Hydromechanics 7.1 Dimensional Analysis, Basic Estimation, and Model Testing 137
Part A | 7.1
0
0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
–0.1
0.5
1
1.5
2
U
~
H 1
T 1
B 1
T 2
B 2
T 3
B 3
T 4
B 4
Stream-wise x/δ
x (m)
Wall-normal z/δ
z (m)
Flow
Top cameras with
lower magnification
Bottom cameras with
higher magnification
0.8m
0.5m
21m from trip
Field of
view
200mm lens
Bellows
Extension rings
a)
b)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.5
0.4
0.3
0.2
0.1
0
12
10
8
6
z
y
x
Fig. 7.8a,b Experimental measurement of a flat plat turbulent boundary
layer. (a) Experimental setup; (b) spatial distribution of e U, the instantaneous
velocity component in the streamwise
direction (in meters per second). Here
ı is the boundary layer thickness and
the dashed rectangles represent the
areas imaged by each of the cameras
in (a). The rectangle H 1 indicates
an area where high resolution was
used to image the near-wall region
of the boundary layer (after [7.18],
courtesy of AIP publishing)
only
C F .Re/ Á
F v
1
2
U 2 S
;
(7.18)
where S is the wetted surface area of the hull, and
a residuary drag component F R , which is a function of
the Froude number only
C R .Fr/ Á
F R
1
2
U 2 S
:
(7.19)
The residuary part of the drag is assumed to arise from
normal pressure forces acting perpendicularly to the
hull (wave drag, viscous pressure resistance, and flow
separation) and the frictional component is assumed to
arise from shear stresses acting tangentially along the
hull (skin friction). The total drag is simply the sum of
the frictional and residuary components
C T .Re; Fr/ D C F .Re/ C C R .Fr/ :
(7.20)
Let a subscript m refer to the model hull and the subscript p refer to the corresponding prototype hull. The
procedure for using Froude’s hypothesis is as follows:
1. Use Froude scaling (7.10) to determine the model
test speed from the anticipated operational speed of
the prototype. Using the known model and prototype lengths, match the model and prototype Froude
numbers and solve for the required test speed
Fr m D
U m
p
gL m
D
U p
p
gL p
D Fr p ! U m D U p
s
L m
L p
:
Generally, the length on the waterline of the model
and prototype are used as the reference lengths L m
and L p , respectively.
2. The total drag on the model is experimentally measured and used to determine the total drag coefficient C Tm .
3. The coefficient of residuary resistance of the
model C Rm is assumed to be equal to the coefficient
of residuary resistance of the prototype C Rp when
the model and prototype Froude numbers are equal.
The residuary resistance of the model is determined
from Froude’s hypothesis as C Rm D C Tm C Fm D
C Rp .
4. The coefficient of friction of both the model and
prototype are determined using Reynolds number
dependent empirical relations for the drag on a flat
plate. In practice, the 1957 ITTC (International
Towing Tank Conference) line [7.19] is often used
C F D
0:075
Œlog 10 .Re/ 2:0 2 :
(7.21)
Part A | 7.1
0
0
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
–0.1
0.5
1
1.5
2
U
~
H 1
T 1
B 1
T 2
B 2
T 3
B 3
T 4
B 4
Stream-wise x/δ
x (m)
Wall-normal z/δ
z (m)
Flow
Top cameras with
lower magnification
Bottom cameras with
higher magnification
0.8m
0.5m
21m from trip
Field of
view
200mm lens
Bellows
Extension rings
a)
b)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.5
0.4
0.3
0.2
0.1
0
12
10
8
6
z
y
x
Fig. 7.8a,b Experimental measurement of a flat plat turbulent boundary
layer. (a) Experimental setup; (b) spatial distribution of e U, the instantaneous
velocity component in the streamwise
direction (in meters per second). Here
ı is the boundary layer thickness and
the dashed rectangles represent the
areas imaged by each of the cameras
in (a). The rectangle H 1 indicates
an area where high resolution was
used to image the near-wall region
of the boundary layer (after [7.18],
courtesy of AIP publishing)
only
C F .Re/ Á
F v
1
2
U 2 S
;
(7.18)
where S is the wetted surface area of the hull, and
a residuary drag component F R , which is a function of
the Froude number only
C R .Fr/ Á
F R
1
2
U 2 S
:
(7.19)
The residuary part of the drag is assumed to arise from
normal pressure forces acting perpendicularly to the
hull (wave drag, viscous pressure resistance, and flow
separation) and the frictional component is assumed to
arise from shear stresses acting tangentially along the
hull (skin friction). The total drag is simply the sum of
the frictional and residuary components
C T .Re; Fr/ D C F .Re/ C C R .Fr/ :
(7.20)
Let a subscript m refer to the model hull and the subscript p refer to the corresponding prototype hull. The
procedure for using Froude’s hypothesis is as follows:
1. Use Froude scaling (7.10) to determine the model
test speed from the anticipated operational speed of
the prototype. Using the known model and prototype lengths, match the model and prototype Froude
numbers and solve for the required test speed
Fr m D
U m
p
gL m
D
U p
p
gL p
D Fr p ! U m D U p
s
L m
L p
:
Generally, the length on the waterline of the model
and prototype are used as the reference lengths L m
and L p , respectively.
2. The total drag on the model is experimentally measured and used to determine the total drag coefficient C Tm .
3. The coefficient of residuary resistance of the
model C Rm is assumed to be equal to the coefficient
of residuary resistance of the prototype C Rp when
the model and prototype Froude numbers are equal.
The residuary resistance of the model is determined
from Froude’s hypothesis as C Rm D C Tm C Fm D
C Rp .
4. The coefficient of friction of both the model and
prototype are determined using Reynolds number
dependent empirical relations for the drag on a flat
plate. In practice, the 1957 ITTC (International
Towing Tank Conference) line [7.19] is often used
C F D
0:075
Œlog 10 .Re/ 2:0 2 :
(7.21)