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Wind Induced Vibration of Cable Stay Bridges Workshop, April 25-27, 2006
Mechanism of Wind & Rain/Wind Induced Cable Vibrations
The Generation Mechanism of Inclined Cable Aerodynamics-Rain-Wind Induced Vibration and
Dry-State GallopingMasaru MATSUMOTO
Kyoto UniversityKyoto, JAPAN
Inclined Cable aerodynamicsHigh Speed Vortex Induced Vibration
Velocity-Restricted ResponseGalloping Instability
Divergent ResponseVelocity Restricted Response
?
Cable Attitude to Wind
wind
wind
β*=arcsin(cosα�sinβ)β :yaw angleβ* :effective yaw angle
α :vertical angle
β
90°-β*
γ =90°-arccos(sinα�sinβ/(sin2α+cos2α�cos2β)1/2
vibrating plane
cable in-planecable vertical axis�(cable in-plane)
vibration plane
horizontal axis
front stagnation point
normal to cable in-plane
γ
γ
γ
γα
90°-β
Inclined Cable Inclined Cable AerodynamicsAerodynamics
The substantial factors
The upper water rivulet formation on a cable surfaceThe axial flow in a near wakeCritical Reynolds number
Cable Aerodynamics in related to KarmanVortex
Karman Vortex
The Karman Vortex Street (KV)
Theodore von Karman,
1881~19631)/(sinh =ahπ
Stable Condition of Vortex Street in Potential Flow, [Theodore von Karman, 1911.]
Karman Vortex�(KV)
Karman Vortex�(KV)
Karman vortex (KV)
Karman vortex can be sensitively influenced by
various factors.
Lock-in Phenomena? (KV)
At near resonance velocity to Karmanvortex, the large amplitude response might be excited by motion-induced vortex, but not by Karman vortex. (Lanivier, Zaso etc.)
Splitter Plate with suitable OR (KV)
Cable model
Splitter plate
Wind
D=50mm L=700mm
Opening ratio(OR) of Splitter plate was changed to suitably control the Karman vortex
Change of Opening Ratio(OR) of Splitter Plate (KV)
OR:90%OR:40%OR:10%
OR is changed from 0% to 100%
Fluctuating Lift Force v.s. OR (KV)
0 10 20 30 40 50 60 70 80 90 1000
0.05
0.1
0.15
0.2
Open ratio of splitter plate [%]
Fluc
tuat
ing
win
d fo
rce
coef
ficie
nt C
L′
CL’
β=0˚, U=6.0m/s
The intensity of Karman Vortex becomes significantly weak if OR decreases.
0 10 20 30 40 50 60 70 80 90 1000
510
1520
2530
3540
45
0
0.005
0.01
Open ratio of splitter plate [%]Frequency [Hz]
P.S.
D. [
(N/m
)2 /Hz]
PSD of fluctuating lift force
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.561Hzm=1.477kg/mδ(2A=10mm)=0.00242Sc(2A=10mm)=2.4337
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.554Hzm=1.543kg/mδ(2A=10mm)=0.00271Sc(2A=10mm)=2.8748
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.571Hzm=1.377kg/mδ(2A=10mm)=0.00215Sc(2A=10mm)=2.0164
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.559Hzm=1.450kg/mδ(2A=10mm)=0.00270Sc(2A=10mm)=2.6912
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.559Hzm=1.496kg/mδ(2A=10mm)=0.00234Sc(2A=10mm)=2.4069
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.550Hzm=1.467kg/mδ(2A=10mm)=0.00234Sc(2A=10mm)=2.3515
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.559Hzm=1.467kg/mδ(2A=10mm)=0.00215Sc(2A=10mm)=2.1635
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.563Hzm=1.418kg/mδ(2A=10mm)=0.00272Sc(2A=10mm)=2.6259
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.559Hzm=1.467kg/mδ(2A=10mm)=0.00272Sc(2A=10mm)=2.7170
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.549Hzm=1.521kg/mδ(2A=10mm)=0.00254Sc(2A=10mm)=2.6327
0 10 20 30 40 50 60 70
0
0.5
1
1.5
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/D 2A [mm]
U [m/s]U/fD
f=3.568Hzm=1.353kg/mδ(2A=10mm)=0.00233Sc(2A=10mm)=2.141
0 10 20 30 40 50 60 70
0
0.5
1
1.5
Free Vibration Test (KV)
0%10%20%30%40%50%60%70%80%90%100%OR
Enlarged maximum amplitude
Vortex induced vib. terminates at near U/fD=15and another steady response appears
β=0˚
External Stimulation v.s. KarmanVortex (KV)
Longitudinally Fluctuating Stimulation(LFS) (KV)
In-phase stimulation at up and down sides separation points
0 5 10 15 20 25 30 35 40 45 500 5 10 15 20 25 30 35
40 45 50
0
0.5
1
Pulsator Frequency [Hz]Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
Due to Karman vortex
0 5 10 15 20 25 30 35 40 45 500 5 10 15 20
25 30 35 40 45 50
0
0.01
0.02
0.03
Pulsator Frequency [Hz]Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
0 5 10 15 20 25 30 35 40 45 500 5 10 15 20
25 30 35 40 45 50
0
0.02
0.04
Pulsator Frequency [Hz]Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
PSD of Velocity in a Wake (by LFS) (KV)
OR10% OR70% OR100%
β=0˚, U=6.0m/s, measured point�1.5D downr stream, 1.0D beneath
weak intensiveIntensity of Karman vortex
No appearance of stimulating frequency peak
Vertically Sinusoidal Stimulation(VSS) (KV)
Out-phase stimulation at up and down separation points
123456789100%
Wavelet Analysis of Fluctuating Lift Force(by VSS) (KV)
Wing frequency�3HzU=6.0m/s
OR
Wing frequency
Karman vortex frequency
β=0˚
WindWind
RainRain
Water rivuletWater rivulet
Axial flowAxial flow
Water rivulet and axial flowWater rivulet and axial flow
Inclined Cable Inclined Cable AerodynamicsAerodynamics
The substantial factors
The upper water rivulet formation on a cable surfaceThe axial flow in a near wakeCritical Reynolds number
Axial Flow
Visualized axial flow by light strings for a proto-type cable
(AX)
wind
Axial flow velocity -approaching wind velocity diagram of a proto-type stay-cable
(AX)
((ββ*=40*=40°° --5050°°, where , where ββ*: equivalent yawing angle)*: equivalent yawing angle)
0 2 4 6 80
2
4
6
8
Mean wind velocity [m/s]
Va
[m/s
]
Visualized Flow Field around an Visualized Flow Field around an Inclined Cable Inclined Cable (AX)(AX)
Visualized axial flow by flags in a wake
of yawed cable (β=45°, V=1m/s)
Visualized intermittently Karman vortices
and axial vortices by fluid paraffinof yawed cable (β=45°, V=1m/s)
Wind tunnel tests (AX)
Top view of wind tunnel
Wind
Wind tunnel walls
β
α =0º
wind tunnel testswind tunnel tests((αα=0=0°°, , ββ=45=45°°) )
Typical Response (β=45°, in smooth flow, without water rivulet) (AX)
0 1 2 3 4 5 6 7 8 9 1011121314 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=2.096Hzm=0.509kg/mδ (2A=10mm)=0.00373Sc(2A=10mm)=1.138
- 0.45
- 0.4
- 0.35
- 0.3
- 0.25
- 0.2
- 0.15
- 0.1
- 0.05
0
0 20 40 60 80 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
V/ fD
2A/D
V-A diagram V-A-δ diagram
Galloping Latent vortex
Experimental set up (three end conditions)Experimental set up (three end conditions)(without rivulet) (yawing angle=45deg) (without rivulet) (yawing angle=45deg)
(AX)(AX)
three end conditions1.without wall2.without wall and with end plates3.with walls installed windows (φ=170mm)
wind tunnel(without wall)
wind
walls and windows
Karman vortex v.s. low frequency component of lift and fluctuating velocity
in wake (AX)
If Karman vortex would be mitigated, low frequency lift/velocity appears.
