Fundamentals of digital piv
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Fundamentals of Digital PIV. Partially in reference to J. Westerweel ‘s presentation. Historical development. Quantitative velocity data from particle streak photographs (1930) Laser speckle velocimetry; Young’s fringes analysis (Dudderar & Simpkins 1977)

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Fundamentals of Digital PIV

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Fundamentals of digital piv

Fundamentals of Digital PIV

Partially in reference to J. Westerweel ‘s presentation


Historical development

Historical development

  • Quantitative velocity data from particle streak photographs (1930)

  • Laser speckle velocimetry; Young’s fringes analysis (Dudderar & Simpkins 1977)

  • Particle image velocimetry

  • Interrogation by means of spatial correlation

  • ‘Digital’ PIV

  • Stereoscopic PIV; holographic PIV


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

Why use imaging?

After: A.K. Prasad, Lect. Notes short-course on PIV, JMBC 1997


Coherent structures in a tbl

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).


Piv principle

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 (<5%).

  • Typically, displacement computed through cross-correlation of IS of the two exposures.


The displacement field

Particle

trajectory

Fluid

pathline

After: Adrian, Adv. Turb. Res. (1995) 1-19

The displacement field

  • The fluid motion is represented as a displacement field


Inherent assumptions

Inherent assumptions

  • Tracer particles follow the fluid motion

  • Tracer particles are distributed homogeneously

  • Uniform displacement within interrogation region


Multiple exposure piv image

Multiple-exposure PIV image


Piv result

PIV result

Turbulent pipe flow

Re = 5300

100×85 vectors

“Hairpin” vortex


Instantaneous vorticity fields

Instantaneous vorticity fields


Visualization vs measurement

Visualization vs. Measurement


Ingredients

“Ingredients”

FLOW

sampling

seeding

quantization

Pixelization

illumination

enhancement

Acquisition

imaging

selection

registration

correlation

Interrogation

estimation

RESULT

analysis

validation


Piv optical configuration

PIV optical configuration


Piv laser

PIV Laser


Light sheet optics

Light sheet optics

(positive) cylindrical lens

(negative) cylindrical lens

(positive) spherical lens

f

f

- To obtained the desired light sheet thickness


Dpiv data processing

DPIV Data Processing


How dense should the seeding be

How dense should the seeding be?

  • Source density:

Ctracer concentration [m-3]

Dz0light-sheet thickness [m]

M0image magnification [-]

dtparticle-image diameter [m]

DIinterrogation-spot diameter [m]

Ns <1 : individual partical image

Ns > 1 : speckle pattern

  • Image density:

The image density represents the mean number of particle images in an interrogation region.For a successful PIV measurementNI > 10 - 15


Two modes of extracting velocity from tracer motion

Two modes of extracting velocity from tracer motion

Low image density

NI << 1

Particle tracking velocimetry

High image density

NI >> 1

Particle image velocimetry


Evaluation at high image density

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.


Double exposure piv recording strategies

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


Piv measurement example

PIV measurement example

Interrogation Cell

1.6mm x 1.6mm (32x32 pixels)

Correlation gives an average displacement vector.

Image Window (4x4 cm2)


Piv interrogation analysis

PIV Interrogation analysis

RP

RD+

RD-

RC+RF

Double-exposure

image

Interrogation

cell

Auto-

correlation


Spatial correlation

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


Mean intensity should be subtracted before correlation

Mean intensity should be subtracted before correlation

When mean intensity <I> is subtracted,

RC = RF =0

The mean image intensity contains no information with respect to the displacement of the particle images.


Illustration of correlation principle 1d

D

Illustration of correlation principle (1D)

Shift direction

R(s)

Shift (a variable)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Correlation peak location corresponds to the separation of the two images

Correlation peak location corresponds to the separation of the two images

D

R(s)

s

D


Illustration of correlation principle 2d

P-I

P-II

Illustration of correlation principle (2D)

R(s)

Shift in 2D

s


Fundamentals of digital piv

P-I

P-II

Match perfectly


Fundamentals of digital piv

P-I

P-II

Match perfectly

R


Fundamentals of digital piv

P-I

P-II

Partially Matched


Fundamentals of digital piv

P-I

P-II

Partially Matched

R


Fundamentals of digital piv

P-I

P-II

With Noise


Fundamentals of digital piv

P-I

P-II

With Noise

R


Sketch of cross correlation

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


Sketch of cross correlation1

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


Definition of similarity of two patterns

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)


Find velocity from double exposure images

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.


Cross correlation through fft

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.


Fundamentals of digital piv

Select interrogation window

f(m,n)

F(u,v)

FFT


Fundamentals of digital piv

Select interrogation window

f(m,n)

F(u,v)

FFT

g(m,n)

G(u,v)

FFT


Fundamentals of digital piv

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


Fundamentals of digital piv

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


Fundamentals of digital piv

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


Displacement correlation peak

Displacement-correlation peak

“random

correlations”

displacement-

correlation peak


Auto correlation

Auto-Correlation


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s


Fundamentals of digital piv

R(s)

s

Second correlation peak location corresponds to the

separation of the two images

Directional Ambiguity


Correlation peak location corresponds to the separation of the two images1

Correlation peak location corresponds to the separation of the two images

D

R(s)

s

D


Correlation peaks in different schemes

Correlation Peaks in Different Schemes

Cross-Correlation

Auto-Correlation

(Double-exposure)

Auto-Correlation

(Multi-exposure)


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