Abstract

Recent simulations and experiments have shown that the viscous-range temperature spectrum in water can be well described by the Kraichnan spectral model. Motivated by this, a tractable expression is developed for the underwater temperature spectrum that is consistent with both the Obukhov-Corrsin law in the inertial range and the Kraichnan model in the viscous range. In analogy with the temperature spectrum, the formula for the salinity spectrum and the temperature-salinity co-spectrum are also derived. The linear combination of these three scalar spectra constitutes a new form of the refractivity spectrum. This spectral model predicts a much stronger optical scintillation than the previous model.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  36. X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. 41(11), 2155–2167 (2011).
    [Crossref]
  37. D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  40. R. C. Mjolsness, “Diffusion of a passive scalar at large Prandtl number according to the abridged Lagrangian interaction theory,” Phys. Fluids 18(10), 1393–1394 (1975).
    [Crossref]

2018 (2)

M. C. Gokce and Y. Baykal, “Aperture averaging and BER for Gaussian beam in underwater oceanic turbulence,” Opt. Commun. 410, 830–835 (2018).
[Crossref]

M. C. Gokce and Y. Baykal, “Aperture averaging in strong oceanic turbulence,” Opt. Commun. 413, 196–199 (2018).
[Crossref]

2017 (7)

2016 (2)

2015 (4)

2014 (2)

2012 (2)

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

2011 (1)

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. 41(11), 2155–2167 (2011).
[Crossref]

2000 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

1997 (1)

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

1992 (3)

V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. 238(-1), 683–698 (1992).
[Crossref]

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49(16), 1494–1509 (1992).
[Crossref]

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

1990 (1)

J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37(1), 13–16 (1990).
[Crossref]

1978 (3)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(3), 541–562 (1978).
[Crossref]

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68(8), 1067–1072 (1978).
[Crossref]

1977 (2)

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83(03), 547–567 (1977).
[Crossref]

1975 (1)

R. C. Mjolsness, “Diffusion of a passive scalar at large Prandtl number according to the abridged Lagrangian interaction theory,” Phys. Fluids 18(10), 1393–1394 (1975).
[Crossref]

1968 (2)

R. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11(5), 945–953 (1968).
[Crossref]

C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids 11(8), 1612–1617 (1968).
[Crossref]

1965 (1)

Y.-H. Pao, “Structure of turbulent velocity and scalar fields a large wave-numbers,” Phys. Fluids 8(6), 1063–1075 (1965).
[Crossref]

1964 (1)

S. Corrsin, “Further generalization of Onsager’s model for turbulent spectra,” Phys. Fluids 7(8), 1156–1159 (1964).
[Crossref]

1959 (1)

G. K. Batchelor, “Small scale variation of convected quantities like temperature in a turbulent fluid,” J. Fluid Mech. 5(01), 113–133 (1959).
[Crossref]

1951 (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22(4), 469–473 (1951).
[Crossref]

1949 (1)

A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geofr. I Geofiz. 13(1), 58–69 (1949).

Abdallah, M.

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

Ata, Y.

Batchelor, G. K.

G. K. Batchelor, “Small scale variation of convected quantities like temperature in a turbulent fluid,” J. Fluid Mech. 5(01), 113–133 (1959).
[Crossref]

Baykal, Y.

Bogucki, D. J.

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

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

Cai, Y.

Champagne, F. H.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Churnside, J. H.

J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37(1), 13–16 (1990).
[Crossref]

Corrsin, S.

S. Corrsin, “Further generalization of Onsager’s model for turbulent spectra,” Phys. Fluids 7(8), 1156–1159 (1964).
[Crossref]

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22(4), 469–473 (1951).
[Crossref]

de Bruyn Kops, S. M.

Domaradzki, A. J.

D. J. Bogucki, A. J. Domaradzki, and 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. A.

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

Dubovikov, M. M.

V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. 238(-1), 683–698 (1992).
[Crossref]

Elamassie, M.

Farwell, N.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

Forcat, F.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. 41(11), 2155–2167 (2011).
[Crossref]

Frehlich, R. G.

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49(16), 1494–1509 (1992).
[Crossref]

Friehe, C. A.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Gerçekcioglu, H.

H. Gerçekcioglu, “BER of annular beams in weak oceanic turbulence,” Selcuk University Journal of Engineering, Science and Technology 5(3), 262–273 (2017).
[Crossref]

H. Gerçekcioğlu, “Bit error rate of focused Gaussian beams in weak oceanic turbulence,” J. Opt. Soc. Am. A 31(9), 1963–1968 (2014).
[Crossref] [PubMed]

Gokce, M. C.

M. C. Gokce and Y. Baykal, “Aperture averaging and BER for Gaussian beam in underwater oceanic turbulence,” Opt. Commun. 410, 830–835 (2018).
[Crossref]

M. C. Gokce and Y. Baykal, “Aperture averaging in strong oceanic turbulence,” Opt. Commun. 413, 196–199 (2018).
[Crossref]

Golmohammady, S.

Hill, R. J.