P.S.D. and wavelet analysis of P.S.D. and wavelet analysis of the the unsteady lift forceunsteady lift force on stationary cable on stationary cable
model (without rivulet) model (without rivulet) (AX)(AX)
0 5 10 15 20 250
0.5
1
1.5
2 x 10- 3
Frequency [Hz]
P.S.
D. [
N2 /Hz]
40 20 10 8 7 6 5 4V/ fD
β=0°, in smooth flow, V=5.0m/s
0 4 8 12 16 200
0.5
1 x 10- 4
Frequency [Hz]
P.S.
D. [
N2 /Hz]
40 20 10 8 7 5 4V/ fD
β=45°, in smooth flow, V=4.0m/s
Wavelet analysis ofWavelet analysis of fluctuating wind velocityfluctuating wind velocityin the wake of stationary yawed cable model in the wake of stationary yawed cable model
(AX)(AX)
(without rivulet, β=45°, V=4m/s, in smooth flow)
Axial Flow v.s. Intensity of Karman Vortex
Axial flow velocityAxial flow velocity (Va) in a wake (Va) in a wake (AX)(AX)
0 4 8 12 16 20 240
0.2
0.4
0.6
0.8
1
1.2
X/ D
Va/V Wind
X
Y
Hot wire
(1) without wall (2) without wall and with end plates
(3) with walls installedwindows (φ=170mm)
((VV=8m/s, =8m/s, ββ=45=45°°))
Definition of coordinates, X, Y, Z Definition of coordinates, X, Y, Z (AX)(AX)
Cable model
Wind
Y
Hot wire elementX
Cable model
Z
Wind
Wavelet analysis of Wavelet analysis of fluctuating wind velocityfluctuating wind velocitymeasured at various points in a wake measured at various points in a wake (AX)(AX)
X/D=7.7
X/D=12.7
X/D=17.6
(1) without wall (2) without wall and with end plates
(3) with walls installedwindows (φ=170mm)
((ββ=45=45°, VV=8.0m/s, =8.0m/s, Y/DY/D=1.8, =1.8, Z/DZ/D=0.5)=0.5)
Experimental set up (three end conditions)Experimental set up (three end conditions)(without rivulet) (yawing angle=45deg) (without rivulet) (yawing angle=45deg)
(AX)(AX)
three end conditions1.without wall2.without wall and with end plates3.with walls installed windows (φ=170mm)
wind tunnel(without wall)
wind
walls and windows
Time history and the wavelet analysis of the unsteady Time history and the wavelet analysis of the unsteady crosscross--flow displacement (flow displacement (unsteady gallopingunsteady galloping))
0 10 20 30 40 50-0.01
-0.005
0
0.005
0.01
Time [sec]
Dis
plac
emen
t [m
]
18 18.2 18.4 18.6 18.8 19-4
-2
0
2
4 x 10-3
Time [sec]
Dis
plac
emen
t [m
]
40 40.2 40.4 40.6 40.8 41-4
-2
0
2
4 x 10-3
Time [sec]
Dis
plac
emen
t [m
]
free cablefree cable--end condition (without end condition (without wall, without endwall, without end--plates)plates)
((VV=5.0m/s, =5.0m/s, ff00=2.87Hz, =2.87Hz, ffkk=8.6Hz)=8.6Hz)0 10 20 30 40 50-0.01
-0.005
0
0.005
0.01
Time [s]
Dis
plac
emen
t [m
]
Intensity of Karman vortex shedding (AX)
� The most intensive: with end plates and without walls (no galloping appears)
� Weak: without end plates without walls (unstable galloping appears)
� The weakest : with walls installed windows (galloping appears)
Upper water rivulet formation
Rain Effect (WR)Water rivulet formation
The effect of rivulet on response and stationary force (its location)
Pa+(σ /R)=P0
Pa:the wind pressure on rivulet surface
σ :the surface tensile force of water rivulet
R :the local curvature radius of water rivulet
P0 :the hydro-pressure of water rivulet
The formation of the upper water rivulet and its location on an inclined surface can be determined by the following formula
The Relation of the Position of Upper Water Rivulet and its Response (WR)
θl,θw
θlθwθs
Windward EdgeHorizontal Line
Stagnation point
Leeward Edge
Rivulet
Artificial rivulet (WR)
wind
Cable diam.:50mm
Artificial rivulet
cable length:1100mm
Width:3.6mm�����thickness:1.6mm
The detail stationary lift and dragThe detail stationary lift and dragforce coefficient subject to rivulet position force coefficient subject to rivulet position
(WR)(WR)
-0.3-0.2-0.1
00.10.20.30.40.5
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180upper rivulet position [deg.] (θ )
C L
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
upper rivulet position [deg.] (θ )
CD
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
upper rivulet position [deg.] (θ )
C L0
0.10.20.30.40.50.60.70.8
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
upper rivulet position [deg.] (θ )C
D
for nonfor non--yawed yawed circular cylindercircular cylinder((ββ=0=0°°))
for yawed for yawed circular cylindercircular cylinder((ββ=45=45°°))
Rivulet position effect on unsteady lift force and Strouhal number induced by Karman vortex
shedding (WR)
0
0.050.1
0.150.2
0.25
0.30.35
0.4
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
upper rivulet position [deg.] (θ )
L f (L
ift fo
rce
ampl
itude
) [N
]
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
upper rivulet position [deg.] (θ )
L f (L
ift fo
rce
ampl
itude
) [N
]
for nonfor non--yawed yawed circular cylindercircular cylinder((ββ=0=0°°))
for yawed for yawed circular cylindercircular cylinder((ββ=45=45°°))
0.1
0.12
0.14
0.16
0.18
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180upper rivulet position [deg.]
