Fundamentals of Digital PIV

1. Fundamentals of Digital PIV Partially in reference to J. Westerweel ‘s presentation

2. 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

3. 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

4. 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).

5. 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.

6. Particle trajectory Fluid pathline After: Adrian, Adv. Turb. Res. (1995) 1-19 The displacement field • The fluid motion is represented as a displacement field

7. Inherent assumptions • Tracer particles follow the fluid motion • Tracer particles are distributed homogeneously • Uniform displacement within interrogation region

8. Multiple-exposure PIV image

9. PIV result Turbulent pipe flow Re = 5300 100×85 vectors “Hairpin” vortex

10. Instantaneous vorticity fields

11. Visualization vs. Measurement

12. “Ingredients” FLOW sampling seeding quantization Pixelization illumination enhancement Acquisition imaging selection registration correlation Interrogation estimation RESULT analysis validation

13. PIV optical configuration

14. PIV Laser

15. Light sheet optics (positive) cylindrical lens (negative) cylindrical lens (positive) spherical lens f f - To obtained the desired light sheet thickness

16. DPIV Data Processing

17. 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

18. Two modes of extracting velocity from tracer motion Low image density NI << 1 Particle tracking velocimetry High image density NI >> 1 Particle image velocimetry

19. 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.

20. 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

21. PIV measurement example Interrogation Cell 1.6mm x 1.6mm (32x32 pixels) Correlation gives an average displacement vector. Image Window (4x4 cm2)

22. PIV Interrogation analysis RP RD+ RD- RC+RF Double-exposure image Interrogation cell Auto- correlation

23. 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

24. 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.

25. D Illustration of correlation principle (1D) Shift direction R(s) Shift (a variable) s

26. R(s) s

27. R(s) s

28. R(s) s

29. R(s) s

30. R(s) s

31. R(s) s

32. Correlation peak location corresponds to the separation of the two images D R(s) s D

33. P-I P-II Illustration of correlation principle (2D) R(s) Shift in 2D s

34. P-I P-II Match perfectly

35. P-I P-II Match perfectly R

36. P-I P-II Partially Matched

37. P-I P-II Partially Matched R

38. P-I P-II With Noise

39. P-I P-II With Noise R

40. 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

41. 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

42. 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)

43. 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.

44. 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.

45. Select interrogation window f(m,n) F(u,v) FFT

46. Select interrogation window f(m,n) F(u,v) FFT g(m,n) G(u,v) FFT

47. 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

48. 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

49. 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

50. Displacement-correlation peak “random correlations” displacement- correlation peak