Part A | 7.1
132 Part A Fundamentals
downstream side, leading to a backward flow and
flow separation from the aft surface of the body.
The detached boundary layer rolls up aft of the body
into two coherent counter-rotating vortices that are
shed from the body once they reach a certain size for
a given local convective flow. The vortices are typically shed asymmetrically from the two sides of the
body and continue to form and get shed alternately
at a frequency f , leading to a staggered array of
counter-rotating vortices, known as a von Karman
vortex street, in the wake of the body. The vortex
shedding induces an alternating lift force normal to
the direction of the flow and the body axis. The lift
force per unit span of the bluff body may be estimated as
l.t/ D U .t/ ;
where is the density of the fluid and .t/ is the
instantaneous circulation around the body. is related to the net vorticity shed from body at any
instance. l.t/ therefore fluctuates at a frequency f
and amplitude l max as .t/ varies between ˙ max ,
where max is the amplitude of the fluctuating circulation. The lift coefficient and the nondimensional
frequency of the fluctuation
C l max .Re/ D
l max
0:5U 2 d
D
max
0:5Ud
;
S.Re/ D
fd
U
;
are functions of the Reynolds number (and the
shape of the body). For a cylinder S has been shown
to be approximately S 0:220:27 for Reynolds
numbers between 10
4 and 10
7 [7.9]. The lift coefficient is approximately given by C l max 0:5 [7.10].
If the natural frequency of vibration of the body is
close to f , then the body undergoes transverse resonant vibrations [7.10]. Such vibrations are called
vortex-induced vibrations (VIV). Strumming oscillations of cables in crosswinds and transverse
vibrations of pilings are examples of the phenomena. Use of a splitter plate at the back of the cylinder
or of helical ribbon fairings on the cylinder can be
made to minimize VIV.
Example 7.1
Dispersion of surface waves on water: assume that the
wave speed c depends on the other five parameters listed
in Table 7.1. Find the ˘ groups, note that the wavenumber is defined as k Á 2˘==, where is the wavelength
of a wave.
Table 7.1 Potential dispersion parameters of interest
Parameter
Dimension
c Wave speed
Œc D
L
T
g Gravity
Œg D
L
T 2
Surface tension
ŒŒ D
M
T 2
k Wave number
Œk D
1
L
h Water depth
Œh D L
Density
ŒŒ D
M
L 3
Solution 7.1
6 variables, 3 dimensions ! 3˘ -groups.
Eliminate length:
Œck D
1
T
; Œgk D
1
T 2 ;
ŒŒ D
M
T 2 ; Œhk D 1 ; ŒŒk
3
D M :
Eliminate time:
Ä ck
p
gk
D
"
c
s
k
g
#
D 1 ;
Ä
gk
D M ;
k
3
D M :
Eliminate mass:
Ä k
2
g
D 1
) c
s
k
g
D '
Â
kh;
k
2
pg
Ã
:
This can be rewritten as
c
2
D
g
k
'
2
Â
kh;
k
2
g
Ã
:
As shown in [7.11], the phase speed of deep water
waves (gravity is the restoring force) is
c
2
D
g
k
tanh.kh/ ;
and the phase speed of capillary waves (surface tension
is the restoring force) is
c
2
D
g
k
Â
1 C
k
2
g
Ã
tanh.kh/ :
132 Part A Fundamentals
downstream side, leading to a backward flow and
flow separation from the aft surface of the body.
The detached boundary layer rolls up aft of the body
into two coherent counter-rotating vortices that are
shed from the body once they reach a certain size for
a given local convective flow. The vortices are typically shed asymmetrically from the two sides of the
body and continue to form and get shed alternately
at a frequency f , leading to a staggered array of
counter-rotating vortices, known as a von Karman
vortex street, in the wake of the body. The vortex
shedding induces an alternating lift force normal to
the direction of the flow and the body axis. The lift
force per unit span of the bluff body may be estimated as
l.t/ D U .t/ ;
where is the density of the fluid and .t/ is the
instantaneous circulation around the body. is related to the net vorticity shed from body at any
instance. l.t/ therefore fluctuates at a frequency f
and amplitude l max as .t/ varies between ˙ max ,
where max is the amplitude of the fluctuating circulation. The lift coefficient and the nondimensional
frequency of the fluctuation
C l max .Re/ D
l max
0:5U 2 d
D
max
0:5Ud
;
S.Re/ D
fd
U
;
are functions of the Reynolds number (and the
shape of the body). For a cylinder S has been shown
to be approximately S 0:220:27 for Reynolds
numbers between 10
4 and 10
7 [7.9]. The lift coefficient is approximately given by C l max 0:5 [7.10].
If the natural frequency of vibration of the body is
close to f , then the body undergoes transverse resonant vibrations [7.10]. Such vibrations are called
vortex-induced vibrations (VIV). Strumming oscillations of cables in crosswinds and transverse
vibrations of pilings are examples of the phenomena. Use of a splitter plate at the back of the cylinder
or of helical ribbon fairings on the cylinder can be
made to minimize VIV.
Example 7.1
Dispersion of surface waves on water: assume that the
wave speed c depends on the other five parameters listed
in Table 7.1. Find the ˘ groups, note that the wavenumber is defined as k Á 2˘==, where is the wavelength
of a wave.
Table 7.1 Potential dispersion parameters of interest
Parameter
Dimension
c Wave speed
Œc D
L
T
g Gravity
Œg D
L
T 2
Surface tension
ŒŒ D
M
T 2
k Wave number
Œk D
1
L
h Water depth
Œh D L
Density
ŒŒ D
M
L 3
Solution 7.1
6 variables, 3 dimensions ! 3˘ -groups.
Eliminate length:
Œck D
1
T
; Œgk D
1
T 2 ;
ŒŒ D
M
T 2 ; Œhk D 1 ; ŒŒk
3
D M :
Eliminate time:
Ä ck
p
gk
D
"
c
s
k
g
#
D 1 ;
Ä
gk
D M ;
k
3
D M :
Eliminate mass:
Ä k
2
g
D 1
) c
s
k
g
D '
Â
kh;
k
2
pg
Ã
:
This can be rewritten as
c
2
D
g
k
'
2
Â
kh;
k
2
g
Ã
:
As shown in [7.11], the phase speed of deep water
waves (gravity is the restoring force) is
c
2
D
g
k
tanh.kh/ ;
and the phase speed of capillary waves (surface tension
is the restoring force) is
c
2
D
g
k
Â
1 C
k
2
g
Ã
tanh.kh/ :