S t0.15
0.17
0.19
0.21
0.23
0.25
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
upper rivulet position [deg.]
S t
Aerostatic/fluctuating force v.s. rivulet position �β=0˚�(WR)
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
Position of water rivulet (deg.)
Stea
dy w
ind
forc
e co
effic
ient
CL ′
CL CD CL’
0 20 40 60 80 100 120 140 160 180-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Position of water rivulet (deg.)
Stea
dy w
ind
forc
e co
effic
ient
CL
0 20 40 60 80 100 120 140 160 1800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Position of water rivulet (deg.)
Stea
dy w
ind
forc
e co
effic
ient
CD
At near 50deg., Karman vortex becomes weak,then drag decreases and stationary lift appears.
Scanlan’s Flutter Derivatives(R. H. Scanlan,
1974)( )
⎭⎬⎫
⎩⎨⎧
+++=b
HkHkVbkH
VkHVbL ηφφηρ *
42*
32*
2*
122
21 &&
⎭⎬⎫
⎩⎨⎧
+++=b
AkAkVbkA
VkAVbM ηφφηρ *
42*
32*
2*
122 )2(
21 &&
2DOF
L/M : the unsteady lift force/pitching momentper unit span length
η/φ : the heaving/torsional displacementsρ : the air densityb : a half chord lengthHi*, Ai* (i=1-4) : the flutter derivativesk : the reduced frequency (k=bω/V)ω : flutter circular frequency
1DOF η, φ motion
2DOF η, φ coupled motion
superposition
1914~2001Scanlan
-150
-100
-50
0
50
100
0 20 40 60 80 100
V/fD
y
ƒ Æ=54° ƒ Æ=56° ƒ Æ=58°
ƒ Æ=60° ƒ Æ=70° ƒ Æ=90°
Aerodynamic derivative H1*�β=0˚�
at sub-cr. Re number (WR)
-150
-100
-50
0
50
100
0 20 40 60 80 100
V/fD
y
without wivulet ƒ Æ=40° ƒ Æ=46°ƒ Æ=48° ƒ Æ=50° ƒ Æ=52°
unstableunstable
Aerody. unstable rivulet position
HH11**
U/fD U/fD
P.S.D. of the P.S.D. of the fluctuating lift forcefluctuating lift force on on stationary cable model with artificial stationary cable model with artificial
upper rivulet upper rivulet (WR)(WR)
0 20 40 60 80 100 120 140 160 180
010
2030
4050
0
0.02
0.04
without rivulet
Rivulet position θ [deg.]Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
0 20 40 60 80 100 120 140 160 180
0
10
20
30
40
0
1
2x 10-3
Rivulet position θ [deg.]
without rivulet
Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
for nonfor non--yawed yawed circular cylindercircular cylinder((ββ=0=0°°))
for yawed for yawed circular cylindercircular cylinder((ββ=45=45°°))
Rivulet positionRivulet position effect on velocityeffect on velocity--amplitude amplitude diagrams of nondiagrams of non--yawed cable model (yawed cable model (ββ=0=0°°) )
(WR)(WR)
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=1.734Hzm=0.823kg/mδ(2A=10mm)=0.003745Sc(2A=10mm)=2.047
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5
f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047
θ=47°
θ=54°
θ=58°
0 20 40 60 80 100 120 140 160 180
010
2030
4050
0
0.02
0.04
without rivulet
Rivulet position θ [deg.]Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
P.S.D. of the fluctuating lift force on stationary cable model
θ=56°
θ=66°
θ=94°
0 1 2 3 4 5 6 7 8 9 1011121314 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5
f=2.089Hzm=0.588kg/mδ (2A=10mm)=0.00336Sc(2A=10mm)=1.178
0 1 2 3 4 5 6 7 8 9 1011121314 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5
f=2.079Hzm=0.531kg/mδ (2A=10mm)=0.0035Sc(2A=10mm)=1.102
0 1 2 3 4 5 6 7 8 9 1011121314 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=2.082Hzm=0.519kg/mδ (2A=10mm)=0.0036Sc(2A=10mm)=1.128
Rivulet positionRivulet position effect on velocityeffect on velocity--amplitude amplitude diagrams of nondiagrams of non--yawed cable model (yawed cable model (ββ=45=45°°) )
(WR)(WR)
0 20 40 60 80 100 120 140 160 180
0
10
20
30
40
0
1
2x 10-3
Rivulet position θ [deg.]
