Title: Fundamentals of Digital PIV
1Fundamentals of Digital PIV
- Partially in reference to J. Westerweel s
presentation
2Historical development
- Quantitative velocity data from particle streak
photographs (1930) - Laser speckle velocimetry Youngs fringes
analysis (Dudderar Simpkins 1977) - Particle image velocimetry
- Interrogation by means of spatial correlation
- Digital PIV
- Stereoscopic PIV holographic PIV
3Why use imaging?
- Conventional methods
- (HWA, LDV)
- Single-point measurement
- Traversing of flow domain
- Time consuming
- Only turbulence statistics
- Particle image velocimetry
- Whole-field method
- Non-intrusive (seeding)
- Instantaneous flow field
After A.K. Prasad, Lect. Notes short-course on
PIV, JMBC 1997
4Coherent structures in a TBL
Kim, H.T., Kline, S.J. Reynolds, W.C. J. Fluid
Mech. 50 (1971) 133-160.
Smith, C.R. (1984) A synthesized model of the
near-wall behaviour in turbulent boundary
layers. In Proc. 8th Symp. on Turbulence
(eds. G.K. Patterson J.L. Zakin) University of
Missouri (Rolla).
5PIV principle
- Flow to be measured is seeded with particles
- Light sheet
- Camera captures two successive light pulses
(small Dt) - Double-exposed image provides a 2D displacement
record of the particles within measurement plane - PIV images are analyzed over a pointwise grid of
local interrogation spots (IS). - Size of IS large enough to include a sufficient
number of particle image pairs, but small enough
so there is little variation in velocity across
IS (lt5). - Typically, displacement computed through
cross-correlation of IS of the two exposures.
6The displacement field
- The fluid motion is represented as a displacement
field
7Inherent assumptions
- Tracer particles follow the fluid motion
- Tracer particles are distributed homogeneously
- Uniform displacement within interrogation region
8Multiple-exposure PIV image
9PIV result
Turbulent pipe flow Re 5300 10085 vectors
Hairpin vortex
10Instantaneous vorticity fields
11Visualization vs. Measurement
12Ingredients
FLOW
sampling
seeding
quantization
Pixelization
illumination
enhancement
Acquisition
imaging
selection
registration
correlation
Interrogation
estimation
RESULT
validation
analysis
13PIV optical configuration
14PIV Laser
15Light sheet optics
(negative) cylindrical lens
(positive) cylindrical lens
(positive) spherical lens
f
f
- To obtained the desired light sheet thickness
16DPIV Data Processing
17How dense should the seeding be?
C tracer concentration m-3 Dz0 light-sheet
thickness m M0 image magnification
- dt particle-image diameter m DI interrogatio
n-spot diameter m
Ns lt1 individual partical image Ns
gt 1 speckle pattern
The image density represents the mean number of
particle images in an interrogation region. For
a successful PIV measurement NI gt 10 - 15
18Two modes of extracting velocity from tracer
motion
Low image density
NI ltlt 1
Particle tracking velocimetry
High image density
NI gtgt 1
Particle image velocimetry
19Evaluation at high image density
At high image density, corresponding particle
image cannot be identified by means of
proximity. Consider a single particle image, and
determine the distance histogram of all possible
match candidates. Each match has an equal
probability, but only one match will be
correct. When this is done for all particle
images, only the matching particle-images pairs
will add up, whereas the random unrelated
particles will not, and a sharp peak will appear
that reflects the displacement of the
particle-image pattern. The histogram analysis is
equivalent to the spatial correlation. The
histogram analysis has actually been proposed for
analysis, but it is not as effective as the
spatial correlation analysis.
20Double-exposure PIV Recording Strategies
- Double exposures on a single frame
auto-correlation - - No need to transfer data within Dt
- - Directional ambiguity of displacement
- - Cannot detect small displacements
- Single exposures on separate frames
cross-correlation - - Fast data transfer, or use cross-correlation
camera - - No directional ambiguity
- - Small displacements detectable
-
21PIV measurement example
Interrogation Cell 1.6mm x 1.6mm (32x32
pixels) Correlation gives an average displacement
vector.
