Aeroelasticity & Experimental Aerodynamics (AERO0032-1)
Lecture 6 Vortex Induced Vibrations
T. Andrianne
2017-2018 Aeroelasticity
= Study of the interaction of inertial, structural and aerodynamic forces
Applications on aircrafts, buildings, surface vehicles etc.
Inertial Forces
Structural dynamics Flight Dynamics Dynamic Aeroelasticity Collar’s triangle Structural Forces Aerodynamic Forces Static Aeroelasticity
1 Flow separation
Airfoil/wing aeroelasticity • Divergence • Flutter
Bluff-body aeroelasticity • Vortex Induced Vibration • Galloping
2
VIV
Vortices …
Flow visualiza on of vor ces shed behind a cylinder
3 VIV
… induce vibrations
Vortex-induced vibra ons 4
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
5 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
6 Dimensional analysis
Non-dimensional variables
• Geometry l = length / width D
• Dimensionless amplitude A vibration amplitude / diameter D
U Large Ur à Quasi-steady assumption • Reduced velocity U ∞ r = fD Small Ur (< 10) à Strong fluid-structure interaction
U∞ f D U A ∞ m
7 Dimensional analysis
m m • Mass ratio m = S = S = susceptibility of flow induced r m D2 / 4 F ρπ vibrations
mS = mass (p.u. length) of the structure [kg/m] ρ = fluid density [kg/m3]
U D • Reynolds number Re = ∞ = inertial / viscous forces ν 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
U Ma ∞ • Mach number = c = tendency of the flow to compress when it encounters a structure In this course : incompressible flows à Ma < 0.3 8 Dimensional analysis
energy dissipated per cycle • Damping factor (classically) η = S 4π × total energy of the structure
YN ln(YN /YN+1) = 2πηS YN (linear system) m
When Fluid-Structure interactions are concerned
à “reduced damping” or Scruton number ≈ mr ×ηS
9 Dimensional analysis
Motion of a linear structure in a subsonic, steady flow
Described by : → A = F l ,U , Re, m ,η D ( D r r S ) • Geometry l D
• Reduced velocity Ur • Dimensionless amplitude A D
• Mass ratio mr • Reynolds number Re
• Damping factor ηS
Source FIV1977 VIV Galloping (Lecture 2) 10 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
11 VIV examples – in air
• Bridges
Great-Belt bridge (Denmark) • Chimneys
• High rise buildings
Burj Khalifa (Dubai) 12
VIV examples – in water / 2 phases flows
• Risers
source : Imperial college • Heat exchangers
source : Sandvik
13 VIV examples : Great-Belt bridge
Unacceptable VIV motion (~ 40cm) during final construction stage at low (and frequent) wind speeds (~8m/s)
(Larsen 2000)
14 VIV examples : Great-Belt bridge
(Frandsen 2001) (Larsen 2000)
15 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
Structural motion takes the
control of aerodynamics
Sumer 2006 16 Dimensional analysis
Motion of a 2D linear structure in a subsonic, steady flow
Described by : A F U , Re, m , → D = ( r r ηS ) Great Belt bridge
D
U∞ m
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 17 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
18 Flow around a static cylinders
Lifting cylinder (potential flow theory) * 2 - 2 2Γsinθ $ Γ ' à Steady lift force C =1−,4sin θ + +& ) / p RV 2 RV +, π ∞ % π ∞ ( ./ à No drag (D’Alembert Paradox) Γ cl = RV∞
Viscous effects à Flow separation 19 Flow around a static cylinders
Creeping flow : vorticity created in the BL is totally dissipated near the body
Transition to turbulence Increasing Re = inertial / viscous forces
Effect of viscosity in the vicinity of the body. Large amount of viscosity not dissipated near the body
20 Flow around a static cylinders How vortex shedding appears ? Flow separation
• 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) 21 Flow around a static cylinders
Strouhal number 600
400 fVS D S VS St ≡ f f U∞ 200 f = shedding frequency VS 0 0 5 10 15 20 25 30 D = cross dimension U ∞ 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 !
