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
Lecture 25. Correlation Interrogation
& FFT Acceleration
2
Correlation Interrogation & FFT Acceleration
Correlation interrogation basics
Interrogation grid & interrogation window
PIV recording
Mg
Interrogation grid (MgNg)
M
Ng
N
Interrogation window (MN)
3
Correlation Interrogation & FFT Acceleration
Correlation interrogation basics
Evaluation sample
PIV recording
Evaluation sample
4
Correlation Interrogation & FFT Acceleration
Correlation interrogation basics
Coordinate systems
y
PIV recording (nxny pixels)
Evaluation sample (MN pixels)
G(x,y)
j
g(i,j)
o
o
i
x=1,2,•••,nx
i=1,2,•••,M
y=1,2,•••,ny
j=1,2,•••,N
x
5
Correlation Interrogation & FFT Acceleration
Correlation interrogation basics
Interrogation window overlap
-
Grid distance smaller than window side length
-
Enable reuse of particle images
-
Over sampling requires too much evaluation time
ox=oy=50%
ox  1 
Mg
oy  1 
Ng
M
N
ox=oy= 75%
6
Correlation Interrogation & FFT Acceleration
Evaluation function
Auto-correlation function
-
One double exposed evaluation sample, i.e. g(i,j)=g2(i,j)=g1(i,j)
-
Displacement determined by positions of the secondary maxima
-
Two possible velocity directions
M
N
1
 m, n     g i, j   g i  m, j  n 
0.8
(m,n)
g(i,j)
i 1 j 1
0.6
0.4
0.2
0
-30
-30
-20
-20
-10
-10
n
0
0
10
10
20
30
(m*=10, n*=-5)
m
20
30
(m*=-10, n*=5)
7
Correlation Interrogation & FFT Acceleration
Evaluation function
Cross-correlation function
-
Two single exposed evaluation samples, i.e. g2(i,j) & g1(i,j)
-
Displacement determined by position of the maximum
-
Velocity direction clear
M
g1(i,j)
N
 m, n     g1 i, j   g 2 i  m, j  n 
i 1 j 1
1
(m,n)
0.8
0.6
0.4
0.2
0
-30
-30
-20
g2(i,j)
-20
-10
-10
n
0
0
10
10
20
m
20
30
30
(m*=3, n*=5)
8
Correlation Interrogation & FFT Acceleration
Fast computation of evaluation function
Acceleration with radix-2 based FFT algorithm
- Side length of interrogation window selected to be powers of 2
- Evaluation sample assumed to be periodically distributed
M
N
 m, n     g1 i, j   g 2 i  m, j  n 
i 1 j 1
g1 i, j 
FFT
g 2 i, j 
FFT
gˆ1 u, v
gˆ 2 u, v
Complex conjugate
gˆ 2* u, v
Changing the sign of the image part
m, n
FFT-1
ˆ u, v  gˆ1 u, vgˆ 2* u, v

9
Correlation Interrogation & FFT Acceleration
Fast computation of evaluation function
Acceleration with radix-2 based FFT algorithm
- Periodical reconstruction of the correlation function
n
N
M
g(i,j)
N
 m, n     g i, j   g i  m, j  n 
i 1 j 1
-M
M
m
m, n   m  kM , n  kN 
-N
10
Class project
Project warm-up
•
Write a Matlab program to read gray values at the center and 4 corners of
image: http://lcgui.net/lectures/lecture02/image01.bmp
•
Try to use pixel or filter operations to process the digital image
– A 124×124-pixel uncompressed 8-bit gray-scale BMP file
File header:
Bitmap header:
Color palette: 1024
Bitmap data:
Total size:
14
40
bytes
15376
16454
bytes
bytes
y
G(x,y)=A(ny-y+1,x)
bytes
bytes
– Simplest Matlab program
A=imread('image01.bmp');
66 (top left)
A(1,1)
71 (top right)
A(1,124)
83 (bottom left)
A(124,1)
A(124,124) 63 (bottom right)
87
A(62,62)
imshow(A);
A(line,column)
x
11
Class project
Project option #1
- Write a computer program to evaluate a double-exposed PIV recording with auto-correlation algorithm
The image in Case D of PIV Challenge 2001 is used here as a sample recording. It is a double-exposure provided by the research group
of Professor Adrian for a near-wall turbulent pip flow experiment. The measurement domain is 6-mm x 6-mm (beginning at the wall)
in a 137-mm diameter pipe, and the flow was directed to right. The airflow was seeded with 1-micron olive oil droplets and
illuminated with a pulsed YAG laser (~200 mJ/pulse). The particles were imaged with a Nikkor 80-mm f/5.6 Lens and recorded in a
4"x5" photographic film with resolution of 125 lines/mm. The time interval between the laser double pulses is 18 µs. The
photographic PIV recording is digitized in size of 1024x1024 pixels.
http://www.edpiv.com/zips/1.ZIP
12
Class project
Project option #2
- Write a computer program to evaluate a single-exposed PIV recording pair with cross-correlation algorithm
A pair of PIV recordings, which is one example out of several thousand PIV recordings recorded by Kaehler (2001) in DLR within the EC
funded EUROWAKE project dedicated to the investigation of wake vortices behind a transport aircraft, is used here to demonstrate
the procedures for evaluating a single-exposed PIV recording pair with the EDPIV software. The measurement was conducted at 1.64
m behind the wing tip, and the field of view of 170-mm x 140-mm was imaged with a digital resolution of 1280x1024 pixels. An
evaluation procedure is suggested as follows.
http://www.edpiv.com/zips/2.ZIP
13