R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. 68(8), 1067–1072 (1978).
[Crossref]

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(3), 541–562 (1978).
[Crossref]

Karyakin, M. Yu.

V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. 238(-1), 683–698 (1992).
[Crossref]

Kashani, F. D.

Korotkova, O.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

Kraichnan, R.

R. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids 11(5), 945–953 (1968).
[Crossref]

LaRue, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Leith, C. E.

C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids 11(8), 1612–1617 (1968).
[Crossref]

Li, Y.

Li, Z.

Liu, L.

Liu, Z.

Lou, H.

D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. 42(10), 1717–1728 (2012).
[Crossref]

Lu, C.

Mashal, A.

Mjolsness, R. C.

R. C. Mjolsness, “Diffusion of a passive scalar at large Prandtl number according to the abridged Lagrangian interaction theory,” Phys. Fluids 18(10), 1393–1394 (1975).
[Crossref]

Muschinski, A.

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Oboukhov, A. M.

A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geofr. I Geofiz. 13(1), 58–69 (1949).

Pao, Y.-H.

Y.-H. Pao, “Structure of turbulent velocity and scalar fields a large wave-numbers,” Phys. Fluids 8(6), 1063–1075 (1965).
[Crossref]

Paulson, C. A.

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83(03), 547–567 (1977).
[Crossref]

Peng, X.

Planella, J.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. 41(11), 2155–2167 (2011).
[Crossref]

Praskovsky, A. A.

V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. 238(-1), 683–698 (1992).
[Crossref]

Qaraqe, K.

Roget, E.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. 41(11), 2155–2167 (2011).
[Crossref]

Sanchez, X.

X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. 41(11), 2155–2167 (2011).
[Crossref]

Shchepakina, E.

O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22(2), 260–266 (2012).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. 238(-1), 683–698 (1992).
[Crossref]

Uysal, M.

Wang, R.

Wang, X.

Wang, Y.

Williams, R. M.

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83(03), 547–567 (1977).
[Crossref]

Wynagaard, J. C.

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

Yao, J.

Yeung, P. K.

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

Yi, X.

Yousefi, M.

Yu, L.

Zhang, Y.

Zhao, D.

Appl. Opt. (3)

Int. J. Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Izv. Akad. Nauk SSSR, Ser. Geofr. I Geofiz. (1)

A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geofr. I Geofiz. 13(1), 58–69 (1949).

J. Appl. Phys. (1)

S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. 22(4), 469–473 (1951).
[Crossref]

J. Atmos. Sci. (2)

F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. 34(3), 515–530 (1977).
[Crossref]

R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49(16), 1494–1509 (1992).
[Crossref]

J. Fluid Mech. (5)

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88(3), 541–562 (1978).
[Crossref]

R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. 83(03), 547–567 (1977).
[Crossref]

G. K. Batchelor, “Small scale variation of convected quantities like temperature in a turbulent fluid,” J. Fluid Mech. 5(01), 113–133 (1959).
[Crossref]

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

V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. 238(-1), 683–698 (1992).
[Crossref]

J. Mod. Opt. (2)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. 37(1), 13–16 (1990).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014).
[Crossref] [PubMed]

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A. Muschinski and S. M. de Bruyn Kops, “Investigation of Hill’s optical turbulence model by means of direct numerical simulation,” J. Opt. Soc. Am. A 32(12), 2423–2430 (2015).
[Crossref] [PubMed]

J. Phys. Oceanogr. (2)

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

Opt. Express (2)

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Figures (5)

Fig. 1
Fig. 1 Normalized scalar spectra comparison of Batchelor spectrum, Kraichnan spectrum, Nikishov spectrum, and Hill numerical spectrum: (a) temperature and (b) salinity spectra.
Fig. 2
Fig. 2 Normalized scalar dissipation spectra ( κ η B ) 2 E T ( κ η B ) / ( χ T ( v / ε ) 1 / 2 η B ) for Pr T 7 . (a) Comparison of Kraichnan spectrum, Nikishov spectrum with Q 1 = 2.35 , the proposed spectrum, and the DNS results; (b) Comparison of Batchelar spectrum, Nikishov spectrum with Q 1 = 4.6 , the proposed spectrum, and the DNS results.
Fig. 3
Fig. 3 Normalized scalar spectra comparison of Kraichnan spectrum, Nikishov spectrum, Hill numerical spectrum, and the proposed spectrum: (a) temperature and (b) salinity spectra.
Fig. 4
Fig. 4 Spectra of E n for different ω and d r : (a) ω =-3 , d r = 5.4 ; (b) ω =-1 , d r = 1 ; (c) ω =-0 .65 , d r =0 .355 ; (d) ω =-0 .25 , d r = 0.037 .
Fig. 5
Fig. 5 Scintillation index of plane and spherical waves vs link length: (a) plane; (b) spherical waves.

Tables (1)

Tables Icon

Table 1 Four Hill’s Models.

