13.42 Lecture 2:Vortex Induced

Vibrations

Prof. A. H. Techet28 April 2005

• 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

Will

iam

son

and

Ros

hko

(198

8)

A/d

f* = fd/UVr = U/fd

End Force Cross-Correlation

Uniform Cylinder

Tapered Cylinder

Hov

er, T

eche

t, Tr

iant

afyl

lou

(JFM

199

8)

Fc = Force Correlation Coefficient, µ = correlation standard deviation

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.

Three Dimensional Effects

Shear layer instabilities as well as longitudinal (braid) vortices lead to transition from laminar to turbulent flow in cylinder wakes.

Longitudinal vortices appear at Rd = 230.

Longitudinal Vortices

C.H.K. Williamson (1992)

The presence of longitudinal vortices leads to rapid breakdown of the wake behind a cylinder.

Strouhal Number for the tapered cylinder:

St = fd / U

where d is the average cylinder diameter.

Oscillating Tapered Cylinder

x

d(x)U(x

) = U

o

Spanwise Vortex Shedding from 40:1 Tapered Cylinder

Tech

et, e

t al (

JFM

199

8)

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

= 22

.9z/

d =

7.9

Evolution of the Hybrid Shedding Mode

‘2P’ ‘2S’

Rd = 1500; St = 0.198; A/d = 0.5

z/d = 22.9z/d = 7.9

Evolution of the Hybrid Shedding Mode

‘2S’‘2P’

Rd = 1500; St = 0.198; A/d = 0.5

z/d = 22.9z/d = 7.9

Evolution of the Hybrid Shedding Mode

‘2S’‘2P’

Rd = 1500; St = 0.198; A/d = 0.5

z/d = 22.9z/d = 7.9

Evolution of the Hybrid Shedding Mode

‘2S’‘2P’

Rd = 1500; St = 0.198; A/d = 0.5

z/d = 22.9z/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 suppressiondevice.

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.

Nominal Reduced Velocity, Vrn

Freq

uenc

y R

atio

, fx/f

y

2-DOF Cylinder Motion CYLINDER ORBITAL PLOTS

Rigid cylinder free to move in both transverse (y) and longitudinal (x) directions. High length:diameter ratio.

FFT Magnitude Contours For Frequency Ratio fx/fy = 1.0

Transverse Position In-Line Position

FFT Magnitude Contours For Frequency Ratio fx/fy = 2.0

Transverse Position In-Line Position

Power Balance Along a Riser

Overview & Aim• Obtain power balance along a riser

– We need power input from the fluid to the riser as a function of z.

– Input:

– Output:

– Power balance is correlated with Clv(z).

212

Cross-flow displacement: Lift force:

Normalized Lift force:

Incoming Velocity Profile: ( )

y(z,t)L(z,t)

L(z,t)f(z,t) dsU

U zρ

=

m( , ) & ( )mlv lvC z C zω

Fourier Decomposition• Obtain the Fourier decomposition of force and

displacement as:

• Fourier Coefficients

1

1

ˆRe

ˆRe

n

n

Ni t

nn

Ni t

nn

y(z,t) y e

f(z,t) f e

ω

ω

=

=

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

∑

∑

ˆ ˆˆ ˆ ( , ) & ( , )n n n n n ny y z f f zω ω= =

Choose a Set of Important Frequencies

• Choose the set of frequencies ωm which represents– peaks in the span-averaged Fourier coefficients:

ˆˆ ( , ) & ( , )m m m my z f zω ω

ˆˆ ,m my f

Obtaining Global Clv• Obtain the multi-frequency lift coefficients in phase with velocity and acceleration as:

• Global lift coefficient in phase with velocity:

• Clv is very important as it expresses energy balance.

• Clv*U(z)2 is representative of the power input from the fluid to the riser.

( ) ( )( )

( ) ( )( )

m

m m

m

m

Lift coefficient in phase with velocity for :

ˆ ˆ ˆ( , ) ( , ) sin arg arg

Lift coefficient in phase with acclera tion for :

ˆ ˆ ˆ( , ) - ( , ) cos arg arg

m

m

lv m m m

la m m m m

C z f z f y

C z f z f y

ω

ω ω

ω

ω ω

= −

= −

m

2m

ˆ

( )ˆ

mlv mm

lv

mm

C yC z

y

ω

ω

⎛ ⎞⎜ ⎟⎜ ⎟= − ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑

Input Sheared Velocity Profile

Selected Frequencies

• Chosen set of frequencies ωm :

ˆˆ ,m my f

( )2 0.1733 0 .1831 0 .1929 0 .2026 0 .2124 0 .2222 0 .2295 0 .2393mω π=

Global Clv & Cm

• Clv*U(z)2 is representative of the power input from the fluid to the riser.

• One can clearly observe that there is a positive input of power from the fluid where the incoming velocity is high.

• The power is transferred to the low incoming velocity region, as traveling waves in the riser.

Oscillating Cylinders

d

y(t)

y(t) = a cos ωt

Parameters:Re = Vm d / ν

Vm = a ω

y(t) = -aω sin(ωt).

b = d2 / νT

KC = Vm T / d

St = fv d / Vm

ν = µ/ ρ ; Τ = 2π/ω

Reducedfrequency

Keulegan-Carpenter #

Strouhal #

Reynolds #

Reynolds # vs. KC #

b = d2 / νT

KC = Vm T / d = 2π a/d

Re = Vm d / ν = ωad/ν = 2π a/d d /νΤ

2)(( )

Re = KC * b

Also effected by roughness and ambient turbulence

Forced Oscillation in a Current

θ

U

y(t) = a cos ωtω = 2 π f = 2π / T

Parameters: a/d, ρ, ν, θ

Reduced velocity: Ur = U/fd

Max. Velocity: Vm = U + aω cos θ

Reynolds #: Re = Vm d / ν

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).

Vα

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

If ωn << 2π fv then the wake is quasi-static.

Lift Force, Y(α) Y(t)

Vα

Cy = Y(t)

1/2 ρ U2 Ap

α

Cy Stable

Unstable

Galloping motion

m

z(t), z(t).L(t)

U-z(t).

Vα

b k

a mz + bz + kz = L(t).. .

L(t) = 1/2 ρ U2 a Clv - ma y(t)..

Cl(α) = Cl(0) + ∂ Cl (0)∂ α

+ ...

Assuming small angles, α:

α ~ tan α = - zU

.β = ∂ Cl (0)

∂ α V ~ U

Instability Criterion

(m+ma)z + (b + 1/2 ρ U2 a )z + kz = 0.. .βU

~

b + 1/2 ρ U2 a βU < 0If

Then the motion is unstable!This is the criterion for galloping.

β is shape dependent

U

1

1

1

2

12

14

Shape ∂ Cl (0)∂ α

-2.7

0

-3.0

-10

-0.66

b1/2 ρ a ( )

Instability:

β = ∂ Cl (0)∂ α

<b

1/2 ρ U a

Critical speed for galloping:

U > ∂ Cl (0)∂ α

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.