m m s m
s m
L
m
L
V L V L
m
V L V L m
m
, it is ,
and Re Re
m s m s
Fr Fr
almost impossible to satisfy both
20. 1. In order to overcome this fundamental difficulty to satisfy
the similarity laws, a major (first) assumption was made
by Froude that the frictional and the wave-making
resistances are independent, and the frictional-resistance
coeff. depends only on the Reynolds #. The wave-making
or residual resistance coeff. depends only on the Froude # .
1 2
2
1
2
1
2
1
2
2
2
1
2
Frictional Resistance:
Wave-making Resistance:
T F R
F
F
R
R
R VL V
C C C f f
SV gL
R VL
C f
V S
R V
C f
V S gL
21. 2. It is also assumed that the frictional resistance coeff. of a ship
(or a model) is the same as that of a smooth flat plate with
the same length and wetted surface area as the ship (or the
model). Therefore, CF or RF of a ship (or a model) can be
computed given the length according to the half-analytically &
half-empirically friction formulas.
3. Based on these two assumptions, we may determine the
resistance of a ship at a constant velocity given the results of
model resistance test. The steps are detailed below.
2
1
2
a. At , the total resistance of a model, , can be measured.
Thus ,
where is the model's wetted surface area.
m Tm
Tm
Tm
m m
m
V R
R
C
S V
S
22. nd
b. According to the 2 assumption, , can be computed given
the length of model according to a friction coefficient formula.
c. Computing the model's resistance coefficient
Fm
C
residual
2
.
d. If , namely, , then
,
the ship's residual resistance coefficient is computed.
e. Same as in Step b, can be comput
Rm Tm Fm
s m s s
m m
s m
Rm RS
FS
C C C
V V V L
m
V L
gL gL
V
C C f
gL
C
ed given the ship's length.
f. The total resistance coeff. of a ship is given by,
.
TS FS RS
FS Rm FS Tm Fm Tm Fm FS
C C C
C C C C C C C C
23. 2
1
2
g. The total resistance of a naked ship (excluding appendages)
can be obtained, , at . When
two geometrically similar ships are running at speeds which
conform to the F
S TS S s S m
R C S V V mV
2
2
roude Law, , they are said to be running
at . It is noticed that, .
rs rm
s s
m m
F F
S L
m
S L
corresponding speeds
In most cases, the total resistance of a ship can be determined
accurately based on the model test results using the above method.
However, the method is based on the 2 major assumptions (a. CF
& CR are independent, b. CFS of a ship is equal to that of a flat
plate with the same length). Sometimes the errors due to the
approximations may be significant. We will study the frictional,
wave-making and eddy-making resistances in detail, for
understanding the computation using the method & its validity.
24. 5.5 Frictional Resistance
• Laminar and Turbulent Flow (review of CVEN 311)
Laminar flow: the fluid appears to move by the sliding of
laminations of the infinitesimal thickness relative to adjacent
layers.
Turbulent flow: is characterized by fluctuations in velocity
at all points of the flow field and these fluctuations with no
definite frequency.
Whether a flow is laminar or turbulent flow depends mainly
on its Reynolds #. For a plate flow,
6
8
6 8
when Re < 10 the flow is laminar,
Re > 10 the flow is turbulent,
10 < Re < 10 the flow is transitional
25. • Friction formulas for a flat plate
The following formulas are commonly used.
1
5
5 1.5
2
1
2
10
1) Blasius formula. (Laminar flow)
1.32/ Re, Re 4.5 10 . Re , , thus, .
2) Prandtl and von Karman formula (turbulent flow)
log Re , 0.074( ) , thus,
F
F F F
F F N F
F
R
VL
C C R V
SV
A
C M C R R
C
1.8
8
10
.
3) Schoenherr formula (1947 ATTC line, derived based on 2))
0.242
log Re , for Re 4.5 10 .
4) 1957 ITTC line formula (known as ship-model correlation line
not a friction coef
F
F
V
C
C
7
2
10
ficient for a flat plate, turbulent flow)
0.075
, for Re 10 .
log Re 2
F
C
26.
27.
