Abstract

In a large number of physical systems formed of discrete particles, a key parameter is the relative distance between the objects, as for example in studies of spray evaporation or droplets micro-explosion. This paper is devoted to the presentation of an approach where the relative 3D location of particles in the control volume is accurately extracted from the interference patterns recorded at two different angles. No reference beam is used and only ten (2 + 8) 2D-FFT have to be computed.

© 2011 OSA

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References

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  1. V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34(7), 837–857 (2003).
    [CrossRef]
  2. G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
    [CrossRef]
  3. G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
    [CrossRef]
  4. C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
    [CrossRef]
  5. S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
    [CrossRef]
  6. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42(5), 827–833 (2003).
    [CrossRef] [PubMed]

2009 (1)

C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
[CrossRef]

2006 (1)

G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
[CrossRef]

2005 (1)

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

2003 (2)

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42(5), 827–833 (2003).
[CrossRef] [PubMed]

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34(7), 837–857 (2003).
[CrossRef]

2002 (1)

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Allano, D.

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Atthasit, A.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Biscos, Y.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Castanet, G.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Cen, K. F.

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Devarakonda, V.

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34(7), 837–857 (2003).
[CrossRef]

Elsinga, G. E.

G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
[CrossRef]

Haigermoser, C.

C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
[CrossRef]

Lavergne, G.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Lavieille, P.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Lebouché, M.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Lebrun, D.

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Lemoine, F.

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

Malek, M.

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Meng, H.

Onorato, M.

C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
[CrossRef]

Oudheusden, B. W.

G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
[CrossRef]

Pan, G.

Patte-Rouland, B.

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Pu, S. L.

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Ray, A. K.

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34(7), 837–857 (2003).
[CrossRef]

Scarano, F.

C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
[CrossRef]

G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
[CrossRef]

Wieneke, B.

G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
[CrossRef]

Appl. Opt. (1)

Exp. Fluids (3)

G. E. Elsinga, F. Scarano, B. Wieneke, and B. W. Oudheusden, “Tomographic particle image velocimetry,” Exp. Fluids 41(6), 933–947 (2006).
[CrossRef]

C. Haigermoser, F. Scarano, and M. Onorato, “Investigation of the flow in a circular cavity using stereo and tomographic particle image velocimetry,” Exp. Fluids 46(3), 517–526 (2009).
[CrossRef]

S. L. Pu, D. Allano, B. Patte-Rouland, M. Malek, D. Lebrun, and K. F. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids 39(1), 1–9 (2005).
[CrossRef]

Int. J. Heat Mass Transfer (1)

G. Castanet, P. Lavieille, F. Lemoine, M. Lebouché, A. Atthasit, Y. Biscos, and G. Lavergne, “Energetic budget on an evaporating monodisperse droplet stream using combined optic methods evaluation of the convective heat transfer,” Int. J. Heat Mass Transfer 45(25), 5053–5067 (2002).
[CrossRef]

J. Aerosol Sci. (1)

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34(7), 837–857 (2003).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Configuration under study.

Fig. 2
Fig. 2

Scattering patterns from one and two particles and associated 2D-FFT maps, the detector is a square of 512 x 512 pixels, located at 1 meter form the coordinate system center with a collecting angle θ 0 equal to 35° ± 5°. The incident wavelength is equal to 0.532 µm. a: the scattering from one particle (d = 30 µm, m = 1.5-0.0i, (x = 0, y = 0, z = 0)); b: the scattering from two particles (the first one and a second identical to the first but located at (x = 150, y = 150, z = 150); c: the 2D-FFT with a Gaussian windowing associated to Fig. 2(a); d: The 2D-FFT with Gaussian windowing associated to Fig. 2(b).

Fig. 3
Fig. 3

Detail of a complex spot in the 2D-FFT space. The value of coordinate α used is the mathematical average of the coordinates α of the elementary spots.

Fig. 4
Fig. 4

Comparison between numerical and analytical values of Eq. (28) coefficients.

Fig. 5
Fig. 5

Images simulated for six particles located in the control volume. The left image corresponds to a recording at θ 0 = 40° while the right image corresponds to a recording at θ 0 = 90°.

Fig. 6
Fig. 6

The 2D-FFT associated to images of Fig. 5. First line displays a 2D-FFT with Harris windowing while the second line displays the 2D-FFT with Gaussian windowing.

Fig. 7
Fig. 7

Identification and spot numbering.

Fig. 8
Fig. 8

A group of three particles interacting two by two. Left at θ 0 = 40°, right at θ 0 = 90°.

Fig. 9
Fig. 9

Three groups of three particles view at θ 0 = 40°. Group B has a common interaction with group A. Group C has a common interaction with group B and one with group A.

Fig. 10
Fig. 10

The group of four particles interacting two by two determined from three groups of three particles interacting two by two.

Fig. 11
Fig. 11

a) Interaction for a group of five particles interacting two by two. b) Interaction for one particle interacting with all the others five particles in the control volume.

Fig. 12
Fig. 12

The 40° 2D-FFT computed from the extracted 3D Location. These 2D-FFT are compared to the one displayed in Fig. 6.

Fig. 13
Fig. 13

The fringes field reconstructed from the selected extracted particle fields.

