On the application of image
analysis techniques to fluid
mechanics experiments
John H. Cushman1, Natalie Kleinfelter1, Monica Moroni2
1Department
of Earth and Atmospheric Sciences and
Department of Mathematics, Purdue University, West Lafayette
2Department of Hydraulics, Transportations and Roads,
University of Rome “La Sapienza” Rome (Italy)
Purpose
• To introduce image analysis techniques
– Non-intrusive
– Flexible
– Lagrangian description->Eulerian description
• Quantities can be derived
• Phenomena can be investigated
Tables of contents
Image analysis
– digital image processing
– different algorithms for image analysis
Applications
– Porous media
– Convective boundary layer
– Subduction
– Multi-dune channel
– Fully developed turbulent channel
“Ingredients” for image analysis
The fluid under investigation and the test section have to be
transparent: mono-phase and multi-phase systems
The fluid has to be seeded with tracer particles with the
following features: neutrally buoyant and highly reflecting
One or more cameras, a high power light source, an
acquisition and digitalization system and image analysis
system are required
Image analysis
Methods and applications
Images: formation and representation
Digital image processing is concerned with the computer
processing of pictures or, more generally, images that have
been converted into a numeric format.
An image is a picture, photograph, display, or other form
giving a visual representation of an object or scene.
However, in digital image processing, an image is a set of K
two dimensional arrays of numbers of N-lines (rows) and Msamples (columns).
NxM
K
K=1 
K=3 
image resolution
defines image bands
black and white image
color image
Images: formation and representation
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The figure in an image of a simple geometric pattern (left-hand side)
and its corresponding digital image (right-hand side). K=1. The digital
image has 11 lines and 11 samples per line. Each number in the
matrix corresponds to one small area of the visual image and the
number gives the level of darkness or lightness of the area
Images: formation and representation
We assume the higher the number, the lighter the area, so
zero is black, the maximum value is white, and intermediate
values are shades of grey (this is arbitrary).
Each small area to which a number is assigned is called
pixel (picture element). The size of the physical area
represented by a pixel is called the spatial resolution of the
pixel. Each pixel has its value, plus a line coordinate and a
sample coordinate.
The minimum value a pixel can have is typically zero, and
the maximum depends on how the number is stored in the
computer. It is common to store each pixel as a byte (8 bits).
Maximum grey level value? 255=2^8-1
0255  grey level scale
Images: display as a matrix
Images: display as a 3D surface
Images: display as an ‘image’
High quality image
Zoom in previous image
Further zoom in previous image
Low quality image
Sequence of images: file .avi
Do other kinds of images exist?
Differences with previous images:
- color image
- no tracer particles
Do images contain useful information?
The analysis of those images allows
the convective boundary to be
determined. It is localized at the
interface between the image light
and dark regions
Buffer Red
Buffer Green
Buffer Blue
Ratio pixel/cm
328 pixels
pixel/cm=328/1
Particle Tracking Algorithms
Particle Tracking algorithms require:
• seeding the fluid with small highly reflecting particles at a given
particle density;
• illuminating as uniformly as possible the flow field with a light
sheet;
• acquiring images of the particles located in this sheet with an
imaging rate fast enough to maintain good resolution.
In contrast to standard PIV (Particle Image Velocimetry) where
the mean displacement of a small group of particles is sought,
with Particle Tracking algorithms pathlines of individual
particles are obtained.
PTV (Particle Tracking Velocimetry)
Main steps of Particle Tracking Velocimetry (PTV):
- pre-processing the images to eliminate background noise;
- determining the particle centroid coordinates in each
frame with sub-pixel accuracy:
• grey
level “binarization”;
• pixel labeling;
• centroid determination;
- tracking particle centroids frame by frame.
PTV: image pre-processing – background
removal
‘Raw’ images
Background removal effect
n=17 rows
Pixel under investigation
1
I (i, k )  2
n
n=17 columns
n
n
 I (i, k )
i 1 k 1
!!! n has to be odd
I filtered (i, k )  I (i, k )  I (i, k )
PTV: centroid determination - binarization
Count
foreground
Threshold
level for
background
pixel brightness:
i.e. 100
Grey levels
0
Ideal threshold
255
PTV: centroid determination
Threshold level for
pixel brightness:
i.e. 100
Options for computing the centroid position:
- Arithmetic average of pixel coordinates
- Weighted average of pixel coordinates and grey levels (weights)
- Average over a gaussian distribution
PTV: trajectories reconstruction
t3
t2
t1
toll
t0
Dmax
Tracer density???
Problem
What if tracer density is high?
Feature tracking
The main steps in FT are:
• seeding the fluid with small highly reflecting particles
at any particle density;
• illuminating as uniformly as possible the flow field with
a light sheet;
• acquiring images of the particles located in this sheet
with an imaging rate fast enough to maintain good
resolution.
Feature Tracking
Continuity equation for the “optical flow” (“image brightness
constancy constrain” (BCC))
DI I
I
I I

u
v

  I T  U  I t  uI x  vI y  0
Dt t
x
 y t
If the equation is computed at a single point, it only provides one
equation for two unknowns, the velocity components. It is only when
the equation is evaluated at each point in a region W surrounding
the one under investigation (feature), that it provides sufficient
information on U
The cost function SSD (Sum of Squared Differences) over a
window W surrounding the feature under investigation representing
the dissimilarity between the image I at time tA (IA) and at
successive time tB (IB), can be written
2

