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Aeroelasticity & Experimental Aerodynamics (AERO0032-1)

Lecture 6 Vortex Induced Vibrations

T. Andrianne

2015-2016

Topic of Lectures 1 to 5 à Airfoil and wing aerodynamics

à Attached flow conditions à Quasi-steady and unsteady analytical models

Topics of Lectures 6 (VIV) and 7 (galloping) à Bluff-body aerodynamics à Flow separation à Empirical models

From previous lectures

1

Today’s lecture

Dimensional analysis Examples of VIV Flow around static cylinders Effect of a prescribed motion of the cylinder Free vibration: VIV Modelling VIV Lock-in mechanism

2

Today’s lecture


3

Dimensional analysis

Non-dimensional variables •  Geometry = length / width

•  Dimensionless amplitude vibration amplitude / diameter

•  Reduced velocity

4

lD

AD

Ur =U∞

fD

m U∞

D U∞

f

Large Ur à Quasi-steady assumption Small Ur (< 10) à Strong fluid-structure interaction

A


•  Mass ratio = susceptibility of flow induced vibrations mS = mass (p.u. length) of the structure [kg/m] ρ = fluid density [kg/m3]

•  Reynolds number = inertial / viscous forces

ν = kinematic viscosity of the fluid [m2/s] à Measure of the boundary layer thickness à Measure of the transition between laminar and turbulent flows

•  Mach number = tendency of the flow to compress

when it encounters a structure In this course : incompressible flows à Ma < 0.3 5

mr =mS

mF

=mS

ρπD2 / 4

Re =U∞Dν

Ma =U∞c


•  Damping factor (classically)

When Fluid-Structure interactions are concerned à “reduced damping” or Scruton number

6

ηS =energy dissipated per cycle

4π × total energy of the structure

m

YN YN ln YN /YN+1( ) = 2πηS

(linear system)

≈ mr ×ηS

Source FIV1977


Motion of a linear structure in a subsonic, steady flow Described by :

7

•  Geometry •  Reduced velocity

•  Dimensionless amplitude

•  Mass ratio

•  Reynolds number

•  Damping factor

lD

AD

Ur

mr

Re

ηS

→ AD = F l

D,Ur, Re,mr,ηS( )

VIV Galloping (Lecture 7)

Today’s lecture


8

VIV examples – in air

•  Bridges

•  Chimneys

•  High rise buildings

9

Great-Belt bridge (Denmark)

Burj Khalifa (Dubai)

VIV examples – in water / 2 phases flows

•  Risers •  Heat exchangers

10

source : Imperial college

source : Sandvik

VIV examples : Great-Belt bridge

11

(Larsen 2000)

Unacceptable VIV motion (~ 40cm) during final construction stage at low (and frequent) wind speeds (~8m/s)

VIV examples : Great-Belt bridge

12

(Larsen 2000) (Frandsen 2001)

From 3D to 2D

•  Vortex shedding is essentially a 3D phenomenon à Flow components in the Z-direction à Geometric variation of the structure à Dynamic behaviour of the structure •  2D analysis as a starting point to understand VIV :

Even on 3D structures, 2D approaches are carried out (flow visualization, load analysis)

•  A/D >> à 2D dominated

13 Sumer 2006

Structural motion takes the control of aerodynamics


Motion of a 2D linear structure in a subsonic, steady flow Described by : Other parameters: •  Surface roughness of the structure (for rounded shapes) •  Turbulence intensity of the upstream flow

à Effect on the wake, hence of the lift and on the VIV 14

→ AD = F Ur, Re,mr,ηS( )

m U∞

D

Great Belt bridge

Today’s lecture


15

Viscous effects à Flow separation

Flow around a static cylinders

Lifting cylinder (potential flow theory)

16

Cp =1− 4sin2θ +

2ΓsinθπRV∞

+Γ

2πRV∞

$

%&

'

()

2*

+,,

-

.//

cl =ΓRV∞

à Steady lift force à No drag (D’Alembert Paradox)


17

Transition to turbulence Increasing Re = inertial / viscous forces

Creeping flow : vorticity created in the BL is totally dissipated near the body

Effect of viscosity in the vicinity of the body. Large amount of viscosity not dissipated near the body

Flow around a static cylinders How vortex shedding appears ?