22 Flow around a static cylinders
Aerodynamic forces
à Impact of the shed vortices on the aerodynamic forces
L
D
à Lift force varies at fVS
à Drag force varies at ~ 2 fVS
Sumer 2006 23 Flow around a static cylinders Aerodynamic forces
• Lift coefficient is an image of the fluctuating vorticity in the wake
(where the dominant frequency is fVS) à C (t) is characterized by f and C rms L L L f D à Dimensionless Strouhal number St = L U∞
L(t) rms rms St 2πU∞ CL (t) = 2 = 2CL sin(ωLt) = 2CL sin( t) 1/ 2ρU∞D D
• Drag coefficient : typically fD ~ 2 fVS
D(t) rms rms rms St 2πU∞ CD (t) = 2 = 2CD sin(ωDt) = 2CD sin(2ωLt) = 2CD sin(2 t) 1/ 2ρU∞D D
24 Flow around a static cylinders Aerodynamic forces
Strongly Reynolds dependent !
L(t) rms CL (t) = 2 = 2CL sin(ωLt) 1/ 2ρU∞D
rms St 2πU = 2C sin( ∞ t) L D
D(t) rms rms CD (t) = 2 = 2CD sin(ωDt) = 2CD sin(2ωLt) 1/ 2ρU∞D
rms St 2πU = 2C sin(2 ∞ t) D D 25 FIV2011 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
26 Oscillating cylinder
à Cylinder forced to move
Imposed motion : y(t) = Asin(2π f f t)
A = Amplitude ff = Forcing frequency
Order of magnitude of the diameter (D) Varying around the sta c cylinder shedding 0 à A/D ~ 1 frequency (fVS ) à 0 0< ff/fVS < 2 0 fVS = vortex shedding frequency that would be observed with cylinder at rest (i.e. following the Strouhal rela on)
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
27 Oscillating cylinder : Patterns
Williamson 2004
• 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” 28
Oscillating cylinder : Frequency
0 Around ff / fVS ~ 1 à No change in pattern
à But effect on the shedding frequency fVS
f D i.e. f does not follow anymore the Strouhal relation St = VS VS U ∞ For a given Reynolds number and amplitude of imposed motion (A) 0 fVS /fVS fVS = ff
ff is increased 1 fVS is measured
0 Around ff / fVS ~ 1 à fVS = ff
= Lock-in phenomenon 1 f /f 0 f VS (wake capture) !! Different from lock-in in VIV !! 29 Oscillating cylinder : Frequency
Lock-in range mainly depends on the imposed amplitude (ratio A / D)
FIV2011 A/D ì à Range ì
30 Oscillating cylinder : Lift force
Imposed motion: y(t) = Asin(2π f f t)
à The resulting lift force L(t) is not necessarily harmonic (idem static cylinder)
But through a standard Fourier analysis, it can approximated by
L (t) = L0 si n(2π f f t + ϕ ) = component of lift @ the forcing frequency (ff)
0 0 0 Driving parameters: A/D and ff /fVS à L0(A/D, ff /fVS ) and φ(A/D, ff /fVS )
Three different representations of the lift force : 1. Modulus and phase 2. Phased lift coefficients 3. Inertia and drag coefficients
31 Oscillating cylinder : Lift force
1. Modulus and phase 2. Phased lift coefficients C for A/D = 0.5 3. Inertia and drag coefficients L 5
CL (t) = CL sin(2π f f t + ϕ) 4
3 modulus phase 2 1 0 à both function of A/D and ff /fVS 0 0 ff /fVS 0 .5 .75 1 1.25 1.5 1.75 max(CL ) for f /f = 1 f VS CL (A = 0) ϕ
2 270 180
