Abstract

An incoherent dynamic light-scattering technique is developed to measure the local velocity and its statistics. By employing two parallel laser beams of different colors, the technique measures the cross-correlation function of the scattered intensities from two separate illuminating volumes. Because there is no phase coherence between the two laser beams, the measured cross-correlation function is sensitive only to the intensity fluctuations caused by a seed particle that crosses the two beams in succession. The flow velocity is obtained from the measured particle transit time. We frame the scattering theory so as to account for the two-beam scattering geometry. Our experiment verifies the calculation and demonstrates applications of the technique. The method has the unique feature of being able to measure simultaneously the local velocity in two opposite directions perpendicular to the incident laser beams. Its advantages are high spatial resolution and accuracy, fast temporal response, and ease of use. The technique is useful in studies of turbulent flows, sedimentation of heavy particles, and flow phenomena in complex fluids.

© 1995 Optical Society of America

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References

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  1. E. D. Siggia, “High Rayleigh number convection,” Ann. Rev. Fluid Mech. 26, 137–168 (1994).
    [CrossRef]
  2. J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
    [CrossRef] [PubMed]
  3. T. C. Van Vechten, C. Franck, “Relative importance of convection and diffusion in binary liquid systems subject to small horizontal temperature gradients,” Phys. Rev. E 48, 3635–3642 (1993).
    [CrossRef]
  4. F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry, 2nd ed. (Academic, New York, 1981).
  5. P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9265 (1993).
    [CrossRef]
  6. R. Schodl, “Laser-two-focus (L2F) for use in aero engines,” AGARD Lect. Ser. 90, 4.1–4.34 (1977).
  7. L. Lading, A. Skov Jensen, C. Fog, H. Andersen, “Time-of-flight laser anemometer for velocity measurements in the atmosphere,” Appl. Opt. 17, 1486–1488 (1978).
    [CrossRef]
  8. R. Schodl, “Laser-two-focus velocimetry,” AGARD Conf. Proc. 399, 7.1–7.31 (1986).
  9. L. Lading, “The time-of-flight laser anemometer,” AGARD Conf. Proc. 193, 23.1–23.20 (1976).
  10. K. G. Bartlett, C. Y. She, “Single-particle correlated time-of-flight velocimeter for remote wind-speed measurement,” Opt. Lett. 1, 175–177 (1977).
    [CrossRef] [PubMed]
  11. C. Y. She, R. F. Kelley, “Photon-burst correlation techniques for atmosphere crosswind measurements,” Appl. Phys. B 33, 195–204 (1984).
    [CrossRef]
  12. B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
  13. P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977).
  14. P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
    [CrossRef]
  15. See, e.g., Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977).
  16. G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
    [CrossRef]
  17. P. Tong, W. I. Goldburg, “Experimental study of relative velocity fluctuations in turbulence,” Phys. Lett. A 127, 147–150 (1988); W. I. Goldburg, P. Tong, H. K. Pak, “A light scattering study of turbulence,” Physica D 38, 134–140 (1989).
    [CrossRef]
  18. N. A. Clark, B. J. Ackerson, A. J. Hurd, “Multidetector scattering as a probe of local structure in disordered phases,” Phys. Rev. Lett. 50, 1459–1462 (1983).
    [CrossRef]
  19. B. J. Ackerson, T. W. Taylor, N. A. Clark, “Characterization of the local structure of fluids by apertured cross-correlation functions,” Phys. Rev. A 31, 3183–3193 (1985).
    [CrossRef] [PubMed]
  20. D. W. Schaefer, B. J. Berne, “Number fluctuation spectroscopy of motile microorganisms,” Biophys. J. 15, 785–794 (1975).
    [CrossRef] [PubMed]
  21. D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Oxford, London, 1988).

1994 (1)

E. D. Siggia, “High Rayleigh number convection,” Ann. Rev. Fluid Mech. 26, 137–168 (1994).
[CrossRef]

1993 (2)

T. C. Van Vechten, C. Franck, “Relative importance of convection and diffusion in binary liquid systems subject to small horizontal temperature gradients,” Phys. Rev. E 48, 3635–3642 (1993).
[CrossRef]

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9265 (1993).
[CrossRef]

1992 (1)

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

1988 (1)

P. Tong, W. I. Goldburg, “Experimental study of relative velocity fluctuations in turbulence,” Phys. Lett. A 127, 147–150 (1988); W. I. Goldburg, P. Tong, H. K. Pak, “A light scattering study of turbulence,” Physica D 38, 134–140 (1989).
[CrossRef]

1986 (1)

R. Schodl, “Laser-two-focus velocimetry,” AGARD Conf. Proc. 399, 7.1–7.31 (1986).

1985 (1)

