introduction of particle image · pdf fileparticle image velocimetry (piv): ......

Introduction of Particle Image Velocimetry

Slides largely generated by J. Westerweel & C. Poelma of Technical University of Delft

Adapted by K. Kiger

Ken Kiger

Burgers Program For Fluid DynamicsTurbulence School

College Park, Maryland, May 24-27

IntroductionParticle Image Velocimetry (PIV):

Imaging of tracer particles, calculate displacement: local fluid velocity

Twin Nd:YAGlaser CCD camera

Light sheetoptics

Frame 1: t = t0

Frame 2: t = t0 + t

Measurementsection

IntroductionParticle Image Velocimetry (PIV)

992

1004

32

32

• divide image pair ininterrogation regions

• small region:~ uniform motion

• compute displacement• repeat !!!


Instantaneous measurement of 2 components in a plane

conventional methods(HWA, LDV)

• single-point measurement• traversing of flow domain• time consuming• only turbulence statistics

z

particle image velocimetry

• whole-field method• non-intrusive (seeding• instantaneous flow field


Instantaneous measurement of 2 components in a plane


• whole-field method• non-intrusive (seeding• instantaneous flow field

instantaneous vorticity field

Example: coherent structures


Turbulent pipe flowRe = 5300100×85 vectors“hairpin”

vortex


Van Doorne, et al.

Overview

PIV components:

- tracer particles- light source- light sheet optics- camera

- measurement settings

- interrogation- post-processing

Hardware (imaging)

Software (image analysis)

Tracer particles

Assumptions:

- homogeneously distributed- follow flow perfectly- uniform displacement within interrogation region

Criteria:

-easily visible-particles should not influence fluid flow!

small, volume fraction < 10-4

Image density

NI << 1

NI >> 1

particle tracking velocimetry


low image density

high image density

Assumption: uniform flow in “interrogation area”

Evaluation at higher density

High NI : no longer possible/desirable to follow individual tracer particles

Particle can be matched with a number of candidates

Possible „matches‟

Repeat process for other particles, sum up: “wrong” combinations will lead to noise, but “true” displacement will dominate

Sum of all possibilities

Statistical estimate of particle motion

Statistical correlations used to find

average particle displacement1-d image @ t=t0

1-d image @ t=t1

R(i, j)

Ia (k,l) I a Ib (k i,l j) I bl 1

By

k 1

Bx

Ia (k,l) I a2

Ib (k i,l j) I b2

l 1

By

k 1

Bx

l 1

By

k 1

Bx

1

2

x yB

k

B

l

a

yx

a lkIBB

I1 1

),(1

Cross-correlation

1-D cross-correlation example

R(i)

Ia (k) I a Ib (k i) I bk 1

Bx

Ia (k) I a2

Ib (k i) I b2

k 1

Bx

k 1

Bx

1

2

Ia

Ib

Ia(x) normalized

Ib(x+ x) normalized

Ia(x)*Ib(x+ x)

Finding the maximum displacement

-Shift 2nd window with respect to the first

- Calculate “match”

- Repeat to find best estimate

Typically 16x16 or 32x32 pixels

Good indicator: R(i, j)


By

k 1

Bx

Ia (k,l) I a2

Ib (k i,l j) I b2

l 1

By

k 1

Bx

l 1

By

k 1

Bx

1

2

x yB

k

B

l

a

yx

a lkIBB

I1 1

),(1


Bad match: sum of product of intensities low







By

k 1

Bx

Ia (k,l) I a2

Ib (k i,l j) I b2

l 1

By

k 1

Bx

l 1

By

k 1

Bx

1

2

x yB

k

B

l

a

yx

a lkIBB

I1 1

),(1


Good match: sum of product of intensities high




Can be implemented as 2D FFT for digitized data

o Impose periodic conditions on interrogation region…causes bias error if not treated properly.