without rivulet
Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
P.S.D. of the fluctuating lift force on stationary cable model
CFD analysis (3D LES) (WR)
Flow field and stationary and unsteady forces of non-yawed cable with upper rivulet
2000.0021000020D×20D×0.2D189×131×3140°
2000.0021000020D×20D×0.2D189×131×3110°
2000.0021000020D×20D×0.2D189×131×390°
2000.0021000020D×20D×0.2D189×131×380°
2000.0021000020D×20D×0.2D189×131×370°
2000.0021000020D×20D×0.2D189×131×366°
2000.0021000020D×20D×0.2D189×131×360°
2000.0021000020D×20D×0.2D189×131×356°
2000.0021000020D×20D×0.2D189×131×352°
2000.0021000020D×20D×0.2D189×131×350°
2000.0021000020D×20D×0.2D189×131×346°
2000.0021000020D×20D×0.2D189×131×340°
2000.0021000020D×20D×0.2D189×131×3no rivulet
Total timeTime stepReDomain Grid Rivulet position
Computational cases of large eddy simulation of flow around a cable model (WR)
�
Entire domain�������������Grid around cable modelComputational grid (WR)
Strouhal number
0
0.05
0.1
0.15
0.2
0.25
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Rivulet position
St
Drag force coefficient
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Rivulet position
Cd
Lift force coefficient
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Rivulet position
Cl
Amplitude of lift force
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Rivulet position
Amp. of L
Strouhal number (by fluctuating lift force,■: no rivulet) (WR)
Drag force and lift force coefficient(■: no rivulet) (WR)
Amplitude of fluctuating lift force (■: no rivulet) (WR)
0T/4 T/4 2T/4 3T/4 4T/4
L
t Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4Pressure contour around cable during one period of fluctuating lift force (no rivulet) (WR)
0T/4 T/4 2T/4 3T/4 4T/4
L
t
Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4Velocity vector around cable during one period of fluctuating lift force (no rivulet) (WR)
Rivulet position 60 degree
0T/4 T/4 2T/4 3T/4 4T/4
L
t
Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4Velocity vector around cable during one period of fluctuating lift force (θ=60°) (WR)
0T/4 T/4 2T/4 3T/4 4T/4
L
t
Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4Pressure contour around cable during one period of fluctuating lift force (θ=66°) (WR)
0T/4 T/4 2T/4 3T/4 4T/4
L
t Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4 Velocity vector around cable during one period of fluctuating lift force (θ=66°) (WR)
High Speed Vortex Excitation
3D Karman Vortex Shedding along Cable Axis (WR)
Instantaneous water rivulet Instantaneous water rivulet distribution distribution (WR)(WR)
V=10m/s
Wind Tunnel Tests (WR)
rivuletrivulet
aa aabb
Position of artificial rivuletPosition of artificial rivulet
rigid cable modelrigid cable model((DD=0.05m)=0.05m)
windwind
rivuletrivulet0.0320.032DD0.0720.072DD
θ
Position Position and length of artificial rivuletand length of artificial rivulet
Wind tunnelWind tunnel
P.S.D. of fluctuating velocity in a wake (a:θ=65°(a/l=0.35), b:θ=55°(b/l=0.24), in
smooth flow, V=8.0m/s) (WR)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
40 20 10 8 7 6 5 4V/ fD
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
40 20 10 8 7 6 5 4V/ fD
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
40 20 10 8 7 6 5 4V/ fD
2)2) θ θ =55=55°°/65/65°°
1)1) θ θ =65=65°° (St:0.167)(St:0.167)
3)3) θθ =55=55°° (St:0.189)(St:0.189)
11 22 33
windwindaa aabb
windwind1.51.5DD
0.40.4DD
hot wirehot wire
P.S.D. & Wavelet analysis of fluctuating lift force
(a:θ=65°(a/l=0.35), b:θ=55°(b/l=0.24), in smooth flow, V=8.0m/s) (WR)
0 10 20 30 40 50
10- 5
100
Frequency [Hz]
P.S.
D. [
N2 /Hz]
40 20 10 8 7 5 4V/ fD
P.S.D.P.S.D. Wavelet analysisWavelet analysis
not so clearnot so clear
intensive unsteady power appears intensive unsteady power appears at high reduced velocity rangeat high reduced velocity range
V-A diagrams (uniform rivulet θ=47°, 54°, 58°, in smooth flow)
(WR)
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5
θ θ =54=54°°
Galloping (G)Galloping (G)
f=1.734Hzm=0.823kg/mδ (2A=10mm)=0.003745Sc(2A=10mm)=2.047
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=1.734Hzm=0.823kg/mδ (2A=10mm)=0.003745Sc(2A=10mm)=2.047
θ θ =47=47°°
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5 f=1.734Hzm=0.823kg/mδ (2A=10mm)=0.003745Sc(2A=10mm)=2.047
θ θ =58=58°°
Non Non -- Galloping Galloping (NG)(NG)
Non Non -- Galloping Galloping (NG)(NG)
V-A diagrams (in smooth flow) (WR)
a:a:θθ=58=58°°(a/l=0.31), (a/l=0.31), b:b:θθ=47=47°°(b/l=0.33)(b/l=0.33)
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5 f=1.734Hzm=0.823kg/mδ (2A=10mm)=0.003745Sc(2A=10mm)=2.047
a:a:θθ=58=58°°(a/l=0.31),(a/l=0.31), b:b:θθ=54=54°°(b/l=0.33)(b/l=0.33)
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5 f=1.734Hzm=0.823kg/mδ (2A=10mm)=0.003745Sc(2A=10mm)=2.047
(NG)(NG) (NG)(NG) (NG)(NG) (G)(G)
V-A diagrams (in smooth flow) (WR)
a:a:θθ=65=65°°(a/l=0.41), (a/l=0.41), b:b:θθ=55=55°°(a/l=0.12)(a/l=0.12)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/ D2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5 f=2.882Hzm=1.838kg/mδ (2A=10mm)=0.001928Sc(2A=10mm)=2.354
a:a:θθ=50=50°°(a/l=0.31), (a/l=0.31), b:b:θθ=55=55°°(b/l=0.33)(b/l=0.33)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5 f=2.874Hzm=1.779kg/mδ (2A=10mm)=0.001947Sc(2A=10mm)=2.300
a:a:θθ=65=65°°(a/l=0.36), (a/l=0.36), b:b:θθ=55=55°°(b/l=0. 24)(b/l=0. 24) a:a:θθ=65=65°°(a/l=0.31), (a/l=0.31), b:b:θθ=55=55°°(b/l=0.33)(b/l=0.33)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5 f=2.871Hzm=1.759kg/mδ (2A=10mm)=0.001809Sc(2A=10mm)=2.114
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5 f=2.874Hzm=1.784/mδ (2A=10mm)=0.00169Sc(2A=10mm)=2.00
V-A & V-A-δ diagrams (in smooth flow) (WR)
- 0.03
- 0.02
- 0.01
0
0.01
0.02
0.03
0 10 20 30 40 50 60 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
V/ fD
2A/D
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5 f=2.874Hzm=1.784/mδ (2A=10mm)=0.00169Sc(2A=10mm)=2.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5 f=2.874Hzm=1.779kg/mδ (2A=10mm)=0.00195Sc(2A=10mm)=2.30
- 0.03
- 0.02
- 0.01
0
0.01
0.02
0.03
0 10 20 30 40 50 60 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
V/ fD
2A/D
a:a:θθ=50=50°°(a/l=0.31), (a/l=0.31), b:b:θθ=55=55°°(b/l=0.33)(b/l=0.33) a:a:θθ=65=65°°(a/l=0.36), (a/l=0.36), b:b:θθ=55=55°°(b/l=0.24)(b/l=0.24)
Movement Effect of Upper Water Rivulet
Observed water rivulet movement on prototype scale cable model during rain-wind induced vibration (MR)
The time history of cable vibration and rivulet movement (MR)
0 5 10 15 20Time [s]
Cab
le m
otio
n
0 5 10 15 20Time [s]
Riv
ulet
mot
ion
0 5 10 15 20Time [s]
Cab
le m
otio
n
0 5 10 15 20Time [s]
Riv
ulet
mot
ion
(large cable amplitude)(large cable amplitude) (small cable amplitude)
Rivulet movement (MR)
Rivulet movement behaves non-uniformly and non-stationary along
cable axis even stationary cross-flow vibration.