Equations (35)

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χ = 2 D 0 κ 2 E ( κ ) d κ .
η ( v 3 / ε ) 1 / 4 , η B ( v D 2 / ε ) 1 / 4 ;
η = Pr 1 / 2 η B .
1 / L 0 κ 1 / η ,
1 / η κ 1 / η B ,
κ 1 / η B .
E ( κ ) = β χ ε 1 / 3 κ 5 / 3 .
E ( κ ) = χ ( v / ε ) 1 / 2 q B κ 1 exp [ q B ( κ η B ) 2 ] ,
E ( κ ) = χ ( v / ε ) 1 / 2 q K κ 1 [ 1 + ( 6 q K ) 1 / 2 κ η B ] exp [ ( 6 q K ) 1 / 2 κ η B ] ,
T ( κ ) = 2 D κ 2 E ( κ ) ,
T ( κ ) F ( κ ) κ .
F ( κ ) κ = 2 D κ 2 E ( κ ) .
F ( κ ) = s ( κ ) E ( κ ) ,
F ( κ ) = D s ( κ ) Q s ( κ ) κ ,
E ( κ ) = β χ ε 1 / 3 κ 5 / 3 [ 1 + Q 1 ( κ η ) 2 / 3 ] exp { 1.5 β Pr 1 ( κ η ) 4 / 3 β Pr 1 Q 1 ( κ η ) 2 } .
f ( κ η ) = q B ( κ η ) 2 / 3 exp [ q B Pr 1 ( κ η ) 2 ] ,
f ( κ η ) = q K ( κ η ) 2 / 3 [ 1 + ( 6 q K Pr 1 ) 1 / 2 κ η ] exp [ ( 6 q K Pr 1 ) 1 / 2 κ η ] .
E ( κ ) = β χ ε 1 / 3 κ 5 / 3 g ( κ η , Pr ) ,
0 x 1 / 3 g ( x , Pr ) d x = Pr 2 β ,
g ( x , Pr ) = [ 1 + n = 1 N a n x n ] exp ( δ x ) ,
a 1 = 5.1615 , a 2 = 67.1941 , a 3 = 45.918 , a 4 = 53.3589 , a 5 = 1.2064 , δ = 3.3927.
( κ η B ) 2 E ( κ ) χ ( v / ε ) 1 / 2 η B = q B ( κ η B ) exp [ q B ( κ η B ) 2 ] .
( κ η B ) 2 E ( κ ) χ ( v / ε ) 1 / 2 η B = q K ( κ η B ) [ 1 + ( 6 q K ) 1 / 2 κ η B ] exp [ ( 6 q K ) 1 / 2 κ η B ] .
( κ η B ) 2 E ( κ ) χ ( v / ε ) 1 / 2 η B = β Pr 1 / 3 ( κ η B ) 1 / 3 [ 1 + Q 1 Pr 1 / 3 ( κ η B ) 2 / 3 ] . × exp { 1.5 β Pr 1 / 3 ( κ η B ) 4 / 3 β Q 1 ( κ η B ) 2 }
( κ η B ) 2 E ( κ ) χ ( v / ε ) 1 / 2 η B = β Pr 1 / 3 ( κ η B ) 1 / 3 [ 1 + n = 1 N a n ( Pr 1 / 2 κ η B ) n ] exp ( δ Pr 1 / 2 κ η B ) .
a 1 = 7.6156 , a 2 = 1.4268 , a 3 = 0.2239 , a 4 = 0.0105 , a 5 = 0.0026 , δ = 0.3837
a 1 = 6.968 , 4 a 2 = 40.4984 , a 3 = 21.7085 , a 4 = 21.4159 , a 5 = 0.0503 , δ = 2.4778
χ n = Α 2 χ T + Β 2 χ S 2 Α Β χ T S ,
χ S = d r ( Α Β ω ) 2 χ T , χ T S = Α 2 Β ω ( 1 + d r ) χ T ,
R F { | ω | | ω | ( | ω | 1 ) 1 1.85 0.85 | ω | 1 1 0.15 , | ω | 1 , 0.5 | ω | 1 , | ω | < 0.5
E n ( κ ) = Α 2 E T ( κ ) + Β 2 E S ( κ ) 2 Α Β E T S ( κ ) .
E n ( κ ) = Α 2 β χ T ε 1 / 3 κ 5 / 3 [ g ( κ η , Pr T ) + d r ω 2 g ( κ η , Pr S ) ( 1 + d r ) ω g ( κ η , Pr T S ) ] ,
Φ n ( κ ) = ( 4 π κ 2 ) 1 E n ( κ ) = ( 4 π ) 1 Α 2 β χ T ε 1 / 3 κ 11 / 3 [ g ( κ η , Pr T ) + d r ω 2 g ( κ η , Pr S ) ( 1 + d r ) ω g ( κ η , Pr T S ) ] .
σ I , p l 2 = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) { 1 cos [ L κ 2 ξ k ] } d κ d ξ ,
σ I , s p 2 = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) { 1 cos [ L κ 2 k ξ ( 1 ξ ) ] } d κ d ξ ,

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