Lock-in/Synchronization in Vortex Induced Vibrations:
Department of Aerospace EngineeringIIT Kanpur
Navrose, Mohd. Furquan, Tulsi Ram Sahu, Sanjay Mittal
Acknowledgement: DST (Department of Science & Technology)
Flow induced vibrations
Tacoma Narrows BridgeFerrybridge power station, UK (1960)
Fluid Structure Interactions:Flexible Splitter Plates
The Classic Benchmark of Turek and Hron1:
Flow past a flexible splitter plate in a channel:
Re=100, =10.0, Ae=1400, s=0.4
1Turek S, Hron J, (2006), Fluid–structure interaction. Modelling, simulation and optimization, Lecture notes in computational science and engineering
Vorticity Field Mesh movement
Structure Model UsedMomentum balance in total Lagrangian form:
Constitutive relation for St-Venant-Kirchhoff's material:
Solved using standard Galerkin approach and coupled to fluid via block-iterative coupling.Movement of mesh is modeled by a linear elastic pseudo-solid (with Jacobian based stiffening).New non-dimensional parameters are:
, and .
.
Fluid Structure Interactions: Flexible Splitter Plates
Incompressible flow equations
Finite Element Formulation (DSD/SST)Deforming Spatial Domain/Stabilized Space Time
Flow past two square cylinders with flexible splitter plates
Fluid Structure Interactions:Effects of Flexible Splitter Plates
Vorticity Field for the Ae=4✕105, s=2 case
Two square cylinders with flexible splitter plates:
Time evolution of phase difference Ae=8.05051 X 105, s=2
Two square cylinders with flexible
splitter plates: Parametric study (displacement)
Two square cylinders with flexible
splitter plates: Parametric study (lift coefficient)
Variation of frequency and amplitude with flexibility
Two square cylinders with flexible
splitter plates: Summary
At lock-in/synchronization, high amplitude oscillations & vibration frequency is close to natural frequency of plate in vacuum
non-dimensional frequency:
Reduced Velocity:
Mass ratio:
Vortex Induced Vibrations
Natural Frequencyin vacuum
Fluid Force Coefficient: CL(t)
Frequency ratio: f * = f/fn
Strouhal Number: St = f D/U
non-dimensional frequency:
Vortex Induced Vibrations (expected)
f
U
fn
fv0
Vortex shedding frequencyfor stationary cylinder
Natural frequency(in vacuum)
Expect Linear Resonance
non-dimensional frequency:
Vortex Induced Vibrations (Reality)
f
U
fn
fv0
Vortex shedding frequencyfor stationary cylinder
Natural frequency(in vacuum)
Expect Linear Resonance
frequencyfor vibrating cylinder
The vortex shedding frequency is altered over a large regime of U
Vortex Induced Vibrations
Steady flow
Vortex shedding
Vortex induced vibrations
Initial excitation branch
Lower branch
Upper branch
U*
Low Re(CFD)
Moderate Re(Experiments)
VIV: low Re v/s moderate Re
Response and flow: qualitatively different in two regimes
Critical Re for appearance of upper branch?
Maximum amplitude in two regimes is different
Initial excitation branch
Lower branch
U*
• Large amplitude vibration of the cylinder
• Matching of the frequency of cylinder vibration & fluid force
Sarpkaya (1995), Khalak and Williamson (1999)
Lock-in: accepted definition
Matching of the frequency of cylinder vibration & fluid forceto the natural frequency in vacuum.
Holds only for large m*
Earlier definition
Bishop and Hassan (1964), Feng (1968), Blevins (1990)
Lock-in: Re = 60, m* = 20
Lock-in
U* = 10.0U* = 5.0
U* = 7.5
Lock-in
fn= natural frequency of the oscillator in vacuum
Lock-in: Re = 60, m* = 20
Frequency ratio: f * = f/fn
fvo
= vortex shedding frequency for a stationary cylinder
f * ~ 1 at lock-in
Question: What determines the frequency at lock-in?
