introduction of particle image velocimetry...particle can be matched with a number of candidates...
TRANSCRIPT
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Introduction of Particle Image Velocimetry
Slides largely generated by J. Westerweel & C. Poelma of Technical University of Delft
Adapted by K. Kiger
Ken Kiger
Burgers Program For Fluid DynamicsTurbulence School
College Park, Maryland, May 24-27
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IntroductionParticle Image Velocimetry (PIV):
Imaging of tracer particles, calculate displacement: local fluid velocity
Twin Nd:YAGlaser CCD camera
Light sheetoptics
Frame 1: t = t0
Frame 2: t = t0 + t
Measurementsection
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IntroductionParticle Image Velocimetry (PIV)
992
1004
32
32
• divide image pair ininterrogation regions
• small region:~ uniform motion
• compute displacement• repeat !!!
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IntroductionParticle Image Velocimetry (PIV):
Instantaneous measurement of 2 components in a plane
conventional methods(HWA, LDV)
• single-point measurement• traversing of flow domain• time consuming• only turbulence statistics
z
particle image velocimetry
• whole-field method• non-intrusive (seeding• instantaneous flow field
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IntroductionParticle Image Velocimetry (PIV):
Instantaneous measurement of 2 components in a plane
particle image velocimetry
• whole-field method• non-intrusive (seeding• instantaneous flow field
instantaneous vorticity field
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Example: coherent structures
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Example: coherent structures
Turbulent pipe flowRe = 5300100×85 vectors“hairpin”
vortex
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Example: coherent structures
Van Doorne, et al.
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Overview
PIV components:
- tracer particles- light source- light sheet optics- camera
- measurement settings
- interrogation- post-processing
Hardware (imaging)
Software (image analysis)
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Tracer particles
Assumptions:
- homogeneously distributed- follow flow perfectly- uniform displacement within interrogation region
Criteria:
-easily visible-particles should not influence fluid flow!
small, volume fraction < 10-4
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Image density
NI > 1
particle tracking velocimetry
particle image velocimetry
low image density
high image density
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Assumption: uniform flow in “interrogation area”
Evaluation at higher density
High NI : no longer possible/desirable to follow individual tracer particles
Particle can be matched with a number of candidates
Possible „matches‟
Repeat process for other particles, sum up: “wrong” combinations will lead to noise, but “true” displacement will dominate
Sum of all possibilities
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Statistical estimate of particle motion
Statistical correlations used to find
average particle displacement1-d image @ t=t0
1-d image @ t=t1
R(i, j)
Ia (k,l) I a Ib (k i,l j) I bl 1
By
k 1
Bx
Ia (k,l) I a2
Ib (k i,l j) I b2
l 1
By
k 1
Bx
l 1
By
k 1
Bx
1
2
x yB
k
B
l
a
yx
a lkIBB
I1 1
),(1
Cross-correlation
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1-D cross-correlation example
R(i)
Ia (k) I a Ib (k i) I bk 1
Bx
Ia (k) I a2
Ib (k i) I b2
k 1
Bx
k 1
Bx
1
2
Ia
Ib
Ia(x) normalized
Ib(x+ x) normalized
Ia(x)*Ib(x+ x)
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Finding the maximum displacement
-Shift 2nd window with respect to the first
- Calculate “match”
- Repeat to find best estimate
Typically 16x16 or 32x32 pixels
Good indicator: R(i, j)Ia (k,l) I a Ib (k i,l j) I b
l 1
By
k 1
Bx
Ia (k,l) I a2
Ib (k i,l j) I b2
l 1
By
k 1
Bx
l 1
By
k 1
Bx
1
2
x yB
k
B
l
a
yx
a lkIBB
I1 1
),(1
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Finding the maximum displacement
Bad match: sum of product of intensities low
-Shift 2nd window with respect to the first
- Calculate “match”
- Repeat to find best estimate
Typically 16x16 or 32x32 pixels
Good indicator: R(i, j)Ia (k,l) I a Ib (k i,l j) I b
l 1
By
k 1
Bx
Ia (k,l) I a2
Ib (k i,l j) I b2
l 1
By
k 1
Bx
l 1
By
k 1
Bx
1
2
x yB
k
B
l
a
yx
a lkIBB
I1 1
),(1
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Finding the maximum displacement
Good match: sum of product of intensities high
-Shift 2nd window with respect to the first
- Calculate “match”
- Repeat to find best estimate
Can be implemented as 2D FFT for digitized data
o Impose periodic conditions on interrogation region…causes bias error if not treated properly.
