Advanced PIV algorithms Why and when advanced PIV algorithms?

Transcription

1 Advanced PIV algorithms Why and when advanced PIV algorithms? Speaker: Antonio Lecuona, Prof. Contributors: Dr. José I. Nogueira, Dr. Ángel Velázquez, A. Acosta, D. Santana, Prof. P. A. Rodríguez, Dr. U. Ruiz-Rivas, B. Méndez. Universidad Carlos III de Madrid, PIVNET 2/Ercoftac SIG 32 workshop Lisbon, 5 th and 6 th July

2 Contents I. Standard PIV algorithms II. III. Image distortion algorithms Advanced post-processing of PIV data based on the Navier-Stokes equations 2

3 I. Standard (correlation) PIV 1. Interrogation window 2. Cross-correlation using FFT or DC 3. Peak detection 4. Subpixel peak fitting s 3

4 I. Standard PIV (cont.) Basic parameters: ( for physical variables) Image sampling: pixel size p Group sampling: Window size F Grid distance, overlapping o = /F ) Flow sampling: particle diameter d and average distance δ Time sampling: inter-image time delay t = 1 Flowfield spatial frequency: λ gradients Usually p is the measuring unit s = f (F,, d, δ, λ,addit. parameters) Additional parameters: Imaging noise (interlacing, pattern, pixel blinds, blooming, ), Optical noise (laser interferences, stray light, reflections and shades), Optical deformations and aberrations, Laser plane non coincidence, Out of plane pair losses, Particle seeding inhomogeneities, Numerical noise. 4

5 I. Standard PIV (cont.) Peak locking: Sources d < 2, particle image undersampling interchangeable with random error (unavoidable). F 1 Particle truncation at window borders, relevant for extreme multigrid (avoidable ). Effects Particularly relevant for small s. Creates non rotating star-like structures with 4, 8.. arms in vortices. Peak fitting accuracy and high flow field sampling (small δ) call for small d near peak locking operation. 5

6 I. Standard PIV (cont.) First advancement: Window shift: compensation of the average s s < 0,5 increase in s/n and accuracy, if in the correct direction). First need of iteration! 6

7 I. Standard PIV (cont.) Standard PIV limits: Physical: δ > d; Nyquist λ > 2δ Amplitude response: δ F λ 1,7 F (in figure 3) Detectability: δ (π N min /4) -1/2 F N min = minimum # particles for peak detection N min Multigrid Aplicable λ Multigrid methods progressively reduce w to increase spatial frequency resolution overcoming amplitude response criterion. N min (F/λ) n detectability limits could arrive first when resolving λ. Line: F λ 7

8 I. Standard PIV (cont.) Aggregation of particle pair peaks: I. A pair peak contributes to the main peak if s m = main peak displacement. s p = pair peak displacement. sm -s p < R + wp min = 3 w p R = radius considered in the peak fitting algorithm. R w p = pair peak waist d/ 2 Consequences: s m -s p I. Large particles and low disp. differences window moving averaging II. III. Small particles and high disp. differences no averaging Intermediate case non linear amplitude response and group locking IV. The most distant peaks are dropped non linear response 8

9 I. Standard PIV (cont.) Examples of group locking I. 1D Linear gradient (spurious peaks eliminated) Grad F/2 Displacement 4 particle peaks Many particles peaks Actually, there is no group locking in this example but uncertainty, unless there is particle a region with the same displacement. 1D Sinusoidal displacement Grad F Group locking Displacement 4 particle peaks Many particles peaks 9

10 I. Standard PIV (cont.) Examples of group locking Correlation plane of 1D sinusoidal displacement, d = 2,5; R = 1,5; F = s = 4 s = 4 s =16 Noiseless gaussian particles randomly located 10

11 I.Standard PIV (cont.) Group locking: a local detection algorithm 1 G = > s s x y s s x y + + x x y y When in a window G is large, it signals the displacement group locking. For 1D sinusoidal displacement Deep areas indicate group lockers Clear areas do not contribute to the peak unless s m -s p < R + w p This effect is negligible for F/λ << 1 because s m -s p vanishes (windows will have the same colour). A qualitative error measure in a window is σ G, but is not computable form PIV data. 11

12 I.Standard PIV (cont.) Examples of group locking (cont.) PIV obtained results 1D Sinusoidal displac. (cont. line input, red line model, points PIV output) Displacement (pixels) Displacement (pixels) Displacement (pixels) Location (pixels) Location (pixels) Location (pixels) λ/f = λ/f = 2 λ/f = 1 Synthetic vortex (G in yellow, int. win. in blue) = 4; d = 2,5; δ = 2; λ= 65,6; A = 4,5. s = Acos(2 πx/ λ)sin(2 π y/ λ) x s = Asin(2 πx/ λ)cos(2 π y/ λ) y F = 16 F = 32 12

