13.42 lecture 2: vortex induced vibrations - mitweb.mit.edu/13.42/www/handouts/viv-2.pdf · viv in...
TRANSCRIPT
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13.42 Lecture 2:Vortex Induced
Vibrations
Prof. A. H. Techet28 April 2005
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• Shedding patterns in the wake of oscillating cylinders are distinct and exist for a certain range of heave frequencies and amplitudes.
• The different modes have a great impact on structural loading.
Wake Patterns Behind Heaving Cylinders
‘2S’ ‘2P’f , A
f , A
UU
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Transition in Shedding Patterns
Will
iam
son
and
Ros
hko
(198
8)
A/d
f* = fd/UVr = U/fd
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End Force Cross-Correlation
Uniform Cylinder
Tapered Cylinder
Hov
er, T
eche
t, Tr
iant
afyl
lou
(JFM
199
8)
Fc = Force Correlation Coefficient, µ = correlation standard deviation
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VIV in the Ocean• Non-uniform currents
effect the spanwise vortex shedding on a cable or riser.
• The frequency of shedding can be different along length.
• This leads to “cells” of vortex shedding with some length, lc.
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Three Dimensional Effects
Shear layer instabilities as well as longitudinal (braid) vortices lead to transition from laminar to turbulent flow in cylinder wakes.
Longitudinal vortices appear at Rd = 230.
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Longitudinal Vortices
C.H.K. Williamson (1992)
The presence of longitudinal vortices leads to rapid breakdown of the wake behind a cylinder.
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Strouhal Number for the tapered cylinder:
St = fd / U
where d is the average cylinder diameter.
Oscillating Tapered Cylinder
x
d(x)U(x
) = U
o
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Spanwise Vortex Shedding from 40:1 Tapered Cylinder
Tech
et, e
t al (
JFM
199
8)
dmax
Rd = 400; St = 0.198; A/d = 0.5
Rd = 1500; St = 0.198; A/d = 0.5
Rd = 1500; St = 0.198; A/d = 1.0
dminNo Split: ‘2P’
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Flow Visualization Reveals: A Hybrid Shedding Mode
• ‘2P’ pattern results at the smaller end
• ‘2S’ pattern at the larger end
• This mode is seen to be repeatable over multiple cycles
Techet, et al (JFM 1998)
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DPIV of Tapered Cylinder Wake
‘2S’
‘2P’
Digital particle image velocimetry (DPIV) in the horizontal plane leads to a clear picture of two distinct shedding modes along the cylinder.
Rd = 1500; St = 0.198; A/d = 0.5
z/d
= 22
.9z/
d =
7.9
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Evolution of the Hybrid Shedding Mode
‘2P’ ‘2S’
Rd = 1500; St = 0.198; A/d = 0.5
z/d = 22.9z/d = 7.9
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Evolution of the Hybrid Shedding Mode
‘2S’‘2P’
Rd = 1500; St = 0.198; A/d = 0.5
z/d = 22.9z/d = 7.9
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Evolution of the Hybrid Shedding Mode
‘2S’‘2P’
Rd = 1500; St = 0.198; A/d = 0.5
z/d = 22.9z/d = 7.9
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Evolution of the Hybrid Shedding Mode
‘2S’‘2P’
Rd = 1500; St = 0.198; A/d = 0.5
z/d = 22.9z/d = 7.9
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NEKTAR-ALE Simulations
Objectives:• Confirm numerically the existence of a stable,
periodic hybrid shedding mode 2S~2P in the wake of a straight, rigid, oscillating cylinder
Principal Investigator:• Prof. George Em Karniadakis, Division of Applied
Mathematics, Brown University
Approach:• DNS - Similar conditions as the MIT experiment
(Triantafyllou et al.)• Harmonically forced oscillating straight rigid
cylinder in linear shear inflow• Average Reynolds number is 400
Vortex Dislocations, Vortex Splits & Force Distribution in Flows past Bluff Bodies
D. Lucor & G. E. Karniadakis
Results:• Existence and periodicity of hybrid mode
confirmed by near wake visualizations and spectral analysis of flow velocity in the cylinder wake and of hydrodynamic forces
Methodology:• Parallel simulations using spectral/hp methods
implemented in the incompressible Navier- Stokes solver NEKTAR
VORTEX SPLIT
Techet, Hover and Triantafyllou (JFM 1998)
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VIV Suppression•Helical strake
•Shroud
•Axial slats
•Streamlined fairing
•Splitter plate
•Ribboned cable
•Pivoted guiding vane
•Spoiler plates
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VIV Suppression by Helical Strakes
Helical strakes are a common VIV suppressiondevice.
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Flexible Cylinders
Mooring lines and towing cables act in similar fashion to rigid cylinders except that their motion is not spanwise uniform.
t
Tension in the cable must be considered when determining equations of motion
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Flexible Cylinder Motion Trajectories
Long flexible cylinders can move in two directions and tend to trace a figure-8 motion. The motion is dictated bythe tension in the cable and the speed of towing.
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Nominal Reduced Velocity, Vrn
Freq
uenc
y R
atio
, fx/f
y
2-DOF Cylinder Motion CYLINDER ORBITAL PLOTS
Rigid cylinder free to move in both transverse (y) and longitudinal (x) directions. High length:diameter ratio.
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FFT Magnitude Contours For Frequency Ratio fx/fy = 1.0
Transverse Position In-Line Position
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FFT Magnitude Contours For Frequency Ratio fx/fy = 2.0
Transverse Position In-Line Position
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Power Balance Along a Riser
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Overview & Aim• Obtain power balance along a riser
– We need power input from the fluid to the riser as a function of z.