Typically 16x16 or 32x32 pixels
Good indicator: R(i, j)Ia (k,l) I a Ib (k i,l j) I b
l 1
By
k 1
Bx
Ia (k,l) I a2
Ib (k i,l j) I b2
l 1
By
k 1
Bx
l 1
By
k 1
Bx
1
2
x yB
k
B
l
a
yx
a lkIBB
I1 1
),(1
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Cross-correlation
This “shifting” method can formally be expressed as a cross-correlation:
1 2( )R I I ds x x s x
- I1 and I2 are interrogation areas (sub-windows) of the total frames- x is interrogation location- s is the shift between the images
“Backbone” of PIV:-cross-correlation of interrogation areas-find location of displacement peak
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Cross-correlation
RDRF
RCcorrelation of the mean correlation of mean &random fluctuations correlation due
to displacement
peak: meandisplacement
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Influence of NI
NI = 5 NI = 10 NI = 25
More particles: better signal-to-noise ratio
Unambiguous detection of peak from noise:NI=10 (average), minimum of 4 per area in 95% of areas(number of tracer particles is a Poisson distribution)
R N NC z
MDD D I I I( ) ~s
0
0
2
2
C particle concentrationz0 light sheet thickness
DI int. area sizeM0 magnification
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Influence of NI
NI = 5 NI = 10 NI = 25
PTV: 1 particle used for velocity estimate; error ePIV: error ~ e/sqrt(NI)
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Influence of in-plane displacement
X / DI = 0.00FI = 1.00
0.280.64
0.560.36
0.850.16
II
IIIDDD
Y
D
XYXFFNR 11),(~)(s
X,Y-Displacement<quarter of window size
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Influence of out-of-plane displacement
Z / z0 = 0.00FO = 1.00
0.250.75
0.500.50
0.750.25
R N F F F zz
zD D I I O O( ) ~ ( )s 1
0
Z-Displacement<quarter of light sheet thickness( z0)
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Influence of gradients
Displacement differences <3-5% of int. area size, DI
Displacement differences <Particle image size, d
a / DI = 0.00a / d = 0.00
0.050.50
0.101.00
0.151.50
R N F F F F a a dD D I I O( ) ~ ( ) exp( / )s2 2
a M0 u t
R.D. Keane & R.J. Adrian
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PIV “Design rules”
• image density NI >10• in-plane motion | X| < ¼ DI• out-of-plane motion | z| < ¼ z0• spatial gradients M0 u t < d
Obtained by Keane & Adrian (1993) using synthetic data
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Window shifting
• in-plane motion | X| < ¼ DI
strongly limits dynamic range of PIV
large window size: too much spatial averaging
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Window shifting
• in-plane motion | X| < ¼ DI
strongly limits dynamic range of PIV
small window size: too much in-plane pair loss
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Window shifting
• in-plane motion | X| < ¼ DI
strongly limits dynamic range of PIV
Multi-pass approach:start with large windows,use this result as „pre-shift‟for smaller windows…
No more in-plane pair loss limitations!
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Window shifting: Example
fixed windows matched windows
Grid turbulence
windows at same location windows at 7px „downstream‟
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Window shifting: Example
Vortex ring, decreasing window sizes
Raffel,Willert andKompenhans
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Sub-pixel accuracy
Maximum in the correlation plane: single-pixel resolution of displacement?
But the peak contains a lot more information!