Fig. 14
Fig. 14

The original locations (in red) and the extracted locations (in yellow and blue).

Tables (3)

Tables Icon

Table 1 The Particle Computation Parameters

Tables Icon

Table 2 Original and Extracted 3D Particle Locations

Tables Icon

Table 3 Statistical Errors on the Relative Distance Between Particles

Equations (32)

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E r s , j = E 0 cos φ j n = 1 i n + 1 ( 1 ) n 2 n + 1 n ( n + 1 ) a n [ ξ n ' ' ( k r j ) + ξ n ( k r j ) ] P n 1 ( cos θ j )
E θ s , j = E 0 k r j cos φ j n = 1 i n + 1 ( 1 ) n 2 n + 1 n ( n + 1 ) [ a n ξ n ' ( k r j ) τ n ( cos θ j ) i b n ξ n ( k r j ) π n ( cos θ j ) ]
E φ s , j = E 0 k r j sin φ j n = 1 i n + 1 ( 1 ) n 2 n + 1 n ( n + 1 ) [ a n ξ n ' ( k r j ) π n ( cos θ j ) i b n ξ n ( k r j ) τ n ( cos θ j ) ]
H r s , j = E 0 ( ε μ ) 1 / 2 sin φ j n = 1 i n + 1 ( 1 ) n 2 n + 1 n ( n + 1 ) b n [ ξ n ' ' ( k r j ) + ξ n ( k r j ) ] P n 1 ( cos θ j )
H θ s , j = E 0 k r j ( ε μ ) 1 / 2 sin φ j n = 1 i n + 1 ( 1 ) n 2 n + 1 n ( n + 1 ) [ a n ξ n ' ( k r j ) π n ( cos θ j ) + i b n ξ n ( k r j ) τ n ( cos θ j ) ]
H φ s , j = E 0 k r j ( ε μ ) 1 / 2 cos φ j n = 1 i n + 1 ( 1 ) n 2 n + 1 n ( n + 1 ) [ i a n ξ n ' ( k r j ) τ n ( cos θ j ) + b n ξ n ( k r j ) π n ( cos θ j ) ]
π n ( cos θ ) = P n 1 ( cos θ ) / sin θ
τ n ( cos θ ) = d P n 1 ( cos θ ) / d θ
P n 1 ( cos θ ) = sin θ d P n ( cos θ ) d cos θ
ξ n ( k r ) = ψ n ( k r ) + i χ n ( k r )
ψ n ( k r ) = ( π k r 2 ) 1 / 2 J n + 1 / 2 ( k r )
χ n ( k r ) = ( 1 ) n ( π k r 2 ) 1 / 2 J n 1 / 2 ( k r )
a n = ψ n ( α ) ψ ' n ( β ) m ψ ' n ( α ) ψ n ( β ) ξ n ( α ) ψ ' n ( β ) m ξ ' n ( α ) ψ n ( β )
b n = m ψ n ( α ) ψ ' n ( β ) ψ ' n ( α ) ψ n ( β ) m ξ n ( α ) ψ ' n ( β ) ξ ' n ( α ) ψ n ( β )
V w t = j = 1 N Y w s , j
S = 1 2 R e [ E t   .   H t * ]
E 1 = E 0 r 1 e x p j ( k r 1 ω t + φ 1 )
E 2 = E 0 r 2 e x p j ( k r 2 ω t + φ 2 )
r 1 = ( x M x 1 ) 2 + ( y M y 1 ) 2 + ( z M z 1 ) 2
r 2 = ( x M x 2 ) 2 + ( y M y 2 ) 2 + ( z M z 2 ) 2
I M = | E 0 2 r 1 r 2 ( 1 + 2   c o s { k [ r 2 r 1 ] + φ 2 φ 1 } ) |
r 1 = [ ρ M G 1 2 [ 1 + ( y M y 1 ) 2 ρ M G 1 2 ] ] 1 / 2
r 2 = [ ρ M G 2 2 [ 1 + ( y M y 2 ) 2 ρ M G 2 2 ] ] 1 / 2
r 1 ρ M G 1 + ( y M y 1 ) 2 2 ρ M G 1
r 2 ρ M G 2 + ( y M y 2 ) 2 2 ρ M G 2
A =   ρ M G 2 ρ M G 1
B =   ρ M G 1 ( y M y 2 ) 2 ρ M G 2 ( y M y 1 ) 2 2   ρ M G 1   ρ M G 2
I η , ξ = | E 0 2 r 1 r 2 ( 1 + 2 c o s ( k ( η M R M ) [ ( x 2 x 1 ) cos θ 0 + ( z 2 z 1 ) sin θ 0 ] + k ξ M R M ( y 2 y 1 ) ) ) |
( X 1 X 2 ) =   ( X 1 X 3 ) + ( X 3 X 2 )
( Y 1 Y 2 ) =   ( Y 1 Y 3 ) + ( Y 3 Y 2 )
( Z 1 Z 2 ) =   ( Z 1 Z 3 ) + ( Z 3 Z 2 )
[ η 1 , 2 ξ 1 , 2 ] = ± [ η 1 , 3 η 2 , 3 ξ 1 , 3 ξ 2 , 3 ]

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