Rivulet movement andcable vibration (MR)
When Karman vortex intensively sheds, cable vibration is mitigated.
Rivulet tends to move in synchronization to cable motion, when Karman vortex shedding becomes weak.
Rivulet movement is controlled by Karman vortex shedding when it becomes intensive.
Simulation of rivulet movement during cross flow vibration (MR)
θ (t) rivulet movement
(actual state)
wind
η cross flow vibration
=
θ (t)=θ0+φ
φ rotational motion
+
(assumed state)
θ0
η cross flow motion
wind wind
The unsteady lift forces can be expressed by use of aerodynamic derivatives (MR)
( ) ( ){ }
( ) ( ){ } 0cossin24
sincos24
*4
*3
*2
0
022
20
*3
*2
0
0*1
2
=⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
++−−
+⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
++−+
ηλληφωρω
ηλληφωρη
HHHDm
D
HHDHm
D&&&
η=η0sinωtφ=φ0sin(ωt+λ)=(φ0sinλ/(η0ω))dη/dt+(φ0cosλ/η0)ηR=(D/2)(φ0/η0)
where
Characteristics of vibration and rivulet motion (Cosenteno et al) (MR)
Time history of rivulet position and cable motion
rivulet positioncable motion
Rivulet position effect on velocity – damping diagrams of yawed cable (MR)
:moving water rivulet:fixed water rivulet
-0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
0 20 40 60 80 100
V/fD
δ
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0 20 40 60 80 100
V/fD
δ-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0 20 40 60 80 100
V/fD
δ
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0 20 40 60 80 100
V/fD
δ
d) θ=72°°c) θ=66°°
a) θ=50°° b) θ=56°°
(β=45°, λ=180°, R=0.0698)
Rivulet Movement Effect (MR)
Wind tunnel test results suggest that the rivulet movement synchronized
to cable motion affects the cable vibration more or less, however, the
fundamental response characteristics of cable vibration can be framed by
the neutral rivulet position.
Reynolds Number Effect
Critical Reynolds Number
Re=UD/ν
ν�1.5x10-5
D=0.15m
if Recr=(3-5)x105, then U=30m/s-50m/s.
Reynolds number effectsReynolds number effects on circular on circular cylinder aerodynamics (cylinder aerodynamics (ScheweSchewe) ) (RE)(RE)
(a) r.m.s. of the lift fluctuations
(b) Strouhal number of the lift fluctuatioSr=fD/u∞
(c) drag coefficient
Reynolds number effectsReynolds number effects on inclined cable on inclined cable aerodynamics (NRCC) aerodynamics (NRCC) (RE)(RE)
lift force coefficientdrag force coefficient
Reynolds number effects (RE)
Surface roughness
High wind velocity
Larger diameter
Styrene foam Max.17m/sD=158mm
Change of Reynolds number
Cable Model with surface roughness (non-yawed state) (RE)
Cable Model (β=0°) with Surface Roughness (Styrene Foam)
Cable Model (β=0°)
Cable Model (β=45°)
Static forces�with surface roughness�(RE)
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CD
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CL
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
Reynolds Number (*105)
Fluc
tuat
ing
win
d fo
rce
coef
ficie
nt CL
′
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
Reynolds Number (*105)
Fluc
tuat
ing
win
d fo
rce
coef
ficie
nt CL
′
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CD
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CL
Wind Velocity (m/s)0 3 6 9 12 15 18
β=0˚
CL CD CL’Decrease of drag
stationary liftDecay of Karman vortex intensity
Critical Re. number range
Increase of dragNon-app. of stationary lift
β=45˚
Super critical Re. number range
Flow field�with surface roughness�β=0 �̊(RE)
0 0.5 1 1.5 2-50-40-30-20-10
01020304050
Reynolds Number (*105)
Diff
eren
ce o
f win
d ve
loci
ty (%
)
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-50-40-30-20-10
01020304050
Reynolds Number (*105)
Diff
eren
ce o
f win
d ve
loci
ty (%
)
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-50-40-30-20-10
01020304050
Reynolds Number (*105)
Diff
eren
ce o
f win
d ve
loci
ty (%
)
Wind Velocity (m/s)0 3 6 9 12 15 18
wind
(a)(�) (�)
0 0.5 1 1.5 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CL
Wind Velocity (m/s)0 3 6 9 12 15 18
(a)
(c)
(b�
Velocity difference at critical Re number range where Karman vortex is suppressed
Non-symmetry flow field
Appearance of Stationary lift
0T/4 T/4 2T/4 3T/4 4T/4
L
t
Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4Velocity vector around cable during one period of fluctuating lift force (θ=60°) (WR)
At critical Re. number range (RE) Decrease of drag
Velocity-difference btw up and down sides
Stationary lift
wind
Relative angle of attack due to vib.
Amplified velocity at down side
exciting force
Aerodynamic derivative v.s. Re number (RE)
Frequency of forced heaving vibration was changed to change
the reduced velocity under the fixed wind velocity to change
Reynolds number.
