Fundamentals of Digital PIV

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Fundamentals of Digital PIV

• Partially in reference to J. Westerweel s
  presentation

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Historical 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

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Why 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
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Coherent 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).
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PIV 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.

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The displacement field

• The fluid motion is represented as a displacement
  field

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Inherent assumptions

• Tracer particles follow the fluid motion
• Tracer particles are distributed homogeneously
• Uniform displacement within interrogation region

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Multiple-exposure PIV image
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PIV result
Turbulent pipe flow Re 5300 10085 vectors
Hairpin vortex
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Instantaneous vorticity fields
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Visualization vs. Measurement
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Ingredients
FLOW
sampling
seeding
quantization
Pixelization
illumination
enhancement
Acquisition
imaging
selection
registration
correlation
Interrogation
estimation
RESULT
validation
analysis
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PIV optical configuration
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PIV Laser
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Light sheet optics
(negative) cylindrical lens
(positive) cylindrical lens
(positive) spherical lens
f
f
- To obtained the desired light sheet thickness
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DPIV Data Processing
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How dense should the seeding be?

• Source density

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

• Image density

The image density represents the mean number of
particle images in an interrogation region. For
a successful PIV measurement NI gt 10 - 15
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Two 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
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Evaluation 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.
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Double-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

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PIV measurement example
Interrogation Cell 1.6mm x 1.6mm (32x32
pixels) Correlation gives an average displacement
vector.
Image Window (4x4 cm2)
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PIV Interrogation analysis
RP
RD
RD-
RCRF
Double-exposure image
Interrogation cell
Auto- correlation
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Spatial 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
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Mean 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.
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Illustration of correlation principle (1D)
Shift direction
R(s)
Shift (a variable)
s
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R(s)
s
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R(s)
s
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R(s)
s
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R(s)
s
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R(s)
s
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R(s)
s
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Correlation peak location corresponds to the
separation of the two images
D
R(s)
s
D
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Illustration of correlation principle (2D)
R(s)
Shift in 2D
s
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Match perfectly
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Match perfectly
R
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Partially Matched
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Partially Matched
R
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With Noise
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With Noise
R
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Sketch 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
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Sketch 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
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Definition 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)

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Find 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.

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Cross-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.

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Select interrogation window
f(m,n)
F(u,v)
FFT
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Select interrogation window
f(m,n)
F(u,v)
FFT
g(m,n)
G(u,v)
FFT
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Select 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
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Select 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
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Select 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
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Displacement-correlation peak
random correlations
displacement- correlation peak
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Auto-Correlation
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Second correlation peak location corresponds to
the separation of the two images
Directional Ambiguity
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Correlation peak location corresponds to the
separation of the two images
D
R(s)
s
D
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Correlation Peaks in Different Schemes
Cross-Correlation
Auto-Correlation (Double-exposure)
Auto-Correlation (Multi-exposure)