By

k 1

Bx

Ia (k,l) I a2

Ib (k i,l j) I b2

l 1

By

k 1

Bx

l 1

By

k 1

Bx

1

2

x yB

k

B

l

a

yx

a lkIBB

I1 1

),(1

Cross-correlation

This “shifting” method can formally be expressed as a cross-correlation:

1 2( )R I I ds x x s x

- I1 and I2 are interrogation areas (sub-windows) of the total frames- x is interrogation location- s is the shift between the images

“Backbone” of PIV:-cross-correlation of interrogation areas-find location of displacement peak

Cross-correlation

RD

RF

RC

correlation of the mean correlation of mean &

random fluctuations correlation due to displacement

peak: meandisplacement

Influence of NI

NI = 5 NI = 10 NI = 25

More particles: better signal-to-noise ratio

Unambiguous detection of peak from noise:NI=10 (average), minimum of 4 per area in 95% of areas(number of tracer particles is a Poisson distribution)

R N NC z

MDD D I I I( ) ~s 0

0

2

2

C particle concentrationz0 light sheet thickness

DI int. area sizeM0 magnification

Influence of NI

NI = 5 NI = 10 NI = 25

PTV: 1 particle used for velocity estimate; error ePIV: error ~ e/sqrt(NI)

Influence of in-plane displacement

X / DI = 0.00FI = 1.00

0.280.64

0.560.36

0.850.16

II

IIIDDD

Y

D

XYXFFNR 11),(~)(s

X,Y-Displacement<quarter of window size

Influence of out-of-plane displacement

Z / z0 = 0.00FO = 1.00

0.250.75

0.500.50

0.750.25

R N F F F zz

zD D I I O O( ) ~ ( )s 1

0

Z-Displacement<quarter of light sheet thickness( z0)

Influence of gradients

Displacement differences <3-5% of int. area size, DI

Displacement differences <Particle image size, d

a / DI = 0.00a / d = 0.00

0.050.50

0.101.00

0.151.50

R N F F F F a a dD D I I O( ) ~ ( ) exp( / )s2 2

a M0 u t

R.D. Keane & R.J. Adrian

PIV “Design rules”

• image density NI >10• in-plane motion | X| < ¼ DI

• out-of-plane motion | z| < ¼ z0

• spatial gradients M0 u t < d

Obtained by Keane & Adrian (1993) using synthetic data

Window shifting

• in-plane motion | X| < ¼ DI

strongly limits dynamic range of PIV

large window size: too much spatial averaging

Window shifting



small window size: too much in-plane pair loss

Window shifting



Multi-pass approach:start with large windows,use this result as „pre-shift‟for smaller windows…

No more in-plane pair loss limitations!

Window shifting: Example

fixed windows matched windows

Grid turbulence

windows at same location windows at 7px „downstream‟

Window shifting: Example

Vortex ring, decreasing window sizes

Raffel,Willert andKompenhans

Sub-pixel accuracy

Maximum in the correlation plane: single-pixel resolution of displacement?

But the peak contains a lot more information!

Gaussian particle images Gaussian correlation peak (but smeared)

Sub-pixel accuracy

Fractional displacement can be obtained using the distribution of gray values around maximum

rX

• peak centroid

• parabolic peak fit

• Gaussian peak fit

R R

R R R

1 1

1 0 1

R R

R R R

1 1

1 1 02 2

ln ln

ln ln ln

R R

R R R

1 1

1 1 02 2

balance

normalization

three-point estimators

Peak locking

“zig-zag” structure, sudden “kinks” in the flow

Sub-pixel Interpolation Errors

• Accuracy depends on:– particle image size

– noise in data (seeding density, camera noise)

– shear rate

• Can exhibit “peak locking”– Interpolation of peak is biased towards a

symmetric data distribution (Integer and 1/2

integer peak locations)

– Polynomials exhibit strong locking when

particle diameter is small

– Gaussian is most commonly used

– Splines are very robust, but expensive to

calculate

• See Particle Image Velocimetry, by Raffel, Willert, and Kompenhans, Springer-Verlag, 1998.