18

•  Flow speed outside wake is much higher than inside •  Vorticity gathers in upper and lower layers •  Induced velocities (due to vortices) causes this perturbation to amplify

à  Origin of the vortex shedding process = shear layer instability (purely inviscid mechanism)

Vortex shedding = global instability : the whole wake is affected = robust : vorticity is continuously produced @ a well defined frequency (see Strouhal)

Flow separation


19

Strouhal number fVS = shedding frequency D = cross dimension U∞ = free stream velocity à Linear dependency of the shedding frequency

with free stream velocity

à St ~ cste in a limited range of Reynolds numbers !

St ≡ fVSDU∞

0 5 10 15 20 25 300

200

400

600

U∞

f Sf VS

Aerodynamic forces à  Impact of the shed vortices on the aerodynamic forces à  Lift force varies at fVS

à  Drag force varies at ~ 2 fVS

Sumer 2006


20

L

D

Aerodynamic forces •  Lift coefficient is an image of the fluctuating vorticity in the wake

(where the dominant frequency is fVS)

à CL(t) is characterized by fL and CLrms

à Dimensionless Strouhal number

•  Drag coefficient : typically fD ~ 2 fVS


21

CL (t) = L(t)1 / 2ρU∞

2D= 2CL

rms sin(ωLt) = 2CLrms sin(St 2πU∞

Dt)

CD (t) = D(t)1 / 2ρU∞

2D= 2CD

rms sin(ωDt) = 2CDrms sin(2ωLt) = 2CD

rms sin(2 St 2πU∞

Dt)

St = fLDU∞


22

CL (t) = L(t)1 / 2ρU∞

2D= 2CL

rms sin(ωLt)

= 2CLrms sin(St 2πU∞

Dt)

CD (t) = D(t)1 / 2ρU∞

2D= 2CD

rms sin(ωDt) = 2CDrms sin(2ωLt)

= 2CDrms sin(2 St 2πU∞

Dt)

Strongly Reynolds dependent !

Aerodynamic forces

FIV2011

Today’s lecture


23

Oscillating cylinder

à Cylinder forced to move

Imposed motion :

A = Amplitude ff = Forcing frequency

Remember : Re has also a strong effect on the shedding process à  Effect of the motion of the cylinder, for a given Re range.

We can observe: Change in the flow pattern and in the shedding frequency

24

y(t) = Asin(2π f f t)

Order of magnitude of the diameter (D) à A/D ~ 1

Varying around the sta?c cylinder shedding frequency (fVS0)

à 0< ff/fVS0 < 2

fVS0 = vortex shedding frequency that would be observed with cylinder at rest (i.e. following the Strouhal rela?on)

Oscillating cylinder : Patterns

•  Limited range of forcing frequency and amplitude: à Two single vortices (“2S mode”)

•  Higher forcing frequencies and amplitudes:

à Two pairs of vortices (“2P mode”) •  Even higher frequencies and amplitudes : non-symmetric “P+S mode”

25

Williamson 2004

Oscillating cylinder : Frequency

Around ff / fVS0 ~ 1 à No change in pattern

à But effect on the shedding frequency fVS

i.e. fVS does not follow anymore the Strouhal relation

26

fVS /fVS0

ff /fVS0 1

1

fVS = ff

For a given Reynolds number and amplitude of imposed motion (A) ff is increased fVS is measured Around ff / fVS

0 ~ 1 à fVS = ff = Lock-in phenomenon

(wake capture) !! Different from lock-in in VIV !!

St = fVSDU∞

FIV2011

Oscillating cylinder : Frequency

Lock-in range mainly depends on the imposed amplitude (ratio A / D) A/D ì à Range ì

27

Oscillating cylinder : Lift force

Imposed motion: à The resulting lift force L(t) is not necessarily harmonic

(idem static cylinder)

But through a standard Fourier analysis, it can approximated by

= component of lift @ the forcing frequency (ff) Driving parameters: A/D and ff /fVS

0 à L0(A/D, ff /fVS0) and φ(A/D, ff /fVS

0)

Three different representations of the lift force : 1.  Modulus and phase 2.  Phased lift coefficients 3.  Inertia and drag coefficients

28


L(t) = L0 sin(2π f f t + ϕ )

