Measurements in Fluid Mechanics 058:180 (ME:5180) Time & Location: 2:30P - 3:20P MWF 3315 SC Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui Phone: (Lab), (Cell)

2 Lecture 33. Peak-locking effect

3 Evaluation Errors Bias & random error for replicated measurement Measuring variable X for N times RMS fluctuation (random error) RMS error Individuale reading of X: Mean value 0

4 Peak-locking Effect Example: PIV test in a thermal convection flow One of PIV recordings32  32-pixel window

5 Peak-locking Effect Example: PIV test in a thermal convection flow One of vector mapsHistogram of U & V

6 Peak-locking Effect Example: PIV test in a thermal convection flow Correlation-based interrogation Correlation-based tracking MQD-tracking Histograms resulting from different algorithms Peak-locking Is the peak-locking an error? Why does the peak-locking exist? How to reduce the peak-locking effect?

7 Histogram for measuring 0.5 pixels    Probability density function (PDF) Source of Peak-locking Probability to get X when measuring X o

8 Distribution density function (DDF) Source of Peak-locking Distribution density function of true value X o in region [a,b]: -  (X o )/(b-a): probability to find true value X o in region [a,b] - Physical truth to be investigated Distribution density function of measured value X: -  (X)/(b-a): probability to get value X when measuring X o in region [a,b] - Investigated phenomenon - Defined in region [- ,+  ]: Histogram of measured variable X: - Number of samples in [X-  /2,X +  /2] - M: average number in 

9 Source of Peak-locking Distribution density function (DDF) Histogram determined by 1)Sample numberM 2)Sub region size  3)Physical truth  (X o ) 4)Bias error  (X o ) 5)Random error  (X o ) Possible sources of peak-locking

10 Bias & Random Error Distribution Simulation of Gaussian particle images Test results with simulated PIV recording pairs - particle image diameter:2  5 pixels - particle image brightness:130  particle image number density: 20 particles in 32  32-pixel window - vector number used for statistics:  15,000 w/o single pixel random noisewith single pixel random noise (CDWS=DWS, CCWS=CWS, FCTR=correlation-base tracking) CDWS – Correlation-based discrete window shift (=DWS) CCWS – Correlation-based continuous window shift (=CWS) FCTR – FFT accelerated correlation-based tracking

11 Peak-locking Factor DDFs and histograms for the test results

12 Response of  to bias and random error distribution   very sensitive to bias error amplitude A    sensitive to random error amplitude A  when >0.02   not sensitive to constant portion of random error  0 Peak-locking Factor Simulation of error distributions: Simulated error distributionsResponse of peak-locking factor

13 Contours of peak-locking factor for  o =0.025  Peaks locked at integer pixels in bright area and at midpixels in dark area  Peak-locking minimum around A  =0  Increasing A  increaes  for A  0 Peak-locking Factor Response of  to bias and random error distribution

14 Influence of particle size on  Test results   increases with incresing particle size by CDWS   descreses with incresing particle size by FCTR & CCWS   increases when particle szie too small by FCTR & CDWS   smallest when particle szie too small by CCWS   generally smallest by FCTR (for Gaussian image profile) Increasing A  when A  >0 for CCWS Peak-locking Factor

15 Influence of particle number density on  Test results   not sensitive to particle image number density   generally smallest by FCTR (for Gaussian image profile) Peak-locking Factor

16 Influence of window size on  Test results   decreases with incresing window size by CDWS   slightly increses with incresing window size by CCWS   slightly decrease with incresing window size by FCTR   generally smallest by FCTR (Gaussian image profile) Peak-locking Factor

17 Image samples of different quality Non-Gaussian Particle Images Influence of particle image profile

18 Application Examples PIV measurement in a thermal convection flow Gray value histogram & evaluation sampleHistogram of particle image displacement - Overexposed particle images - Particle image diameter 3  4 pixels - No peak-locking for CCWS

19 Application Examples PIV measurement in a wake vortex flow Gray value histogram & evaluation sampleHistogram of particle image displacement - Particle image diameter 1 pixels - Least peak-locking for CCWS

20 Application Examples PIV measurement in a micro channel flow Gray value histogram & evaluation sampleHistogram of particle image displacement - Mid-pixel peak-locking for CCWS - Particle image diameter 4  6 pixels

21 Gui and Wereley (2002) A correlation-based continues window shift technique for reducing the peak locking effect in digital PIV image evaluation. Exp Fluids 32: References

Matlab program for showing peak-locking effect A1=imread('A001_1.bmp'); % input image file A2=imread('A001_2.bmp'); % input image file G1=img2xy(A1); % convert image to gray value distribution G2=img2xy(A2); % convert image to gray value distribution Mg=16; % interrogation grid width Ng=16; % interrogation grid height M=32; % interrogation window width N=32; % interrogation window height [nx ny]=size(G1); row=ny/Mg-1; % grid row number col=nx/Mg-1; % grid column number sr=12; % search radius for i=1:col % correlation interrogation begin for j=1:row x=i*Mg; y=j*Ng; g1=sample01(G1,M,N,x,y); g2=sample01(G2,M,N,x,y); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx; V(i,j)=vy; X(i,j)=x; Y(i,j)=y; end end % correlation interrogation end nn=0; % count number of displacements with 0.1 pixel steps for k=-120:120 nn=nn+1; D(nn)=double(k/10); Px(nn)=0; Py(nn)=0; for i=1:col for j=1:row if U(i,j)>= D(nn)-0.05 & U(i,j) < D(nn)+0.05 Px(nn)=Px(nn)+1; end if V(i,j)>= D(nn)-0.05 & V(i,j) < D(nn)+0.05 Py(nn)=Py(nn)+1; end plot(D,Px,'r*-') % make plots hold on plot(D,Py,'b*-') hold off