1

13.42 LectureVIV II

Alex TechetMarch 18, 2003

Vortex Induced Forces

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

Force coefficients:

Cx = Cy = X(t)

1/2 ρ U2 d

Y(t)1/2 ρ U2 d

Force Time Trace

Cx

Cy

DRAG

LIFT

2

VIV

Experiment: cylinder is compliantly mounted and allowed to move only in the y-direction.

Large responses may occur when:

ωv = 2 π fv ~ ωn = km + ma

What is ma a function of?EXPERIMENT: Force motion y(t) and measure force Y(t)

y(t) = a cos(ωt)

Y(t) = Y1 cos(ωt-ψ) + Yr(t)

y(t) = -aω sin(ωt).

..y(t) = -a ω2 cos(ωt)

Lift force is sinusoidal component and residual force. Filteringthe recorded lift data will give the sinusoidal term which can

be subtracted from the total force.

Lift Force:Y1(t) = Y1 cos(ωt-ψ)

= (Y1 cosψ) cos ωt + (Y1 sinψ) sin ωt

Y1aω2= cos ψ y(t) + sin ψ y(t)Y1

aω( () ).. .

Added Mass: Ma(ω,a) =Y1aω2 cos ψ

Lift in-phase with velocity: Y1v(ω,a) = -Y1aω

sin ψ

Y1(t)

3

Total Force:= - Ma(ω,a) y(t) + Y1v(ω,a) y(t).. .Y1(t)

= - (π/4 ρ d2) Cma(ω,a) y(t)

+ (1/2ρdU2)CLv(ω,a) y(t)

..

.

If CLv >0 then the fluid force amplifies the motion instead of opposing it. This is self-excited oscillation.

Cma, CLv are dependent on ω and a.

Lift in phase with velocity

Gopalkrishnan (1993)

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

4

Amplitude Estimation

ζ = b

2 k(m+ma*)

ma* = ρ V Cma; where Cma = 1.0

Blevins (1990)

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

SG=2 π fn2 2m (2πζ)

ρ d2; fn = fn/fs; m = m + ma

*^^ __

Three Dimensional EffectsShear 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.

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Longitudinal Vortices

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’

6

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

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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

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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.

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Trip Wire ExperimentsTrip wires cause the shear layer to transition early and thus retards separation. The wires can reduce amplitude of vibration but this reduction is very dependant on angle on incoming flow.

Lift Characteristics of Rigid Cylinder

a) mean lift coefficientb) phase angle between oscillating lift force and cylinder motion.

Tandem Rigid Cylinders-Amplitude

Leading cylinder

Trailing cylinder10D

Trailing cylinder5D

∆-single cylinder

-no offset

-1D lateral offset

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∆-single cylinder

-no offset

-1D lateral offset

Leading cylinder

Trailing cylinder10D

Trailing cylinder5D

Tandem Rigid Cylinders-Frequency

Leading cylinder

Trailing cylinder10D

Trailing cylinder5D

∆-single cylinder

-no offset

-1D lateral offset

Tandem Rigid Cylinders-Drag

Tandem Flexible Cylinders-Amplitude

A - Leading Cylinder B-Trailing Cylinder

∇-single inline -tandem inline

-single transverse -tandem transverse

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Tandem Flexible Cylinders-Frequency

A - Leading Cylinder B-Trailing Cylinder

∇-single inline -tandem inline

-single transverse -tandem transverse

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

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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)

.V

α

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

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

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Lift Force, Y(α) Y(t)

Vα

Cy = Y(t)

1/2 ρ U2 Ap

α

Cy Stable

Unstable

Galloping motion

m

y(t), y(t).Y(t)

U-y(t)

.V

α

b k

a my + by + ky = Y(t).. .

Y(t) = 1/2 ρ U2 a Cy - ma y(t)..

Cy(α) = Cy(0) + ∂ Cy (0)∂ α

+ ...

Assuming small angles, α:

α ~ tan α = - yU

.β = ∂ Cy (0)

∂ α V ~ U

Instability Criterion

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

~

b + 1/2 ρ U2 a βU < 0If

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

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β is shape dependent

U

1

1

1

2

12

14

Shape ∂ Cy (0)∂ α

-2.7

0

-3.0

-10

-0.66

b1/2 ρ a ( )

Instability:

β = ∂ Cy (0)∂ α

<b

1/2 ρ U a

Critical speed for galloping:

U > ∂ Cy (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.

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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.