13 I.Standard PIV (cont.) Amplitude response to 1D sinusoidal displacement field First armonic relative amplitude A C B Circles F = 64. Squares F =32. Empty symbols: Clean images. Filled symbols: Noisy images. Continuous line: moving average amplitude response. Discontinuous line squares indicate the displacement distribution within the interrogation window, not to scale. Normalised window size F /λ Figure 3.- The average 1D amplitude responses of single step PIV without window weighting function. The amplitude response is higher than moving average. This is explained by group locking. Regions of negative response Window shift in the opposite direction For F/λ I amplitude response is statistically 0, but could be ±1 owing to group locking high uncertainty. 13

14 I. Standard PIV (cont.) Conclusions: I. Correlation PIV with no image distortion is accurate only when peak overlapping is assured s m -s p < R + w p. It is not equivalent to the rule of thumb ( s max - s min )/F < 5% although it is generally fulfilled. II. Group locking appears in the first stages of multigrid and image distortion PIV. III. Group locking introduces non-linearities that increase SNR. Solutions: I. Multigrid PIV: iterative reduction of F incurring in loss of robustness. II. Image distortion: Iteratively reduces s m -s p so that peak aggregation increases SNR and non-linearities are reduced. Window distortion is a partial application of image distortion. δ Resulting in a wider N min area of application: Aplicable λ 14

15 II. Image distortion algorithms The main purpose is to reduce s m -s p thus increasing SNR and reducing group locking. At the end s 0 would result. Negative amplitude response could induce divergence instability. Solutions to instability: Window weighting (LFC-PIV) effective, but there is a small error. Alliasing reduction compromise between spatial resolution and robustness. Averaging reduces spatial resolution. Its amplitude response is negative for some λ. 15

16 II. Image distortion algorithms (cont.) + LFC-PIV Correlation Image a* + Correlation + Previous measurements Image a Compensation of the particle pattern deformaton Image b* Image a* Image b Compensation of the particle pattern deformaton after several iterations (15 in this case) Image b* Sketch of the LFCPIV iterative procedure. Black dots represent particle images in negative, grid-like distributed in order to show the particle pattern deformation. No error would yield a perfect cross ruled particle pattern after processing. Grey grid is for reference only, showing a rotation in the middle of the image and a shear at the borders after the compensation. Framed images represent actual measured displacement fields. Long horizontal arrows represent LFCPIV processing after 1, 2 and 15 iterations. Grid spacing is 16 pixels. 16

17 II. Image distortion algorithms (cont.) Example of results obtained with LFC-PIV Original image LFC, F = 64, λ = 65,6 17

18 II. Image distortion algorithms (cont.) Example of results obtained with LFC-PIV Vorticity map based on LFC PIV data with grid distance of 8 pixels. DLR provided image. F = 64 18

19 II. Image distortion algorithms (cont.) Conclusions: Image distortion widens the applicability of correlation PIV Reduces: Low pass window effects Group locking Increases: SNR Spatial resolution Computing time Still under development: Instabilities and error growth 19

20 III. Advanced post-processing of PIV data based on the Navier-Stokes equations Normally carried out using statistics based methodologies (filters, interpolators, etc.) An alternate approach is to devise a post-processing methodology that is based on the Physics of the problem (that is: on the Navier-Stokes equations) This alternate approach is very challenging from a technical standpoint but, if successful, the potential benefits for practical industrial applications could be significant. Accuracy of PIV predictions could increase by a sizable margin. 20

21 A 2D Navier-Stokes based PIV Post-processing methodology has to be dependent on a very robust and flexible computational algorithm. A good candidate that fulfils these requirements is a gridless (loose connectivity) formulation. Close up view of the mesh around a cylinder at Re = 500. Results: C d experimental =1.16, C d numerical = Different topologies can be used simultaneously (left) Random placement is feasible (right) 21

22 Reconstruction of the full Navier-Stokes flow field (Re = 100, M = 0) around a cylinder. 200 randomly scattered points outside the boundary layer were used to provide seed velocity information ( simulating tracking PIV) Reconstructed pressure profile θ Reconstructed shear stress C d experimental: 1.46, C d reconstructed: 1.44 Experimental separation point: 110 deg. Reconstructed separation point: 120 deg θ 22

23 To check the numerical robustness of the method, 5 % of the 200 seed points were perturbed ± 10 % in their velocity components. A slight deterioration of the reconstruction was observed. A finer resolution is needed for this case. C dexp = 1.46, C drecons = 1.44 Separation point: exper. = 110 deg, recons. = 120 deg C dexp = 1.46, C drecons = 1.47 Separation point: exper. = 110 deg, recons. = 127 deg 23