Introduction of Particle Image Velocimetry

Transcription

1 Introduction of Particle Image Velocimetry Ken Kiger Burgers Program For Fluid Dynamics Turbulence School College Park, Maryland, May Slides largely generated by J. Westerweel & C. Poelma of Technical University of Delft Adapted by K. Kiger

2 Introduction Particle Image Velocimetry (PIV): Imaging of tracer particles, calculate displacement: local fluid velocity Frame 1: t = t 0 Measurement section Light sheet optics Frame 2: t = t 0 + t Twin Nd:YAG laser CCD camera

3 Introduction Particle Image Velocimetry (PIV) divide image pair in interrogation regions small region: ~ uniform motion compute displacement repeat!!!

4 Introduction Particle Image Velocimetry (PIV): Instantaneous measurement of 2 components in a plane 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

5 Introduction Particle Image Velocimetry (PIV): Instantaneous measurement of 2 components in a plane instantaneous vorticity field particle image velocimetry whole-field method non-intrusive (seeding instantaneous flow field

6 Example: coherent structures

7 Example: coherent structures Turbulent pipe flow Re = vectors hairpin vortex

8 Example: coherent structures Van Doorne, et al.

9 Overview PIV components: - tracer particles - light source - light sheet optics - camera Hardware (imaging) - measurement settings - interrogation - post-processing Software (image analysis)

10 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

11 Image density low image density N I << 1 particle tracking velocimetry high image density N I >> 1 particle image velocimetry

12 Evaluation at higher density High N I : no longer possible/desirable to follow individual tracer particles Possible matches Sum of all possibilities Assumption: uniform flow in interrogation area Particle can be matched with a number of candidates Repeat process for other particles, sum up: wrong combinations will lead to noise, but true displacement will dominate

13 Statistical estimate of particle motion Statistical correlations used to find average particle displacement 1-d t=t 0 B x B y R(i, j) k 1 l 1 I a (k,l) I a I b (k i,l j) I b 1 B x B y B y 2 I a (k,l) I a 2 B x k 1 l 1 k 1 l 1 I b (k i,l j) I b 2 1-d t=t 1 I a 1 B B x y B B x y k 1 l 1 I a ( k, l) Cross-correlation

14 1-D cross-correlation example I a I a (x) normalized I b (x+ x) normalized I a (x)*i b (x+ x) I b B x R(i) k 1 I a (k) I a I b (k i) I b 1 B x B x 2 I a (k) I a 2 k 1 k 1 I b (k i) I b 2

15 Finding the maximum displacement -Shift 2 nd window with respect to the first Typically 16x16 or 32x32 pixels - Calculate match B x B y Good indicator: R(i, j) k 1 l 1 I a (k,l) I a I b (k i,l j) I b 1 B x B y B y 2 I a (k,l) I a 2 B x k 1 l 1 k 1 l 1 I b (k i,l j) I b 2 - Repeat to find best estimate I a 1 B B x y B B x y k 1 l 1 I a ( k, l)

16 Finding the maximum displacement -Shift 2 nd window with respect to the first Typically 16x16 or 32x32 pixels - Calculate match B x B y Good indicator: R(i, j) k 1 l 1 I a (k,l) I a I b (k i,l j) I b 1 B x B y B y 2 I a (k,l) I a 2 B x k 1 l 1 k 1 l 1 I b (k i,l j) I b 2 - Repeat to find best estimate I a 1 B B x y B B x y k 1 l 1 I a ( k, l) Bad match: sum of product of intensities low

17 Finding the maximum displacement -Shift 2 nd window with respect to the first Typically 16x16 or 32x32 pixels - Calculate match B x B y Good indicator: R(i, j) k 1 l 1 I a (k,l) I a I b (k i,l j) I b 1 B x B y B y 2 I a (k,l) I a 2 B x k 1 l 1 k 1 l 1 I b (k i,l j) I b 2 - Repeat to find best estimate I a 1 B B x y B B x y k 1 l 1 I a ( k, l) 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.