1
1 
1   I
 DI  

2
T
I B  I A  dS      dS     I  U  dS
SSD 
2 
W W
W W   t
Wt W

 Dt  t A 

2
Feature Tracking
To obtain a least squares estimation of U(x), the derivative of the cost
function with respect to U is evaluated:
2

 SSD 2
I 
 I x
 T

 I   I  U  dS  2  

U
WW
t 

W
 I y I x
Setting the equation to zero:
 I x2 dS
 W
 I I dS
 y x
W
 I x I t dS 
I y dS 




W
W

U

0



2
I I dS
W I y dS 
 y t 
W

I
x
or more simply:
G U b  0
IxIy 
 I x I t 


U

 I I  dS
2 
I y 
 y t 

Feature Tracking
The square matrix G is invertible if both eigenvalues 1 and 2 are
non zero. Three different cases are possible:
Ix=0 and Iy=0, i.e. uniform intensity distribution in both
directions (case a): both eigenvalues are zero
Ix=0 and Iy≠0, i.e. uniform intensity distribution in the x direction
(case b) or Ix≠0 and Iy=0, i.e. uniform intensity distribution in the
y direction (case c): one eigenvalue is null (1>0; 2=0 )
Ix≠0 and Iy≠0, i.e. not uniform intensity distribution in both
directions (case d): both eigenvalues are positive (1>0; 2>0)
y
a)
y
x
b)
y
y
x
x c)
x d)
FT vs. PTV
Highlights on PTV
–Best buffer identification
and choice of the
threshold value;
–Centroid identification
with sub-pixel accuracy;
–Trajectory reconstruction:
the nearest principle is
employed. The algorithm
applies the criterium of
“minimum acceleration”
when ambiguities arise.
Highlights on FT in a pure
traslational model
– Image features selection
– Features
tracking on
from frame
No constraints
to frame.
image particle
The
matching measure
density
introduced to follow a feature
(and its interrogation window)
and
“most similar”
region at
Noitssubjective
parameters
thehave
successive
time is the
to be provided
“Sum of Squared Differences”
(SSD) among intensity values:
the displacement is defined as
the one minimizing the SSD.
Problem
What if the flow is 3D?
The state of art on 3D techniques
There exist a number of imaging-based measurement
techniques for determining 3D velocity fields in an
observation volume. Among these are:
• scanning techniques (Guezennec et al. 1994, Moroni
and Cushman, 2001a, b);
• holographic techniques (Hinsch and Hinrichs 1996,
Katz 1999);
• defocusing techniques (Willert and Gharib 1992);
• photogrammetric techniques (Maas 1992, Kasagi and
Nishino 1990).
Main features of scanning and
photogrammetric 3D-PTV
“Scanning” methods share the following basic steps (Moroni &
Cushman, 2001):
1.
2.
3.
4.
set-up calibration
determination of centroids in 2-D;
trajectories reconstruction in 2-D;
trajectories matching  3-D.
“Photogrammetric” methods share the following basic steps
(Papantoniou and Dracos, 1990):
1. stereoscopic calibrated imaging and recording of a suitably
illuminated particle flow;
2. photogrammetric analysis of the resulting images to derive the
instantaneous 3-D particle positions;
3. tracking of the 3-D coordinate sets in time to derive the tracer
trajectories.
Scanning 3D-PTV
optical rays
are supposed to be
parallel
Three dimensional trajectory construction
from two 2-D projected trajectories
Photogrammetric 3D reconstruction
Y'

Pi
Z
Assumptions:

Zi
– model of imaging: pinhole camera
modeled by its
Y
P'

optical center C and the image plane R
C

– the theorem of photogrammetry holds
Xi
i
y
c

x
Yi
Z0
Introduce the following reference frames:
X0
z
Y
• world reference
frame X,Y,Z;
X'
• 0image reference frame ;
X 
•Z' camera standard reference frame X',Y',Z'
0
3D trajectory reconstruction
In formulas:
  0  c
r11 ( X  X 0 )  r21 (Y  Y0 )  r31 ( Z  Z 0 )
r13 ( X  X 0 )  r23 (Y  Y0 )  r33 ( Z  Z 0 )
r12 ( X  X 0 )  r22 (Y  Y0 )  r32 ( Z  Z 0 )
  0  c
r13 ( X  X 0 )  r23 (Y  Y0 )  r33 ( Z  Z 0 )
(X,Y,Z)
(X0,Y0,Z0)
R= rij
(0,0)
c
R
: object point coordinates
: camera projective center coordinates
: elements of 33 rotation matrix with angles , , 
: image principle point
: image principle distance
cos  cos 


  cos sen  sensen cos 
 sensen  cos sen cos 

 cos sen
cos  cos   sensensen
sen cos   cos sensen
sen 

 sen cos  
cos  cos  
3D trajectory reconstruction
Methods for detecting 3D trajectories:
Geometric method
Radiometric method
Acquisition system
calibration
2D centroid locations
Tracking in 2D
Correspondence
Structure from stereo
3D location
coordinates
Tracking in 3D
Calibration
Aim: 9 parameters to be determined: 0, 0, c,
, , , X0, Y0 and Z0. The camera constructor
usually insures 0 and 0 to be equal to zero.
Method: given r observations (r>u)
li  ai1 x1  ai 2 x2  ...  aiu xu
where
xk are unknowns (equal to u);
li is the observation;
aik are coefficients.
Since r>u
s  Axˆ  l
linear
equation
Corrections
ˆx  ( A T A) 1 A T l
Calibration (2)
non-linear
equation
li  f i ( x1 , x2 , ... , xu )
A Taylor expansion is required, with starting values.
Each point Pi of unknown or known coordinates (Xi,Yi,Zi)
and known image coordinates ( i ,  i ), provides an equation
as:
0
0
0
  
  
  
 
 


 dX 0  
 dY0  
 dZ 0    dc  
si  
 d 
 c 
  
 X 0 
 Y0 
 Z 0 
0
  
  
  d    d  ( ij   ij0 )
  
  
si  ... (analogous)
0
0
0
Calibration (3)
R
Y
Z
G
X
B
Calibration (4)
N
points
st
st
st
Red
camera
342
90
300
0
Green
camera
349
90
0
0
Blu
camera
360
90
60
0
Iterative procedure
Calibration (5)
Camera
X0
(cm)
Y0
(cm)
Z0
(cm)
c
(cm)