Oscillating cylinder : Lift force 1.  Modulus and phase 2.  Phased lift coefficients 3.  Inertia and drag coefficients

29

CL (t) =CL sin(2π f f t + ϕ )

modulus phase

à both function of A/D and ff /fVS0

ff /fVS0

.5

1

CL for A/D = 0.5

2

3

4

5

0 .75 1 1.25 1.5 1.75

adapted from Carberry et al. (2005) ff /fVS

0 .5

-90

ϕ

0

90

180

270

-180 .75 1 1.25 1.5 1.75

A/D 0

max CL( )CL (A = 0)

for ff /fVS0 = 1

1

2

0 .5 1 1.5 2


30

CL (t) = CL cosϕ( )sin(2π f f t)+ CL sinϕ( )cos(2π f f t)

à both function of A/D and ff /fVS0

ff /fVS0

.5

for A/D = 0.5

0

1

.75 1 1.25 1.5 1.75 adapted from Sarpkaya (1979)

ff /fVS0

.5

0

1

.75 1 1.25 1.5 1.75

in phase with (and - ) Conservative forces

y !!y in phase with Non-conservative forces à might lead to positive or

negative work

!y

CL cosϕ CL sinϕ



31

CL (t) = −ρ πD2

4"

#$

%

&'CM !!y−

12ρDCD !y !y

adapted from Sarpkaya (1979)

Inertia coefficient (added mass) Link with modulus :

Drag coefficient

CM =CL cosϕUr2

2π 3 y0 D( )

As expected for Ur à 0, CM à 1 (motion in still fluid)

Today’s lecture


32

Free vibration: VIV Consider a single dof system : a simple linear oscillator Remember from the dimensional analysis : In the case of free vibration, special interest for : fS = frequency of the free motion ( ‘S’ stands for Structural) fS0 = frequency of the free motion (in still fluid, i.e. U∞ = 0) A = amplitude of the free motion fVS = frequency of the vortex shedding process (with motion)

33

m U∞

D

AD = F Ur, Re,mr,ηS( )

→AD, fSfS0 ,fVSfS0

"

#$

%

&'= F Ur, Re,mr,ηS( ) Ur =

U∞

fS0D

with fS0 independent of U∞ (but not of the fluid properties)

Structural part

Fluid part

Free vibration: VIV Fixed structural parameters : D, fS0, mr, and η Fixed fluid properties : ν, ρ

34

AD, fSfS0 ,fVSfS0

!

"#

$

%&= F Ur, Re,mr,ηS( )

Airspeed effect (U∞)

For small variations of U∞ in a Re range à Neglecting the Re effect

à Ur only

FSI2011

Frequency of motion Frequency of vortex shedding

Free vibration: VIV

35

FIV1977 FSI2011

Amax Lock-‐in range

in WATER

in AIR

Free vibration: VIV Two key quantities to characterize VIV (practical interest) : •  Amax : max amplitude of the y motion, reached at Ur=1/St •  Lock-in range : also occurs around Ur=1/St From the dimensional analysis :

36

AD, fSfS0 ,fVSfS0

!

"#

$


→AMaxD,Lockin Ur( )

"

#$%

&'= F Re,mr,ηS( )

à Neglecting the Re effect

à mr and ηS only

à Scruton number ( ~ reduced damping) Sc = π2(1+mr )ηS

à Skop-Griffin number : SG = 4π 2St2Sc

Free vibration: VIV Amax = Function(SG) Lock-in range = Function(SG)

37

à Amax < 1 or 2 diameters (even for very low mass-damping combination)

à  Amax ~ 0 for heavy/damped structures à  Lock-in range ì if SG î

à  VIV = Self-limited phenomenon

Today’s lecture


38

Modelling VIV Objective of the model : à  Estimation of Amax and Lock-in range in a given Re range (Critical airspeed from the St number (from literature or wind tunnel tests)) •  “Empirical” models, i.e. based on static and oscillating experimental data •  Many applications of VIV (in different fluids) à Tons of models (1960’s) •  In this course : 2D models only •  Classification:

39

FFluid = −mA!!y+F

Inviscid added mass Total force – added mass effect

F(t) U∞

WAKE

y

Type A: forced system

F(y(t),t) U∞

WAKE

y

Type B: fluid elastic

F(q)

U∞ WAKE y G(y)

