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)

• 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

z

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

Coherent structures in a TBLKim, 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

•Flow to be measured is seeded with particles •Light sheet•Camera captures two successive light pulses (small t)•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.

D X t t u X t t dt

t

t

( ; , ) ( ),

X ti ( )2

X ti ( )1

D

Particletrajectory

Fluidpathline

u X t( , )

v ti ( )

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

The displacement field

• The fluid motion is represented as a displacement field

Inherent assumptions

• Tracer particles follow the fluid motion

• Tracer particles are distributed homogeneously

• Uniform displacement within interrogation region

Multiple-exposure PIV image

PIV result

“Hairpin” vortex

Turbulent pipe flowRe = 5300100×85 vectors

Instantaneous vorticity fields

Visualization vs. Measurement

FLOW

RESULT

seeding

illumination

imaging

registration

sampling

quantization

enhancement

selection

correlation

estimation

validationanalysis

Interrogation

Acquisition

Pixelization

“Ingredients”

PIV optical configuration

PIV Laser

Light sheet optics

f f

(negative) cylindrical lens (positive) cylindrical lens

(positive) spherical lens

- To obtained the desired light sheet thickness

DPIV Data Processing

How dense should the seeding be?

• Source density: NC z

MdS

0

02

2

4

NC z

MDI I

0

02

2• Image density:

C tracer concentration [m-3]z0 light-sheet thickness [m]M0 image magnification [-]d particle-image diameter [m]DI interrogation-spot diameter [m]

The image density represents the mean number of particle images in an interrogation region.

For a successful PIV measurement NI > 10 - 15

Ns <1 : individual partical image Ns > 1 : speckle pattern

NI << 1

NI >> 1

Particle tracking velocimetry

Particle image velocimetry

Low image density

High image density

Two modes of extracting velocity from tracer motion

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 exposures on a single frame – auto-correlation - No need to transfer data within t- 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

Interrogation Cell

1.6mm x 1.6mm (32x32 pixels)

Correlation gives an average displacement vector.

Image Window (4x4 cm2)

PIV Interrogation analysis

Double-exposureimage

Interrogationcell

Auto-correlation

RP

RD+RD-

RC+RF

Spatial Correlation

XsXXs dIIR )()(ˆ)( 21

I I I The image intensities are separated into:

Mean intensity intensity fluctuation

R R R RC F D( ) ( ) ( ) ( )s s s s 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

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)

R(s)

s

XsXXs dIIR )()(ˆ)( 21

)(1 XI

)(2 XI

Shift (a variable)

Shift direction

R(s)

s

)(1 XI

)(2 sI X

0s

R(s)

s

)(1 XI

)(2 sI X

R(s)

s

)(1 XI

)(2 sI X

R(s)

s

)(1 XI

)(2 sI X

R(s)

s

)(1 XI

)(2 sI X

R(s)

s

)(1 XI

)(2 sI X

R(s)

s

Correlation peak location corresponds to the separation of the two images

)(1 XI

)(2 XI

Illustration of correlation principle (2D)

R(s)

s

XsXXs dIIR )()(ˆ)( 21

)(1 XI )(2 XI

Shift in 2D

P-I P-II

P-I

P-II

Match perfectly

P-IP-II

Match perfectly

R

P-II

Partially Matched

P-I

P-II

Partially Matched

P-I

R

P-II

With Noise

P-I

R

With Noise

P-IIP-I

Sketch of Cross-correlation

P-I

P-II

• 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

Sketch of Cross-correlation

P-I

P-II

• 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

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)

r

f g

f g

ij ijj

M

i

N

ij ijj

M

i

N

j

M

i

N

11

2 2

1111

N

i

N

iii

N

iii

gf

gfgfr

1 1

22

1

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

• Direct cross-correlation (in space domain)

– (m,n) is the displacement

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

N

k

M

l

N

k

M

l

N

k

M

lfg

lkglkf

nlmkglkfnm

1 1

2

1 1

2

1 1

),(),(

),(),(),(

f(m,n)FFT

F(u,v)

Select interrogation window

f(m,n)FFT

F(u,v)

g(m,n)FFT

G(u,v)

Select interrogation window

f(m,n)FFT

F(u,v)

g(m,n)FFT

G(u,v)

FT of Cross-correlation

’(u,v)=F(u,v)G*(u,v)

Select interrogation window

f(m,n)FFT

F(u,v)

g(m,n)FFT

G(u,v)

FT ofCross-correlation

’(u,v)=F(u,v)G*(u,v)

Select interrogation window

’(u,v)

FFT-1

f(m,n)FFT

F(u,v)

g(m,n)FFT

G(u,v)

FT ofCross-correlation

’(u,v)=F(u,v)G*(u,v)

’(u,v)

FFT-1

Find x, y then convert

to velocity

Select interrogation window

’(m,n) = f(m,n) g(m,n)

Peak detection

Displacement-correlation peak

R s R s R sD D D( ) ( ) ( )

displacement-correlation peak

“random correlations”

Auto-Correlation

)(1 XI

)(2 sI X

0s

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

)(1 XI

)(2 sI X

R(s)

s0

Second correlation peak location corresponds to the separation of the two images

Directional Ambiguity

R(s)

s

Correlation peak location corresponds to the separation of the two images

)(1 XI

)(2 XI

Correlation Peaks in Different Schemes

Cross-Correlation Auto-Correlation(Double-exposure)

Auto-Correlation(Multi-exposure)