mean
(pixel)
st dev
(pixel)
Red
-55.39
-17.29
14.55
2.72
89.47
299.35
-0.31
7.25
3.19
Green
12.09
-54.58
14.42
2.54
89.89
0.00
0.24
7.06
3.21
Blu
79.00
-18.19
12.64
2.62
92.86
59.49
-2.65
17.91
7.82
Need for a calibration procedure with an easier implementation
Structure from stereo
Correspondences using the epipolar lines
Pf
Pe
Pd
I3
E ( 23 ) i
P
Pa Pb
Pc
E12
I1
I2
Test on synthetic data
The algorithm was tested on the basis of a synthetically
generated data set simulating curling trajectories with a
starting location randomly distributed in the world reference
system.
The effect of an increasing number of particle seeding the
measurement volume, i.e. of trajectories, was tested.
The effect of a wrong positioning of camera’s projective
centers and rotation angles was tested as well.
Test on synthetic data (2)
Up to 1200 spots per frame, 100% of particles
were matched
With a 10% error in calibration parameters and
1200 spots per frame, 95% of particles were
matched
Applications
•
•
•
•
•
Porous media
Convective boundary layer
Subduction
Multi-dune channel
Fully developed turbulent channel
Mapping flow during retreating
subduction
Purpose
• To model the large-scale mantle circulation
induced by subduction of a laterally migrating
slab in three-dimensional dynamically consistent
laboratory models
• To compare experimental results and numerical
simulation outcomes
• To predict the path of melted material, the
distribution of geochemical anomalies, the
formation of back-arc basins
Introductions
• The Earth's interior consists
of rock and metal. It is made
up of four main layers:
– the inner core: a solid metal
core made up of nickel and
iron (1200 km diameter)
– the outer core: a liquid molten
core of nickel and iron
– the mantle: dense and mostly
solid silicate rock
– the crust: thin silicate rock
material
Image of Earth and the interior
layers
Introductions (2)
Plate tectonics is a theory of geology developed to explain the
phenomenon of continental drift and is currently the theory
accepted by the vast majority of scientists working in this area. In
the theory of plate tectonics the outermost part of the Earth's
interior is made up of
two layers: the
lithosphere
comprising the crust
and the solidified
uppermost part of
the mantle. Below
the lithosphere lies
the asthenosphere
which comprises the
inner viscous part of
the mantle.
Introductions (3)
The lithosphere is broken into giant plates that fit around
the globe like puzzle pieces. These puzzle pieces move a
little bit each year as they slide on the asthenosphere.
The asthenosphere is ductile and can be pushed and
deformed like putty in response to the warmth of the
Earth.
Subduction is the process in which one plate is pushed
downward beneath another plate into the underlying
mantle when plates move towards each other. The plate
that is denser will slide under the thicker, less dense
plate.
Experimental set-up: assumptions
• Viscous rheology
– the Earth system is simulated using viscous rheologies
• Self-Consistent Subduction
– Slab pull is the only active force within the system. This ensures
that the experimental subduction process is a self-consistent
response to the dynamic interaction between slab and mantle.
• Convectively Neutral Mantle
– Flow is generated only by subduction. Thermal convection and
global or local background flow that is not generated by the
plate/slab system are neglected.
• Isothermal Experiments
– Thermal effects during the subduction process are neglected.
• No Overriding Plate
– The overriding plate is not modeled. We assume the plate is
completely surrounded by fault zones whose viscosity is the
same as the upper mantle one.
Experimental set-up: materials
• Silicone putty (Rhodrosil Gomme, PBDMS+ iron fillers) and
glucose syrup are used as analogue of the lithosphere and upper
mantle, respectively.
• Silicone putty is a viscoelastic material with purely viscous
behavior at experimental strain rate (= 1480 kg/m3, = 3.6 105
Pa s)
• Glucose syrup is a transparent Newtonian low-viscosity and highdensity fluid (fluid #1: = 1415 kg/m3, = 30 Pa s; fluid #2: =
1382 kg/m3, = 3 Pa s)
• The bottom of the test section plays the rule of the 660 km
discontinuity (interface between upper and lower mantle).
Scaling factors:
1 cm in the laboratory model corresponds to 60 km in Nature
1 Myr in Nature corresponds to  1 min
The experimental apparatus
Camera
(top view)
Camera
(lateral view)
x
y
Plate width
- 30 cm
- 20 cm
- 10 cm
z
Experimental set-up: procedure
• Glucose syrup is seeded with neutrally buoyant, highly
reflecting air microbubbles acting as passive tracers. Air
bubbles negligibly influence density and viscosity of the
mantle.
• The subduction process is manually started in all the
experiments by forcing downward the leading edge of the
silicone plate into the glucose to a depth of 3 cm (180 km in
nature).
• Each experiment is monitored over its entire duration by two
black and white progressive scan cameras imaging the lateral
and top views.
• Two neon lights produce a planar uniform radiation focused
onto two normal cross-sections through the system.
FT: Trajectories
Fluid #2, plate width= 20 cm, lateral view
FT: Trajectories
Fluid #2, plate width= 20 cm, top view
FT: Trajectories
Fluid #2, plate width= 10 cm, top view
FT: Velocity field
Time-averaged velocity field - fluid #2, plate width= 20 cm - top view
Results: evolution of subduction
We observe that the subduction process evolves
in three main stages:
1. initial transient sinking into the upper mantle
2. interaction with the 660 km discontinuity
3. steady state subduction regime
Evolution of subduction: first stage
At the beginning of the experiment, the trench retreats progressively
accelerating and the slab dip increases.
Both poloidal and
toroidal advection
cells can be
Poloidal
recognized in the
cell
velocity field
since this initial
transient stage.
Images show
mass exchange
of mantle from
the ocean to the
wedge side of the
plate.
Evolution of subduction: first stage
Top view
Toroidal
cell
Evolution of subduction: second stage
After the bottom of the test section is reached, the trench velocity
significantly diminished while the tip of the slab folds and deforms in
correspondence of the 660 km discontinuity. Mantle circulation
slows down as well.
Evolution of subduction: third stage
Once the leading edge of the subducting plate has reached a stable
arrangement at the bottom of the box, mantle circulation in a steady
state regime establishes. The steady subduction velocity is a direct
consequence of the constant slab pull force applied to the constant
portion of subducted lithosphere.
Less vigorous
poloidal cell
Results: influence of plate width
•The subducting plate width (w) was changed preserving laterally
unconstrained boundary conditions. The plate width strongly
influences the subduction process. Increasing w from 10 cm to 30 cm,
the trench velocity and consequently also mantle velocity decreases.
The increase in w also strongly affects the vigor of mantle circulation.
In particular, under the same density/viscosity mantle properties, a
wider plate moves a larger amount of mantle material
Conclusions
• Our experiments confirm previous results in terms of subduction
kinematics, identifying the presence of a typical sequence of
stages in the kinematic evolution of a retreat subduction process:
• the sinking of the slab into the upper mantle;
• the interaction with the 660 discontinuity;
• the steady state stage with the slab lying at the upper/lower
mantle transition zone.
• The dependence of the plate width and the subduction
kinematics is confirmed.
• Feature Tracking is a suitable technique to map and to
quantitatively estimate the pattern of flow triggered in the mantle
by subduction.
• Rollback subduction generates a complex 3-D time dependent
circulation pattern characterized by the presence of poloidal and
toroidal components, both active since the beginning of the
subduction process and evolving according to kinematic stages.
Applications
•
•
•
•
•
Porous media
Convective boundary layer
Subduction
Multi-dune channel
Fully developed turbulent channel
Turbulent mixing layer growth and
internal waves formation:
laboratory simulations
Motivations and purposes
The flux through the interface between the mixing layer
and the stable layer plays a fundamental role in
characterizing and forecasting the quality of water in
stratified lakes and in the upper oceans, and the quality
of air in the atmosphere.
General aims of the investigation:
• predicting mixing layer growth as a function of initial and
boundary conditions
• understanding the interaction between the mixing layer and
the stable layer -> internal waves
• describing the fate of a contaminant dissolved within the
fluid phase
Novelty of this contribution
Enhanced equipment for measuring the fluid temperature
providing a larger time resolution
Large database of experiments run under several initial and
boundary conditions
Image analysis performed with the classical Particle
Tracking (PTV) algorithm and the Feature Tracking (FT)
algorithm
Penetrative convection in the ABL
z

