lisbon 2010
TRANSCRIPT
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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