Type C: wake oscillator

mA =ρπD2

4

Modelling VIV Type A : Forced system models From static cylinder : = Classical linear vibration problem

40

F(t) U∞

WAKE

y

Type A

CL (t) = 2CLrms sin St 2πU∞

Dt

"

#$

%

&'

F(t) = L(t) =1/ 2ρU∞2D 2CL

rms sin St 2πU∞

Dt

"

#$

%

&'

FIV2011

à  At resonance (Ur=1/St) : fVS=fS0

•  Amplitude of motion inversely •  proportional to damping (ζ ) •  Phase jump from 0 to π •  fVS follows the Strouhal relation

mS +mA( ) !!y+ 2ηS mS +mA( )ωS0 !y+ mS +mA( ) ωS

0( )2y = L(t)

Modelling VIV Type A : Forced system models OK @ large SG numbers KO @ low SG numbers No prediction of Lock-in 41


0( )2y = L(t) F(t) U∞

WAKE

y

Type A

y(t)D

=12π 3

CLUr2

(1+m*) 1−Ur2St2( )

2+ 4ηS

2Ur2St2( )

1/2 sin 2πStUDt +ϕ

"

#$

%

&'

AD=

CL

4π 3(1+m*)St2ηS2 =

CL

2SG

at resonance with CL=0.35

FIV2011

tanϕ = 2ηSStUr1− St2Ur2

F(y(t),t) U∞

WAKE

y

Type B

Modelling VIV Type B : Fluid elastic models à  Force dependent of motion (Amplitude, displacement, velocity, acceleration, combination of them …)

Similar to galloping models (Lecture 7) à  Several types exist : 1.  Modified forcing model (Blevins, 1990) :

42

L(t) =1/ 2ρU∞2DCL

AD"

#$

%

&'sin St 2πU∞

Dt

"

#$

%

&'

A/D 0

max CL( )CL (A = 0) for ff /fS0 = 1

1

2

0 .5 1 1.5 2

→CLAD"

#$

%

&'=CL

0 +αAD"

#$

%

&'+β

AD"

#$

%

&'2

From oscillating cylinder (forced motion)

CL0 α and β from experiments

F(y(t),t) U∞

WAKE

y

Type B

Modelling VIV

43


AD"

#$

%

&'sin St 2πU∞

Dt

"

#$

%

&'


0( )2y = L(t)

CLAD!

"#

$

%&=CL

0 +αAD!

"#

$

%&+β

AD!

"#

$

%&2

AD=CL

AD!

"#

$

%&

2SG

at resonance (same than Type A)

→CL0 = α − 2SG( ) A

D#

$%

&

'(+β

AD#

$%

&

'(2

FIV2011

OK @ large SG numbers Better than Type A @ low SG numbers (underestimate !) Still no prediction of Lock-in

2nd order equation to solve for A/D à

Modelling VIV 2.  Advanced forcing models:

à Amplitude and phase of the force depend on amplitude and reduced frequency of the motion à Concept of phased lift coefficients (oscillating cylinder)

Identified from imposed motion à Ur=U∞/(fD)

is valid outside coincidence

At coincidence :

44

F(y(t),t) U∞

WAKE

y

Type B

L AD,Ur, t

!

"#

$

%&=1/ 2ρU∞

2 CL cosϕAD,Ur

!

"#

$

%&sin ωt( )+CL sinϕ

AD,Ur

!

"#

$

%&cos ωt( )

(

)*

+

,-

mS!!y+ 2ηSmSωS0 !y+mS ωS

0( )2y = L A

D,Ur, t

!

"#

$

%&

AD=−CL sinϕ A /D,Ur =1/ St( )

2SG

F(y(t),t) U∞

WAKE

y

Type B

Modelling VIV 1.  Modified forcing model (Blevins, 1990) :

2.  Advanced forcing models :

3.  Time domain fluidelastic models:

à Explicit use of the time dependence of time (Chen et al.1995)

45


AD"

#$

%

&'sin St 2πU∞

Dt

"

#$

%

&'

L AD,Ur, t

!

"#

$

%&=1/ 2ρU∞

2 CL cosϕAD,Ur

!

"#

$


AD,Ur

!

"#

$

%&cos ωt( )

(

)*

+

,-

L AD,Ur, y, !y, !!y

!