T
T0
0
z
y
x
T,
Penetrative convection in lakes
Z
H
EPILIMNIO
T
TERMOCLINO
Entrainment
(water flow,
nutrients and
contaminants)
IPOLIMNIO
TEMPERATURA
Mixing layer visualization through LIF
Experimental set-up
Measuring techniques: velocity field
•
•
•
•
Seeding the flow (100 m pollen particles);
Illuminating the test section (500 W lamps);
Acquiring images (2-CCD camera, 25 fps);
Processing images through Classical Particle Tracking
Velocimetry (PTV) and Feature Tracking (FT)
Mixing layer: PTV (low seeding density)
Mixing layer: FT (large seeding density)
Internal waves: FT (large seeding
density)
Measuring techniques: Temperature
Thermocouples are placed within the test section:
- along a vertical line (to detect temperature profiles)
- on the lower boundary (to test horizontal
homogeneity)
Features of a subset of experiments
Experiment #
1
2
3
4
5
6
7
8
9
10
11
Tb0
(K)
288.15
287.05
287.15
288.39
285.15
286.66
285.15
293.76
294.32
294.00
287.54
TbC
(K)
292.65
298.15
298.65
305.15
293.85
292.72
294.48
300.00
300.20
299.05
296.71
  (T / z )
(K/m)
24.5
51.5
53.0
40.5
82.0
47.4
68.0
41.1
55.1
29.5
70.3
PTV
OF
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Temperature profile before heating starts
0.20
0.18
0.16
Height (m)
0.14
0.12
Exp #1
Exp #2
Exp #3
Exp #4
Exp #5
Exp #6
Exp #7
Exp #8
Exp #9
Exp #10
Exp #11
0.10
0.08
0.06
0.04
0.02
0.00
282
284
286
288
290
292
294
Temperature (K)
296
298
300
302
304
Three methods for detecting the mixing layer
height
- Mean temperature profiles
- Theoretical height according to the mean
temperature
- Velocity standard deviation profiles
All methods assume horizontal homogeneity. The
first two employ temperature data detected through
the thermocouples. The last one employs velocity
data reconstructed through PTV and OF.
Mean temperature profiles for exp #3
z
  (T / z )
h(t)
T0
0.12
2 min
0.1
6 min
10 min
T(t)
T
14 min
18 min
Theoretical height
according to the
mean temperature
in the mixing layer
h(t )  0 (T (t )  T0 )
Height (m)
0.08
22 min
0.06
26 min
30 min
0.04
34 min
0.02
0
282.0
283.0 284.0
285.0 286.0 287.0 288.0
Temperature (K)
289.0 290.0
291.0 292.0
Vertical velocity standard deviation for
exp #3
0.16
0.25 min
2.25 min
4.25 min
6.25 min
8.25 min
10.25 min
14.25 min
18.25 min
24.25 min
30.25 min
0.14
Height (m)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.0005
0.001
0.0015
sw (m/s)
0.002
0.0025
Height of the mixing layer for exp #3
Transilient matrix