B. J. Ackerson, T. W. Taylor, N. A. Clark, “Characterization of the local structure of fluids by apertured cross-correlation functions,” Phys. Rev. A 31, 3183–3193 (1985).
[CrossRef] [PubMed]

1984 (1)

C. Y. She, R. F. Kelley, “Photon-burst correlation techniques for atmosphere crosswind measurements,” Appl. Phys. B 33, 195–204 (1984).
[CrossRef]

1983 (1)

N. A. Clark, B. J. Ackerson, A. J. Hurd, “Multidetector scattering as a probe of local structure in disordered phases,” Phys. Rev. Lett. 50, 1459–1462 (1983).
[CrossRef]

1980 (1)

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
[CrossRef]

1979 (1)

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

1978 (1)

1977 (2)

1976 (1)

L. Lading, “The time-of-flight laser anemometer,” AGARD Conf. Proc. 193, 23.1–23.20 (1976).

1975 (1)

D. W. Schaefer, B. J. Berne, “Number fluctuation spectroscopy of motile microorganisms,” Biophys. J. 15, 785–794 (1975).
[CrossRef] [PubMed]

Ackerson, B. J.

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9265 (1993).
[CrossRef]

B. J. Ackerson, T. W. Taylor, N. A. Clark, “Characterization of the local structure of fluids by apertured cross-correlation functions,” Phys. Rev. A 31, 3183–3193 (1985).
[CrossRef] [PubMed]

N. A. Clark, B. J. Ackerson, A. J. Hurd, “Multidetector scattering as a probe of local structure in disordered phases,” Phys. Rev. Lett. 50, 1459–1462 (1983).
[CrossRef]

Andersen, H.

Bartlett, K. G.

Berne, B. J.

D. W. Schaefer, B. J. Berne, “Number fluctuation spectroscopy of motile microorganisms,” Biophys. J. 15, 785–794 (1975).
[CrossRef] [PubMed]

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Chaikin, P. M.

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

Clark, N. A.

B. J. Ackerson, T. W. Taylor, N. A. Clark, “Characterization of the local structure of fluids by apertured cross-correlation functions,” Phys. Rev. A 31, 3183–3193 (1985).
[CrossRef] [PubMed]

N. A. Clark, B. J. Ackerson, A. J. Hurd, “Multidetector scattering as a probe of local structure in disordered phases,” Phys. Rev. Lett. 50, 1459–1462 (1983).
[CrossRef]

Durst, F.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry, 2nd ed. (Academic, New York, 1981).

Fog, C.

Franck, C.

T. C. Van Vechten, C. Franck, “Relative importance of convection and diffusion in binary liquid systems subject to small horizontal temperature gradients,” Phys. Rev. E 48, 3635–3642 (1993).
[CrossRef]

Fuller, G. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
[CrossRef]

Goldburg, W. I.

P. Tong, W. I. Goldburg, “Experimental study of relative velocity fluctuations in turbulence,” Phys. Lett. A 127, 147–150 (1988); W. I. Goldburg, P. Tong, H. K. Pak, “A light scattering study of turbulence,” Physica D 38, 134–140 (1989).
[CrossRef]

Herbolzheimer, E.

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

Hurd, A. J.

N. A. Clark, B. J. Ackerson, A. J. Hurd, “Multidetector scattering as a probe of local structure in disordered phases,” Phys. Rev. Lett. 50, 1459–1462 (1983).
[CrossRef]

Kelley, R. F.

C. Y. She, R. F. Kelley, “Photon-burst correlation techniques for atmosphere crosswind measurements,” Appl. Phys. B 33, 195–204 (1984).
[CrossRef]

Lading, L.

Leal, L. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
[CrossRef]

Melling, A.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry, 2nd ed. (Academic, New York, 1981).

Pecora, R.

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

Pusey, P. N.

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977).

Rallison, J. M.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
[CrossRef]

Russel, W. B.

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

Rutgers, M. A.

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

Schaefer, D. W.

D. W. Schaefer, B. J. Berne, “Number fluctuation spectroscopy of motile microorganisms,” Biophys. J. 15, 785–794 (1975).
[CrossRef] [PubMed]

Schmidt, R. L.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
[CrossRef]

Schodl, R.

R. Schodl, “Laser-two-focus velocimetry,” AGARD Conf. Proc. 399, 7.1–7.31 (1986).

R. Schodl, “Laser-two-focus (L2F) for use in aero engines,” AGARD Lect. Ser. 90, 4.1–4.34 (1977).

She, C. Y.