18 Cross-correlation This shifting method can formally be expressed as a cross-correlation: R() s I x I x s dx I 1 and I 2 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

19 Cross-correlation peak: mean displacement correlation of the mean R C R F correlation of mean & random fluctuations R D correlation due to displacement

20 Influence of N I R ( ) ~ N N C z 2 M 0 D 2 D sd I I I 0 C z 0 D I M 0 particle concentration light sheet thickness int. area size magnification N I = 5 N I = 10 N I = 25 More particles: better signal-to-noise ratio Unambiguous detection of peak from noise: N I =10 (average), minimum of 4 per area in 95% of areas (number of tracer particles is a Poisson distribution)

21 Influence of N I N I = 5 N I = 10 N I = 25 PTV: 1 particle used for velocity estimate; error e PIV: error ~ e/sqrt(n I )

22 Influence of in-plane displacement X,Y-Displacement < quarter of window size X / D I = 0.00 F I = X RD (s D) ~ N I FI FI ( X, Y) 1 1 D I D Y I

23 Influence of out-of-plane displacement Z-Displacement < quarter of light sheet thickness ( z 0 ) Z / z 0 = 0.00 F O = R ( s ) ~ N F F F ( z) 1 D D I I O O z z 0

24 Influence of gradients a M 0 u t Displacement differences < 3-5% of int. area size, D I Displacement differences < Particle image size, d a / D I = 0.00 a / d = R ( s ) ~ N F F F F ( a) exp( a / d ) D D I I O R.D. Keane & R.J. Adrian

25 PIV Design rules image density N I >10 in-plane motion X < ¼ D I out-of-plane motion z < ¼ z 0 spatial gradients M 0 u t < d Obtained by Keane & Adrian (1993) using synthetic data

26 Window shifting in-plane motion X < ¼ D I strongly limits dynamic range of PIV large window size: too much spatial averaging

27 Window shifting in-plane motion X < ¼ D I strongly limits dynamic range of PIV small window size: too much in-plane pair loss

28 Window shifting in-plane motion X < ¼ D I strongly limits dynamic range of PIV Multi-pass approach: start with large windows, use this result as pre-shift for smaller windows No more in-plane pair loss limitations!

29 Window shifting: Example Grid turbulence fixed windows matched windows windows at same location windows at 7px downstream

30 Window shifting: Example Vortex ring, decreasing window sizes Raffel, Willert and Kompenhans

31 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)

32 Sub-pixel accuracy X r Fractional displacement can be obtained using the distribution of gray values around maximum

33 three-point estimators peak centroid R R 1 1 R R R parabolic peak fit R R R R 2R Gaussian peak fit ln R ln R ln R ln R 2ln R balance normalization

34 Peak locking zig-zag structure, sudden kinks in the flow

35 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, Interpolation error (pixels) Interpolation error (pixels) Interpolation error (pixels) Actual peak location (pixels) Polynomial fit Gaussian fit Polynomial fit Gaussian fit Actual peak location (pixels) Polynomial fit Gaussian fit Actual peak location (pixels) Pixel Radius R = 0.5 R = 1.0 R = 1.5

36 Peak locking Histogram of velocities in a turbulent flow centroid Gaussian peak fit Even with Gaussian peak fit: particle image size too small peak locking (Consider a point particle sampled by discrete pixels)

37 Sub-pixel accuracy optimal resolution: particle image size: ~2 px Smaller: particle no longer resolved total error bias errors random errors Larger: random noise increase three-point estimators: d / d r Peak centroid Parabolic peak fit Gaussian peak fit... Theoretical: px In practice px Main difference: sensitivity to peak locking or pixel lock-in, bias towards integer displacements

38 displacement measurement error

39 window matching fixed windows matched windows F I ~ 0.75 F I ~ 1 velocity pdf measurement error u 2 C 2 u 2 / C 2 signal noise SNR u 2 4C 2 u 2 1 / 4C 2