Wo. Surface Roughness
0 2 4 6 8 10 12 14 16 18 0
20
40
60
80
2A/D 2A [mm]
U [m/s]U/fD
f=3.221Hzm=1.863kg/mδ(2A=10mm)=0.00299Sc(2A=10mm)=0.3768
0 5 10 15 20 25 30 35
0
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12 14 16 18 0
20
40
60
80
2A/D 2A [mm]
U [m/s]U/fD
f=3.279Hzm=1.985kg/mδ(2A=10mm)=0.00375Sc(2A=10mm)=0.4999
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
Free vib. test�β=0˚�(RE)
w. Surface roughness
Response with unsteady amp. appears at cr. Re.
number range
Aerodynamic derivative H1* �w. SR�β=0˚,
without rivulet �(RE)
-70
-60
-50
-40
-30
-20
-10
0
0 20 40 60 80 100 120
U/fD
H1*
Re=47000 Re=62000 Re=93000
-700-600-500-400-300-200-100
0100200300
0 50 100 150 200
U/fD
H1*
Re=109000 Re=120000 Re=130000
Re=135000 Re=140000 Re=166000
Sub-crtical Re. number range Cr. Re number range
Super cr. Re number range
unstable
Summary
The Role of KarmanVortex on Inclined Cable Aerodynamics
Karman Vortex Mitigation by
Axial FlowRivulet formationCritical Re numberTurbulenceThe others
The Role of Karman Vortex
Axial Flow(AF) Mitigation of
Water Rivulet (WR) ►► KarmanVortex
Reynolds Number (RE) ▼▼
GallopingHigh Spewed Vortex Exc.
If Karman Vortex is controlled,
Low frequency fluctuation component of flow field and lift appears. Which would excite the galloping or high speed vortex exc..
Non-symmetrical flow field appears, then stationary lift force can be generated. Which would excite the galloping.
Future Subjects
How to Control the Cable Vib.?
Mechanical Countermeasure ?Aerodynamic Countermeasure ?
How to encourage the 2D KarmanVortex?
Thank you for your
kind attention.
P.S.D. and wavelet analysis of P.S.D. and wavelet analysis of the the unsteady lift forceunsteady lift force on stationary cable on stationary cable
model (without rivulet) model (without rivulet) (AX)(AX)
0 5 10 15 20 250
0.5
1
1.5
2 x 10- 3
Frequency [Hz]
P.S.
D. [
N2 /Hz]
40 20 10 8 7 6 5 4V/ fD
β=0°, in smooth flow, V=5.0m/s
0 4 8 12 16 200
0.5
1 x 10- 4
Frequency [Hz]
P.S.
D. [
N2 /Hz]
40 20 10 8 7 5 4V/ fD
β=45°, in smooth flow, V=4.0m/s
Wavelet analysis ofWavelet analysis of fluctuating wind velocityfluctuating wind velocityin the wake of stationary yawed cable model in the wake of stationary yawed cable model
(AX)(AX)
(without rivulet, β=45°, V=4m/s, in smooth flow)
Aerostatic/fluctuating force v.s. rivulet position �β=0˚�(WR)
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
Position of water rivulet (deg.)
Stea
dy w
ind
forc
e co
effic
ient
CL ′
CL CD CL’
0 20 40 60 80 100 120 140 160 180-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Position of water rivulet (deg.)
Stea
dy w
ind
forc
e co
effic
ient
CL
0 20 40 60 80 100 120 140 160 1800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Position of water rivulet (deg.)
Stea
dy w
ind
forc
e co
effic
ient
CD
At near 50deg., Karman vortex becomes weak,then drag decreases and stationary lift appears.
Rivulet positionRivulet position effect on velocityeffect on velocity--amplitude amplitude diagrams of nondiagrams of non--yawed cable model (yawed cable model (ββ=0=0°°) )
(WR)(WR)
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=1.734Hzm=0.823kg/mδ(2A=10mm)=0.003745Sc(2A=10mm)=2.047
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5
f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047
0 1 2 3 4 5 6 7 8 9 10 11 12 0
20
40
60
802A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=1.734Hzm=0.823kg/mδ 2A=10mm)=0.003745Sc(2A=10mm)=2.047
θ=47°
θ=54°
θ=58°
0 20 40 60 80 100 120 140 160 180
010
2030
4050
0
0.02
0.04
without rivulet
Rivulet position θ [deg.]Frequency [Hz]
P.S.
D. [
(m/s
)2 /Hz]
P.S.D. of the fluctuating lift force on stationary cable model
0T/4 T/4 2T/4 3T/4 4T/4
L
t
Fluctuating lift force about 0T/4
�
about T/4 about 2T/4
about 3T/4 about 4T/4Velocity vector around cable during one period of fluctuating lift force (θ=60°) (WR)
Experimental set up (three end conditions)Experimental set up (three end conditions)(without rivulet) (yawing angle=45deg) (without rivulet) (yawing angle=45deg)
(AX)(AX)
three end conditions1.without wall2.without wall and with end plates3.with walls installed windows (φ=170mm)
wind tunnel(without wall)
wind
walls and windows
Time history and the wavelet analysis of the unsteady Time history and the wavelet analysis of the unsteady crosscross--flow displacement (flow displacement (unsteady gallopingunsteady galloping))
0 10 20 30 40 50-0.01
-0.005
0
0.005
0.01
Time [sec]
Dis
plac
emen
t [m
]
18 18.2 18.4 18.6 18.8 19-4
-2
0
2
4 x 10-3
Time [sec]
Dis
plac
emen
t [m
]
40 40.2 40.4 40.6 40.8 41-4
-2
0
2
4 x 10-3
Time [sec]
Dis
plac
emen
t [m
]
free cablefree cable--end condition (without end condition (without wall, without endwall, without end--plates)plates)
((VV=5.0m/s, =5.0m/s, ff00=2.87Hz, =2.87Hz, ffkk=8.6Hz)=8.6Hz)0 10 20 30 40 50-0.01
-0.005
0
0.005
0.01
Time [s]
Dis
plac
emen
t [m
]
Observed water rivulet movement on prototype scale cable model during
rain-wind induced vibration
The time history of cable vibration and The time history of cable vibration and rivulet movementrivulet movement
0 5 10 15 20Time [s]
Cab
le m
otio
n
0 5 10 15 20Time [s]
Riv
ulet
mot
ion
0 5 10 15 20Time [s]
Cab
le m
otio
n
0 5 10 15 20Time [s]
Riv
ulet
mot
ion
(intensive cable response)(intensive cable response) (weak cable response)
Static forces�with surface roughness�(RE)
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CD
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CL
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
Reynolds Number (*105)
Fluc
tuat
ing
win
d fo
rce
coef
ficie
nt CL
′
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
Reynolds Number (*105)
Fluc
tuat
ing
win
d fo
rce
coef
ficie
nt CL
′
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CD
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CL
Wind Velocity (m/s)0 3 6 9 12 15 18
β=0˚
CL CD CL’Decrease of drag
stationary liftDecay of Karman vortex intensity
Critical Re. number range
Increase of dragNon-app. of stationary lift
β=45˚
Super critical Re. number range
Flow field�with surface roughness�β=0 �̊(RE)
0 0.5 1 1.5 2-50-40-30-20-10
01020304050
Reynolds Number (*105)
Diff
eren
ce o
f win
d ve
loci
ty (%
)
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-50-40-30-20-10
01020304050
Reynolds Number (*105)
Diff
eren
ce o
f win
d ve
loci
ty (%
)
Wind Velocity (m/s)0 3 6 9 12 15 18
0 0.5 1 1.5 2-50-40-30-20-10
01020304050
Reynolds Number (*105)
Diff
eren
ce o
f win
d ve
loci
ty (%
)
Wind Velocity (m/s)0 3 6 9 12 15 18
wind
(a)(�) (�)
0 0.5 1 1.5 2-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Reynolds Number (*105)
Stea
dy w
ind
forc
e co
effic
ient
CL
Wind Velocity (m/s)0 3 6 9 12 15 18
(a)
(c)
(b�
Velocity difference at critical Re number range where Karman vortex is suppressed
Non-symmetry flow field
Appearance of Stationary lift
At critical Re. number range (RE) Decrease of drag
Velocity-difference btw up and down sides
Stationary lift
wind
Relative angle of attack due to vib.
