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

Standing wave

Small amplitude wave:

Van Dyke, “An Album of Fluid Motion”

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

Experimental setup

CW Laser

PW Laser

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

2D k-spectrum of the parametric waves in a turbulent mode averaged

over 100 measurements

Distribution in random flow (wave turbulence)

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.

How waves move small particles?

• Stokes drift (1847):

• Kundt’s tube stiration in a sound waves (King, 1935):

022~ kzekav

3

1

21

3/)1(2

)2sin(4 3

F

kxFEkaP

E – the mean energy density, 0

1

0

1 ,c

c