Sergei Lukaschuk, Petr Denissenko Grisha Falkovich The University of Hull, UK The Weizmann Institute...
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Transcript of Sergei Lukaschuk, Petr Denissenko Grisha Falkovich The University of Hull, UK The Weizmann Institute...
Sergei Lukaschuk, Petr Denissenko
Grisha Falkovich
The University of Hull, UK
The Weizmann Institute of Science, Israel
Clustering and Mixing of Floaters by Waves
Warwick Turbulent Symposium. December 8, 2005.
Effect of surface tensionCapillarity breaks Archimedes’ law
Two bodies of the same weight displace different amount of water depending on their material (wetting conditions)
• Hydrophilic particles are lighter
• Hydrophobic particles are heavier than displaced fluid
Small hydrophilic particles climb up,and hydrophobic particles slide down along inclined surface.
Similar particles attract each other and form clusters.
A repulsion may exist in the case of non-identical particles
Cheerious effect
Equation for the depth of the submerged part, :
M – p. mass, md – mass of displaced fluid, Fc – capillary force, v - friction coefficient( )
Equation of motion for horizontal projection:
For the long gravity waves when
Working liquid: water surface tension: 71.6 mN/m refraction index: 1.33
Particles: glass hollow spheres average size 60 m density 0.6 g/cm3
Measurement System Cell geometry: 9.6 x 58.3 x 10 mm, 50 x 50 x 10 mm Boundary conditions: pinned meniscus = flat surface Acceleration measurements: Single Axis Accelerometer,
ADXL150 (Resolution 1 mg / Hz1/2 , Range 25 g, 16-bit A-to-D, averaging ~ 10 s, Relative error ~ 0.1%)
Temperature control: 0.2ºC Vibrations: Electromagnetic shaker controlled by digital
waveform generator. Resonant frequency > 1 kHz Illumination: expanded beam
CW Laser to characterise particles concentration, wave configuration and the amplitude
PIV pulsed (10 nsec) Yag laser for the particle motion Imaging
3 PIV cameras synchronized with shaker oscillation
Measurement methods
• Particle Concentration off-axis imaging synchronized with zero-phase of the
surface wave measuring characteristic – light intensity, its dispersion and
moments averaged over area of different size• Wave configuration:
shadowgraph technique 2D Fourier transform in space to measure averaged k-vector
• Wave amplitude measurement refraction angle of the light beam of 0.2 mm
diam. dispersion of the light intensity
Standing wave : Particle concentration and Wave amplitude are characterized by the dispersion of the light intensity
F=100.9 Hz, l=8 mm, s=5 mm, A=0.983 g
T1
5000
1 104
1.5 104
2 104
2.5 104
3 104
3.5 104
0.96 0.98 1 1.02 1.04 1.06 1.08
y = -2.7482e+05 + 2.8997e+05x R= 0.99641
Av
^2
, [a
.u.]
A , [g]
Wave Amplitude vs Acceleration F= 100.9 Hz Cell: 58.3 x 9.6 mm
Ac=0.965 0.01
Balkovsky, Fouxon, Falkovich, Gawedzki, Bec, Horvai
∑λ<0 → singular (fractal) distribution – Sinai-Ruelle-Bowen measure
multi-fractal measure
Moments of concentrations 2,3,4,5 and 6th versus the scale of coarse graining. Inset: scaling exponent of the moments of particle number versus moment number.
1 1.5 2 2.5 30
1
2
3
4
5
6
7
8
<lo
g 10 M
i>
log10
(BinSize in pixels)
2 4 6
4
6
Moment No.1
Mo
me
nt
Ex
po
ne
nt
0 200 400 600 800 1000 12001300
1400
1500
1600
1700
1800
1900
2000The number of paricles in frames (1:1050)
Frame number
Me
an
(n)=
16
32
.86
1
Std
(n)=
11
0.1
25
3
Random particle distribution
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-28
-26
-24
-22
-20
-18
-16
-14
-12
-10
-8
Log
of m
omen
ts a
vera
ged
over
100
imag
es
Moments of particle number as a function of bin size for simulated random particle distribution.
log10(boxsize)
The number of particles1979
M8M6M4M2
n=2000 in the AOI, std(n)=39
PDF of the number of particles in a bin 128x128
0 20 40 60 80 1000
20
40
60
80
100
120PDF of the Number of Particles in the area 128x128 bins
Number of particles0 20 40 60 80
10-4
10-3
10-2
10-1
100
PDF of the number of particles for the bins 128x128
Number of paricles
0 50 100 150 2000
1
2
3
4
5
6
7
8
9PDF of the Number of Particles in the bins 256x256
Particle number
PDF of the number of particles in a bin 256 x 256
0 50 100 150
10-4
10-3
10-2
10-1
100
PDF of the number of particles for the bins 256x256
Number of paricles
0 10 20 30 40 50 60
10-4
10-3
10-2
10-1
100
PDF of the number of particles for the bins 64x64
Number of paricles0 100 200 300 400
10-4
10-3
10-2
10-1
100
PDF of the number of particles for the bins 512x512
Number of paricles
Conclusion
Small floaters are inertial →
they drift and form clusters in a standing wave wetted particles form clusters in the nodesunwetted - in the antinodesclustering time is proportional to A2
they create multi-fractal distribution in random waves.