introduction of particle image · pdf fileparticle image velocimetry (piv): ......
TRANSCRIPT
![Page 1: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/1.jpg)
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
![Page 2: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/2.jpg)
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
![Page 3: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/3.jpg)
IntroductionParticle Image Velocimetry (PIV)
992
1004
32
32
• divide image pair ininterrogation regions
• small region:~ uniform motion
• compute displacement• repeat !!!
![Page 4: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/4.jpg)
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
![Page 5: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/5.jpg)
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
![Page 6: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/6.jpg)
Example: coherent structures
![Page 7: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/7.jpg)
Example: coherent structures
Turbulent pipe flowRe = 5300100×85 vectors“hairpin”
vortex
![Page 8: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/8.jpg)
Example: coherent structures
Van Doorne, et al.
![Page 9: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/9.jpg)
Overview
PIV components:
- tracer particles- light source- light sheet optics- camera
- measurement settings
- interrogation- post-processing
Hardware (imaging)
Software (image analysis)
![Page 10: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/10.jpg)
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
![Page 11: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/11.jpg)
Image density
NI << 1
NI >> 1
particle tracking velocimetry
particle image velocimetry
low image density
high image density
![Page 12: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/12.jpg)
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
![Page 13: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/13.jpg)
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
![Page 14: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/14.jpg)
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)
![Page 15: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/15.jpg)
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 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
![Page 16: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/16.jpg)
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 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
![Page 17: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/17.jpg)
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 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
![Page 18: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/18.jpg)
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
![Page 19: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/19.jpg)
Cross-correlation
RD
RF
RC
correlation of the mean correlation of mean &
random fluctuations correlation due to displacement
peak: meandisplacement
![Page 20: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/20.jpg)
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
![Page 21: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/21.jpg)
Influence of NI
NI = 5 NI = 10 NI = 25
PTV: 1 particle used for velocity estimate; error ePIV: error ~ e/sqrt(NI)
![Page 22: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/22.jpg)
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
![Page 23: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/23.jpg)
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)
![Page 24: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/24.jpg)
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
![Page 25: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/25.jpg)
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
![Page 26: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/26.jpg)
Window shifting
• in-plane motion | X| < ¼ DI
strongly limits dynamic range of PIV
large window size: too much spatial averaging
![Page 27: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/27.jpg)
Window shifting
• in-plane motion | X| < ¼ DI
strongly limits dynamic range of PIV
small window size: too much in-plane pair loss
![Page 28: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/28.jpg)
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!
![Page 29: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/29.jpg)
Window shifting: Example
fixed windows matched windows
Grid turbulence
windows at same location windows at 7px „downstream‟
![Page 30: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/30.jpg)
Window shifting: Example
Vortex ring, decreasing window sizes
Raffel,Willert andKompenhans
![Page 31: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/31.jpg)
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)
![Page 32: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/32.jpg)
Sub-pixel accuracy
Fractional displacement can be obtained using the distribution of gray values around maximum
rX
![Page 33: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/33.jpg)
• 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
![Page 34: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/34.jpg)
Peak locking
“zig-zag” structure, sudden “kinks” in the flow
![Page 35: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/35.jpg)
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
![Page 36: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/36.jpg)
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
![Page 37: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/37.jpg)
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
![Page 38: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/38.jpg)
displacement measurement error
![Page 39: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/39.jpg)
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
![Page 40: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/40.jpg)
fixed windows matched windows
X = 7 px u’/U = 2.5%
application example:grid-generated turbulence
![Page 41: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/41.jpg)
Data Validation
“article” “lab”
Spurious or “Bad” vectors
![Page 42: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/42.jpg)
Spurious vectors
Three main causes:
- insufficient particle-image pairs
- in-plane loss-of-pairs, out-of-plane loss-of-pairs
- gradients
![Page 43: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/43.jpg)
Effect of tracer density
NI= 1NI= 3NI= 5NI=11NI=20NI=45NI=80
![Page 44: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/44.jpg)
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
![Page 45: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/45.jpg)
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
![Page 46: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/46.jpg)
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
![Page 47: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/47.jpg)
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
![Page 48: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/48.jpg)
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
![Page 49: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/49.jpg)
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)
![Page 50: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/50.jpg)
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
![Page 51: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/51.jpg)
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
![Page 52: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/52.jpg)
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…
![Page 53: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/53.jpg)
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 p
D3
6, the particle mass
m f f
D3
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
![Page 54: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/54.jpg)
Simple thought experiment
• Lets see how a particle responds to a step change in velocity– Only consider viscous drag
mp
dr v p
dt3 D
r u
r v p u
0 for t < 0
U for t 0
dvp
dt
18
pD2U vp
1
p
U vp
v p v p particularv p homogeneous
U C1 exp t p
v p1
p
v p
1
p
U
v p U 1 exp t p
u, vp
t
U
![Page 55: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/55.jpg)
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 f
0
exp i t d
vp t p
0
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
2pD2
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
![Page 56: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/56.jpg)
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 / f
tan 1 A(1 B)
A2 B
![Page 57: Introduction of Particle Image · PDF fileParticle Image Velocimetry (PIV): ... “Backbone” of PIV:-cross-correlation of interrogation areas ... particle image size too small peak](https://reader034.vdocuments.site/reader034/viewer/2022050807/5a78fd347f8b9a523d8ba51e/html5/thumbnails/57.jpg)
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