1. Properties of 2D discrete FFT  Fixed sample size Size of window has to be a base-2 dimension, 32x32, or 64x64  Periodicity assumption Particle image is assumed to be periodic  Aliasing Correlation data is periodic, peak located at area outside of IC window will be found on the opposite side  Bias error Large shift (less overlapping pattern) leads to smaller and biased peak

2. Periodicity assumption  Applying FFT to a domain assumes the domain is periodic.

3. Periodicity assumption  Applying FFT to a domain assumes the domain is periodic. Periodic condition for the domain

4. Periodicity assumption  Applying FFT to a domain assumes the domain is periodic. Periodic condition for the domain Particle image does not satisfy this condition

5. Periodicity assumption  Applying FFT to a domain assumes the domain is periodic. Periodic condition for the domain Particle image does not satisfy this condition Bias error Shift (  s) Correlation Bias error True peak Measured peak 1

6. Solution for the bias error 1D example: f 0 1 0N g 0 1 0N convolving 0 1 -N/2N/2 0 Center 0: 1; ±N/2: 1/2 Reduce bias error Multiplying weighting factors Adjusting correlation coefficients Re-scale to [1/2, 1] Image is convolved with itself

7. Subpixel interpolation Reason shift correlation Calculating in digital form shift correlation Solution : subpixel interpolation -curve fitting from 3 points near peak - finding the peak in the curve shift correlation Curve fitting

8. Schemes for subpixel interpolation Remark: -Gaussian fitting is more commonly used -Parabolic fitting has a divided-by-zero problem when R i-1 = R i = R i+1

9. Advanced techniques - multiple pass interrogation algorithm  Principle Crosscorrelating one IC window with another IC window with known shift Determination of known shift – normal crosscorrelation  Implementation Using normal crosscorrelation method to get the velocity map; Detecting bad vectors and replacing with interpolated vectors; Computing crosscorrelation coefficients by shifting another IC window with the displacement determined by the obtained velocity;

10. Advanced techniques - multiple pass interrogation algorithm P-I P-II Normal crosscorrelation P-I P-II multiple pass interrogation IC

11. Advanced techniques - multiple pass interrogation algorithm P-I P-II Normal crosscorrelation P-I P-II multiple pass interrogation IC

12. Advanced techniques - multiple pass interrogation algorithm P-I P-II Normal crosscorrelation P-I P-II multiple pass interrogation IC

13. Advanced techniques – hierarchical approach  Principle Similar to the multiple pass algorithm but the sampling grid system is gradually refined with reducing IC size simultaneously  Advantage Able to capture complex structure;  Implementation Run multiple pass algorithms under adaptive grid systems and correspondingly changed IC window

14. Example of hierarchical approach

15. Multiple peak detection  Reason The strongest peak is not always associated with the correct shift, especially in the area of complex structure, e.g., strong vorticity, strong shear  Implementation Detecting local maximums and keeping them as “peak candidates”; Compare velocity with surrounding velocities, ruling out the unreasonable “peak candidates” until the searching status is not changed

16. Example of multiple peak detection Multiple peak detection Single peak detection

17. Data validation  Reason of the occurrence of bad vectors Inhomogeneous seeding Noise Complex flow structure  Validation method Direct comparison Median Filter Mean Filter

18. Data validation (cont’d) U 1 =U i+1,j U 2 =U i+1,j-1 U 8 =U i+1,j+1 U 9 =U i,j U 3 =U i,j-1 U 7 =U i,j+1 U 5 =U i-1,j U 4 =U i-1,j-1 U 6 =U i-1,j+1 Direct comparison Median Filter Mean Filter

19. Interpolation  Linear interpolation xx yy x1,y1x1,y1 x1,y1x1,y1 x3,y3x3,y3 x4,y4x4,y4 x0,y0x0,y0 xc,ycxc,yc

20. Read Tecplot data file  Format of head before data TITLE = "Import0 in Mae513 Velocity vectors [positions in cm] [velocities in cm/s]" VARIABLES = " Position x "," Position y "," Velocity u "," Velocity v " ZONE T="Data", I=34, J=34, F=POINT DT=( SINGLE, SINGLE, SINGLE, SINGLE ) -3.200000000e+001 -3.200000000e+001 0.000000000e+000 0.000000000e+000 0.000000000e+000 -3.200000000e+001 0.000000000e+000 0.000000000e+000 x y u v

21. Program list (Matlab) FileName = 'Take_01000.dat'; % TecPlot data file which saves velocity data fid = fopen(FileName,'r'); Line = fgetl( fid ); Line = fgetl( fid ); Line = fgetl( fid ); Dim = sscanf( Line, 'ZONE T="Data", I=%d, J=%d, F=POINT'); Line = fgetl( fid ); nx = Dim(1);% Grid number in x direction ny = Dim(2);% Grid number in y direction u = zeros(nx, ny); v = zeros(nx, ny); x = zeros(nx, 1); y = zeros(1, ny); temp = fscanf(fid, '%e'); s = 1; for j = 1:ny for i = 1:nx x(i, 1) = temp(s); y(1, j) = temp(s+1); u(i,j) = temp(s+2); v(i,j) = temp(s+3); s = s + 4; end fclose(fid); % now you have x(1:nx, 1), y(1, 1:ny), u(1:nx, 1:ny), v(1:nx,1:ny) % you can do further processing of x, y, u, v

22. Program list (Fortran) !! maximum dimension for u and v parameter (mx = 101, my = 101) real x(mx), y(my), u(mx,my), v(mx,my) character*32 FileName /'Take_01000.dat'/ character*80 Line open(100, file=FileName, status = 'old') read(100, *); read(100, *); read(100, '(A80)') Line; read(100, *) !! The line contains the string of !! 'ZONE T="Data", I=%d, J=%d, F=POINT' ie1 = index(Line, 'I=') + 2 ie2 = index(Line, ', J=') - 1 ie3 = index(Line, 'J=') + 2 ie4 = index(Line, ', F=POINT') - 1 read(Line(ie1:ie2), '(I)') nx read(Line(ie3:ie4), '(I)') ny do j=1, ny do i=1, nx read(100, *) x(i), y(j), u(i,j), v(i,j) enddo close(100)

23. Program list (C/C++) #include // maximum dimension for x and y directions #define MX101 #define MY101 int main() { char FileName[32]; int nx, ny; char Line[81]; float x[MX], y[MY], u[MY][MX], v[MY][MX]; strcpy(FileName, "Take_01000.dat"); FILE *fp = fopen(FileName, "r"); if ( !fp ) return -1; fgets(Line, 80, fp); fgets(Line, 80, fp); fgets(Line, 80, fp); fgets(Line, 80, fp); // printf("%s", Line); sscanf(Line, "ZONE T=\"Data\", I=%d, J=%d, F=POINT\n", &nx, &ny); printf("nx=%d ny=%d\n", nx, ny); fgets(Line, 80, fp); for (int j=0; j<ny; j++) for (int i=0; i<nx; i++) { fscanf(fp, "%e %e %e %e", &x[i], &y[j], &u[j][i], &v[j][i]); } fclose(fp); }