– Input:
– Output:
– Power balance is correlated with Clv(z).
212
Cross-flow displacement: Lift force:
Normalized Lift force:
Incoming Velocity Profile: ( )
y(z,t)L(z,t)
L(z,t)f(z,t) dsU
U zρ
=
m( , ) & ( )mlv lvC z C zω
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Fourier Decomposition• Obtain the Fourier decomposition of force and
displacement as:
• Fourier Coefficients
1
1
ˆRe
ˆRe
n
n
Ni t
nn
Ni t
nn
y(z,t) y e
f(z,t) f e
ω
ω
=
=
⎡ ⎤= ⎢ ⎥
⎢ ⎥⎣ ⎦
⎡ ⎤= ⎢ ⎥
⎢ ⎥⎣ ⎦
∑
∑
ˆ ˆˆ ˆ ( , ) & ( , )n n n n n ny y z f f zω ω= =
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Choose a Set of Important Frequencies
• Choose the set of frequencies ωm which represents– peaks in the span-averaged Fourier coefficients:
ˆˆ ( , ) & ( , )m m m my z f zω ω
ˆˆ ,m my f
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Obtaining Global Clv• Obtain the multi-frequency lift coefficients in phase with velocity and acceleration as:
• Global lift coefficient in phase with velocity:
• Clv is very important as it expresses energy balance.
• Clv*U(z)2 is representative of the power input from the fluid to the riser.
( ) ( )( )
( ) ( )( )
m
m m
m
m
Lift coefficient in phase with velocity for :
ˆ ˆ ˆ( , ) ( , ) sin arg arg
Lift coefficient in phase with acclera tion for :
ˆ ˆ ˆ( , ) - ( , ) cos arg arg
m
m
lv m m m
la m m m m
C z f z f y
C z f z f y
ω
ω ω
ω
ω ω
= −
= −
m
2m
ˆ
( )ˆ
mlv mm
lv
mm
C yC z
y
ω
ω
⎛ ⎞⎜ ⎟⎜ ⎟= − ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
∑
∑
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Input Sheared Velocity Profile
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Selected Frequencies
• Chosen set of frequencies ωm :
ˆˆ ,m my f
( )2 0.1733 0 .1831 0 .1929 0 .2026 0 .2124 0 .2222 0 .2295 0 .2393mω π=
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Global Clv & Cm
• Clv*U(z)2 is representative of the power input from the fluid to the riser.
• One can clearly observe that there is a positive input of power from the fluid where the incoming velocity is high.
• The power is transferred to the low incoming velocity region, as traveling waves in the riser.
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Oscillating Cylinders
d
y(t)
y(t) = a cos ωt
Parameters:Re = Vm d / ν
Vm = a ω
y(t) = -aω sin(ωt).
b = d2 / νT
KC = Vm T / d
St = fv d / Vm
ν = µ/ ρ ; Τ = 2π/ω
Reducedfrequency
Keulegan-Carpenter #
Strouhal #
Reynolds #
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Reynolds # vs. KC #
b = d2 / νT
KC = Vm T / d = 2π a/d
Re = Vm d / ν = ωad/ν = 2π a/d d /νΤ
2)(( )
Re = KC * b
Also effected by roughness and ambient turbulence
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Forced Oscillation in a Current
θ
U
y(t) = a cos ωtω = 2 π f = 2π / T
Parameters: a/d, ρ, ν, θ
Reduced velocity: Ur = U/fd
Max. Velocity: Vm = U + aω cos θ
Reynolds #: Re = Vm d / ν
Roughness and ambient turbulence
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Wall Proximity
e + d/2
At e/d > 1 the wall effects are reduced.
Cd, Cm increase as e/d < 0.5
Vortex shedding is significantly effected by the wall presence.
In the absence of viscosity these effects are effectively non-existent.
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GallopingGalloping is a result of a wake instability.
m
y(t), y(t).Y(t)
U-y(t).
Vα
Resultant velocity is a combination of the heave velocity and horizontal inflow.
If ωn << 2π fv then the wake is quasi-static.
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Lift Force, Y(α) Y(t)
Vα
Cy = Y(t)
1/2 ρ U2 Ap
α
Cy Stable
Unstable
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Galloping motion
m
z(t), z(t).L(t)
U-z(t).
Vα
b k
a mz + bz + kz = L(t).. .
L(t) = 1/2 ρ U2 a Clv - ma y(t)..
Cl(α) = Cl(0) + ∂ Cl (0)∂ α
+ ...
Assuming small angles, α:
α ~ tan α = - zU
.β = ∂ Cl (0)
∂ α V ~ U
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Instability Criterion
(m+ma)z + (b + 1/2 ρ U2 a )z + kz = 0.. .βU
~
b + 1/2 ρ U2 a βU < 0If
Then the motion is unstable!This is the criterion for galloping.
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β is shape dependent
U
1
1
1
2
12
14
Shape ∂ Cl (0)∂ α
-2.7
0
-3.0
-10
-0.66
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b1/2 ρ a ( )
Instability:
β = ∂ Cl (0)∂ α
<b
1/2 ρ U a
Critical speed for galloping:
U > ∂ Cl (0)∂ α
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Torsional Galloping
Both torsional and lateral galloping are possible.FLUTTER occurs when the frequency of the torsional
and lateral vibrations are very close.
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Galloping vs. VIV
• Galloping is low frequency
• Galloping is NOT self-limiting
• Once U > Ucritical then the instability occurs irregardless of frequencies.
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References
• Blevins, (1990) Flow Induced Vibrations, Krieger Publishing Co., Florida.