Hydromechanics 7.1 Dimensional Analysis, Basic Estimation, and Model Testing 131
Part A | 7.1
c) Froude number.(FR)
The ratio of the inertial to gravity forces is the
square of the Froude number
Fr
2
D
F i
F g
U
2 L
2
gL 3 D
U
2
gL
! Fr D
U
p
gL
:
(7.10)
The Froude number is most often used to characterize the flow at a free surface and is related
to the celerity (phase speed) of a gravity wave.
It can also be used to predict when wave drag
is important for ships and other vessels. As the
equilibrium position of subsurface buoys represents
a balance between buoyancy, weight, and drag,
which varies as the square of the freestream flow
speed, Froude scaling can also be used to model
their dynamic responses, even when they are deeply
submerged [7.2].
d) Weber number.
The Weber number is the ratio of inertial forces to
surface tension forces
We D
U
2 L
2
T s L
D
U
2 L
T s
;
(7.11)
where T s is the surface tension (T s D 0:073 N m
1
for water). The Weber number can be used to estimate the size of droplets in a spray, the thickness
of a spray sheet (such as at the chine of a planing
hulled vessel), and is also important in formation
of ventilated cavities behind surface piercing propellers [7.3, 4]. Note that it is not generally possible
to match both the Weber number and the Froude
number in a flow.
e) Cavitation number.
This is the ratio of the difference between the local
pressure in a flow and the vapor pressure and the
dynamic pressure of the flow
1
2
U
2
Á
p 0 p v
1
2
U 2 ;
(7.12)
where the vapor pressure of water at 20
ı C is p v D
2:336 kPa. Cavitation may generally occur when
< 0:1 and the inception will tend to depend on
a number of factors including water purity, water
depth, and background pressure – which can necessitate the use of pressured model test facilities
in practice. As the local pressure over a submerged
surface approaches the local vapor pressure of the
water, the water essentially starts to boil and small
pockets of water vapor will form over the surface.
The process is unsteady and when the air pockets begin to collapse they can damage the surface.
Cavitation differs from ventilation, in which there
is a connection or small tunnel of air between
the submerged surface and the free surface of the
water. Both cavitation and ventilation can lead to
a loss of lift on submerged hydrofoils or propeller
blades.
f) Force coefficients.
The two-dimensional (2-D) drag coefficient is typically defined using the drag per unit span of a wing,
d, whereas the three-dimensional (3-D) drag coefficient is defined using the total drag on a wing, D
2-D
C d Á
d
U 2 L
;
(7.13)
3-D
C D Á
D
U 2 L 2 :
(7.14)
The lift coefficients are defined similarly, where l is
the drag per unit span and L is the total lift
2-D
C l Á
l
U 2 L
;
(7.15)
3-D
C L Á
L
U 2 L 2 :
(7.16)
g) Strouhal number.
The Strouhal number represents the ratio of a characteristic flow time to a characteristic oscillation
time. For example, if the frequency of vortex shedding in the wake of a cylinder of diameter L in a flow
with a speed U is f , then the Strouhal number will
be
Sr Á
char. flow time
char. oscillation time
L=U
1=f
D
fL
U
;
(7.17)
the Strouhal number relates the frequency of vortex shedding in a flow and is important for ocean
engineering problems, such as the strumming of
underwater cables and structures in a current and
the swimming of marine animals [7.5–7]. Incidentally, the Strouhal number is proportional to the tip
speed ratio (TSR) of underwater ocean energy extraction turbines [7.2, 8] and varies as the inverse of
the advance ratio J (often used to characterize propellers) [7.5].
In the motion of a bluff body such as a cylinder
of diameter d, accelerating from rest to moving at
a constant speed U normal to its axis, the body
experiences an adverse pressure gradient on its
Part A | 7.1
c) Froude number.(FR)
The ratio of the inertial to gravity forces is the
square of the Froude number
Fr
2
D
F i
F g
U
2 L
2
gL 3 D
U
2
gL
! Fr D
U
p
gL
:
(7.10)
The Froude number is most often used to characterize the flow at a free surface and is related
to the celerity (phase speed) of a gravity wave.
It can also be used to predict when wave drag
is important for ships and other vessels. As the
equilibrium position of subsurface buoys represents
a balance between buoyancy, weight, and drag,
which varies as the square of the freestream flow
speed, Froude scaling can also be used to model
their dynamic responses, even when they are deeply
submerged [7.2].
d) Weber number.
The Weber number is the ratio of inertial forces to
surface tension forces
We D
U
2 L
2
T s L
D
U
2 L
T s
;
(7.11)
where T s is the surface tension (T s D 0:073 N m
1
for water). The Weber number can be used to estimate the size of droplets in a spray, the thickness
of a spray sheet (such as at the chine of a planing
hulled vessel), and is also important in formation
of ventilated cavities behind surface piercing propellers [7.3, 4]. Note that it is not generally possible
to match both the Weber number and the Froude
number in a flow.
e) Cavitation number.
This is the ratio of the difference between the local
pressure in a flow and the vapor pressure and the
dynamic pressure of the flow
1
2
U
2
Á
p 0 p v
1
2
U 2 ;
(7.12)
where the vapor pressure of water at 20
ı C is p v D
2:336 kPa. Cavitation may generally occur when
< 0:1 and the inception will tend to depend on
a number of factors including water purity, water
depth, and background pressure – which can necessitate the use of pressured model test facilities
in practice. As the local pressure over a submerged
surface approaches the local vapor pressure of the
water, the water essentially starts to boil and small
pockets of water vapor will form over the surface.
The process is unsteady and when the air pockets begin to collapse they can damage the surface.
Cavitation differs from ventilation, in which there
is a connection or small tunnel of air between
the submerged surface and the free surface of the
water. Both cavitation and ventilation can lead to
a loss of lift on submerged hydrofoils or propeller
blades.
f) Force coefficients.
The two-dimensional (2-D) drag coefficient is typically defined using the drag per unit span of a wing,
d, whereas the three-dimensional (3-D) drag coefficient is defined using the total drag on a wing, D
2-D
C d Á
d
U 2 L
;
(7.13)
3-D
C D Á
D
U 2 L 2 :
(7.14)
The lift coefficients are defined similarly, where l is
the drag per unit span and L is the total lift
2-D
C l Á
l
U 2 L
;
(7.15)
3-D
C L Á
L
U 2 L 2 :
(7.16)
g) Strouhal number.
The Strouhal number represents the ratio of a characteristic flow time to a characteristic oscillation
time. For example, if the frequency of vortex shedding in the wake of a cylinder of diameter L in a flow
with a speed U is f , then the Strouhal number will
be
Sr Á
char. flow time
char. oscillation time
L=U
1=f
D
fL
U
;
(7.17)
the Strouhal number relates the frequency of vortex shedding in a flow and is important for ocean
engineering problems, such as the strumming of
underwater cables and structures in a current and
the swimming of marine animals [7.5–7]. Incidentally, the Strouhal number is proportional to the tip
speed ratio (TSR) of underwater ocean energy extraction turbines [7.2, 8] and varies as the inverse of
the advance ratio J (often used to characterize propellers) [7.5].
In the motion of a bluff body such as a cylinder
of diameter d, accelerating from rest to moving at
a constant speed U normal to its axis, the body
experiences an adverse pressure gradient on its