Abstract

We propose an optical method that uses phase data of a laser beam obtained from a Shack–Hartmann sensor to estimate both the inner and outer scales of turbulence. The method is based on the sequential analysis of normalized correlation functions of Zernike coefficients. It allows the exclusion Cn2 from the analysis and reduces the solution of a two-parameter problem to a sequential solution of two single-parameter problems. The method has been applied to estimate the outer and inner scales of turbulence induced in the water cell.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  2. G. C. Valley and S. M. Wandzura, “Spatial correlation of phase-expansion coefficients for propagation through atmospheric turbulence,” J. Opt. Soc. Am. 69, 712–717 (1979).
    [CrossRef]
  3. A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds number,” Sov. Phys. Usp. 10, 734–738 (1968).
    [CrossRef]
  4. P. H. Hu, J. Stone, and T. J. Stanley, “Application of Zernike polynomials to atmospheric propagation problems,” J. Opt. Soc. Am. A 6, 1595–1608 (1989).
    [CrossRef]
  5. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  6. T. von Kármán, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. USA 34, 530–539 (1948).
    [CrossRef]
  7. N. Takato and I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 958–963 (1995).
    [CrossRef]
  8. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
    [CrossRef]
  9. A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
    [CrossRef]
  10. G. C. Valley, “Long- and short-term Strehl ratios for turbulence with finite inner and outer scales,” Appl. Opt. 18, 984–987 (1979).
    [CrossRef]
  11. A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. 44, 1111–1126 (1997).
    [CrossRef]
  12. M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izvestiya VUZ. Radiofizika 8, 711–724 (1965).
    [CrossRef]
  13. M. E. Gracheva, “Investigation of the statistical properties of strong fluctuations in the intensity of light propagated through the atmosphere near the earth,” Radiophys. Quantum Electron. 10, 424–433 (1967).
    [CrossRef]
  14. R. H. Kleen and G. R. Ochs, “Measurements of the wavelength dependence of scintillation in strong turbulence,” J. Opt. Soc. Am. 60, 1695–1697 (1970).
    [CrossRef]
  15. G. W. Reinhardt and S. A. Collins, “Outer scale effects in turbulence-degraded light-beam spectra,” J. Opt. Soc. Am. 62, 1526–1528 (1972).
    [CrossRef]
  16. M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, “Atmospheric turbulence effects on the phase of laser beams,” Appl. Opt. 13, 1582–1585 (1974).
    [CrossRef]
  17. J. L. Bufton, “Comparison of vertical profile turbulence structure with stellar observations,” Appl. Opt. 12, 1785–1793 (1973).
    [CrossRef]
  18. C. E. Coulman, J. Vernin, Y. Coqueugniot, and J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive index structure profiles,” Appl. Opt. 27, 155–160 (1988).
    [CrossRef]
  19. A. Consortini and K. A. O’Donnell, “Measuring the inner scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Waves Random Media 3, 85–92 (1993).
    [CrossRef]
  20. R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” Appl. Opt. 27, 2194–2198 (1988).
    [CrossRef]
  21. G. R. Ochs and R. J. Hill, “Optical-scintillation method of measuring turbulence inner scale,” Appl. Opt. 24, 2430–2432 (1985).
    [CrossRef]
  22. A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Waves Random Media 1, S11–S28 (1991).
    [CrossRef]
  23. V. V. Voitsekhovich, “Outer scale of turbulence: comparison of different model,” J. Opt. Soc. Am. A 12, 1346–1353 (1995).
    [CrossRef]
  24. V. V. Voitsekhovich and S. Cuevas, “Adaptive optics and the outer scale of turbulence,” J. Opt. Soc. Am. A 12, 2523–2531 (1995).
    [CrossRef]
  25. M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
    [CrossRef]
  26. S. Hippler, F. Hormuth, D. J. Butler, W. Brandner, and T. Henning, “Atmosphere-like turbulence generation with surface-etched phase-screens,” Opt. Express 14, 10139–10148(2006).
    [CrossRef]
  27. M. J. Curley, B. H. Peterson, J. C. Wang, S. S. Sarkisov, S. S. Sarkisov, G. R. Edlin, R. A. Snow, and J. F. Rushing, “Statistical analysis of cloud-cover mitigation of optical turbulence in the boundary layer,” Opt. Express 14, 8929–8946 (2006).
    [CrossRef]
  28. A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
    [CrossRef]
  29. L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt. 16, 2242–2251 (1977).
    [CrossRef]
  30. R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack–Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. 337, 103–108 (2002).
    [CrossRef]
  31. T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. 369, 835–845 (2006).
    [CrossRef]
  32. M. Goodwin, C. Jenkins, and A. Lambert, “Improved detection of atmospheric turbulence with SLODAR,” Opt. Express 15, 14844–14860 (2007).
    [CrossRef]
  33. J. W. Deardorff and G. E. Willis, “Investigation of turbulent thermal convection between horizontal plates,” J. Fluid Mech. 28, 675–704 (1967).
    [CrossRef]
  34. W. Hou, “A simple underwater imaging model,” Opt. Lett. 34, 2688–2690 (2009).
    [CrossRef]
  35. D. J. Bogucki, J. A. Domaradzki, C. Anderson, H. W. Wijesekera, J. R. V. Zaneveld, and C. Moore, “Optical measurement of rates of dissipation of temperature variance due to oceanic turbulence,” Opt. Express 15, 7224–7230 (2007).
    [CrossRef]
  36. G. W. Carhart and M. A. Vorontsov, “Synthetic imaging: nonadaptive anisoplanatic image correction in atmospheric turbulence,” Opt. Lett. 23, 745–747 (1998).
    [CrossRef]
  37. M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993).
    [CrossRef]
  38. V. P. Lukin, “Influence of the source spectrum on the optical measurements of turbulence,” Appl. Opt. 48, A93–A97 (2009).
    [CrossRef]