C. Y. She, R. F. Kelley, “Photon-burst correlation techniques for atmosphere crosswind measurements,” Appl. Phys. B 33, 195–204 (1984).
[CrossRef]

K. G. Bartlett, C. Y. She, “Single-particle correlated time-of-flight velocimeter for remote wind-speed measurement,” Opt. Lett. 1, 175–177 (1977).
[CrossRef] [PubMed]

Siggia, E. D.

E. D. Siggia, “High Rayleigh number convection,” Ann. Rev. Fluid Mech. 26, 137–168 (1994).
[CrossRef]

Skov Jensen, A.

Taylor, T. W.

B. J. Ackerson, T. W. Taylor, N. A. Clark, “Characterization of the local structure of fluids by apertured cross-correlation functions,” Phys. Rev. A 31, 3183–3193 (1985).
[CrossRef] [PubMed]

Tong, P.

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9265 (1993).
[CrossRef]

P. Tong, W. I. Goldburg, “Experimental study of relative velocity fluctuations in turbulence,” Phys. Lett. A 127, 147–150 (1988); W. I. Goldburg, P. Tong, H. K. Pak, “A light scattering study of turbulence,” Physica D 38, 134–140 (1989).
[CrossRef]

Tritton, D. J.

D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Oxford, London, 1988).

Van Vechten, T. C.

T. C. Van Vechten, C. Franck, “Relative importance of convection and diffusion in binary liquid systems subject to small horizontal temperature gradients,” Phys. Rev. E 48, 3635–3642 (1993).
[CrossRef]

Whitelaw, J. H.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry, 2nd ed. (Academic, New York, 1981).

Xia, K.-Q.

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9265 (1993).
[CrossRef]

Xue, J.-Z.

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

AGARD Conf. Proc. (2)

R. Schodl, “Laser-two-focus velocimetry,” AGARD Conf. Proc. 399, 7.1–7.31 (1986).

L. Lading, “The time-of-flight laser anemometer,” AGARD Conf. Proc. 193, 23.1–23.20 (1976).

AGARD Lect. Ser. (1)

R. Schodl, “Laser-two-focus (L2F) for use in aero engines,” AGARD Lect. Ser. 90, 4.1–4.34 (1977).

Ann. Rev. Fluid Mech. (1)

E. D. Siggia, “High Rayleigh number convection,” Ann. Rev. Fluid Mech. 26, 137–168 (1994).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

C. Y. She, R. F. Kelley, “Photon-burst correlation techniques for atmosphere crosswind measurements,” Appl. Phys. B 33, 195–204 (1984).
[CrossRef]

Biophys. J. (1)

D. W. Schaefer, B. J. Berne, “Number fluctuation spectroscopy of motile microorganisms,” Biophys. J. 15, 785–794 (1975).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

P. Tong, K.-Q. Xia, B. J. Ackerson, “Incoherent cross-correlation spectroscopy,” J. Chem. Phys. 98, 9256–9265 (1993).
[CrossRef]

J. Fluid. Mech. (1)

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, “The measurement of velocity gradients in laminar flow by homodyne light-scattering spectroscopy,” J. Fluid. Mech. 100, 555–575 (1980).
[CrossRef]

J. Phys. A (1)

P. N. Pusey, “Number fluctuations of interacting particles,” J. Phys. A 12, 1805–1818 (1979).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

P. Tong, W. I. Goldburg, “Experimental study of relative velocity fluctuations in turbulence,” Phys. Lett. A 127, 147–150 (1988); W. I. Goldburg, P. Tong, H. K. Pak, “A light scattering study of turbulence,” Physica D 38, 134–140 (1989).
[CrossRef]

Phys. Rev. A (1)

B. J. Ackerson, T. W. Taylor, N. A. Clark, “Characterization of the local structure of fluids by apertured cross-correlation functions,” Phys. Rev. A 31, 3183–3193 (1985).
[CrossRef] [PubMed]

Phys. Rev. E (1)

T. C. Van Vechten, C. Franck, “Relative importance of convection and diffusion in binary liquid systems subject to small horizontal temperature gradients,” Phys. Rev. E 48, 3635–3642 (1993).
[CrossRef]

Phys. Rev. Lett. (2)

J.-Z. Xue, E. Herbolzheimer, M. A. Rutgers, W. B. Russel, P. M. Chaikin, “Diffusion, dispersion, and settling of hard spheres,” Phys. Rev. Lett. 69, 1715–1718 (1992).
[CrossRef] [PubMed]

N. A. Clark, B. J. Ackerson, A. J. Hurd, “Multidetector scattering as a probe of local structure in disordered phases,” Phys. Rev. Lett. 50, 1459–1462 (1983).
[CrossRef]

Other (5)

D. J. Tritton, Physical Fluid Dynamics, 2nd ed. (Oxford, London, 1988).

See, e.g., Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977).

