Abstract

Turbulent inhomogeneities of fluid flow have the effect of scattering light in near-forward angles, thus providing an opportunity to use optics to quantify turbulence. Here we report measurements of the volume-scattering function in the range of 10-7 to 10-3 rad using a wave-front sensing technique. The total scattering coefficient b, due to scattering on turbulent inhomogeneities, is between 1 and 10 m-1 under typical oceanographic conditions. The numerical calculations of turbulent volume-scattering functions compare well with the laboratory measurement. These results suggest that optical measurements at small angles are affected by turbulence-related scattering, and their effects can be well modeled with numerical calculations.

© 2004 Optical Society of America

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References

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  1. W. H. Wells, “Theory of small-angle scattering,” in Electromagnetics of the Sea (Advisory Group for Aerospace Research and Development, North Atlantic Treaty Organization, Neuilly-Sur-Seine, France, 1973), pp. 35–50.
  2. R. V. Ozmidov, “On the turbulent exchange in a stably stratified ocean,” Izv. Ross. Akad. Nauk Atmos. Oceanic Phys. 1, 493–497 (1965).
  3. D. Bogucki, J. A. Domaradzki, D. Stramski, R. Zaneveld, “Comparison of near-forward scattering on turbulence and particles,” Appl. Opt. 37, 4669–4677 (1998).
    [CrossRef]
  4. V. I. Tatarski, in Wave Propagation in Turbulent Media (McGraw-Hill, New York, 1961), p. 520.
  5. R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54, 52–61 (1963).
    [CrossRef]
  6. T. M. Dillon, D. R. Caldwell, “The Batchelor spectrum and dissipation in the upper ocean,” J. Geophys. Res. 85, 1910–1916 (1980).
    [CrossRef]
  7. D. Bogucki, A. Domaradzki, P. K. Yeung, “Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
    [CrossRef]
  8. R. C. Millard, G. Seaver, “An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength,” Deep-Sea Res. 37, 1909–1926 (1990).
    [CrossRef]
  9. Z. A. Daya, R. E. Ecke, “Prandtl-number dependence of interior temperature and velocity fluctuations in turbulent convection,” Phys. Rev. E 66, 45301–45303 (2002).
    [CrossRef]
  10. G. Willis, J. Deardorff, “Investigation of turbulent thermal convection between horizontal plates,” J. Fluid Mech. 28, 675–704 (1967).
    [CrossRef]
  11. F. Heslot, B. Castaing, A. Libchaber, “Transitions to turbulence in Helium gas,” Phys. Rev. A 36, 5870–5873 (1987).
    [CrossRef] [PubMed]
  12. D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
    [CrossRef]
  13. R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, New York, 1994).
  14. T. Eidson, “Numerical simulation of the turbulent Rayleigh-Bénard problem,” J. Fluid Mech. 158, 245–253 (1985).
    [CrossRef]
  15. J. Domaradzki, R. Metcalfe, “Direct numerical simulations of the effects of shear on turbulent Rayleigh-Bénard convection,” J. Fluid Mech. 193, 499–501 (1988).
    [CrossRef]
  16. S. Christie, J. Domaradzki, “Numerical evidence for nonuniversality of the soft/hard turbulence classification for thermal convection,” Phys. Fluids 5, 412–415 (1993).
    [CrossRef]
  17. R. Kerr, “Rayleigh number scaling in numerical convection,” J. Fluid Mech. 310, 139–158 (1996).
    [CrossRef]
  18. B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
    [CrossRef]

2002 (1)

Z. A. Daya, R. E. Ecke, “Prandtl-number dependence of interior temperature and velocity fluctuations in turbulent convection,” Phys. Rev. E 66, 45301–45303 (2002).
[CrossRef]

1998 (1)

1997 (1)

D. Bogucki, A. Domaradzki, P. K. Yeung, “Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[CrossRef]

1996 (1)

R. Kerr, “Rayleigh number scaling in numerical convection,” J. Fluid Mech. 310, 139–158 (1996).
[CrossRef]

1993 (1)

S. Christie, J. Domaradzki, “Numerical evidence for nonuniversality of the soft/hard turbulence classification for thermal convection,” Phys. Fluids 5, 412–415 (1993).
[CrossRef]

