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Fundamentals of Digital PIV

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Fundamentals of Digital PIV

Partially in reference to J. Westerweel ‘s presentation

- 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

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

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).

- 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.

Particle

trajectory

Fluid

pathline

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

- The fluid motion is represented as a displacement field

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

Turbulent pipe flow

Re = 5300

100×85 vectors

“Hairpin” vortex

FLOW

sampling

seeding

quantization

Pixelization

illumination

enhancement

Acquisition

imaging

selection

registration

correlation

Interrogation

estimation

RESULT

analysis

validation

(positive) cylindrical lens

(negative) cylindrical lens

(positive) spherical lens

f

f

- To obtained the desired light sheet thickness

DPIV Data Processing

- 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

Low image density

NI << 1

Particle tracking velocimetry

High image density

NI >> 1

Particle image velocimetry

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 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

Interrogation Cell

1.6mm x 1.6mm (32x32 pixels)

Correlation gives an average displacement vector.

Image Window (4x4 cm2)

RP

RD+

RD-

RC+RF

Double-exposure

image

Interrogation

cell

Auto-

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

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.

D

Shift direction

R(s)

Shift (a variable)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

D

R(s)

s

D

P-I

P-II

R(s)

Shift in 2D

s

P-I

P-II

Match perfectly

P-I

P-II

Match perfectly

R

P-I

P-II

Partially Matched

P-I

P-II

Partially Matched

R

P-I

P-II

With Noise

P-I

P-II

With Noise

R

- 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

- 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

- 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)

- 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.

Direct cross-correlation (in space domain)

(m,n) is the displacement

Correlation via FFT (in frequency domain). Advantage: reduce the computation time.

Select interrogation window

f(m,n)

F(u,v)

FFT

Select interrogation window

f(m,n)

F(u,v)

FFT

g(m,n)

G(u,v)

FFT

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

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

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

“random

correlations”

displacement-

correlation peak

R(s)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

R(s)

s

Second correlation peak location corresponds to the

separation of the two images

Directional Ambiguity

D

R(s)

s

D

Cross-Correlation

Auto-Correlation

(Double-exposure)

Auto-Correlation

(Multi-exposure)