Dispersion phenomena occurring within the mixing layer
are intrinsically non-local.
An approach different than the ADE is needed
Although the system is deterministic, the transilient matrix
can be regarded as describing the set of probabilities that
a tracer particle originates in one subvolume and ends up
in another
The transilient matrix is referred to as the conditional
probability P(r,t|r’,t’) that a particle at position r’ at t’ ends
up at position r at time t.
Transilient matrix
6
i
5
4
3
2
1
j
Cij
Transilient matrix: physics
destination
FAST MIXING
UPWARD
Fast upward
mixing
SLOW MIXING
UPWARD
Slow upward
mixing
NO MIXING
SLOW MIXING
DOWNWARD
FAST MIXING
DOWNWARD
source
Scaling parameters
w*  g  qs zi
3
Height of the mixing layer
zi
t* 
Convective velocity
(g: acceleration of gravity, :
thermal expansion coefficient,
qs: surface kinematic heat flux)
zi
w*
Convective time
The phenomena occurring in the mixing layer are not steady: its
height and mean temperature increase with time. Thus the scaling
parameters are functions of time. In order to compare data not
acquired simultaneously quasi-steadiness has to be assumed.
Hence, after normalization, it can be considered time
independent.
Transilient matrix
Transilient matrix
Transilient matrix
Transilient matrix
Transilient matrix
Transilient matrix
Cross section through the transilient matrix
Concentration begins as a  function at the source depth, marked with
a continuous line, and progressively more disperse curves correspond
to later times. The dotted line marks the boundary of convection zone.
Δt*= 0.01
Δt*= 0.05
Δt*= 0.07
Δt*= 0.08
Δt*= 0.12
Δt*= 0.16
Δt*= 0.3
Δt*= 0.5
Δt*= 0.7
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
Destination depth
1.5
Internal waves
1
0.8
0.6
0.4
0.2
0
-0.2 0
20
40
60
80
100
Time (s)
-0.4
Correlation coefficient for w for experiment #2
  T
N   g 
  z

 

