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 measur em ent 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

Coherent structures in a TBL Kim, H.T., Kline, S.J. & Reynolds, W.C. J. Fluid Mech. 50 (1971) 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.

Particle trajectory Fluid pathline 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 flow Re = ×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: Image density: C tracer concentration [m -3 ]  z 0 light-sheet thickness [m] M 0 image magnification [-] d  particle-image diameter [m] D I interrogation-spot diameter [m] The image density represents the mean number of particle images in an interrogation region. For a successful PIV measurement N I > Ns <1 : individual partical image Ns > 1 : speckle pattern

N I << 1 N I >> 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 cm 2 )

PIV Interrogation analysis Double-exposure image Interrogation cell Auto- correlation RPRP RD+RD+ RD-RD- R C +R F

Spatial Correlation The image intensities are separated into : Mean intensityintensity fluctuation The spatial correlation can be separated into three terms: R C -- mean background correlation R F -- correlation between mean intensity and intensity fluctuations R D -- correlation of image fluctuations

Mean intensity should be subtracted before correlation When mean intensity is subtracted, R C = R F =0 The mean image intensity contains no information with respect to the displacement of the particle images.

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

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R(s)R(s) s  Correlation peak location corresponds to the separation of the two images 

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

P-I P-II Match perfectly

P-I P-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-II P-I

Sketch of Cross-correlation P-I P-II Form a pattern in the 1 st image (P-I) Form a number of patterns within the selected domain in the 2 nd 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 1 st image (P-I) Form a number of patterns within the selected domain in the 2 nd 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 1 st image and 2 nd image, respectively. (N and M are the width and height of the patterns)

Find velocity from double-exposure images Select a window (pattern) P-I in the 1 st image. Select a domain in the 2 nd 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.

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

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

f(m,n)f(m,n) FFT F(u,v)F(u,v) g(m,n)g(m,n) G(u,v)G(u,v) FT of Cross-correlation  ’(u,v) =F(u,v)G*(u,v) Select interrogation window

f(m,n)f(m,n) FFT F(u,v)F(u,v) g(m,n)g(m,n) G(u,v)G(u,v) FT of Cross-correlation  ’(u,v) =F(u,v)G*(u,v) Select interrogation window  ’(u,v) FFT -1

f(m,n)f(m,n) FFT F(u,v)F(u,v) g(m,n)g(m,n) G(u,v)G(u,v) FT of Cross-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 displacement- correlation peak “random correlations”

Auto-Correlation

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R(s)R(s) s Second correlation peak location corresponds to the separation of the two images Directional Ambiguity

R(s)R(s) s  Correlation peak location corresponds to the separation of the two images 

Correlation Peaks in Different Schemes Cross-CorrelationAuto-Correlation (Double-exposure) Auto-Correlation (Multi-exposure)