"#

$

%&= −m A

D,Ur

!

"#

$

%&!!y− ρU∞

2c AD,Ur

!

"#

$

%& !y− ρU∞

2k AD,Ur

!

"#

$

%&y

)

*+

,

-.+1/ 2ρU∞

2CL sin ωVSt( )

Classical fluidelastic forces (added mass, damping and stiffness) forcing term, at the frequency of vortex shedding

Modelling VIV

Problem of the Chen’s model : harmonic motion assumed, because of the

dependence of m, c and k on Ur and A/D

(identification based on oscillating cylinder experiments)

Another (much more complex) model, proposed by Simiu & Scanlan (1996)

46

L AD,Ur, y, !y, !!y

!

"#

$

%&= −m A

D,Ur

!

"#

$

%&!!y− ρU∞

2c AD,Ur

!

"#

$

%& !y− ρU∞

2k AD,Ur

!

"#

$

%&y

)

*+

,

-.+1/ 2ρU∞

2CL sin ωVSt( )

L y, !y, t( ) =1/ 2ρU∞2D c(Ur ) 1−ε

y2

D2

#

$%

&

'(!yU∞

+ k(Ur )yU∞

+CL (Ur )sin ωVSt( ))

*+

,

-.

depends on Ur only à not especially harmonic

non linear term to account for amplitude effect

Modelling VIV

47

F(t) =1/ 2ρU∞2D 2CL

rms sin St 2πU∞

Dt

"

#$

%

&'

F(t) =1/ 2ρU∞2DCL

AD"

#$

%

&'sin St 2πU∞

Dt

"

#$

%

&'

F AD,Ur, t

!

"#

$

%&=1/ 2ρU∞

2 CL cosϕAD,Ur

!

"#

$


AD,Ur

!

"#

$

%&cos ωt( )

(

)*

+

,-

F AD,Ur, y, !y, !!y

!

"#

$

%&= −m A

D,Ur

!

"#

$

%&!!y− ρU∞

2c AD,Ur

!

"#

$

%& !y− ρU∞

2k AD,Ur

!

"#

$

%&y

)

*+

,

-.+1/ 2ρU∞

2CL sin ωVSt( )

F y, !y, t( ) =1/ 2ρU∞2D c(Ur ) 1−ε

y2

D2

#

$%

&

'(!yU∞

+ k(Ur )yU∞

+CL (Ur )sin ωVSt( ))

*+

,

-.

F(t) U∞

WAKE

y

Type A : forced system

F(y(t),t) U∞

WAKE

y

Type B : Fluid elastic

F(q)

U∞ WAKE y G(y)

Type C : wake oscillator

Modelling VIV

Limitations of Types A and B :

1.  Limited to harmonic motions (except at the price of cumbersome models (Simiu & Scanlan 1996))

à  Difficult to generalize to more complex motions (multifrequencies …) à  Not a major limitation, because VIV often results in quasi-harmonic

motions

2.  More important: no physical interpretation of wake as a global mode

48

F(t) U∞

WAKE

y


F(y(t),t) U∞

WAKE

y


F(q)

U∞ WAKE y G(y)


Modelling VIV Type C : Wake oscillator models “Fluid force is the result of the wake dynamics, itself influenced by the cylinder motion” à Coupling of two systems : CYLINDER (y variable) and WAKE (q variable) Several formulations exist for F(q), G(y), W[q(t)] and the choice for q(t) In all cases, the parameters of the models come from experiments on : •  Static cylinder à Lift force à F(q)

Wake dynamics à W[q(t)] •  Oscillating cylinder à Wake dynamics à G(y)

49

F(q)

U∞ WAKE y G(y)

Type C

m!!y+... = F(q)W q(t)[ ] =G(y,...)

!"#

$#

F(q) = effect of the wake on the cylinder G(y) = effect of the cylinder on the wake W[q(t)] = dynamics of the free wake

Modelling VIV Type C : Wake oscillator models Most natural choice of q(t) by Hartlen and Currie (pioneers) : à  Directly :

W[q(t)] must respect two essential features of the wake as a global mode :

1.  The oscillating wake results from a self-sustained flow instability, as linear unstable mode

2.  This instability is self-limited, as a nonlinear global mode A Rayleigh equation capture these features : (a van der Pol oscillator does it too) Finally, (rather arbitrary choice by H&C) 50

m!!y+... = F(q)W q(t)[ ] =G(y,...)

