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.