Gaussian particle images Gaussian correlation peak (but smeared)
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Sub-pixel accuracy
Fractional displacement can be obtained using the distribution of gray values around maximum
rX
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• peak centroid
• parabolic peak fit
• Gaussian peak fit
R R
R R R
1 1
1 0 1
R R
R R R
1 1
1 1 02 2
ln ln
ln ln ln
R R
R R R
1 1
1 1 02 2
balance
normalization
three-point estimators
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Peak locking
“zig-zag” structure, sudden “kinks” in the flow
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Sub-pixel Interpolation Errors
• Accuracy depends on:– particle image size
– noise in data (seeding density, camera noise)
– shear rate
• Can exhibit “peak locking”– Interpolation of peak is biased towards a
symmetric data distribution (Integer and 1/2
integer peak locations)
– Polynomials exhibit strong locking when
particle diameter is small
– Gaussian is most commonly used
– Splines are very robust, but expensive to
calculate
• See Particle Image Velocimetry, by Raffel, Willert, and Kompenhans, Springer-Verlag, 1998.
-0.5 -0.25 0 0.25 0.5-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Actual peak location (pixels)
Inte
rpola
tion e
rror
(pix
els
)
Polynomial fitGaussian fit
-0.5 -0.25 0 0.25 0.5-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Actual peak location (pixels)
Inte
rpola
tion e
rror
(pix
els
)
Polynomial fitGaussian fit
-0.5 -0.25 0 0.25 0.5-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Actual peak location (pixels)
Inte
rpola
tion e
rror
(pix
els
)
Polynomial fitGaussian fit
R = 0.5
R = 1.0
R = 1.5
Pixel Radius
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Peak locking
centroid Gaussian peak fitEven with Gaussian peak fit:
particle image size too small peak locking(Consider a „point particle‟ sampled by discrete pixels)
Histogram of velocities in a turbulent flow
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Sub-pixel accuracy
optimal resolution: particle image size: ~2 pxSmaller: particle no longer resolvedLarger: random noise increase
“three-point” estimators:
Peak centroidParabolic peak fitGaussian peak fit... Main difference: sensitivity to “peak locking”
or “pixel lock-in”, bias towards integer displacements
Theoretical: 0.01 – 0.05 pxIn practice 0.05-0.1 px
bias errors random errorstotalerror
d / dr
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displacement measurement error
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fixed windows matched windows
signal
noise
SNR
u’2
C2
u’2 / C2
u’2
4C2u’2
1 / 4C2
FI ~ 0.75 FI ~ 1
velocity pdf
measurementerror
window matching
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fixed windows matched windows
X = 7 px u’/U = 2.5%
application example:grid-generated turbulence
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Data Validation
“article” “lab”
Spurious or “Bad” vectors
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Spurious vectors
Three main causes:
- insufficient particle-image pairs
- in-plane loss-of-pairs, out-of-plane loss-of-pairs
- gradients
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Effect of tracer density
NI= 1NI= 3NI= 5NI=11NI=20NI=45NI=80
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Remedies
• increase NI• practical limitations:
• optical transparency of the fluid• two-phase effects• image saturation / speckle
• detection, removal & replacement• keep finite NI ( ~ 0.05 )
• data loss is small• signal loss occurs in isolated points• data recovery by interpolation
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Detection methods
• human perception• peak height
• amount of correlated signal• peak detectability
• peak height relative to noise• lower limit for SNR
• residual vector analysis• fluctuation of displacement
• multiplication of correlation planes• fluid mechanics
• continuity• fuzzy logic & neural nets
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Residual analysis
• evaluate fluctuation of measured velocity residual • ideally: Uref = true velocity• reference values:
• Uref = global mean velocity• comparable to 2D-histogram analysis• does not take local coherent motion into account• probably only works in homogeneous turbulence
• Uref = local (3×3) mean velocity• takes local coherent motion into account• very sensitive to outliers in the local neighborhood
• Uref = local (3×3) median velocity• almost identical statistical properties as local mean• Strongly suppressed sensitivity to outliers in heighborhood
refUUr
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Example of residual test
0 1
234
5
6 7 8
0 1 2 3 4 5 6 7 8
2.3 2.2 3.0 3.7 3.1 3.2 2.4 3.5 2.7
2.2 2.3 2.4 2.7 3.0 3.1 3.2 3.5 3.7
RMS
0.53
RMS
2.29
2.3 9.7 3.0 3.7 3.1 3.2 2.4 3.5 2.7
2.3 2.4 2.7 3.0 3.1 3.2 3.5 3.7 9.7
Mean
2.9
Mean
3.7
Standard mean and r.m.s. are very sensitive to bad data contamination… need robust measure of fluctuation
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Median test
1 2 3
4 0 5
6 7 8
1 - Calculate reference velocity: median of 8 neighbors
2 – calculate residuals:
3 – Normalize target residual by: median(ri) +
4 – Robust measure found for:
uref median u1,u2,...u8
ri ui uref
r0*
u0 uref
median(ri)
0.1 and r0* 2
Wes
terw
eel &
Sca
rano
, 200
5
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Interpolation
Bilinear interpolation satisfies continuity
For 5% bad vectors, 80% of the vectors are isolated
Bad vector can be recovered without any problems
N.B.: interpolation biases statistics (power spectra, correlation function)Better not to replace bad vectors (use e.g. slotting method)
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Overlapping windows
Method to increase data yield:
Allow overlap between adjacent interrogation areas
aMotivation: particle pairs near edgescontribute less to correlation result;Shift window so they are in the center:additional, relatively uncorrelated result
50% is very common, but beware of oversampling
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A Generic PIV program
Data acquisition
Image pre-processing
PIV cross-correlation
Vector validation
Post-processing
Laser control,Camera settings, etc.
