1. Measurements in Fluid Mechanics 058:180:001 (ME:5180:0001) Time & Location: 2:30P - 3:20P MWF 218 MLH Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui lichuan-gui@uiowa.edu http://lcgui.net Students are encouraged to attend the class. You may not be able to understand by just reading the lecture notes.

2. 2 Lecture 30. Central Difference Interrogation

3. Recording 1 Recording 2 g1(i,j) g2(i,j) Correlation-based Interrogation w/o Window Shift Gray value distribution in a PIV recording pair 2 nd window shifted by S ws =0

4. g 1 (i,j) g 2 (i,j) Correlation-based Interrogation w/o Window Shift Evaluation samples of M  N pixels at point (x m,y m ) Recording 1 Recording 2 2 nd window shifted by S ws =0

5. g 1 (i,j) g 2 (i,j) Correlation-based Interrogation w/o Window Shift Evaluation Function Recording 1 Recording 2 2 nd window shifted by S ws =0

6. g 1 (i,j) g 2 (i,j) Correlation-based Interrogation w/o Window Shift Particle image displacement determined by position of the maximal function value Recording 1 Recording 2 2 nd window shifted by S ws =0 (m*,n*)

7. Correlation-based Interrogation w/o Window Shift Compared to correlation tracking: – higher evaluation speed – insensitive to brightness Problems: – error dependent on displacement – evaluation bias g 1 (i,j) g 2 (i,j) S Recording 1 Recording 2 2 nd window shifted by S ws =0

8. Forward Difference Interrogation (FDI) g1(i,j) g2(i,j) S Recording 1 Recording 2 Gray value distribution in a PIV recording pair 2 nd window shifted by S ws =S Interrogation window shift

9. Forward Difference Interrogation (FDI) g 1 (i,j) f 2 (i,j) S Recording 1 Recording 2 2 nd window shifted by S ws =S Evaluation samples of M  N pixels at point (x m,y m )

10. Forward Difference Interrogation (FDI) g 1 (i,j) f 2 (i,j) S Recording 1 Recording 2 2 nd window shifted by S ws =S Evaluation Function

11. Forward Difference Interrogation (FDI) g 1 (i,j) f 2 (i,j) (m*,n*) Recording 1 Recording 2 2 nd window shifted by S ws =S Particle image displacement

12. Forward Difference Interrogation (FDI) g1(i,j) g2(i,j) Recording 1 Recording 2 2 nd window shifted by S ws =S S‘ Compared to correlation w/o shift: – multi-pass (iterated) – higher reliability – lower evaluation error – periodic error distribution (1 pixel) S = S ws +S ‘

13. Forward Difference Interrogation (FDI) Typical curvature flow

14. Forward Difference Interrogation (FDI) Interrogation point (x m,y m ) (x m,y m )

15. Forward Difference Interrogation (FDI) Flow direction at evaluation point (x m,y m )

16. Forward Difference Interrogation (FDI) True velocity to be determined (S o ) (x m,y m ) SoSo

17. Forward Difference Interrogation (FDI) (x m,y m ) SoSo Interrogation window in the first recording g 1 (i,j) g 1 (i,j)

18. Forward Difference Interrogation (FDI) (x m,y m ) SoSo Matched image pattern in the second recording f 2 (i,j) g 1 (i,j) f 2 (i,j)

19. Forward Difference Interrogation (FDI) (x m,y m ) SoSo g 1 (i,j) f 2 (i,j) S FDI interrogation result (S)

20. Forward Difference Interrogation (FDI) (x m,y m ) SoSo g 1 (i,j) f 2 (i,j) S FDI interrogation bias error (  ) SoSo S 

21. Forward Difference Interrogation (FDI) (x m,y m ) SoSo g 1 (i,j) g 2 (i,j) S SoSo S  Adjusted position (x’ m,y’ m ) FDI interrogation position deviation (x m -x’ m,y m -y’ m )

22. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S‘ S ws2 = S/2 S ws1 = -S/2 S = S ws +S ‘ Central Difference Interrogation (CDI) Gray value distribution in a PIV recording pair Interrogation window shift

23. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S‘ S ws2 = S/2 S ws1 = -S/2 Central Difference Interrogation (CDI) Evaluation samples of M  N pixels at point (x m,y m )

24. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S‘ S ws2 = S/2 S ws1 = -S/2 Central Difference Interrogation (CDI) Evaluation Function

25. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S ws2 = S/2 S ws1 = -S/2 Central Difference Interrogation (CDI) Particle image displacement (m*,n*)

26. True velocity to be determined (S o ) (x m,y m ) SoSo Central Difference Interrogation (CDI)

27. Window shift in the first recording (  -S o /2) (x m,y m ) SoSo Central Difference Interrogation (CDI)  -0.5S o

28. (x m,y m ) SoSo Central Difference Interrogation (CDI)  -0.5S o f 1 (i,j) Interrogation window in the first recording f 1 (i,j)

29. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) Matched image pattern in the second recording f 2 (i,j)

30. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S CDI interrogation result (S)

31. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S SoSo S  CDI interrogation bias error (  )

32. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S SoSo S  Adjusted position (x’ m,y’ m ) CDI interrogation position deviation (x m -x’ m,y m -y’ m )

33. – smaller position deviation – smaller curvature flow bias – same computation speed (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S SoSo S  Adjusted position (x’ m,y’ m ) Compare to FDI

34. Compare FDI and CDI in a four-roll mill test Top view Velocity field Experimental setup and flow velocity distribution

35. Compare FDI and CDI in a four-roll mill test PIV recording frames Animated PIV recordingsOverlapped PIV recordings

36. Compare FDI and CDI in a four-roll mill test FDI and CDI evaluation errors

37. Westerweel J, Dabiri D, Gharib M (1997) The effect of a discrete window offset on the accuracy of cross-correlation analysis of digital PIV recordings. Exp Fluids 23:20–28 Wereley ST Meinhart CD (2001) Second-order accurate particle image velocimetry. Exp Fluids 31:258–268 Gui L and Wereley ST (2002) A correlation-based continues window shift technique for reducing the peak locking effect in digital PIV image evaluation. Exp. Fluids 32: 506-517 Wereley ST and Gui L (2003) A correlation-based central difference image correction (CDIC) method and application in a four-roll-mill flow PIV measurement. Exp. Fluids 34, 42-51 References

38. Matlab program for FDI&CDI Test with iterated window shift clear; A1=imread('lecture30-image01.bmp'); A2=imread('lecture30-image02.bmp'); G1=img2xy(A1); G2=img2xy(A2); Mg=64; % interrogation grid width Ng=64; % interrogation grid height M1=64; % initial window width N1=64; % initial window height M2=32; % final window width N2=32; % final window height NN=20; % iteration number [nx ny]=size(G1); row=ny/Mg-1; col=nx/Ng-1; sr=12; for i=1:col for j=1:row U_FDI(i,j)=0; % initial velocity V_FDI(i,j)=0; U_CDI(i,j)=0; V_CDI(i,j)=0; end for iteration=1:NN M=(iteration-1)*(M2-M1)/(NN-1)+M1; N=(iteration-1)*(N2-N1)/(NN-1)+N1; clear g1 g2 C m n; for i=1:col for j=1:row x=i*Mg; y=j*Ng; X(i,j)=x; Y(i,j)=y; % FDI --------------------------------------------- wsx=U_FDI(i,j); wsy=V_FDI(i,j); g1=sample3(G1,M,N,x,y); g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U_FDI(i,j)=vx+wsx; V_FDI(i,j)=vy+wsy; % CDI ---------------------------------------------- wsx=U_CDI(i,j); wsy=V_CDI(i,j); g1=sample3(G1,M,N,x-wsx/2,y-wsy/2); g2=sample3(G2,M,N,x+wsx/2,y+wsy/2); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U_CDI(i,j)=vx+wsx; V_CDI(i,j)=vy+wsy; end dx=U_FDI+10*(X-512)/512; dy=V_FDI-10*(Y-512)/512; D_FDI(iteration)=sqrt(mean(mean(dx.^2+dy.^2))); dx=U_CDI+10*(X-512)/512; dy=V_CDI-10*(Y-512)/512; D_CDI(iteration)=sqrt(mean(mean(dx.^2+dy.^2))); end plot(D_FDI,'r*-') hold on plot(D_CDI,'b*-') hold off Simulated flow: % RMS error of CDI % RMS error of FDI

39. -*- RMS error of FDI -*- RMS error of CDI Test image pair: http://lcgui.net/ui-lecture/lecture30/lecture30-image01.bmp http://lcgui.net/ui-lecture/lecture30/lecture30-image02.bmp