13.42 Lecture:
Vortex Induced Vibrations
Prof. A. H. Techet
18 March 2004
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.
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
Axial Pressure Force
Base
pressure
(i)
(ii)
i) Potential flow:
-p/w < q < p/2
ii) P ~ PB
p/2  q  3p/2
(for LAMINAR flow)
Reynolds Number Dependency
Rd < 5
5-15 < Rd < 40
40 < Rd < 150
150 < Rd < 300
Transition to turbulence
300 < Rd < 3*105
3*105 < Rd < 3.5*106
3.5*106 < Rd
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
Wake Instability
Classical Vortex Shedding
l
h
Von Karman Vortex Street
Alternately shed opposite signed vortices
Vortex shedding dictated by
the Strouhal number
St=fsd/U
fs is the shedding frequency, d is diameter and U inflow speed
Additional VIV Parameters
• Reynolds Number
UD inertial effects
Re 

v
viscous effects
– subcritical (Re<105) (laminar boundary)
• Reduced Velocity
U
Vrn 
fn D
• Vortex Shedding Frequency
SU
fs 
D
– S0.2 for subcritical flow
Strouhal Number vs. Reynolds
Number
St = 0.2
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.
Vortex Induced Forces
Due to unsteady flow, forces, X(t) and Y(t), vary with time.
Force coefficients:
Cx =
D(t)
1/
2r
U2
d
Cy =
L(t)
1/
2r
U2 d
Force Time Trace
DRAG
Cx
Avg. Drag ≠ 0
LIFT
Cy
Avg. Lift = 0
Alternate Vortex shedding causes
oscillatory forces which induce
structural vibrations
Heave Motion z(t)
z (t )  zo cos wt
z (t )   zow sin wt
z (t )   zow 2 cos wt
LIFT = L(t) = Lo cos (wst+)
Rigid cylinder is now similar
to a spring-mass system with
a harmonic forcing term.
DRAG = D(t) = Do cos (2wst+ )
ws = 2p fs
“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.
Shedding
frequency
Natural frequency
of oscillation
wv = 2p fv = 2p St (U/d)
wn = m +k m
a
Equation of Cylinder Heave due
to Vortex shedding
mz  bz  kz  L(t )
z(t)
L(t )   La z (t )  Lv z (t )
m
mz (t )  bz (t )  kz (t )   La z (t )  Lv z (t )
k
b
(m  La ) z (t )  (b  Lv ) z (t )  k z (t )  0
Added mass term
Restoring force
Damping
If Lv > b system is
UNSTABLE
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: L(t )  Lo cos(wt  o )
if w  wv
L(t )  Lo cos wt cos o  Lo sin wt sin o
L(t ) 
 Lo cos o
Lo sin o
z
(
t
)

z (t )
2
zow
zow
where wv is the frequency of vortex shedding
Lift Force Components:
Two components of lift can be analyzed:
Lift in phase with acceleration (added mass):
M a (w , a) 
Lo
2 cos o
aw
Lift in-phase with velocity:
Lv  
Lo
sin o
aw
Total lift:
L(t )  M a (w, a) z (t )  Lv (w, a) z (t )
(a = zo is cylinder heave amplitude)
Total Force:
L(t )  M a (w, a) z (t )  Lv (w, a) z (t )