1
Experiment #
N
(s-1)

(s-1)
1
0.35
0.33
2
0.30
0.26
2
Conclusions and future works
The experimental apparatus described allows:
• to reproduce a stable stratification;
• to study heat exchanges;
• to investigate the penetrative convection phenomenon;
• to visualize internal waves and to compute their features;
• to describe the dispersion phenomenon within the mixing
layer through the Transilient Matrix.
Once the transilient matrix is known for a given flow, a number of
quantitative measures of nonlocal transport can be obtained from
it directly. For instance matrix moments with respect to the
destination indices at each source level or relative to the source
indices can be computed
Applications
•
•
•
•
•
Porous media
Convective boundary layer
Subduction
Multi-dune channel
Fully developed turbulent channel
Preliminary Tests On Fluid-Dynamic
Features and Plastic Separation
Feasibility of a Hydraulic Separator
Purpose
• To contribute to the issue of plastic materials recycling
because of environmental awareness, need to
conserve materials and energy, and growing demand
to increase production economy.
• To reconstruct a laboratory model of a hydraulic
separator for low specific mass plastic particles (1
g/cm3).
• To test the capability of image analysis techniques to
reconstruct the velocity field within the facility.
• To set up of the best operative conditions for the
apparatus.
Purpose (2)
• Chaotic advection regards the complex behaviour
characterising a passive scalar due to Lagrangian
dynamics of flow. The main issue related to chaotic
advection is the enhancement of transport it produces.
• Laminar flow can give rise to chaotic behaviour of
Lagrangian particle trajectories even though the Eulerian
velocity at any given point in space is fixed or periodic in
time
• Chaotic mixing and complex distribution of material can
be produced in which nearby fluid elements diverge
strongly from each other
The experimental apparatus
300
12
4
15
C1
60
30
30
20
C3
C4
C5
C6
C7
C8
8
8
30
R1
R2
R3
R4
R5
R6
R7
R8
Longitudinal section (dimensions in mm)
Flow out
Flow in
A multi-dune channel is a device constructed from a sequence
of closed parallel cylindrical tubes welded together in plane.
The complex is sliced down its lateral mid-plane and the lower
half is shifted laterally and then fixed relative to the upper half.
The experimental apparatus
I8
O8
I7
O7
I6
O6
I5
O5
C1
C2
C3
C4
C5
C6
C7
200
C8
O4
I4
O3
I3
I2
O2
30
20
8
O1
I1
R1
R2
R3
R4
R5
R6
R7
R8
Upper and lower view of the apparatus (dimensions in mm)
Flow out
Flow in
300
Flow rate
The multi-dune is filled through 4 distinct tanks each linked to
two input nozzles (I1+I2, I3+I4, I5+I6, I7+I8). The experiments
were run for five elevations of the tanks: 2.10 m (test series A),
2.30 m (test series B), 2.50 m (test series C), 3.00 m (test series
D) and 3.50 m (test series E).
TEST
CASE
HYDRAULIC HEAD
AT THE INPUT
NOZZLES
(m)
FLOW
RATE
(l/min)
A
2.1
9.61
B
2.3
9.92
C
2.5
10.56
D
3.0
11.47
E
3.5
12.54
The acquisition system
Green plastic powder
(Ø=0.25 mm)
preconditioned with a
solution of water and sodium
hydroxide to neutralize the
electrostatic charge was
used as a tracer.
A high-speed camera
allowed acquisition of 250
frames per second (spatial
resolution 480420 pixels).
A high wattage lamp
illuminated a light sheet for
image acquisition.
Negative of an acquired image.
The camera imaged chamber #3
FT – Trajectories: lower flow rate
FT - Trajectories: larger flow rate
Results – Velocity field
Case A – Lower flow rate
Case E – Larger flow rate
Horizontal velocity component
Results – Velocity field
Case A – Lower flow rate
Case E – Larger flow rate
Vertical velocity component
Results – Velocity field
Case A – Lower flow rate
Case E – Larger flow rate
Horizontal velocity variance
Results – Velocity field
Case A – Lower flow rate
Case E – Larger flow rate
Vertical velocity variance
Results - Streamlines
2:
Sector
Sector 3:
1: principal
secondary
vorticity
transport
flow. The
vorticity zone.
zone.
IfIt a fluid
thrust
is moves
proportional
takes
place
where
particle
from to
vertical
the
isflow
the velocity
principalvelocity
component
and
in
higher.
A particle
region to
this
region,
conjunction
with
entering
thethe
flow gravity
in
it will have
and
buoyancy
this
region
can back
chance
to come
determines
the destiny
move
one
to the from
previous
of
a particle.
If the thrust
camera
to assuming
another
chamber,
is
larger
thanflow
the net
the
principal
weight,
thenot
particle
will
thrust will
prevent
interact
it to fall with
out. the principal
transport flow and,
consequently, it will be
displaced in the
following chamber.
Applications
•
•
•
•
•
Porous media
Convective boundary layer
Subduction
Multi-dune channel
Fully developed turbulent channel
PTV for the characterization of turbulent
channel flow: comparison of experimental
and simulation approaches
Objectives of this study
• Image Analysis Techniques algorithms need to be tested
with experiments of well-known flow properties.
• Tests can be performed by analysing synthetically
generated images or experimental images
• At the present stage of development, are both PIV, PTV or
FT useful for the study of near-wall turbulence?
• A proper description of turbulent flows requires evaluating
the mean velocity and the root-mean square (rms) of the
fluctuating velocity together with the turbulence scales.
The experimental set-up
Turbulent channel flow (d= 2 cm, x/d = 80, z/d = 10)
Tracers (p/f = 1.06, dP = 40 mm)
FT: effects of parameter mindist
FT: effects of parameter mindist
Synthetic and real images
Composition of 10 real images subtracting 24 pixels between consecutive ones
Synthetic and real images (2)
Composition of 10 synthetic images subtracting 24 pixels between consecutive ones
DNS data
2
Mean velocity profile normalized by
the wall-shear velocity
y/h
1.5
1
0.5
u+
0
0
5
10
15
20
2
y/h
Turbulent intensities and Reynolds
shear stresses normalized by the
wall-shear velocity
1.5
u'+
v'+
1
(u'v')+
0.5
0
-1
0
1
2 u'+,v'+,(u'v')+
3
PTV results
Y/d
1
0.9
0.8
0.7
u+ - Re= 2900
v+ - Re= 2900
u+ Re= 5400
v+ Re= 5400
u - Kim
0.6
0.5
0.4
0.3
0.2
0.1
0
-5
0
5
10
15
u+
20
Plot of the mean velocity as a function of depth
PTV results
1
0.9
0.8
0.7
u'+ - Re=5400
v'+ - Re=5400
u'+ - Re=2900
v'+ - Re=2900
u'+ - Kim
v'+ - Kim
0.6
Y/d 0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
u'+, v'+
Plot of the turbulent intensities as a function of depth
6
PTV results
1
0.9
0.8
(u'v')+ - Re=2900
(u'v')+ - Re=5400
(u'v')+ - Kim
0.7
0.6
Y/d
0.5
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
(u'v')+
Plot of the Reynolds stress as a function of depth
1
FT results
2
1.8
1.6
1.4
u'+ CF
v'+ CF
(u'v')+ CF
u'+ NCF
v'+ NCF
(u'v')+ NCF
1.2
y/h
1
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
1.5
2
2.5
3
u'+,v'+,(u'v')+
3.5
4
Plot of turbulent intensities as a function of depth
PIV results
2
1.8
1.6
1.4
1.2
u'+
v'+
(u'v')+
1
y/h
0.8
0.6
0.4
0.2
0
-1
-0.5
0
0.5
1
1.5
2
2.5 u'+,v'+,(u'v')+
3
Plot of turbulent intensities as a function of depth
Results: Lagrangian framework
Translient matrix at time t= 0.001 s
Results: Lagrangian framework
Translient matrix at time t= 0.017 s
Results: Lagrangian framework
Translient matrix at time t= 0.047 s
Results: Lagrangian framework
Translient matrix at time t= 0.058 s
Conclusions
• PTV is useful for studying near-wall turbulence
both in the Eulerian and in the Lagrangian
frameworks;
• low sensitivity of the results on the image
analysis “parameters”;
• a larger acquisition field is required.
Applications
•
•
•
•
•
Porous media
Convective boundary layer
Subduction
Multi-dune channel
Fully developed turbulent channel
Application of 3D-PTV to Track Particles
Moving Inside Heterogeneous Porous
Media
Motivations and outlines