2009

2007

2006

2002

R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack–Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. 337, 103–108 (2002).
[CrossRef]

1998

1997

A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. 44, 1111–1126 (1997).
[CrossRef]

1995

1993

A. Consortini and K. A. O’Donnell, “Measuring the inner scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Waves Random Media 3, 85–92 (1993).
[CrossRef]

M. I. Charnotskii, “Anisoplanatic short-exposure imaging in turbulence,” J. Opt. Soc. Am. A 10, 492–501 (1993).
[CrossRef]

1991

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Waves Random Media 1, S11–S28 (1991).
[CrossRef]

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
[CrossRef]

1989

1988

1985

1979

1977

A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt. 16, 2242–2251 (1977).
[CrossRef]

1976

1974

1973

1972

1970

1968

A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds number,” Sov. Phys. Usp. 10, 734–738 (1968).
[CrossRef]

1967

M. E. Gracheva, “Investigation of the statistical properties of strong fluctuations in the intensity of light propagated through the atmosphere near the earth,” Radiophys. Quantum Electron. 10, 424–433 (1967).
[CrossRef]

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

1965

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izvestiya VUZ. Radiofizika 8, 711–724 (1965).
[CrossRef]

1948

T. von Kármán, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. USA 34, 530–539 (1948).
[CrossRef]

Anderson, C.

Bertolotti, M.

Bissonnette, L. R.

Bogucki, D. J.

Brandner, W.

Bufton, J. L.

Butler, D. J.

Butterley, T.

T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. 369, 835–845 (2006).
[CrossRef]

Caccia, J. L.

Carhart, G. W.

Carnevale, M.

Charnotskii, M. I.

Collins, S. A.

Consortini, A.

A. Consortini and K. A. O’Donnell, “Measuring the inner scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Waves Random Media 3, 85–92 (1993).
[CrossRef]

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Waves Random Media 1, S11–S28 (1991).
[CrossRef]

A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973).
[CrossRef]

Coqueugniot, Y.

Coulman, C. E.

Cuevas, S.

Curley, M. J.

Dainty, J. C.

A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. 44, 1111–1126 (1997).
[CrossRef]

Deardorff, J. W.

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

Domaradzki, J. A.

Edlin, G. R.

Frehlich, R. G.

Goodwin, M.

Gracheva, M. E.

M. E. Gracheva, “Investigation of the statistical properties of strong fluctuations in the intensity of light propagated through the atmosphere near the earth,” Radiophys. Quantum Electron. 10, 424–433 (1967).
[CrossRef]

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izvestiya VUZ. Radiofizika 8, 711–724 (1965).
[CrossRef]

Gurvich, A. S.

A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izvestiya VUZ. Radiofizika 8, 711–724 (1965).
[CrossRef]

Henning, T.

Hill, R. J.

Hippler, S.

Hormuth, F.

Hou, W.

Hu, P. H.

Jenkins, C.

Kallistratova, M. A.

A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

Kleen, R. H.

Kolmogorov, A. N.

A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds number,” Sov. Phys. Usp. 10, 734–738 (1968).
[CrossRef]

Lambert, A.

Lukin, V. P.

Maccioni, A.

A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. 44, 1111–1126 (1997).
[CrossRef]

Martvel, F. E.

A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

Montera, D.

Moore, C.

Moroder, E.

Muzii, L.

Noll, R. J.

O’Donnell, K.

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Waves Random Media 1, S11–S28 (1991).
[CrossRef]

O’Donnell, K. A.

A. Consortini and K. A. O’Donnell, “Measuring the inner scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Waves Random Media 3, 85–92 (1993).
[CrossRef]

Ochs, G. R.

Peterson, B. H.

Reinhardt, G. W.