1 90 0 -90 0 0 .5 1 1.5 2 A/D -180 0 .5 .75 1 1.25 1.5 1.75 ff /fVS adapted from Carberry et al. (2005) 32 Oscillating cylinder : Lift force
1. Modulus and phase y(t) Asin(2 f t) 2. Phased lift coefficients = π f 3. Inertia and drag coefficients
CL (t) = (CL cosϕ)sin(2π f f t)+(CL sinϕ)cos(2π f f t)
in phase with y (and - ! y! ) in phase with y! Conservative forces Non-conservative forces
0 à both function of A/D and ff /fVS à might lead to positive or negative work for A/D = 0.5 C sin CL cosϕ L ϕ
1 1
0 0
0 f /f 0 .5 .75 1 1.25 1.5 1.75 ff /fVS .5 .75 1 1.25 1.5 1.75 f VS adapted from Sarpkaya (1979) 33 Oscillating cylinder : Lift force
1. Modulus and phase 2. Phased lift coefficients 3. Inertia and drag coefficients
" D2 % 1
CL (t) = −ρ$π 'CM !y!− ρDCD y! y! 4 2 # &
Inertia coefficient Drag coefficient
(added mass) 2 Ur Link with modulus : CM = CL cosϕ 3 2π (y0 D)
As expected for Ur à 0, CM à 1 (motion in still fluid)
adapted from Sarpkaya (1979) 34 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
35 Free vibration: VIV
Consider a single dof system : a simple linear oscillator D U ∞ m
A F U , Re, m , Remember from the dimensional analysis : D = ( r r ηS )
In the case of free vibration, special interest for :
fS = frequency of the free motion ( ‘S’ stands for Structural) 0 fS = frequency of the free motion (in still fluid, i.e. U∞ = 0) Structural part A = amplitude of the free motion fVS = frequency of the vortex shedding process (with motion) Fluid part
" % A fS fVS U 0 → , , = F U , Re, m ,η ∞ fS independent of U∞ $ 0 0 ' ( r r S ) with Ur = 0 #D fS fS & fS D (but not of the fluid properties) 36 Free vibration: VIV
0 Fixed structural parameters : D, fS , mr, and η Fixed fluid properties : ν, ρ Frequency of motion Frequency of vortex shedding ! $ A fS fVS # , 0 , 0 & = F (Ur, Re, mr,ηS ) "D fS fS %
Airspeed effect (U∞)
For small variations of U∞ in a Re range
à Neglecting the Re effect
à Ur only
FSI201137 Free vibration: VIV
Amax Lock-in range in AIR in WATER
FIV1977 FSI2011 38 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
! A f f $ From the dimensional analysis : S VS # , 0 , 0 & = F (Ur, Re, mr,ηS ) D f f " S S %
" AMax % → , Lockin(Ur ) = F (Re, mr,ηS ) #$ D &' à Neglecting the Re effect
à mr and ηS only π à Scruton number ( ~ reduced damping) Sc = (1+ m )η 2 r S à Skop-Griffin number : SG = 4π 2St 2Sc
39 Free vibration: VIV
Amax = Function(SG) Lock-in range = Function(SG)
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
40 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
41 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: FFluid = −mA!y!+ F
ρπ D2 Inviscid added mass m = Total force – added mass effect A 4
Type A: forced system Type B: fluid elastic Type C: wake oscillator
y y y G(y) U∞ F(t) U∞ F(y(t),t) U∞ WAKE WAKE WAKE F(q)
42 Modelling VIV
Type A : Forced system models Type A FIV2011
y From static cylinder : U∞ F(t)
rms " St 2πU∞ % WAKE CL (t) = 2CL sin$ t' # D &