Flow within porous media is three-dimensional

We are interested in describing the dispersion
process in a Lagrangian framework

3-dimensional experimental techniques (3-D PTV)
suitable for studying conservative tracer movement in
porous formations homogeneous or heterogeneous
at the bench scale have to be employed
Simplification
The simplification inherent in the laboratory model
• Use of Pyrex particles of various dimensions and sizes to
form a porous matrix
• Use of glycerol for the fluid so that the matched-index
technique is applicable
• Use of air bubbles for the tracer that move passively with
the flowing glycerol
• Create the flow field with a hydraulic pump of constant
mean, but variable velocity
• Image only interior of the matrix so that boundary effects
are minimized
Simplification (2)
Why choose 3-DPTV
• Tracer dispersion phenomena are naturally described in a
Lagrangian framework as the tracer acts as a tag of the fluid
particles
• 3-dimensional techniques (3-DPTV) suitable for studying
conservative tracer movement in porous formations
homogeneous or heterogeneous at the bench scale have to be
employed
Why choose air bubbles as tracer
• They are passive
• They have a substantially different refractive index that glycerol
• They are easy to use and to re-use
Scanning 3D-PTV – Experimental apparatus
“Scanning” 3D-PTV
Experimental
Set-up for porous media
Carefully calibrating we obtained
the following results:
75% of trajectories where
matched
with tolerance=10 pixels
Photogrammetric 3D-PTV- experimental
apparatus
Photogrammetric 3D-PTV
Experimental set-up
for porous media
Creating heterogeneous media
Het1
Sector2: =1 cm
Sector3: =0.7 cm
Sector1: =1 cm
Het2
Sector2: =0.4 cm
Sector3: =1 cm
Sector1: =0.7 cm
(a)
(b)
Positioning of the heterogeneous medium inside the column through a PVC
shaper with three sectors
Trajectories reconstructed
XZ Plane
YZ Plane
How can we describe particle dispersion?
The effects of heterogeneity on the dispersive process are studied
by examining the evolution of several scattering functions.
For a time stationary velocity field, the self-part of the intermediate
scattering function Gs is defined as
1
Gs ( x, t ) 
N
N
  [x  ( X (t )  X (0))]
i 1
i
i
The relative scattering function, Gr, compares the growth of the
separations of particles from an initial distribution.
1
Gr (x, t ) 
 (x  ( X i (t  t0i )  X j (t  t0 j )

N t ( N t  1) i  j
Results:
(b)
(a)
(c)
Gs (self part of the intermediate scattering
function) for Hom2 at 3, 13, and 26 seconds,
(a), (b), and (c) respectively
Initially the spread about the
mean for the short time scale is
greater in the transverse
direction, but subsequently the
spread of particle displacement
is greater in the longitudinal
direction. The center of mass in
the longitudinal direction at early
times remains near wt, but by 13
s it shows a significant
divergence from wt (w is the
longitudinal mean velocity)
Results:
The relative scattering function is
an indication of the internal
mixing between particles. If the
growth of Gr for one system is
less than another with similar
velocity variances, then there
must be more internal mixing in
the system with little growth in
Gr. Het1 and Het2 are an
example of this, indicating more
internal mixing for particles in
Het1.
(a)
(b)
Gr (relative scattering function) for
Het1 (a) and Het2 (b) at times 3, 6
and 9 seconds
Conclusions:
•It is possible to reconstruct 3D trajectories through matched
refractive index 3D-PTV.
•From these trajectories we can compute: mean square
displacements, velocity covariances, classical dispersion tensor,
intermediate scattering function and generalized dispersion tensor.
•The mean displacement square root clearly highlights the different
behavior of homogeneous and heterogeneous media, even if it can
not be used to quantitatively measure the dispersion coefficient.
•The classical dispersion coefficient formulation is valid only when
the hypothesis of long-time and large-distance is satisfied.
•The Lyapunov exponent theory allows the dispersion process to be
described through the "doubling time" at a certain scale of couples
of particles belonging to the cloud.
Applications
•
•
•
•
•
Porous media
Convective boundary layer
Subduction
Multi-dune channel
Fully developed turbulent channel