1990 (1)

R. C. Millard, G. Seaver, “An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength,” Deep-Sea Res. 37, 1909–1926 (1990).
[CrossRef]

1989 (1)

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

1988 (1)

J. Domaradzki, R. Metcalfe, “Direct numerical simulations of the effects of shear on turbulent Rayleigh-Bénard convection,” J. Fluid Mech. 193, 499–501 (1988).
[CrossRef]

1987 (1)

F. Heslot, B. Castaing, A. Libchaber, “Transitions to turbulence in Helium gas,” Phys. Rev. A 36, 5870–5873 (1987).
[CrossRef] [PubMed]

1985 (1)

T. Eidson, “Numerical simulation of the turbulent Rayleigh-Bénard problem,” J. Fluid Mech. 158, 245–253 (1985).
[CrossRef]

1980 (1)

T. M. Dillon, D. R. Caldwell, “The Batchelor spectrum and dissipation in the upper ocean,” J. Geophys. Res. 85, 1910–1916 (1980).
[CrossRef]

1967 (1)

G. Willis, J. Deardorff, “Investigation of turbulent thermal convection between horizontal plates,” J. Fluid Mech. 28, 675–704 (1967).
[CrossRef]

1965 (1)

R. V. Ozmidov, “On the turbulent exchange in a stably stratified ocean,” Izv. Ross. Akad. Nauk Atmos. Oceanic Phys. 1, 493–497 (1965).

1963 (1)

Bogucki, D.

D. Bogucki, J. A. Domaradzki, D. Stramski, R. Zaneveld, “Comparison of near-forward scattering on turbulence and particles,” Appl. Opt. 37, 4669–4677 (1998).
[CrossRef]

D. Bogucki, A. Domaradzki, P. K. Yeung, “Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[CrossRef]

Caldwell, D. R.

T. M. Dillon, D. R. Caldwell, “The Batchelor spectrum and dissipation in the upper ocean,” J. Geophys. Res. 85, 1910–1916 (1980).
[CrossRef]

Castaing, B.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

F. Heslot, B. Castaing, A. Libchaber, “Transitions to turbulence in Helium gas,” Phys. Rev. A 36, 5870–5873 (1987).
[CrossRef] [PubMed]

Christie, S.

S. Christie, J. Domaradzki, “Numerical evidence for nonuniversality of the soft/hard turbulence classification for thermal convection,” Phys. Fluids 5, 412–415 (1993).
[CrossRef]

Daya, Z. A.

Z. A. Daya, R. E. Ecke, “Prandtl-number dependence of interior temperature and velocity fluctuations in turbulent convection,” Phys. Rev. E 66, 45301–45303 (2002).
[CrossRef]

Deardorff, J.

G. Willis, J. Deardorff, “Investigation of turbulent thermal convection between horizontal plates,” J. Fluid Mech. 28, 675–704 (1967).
[CrossRef]

Dillon, T. M.

T. M. Dillon, D. R. Caldwell, “The Batchelor spectrum and dissipation in the upper ocean,” J. Geophys. Res. 85, 1910–1916 (1980).
[CrossRef]

Domaradzki, A.

D. Bogucki, A. Domaradzki, P. K. Yeung, “Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[CrossRef]

Domaradzki, J.

S. Christie, J. Domaradzki, “Numerical evidence for nonuniversality of the soft/hard turbulence classification for thermal convection,” Phys. Fluids 5, 412–415 (1993).
[CrossRef]

J. Domaradzki, R. Metcalfe, “Direct numerical simulations of the effects of shear on turbulent Rayleigh-Bénard convection,” J. Fluid Mech. 193, 499–501 (1988).
[CrossRef]

Domaradzki, J. A.

Ecke, R. E.

Z. A. Daya, R. E. Ecke, “Prandtl-number dependence of interior temperature and velocity fluctuations in turbulent convection,” Phys. Rev. E 66, 45301–45303 (2002).
[CrossRef]

Eidson, T.

T. Eidson, “Numerical simulation of the turbulent Rayleigh-Bénard problem,” J. Fluid Mech. 158, 245–253 (1985).
[CrossRef]

Gunaratne, G.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

Heslot, F.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

F. Heslot, B. Castaing, A. Libchaber, “Transitions to turbulence in Helium gas,” Phys. Rev. A 36, 5870–5873 (1987).
[CrossRef] [PubMed]

Hufnagel, R. E.