!"#

$#

q(t) =CL (t)

F q(t)( ) = F CL (t)( ) = 12ρU∞

2DCL (t)

W q(t)[ ] = !!q− a !q+ b !q3 +ω 2q = 0with a,b > 0

G(y) = c!y

Modelling VIV Type C : Wake oscillator model by Facchinetti, de Langre & Biolley •  Using the wake dof :

•  Based on a van der Pol oscillator for W[q(t)]

•  Using acceleration for coupling the wake to the structure

51

mS + ρD2 π

4!

"#

$

%&!!y+ cS +

12ρU∞D St CD

!

"#

$

%& !y+ kSy =

14ρU 2

∞DCL0q

!!q+ 2πStU∞

Dε q2 −1( ) !q+ 2πStU∞

D!

"#

$

%&

2

=AD!!y

)

*

++

,

++

q(t) = 2CL (t) /CL0

from force measurement in static cylinder experiments (typical values: CL0=0.3 and CD=1.2)

from wake analysis (hotwire) in static cylinder exp. (typical values : St=0.2 and ε = 0.3) from wake analysis in oscillating cylinder exp. (typical value : A = 12)

Modelling VIV Type C : Wake oscillator model by Facchinetti, de Langre & Biolley Using typical values (+CD=1) à  Strong underestimation of the amplitude à  But good qualitative influence of the SG parameter on the lock-in

range

52

modified forcing model

wake oscillator

forcing model

FIV2011 FIV2011

Modelling VIV Type C : Wake oscillator model by Y. Tamura and G. Matsui (1979)

53

!!Y + 2ηS +mr f +CD( ) ν2πSt

!

"#$

%&!Y +Y = − f mrν

2 α(2πSt)2

!!α − 2ξν 1− 4 f 2

C2L0

α 2!

"#

$

%& !α +ν 2α = −mr

!!Y − 2πSt ν !Y

(

)

**

+

**

m!!y+... = F(q)W q(t)[ ] =G(y,...)

!"#

$#

CL = − f α + 2πSt!Yν

"

#$

%

&'

Y = non-dim vertical displacement α = wake position

from force measurement in static cylinder experiments (typical values: CL0=0.3 and CD=1.2)

from wake analysis (hotwire) in static cylinder exp. (typical values : St=0.2 and f = 1.16) from wake analysis in oscillating cylinder exp.

Today’s lecture


54

Lock-in mechanism à  Link between Lock-in and linear coupled-mode flutter (L 1 & 2) (proposed by de Langre 2006) Wake oscillator model by Facchinetti, de Langre & Biolley (FLB) Non-dimensionalized : Neglecting damping and NL terms : (i.e. not responsible for lock-in)

à Ω is the only driving parameter

55

mS + ρD2 π

4!

"#

$

%&!!y+ cS +

12ρU∞D St CD

!

"#

$

%& !y+ kSy =

14ρU 2

∞DCL0q

!!q+ 2πStU∞

Dε q2 −1( ) !q+ 2πStU∞

D!

"#

$

%&

2

=AD!!y

)

*

++

,

++

!!Y +λ !Y +Y =MΩ2q

!!q+εΩ q2 −1( ) !q+Ω2q = A !!Y

#

$%

&%

t = kSmS +mF

T

Y = y /Dγ =CD / 4πSt( )Ω = St Ur

µ =mS +mF

ρD2

λ = 2ζ +γ /µ

M =CL

0

16π 2St2µ

!!Y +Y =MΩ2q!!q+Ω2q = A !!Y

"#$

%$

Ω = St Ur =fVSf

Lock-in mechanism Solution of this system is straightforward : Feeding the solution into the system equations à  frequency equation :

if AM < 1 (verified in practice) à Two modal frequencies ω1 and ω2 Variation of ω1 and ω2 with Ω ?