Rhoadarmer, T. A.

Roggemann, M. C.

Ronchi, L.

Rushing, J. F.

Sarazin, M.

T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. 369, 835–845 (2006).
[CrossRef]

Sarkisov, S. S.

Sette, D.

Snow, R. A.

Stanley, T. J.

Stone, J.

Takato, N.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Valley, G. C.

Vernin, J.

Voitsekhovich, V. V.

von Kármán, T.

T. von Kármán, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. USA 34, 530–539 (1948).
[CrossRef]

Vorontsov, M. A.

Wandzura, S. M.

Wang, J. C.

Welsh, B. M.

Wijesekera, H. W.

Willis, G. E.

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

Wilson, R. W.

T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. 369, 835–845 (2006).
[CrossRef]

R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack–Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. 337, 103–108 (2002).
[CrossRef]

Winker, D. M.

Yamaguchi, I.

Zaneveld, J. R. V.

Appl. Opt.

G. C. Valley, “Long- and short-term Strehl ratios for turbulence with finite inner and outer scales,” Appl. Opt. 18, 984–987 (1979).
[CrossRef]

M. Bertolotti, M. Carnevale, L. Muzii, and D. Sette, “Atmospheric turbulence effects on the phase of laser beams,” Appl. Opt. 13, 1582–1585 (1974).
[CrossRef]

J. L. Bufton, “Comparison of vertical profile turbulence structure with stellar observations,” Appl. Opt. 12, 1785–1793 (1973).
[CrossRef]

C. E. Coulman, J. Vernin, Y. Coqueugniot, and J. L. Caccia, “Outer scale of turbulence appropriate to modeling refractive index structure profiles,” Appl. Opt. 27, 155–160 (1988).
[CrossRef]

R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” Appl. Opt. 27, 2194–2198 (1988).
[CrossRef]

G. R. Ochs and R. J. Hill, “Optical-scintillation method of measuring turbulence inner scale,” Appl. Opt. 24, 2430–2432 (1985).
[CrossRef]

M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
[CrossRef]

L. R. Bissonnette, “Atmospheric scintillation of optical and infrared waves: a laboratory simulation,” Appl. Opt. 16, 2242–2251 (1977).
[CrossRef]

V. P. Lukin, “Influence of the source spectrum on the optical measurements of turbulence,” Appl. Opt. 48, A93–A97 (2009).
[CrossRef]

Izvestiya VUZ. Radiofizika

M. E. Gracheva and A. S. Gurvich, “Strong fluctuations in the intensity of light propagated through the atmosphere close to the earth,” Izvestiya VUZ. Radiofizika 8, 711–724 (1965).
[CrossRef]

J. Fluid Mech.

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

J. Mod. Opt.

A. Maccioni and J. C. Dainty, “Measurement of thermally induced optical turbulence in a water cell,” J. Mod. Opt. 44, 1111–1126 (1997).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

R. W. Wilson, “SLODAR: measuring optical turbulence altitude with a Shack–Hartmann wavefront sensor,” Mon. Not. R. Astron. Soc. 337, 103–108 (2002).
[CrossRef]

T. Butterley, R. W. Wilson, and M. Sarazin, “Determination of the profile of atmospheric optical turbulence strength from SLODAR data,” Mon. Not. R. Astron. Soc. 369, 835–845 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. USA

T. von Kármán, “Progress in the statistical theory of turbulence,” Proc. Natl. Acad. Sci. USA 34, 530–539 (1948).
[CrossRef]

Radiophys. Quantum Electron.

M. E. Gracheva, “Investigation of the statistical properties of strong fluctuations in the intensity of light propagated through the atmosphere near the earth,” Radiophys. Quantum Electron. 10, 424–433 (1967).
[CrossRef]

A. S. Gurvich, M. A. Kallistratova, and F. E. Martvel, “An investigation of strong fluctuations of light intensity in a turbulent medium at a small wave parameter,” Radiophys. Quantum Electron. 20, 705–714 (1977).
[CrossRef]

Sov. Phys. Usp.

A. N. Kolmogorov, “Local structure of turbulence in an incompressible viscous fluid at very high Reynolds number,” Sov. Phys. Usp. 10, 734–738 (1968).
[CrossRef]

Waves Random Media

A. Consortini and K. A. O’Donnell, “Measuring the inner scale of atmospheric turbulence by correlation of lateral displacements of thin parallel laser beams,” Waves Random Media 3, 85–92 (1993).
[CrossRef]

Waves Random Media

A. Consortini and K. O’Donnell, “Beam wandering of thin parallel beams through atmospheric turbulence,” Waves Random Media 1, S11–S28 (1991).
[CrossRef]

Other

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Dependence of correlation functions on the S / D parameter at various values of l m / D in Tatarskii: (a) for the first mode, (b) fourth mode, (c) seventh mode; and at various values of L 0 / D in the von Karman model, (d) first mode, (e) fourth mode, and (f) seventh mode.