Kadanoff, L.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

Kerr, R.

R. Kerr, “Rayleigh number scaling in numerical convection,” J. Fluid Mech. 310, 139–158 (1996).
[CrossRef]

Libchaber, A.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

F. Heslot, B. Castaing, A. Libchaber, “Transitions to turbulence in Helium gas,” Phys. Rev. A 36, 5870–5873 (1987).
[CrossRef] [PubMed]

McKechnie, T. S.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

Metcalfe, R.

J. Domaradzki, R. Metcalfe, “Direct numerical simulations of the effects of shear on turbulent Rayleigh-Bénard convection,” J. Fluid Mech. 193, 499–501 (1988).
[CrossRef]

Millard, R. C.

R. C. Millard, G. Seaver, “An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength,” Deep-Sea Res. 37, 1909–1926 (1990).
[CrossRef]

Neal, D. R.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

OHern, T. J.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

Ozmidov, R. V.

R. V. Ozmidov, “On the turbulent exchange in a stably stratified ocean,” Izv. Ross. Akad. Nauk Atmos. Oceanic Phys. 1, 493–497 (1965).

Seaver, G.

R. C. Millard, G. Seaver, “An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength,” Deep-Sea Res. 37, 1909–1926 (1990).
[CrossRef]

Shul, R.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

Stanley, N. R.

Stramski, D.

Tatarski, V. I.

V. I. Tatarski, in Wave Propagation in Turbulent Media (McGraw-Hill, New York, 1961), p. 520.

Thomae, S.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

Torczynski, J. R.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

Walker, R. E.

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, New York, 1994).

Warren, M. E.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

Wells, W. H.

W. H. Wells, “Theory of small-angle scattering,” in Electromagnetics of the Sea (Advisory Group for Aerospace Research and Development, North Atlantic Treaty Organization, Neuilly-Sur-Seine, France, 1973), pp. 35–50.

Willis, G.

G. Willis, J. Deardorff, “Investigation of turbulent thermal convection between horizontal plates,” J. Fluid Mech. 28, 675–704 (1967).
[CrossRef]

Wu, X.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

Yeung, P. K.

D. Bogucki, A. Domaradzki, P. K. Yeung, “Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[CrossRef]

Zaleski, S.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

Zanetti, G.

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

Zaneveld, R.

Appl. Opt. (1)

Deep-Sea Res. (1)

R. C. Millard, G. Seaver, “An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength,” Deep-Sea Res. 37, 1909–1926 (1990).
[CrossRef]

Izv. Ross. Akad. Nauk Atmos. Oceanic Phys. (1)

R. V. Ozmidov, “On the turbulent exchange in a stably stratified ocean,” Izv. Ross. Akad. Nauk Atmos. Oceanic Phys. 1, 493–497 (1965).

J. Fluid Mech. (6)

G. Willis, J. Deardorff, “Investigation of turbulent thermal convection between horizontal plates,” J. Fluid Mech. 28, 675–704 (1967).
[CrossRef]

T. Eidson, “Numerical simulation of the turbulent Rayleigh-Bénard problem,” J. Fluid Mech. 158, 245–253 (1985).
[CrossRef]

J. Domaradzki, R. Metcalfe, “Direct numerical simulations of the effects of shear on turbulent Rayleigh-Bénard convection,” J. Fluid Mech. 193, 499–501 (1988).
[CrossRef]

R. Kerr, “Rayleigh number scaling in numerical convection,” J. Fluid Mech. 310, 139–158 (1996).
[CrossRef]

B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X. Wu, S. Zaleski, G. Zanetti, “Scaling of hard turbulence in Rayleigh-Bérnard convection,” J. Fluid Mech. 204, 1–31 (1989).
[CrossRef]

D. Bogucki, A. Domaradzki, P. K. Yeung, “Direct numerical simulations of passive scalars with Pr > 1 advected by turbulent flow,” J. Fluid Mech. 343, 111–130 (1997).
[CrossRef]