Amplified velocity at down side
exciting force
Future Subjects
How to Control the Cable Vib.?
Mechanical Countermeasure ?Aerodynamic Countermeasure ?
How to encourage the 2D KarmanVortex?
Thank you for your
kind attention.
Quasi-Steady Theory for Inclined Cable with Fixed rivulet ?
Wind
Cable model(D=0.05m)
Wind tunnel wall
30% Perforated splitter plate (PSP)
Axial Flow Simulation by a Perforated Splitter Plate(30%) (QS)
Cross-flow Vibration affected by PSP (QS)
0 1 2 3 4 5 6 7 8 9 1011121314 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80 100 120
0
0.5
1
1.5f=2.096Hzm=0.509kg/mδ (2A=10mm)=0.00373Sc(2A=10mm)=1.138
0 1 2 3 4 5 6 7 8 9 1011121314 0
10
20
30
40
502A/ D 2A [mm]
V [m/ s]V/ fD0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
f=4.538Hzm=4.754kg/mδ(2A=10mm)=0.00216Sc(2A=10mm)=7.168
β=0°, with a 30% perforated splitter plateV-A diagram wavelet analysis of the lift force
β=45°, without a splitter plate V-A diagram wavelet analysis of the lift force
Probably PSP for non-yawed cable would be simulated to yawed cable aerodynamics from the point of axial flow effect. (QS)
Observed water rivulet movement on prototype Observed water rivulet movement on prototype scale cable model during rainscale cable model during rain--wind induced wind induced
vibrationvibration
V=10m/s
Observed water rivulet movement on prototype scale cable model during
rain-wind induced vibration
The Problems of the Stationary Lift and Drag Forces in Applying to
Quasi-steady Theory (QS)
Real inclined stay cables
Vθ
axial flowwater rivulet
y.
axial flowwater rivulet
Vrα
θwind
Aerostatic force coefficients test in wind tunnel
Vθ
axial flowwater rivulet
Vrα
y.
θ
axial flowwater rivulet
wind
Wind
Cable model(D=0.05m)
Wind tunnel wall
30% Perforated splitter plate (PSP)
Axial Flow Simulation by a Perforated Splitter Plate(30%) (QS)
Various situations ( Position of water rivulet, axial flow and PSP ) (QS)
PSP
water rivulet
Vθ
y.
wind θ PSP
water rivulet
Vr
α
(a) Wind tunnel tests situation, β=0°, with PSP
PSP
water rivulet
Vθ
y.
wind θPSP
water rivulet
Vr
α
(b) Wind tunnel tests situation, β=0°, with fixed PSP (ε=180°)
y.
water rivulet
Vθwind θ
water rivulet
Vr
α
(c) Wind tunnel tests situation, β=0°, without PSP
PSP
water rivulet
Vθ
y.
windθ-α PSP
water rivulet
Vr
α
(d) Wind tunnel tests situation, β=0°, with fixed water rivulet, with PSP
axial flowwater rivulet
Vθ
y.
wind θ
water rivulet
Vr
α
axial flow
(e) Wind tunnel tests situation, β=45°, without PSP
axial flowwater rivulet
Vθ
y.
wind θ
water rivulet
Vr
α
axial flow
(f) Actual situation
The Stationary Lift and Drag Coefficients (β=0°, in smooth flow, V=8.0m/s ) (QS)
0 20 40 60 80 100 120 140 160 180- 0.5- 0.4- 0.3- 0.2- 0.1
00.10.20.30.40.50.60.7
position of water rivulet θ [deg.]
Stea
dy li
ft fo
rce
coef
ficie
nt C
L
0 20 40 60 80 100 120 140 160 1800.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
position of water rivulet θ [deg.]St
eady
dra
g fo
rce
coef
ficie
nt C
D
::ε ε =176=176°° x:x:εε =178=178°° oo::ε ε =180=180°° ::εε =182=182°° +:+:εε =184=184°°**Position of PSP
Angle of Attack, Drag, Lift and Lateral Forces (QS)
D:drag force
L:lift forceF:lateral force
waterrivulet
V
Vr
wind θ
α=tan-1 (y/V).
dCF/dα diagrams under various situation (QS)
0 10 20 30 40 50 60 70 80 90 100 110120 130140 150160 170180- 4
- 2
0
2
4
position of water rivulet θ [deg.]
dCF/
dα
(a) Wind tunnel tests situation, (a) Wind tunnel tests situation, ββ=0=0°°, with PSP, with PSP(b) Wind tunnel tests situation, (b) Wind tunnel tests situation, ββ=0=0°°, with fixed PSP (, with fixed PSP (εε=180=180°°))(c) Wind tunnel tests situation, (c) Wind tunnel tests situation, ββ=0=0°°, without PSP, without PSP(d) Wind tunnel tests situation, (d) Wind tunnel tests situation, ββ=0=0°°, with fixed water rivulet, with PSP, with fixed water rivulet, with PSP(e) Wind tunnel tests situation, (e) Wind tunnel tests situation, ββ=45=45°°, without PSP, without PSP
0 20 40 60 80 100 120 140 160 180- 0.6- 0.4- 0.2
00.20.40.60.8
11.21.41.6
position of water rivulet θ [deg.]