-0.5 -0.25 0 0.25 0.5-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Actual peak location (pixels)

Inte

rpola

tion e

rror

(pix

els

)

Polynomial fitGaussian fit

-0.5 -0.25 0 0.25 0.5-0.3

-0.2

-0.1

0

0.1

0.2

0.3


Inte

rpola

tion e

rror

(pix

els

)


-0.5 -0.25 0 0.25 0.5-0.3

-0.2

-0.1

0

0.1

0.2

0.3


Inte

rpola

tion e

rror

(pix

els

)


R = 0.5

R = 1.0

R = 1.5

Pixel Radius

Peak locking

centroid Gaussian peak fitEven with Gaussian peak fit:

particle image size too small peak locking(Consider a „point particle‟ sampled by discrete pixels)

Histogram of velocities in a turbulent flow

Sub-pixel accuracy

optimal resolution: particle image size: ~2 pxSmaller: particle no longer resolvedLarger: random noise increase

“three-point” estimators:

Peak centroidParabolic peak fitGaussian peak fit... Main difference: sensitivity to “peak locking”

or “pixel lock-in”, bias towards integer displacements

Theoretical: 0.01 – 0.05 pxIn practice 0.05-0.1 px

bias errors random errorstotalerror

d / dr

displacement measurement error


signal

noise

SNR

u’2

C2

u’2 / C2

u’2

4C2u’2

1 / 4C2

FI ~ 0.75 FI ~ 1

velocity pdf

measurementerror

window matching


X = 7 px u’/U = 2.5%

application example:grid-generated turbulence

Data Validation

“article” “lab”

Spurious or “Bad” vectors

Spurious vectors

Three main causes:

- insufficient particle-image pairs

- in-plane loss-of-pairs, out-of-plane loss-of-pairs

- gradients

Effect of tracer density

NI= 1NI= 3NI= 5NI=11NI=20NI=45NI=80

Remedies

• increase NI

• practical limitations:• optical transparency of the fluid• two-phase effects• image saturation / speckle

• detection, removal & replacement• keep finite NI ( ~ 0.05 )

• data loss is small• signal loss occurs in isolated points• data recovery by interpolation

Detection methods

• human perception• peak height

• amount of correlated signal• peak detectability

• peak height relative to noise• lower limit for SNR

• residual vector analysis• fluctuation of displacement

• multiplication of correlation planes• fluid mechanics

• continuity• fuzzy logic & neural nets

Residual analysis

• evaluate fluctuation of measured velocity residual • ideally: Uref = true velocity• reference values:

• Uref = global mean velocity• comparable to 2D-histogram analysis• does not take local coherent motion into account• probably only works in homogeneous turbulence

• Uref = local (3×3) mean velocity• takes local coherent motion into account• very sensitive to outliers in the local neighborhood

• Uref = local (3×3) median velocity• almost identical statistical properties as local mean• Strongly suppressed sensitivity to outliers in heighborhood

refUUr

Example of residual test

0 1

234

5

6 7 8

0 1 2 3 4 5 6 7 8

2.3 2.2 3.0 3.7 3.1 3.2 2.4 3.5 2.7

2.2 2.3 2.4 2.7 3.0 3.1 3.2 3.5 3.7

RMS

0.53

RMS

2.29

2.3 9.7 3.0 3.7 3.1 3.2 2.4 3.5 2.7

2.3 2.4 2.7 3.0 3.1 3.2 3.5 3.7 9.7

Mean

2.9

Mean

3.7

Standard mean and r.m.s. are very sensitive to bad data contamination… need robust measure of fluctuation

Median test

1 2 3

4 0 5

6 7 8

1 - Calculate reference velocity: median of 8 neighbors

2 – calculate residuals:

3 – Normalize target residual by: median(ri) +

4 – Robust measure found for:

uref median u1,u2,...u8

ri ui uref

r0

*u0 uref

median(ri)

0.1 and r0

* 2

Wes

terw

eel &

Sca

rano

, 200

5

Interpolation

Bilinear interpolation satisfies continuity

For 5% bad vectors, 80% of the vectors are isolated

Bad vector can be recovered without any problems

N.B.: interpolation biases statistics (power spectra, correlation function)Better not to replace bad vectors (use e.g. slotting method)

Overlapping windows

Method to increase data yield:

Allow overlap between adjacent interrogation areas

aMotivation: particle pairs near edgescontribute less to correlation result;Shift window so they are in the center:additional, relatively uncorrelated result

50% is very common, but beware of oversampling

A Generic PIV program

Data acquisition

Image pre-processing

PIV cross-correlation

Vector validation

Post-processing

Laser control,Camera settings, etc.

Reduce non-uniformityof illumination; Reflections

Pre-shift; Decreasingwindow sizes

Vorticity, interpolationof missing vectors, etc.

Median test,Search window

PIV softwareFree

PIVware: command line, linux (Westerweel)JPIV: Java version of PIVware (Vennemann)MatPiv: Matlab PIV toolbox (Cambridge, Sveen)URAPIV: Matlab PIV toolbox (Gurka and Liberzon)DigiFlow (Cambridge), PIV Sleuth (UIUC), MPIV, GPIV, CIV, OSIV,…

Commercial

PIVtec PIVviewTSI InsightDantec FlowmapLaVision DaVisOxford Lasers/ILA VidPIV…

Particle Motion: tracer particle

• Equation of motion for spherical particle:

– Where

– Neglect: non-linear drag (only really needed for high-speed flows), Basset history term (higher order effect)

gmdt

udm

dt

vd

dt

udmvuD

dt

vdm

p

g

pf

p

fp

p

p

1213

mp p

D3

6, the particle mass

m f f

D3

6, fluid mass of same volume as particle

D particle diameter

fluid viscosityr u fluid velocityr v p particle velocity

g fluid density

p particle material density

Added massViscous drag Pressure gradient

buoyancyInertia

Simple thought experiment

• Lets see how a particle responds to a step change in velocity– Only consider viscous drag

mp

dr v p

dt3 D

r u

r v p u

0 for t < 0

U for t 0

dvp

dt

18

pD2U vp

1

p

U vp

v p v p particularv p homogeneous

U C1 exp t p

v p1

p

v p

1

p

U

v p U 1 exp t p

u, vp

t

U

Particle Transfer Function

• Useful to examine steady-state particle response to 1-D oscillating flow of arbitrary sum of frequencies– Represent u as an infinite sum of harmonic functions

– Neglect gravity (DC response, not transient)

u t f

0

exp i t d

vp t p

0

exp i t d

2mp

3 D

dv p

dt2 u v p

m f

mp

mp

3 D

du

dt

dv p

dt2

m f

mp

mp

3 D

du

dt

du t

dti f

0

exp i t d

2pD2

18pi exp i t

0

d 2 f p exp i t0

df

p

pD2

18i 3 f p exp i t d

0

2iSt p 2 f p i St 3 f p

dvp t

dti p

0

exp i t d

Stp

f

pD2

18

f

p

Particle Transfer Function

• This can be rearranged:

• Examine– Liquid in air, gas in water, plastic in water

r v pr u

p p

f f

1 2

A2 B2

A2 1

1 2

23

22

B

StA

tan 1Im p / f

Re p / f

tan 1 A(1 B)

A2 B

Liquid particles in Air

– Liquid drops in air require St~0.3 for 95% fidelity (1 m ~ 15 kHz)

– Size relatively unimportant for near-neutrally buoyant particles

– Bubbles are a poor choice: always overrespond unless quite small

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