B. J. Berne, R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

P. N. Pusey, “Statistical properties of scattered radiation,” in Photon Correlation Spectroscopy and Velocimetry, H. Z. Cummins, E. R. Pike, eds. (Plenum, New York, 1977).

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser-Doppler Anemometry, 2nd ed. (Academic, New York, 1981).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup: M1, M2, and M3, mirrors; L1, L2, and L3, lenses; S, adjustable slit; BS, beam splitter; PM1 and PM2, photomultipliers; and C, scattering cell (drawing not to scale).

Fig. 2
Fig. 2

Intensity cross-correlation functions gc(t) measured in a laminar flow, with V0 = 29.74 cm/s: circles, green cross blue correlation function; squares, blue cross green correlation function. The solid curve is a fit to the green cross blue data. The fitted function is gc(t) = 1 + 0.633 exp{−2[(t − 0.538 ms)/0.253 ms]2} (see text).

Fig. 3
Fig. 3

Measurements of the velocity V0 at different pump settings: circles, from the measured cross-correlation function gc(t); triangles, from the flow rate measurement. The solid line is a linear fit to the triangles.

Fig. 4
Fig. 4

Measured 1/e2 half-width τ of the Gaussian peak as a function of the velocity V0. The solid curve is the fitted function τ = 7.64/V0 (ms). The inset shows the laser beam radius ( obtained from the measured τ at different V0.

Fig. 5
Fig. 5

Measured Gaussian peak height, gc(t0) − 1, as a function of the average number N ¯ of particles in the scattering volume. Solid line, fitted function gc(t0) − 1 = 0.079/ N ¯.

Fig. 6
Fig. 6

Measured velocity profiles across the pipe diameter in a laminar pipe flow (circles, Re = 617) and in a turbulent pipe flow (squares, Re = 3014). The solid curve is a fit to the Poiseuille formula V(r) = V0[1 − (r/R)2] for laminar pipe flows. Here r is the radial distance from the center of the pipe, V0 = 34.3 cm/s is the velocity at r = 0, and R = 1.8 mm is the inner radius of the pipe.

Equations (13)

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g c ( t ) = I b ( t ) I g ( t + t ) I b I g ,
g c ( t ) = E b * ( t ) E b ( t ) E g * ( t + t ) E g ( t + t ) E b * ( t ) E b ( t ) E g * ( t ) E g ( t ) = K I b ( t ) I g ( t ) ,
E ( t ) = j = 1 N 0 a j ( t ) exp [ - i q · r j ( t ) ] ,
K = i , j , k , l N 0 a i b ( t ) a j b ( t ) a k g ( t + t ) a l g ( t + t ) × exp { - i q b · [ r i ( t ) - r j ( t ) ] - i q g · [ r k ( t + t ) - r l ( t + t ) ] } ,
K = i = 1 N 0 A i b ( t ) A i g ( t + t ) + i j N 0 A i b ( t ) A j g ( t + t ) + i j N 0 a i b ( t ) a i g ( t + t ) a j b ( t ) a j g ( t + t ) × F s ( q , t ) 2 ,
K = ( N 0 ) 2 A ¯ b A ¯ g + N 0 δ A 1 b ( t ) A 1 g ( t + t ) + N 0 ( N 0 - 1 ) δ A 1 b ( t ) δ A 2 g ( t + t ) ,
g c ( t ) = 1 + δ A 1 b ( t ) δ A 1 g ( t + t ) N 0 A ¯ b A ¯ g + ( N 0 - 1 ) δ A 1 b ( t ) δ A 2 g ( t + t ) N 0 A ¯ b A ¯ g = 1 + g N S ( t ) + g N T ( t ) .
g N S ( t ) = 1 N 0 1 / V d 3 r d 3 r 0 I b ( r ) I g ( r 0 ) P ( r - r 0 ; t ) ( 1 / V ) 2 I b ( r ) d 3 r I g ( r ) d 3 r ,
I ( r ) = I 0 exp [ - 2 ( r / σ ) 2 ] ,
P ( r - r 0 ; t ) = δ ( x - x 0 - v t ) δ ( y - y 0 ) δ ( z - z 0 ) .
g N S ( t ) = 1 N ¯ exp [ - ( v t - l ) 2 σ 2 ] = 1 N ¯ exp [ - 2 ( t - t 0 2 σ / v ) 2 ] ,
g N S ( t ) = 1 N ¯ d v P ( v ) exp [ - ( v t - l ) 2 σ 2 ] ,
g c ( t ) = I b ( t ) I g ( t + t ) I b ( t ) I g ( t ) = 1 + β G c ( t ) ,

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