2 rms " St 2πU∞ % F(t) = L(t) =1/ 2ρU∞D 2CL sin$ t' # D & 0 0 2 (mS + mA )!y!+ 2ηS (mS + mA )ωS y! +(mS + mA )(ωS ) y = L(t)
= Classical linear vibration problem
0 à At resonance (Ur=1/St) : fVS=fS
• Amplitude of motion inversely proportional to damping (ζ ) • Phase jump from 0 to π
• fVS follows the Strouhal relation 43 Modelling VIV
Type A : Forced system models Type A
2 m m !y! 2 m m 0 y! m m 0 y L(t) y ( S + A ) + ηS ( S + A )ωS +( S + A )(ωS ) = U∞ F(t) WAKE
2 y(t) 1 C U " U % = L r sin 2πSt t +ϕ 3 1/2 $ ' D 2π * 2 2 2 2 2 2 # D & (1+ m ) (1−Ur St ) + 4ηSUr St ( ) 2η StUr tanϕ = S 1− St 2Ur2 FIV2011
at resonance with CL=0.35
A CL CL = 3 * 2 2 = D 4π (1+ m )St ηS 2SG
OK @ large SG numbers KO @ low SG numbers No prediction of Lock-in 44 Modelling VIV
Type B : Fluid elastic models Type B
y à Force dependent of motion U∞ F(y(t),t) (Amplitude, displacement, velocity, acceleration, combination of them …) WAKE
Similar to galloping models (Lecture 2)
à Several types exist :
2 " A % " St 2πU∞ % 1. Modified forcing model (Blevins, 1990) : L(t) =1/ 2ρU DCL $ 'sin$ t' ∞ # D & # D &
max(CL ) C (A = 0) 0 L for ff /fS = 1
2 2 " A % 0 " A % " A % From oscillating cylinder → CL $ ' = CL +α$ '+ β $ ' (forced motion) 1 # D & # D & # D &
0 CL0 α and β from experiments 0 .5 1 1.5 2 A/D 45 Modelling VIV
2 " A % " St 2πU∞ % Type B L(t) =1/ 2ρU DCL $ 'sin$ t' ∞ # D & # D & y 0 0 2 U∞ F(y(t),t) (mS + mA )!y!+ 2ηS (mS + mA )ωS y! +(mS + mA ) ωS y = L(t) ( ) WAKE at resonance (same than Type A) ! A $ C 2 L # & ! A $ 0 ! A $ ! A $ A " D % CL # & = CL +α# &+ β # & = " D % " D % " D % FIV2011 D 2SG
2 0 # A & # A & → CL = (α − 2SG)% (+ β % ( $ D ' $ D ' 2nd order equation to solve for A/D à
OK @ large SG numbers Better than Type A @ low SG numbers (underestimate !) Still no prediction of Lock-in 46 Modelling VIV
2. Advanced forcing models: Type B à Amplitude and phase of the force depend on amplitude and reduced frequency of the motion y U∞ F(y(t),t) WAKE à Concept of phased lift coefficients (oscillating cylinder)
! A $ 2 ( ! A $ ! A $ + L# ,Ur,t& =1/ 2ρU∞ *CL cosϕ # ,Ur &sin(ωt) +CL sinϕ # ,Ur &cos(ωt)- " D % " D % " D % ) ,
Identified from imposed motion à Ur=U∞/(fD)
0 0 2 ! A $ mS !y !+ 2η SmSω S y! + mS ω S y = L# ,U r,t& is valid outside coincidence ( ) " D %
A −C sinϕ A / D,U =1/ St At coincidence : = L ( r ) D 2SG 47
Modelling VIV
1. Modified forcing model (Blevins, 1990) : Type B
2 " A % " St 2πU∞ % L(t) =1/ 2ρU∞DCL $ 'sin$ t' y # D & # D & U∞ F(y(t),t) WAKE
2. Advanced forcing models :
! A $ 2 ( ! A $ ! A $ + L# ,Ur,t& =1/ 2ρU∞ *CL cosϕ # ,Ur &sin(ωt) +CL sinϕ # ,Ur &cos(ωt)- " D % ) " D % " D % ,
3. Time domain fluidelastic models:
à Explicit use of the time dependence of time (Chen et al.1995)
! A $ ) ! A $ 2 ! A $ 2 ! A $ , 2 L# ,Ur, y, y!, !y!& = +−m# ,Ur &!y!− ρU∞c# ,Ur &y! − ρU∞k# ,Ur &y.+1/ 2ρU∞CL sin(ωVSt) " D % * " D % " D % " D % -
Classical fluidelastic forces (added mass, damping and stiffness)
