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Uncovering large-scale coherentstructures in natural and forcedturbulent wakes by combiningPIV, POD and FTLE

Particle Image Velocimetry, Proper Orthogonal

Decomposition, Finite Time Lyapunov Exponent

Leonidas Kourentis and

Efstathios Konstantinidis*

Department of Mechanical Engineering

University of Western Macedonia, Kozani, Greece

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Fluid mechanics principles

Eulerian description

Study of the fluid motion in

a specified control volume

Coordinate system fixed in

space (laboratory frame ofreference)

No information on dynamics

of fluid elements (particles)

Lagrangian description

Study of the motion of fluid

particles moving with the

flow

Coordinate system not fixedin space (moving frame of

reference)

Information on dynamics of

fluid particles depends ontheir initial position

Eduction of coherent structures is afundamental issue in fluid mechanics

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Coherent structures in the wakeof cylinders in cross-flow Primary flow instability

associated with vortexformation and shedding

Important role for the

transport of momentum andheat/mass in natural andtechnological processes

Dye visualization provides agood indication at lowReynolds numbers

At high Reynolds numbers,turbulent dispersion rendersabove method ineffective

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Particle methods

Particle visualization can

provide limited information

on coherent structures

Long-exposure images of

fluorescent particles canenhance description

Tracing the motion of many

individual particles yet

cumbersome

PIV: no information on

Lagrangian dynamicsImages of reflecting particles in the

forced wake of a circular cylinder

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Motion of particles

Perturbation

Deformation tensor Cauchy-Green (flow map)

Lyapunov exponent

Ridges in FTLE field reveal Lagrangian Coherent Structures (LCS)

Developed by Haller (2002) and Shadden et al. (2006)

Integration in time

Backward attractive LCS (aLCS)

Forward repelling LCS (rLCS)

Finite-Time Lyapunov Exponent (FTLE)0 0 0 0

0 0 0 0

( ; , ) ( ( ; , ), )

( ; , )

t t t t t

t t

x x v x x

x x x

0

0 0( ) ( ; , )t

t t t 0x x x d ( , , ) d ( , , )

d d

x t T x t T

x x

T

0( )

maxmax ( ) (0) (0)Tt TT e

xx x x

0 max

1( ) ln ( )T

tT

x

(0) y x x(0)x

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Verification of FTLE methodology

Plots show results derived from

computational fluid dynamics

(CFD) using an in-house code

with weightless particles injectedinto the flow

FTLE integration time = 4 periods

180oU D

Re

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Aims and objectives

Determine FTLE distributions in the cylinder

wake using reconstructed PIV data

Provide an insightful quantitative visualization of

the large-scale coherent structures associatedwith vortex shedding

Understand the effect of forced perturbations on

the mechanism of vortex formation and shedding

in the lock-on regime

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Basic configuration

Inflow with superimposedperiodic velocity oscillations

Circular cylinderperpendicular to

the incident flow

Main parameters:

Present results: 2150 2.02 0.23

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Vortex shedding lock-on

0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

=

InFlow Oscillations

Armstong et al. (1986), Re = 21

500

Barbi et al. (1986), Re = 3

000

Barbi et al. (1986), Re = 40

000

Konstantinidis et al (2003), Re = 2150

Konstantinidis et al, Re = 2150

Cylinder Oscillations

Tatsuno (1972), Re = 100

Tanida et al. (1973), Re = 80

Tanida et al. (1973), Re = 4

000

Griffin & Ramberg (1976), Re = 190

Nishihara et al (2004), Re = 17000

aU

d 2f

ed

primary

lock-on

range

fe/f

o

Reynolds number dependence ?

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Experimental setup

Water tunnel (72mm x 72mm) at Kings College

Pulsating inflow (rotating slot valve, adjustablefrequency and amplitude)

Circular cylinder (aspect ratio = 10, no end-plates)

Phase-referencing (optical shaft encoder)

Laser-Doppler Velocimetry (high temporal resolution)

PIV (2C, high spatial resolution, optimized settings)

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PIV snapshots unforced flow

Uncorrelated instantaneous velocity fields

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PIV snapshots forced flow

Modification of vortex characteristics

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x/d

y/d

0 1 2 3

-1

0

1

Effect of forcing on time-averagedwake structure

x/d

y/d

0 1 2 3

-1

0

1

x/d

y/d

0 1 2 3 4

-1

0

1

U/Um= 0.23

0.36

x/d

y/d

0 1 2 3 4

-1

0

1

U/Um= 0.00

0.16

x/d

y/d

0 1 2 3 4

-1

0

1

U/Um

=0.00

0.094

x/d

y/d

0 1 2 3 4

-1

0

1

U/Um

=0.23

0.20

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Proper Orthogonal Decomposition (POD)

Orthogonal (Empirical) Basis

Functions

Method of Snapshots(Sirovich, 1987)

Basic spatio-temporal modes

ranked according todecreasing kinetic energy

(given by the eigenvalues of

the corresponding modes)

1

0

( , ) ( ) ( )M

i k i k

k

t a t

v x x

C A A

1

0

( ) ( , )M

j j k k

k

a t t

v x

1

( , ) ( , )M

ij i j

k

c t t

v x v x

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Energy spectrum

Threshold for noise corruption (Epps & Techet, 2010)

= 0.036, N =450, M = 4584

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Spatial structure of POD modes

Natural wake Forced wake

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Phase reconstruction

Frequency obtained from time-resolved LDV data

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Reconstructed velocity/vorticitydata and derived FTLE fields

Naturalwake

Forcedwake

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FTLE distributions

Natural wake

Forced wake

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Comparison

Laminar wake

Turbulent wake(low-order dynamics)

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Combination of attracting and repelling

coherent structures in the natural wake

aLCS

rLCS

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Combination of attracting and repelling

coherent structures in the forced wake

aLCS

rLCS

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Conclusions The FTLE method can be successfully employed to uncover large-

scale coherent structures Visualization of the Bernhard von Karman vortex street in both natural and

forced turbulent wakes

Provides additional information on the vortex formation mechanism

The spatio-temporal dynamics characterising the global flow

organization in the cylinder wake can be captured by the two mostenergetic POD modes

Basic vortex structure is not affected by forced excitation (actually enhanced)

but the driving mechanism of the formation process is different

Dynamics of coherent structures in cylinder wakes are remarkably

similar over a wide range of Reynolds numbers from 200 - 200

103

Modelling of the wake dynamics is important for practical applications, e.g.

prediction of vortex-induced vibrations