28. It is noted that CF computed according to the Blasius formula
(laminar) and CF according to turbulent flow formula, say
Schoenherr formulas are quite different. For a small model
there would be a laminar flow over the (at least) forward part
which would develop into turbulent flow further afterwards.
Then it would be inaccurate, if we either use Blasius formula or
Scheonherr formula to compute the frictional coeff. of the model.
To overcome this problem, we have to
1. set a lower limit on the size of model
2. use turbulence stimulating devices at the bow of a model to
stimulate an early transition from a laminar to turbulence flow,
such as a “trip wire” at ½ station after the forward
perpendicular line.
Therefore, the frictional coeff. of a model can be computed
according to a turbulent flow formula.
29. • Influence of Roughness of a plate on CF
The formulas for computing CF are applied to the flat plates with
smooth surface. The rough surface (of a ship) will result in the
increase of CF . Roughness (on the surface of a hull) may be
classified into 3 types.
1. Structural roughness: caused by welded joints, warviness of
shell plating on the hull. A newly-built ship will have
(for Schoenherr formula).
2. Corrosion
3. Fouling: caused by the attachment of marine organisms such as
seaweeds, shells and barnacles.
Corrosion & fouling occur for ships having sailed for a certain
period of time. They will decrease the velocity of the ship. Ship
owner will decide when the ship should go to the dock for cleaning.
0.0004
f
C
30. 5.6 Wave-Making Resistance
• Wave-making resistance is important to
1. a surface ship (negligible for submarine); and
2. its speed is high. Accurately speaking, its Froude # ,
or in U.S. the speed/length ratio, is high.
It is noticed that the speed to length ratio is a dimensional
coefficient, where V is in knots, L in feet.
A nautical mile/hr (knot) = 0.5144 m/s.
R
V
F
gL
V
L
6
1 is equivalent to 0.3
When 0.1, & is negligible.
When 0.45, , is dominant in .
R
R W W
R W W T
V
F
L
F C R
F C V R R
31. • Ways to study or determine wave-making
resistance
1. Experiments with models in towing tank; At present, model test
is still the most important tool for prediction of wave-making
resistance.
2. *Theoretical and numerical computations (CFD): help in
interpreting model test results, reduce the range of model tests,
and guide further research.
32. • Ship Wave Pattern
Lord Kelvin (1887) considered a single pressure point traveling
in a straight line over the surface of the water, sending out
waves which combine to form a characteristic pattern.
Transverse Waves
Divergence Waves
33. • Description of the wave pattern of a moving pressure point
1. A system of transverse waves: the heights of successive crests
diminish when T.W go afterwards w.r.t. the pressure point.
2. A series of divergent waves: the whole pattern is roughly
contained within two straight lines, which start from the
pressure point and make angles of 19˚ 28’ on each side of the
line of the motion.
2
2 2
In deep water, ( ),
wavelength of T.W:
2 /
Wavelength of D.W:
2 cos / ,
=19 28'
TW
DW
h
V g
V g
34. •Ship Wave Pattern
Kelvin wave pattern illustrates and explains many of the
features of ship waves. Ship wave pattern is similar to the
combination of two Kelvin wave systems generated by two
pressure points, with one near the bow and the other near the
stern.
39. • Interference Effects
1. At lower speed (Froude #), waves made by a ship are very small
& wave-making resistance is insignificant.
2. At lower Froude #, divergent waves are the primary wave
system. As the Froude # of a ship increases and the depth of
water decreases, transverse waves are more important.
3. The wavelength of T.W. increases with the speed of a ship.
Thus the position of the T.W’s crest (or trough) w.r.t. the ship
changes.
40. 4.If the trough of the T.W. generated by the bow coincides with that
generated by the stern, then CW becomes very large. If the crest
of T.W generated by the bow coincides with the trough of T.W
generated by the stern, then CW becomes small. This
phenomenon is called bow and stern wave interference, which
accounts for the humps and valley in the CW curves.
5. In order to reduce the
resistance, a ship designer
chooses appropriate L, V
such that CW is at valley
instead of at humps.
(p149-151)