Image Window (4x4 cm2)
22PIV Interrogation analysis
RP
RD
RD-
RCRF
Double-exposure image
Interrogation cell
Auto- correlation
23Spatial Correlation
The image intensities are separated into
Mean intensity
intensity fluctuation
The spatial correlation can be separated into
three terms
RC -- mean background correlation RF --
correlation between mean intensity and intensity
fluctuations RD -- correlation of image
fluctuations
24Mean intensity should be subtracted before
correlation
When mean intensity ltIgt is subtracted, RC RF 0
The mean image intensity contains no information
with respect to the displacement of the particle
images.
25Illustration of correlation principle (1D)
Shift direction
R(s)
Shift (a variable)
s
26R(s)
s
27R(s)
s
28R(s)
s
29R(s)
s
30R(s)
s
31R(s)
s
32Correlation peak location corresponds to the
separation of the two images
D
R(s)
s
D
33Illustration of correlation principle (2D)
R(s)
Shift in 2D
s
34Match perfectly
35Match perfectly
R
36Partially Matched
37Partially Matched
R
38With Noise
39With Noise
R
40Sketch of Cross-correlation
- Form a pattern in the 1st image (P-I)
- Form a number of patterns within the selected
domain in the 2nd image (P-II) - Compare P-I to all P-IIs
- The two most similar patterns are picked up
P-II
P-II
P-I
41Sketch of Cross-correlation
- Form a pattern in the 1st image (P-I)
- Form a number of patterns within the selected
domain in the 2nd image (P-II) - Compare P-I to all P-IIs
- The two most similar patterns are picked up
P-II
P-II
P-I
42Definition of similarity of two patterns
- Similarity of two vectors production of two
vectors - Similarity of two patterns, f and g are gray
level distributions in 1st image and 2nd image,
respectively. (N and M are the width and height
of the patterns)
43Find velocity from double-exposure images
- Select a window (pattern) P-I in the 1st image.
- Select a domain in the 2nd image where the
pattern matching between P-I and P-II is to be
undertaken. - Compare P-I to all P-IIs in the domain, two
patterns that show maximum similarity value are
identical. - Displacement between two centers of two pattern
is the average velocity of the window. - Note
- Selected window is called interrogation window or
interrogation cell - Evaluation of similarity cross-correlation
coefficient - The method needs (NM)2 computation time
inefficient.
44Cross-correlation through FFT
- Direct cross-correlation (in space domain)
- (m,n) is the displacement
- Correlation via FFT (in frequency domain).
Advantage reduce the computation time.
45Select interrogation window
f(m,n)
F(u,v)
FFT
46Select interrogation window
f(m,n)
F(u,v)
FFT
g(m,n)
G(u,v)
FFT
47Select interrogation window
f(m,n)
F(u,v)
FT of Cross-correlation F(u,v) F(u,v)G(u,v)
FFT
g(m,n)
G(u,v)
FFT
48Select interrogation window
f(m,n)
F(u,v)
FT of Cross-correlation F(u,v) F(u,v)G(u,v)
FFT
g(m,n)
G(u,v)
FFT
F(u,v)
FFT-1
49Select interrogation window
f(m,n)
F(u,v)
FT of Cross-correlation F(u,v) F(u,v)G(u,v)
FFT
g(m,n)
G(u,v)
FFT
F(u,v)
f(m,n) f(m,n) ? g(m,n)
FFT-1
Peak detection
Find Dx, Dy then convert to velocity
50Displacement-correlation peak
random correlations
displacement- correlation peak
51Auto-Correlation
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59Second correlation peak location corresponds to
the separation of the two images
Directional Ambiguity
60Correlation peak location corresponds to the
separation of the two images
D
R(s)
s
D
61Correlation Peaks in Different Schemes
Cross-Correlation
Auto-Correlation (Double-exposure)
Auto-Correlation (Multi-exposure)