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