forcing term, at the frequency of vortex shedding 48 Modelling VIV
! A $ ) ! A $ 2 ! A $ 2 ! A $ , 2 L# ,Ur, y, y!, !y!& = +−m# ,Ur &!y!− ρU∞c# ,Ur &y! − ρU∞k# ,Ur &y.+1/ 2ρU∞CL sin(ωVSt) " D % * " D % " D % " D % - 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)
) 2 , 2 # y & y! y L(y, y!,t) =1/ 2ρU∞D+c(Ur )%1−ε 2 ( + k(Ur ) +CL (Ur )sin(ωVSt). $ D 'U U * ∞ ∞ -
depends on Ur only à not especially harmonic
non linear term to account for amplitude effect
49 Modelling VIV Type A : forced system Type B : Fluid elastic Type C : wake oscillator
y y y G(y) U∞ F(t) U∞ F(y(t),t) U∞ WAKE WAKE WAKE F(q)
2 rms " St 2πU∞ % F(t) =1/ 2ρU D 2CL sin$ t' ∞ # D &
2 " A % " St 2πU∞ % F(t) =1/ 2ρU DCL $ 'sin$ t' ∞ # D & # D &
! A $ 2 ( ! A $ ! A $ + F# ,Ur,t& =1/ 2ρU∞ *CL cosϕ # ,Ur &sin(ωt) +CL sinϕ # ,Ur &cos(ωt)- " D % ) " D % " D % ,
! A $ ) ! A $ 2 ! A $ 2 ! A $ , 2 F# ,Ur, y, y!, !y!& = +−m# ,Ur &!y!− ρU∞c# ,Ur &y! − ρU∞k# ,Ur &y.+1/ 2ρU∞CL sin(ωVSt) " D % * " D % " D % " D % - ) 2 , 2 # y & y! y F (y, y!,t) =1/ 2ρU∞D+c(Ur )%1−ε 2 ( + k(Ur ) +CL (Ur )sin(ωVSt). * $ D 'U∞ U∞ -
50 Modelling VIV Type A : forced system Type B : Fluid elastic Type C : wake oscillator
y y y G(y) U∞ F(t) U∞ F(y(t),t) U∞ WAKE WAKE WAKE F(q)
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
51
Modelling VIV
Type C : Wake oscillator models Type C y G(y) U∞ “Fluid force is the result of the wake dynamics, WAKE itself influenced by the cylinder motion” F(q)
à Coupling of two systems : CYLINDER (y variable) and WAKE (q variable)
! F(q) = effect of the wake on the cylinder # m!y!+... = F(q) " G(y) = effect of the cylinder on the wake W q(t) = G(y,...) $# [ ] W[q(t)] = dynamics of the free wake
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)
52
Modelling VIV #! m!y!+... = F(q) Type C : Wake oscillator models " W q(t) = G(y,...) $# [ ]
Most natural choice of q(t) by Hartlen and Currie (pioneers) : q(t) = CL (t)
1 2 à Directly : F (q(t)) = F (CL (t)) = ρU∞DCL (t) 2
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 : W [q(t)] = q!!− aq! + bq!3 +ω 2q = 0 (a van der Pol oscillator does it too) with a, b > 0
Finally, G( y) = cy! (rather arbitrary choice by H&C)
53 Modelling VIV
Type C : Wake oscillator model by Facchinetti, de Langre & Biolley
0 • Using the wake dof : q(t) = 2CL (t) / CL • Based on a van der Pol oscillator for W[q(t)] • Using acceleration for coupling the wake to the structure ) ! 2 π $ ! 1 $ 1 2 0 + #mS + ρD &!y!+#cS + ρU∞D St CD &y! + kS y = ρU ∞DCLq + " 4 % " 2 % 4 * 2 + U∞ 2 ! U∞ $ A + q!!+ 2πSt ε (q −1)q! +#2πSt & = !y! , D " D % D
0 from force measurement in static cylinder experiments (typical values: CL =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)
54 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
FIV2011 FIV2011 forcing model
modified forcing model
wake oscillator
55 Modelling VIV
Type C : Wake oscillator model by Y. Tamura and G. Matsui (1979)
Y = non-dim vertical displacement α = wake position