56

!!Y +Y =MΩ2q!!q+Ω2q = A !!Y

"#$

%$

Y,q( ) = Y0,q0( )eiωt

D(ω) =ω 4 + AM −1( )Ω2 −1#$ %&ω2 +Ω2 = 0

ω12 =121+ 1− AM( )Ω2 + 1+ 1− AM( )Ω2( )

2− 4Ω2#

$%&

'(

ω22 =121+ 1− AM( )Ω2 − 1+ 1− AM( )Ω2( )

2− 4Ω2#

$%&

'(

Lock-in mechanism Depending on the sign of the radicand :

or à ω1 and ω2 are real modal frequencies à  two neutrally stable modes co-exist

(neutrally because damping is neglected here)

The corresponding mode shapes :

57

ω1,22 =

121+ 1− AM( )Ω2 ± 1+ 1− AM( )Ω2( )

2− 4Ω2#

$%&

'(

1+ 1− AM( )Ω2( )2− 4Ω2 ≥ 0 Ω <

11+ AM

Y0q0=MΩ2

1−ω 2 =Ω2 −ω 2

Aω 2

Ω

de Langre 2006

11+ AM

11− AM

à  close to ωR = Ω = Wake mode à  close to ωR = 1 = Structural mode

Ω >1

1− AM

Lock-in mechanism

58

11+ AM

<Ω <1

1− AM

Ω

de Langre 2006

11+ AM

11− AM

ω1,2 =Ω

1+ tan2θ1± i tanθ( )

θ =12tan−1

4Ω2 − 1+ (1− AM )Ω2( )2

1+ (1− AM )Ω2

In the range Two modes exist, but with complex conjugate frequencies

à  Coupled-Mode Flutter solution (CMF) = merging of two frequencies

of the neutral modes

à Range of instability one stable mode and one unstable mode

Lock-in mechanism Coupled-Mode Flutter solution (CMF) one stable mode and one unstable mode In the CMF range : -  No distinction between modes -  Strong deviation of the frequency of the

wake mode from ω = Ω (Strouhal relation), instead ω ~ 1

-  One unstable mode instead of two well separated stable (pure wake and pure structural) modes

à CMF range ~ Lock-in range

59

11+ AM

11− AM

Ω

de Langre 2006

ωi < 0 ωi > 0

unstable

stable

Lock-in mechanism Most unstable mode ? Remember we assumed no damping : ε = λ = 0 Out of the Lock-in range, it is interesting to know which mode is dominant Considering non-zero damping : ε and λ > 0 A first order expansion of the frequency equation yields : if ω0 stands for ωW à ε and λ have a destabilization effect on the wake mode if ω0 stands for ωS à ε and λ have a damping effect on the structural mode

à in both cases the most unstable mode has the frequency of the wake mode 60

ω =ω0 + iε Ωω0

2 1−ω0

2

2(ω0

4 −Ω2 )

#

$%%

&

'((− iλ ω

0

2 Ω2 −ω0

2

2(ω0

4 −Ω2 )

#

$%%

&

'((

Lock-in mechanism à Resulting frequency of oscillation = the one of the dominant mode

i.e. the wake mode

61

AM = 0.05 AM = 0.75

Small Lock-in range Small deviation from Strouhal

Large Lock-in range Large deviation from Strouhal outside CMF

de Langre 2006

Lock-in mechanism Critical mass ratio Lock-in range = limit of CMF range : The extend of Lock-in is significantly affected by the ratio of the cylinder mass to the fluid mass : ≠ The lower Lock-in limit : The upper Lock-in limit :

62

µ =mS +mF

ρD2

11+ AM

<Ω <1

1− AM

m* =mS

ρπD2 / 4m* =

4πµ −1

ΩMIN =1

1+ AM→ St Ur

MIN = 1+ ACL0

4π 3St2 (m* +1)

#

$%%

&

'((

−1

ΩMAX =1

1− AM→ St Ur

MAX = 1− ACL0

4π 3St2 (m* +1)

$

%&&

'

())

−1

Ω

de Langre 2006

Lock-in mechanism Critical mass ratio The lower Lock-in limit : The upper Lock-in limit :

63

UrMIN =

1St

1+ ACL0

4π 3St2 (m* +1)

!

