13.42 lecture: vortex induced vibrations
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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 sin q. - PowerPoint PPT PresentationTRANSCRIPT
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() = 2U sin
P() = 1/2 U()2 = P + 1/2 U2
Cp = {P() - P }/{1/2 U2}= 1 - 4sin2
Axial Pressure Force
i) Potential flow:
-/w < < /2
ii) P ~ PB
/2 3/2
(for LAMINAR flow)
Base pressure
(i) (ii)
Reynolds Number DependencyRd < 5
5-15 < Rd < 40
40 < Rd < 150
150 < Rd < 300
300 < Rd < 3*105
3*105 < Rd < 3.5*106
3.5*106 < Rd
Transition to turbulence
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
Von Karman Vortex Street
lh
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
• Reynolds Number
– subcritical (Re<105) (laminar boundary)
• Reduced Velocity
• Vortex Shedding Frequency
– S0.2 for subcritical flow
Additional VIV Parameters
D
SUfs
effects viscous
effects inertialRe
v
UD
Df
UV
nrn
Strouhal Number vs. Reynolds Number
St = 0.2
Vortex Shedding Generates forces on Cylinder
FD(t)
FL(t)
Uo Both Lift and Drag forces persist on a cylinder in cross flow. Lift is perpendicular to the inflow velocity and drag is parallel.
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 = Cy = D(t)
1/2 U2 d
L(t)1/2 U2 d
Force Time Trace
Cx
Cy
DRAG
LIFT
Avg. Drag ≠ 0
Avg. Lift = 0
Alternate Vortex shedding causes oscillatory forces which induce
structural vibrations
Rigid cylinder is now similar to a spring-mass system with a harmonic forcing term.
LIFT = L(t) = Lo cos (st+)
s = 2 fs
DRAG = D(t) = Do cos (2st+ )
Heave Motion z(t)
2
( ) cos
( ) sin
( ) cos
o
o
o
z t z t
z t z t
z t z t
“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.
v = 2fv = 2St (U/d)
n = km + ma
Shedding frequency
Natural frequencyof oscillation
Equation of Cylinder Heave due to Vortex shedding
Added mass term
Damping If Lv > b system is UNSTABLE
k b
m
z(t)( )mz bz kz L t
( ) ( ) ( )a vL t L z t L z t
( ) ( ) ( ) ( ) ( )a vmz t bz t kz t L z t L z t
( ) ( ) ( ) ( ) ( ) 0a vm L z t b L z t k z t
Restoring force
LIFT FORCE:
Lift Force on a Cylinder
( ) cos( )o oL t L t vif
( ) cos cos sin sino o o oL t L t L t
2
cos sin( ) ( ) ( )o o o o
o o
L LL t z t z t
z z
where v is the frequency of vortex shedding
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 Components:
Lift in phase with acceleration (added mass):
Lift in-phase with velocity:
2( , ) cosoa o
LM a
a
sinov o
LL
a
Two components of lift can be analyzed:
(a = zo is cylinder heave amplitude)
Total lift:
( ) (( , () () , ))a vL t z t L aM za t
Total Force:
• If CLv > 0 then the fluid force amplifies the motion instead of opposing it. This is self-excited oscillation.
• Cma, CLv are dependent on and a.
( ) (( , () () , ))a vL t z t L aM za t
24
212
( , ) ( )
( , ) ( )
ma
Lv
d C a z t
dU C a z t
Coefficient of Lift in Phase with Velocity
Vortex Induced Vibrations are
SELF LIMITED
In air: air ~ small, zmax ~ 0.2 diameter
In water: water ~ large, zmax ~ 1 diameter
Lift in phase with velocity
Gopalkrishnan (1993)
Amplitude Estimation
= b
2 k(m+ma*)
ma* = V Cma; where Cma = 1.0
Blevins (1990)
a/d = 1.29/[1+0.43 SG]3.35~
SG=2 fn2
2m (2d2
; fn = fn/fs; m = m + ma*
^^ __
Drag Amplification
Gopalkrishnan (1993)
Cd = 1.2 + 1.1(a/d)
VIV tends to increase the effective drag coefficient. This increase has been investigated experimentally.
Mean drag: Fluctuating Drag:
Cd occurs at twice the shedding frequency.
~
3
2
1
Cd |Cd|~
0.1 0.2 0.3
fdU
ad = 0.75
Single Rigid Cylinder Results
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
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 bythe tension in the cable and the speed of towing.
• 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
Transition in Shedding Patterns
Wil
liam
son
and
Ros
hko
(198
8)
A/d
f* = fd/UVr = U/fd
Formation of ‘2P’ shedding pattern
End Force Correlation
Uniform Cylinder
Tapered Cylinder
Hov
er, T
eche
t, T
rian
tafy
llou
(JF
M 1
998)
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.
Strouhal Number for the tapered cylinder:
St = fd / U
where d is the average cylinder diameter.
Oscillating Tapered Cylinder
x
d(x)
U(x
) =
Uo
Spanwise Vortex Shedding from 40:1 Tapered Cylinder
Tec
het,
et a
l (JF
M 1
998)
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’
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)
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
= 2
2.9
z/d
= 7
.9
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)
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 suppresiondevice.
Oscillating Cylinders
d
y(t)
y(t) = a cos t
Parameters:Re = Vm d /
Vm = a
y(t) = -a sin(t).
b = d2 / T
KC = Vm T / d
St = fv d / Vm
Reducedfrequency
Keulegan-Carpenter #
Strouhal #
Reynolds #
Reynolds # vs. KC #
b = d2 / T
KC = Vm T / d = 2 a/d
Re = Vm d / ada/d d
2)(( )
Re = KC * b
Also effected by roughness and ambient turbulence
Forced Oscillation in a Current
U
y(t) = a cos t
= 2 f = 2/ T
Parameters: a/d,
Reduced velocity: Ur = U/fd
Max. Velocity: Vm = U + a cos
Reynolds #: Re = Vm d /
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.
GallopingGalloping is a result of a wake instability.
m
y(t), y(t).Y(t)
U
-y(t).
V
Resultant velocity is a combination of the heave velocity and horizontal inflow.
If n << 2 fv then the wake is quasi-static.
Lift Force, Y()
Y(t)
V
Cy = Y(t)
1/2 U2 Ap
CyStable
Unstable
Galloping motion
m
z(t), z(t).L(t)
U
-z(t).
V
b k
a mz + bz + kz = L(t).. .
L(t) = 1/2 U2 a Clv - ma y(t)..
Cl() = Cl(0) + Cl (0)
+ ...
Assuming small angles,
~ tan = -zU
. =
Cl (0)
V ~ U
Instability Criterion
(m+ma)z + (b + 1/2 U2 a )z + kz = 0.. .
U ~
b + 1/2 U2 a U < 0If
Then the motion is unstable!This is the criterion for galloping.
is shape dependent
U
1
1
1
2
12
14
Shape Cl (0)
-2.7
0
-3.0
-10
-0.66
b
1/2 a ( )
Instability:
= Cl (0)
<b
1/2 U a
Critical speed for galloping:
U > Cl (0)
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.