1. 13.42 Lecture: Vortex Induced Vibrations
Prof. A. H. Techet
18 March 2004

2. 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.

3. 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

4. Axial Pressure Force i) Potential flow: -p/w < q < p/2
(ii)
Base
pressure
i) Potential flow:
-p/w < q < p/2
ii) P ~ PB
p/2  q  3p/2
(for LAMINAR flow)

5. Reynolds Number Dependency
Transition to turbulence
300 < Rd < 3*105
3*105 < Rd < 3.5*106
3.5*106 < Rd

6. 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

7. Wake Instability

8. Classical Vortex Shedding
Von Karman Vortex Street
Alternately shed opposite signed vortices

9. Vortex shedding dictated by the Strouhal number
St=fsd/U
fs is the shedding frequency, d is diameter and U inflow speed

10. Additional VIV Parameters
Reynolds Number
subcritical (Re<105) (laminar boundary)
Reduced Velocity
Vortex Shedding Frequency
S0.2 for subcritical flow

11. Strouhal Number vs. Reynolds Number

12. Vortex Shedding Generates forces on CylinderUo
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.

13. Vortex Induced Forces
Due to unsteady flow, forces, X(t) and Y(t), vary with time.
Force coefficients:
D(t)
1/2 r U2 d
L(t)
1/2 r U2 d
Cx =
Cy =

14. Force Time Trace
DRAG Cx
Avg. Drag ≠ 0
LIFT Cy
Avg. Lift = 0

15. 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

16. “Lock-in” wv = 2p fv = 2p St (U/d)
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
wv = 2p fv = 2p St (U/d)
wn = k
m + ma
Natural frequency
of oscillation

17. Equation of Cylinder Heave due to Vortex shedding
z(t) m k b
Added mass term
Restoring force
Damping
If Lv > b system is
UNSTABLE

18. 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

19. 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)

20. 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 w and a.

21. 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

22. Lift in phase with velocity
Gopalkrishnan (1993)

23. a/d = 1.29/[1+0.43 SG]3.35 Amplitude Estimation ~ Blevins (1990) _ _
2m (2pz)
r d2 ^ ^
SG=2 p fn2
; fn = fn/fs; m = m + ma*
z = b
2 k(m+ma*)
ma* = r V Cma; where Cma = 1.0

24. Drag Amplification ~ Cd = 1.2 + 1.1(a/d)
VIV tends to increase the effective drag coefficient. This increase has been investigated experimentally. 3 2 1 Cd
|Cd| ~
0.1
0.2
0.3 fd U a d
= 0.75
Gopalkrishnan (1993)
Mean drag:
Fluctuating Drag: ~
Cd = (a/d)
Cd occurs at twice the
shedding frequency.

25. Single Rigid Cylinder Results
1.0
One-tenth highest transverse oscillation amplitude ratio
Mean drag coefficient
Fluctuating drag coefficient
Ratio of transverse oscillation frequency to natural frequency of cylinder
1.0

26. 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

27. 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.

28. Wake Patterns Behind Heaving Cylinders
f , A U
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.

29. Transition in Shedding Patterns
A/d
Williamson and Roshko (1988)
Vr = U/fd
f* = fd/U

30. Formation of ‘2P’ shedding pattern

31. End Force Correlation Uniform Cylinder Tapered Cylinder
Hover, Techet, Triantafyllou (JFM 1998)
Tapered Cylinder

32. 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.

33. Oscillating Tapered Cylinder
Strouhal Number for the tapered cylinder:
St = fd / U
where d is the average
cylinder diameter. x
d(x)
U(x) = Uo

34. Spanwise Vortex Shedding from 40:1 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

35. 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)

36. 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
z/d = 7.9
‘2P’
Rd = 1500; St = 0.198; A/d = 0.5

37. Principal Investigator:
Vortex Dislocations, Vortex Splits & Force Distribution in Flows past Bluff Bodies D. Lucor & G. E. Karniadakis
Objectives:
Confirm numerically the existence of a stable, periodic hybrid shedding mode 2S~2P in the wake of a straight, rigid, oscillating cylinder
Techet, Hover and Triantafyllou (JFM 1998)
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

38. VIV Suppression Helical strake Shroud Axial slats Streamlined fairing
Splitter plate
Ribboned cable
Pivoted guiding vane
Spoiler plates

39. VIV Suppression by Helical Strakes
Helical strakes are a
common VIV suppresion
device.

40. Oscillating Cylinders
Parameters:
y(t)
Re = Vm d / n
Reynolds # d
b = d2 / nT
Reduced
frequency
y(t) = a cos wt
y(t) = -aw sin(wt) .
KC = Vm T / d
Keulegan-
Carpenter #
Vm = a w
St = fv d / Vm
Strouhal #
n = m/ r ; T = 2p/w

41. Reynolds # vs. KC # )( ( ) Re = KC * b
Re = Vm d / n = wad/n = 2p a/d d /nT 2 )( ( )
KC = Vm T / d = 2p a/d
Re = KC * b
b = d2 / nT
Also effected by roughness and ambient turbulence

42. 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

43. 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.

44. Galloping is a result of a wake instability.m
y(t), y(t) .
Y(t) U
-y(t) V a
Resultant velocity is a combination of the heave velocity and horizontal inflow.
If wn << 2p fv then the wake is quasi-static.

45. Lift Force, Y(a)
Y(t) V a
Y(t)
1/2 r U2 Ap
Cy = Cy
Stable a
Unstable

46. Galloping motion mz + bz + kz = L(t) ..
z(t), z(t) .
L(t) U
-z(t) V a
mz + bz + kz = L(t) .. . a
L(t) = 1/2 r U2 a Clv - ma y(t) .. b k
 Cl (0)
 a
Cl(a) = Cl(0) +
+ ...
Assuming small angles, a:
a ~ tan a = - z U .
b =
 Cl (0)
 a
V ~ U

47. Instability Criterion.. . b U
(m+ma)z + (b + 1/2 r U2 a )z + kz = 0 ~ b U If
b + 1/2 r U2 a
< 0
Then the motion is unstable!
This is the criterion for galloping.

48. b is shape dependent Shape  Cl (0)  a -2.7 U -3.0 -10 -0.66 1 1 2 2U 1 2
-3.0 1 4
-10
-0.66

49. Instability: ( ) U > b  Cl (0) 1/2 r U a b =  a < b 1/2 r a
Critical speed for galloping: b
1/2 r a
U >
( )
 Cl (0)
 a

50. Torsional Galloping Both torsional and lateral galloping are possible.
FLUTTER occurs when the frequency of the torsional
and lateral vibrations are very close.

51. Galloping vs. VIV Galloping is low frequency
Galloping is NOT self-limiting
Once U > Ucritical then the instability occurs irregardless of frequencies.

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