J. Geophys. Res. (1)

T. M. Dillon, D. R. Caldwell, “The Batchelor spectrum and dissipation in the upper ocean,” J. Geophys. Res. 85, 1910–1916 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Phys. Fluids (1)

S. Christie, J. Domaradzki, “Numerical evidence for nonuniversality of the soft/hard turbulence classification for thermal convection,” Phys. Fluids 5, 412–415 (1993).
[CrossRef]

Phys. Rev. A (1)

F. Heslot, B. Castaing, A. Libchaber, “Transitions to turbulence in Helium gas,” Phys. Rev. A 36, 5870–5873 (1987).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Z. A. Daya, R. E. Ecke, “Prandtl-number dependence of interior temperature and velocity fluctuations in turbulent convection,” Phys. Rev. E 66, 45301–45303 (2002).
[CrossRef]

Other (4)

W. H. Wells, “Theory of small-angle scattering,” in Electromagnetics of the Sea (Advisory Group for Aerospace Research and Development, North Atlantic Treaty Organization, Neuilly-Sur-Seine, France, 1973), pp. 35–50.

V. I. Tatarski, in Wave Propagation in Turbulent Media (McGraw-Hill, New York, 1961), p. 520.

D. R. Neal, T. J. OHern, J. R. Torczynski, M. E. Warren, R. Shul, T. S. McKechnie, “Wavefront sensors for optical diagnostics in fluid mechanics: application to heated flow, turbulence and droplet evaporation,” in Optical Diagnostics in Fluid and Thermal Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2005, 194–203 (1993).
[CrossRef]

R. E. Walker, Marine Light Field Statistics (Wiley-Interscience, New York, 1994).

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Figures (5)

Fig. 1
Fig. 1

Experimental setup, including the source of turbulence in the convective tank; the right-most panel illustrates the focal-spot distribution on the linear CCD element.

Fig. 2
Fig. 2

Comparison of VSF for selected runs corresponding to data from Table 1. o, measured; *, calculated values.

Fig. 3
Fig. 3

The raw Hartman wave-front sensor data; scattering angles for a run with ΔT = 7 °C (run-1/2). The streaks on the image are related to the turbulent structures convecting across the light beam.

Fig. 4
Fig. 4

Top, scattering coefficient as a function of ΔT, and the estimated calibration constant Rat(ΔT = 0); Bottom, estimated b from experimental data by use of the simplest linear fit of log[Rat(ΔT)] versus ΔT.

Fig. 5
Fig. 5

Comparison of theoretical and experimental estimates of temperature dissipation rates - χ; o, the literature-derived estimates, *, lower bound using Hölder inequality.

Tables (1)

Tables Icon

Table 1 Experimental Runs, Summarya

Equations (26)

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2+κ2E=-2κ2nrE,
Ex, y, z=ψx, y, zexpiκz,
2iκ ψz+T2ψ+2κ2nψ=0,
ψx, y, z=A0x, yexpiκ 0z nx, y, zdz, zL,
VSFθ= dIθEOdV,
b=2π 0πVSFθsinθdθ.
Trχr, r= χr/χc1/2c/r1/4ν/νc1/4Tcχc, c.
b=lnEz+ΔzEz/Δz,
bχ, =ln1-RatΔT1-RatΔT=0/L,
px=Nx/δxNt,
1=2π 0πVSFθsinθdθ=-ππdϕ 0πVSFsinθ/θdθ.
PdxdyVSFθx, y= 0π pxdx,
P=0xπ0yπ2-x21/2.
VSFθ=-1/2π θπdpx/dxx2-θ21/2dx
px=4 xπθVSFθθ2-x21/2dθ.
VSFθ; ΔT=bΔTVSF θ; ΔT,
PU/d=D2νRa·Nu/d4.
PU/d0.13Ra1.28D2ν/d4.
χ=2Dθ2,
χ=6Dθ/x2.
χ=6Dθ/x2= limts 6D/ts0tsθ/t21/x/t2dt=limts 6D/ts0tsθ/t2ux-2dt
ab |fxgx|dxab |fx|pdx1/p×ab |gx|qdx1/q.
Tz- 12ΔTδT=- ΔTdNu.
Tw=Twmaxz/δT,
PT= 2d0δTTwzTzdz=DNuΔT/d2.
χcore=ccorePT.

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