( ! ν $ 2 α * Y!! 2 m f C Y! Y f m ! +# ηS + r ( + D ) & + = − rν 2 # m!y!+... = F(q) * " 2πSt% (2πSt) " ) ! 2 $ # W [q(t)] = G(y,...) * 4 f 2 2 !! ! $ α!!− 2ξν #1− α &α! +ν α = −mrY − 2πSt ν Y * C2 + " L0 % " Y! % CL = − f $α + 2πSt ' # ν & 0 from force measurement in static cylinder experiments (typical values: CL =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. 56 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
57 Lock-in mechanism
à Link between Lock-in and linear coupled-mode flutter (> L3) (proposed by de Langre 2006) Wake oscillator model by Facchinetti, de Langre & Biolley (FLB) ) ! 2 π $ ! 1 $ 1 2 0 + #mS + ρD &!y!+#cS + ρU∞D St CD &y! + kS y = ρU ∞DCLq + " 4 % " 2 % 4 * k 2 t = S T + U∞ 2 ! U∞ $ A q!!+ 2πSt ε q −1 q! +#2πSt & = !y! mS + mF + D ( ) " D % D , Y = y / D C / 4 St # !! ! 2 γ = D ( π ) Non-dimensionalized : % Y + λY +Y = MΩ q $ 2 2 Ω = St Ur % q!!+εΩ q −1 q! +Ω q = AY!! & ( ) m + m µ = S F ρD2 " 2 $ Y!!+Y = MΩ q λ = 2ζ +γ / µ Neglecting damping and NL terms : # q!!+Ω2q = AY!! 0 (i.e. not responsible for lock-in) %$ CL M = 2 2 16π St µ f à Ω is the only driving parameter Ω = St U = VS r f
58 Lock-in mechanism
iωt Solution of this system is straightforward : (Y, q) = (Y0, q0 )e
" 2 $ Y!!+Y = MΩ q # 2 !! %$ q!!+Ω q = AY
Feeding the solution into the system equations 4 2 2 2 à frequency equation : D(ω) = ω +$#(AM −1)Ω −1&%ω +Ω = 0 if AM < 1 (verified in practice)
à Two modal frequencies ω1 and ω2
2 1 # 2 2 2 2 & ω1 = %1+(1− AM )Ω + (1+(1− AM )Ω ) − 4Ω ( 2$ ' 2 1 # 2 2 2 2 & ω2 = 1+(1− AM )Ω − 1+(1− AM )Ω − 4Ω 2% ( ) ( $ '
Variation of ω1 and ω2 with Ω (= driving parameter) ?
59 Lock-in mechanism
Depending on the sign of the radicand :
2 1 # 2 2 2 2 & ω1,2 = %1+(1− AM )Ω ± (1+(1− AM )Ω ) − 4Ω ( 2$ '
2 1 1 (1 +(1− AM )Ω2 ) − 4Ω 2 ≥ 0 Ω < or Ω > 1+ AM 1− AM de Langre 2006
à ω1 and ω2 are real modal frequencies à two neutrally stable modes co-exist (neutrally because damping is neglected here)
1 1 The corresponding mode shapes : 1+ AM 1− AM Y MΩ2 Ω2 −ω 2 0 = = q 1 2 A 2 0 −ω ω
à close to ωR = Ω = Wake mode à close to ωR = 1 = Structural mode
Ω 60 Lock-in mechanism 1 1 In the range < Ω < 1+ AM 1− AM
Two modes exist, but with complex conjugate frequencies
Ω ω1,2 = 2 (1±itanθ) 1+ tan θ de Langre 2006 2 4Ω2 − 1+ (1− AM )Ω2 1 −1 ( ) θ = tan 2 2 1+ (1 − AM )Ω à Coupled-Mode Flutter solution (CMF) = merging of two frequencies 1 1 1+ AM 1− AM of the neutral modes
à Range of instability
one stable mode and one unstable mode Ω 61
Lock-in mechanism de Langre 2006
Coupled-Mode Flutter solution (CMF) one stable mode and one unstable mode
ωi < 0 ωi > 0
In the CMF range : - No distinction between modes - Strong deviation of the frequency of the wake mode from ω = Ω (Strouhal 1 unstable1 relation), instead ω ~ 1 1+ AM 1− AM - One unstable mode instead of two well separated stable (pure wake and pure structural) modes stable à CMF range ~ Lock-in range
62 Ω
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 :