Reduce non-uniformityof illumination; Reflections
Pre-shift; Decreasingwindow sizes
Vorticity, interpolationof missing vectors, etc.
Median test,Search window
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PIV softwareFree
PIVware: command line, linux (Westerweel)JPIV: Java version of PIVware (Vennemann)MatPiv: Matlab PIV toolbox (Cambridge, Sveen)URAPIV: Matlab PIV toolbox (Gurka and Liberzon)DigiFlow (Cambridge), PIV Sleuth (UIUC), MPIV, GPIV, CIV, OSIV,…
Commercial
PIVtec PIVviewTSI InsightDantec FlowmapLaVision DaVisOxford Lasers/ILA VidPIV…
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Particle Motion: tracer particle
• Equation of motion for spherical particle:
– Where
– Neglect: non-linear drag (only really needed for high-speed flows), Basset history term (higher order effect)
gmdt
udm
dt
vd
dt
udmvuD
dt
vdm
p
g
pf
p
fp
p
p
1213
mp pD3
6, the particle mass
m f fD3
6, fluid mass of same volume as particle
D particle diameter
fluid viscosityr u fluid velocityr v p particle velocity
g fluid density
p particle material density
Added massViscous drag Pressure gradient
buoyancyInertia
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Simple thought experiment
• Lets see how a particle responds to a step change in velocity– Only consider viscous drag
mpd
r v p
dt3 D
r u
r v p u
0 for t < 0
U for t 0
dvp
dt
18
pD2
U vp1
p
U vp
v p v p particularv p homogeneous
U C1 exp t p
v p1
p
v p1
p
U
v p U 1 exp t p
u, vp
t
U
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Particle Transfer Function
• Useful to examine steady-state particle response to 1-D oscillating flow of arbitrary sum of frequencies– Represent u as an infinite sum of harmonic functions
– Neglect gravity (DC response, not transient)
u t f0
exp i t d
vp t p0
exp i t d
2mp
3 D
dv p
dt2 u v p
m f
mp
mp
3 D
du
dt
dv p
dt2
m f
mp
mp
3 D
du
dt
du t
dti f
0
exp i t d
2pD
2
18pi exp i t
0
d 2 f p exp i t0
df
p
pD2
18i 3 f p exp i t d
0
2iSt p 2 f p i St 3 f p
dvp t
dti p
0
exp i t d
Stp
f
pD2
18
f
p
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Particle Transfer Function
• This can be rearranged:
• Examine– Liquid in air, gas in water, plastic in water
r v pr u
p p
f f
1 2
A2 B2
A2 1
1 2
23
22
B
StA
tan 1Im p / f
Re p / ftan 1
A(1 B)
A2 B
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Liquid particles in Air
– Liquid drops in air require St~0.3 for 95% fidelity (1 m ~ 15 kHz)
– Size relatively unimportant for near-neutrally buoyant particles
– Bubbles are a poor choice: always overrespond unless quite small