Stea
dy w
ind
forc
e co
effic
ient
sCD
CL
Steady Wind Force Coefficients (QS)
o:o:ββ=0=0°°, without windows, with PSP, without windows, with PSP::ββ=0=0°°, without windows , without windows
x:x:ββ=45=45°°, with 110mm windows, with 110mm windows
dCF/dα Diagram (β=0°, with PSP) (QS)
0 10 20 30 40 50 60 70 80 90 100 110120 130140 150160 170180- 4
- 2
0
2
4
position of water rivulet θ [ �]
dCF/
dα
Galloping Galloping dCdCFF/d/dαα < < 00Quasi-steady theory
Quasi-steady Approach
Wind tunnel tests(ex. β=45° situation)
Using yawed cable model Using yawed cable model (QS)(QS)
Wind
Actual situation
RelativeWind
Wind
Quasi-steady Approach (QS)
Wind tunnel tests(Correct situation)Actual situation
RelativeWind
The non-linear analysis based on the quasi-steady theory (QS)
CF=A1+A2α+A3α2+A4α3+A5α4+A6α5+A7α6+A8α7
α= tan-1 (y/V)
=(y/V)-(1/3)(y/V)3+(1/5)(y/V)5-(1/7)(y/V)7
α : angle of attack
y : the cross-flow response velocity
Comparison of V-A diagrams ; wind tunnel tests (β=45°) and the quasi-steady analysis
(CF:β=0°,with PSP) (QS)
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
(a) without water rivulet
f=2.096Hzm=0.509kg/mδ (2A=10mm)=0.00373Sc(2A=10mm)=1.138
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
(b) θ=0°
f=2.076Hzm=0.523kg/mδ (2A=10mm)=0.00339Sc(2A=10mm)=1.046
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
(d) θ=66°
f=2.085Hzm=0.506kg/mδ (2A=10mm)=0.00369Sc(2A=10mm)=1.107
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
(f) θ=180°
f=2.075Hzm=0.523kg/mδ (2A=10mm)=0.00348Sc(2A=10mm)=1.079
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
(c) θ=56°
f=2.089Hzm=0.588kg/mδ (2A=10mm)=0.00336Sc(2A=10mm)=1.178
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5f=2.081Hzm=0.516kg/mδ (2A=10mm)=0.00373Sc(2A=10mm)=1.148
(e) θ=72°
OO : wind tunnel : wind tunnel test (test (ββ=45=45°°))
OO : quasi : quasi –– steady steady analysis analysis ((CCF:F:ββ=0=0°°,,withPSPwithPSP))
X : limit cycleX : limit cycle
ΔΔ: : VrVrcrcr
(Parkinson)(Parkinson)
: : VrVrcrcr
(Den Hartag)(Den Hartag)**
0 2 4 6 8 10 12 14 16 18 20 22 24 0
20
40
60
80
100
1202A/ D2A [mm]
V [m/ s]
V/ fD0 100 200 300 400 500
0
0.5
1
1.5
2
e) Sc=248
f=0.833Hzm=1.894kg/mδ(2A=10mm)=0.231Sc(2A=10mm)=248
0 1 2 3 4 5 6 7 8 9 10 0
20
40
60
80
100
1202A/ D 2A [mm]
V [m/ s]
V/ fD0 50 100 150 200
0
0.5
1
1.5
2
b) Sc=46.2
f=0.827Hzm=1.888kg/mδ(2A=10mm)=0.0431Sc(2A=10mm)=46.2
0 1 2 3 4 5 6 7 8 9 1011121314 0
20
40
60
80
100
1202A/ D 2A [mm]
V [m/ s]
V/ fD0 50 100 150 200 250 300
0
0.5
1
1.5
2
c) Sc=125
f=0.828Hzm=1.867kg/mδ (2A=10mm)=0.117Sc(2A=10mm)=125
0 2 4 6 8 10 12 14 16 0
20
40
60
80
100
1202A/ D2A [mm]
V [m/ s]
V/ fD0 100 200 300
0
0.5
1
1.5
2
d) Sc=169
f=0.825Hzm=1.903kg/mδ(2Α=10μμ)=0.155Sc(2A=10mm)=169
0 1 2 3 4 5 6 7 8 9 10 0
20
40
60
80
100
1202A/ D 2A [mm]
V [m/ s]
V/ fD0 50 100 150 200
0
0.5
1
1.5
2
a) Sc=12.3
f=0.826Hzm=1.922kg/mδ(2Α=10μμ)=0.0113Sc(2A=10mm)=12.3
Comparison of V-A diagrams ; wind tunnel tests (β=45°,without rivulet) and the quasi-
steady analysis (CF:β=0°,with PSP) (QS)
OO : wind tunnel test : wind tunnel test
((ββ=45=45°°))
OO : quasi: quasi--steady analysissteady analysis
((CCFF::ββ=0=0°°,,with PSPwith PSP))
Comparison of V-A diagrams ; wind tunnel tests (β=45°) and the quasi-steady
analysis (CF:β=45°,without PSP) (QS)
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
f=2.085Hzm=0.506kg/mδ(2A=10mm)=0.00369Sc(2A=10mm)=1.107
(b) θ=66°
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5f=2.089Hzm=0.588kg/mδ(2A=10mm)=0.00336Sc(2A=10mm)=1.178
(a) θ=56°
0 1 2 3 4 5 6 7 8 9 10 11 0
20
40
60
80
1002A/ D 2A [mm]
V [m/ s]
V/ fD0 20 40 60 80
0
0.5
1
1.5
f=2.081Hzm=0.516kg/mδ(2A=10mm)=0.00373Sc(2A=10mm)=1.148
(c) θ=72°
OO : wind tunnel test: wind tunnel test ((ββ=45=45°°))
OO : quasi: quasi--steady analysis (steady analysis (CCFF::ββ=45=45°°,,without PSPwithout PSP))
ΔΔ : : VrVrcr cr (Parkinson)(Parkinson) , : : VrVrcr cr (Den (Den HartagHartag))**
Quasi-Steady Theory (QS)
Still we have to discuss more in application of Quasi-Steady Theory.