13.42 Lecture: Vortex Induced Vibrations

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13.42 LectureVortex Induced Vibrations

• Prof. A. H. Techet
• 18 March 2004

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Classic VIV Catastrophe
If ignored, these vibrations can prove
catastrophic to structures, as they did in the
case of the Tacoma Narrows Bridge in 1940.
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Potential Flow
U(q) 2U? sinq
P(q) 1/2 r U(q)2 P? 1/2 r U?2
Cp P(q) - P ?/1/2 r U?2 1 - 4sin2q
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Axial Pressure Force
(i)
(ii)
Base pressure
i) Potential flow -p/w lt q lt p/2 ii) P
PB p/2 ? q ? 3p/2 (for LAMINAR flow)
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Reynolds Number Dependency
Rd lt 5
5-15 lt Rd lt 40
40 lt Rd lt 150
150 lt Rd lt 300
Transition to turbulence
300 lt Rd lt 3105
3105 lt Rd lt 3.5106
3.5106 lt Rd
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Shear layer instability causes vortex roll-up

• Flow speed outside wake is much higher than
  inside
• Vorticity gathers at downcrossing points in upper
  layer
• Vorticity gathers at upcrossings in lower layer
• Induced velocities (due to vortices) causes this
  perturbation to amplify

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Wake Instability
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Classical Vortex Shedding
l
h
Von Karman Vortex Street
Alternately shed opposite signed vortices
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Vortex shedding dictated by the Strouhal number
Stfsd/U
fs is the shedding frequency, d is diameter and U
inflow speed
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Additional VIV Parameters

• Reynolds Number
• subcritical (Relt105) (laminar boundary)
• Reduced Velocity
• Vortex Shedding Frequency
• S?0.2 for subcritical flow

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Strouhal Number vs. Reynolds Number
St 0.2
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Vortex Shedding Generates forces on Cylinder
Uo
Both Lift and Drag forces persist on a cylinder
in cross flow. Lift is perpendicular to the
inflow velocity and drag is parallel.
FL(t)
FD(t)
Due to the alternating vortex wake (Karman
street) the oscillations in lift force occur at
the vortex shedding frequency and oscillations in
drag force occur at twice the vortex shedding
frequency.
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Vortex Induced Forces
Due to unsteady flow, forces, X(t) and Y(t), vary
with time.
Force coefficients
Cx
Cy
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Force Time Trace
DRAG
Cx
Avg. Drag ? 0
LIFT
Cy
Avg. Lift 0
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Alternate Vortex shedding causes oscillatory
forces which induce structural vibrations
Heave Motion z(t)
LIFT L(t) Lo cos (wst?)
DRAG D(t) Do cos (2wst ?)
Rigid cylinder is now similar to a spring-mass
system with a harmonic forcing term.
ws 2p fs
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Lock-in
A cylinder is said to be locked in when the
frequency of oscillation is equal to the
frequency of vortex shedding. In this region the
largest amplitude oscillations occur.
wv 2p fv 2p St (U/d)
Shedding frequency
Natural frequency of oscillation
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Equation of Cylinder Heave due to Vortex shedding
z(t)
m
b
k
Added mass term
Restoring force
If Lv gt b system is UNSTABLE
Damping
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Lift Force on a Cylinder
Lift force is sinusoidal component and residual
force. Filtering the recorded lift data will give
the sinusoidal term which can be subtracted from
the total force.
LIFT FORCE
where wv is the frequency of vortex shedding
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Lift Force Components
Two components of lift can be analyzed
Lift in phase with acceleration (added mass)
Lift in-phase with velocity
Total lift
(a zo is cylinder heave amplitude)
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Total Force

• If CLv gt 0 then the fluid force amplifies the
  motion instead of opposing it. This is
  self-excited oscillation.
• Cma, CLv are dependent on w and a.

