PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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, lecuona@ing.uc3m.es
PIVNET 2/Ercoftac SIG 32 workshop
Lisbon, 5th and 6th July 2002.
1
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
Contents
I.
Standard PIV algorithms
II.
Image distortion algorithms
III. Advanced post-processing of PIV data based
on the Navier-Stokes equations
2
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I. Standard (correlation) PIV
1. Interrogation window
2. Cross-correlation using FFT or DC
3. Peak detection
s’
4. Subpixel peak fitting
3
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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.
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.
Standard PIV (cont.)
Peak locking:
Sources
•
d < 2, particle image undersampling interchangeable with random
•
F ∼ 1 Particle truncation at window borders, relevant for extreme
error (unavoidable) .
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
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.
Standard PIV (cont.)
Standard PIV limits:
δ
F
Physical: δ >≈ d; Nyquist λ > 2δ
Amplitude response:
λ ≥ 1,7⋅F (in figure 3)
Nmin
Multigrid
Detectability: δ ≤ (π⋅Nmin /4)-1/2 F
Nmin = minimum # particles for
peak detection
Multigrid methods progressively reduce w to increase
spatial frequency resolution overcoming amplitude
response criterion.
Nmin ∝ (F/λ)n→ detectability limits could arrive first when
resolving λ. Line:
Aplicable
λ
F
λ
7
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.
Standard PIV (cont.)
Aggregation of particle pair peaks:
I.
A pair peak contributes to the main peak if
sm = main peak displacement.
sp = pair peak displacement.
sm - s p < R + w p
14243
min = 3
wp
R = radius considered in the peak fitting algorithm.
R
wp = pair peak waist ≈ d/√2
Consequences:
I.
sm-sp
Large particles and low disp. differences → window moving averaging
II.
Small particles and high disp. differences → no averaging
III.
Intermediate case → non linear amplitude response and group locking
IV.
The most distant peaks are dropped → non linear response
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.
Standard PIV (cont.)
Examples of group locking
I.
1D Linear gradient (spurious peaks eliminated)
Grad⋅F/2
Grad⋅F
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
Group locking
Displacement
4 particle peaks
Many particles peaks
9
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.
Standard PIV (cont.)
Examples of group locking
Correlation plane of 1D sinusoidal displacement, d = 2,5; R = 1,5; F = 32
0.74
0.72-0.74
0.7-0.72
0.68-0.7
0.66-0.68
0.64-0.66
0.62-0.64
0.6-0.62
0.72
0.7
0.68
0.66
0.64
0.62
0.6
s=4
s=4
s =16
Noiseless gaussian particles randomly located
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.Standard PIV (cont.)
Group locking: a local detection algorithm
G=
1
>0
 ∂s 2  ∂s  2   ∂s 2  ∂s  2 
 x  +  y    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 sm-sp < R + wp
This effect is negligible for F/λ << 1 because sm-sp vanishes (windows will have
the same colour).
A qualitative error measure in a window is σG, but is not computable form PIV
data.
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.Standard PIV (cont.)
Examples of group locking (cont.) PIV obtained results
10
10
8
8
8
6
6
6
4
2
0
-2
-4
-6
Displacement (pixels)
10
Displacement (pixels)
Displacement (pixels)
1D Sinusoidal displac. (cont. line input, red line model, points PIV output)
4
2
0
-2
-4
-6
-8
-8
32
64
96
128
160
192
224
256
288
Location (pixels)
λ/F =
2
0
-2
-4
-6
-8
-10
-10
4
-10
32
64
96
128
160
192
224
Location (pixels)
λ/F = 2
256
288
32
64
96
128
160
Location (pixels)
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 x = − A cos(2π x / λ ) sin(2π y / λ )
s y = A sin(2π x / λ ) cos(2π y / λ )
F = 16
F = 32
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I.Standard PIV (cont.)
Amplitude response to 1D sinusoidal displacement field
First armonic relative amplitude
1.0
Circles F = 64. Squares F = 32.
0.8
0.6
Empty symbols: Clean images.
0.4
0.2
0.0
A
-0.2
C
B
-0.4
-0.6
-0.8
-1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
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
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
I. Standard PIV (cont.)
Conclusions:
I. Correlation PIV with no image distortion is accurate only when peak
overlapping is assured → sm-sp < R + wp. It is not equivalent to the rule of
thumb (smax  -  smin )/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 sm-sp so that peak aggregation increases
SNR and non-linearities are reduced. Window distortion is a partial application
of image distortion.
δ
Nmin
Resulting in a wider
area of application:
Aplicable
λ
14
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
II. Image distortion algorithms
•
The main purpose is to reduce sm-sp 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.
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
II. Image distortion algorithms (cont.)
+
LFC-PIV
+
Correlation
Image a*
Image a
Correlation
+ Previous
measurements
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.
16
Grid spacing is 16 pixels.
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
II. Image distortion algorithms (cont.)
Example of results obtained with LFC-PIV
Original image
LFC, F = 64, λ = 65,6
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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
PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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: Cd experimental =1.16, Cd numerical = 1.18.
• Different topologies can be used simultaneously (left)
• Random placement is feasible (right)
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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).
1
0.8
0.6
Reconstructed pressure profile
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
0
1
θ
2
3
0.009
0.008
0.007
Reconstructed
shear stress
0.006
0.005
0.004
0.003
Cd experimental: 1.46, Cd reconstructed: 1.44
Experimental separation point: 110 deg.
Reconstructed separation point: 120 deg.
0.002
0.001
0.000
-0.001
0.0
1.0
θ
2.0
3.0
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PIVNET2/ERCOFTAC SIG32 workshop, Lisbon, 5th July 2002, Advanced PIV algorithms
A. Lecuona et al. Universidad Carlos III de Madrid
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.
Cdexp = 1.46, Cdrecons = 1.44
Separation point:
exper. = 110 deg, recons. = 120 deg
Cdexp = 1.46, Cdrecons = 1.47
Separation point:
exper. = 110 deg, recons. = 127 deg23