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13.42 Lecture: Vortex Induced Vibrations

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13.42 Lecture:Vortex Induced Vibrations

Prof. A. H. Techet

18 March 2004

If ignored, these vibrations can prove catastrophic to structures, as they did in the case of the Tacoma Narrows Bridge in 1940.

U(q) = 2Usinq

P(q) = 1/2 r U(q)2 = P + 1/2 r U2

Cp = {P(q) - P }/{1/2 r U2}= 1 - 4sin2q

(i)

(ii)

Base

pressure

i) Potential flow:

-p/w < q < p/2

ii) P ~ PB

p/2 q 3p/2

(for LAMINAR flow)

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

- 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

l

h

Von Karman Vortex Street

Alternately shed opposite signed vortices

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

St = 0.2

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.

L(t)

D(t)

1/2 r U2 d

1/2 r U2 d

Due to unsteady flow, forces, X(t) and Y(t), vary with time.

Force coefficients:

Cx =

Cy =

DRAG

Cx

Avg. Drag ≠ 0

LIFT

Cy

Avg. Lift = 0

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

k

wn =

m + ma

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

z(t)

m

b

k

Added mass term

Restoring force

If Lv > b system is

UNSTABLE

Damping

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 wvis the frequency of vortex shedding

Two components of lift can be analyzed:

Lift in phase with acceleration (added mass):

Lift in-phase with velocity:

Total lift:

(a = zois cylinder heave amplitude)

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

Vortex Induced Vibrations are

SELF LIMITED

In air: rair ~ small, zmax ~ 0.2 diameter

In water: rwater ~ large, zmax ~ 1 diameter

Gopalkrishnan (1993)

b

2 k(m+ma*)

z =

ma* = r V Cma; where Cma = 1.0

a/d = 1.29/[1+0.43 SG]3.35

~

Blevins (1990)

_

_

2m (2pz)

r d2

^

^

; fn = fn/fs; m = m + ma*

SG=2 p fn2

~

Cd

|Cd|

3

2

a

d

= 0.75

1

fd

U

0.1

0.2

0.3

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.

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

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.

f , A

U

U

f , A

‘2P’

‘2S’

- 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.

A/d

Williamson and Roshko (1988)

f* = fd/U

Vr = U/fd

Formation of ‘2P’ shedding pattern

Uniform Cylinder

Hover, Techet, Triantafyllou (JFM 1998)

Tapered Cylinder

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.

x

d(x)

Strouhal Number for the tapered cylinder:

St = fd / U

where d is the average

cylinder diameter.

U(x) = Uo

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

- ‘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)

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

Techet, Hover and Triantafyllou (JFM 1998)

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 2S~2P 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

- Helical strake
- Shroud
- Axial slats
- Streamlined fairing
- Splitter plate
- Ribboned cable
- Pivoted guiding vane
- Spoiler plates

Helical strakes are a

common VIV suppresion

device.

.

y(t) = -aw sin(wt)

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

Re = Vm d / n = wad/n = 2p a/d d /nT

(

)(

)

2

KC = Vm T / d = 2pa/d

Re = KC * b

b = d2/nT

Also effected by roughness and ambient turbulence

Parameters:

a/d, r, n, q

y(t) = a cos wt

w = 2 p f = 2p / T

q

U

Reduced velocity: Ur = U/fd

Max. Velocity: Vm = U + aw cos q

Reynolds #: Re = Vm d / n

Roughness and ambient turbulence

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.

Y(t)

.

y(t), y(t)

U

.

-y(t)

m

a

V

Galloping is a result of a wake instability.

Resultant velocity is a combination of the heave velocity and horizontal inflow.

If wn << 2p fv then the wake is quasi-static.

Y(t)

1/2 r U2 Ap

Y(t)

V

a

Stable

Cy =

Cy

a

Unstable

L(t)

.

z(t), z(t)

U

..

L(t) = 1/2r U2 a Clv - ma y(t)

.

-z(t)

m

a

V

Cl (0)

a

Cl (0)

a

..

.

mz + bz + kz = L(t)

.

a ~ tan a = -

b =

z

U

a

b

k

Cl(a) = Cl(0) +

+ ...

Assuming small angles, a:

V ~ U

b

U

b

U

..

.

~

(m+ma)z + (b + 1/2r U2 a )z + kz = 0

b + 1/2r U2 a

< 0

If

Then the motion is unstable!

This is the criterion for galloping.

1

1

1

2

2

Cl (0)

a

1

4

1

Shape

-2.7

0

U

-3.0

-10

-0.66

Cl (0)

a

Cl (0)

a

b

1/2r U a

<

b =

Critical speed for galloping:

b

1/2r a

U >

( )

Both torsional and lateral galloping are possible.

FLUTTER occurs when the frequency of the torsional

and lateral vibrations are very close.

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

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