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Coefficient of Lift in Phase with Velocity
Vortex Induced Vibrations are SELF LIMITED
In air rair small, zmax 0.2 diameter In
water rwater large, zmax 1 diameter
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Lift in phase with velocity
Gopalkrishnan (1993)
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Amplitude Estimation
Blevins (1990)
_
_
2m (2pz) r d2


fn fn/fs m m ma
SG2 p fn2
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Drag Amplification
VIV tends to increase the effective drag
coefficient. This increase has been investigated
experimentally.
Gopalkrishnan (1993)
Fluctuating Drag
Mean drag

Cd 1.2 1.1(a/d)
Cd occurs at twice the shedding frequency.
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Single Rigid Cylinder Results
1.0

1. One-tenth highest transverse oscillation
   amplitude ratio
2. Mean drag coefficient
3. Fluctuating drag coefficient
4. Ratio of transverse oscillation frequency to
   natural frequency of cylinder

1.0
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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 by the tension in the
cable and the speed of towing.
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Wake Patterns Behind Heaving Cylinders

• 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.

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Transition in Shedding Patterns
A/d
Williamson and Roshko (1988)
f fd/U
Vr U/fd
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Formation of 2P shedding pattern
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End Force Correlation
Uniform Cylinder
Hover, Techet, Triantafyllou (JFM 1998)
Tapered Cylinder
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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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Oscillating Tapered Cylinder

• Strouhal Number for the tapered cylinder
• St fd / U
• where d is the average
• cylinder diameter.

U(x) Uo
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Spanwise Vortex Shedding from 401 Tapered
Cylinder
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
dmax
Techet, et al (JFM 1998)
No Split 2P
dmin
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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
Digital particle image velocimetry (DPIV) in the
horizontal plane leads to a clear picture of two
distinct shedding modes along the cylinder.
2S
z/d 22.9
2P
z/d 7.9
Rd 1500 St 0.198 A/d 0.5
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Vortex Dislocations, Vortex Splits Force
Distribution in Flows past Bluff BodiesD. Lucor
G. E. Karniadakis

• Objectives
• Confirm numerically the existence of a stable,
  periodic hybrid shedding mode 2S2P in the wake
  of a straight, rigid, oscillating cylinder

• 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 SPLIT

• Methodology
• Parallel simulations using spectral/hp methods
  implemented in the incompressible Navier- Stokes
  solver NEKTAR

NEKTAR-ALE Simulations

• 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

• Principal Investigator
• Prof. George Em Karniadakis, Division of Applied
  Mathematics, Brown University

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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
suppresion device.
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Oscillating Cylinders
Parameters
y(t)
Re Vm d / n
Reynolds
d
b d2 / nT
Reduced frequency
y(t) a cos wt
Keulegan- Carpenter
KC Vm T / d
Vm a w
St fv d / Vm
Strouhal
n m/ r T 2p/w
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Reynolds vs. KC
KC Vm T / d 2p a/d
Re KC b
b d2 / nT
Also effected by roughness and ambient turbulence
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Forced Oscillation in a Current
y(t) a cos wt
q
w 2 p f 2p / T
U
Reduced velocity Ur U/fd
Max. Velocity Vm U aw cos q
Reynolds Re Vm d / n
Roughness and ambient turbulence
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Wall Proximity
e d/2
At e/d gt 1 the wall effects are reduced. Cd, Cm
increase as e/d lt 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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Galloping
Galloping is a result of a wake instability.
Resultant velocity is a combination of the heave
velocity and horizontal inflow.
If wn ltlt 2p fv then the wake is quasi-static.
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Lift Force, Y(a)
Y(t)
V
a
Cy
Stable
Cy
a
Unstable
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Galloping motion
a
b
k
Cl(a) Cl(0)
...
Assuming small angles, a
V U
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Instability Criterion
..
.

(mma)z (b 1/2 r U2 a )z kz 0
b 1/2 r U2 a
lt 0
If
Then the motion is unstable! This is the
criterion for galloping.
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b is shape dependent
Shape
-2.7
0
U
-3.0
-10
-0.66
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Instability
lt
b
Critical speed for galloping
b 1/2 r a
U gt
( )
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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 gt Ucritical then the instability occurs
  irregardless of frequencies.

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References

• Blevins, (1990) Flow Induced Vibrations, Krieger
  Publishing Co., Florida.