p
2
r
d
 Cma (w, a) z (t )
4
1
r dU  CLv (w , a) z (t )
2
2
• If CLv > 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.
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
Lift in phase with velocity
Gopalkrishnan (1993)
Amplitude Estimation
Blevins (1990)
a/ =~ 1.29/[1+0.43 S ]3.35
G
d
_
_
^ 2 2m (2pz ; f^ = f /f ; m
*
=
m
+
m
SG=2 p fn
n
n
s
a
r d2
b
z=
2 k(m+ma*)
ma* = r V Cma; where Cma = 1.0
Drag Amplification
VIV tends to increase the effective drag coefficient. This increase
has been investigated experimentally.
~
Cd
|Cd|
Gopalkrishnan (1993)
3
a
= 0.75
d
2
1
0.1
0.2
0.3
fd
U
Fluctuating Drag:
Mean drag:
Cd = 1.2 + 1.1(a/d)
~
Cd occurs at twice the
shedding frequency.
Single Rigid Cylinder Results
1.0
a)
One-tenth highest
transverse
oscillation amplitude
ratio
b)
Mean drag
coefficient
c)
Fluctuating drag
coefficient
d)
Ratio of transverse
oscillation frequency
to natural frequency
of cylinder
1.0
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
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.
Wake Patterns Behind
Heaving Cylinders
f,A
U
f,A
‘2S’
U
‘2P’
• 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.
Transition in Shedding Patterns
Vr = U/fd
f* = fd/U
Williamson and Roshko (1988)
A/d
Formation of ‘2P’ shedding pattern
End Force Correlation
Hover, Techet, Triantafyllou (JFM 1998)
Uniform Cylinder
Tapered Cylinder
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.
Oscillating Tapered Cylinder
U(x) = Uo
x
Strouhal Number for the tapered
cylinder:
St = fd / U
d(x)
where d is the average
cylinder diameter.
Spanwise Vortex Shedding from
40:1 Tapered Cylinder
Techet, et al (JFM 1998)
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
No Split: ‘2P’
dmax
dmin
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)
z/d = 7.9
z/d = 22.9
DPIV of Tapered Cylinder Wake
Digital particle image
‘2S’ velocimetry (DPIV)
in the horizontal plane
leads to a clear
picture of two distinct
shedding modes along
‘2P’
the cylinder.
Rd = 1500; St = 0.198; A/d = 0.5
Vortex Dislocations, Vortex Splits & Force
Distribution in Flows past Bluff Bodies
D. Lucor & G. E. Karniadakis
Techet, Hover and Triantafyllou (JFM 1998)
Objectives:
•
Confirm numerically the existence of a stable,
periodic hybrid shedding mode 2S~2P in the
wake of a straight, rigid, oscillating cylinder
Approach:
•
VORTEX SPLIT
•
•
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
Methodology:
•
NEKTAR-ALE Simulations
Parallel simulations using spectral/hp methods
implemented in the incompressible Navier- Stokes
solver NEKTAR
Principal Investigator:
Results:
•
•
Prof. George Em Karniadakis, Division of Applied
Mathematics, Brown University
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
VIV Suppression
•Helical strake
•Shroud
•Axial slats
•Streamlined fairing
•Splitter plate
•Ribboned cable
•Pivoted guiding vane
•Spoiler plates
VIV Suppression by Helical Strakes
Helical strakes are a
common VIV suppresion
device.
Oscillating Cylinders
Parameters:
y(t)
d
y(t) = a cos wt
.
y(t) = -aw sin(wt)
Vm = a w
n  m/ r ; T  2p/w
Re = Vm d / n
b=
d2
/ nT
Reynolds #
Reduced
frequency
KC = Vm T / d
KeuleganCarpenter #
St = fv d / Vm
Strouhal #
Reynolds # vs. KC #
Re = Vm d / n 
wad/
n
 2p
a/
d 2/
( d)(
nT
)
KC = Vm T / d = 2p a/d
Re = KC * b
b = d2
/ nT
Also effected by roughness and ambient turbulence
Forced Oscillation in a Current
y(t) = a cos wt
q
w = 2 p f = 2p / T
U
Parameters: a/d, r, n, q
Reduced velocity: Ur = U/fd
Max. Velocity: Vm = U + aw cos q
Reynolds #: Re = Vm d / n
Roughness and ambient turbulence
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.
Galloping
Galloping is a result of a wake instability.
Y(t) .
y(t), y(t)
U
a
V
.
-y(t)
m
Resultant velocity is a combination of the
heave velocity and horizontal inflow.
If wn << 2p fv then the wake is quasi-static.
Lift Force, Y(a)
Y(t)
V
a
Cy =
Y(t)
1/
2r
U2
Ap
Cy
Stable
a
Unstable
Galloping motion
L(t) .
z(t), z(t)
U
a
V
.
-z(t)
a
m
b
k
..
.
mz + bz + kz = L(t)
L(t) =
1/
2
r
U2
 Cl (0)
Cl(a) = Cl(0) +
+ ...
a
Assuming small angles, a:
.
z
 Cl (0)
a ~ tan a = b=
U
a
..
a Clv - ma y(t)
V~U
Instability Criterion
..
(m+ma)z + (b +
1/
2
r
U2
b+
1/
2
r
U2
If
a
b
U
a
b
<0
U
.
)z + kz =~ 0
Then the motion is unstable!
This is the criterion for galloping.
b is shape dependent
 Cl (0)
a
Shape
1
1
-2.7
1
0
2
U
2
-3.0
1
4
1
-10
-0.66
Instability:
b=
b
 Cl (0)
< 1/ r U a
a
2
Critical speed for galloping:
b
U >
1/
2
r a
(
 Cl (0)
a
)
Torsional Galloping
Both torsional and lateral galloping are possible.
FLUTTER occurs when the frequency of the torsional
and lateral vibrations are very close.
Galloping vs. VIV
• Galloping is low frequency
• Galloping is NOT self-limiting
• Once U > Ucritical then the instability occurs
irregardless of frequencies.
References
• Blevins, (1990) Flow Induced Vibrations,
Krieger Publishing Co., Florida.