Lighthill’s decomposition (1986):
Equation of motion of the oscillator:
m = mass of the cylinderc = damping coefficientk = stiffness of the spring
Force due to vortex shedding
Added massterm
CA for a circular cylinder in ideal fluid = 1.0 (Brennen 1982)In real fluid, CA depends on the flow regime and history of the flow
(Sarpkaya 2004, Vikestad et al. 2000)
Added mass coefficient = CA = ma / md
md : mass of the fluid displaced by the cylinder
Added mass
Lock-in
In the lock-in regime CA achieves low values
Added mass modifies natural frequency of the oscillator in fluid
Added mass during lock-in
Question: What is the natural frequency of the fluid-structure system
force balance
mass conservation
Governing equations
Inertial frame:
Moving frame attached to the cylinder:
: velocity of the cylinder
: coefficient of fluid force,
: flow velocity, : stress
Flow equations
Structural equations
: structural damping,: reduced natural frequency,
: mass ratio,
: displacement of the cylinder
: density,
Decomposition:
Steady state:
flow
structure
Governing equations for coupled fluid-structure system
Linear Stability Analysis
flow
structure
Linearized Disturbance Equations
: growth rate: frequency
global eigen-mode:
eigen-value:
flow
structure
Linear Stability Analysis
m* = 20 (decoupled) m* = 5 (coupled)
SM
SM
SM
WM
WM
WM
AEMII
AEMII
AEMII
AEMI
AEMI
AEMI
Re=60: Two eigenmodes
m* = 20 (decoupled)
m* = 5 (coupled)
SM
WM
AEMI
AEMII
U* = 5.5 U* = 7.0 U* = 8.0
Re=60: Two eigenmodes
Reo : critical Reynolds number for the onset of vortex shedding ~ 47
Re/Reo = 0.8, m* = 75
Re/Reo = 1.0, m* = 75
Re/Reo = 1.2, m* = 75
Regime of coupled and decoupled modes:
f* from DTI and LSA: Re = 60, m* = 20
LSA
DTI
Linear Analysis does not explain lock-in frequency!
What does LSA (not) tell us?
Largest growth rate does not implylargest response amplitude
Instability does not imply lock-in
When two modes are unstable whichone leads to lock-in
What determines amplitude in the limit-cycle
What determines the final frequency
Lock-in
WM
SM
Re = 60, m* = 20
Energy of an eigenmodeEnergy associated with an eigenmode: Es + Ef
Es : energy of the oscillator via kinetic and potential energyEf : kinetic energy of the fluid
Evolution of disturbances:
Energy ratio:
Evolution of energy:
U* = 8.0lock-in
U* = 5.5no lock-in
Lock-in (via DTI) v/s instability (via LSA)Re = 60, m* = 20
Es = energy of the structure (kinetic energy + potential energy)
Tnl
Tnl
DTI initiated with unstable WM
Method to estimate Tnl Computations are initiated with an unstable eigenmode
Order of the convection term in base flow
Non-linearities arise from the convection terms
When do disturbances become large ?
Evolution of disturbance in linear regime:
At the end of linear regime:
Re=60, m* = 20, U* = 8.0
Tnl
Which eigenmodes lead to lock-in?
All eigenmodes with energy ratio greater than a threshold lead to lock-in in limit cycle
Lock-in
WM
SM
Re = 60, m* = 20
Ert
Which eigenmodes lead to lock-in?
All eigenmodes with energy ratio greater than a threshold lead to lock-in in limit cycle
Lock-in
WM
SM
Re = 60, m* = 20
Ert
Which mode causes lock-inRe = 60, m* = 20
SM WMWMWM
SM WMWMWM
Which eigenmodes lead to lock-in?
All linearly unstable modes do not lead to lock-in
Non-linear effects saturate the response and alter the natural frequency/added mass
Most growth in the linear regime
Energy ratio: relative energy of an eigenmode in the structure
Lock-in occurs if energy ratio larger than a threshold
Natural frequency at lock-in closer to that in vacuum (non-linear effects)
Phase angle, between CL & Y
Re = 60, m* = 20 DTI:suffers a jump from during lock-in; Prasanth & Mittal (2008):
U* = 6.0 U* = 8.0
No phase jump in LSA Phase jump in DTI occurs when phase from LSA crosses 90 deg. Non-linear effects
Conclusions
Linear Stability Analysis provides some answers
However, bulk of amplitude and frequency saturationtake place via non-linearities
Still not clear, why the fluid-structure system has affinity for close to zero added mass