40 application example: grid-generated turbulence X = 7 px u /U = 2.5% fixed windows matched windows

41 Data Validation article lab Spurious or Bad vectors

42 Spurious vectors Three main causes: - insufficient particle-image pairs - in-plane loss-of-pairs, out-of-plane loss-of-pairs - gradients

43 Effect of tracer density NNII=80 =20 =45 =11 = 153

44 Remedies increase N I practical limitations: optical transparency of the fluid two-phase effects image saturation / speckle detection, removal & replacement keep finite N I ( ~ 0.05 ) data loss is small signal loss occurs in isolated points data recovery by interpolation

45 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

46 Residual analysis evaluate fluctuation of measured velocity residual ideally: U ref = true velocity reference values: U ref = global mean velocity comparable to 2D-histogram analysis r U does not take local coherent motion into account probably only works in homogeneous turbulence U ref = local (3 3) mean velocity takes local coherent motion into account very sensitive to outliers in the local neighborhood U ref = local (3 3) median velocity almost identical statistical properties as local mean Strongly suppressed sensitivity to outliers in heighborhood U ref

47 Example of residual test Mean 2.9 RMS Mean 3.7 RMS 2.29 Standard mean and r.m.s. are very sensitive to bad data contamination need robust measure of fluctuation

48 Median test Calculate reference velocity: median of 8 neighbors u ref median u 1,u 2,...u calculate residuals: r i u i u ref Normalize target residual by: median(r i ) + r 0 * u 0 u ref median(r i ) 4 Robust measure found for: 0.1 and r 0 * 2 Westerweel & Scarano, 2005

49 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)

50 Overlapping windows Method to increase data yield: Allow overlap between adjacent interrogation areas a Motivation: particle pairs near edges contribute less to correlation result; Shift window so they are in the center: additional, relatively uncorrelated result 50% is very common, but beware of oversampling

51 A Generic PIV program Reduce non-uniformity of illumination; Reflections Median test, Search window Vorticity, interpolation of missing vectors, etc. Data acquisition Image pre-processing PIV cross-correlation Vector validation Post-processing Laser control, Camera settings, etc. Pre-shift; Decreasing window sizes

52 PIV software Free 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 TSI Dantec LaVision Oxford Lasers/ILA DaVis PIVview Insight Flowmap VidPIV

53 Particle Motion: tracer particle Equation of motion for spherical particle: m p dv dt Where p Inertia 3 D u v p Viscous drag 1 2 m f du dt D 3 m p p, the particle mass 6 dv dt p m D 3 m f f, fluid mass of same volume as particle 6 D particle diameter fluid viscosity r u fluid velocity r v p particle velocity g fluid density p particle material density Added mass f du dt Pressure gradient m p 1 g p buoyancy g Neglect: non-linear drag (only really needed for high-speed flows), Basset history term (higher order effect)

54 Simple thought experiment Lets see how a particle responds to a step change in velocity Only consider viscous drag m p d r v p dt 3 D r u r v p u 0 for t < 0 U for t 0 U u, v p dv p dt 18 pd 2 U v p 1 p U v p t v p 1 p v p 1 p U v p v p particular v p homogeneous U C 1 exp t p v p U 1 exp t p

55 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 0 v p t p 0 f exp i t d exp i t d du t dt dv p t dt 0 i i 0 f p exp i t d exp i t d 2m p 3 D dv p dt 2 u v p m f m p m p 3 D du dt dv p dt 2 m f m p m p 3 D du dt 0 2 pd2 18 p i exp i t d 2 f p exp i t 0 d 0 f p p D2 18 i 3 f p exp i t d 2iSt p 2 f p i St 3 f p St p f p D2 18 f p

56 Particle Transfer Function This can be rearranged: r v p r u p p tan 1 Im p / f Re p / f f f 1 2 A 2 B 2 A A(1 B) tan 1 A 2 B A B St Examine Liquid in air, gas in water, plastic in water

57 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