Fig. 2.
Fig. 2.

Water cell.

Fig. 3.
Fig. 3.

Direction of aperture shifting ( y axis).

Fig. 4.
Fig. 4.

Correlation functions for Zernike modes ( j = 1 , 2 ) with aperture shift in x direction (curves with square centers) or y direction (curves with circle centers) and no flow. (a)  C 2 ( x axis) and C 1 ( y axis), Δ t = 10 ° C , (b)  C 1 ( x axis) and C 2 ( y axis), Δ t = 10 ° C , (c)  C 2 ( x axis) and C 1 ( y axis), Δ t = 15 ° C , and (d)  C 1 ( x axis) and C 2 ( y axis), Δ t = 15 ° C .

Fig. 5.
Fig. 5.

Correlation functions for Zernike modes ( j = 1 , 2 ) with aperture shift in x direction (curves with square centers) or y direction (curves with circle centers) and no flow. (a)  C 2 ( x axis) and C 1 ( y axis), Δ t = 20 ° C , (b)  C 1 ( x axis) and C 2 ( y axis), Δ t = 20 ° C , (c)  C 2 ( x axis) and C 1 ( y axis), Δ t = 25 ° C , and (d)  C 1 ( x axis) and C 2 ( y axis), Δ t = 25 ° C .

Fig. 6.
Fig. 6.

Correlation functions for Zernike modes ( j = 1 , 2 ) with aperture shift in x direction (curves with square centers) or y direction (curves with circle centers) and in the presence of a flow. (a)  C 2 ( x axis) and C 1 ( y axis), Δ t = 10 ° C , (b)  C 1 ( x axis) and C 2 ( y axis), Δ t = 10 ° C , (c)  C 2 ( x axis) and C 1 ( y axis), Δ t = 15 ° C , and (d)  C 1 ( x axis) and C 2 ( y axis), Δ t = 15 ° C .

Fig. 7.
Fig. 7.

Comparison of experimental correlations for Zernike modes of a collimated beam obtained at different values of aperture shift S with aperture size D in the presence or absence of a flow at temperature difference values Δ t = 10 ° C , 15°C, 20°C, and 25°C. (a)  j = 6 , (b)  j = 7 , (c)  j = 8 , and (d)  j = 9 .

Fig. 8.
Fig. 8.

Sequential estimation of characteristic scales of turbulence. (a) Estimation of l m from C 7 , (b) estimation of l m from C 8 , (c) estimation of L 0 from C 1 , and (d) estimation of L 0 from C 2 .

Fig. 9.
Fig. 9.

Experimental correlations for Zernike modes obtained at Δ t = 10 ° C with no flow, their approximations with Tatarskii model, D = 1.5 cm : (a)  j = 6 , l m not defined, (b)  j = 7 , l m = 13 16 mm , (c)  j = 8 , l m = 13 16 mm , and (d)  j = 9 , l m = 18 20 mm .

Tables (2)

Tables Icon

Table 1. Rayleigh and Prandtl Numbers for Experiments

Tables Icon

Table 2. Estimation Results

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

φ ( R ρ , θ ) = a j Z j ( ρ , θ ) ,
φ 1 = φ ( R ρ 1 , θ ) ,
φ 2 = φ ( R ( ρ 2 + 2 S ) , θ ) ,
a i j = φ i Z j ( ρ i ) W ( ρ i ) d ρ i ,
W ( ρ ) = { 1 / π | ρ | 1 0 | ρ | > 1 ,
a 1 j a 2 j * ( S ) = d ρ 1 d ρ 2 C [ R ( ρ 2 + 2 S ρ 1 ) ] × Z 1 j ( ρ 1 ) W ( ρ 1 ) Z 2 j * ( ρ 2 ) W * ( ρ 2 ) ,
C [ R ( ρ 2 + 2 S ρ 1 ) ] = φ 1 φ 2 * .
a 1 j a 2 j * ( S ) = d κ 1 R 2 exp [ 2 π i k ( 2 S ) ] Φ S ( k R ) Z ˜ 1 j ( k ) Z ˜ 2 j * ( k ) ,
Φ ( k ) = A exp { ( k / k m ) 2 } k ( k 2 + k 0 2 ) 11 / 6 ,
a 1 j a 2 j ( S ) = A ( n + 1 ) 0 exp ( k 2 k m 2 ) × ( J 0 ( 2 Sk ) + p J 2 m ( 2 Sk ) ) J n + 1 2 ( 2 Sk ) k ( k 2 + k 0 2 ) 11 / 6 d k ,
C = a 1 j a 2 j a 1 2 ,
Ra = g α Δ T l 3 ν χ , Pr = α ν ,
Re = V L / ν ,

Metrics