"##

$

%&&

−1

UrMAX =

1St1− ACL

0

4π 3St2 (m* +1)

"

#$$

%

&''

−1

de Langre 2006 Fixed St = 0.2 2 values for CL

0

CL0 = 0.6

CL0 = 0.3

Depend on St, m* and CL0 only

à Depending on the value of CL0, a critical value of m* exists

for which lock-in persists forever

mCRIT* =

ACL0

4π 3St2−1

Today’s lecture

Dimensional analysis Examples of VIV Flow around static cylinders Effect of a prescribed motion of the cylinder Free vibration : VIV Modelling VIV Lock-in mechanism

64

Summary •  Static cylinder à strong dependence on Re •  Oscillating cylinder

Strong dependence on ff/fVS0 forcing frequency

A/D forcing amplitude à Effects on the flow pattern (2S, 2P S+P modes) & vortex shedding frequency

Three different representations of the lift force : Modulus and phase Phased lift coefficients Inertia and drag coefficients

65

à lock-in

à function of A/D and ff /fVS0

Summary •  Free cylinder à VIV

66

AD, fSfS0 ,fVSfS0

!

"#

$


Amax = Function(SG) Lock-in range = Function(SG) à  VIV = Self-limited phenomenon

à  3 types of models: - forced system models - fluid elastic models - wake oscillator models

SG4πSt2

SG

Summary

67

F(t) U∞

WAKE

y


F(y(t),t) U∞

WAKE

y


F(q)

U∞ WAKE y G(y)


F(t) =1/ 2ρU∞2D 2CL

rms sin St 2πU∞

Dt

"

#$

%

&' F A

D,Ur, y, !y, !!y

!

"#

$

%& m!!y+... = F(q)

W q(t)[ ] =G(y,...)

!"#

$#

modified forcing model

wake oscillator

forcing model

Type A B and C are empirical models Type C models the cause of the fluid forces (that types A & B take for granted)

Summary •  Lock-in mechanism through linear analysis

à Coupled-Mode Flutter (CMF) •  Critical mass ratio à infinite lock-in range

•  Link between Airfoil flutter and Lock-in

pitch (α) and plunge (h) transverse displacement (y) and wake variable (q)

68

CL(t)

related to position of the separation point

mS + ρD2 π

4!

"#

$

%&!!y+ cS +

12ρU∞D St CD

!

"#

$

%& !y+ kSy =

14ρU 2

∞DCL0q

!!q+ 2πStU∞

Dε q2 −1( ) !q+ 2πStU∞

D!

"#

$

%&

2

=AD!!y

)

*

++

,

++

!!Y +Y =MΩ2q!!q+Ω2q = A !!Y

"#$

%$

References

•  FIV1977 ‘Flow-induced vibration’, R.D. Blevins •  FSI2011 ‘Fluid-structure Interactions : cross-flow induced instabilities’, M.P. Paidoussis,

S. J. Price, E. de Langre. Cambridge University Press •  Williamson 2004 : C.H.K. Williamson and R. Govardhan, ‘Vortex-Induced Vibrations’,

ann. rev. Fluid Mech 2004, 36:413-55 •  Larsen 2000 : A. Larsen et al. ‘Storebelt suspension bridge - vortex shedding excitation

and mitigation by guide vanes’, J. Wind Engineering and Industrial Aerodynamics 2000, 283-296

•  Frandsen 2001 J.B. Frandsen, ‘Simultaneous pressures and accelerations measured full-scale on the Great Belt East suspension bridge’, , J. Wind Engineering and Industrial Aerodynamics 2001, 95-129

•  Carberry et al. 2005 Carberry, J., Sheridan, J. & Rockwell, D. ‘Controlled oscillations of a cylinder: Forces and wake modes’, J.l of Fluid Mechanics 2005 538, 31–69.

•  Sarpkaya 1979 Sarpkaya, T. ‘ Vortex-induced oscillations – a selective review’ J. of Applied Mechanics 1979, 46, 241–258.

•  Sumer 2006 B.M. Sumer J. Fredsoe ‘Hydrodynamics around cylindrical structures’, Advanced series on ocean engineering, Vol 26, World Scientific

•  de Langre 2006 E. de Langre ‘Frequency lock-in is caused by coupled-mode flutter’, J. of Fluids and structures 2006, 22, 783-791

•  Y. Tamura and G. Matsui ‘Wake oscillator model of vortex induced oscillation of circular cylinder’ Proceedings of the 5th International Conference on Wind Engineering, Vol.2, Fort Collins, Colorado, USA, July 1979, pp. 1085-1092

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