# 1−ω 2 & # Ω2 −ω 2 & ω = ω +iε%Ωω 2 0 (−iλ%ω 2 0 ( 0 0 2(ω 4 −Ω2 ) 0 2(ω 4 −Ω2 ) $% 0 '( $% 0 '(
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 63 Lock-in mechanism
à Resulting frequency of oscillation = the one of the dominant mode i.e. the wake mode AM = 0.05 AM = 0.75 de Langre 2006
Small Lock-in range Large Lock-in range Small deviation from Strouhal Large deviation from Strouhal outside CMF
64 Lock-in mechanism de Langre 2006
Critical mass ratio
Lock-in range = limit of CMF range : 1 1 < Ω < 1+ AM 1− AM
Ω The extend of Lock-in is significantly affected by the ratio of the cylinder
* mS mS + mF mass to the fluid mass : m = ≠ µ = 2 ρπ D2 / 4 ρD 4 m* = µ −1 π
−1 # 0 & MIN 1 MIN ACL The lower Lock-in limit : Ω = → St Ur = %1+ 3 2 * ( 1+ AM $% 4π St (m +1) '( −1 1 $ AC0 ' The upper Lock-in limit : MAX MAX L Ω = → St Ur = &1− 3 2 * ) 1− AM & 4π St (m +1) ) % ( 65 Lock-in mechanism
Critical mass ratio −1 ! 0 $ MIN 1 ACL The lower Lock-in limit : Ur = #1+ 3 2 * & St # 4π St (m +1) & " % 0 −1 Depend on St, m* and CL only " 0 % MAX 1 ACL The upper Lock-in limit : Ur = $1− 3 2 * ' St $ 4π St (m +1) ' # &
Fixed St = 0.2 de Langre 2006 0 2 values for CL AC0 C 0 = 0.6 m* = L −1 L CRIT 4π 3St 2 0 CL = 0.3
0 à Depending on the value of CL , a critical value of m* exists for which lock-in persists forever 66 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
67 Summary
• Static cylinder à strong dependence on Re • Oscillating cylinder 0 Strong dependence on ff/fVS forcing frequency A/D forcing amplitude
à Effects on the flow pattern (2S, 2P S+P modes) & vortex shedding frequency à lock-in
Three different representations of the lift force : Modulus and phase Phased lift coefficients Inertia and drag coefficients 0 à function of A/D and ff /fVS 68 Summary ! $ A fS fVS • Free cylinder à VIV # , 0 , 0 & = F (Ur, Re, mr,ηS ) "D fS fS % Amax = Function(SG) Lock-in range = Function(SG) à VIV = Self-limited phenomenon
SG SG 4πSt 2 à 3 types of models: - forced system models - fluid elastic models - wake oscillator models
69 Summary Type A : forced system Type B : Fluid elastic Type C : wake oscillator
y y y G(y) U∞ F(t) U∞ F(y(t),t) U∞ WAKE WAKE WAKE F(q)
2 rms " St 2πU % ! A $ ∞ F ,U , y, y!, !y! F(t) =1/ 2ρU∞D 2CL sin$ t' # r & ! # D & " D % # m!y!+... = F(q) " # W [q(t)] = G(y,...) forcing model $ modified forcing model Type A B and C are empirical models
Type C models the cause of the fluid wake oscillator forces (that types A & B take for granted)
70 Summary
• Lock-in mechanism through linear analysis
" 2 $ Y!!+Y = MΩ q # 2 !! %$ q!!+Ω q = AY 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) ) ! 2 π $ ! 1 $ 1 2 0 + #mS + ρD &!y!+#cS + ρU∞D St CD &y! + kS y = ρU ∞DCLq + " 4 % " 2 % 4 * 2 C (t) + U∞ 2 ! U∞ $ A L + q!!+ 2πSt ε (q −1)q! +#2πSt & = !y! , D " D % D related to position of the separation point
71 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 72