Abstract

Optical distortions caused by turbulent airflow surrounding an aircraft, known as aero-optical phenomena, are a major impediment to applications of airborne laser systems. To better understand the spectral properties of aero-optical distortions, a general expression for the wavenumber spectrum of the refractive index is derived from the ideal-gas law and Gladstone-Dale relation. The derived index-of-refraction spectrum accounts for changes in air density due to both temperature and pressure fluctuations and is used to calculate the phase-distortion spectrum of an optical beam propagating through a weakly compressible, turbulent flow field. Numerical simulations of weakly compressible, temporally evolving shear layers are used to verify theoretical results and confirm that if the log slope of the one-dimensional density spectrum in the inertial subrange is −mρ, the optical phase distortion spectral slope is given by −(mρ + 1). The value of mρ is shown to be dependent on the ratio of shear-layer free-stream densities and bounded by the spectral slopes of temperature and pressure fluctuations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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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]

2017 (2)

2016 (2)

E. J. Jumper and S. Gordeyev, “Physics and measurement of aero-optical effects: past and present,” Ann. Rev. Fluid Mech. 49, 419–441 (2016).
[Crossref]

K. Wang and M. Wang, “Computational analysis of aero-optical distortions by flow over a cylindrical turret,” AIAA J. 54(5), 1461–1471 (2016).
[Crossref]

2014 (1)

S. Gordeyev, A. E. Smith, J. A. Cress, and E. J. Jumper, “Experimental studies of aero-optical properties of subsonic turbulent boundary layers,” J. Fluid Mech. 740, 214–253 (2014).
[Crossref]

2012 (4)

K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech. 696, 122–151 (2012).
[Crossref]

S. Gordeyev and E. J. Jumper, “Aero-optical effects of supersonic boundary layers,” AIAA J. 50(3), 682–690 (2012).
[Crossref]

Q. Gao, S. Yi, Z. Jiang, L. He, and Y. Zhao, “Hierarchical structure of the optical path length of the supersonic turbulent boundary layer,” Opt. Express,  20(15), 16494–16503 (2012).
[Crossref]

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Ann. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

2010 (1)

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[Crossref]

2008 (1)

A. P. Freeman and H. J. Catrakis, “Direct reduction of aero-optical aberrations by large structure suppression control in turbulence,” AIAA J. 46(10), 2582–2590 (2008).
[Crossref]

2007 (1)

D. You and P. Moin, “A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries,” Phys. Fluids,  19(6), 065110 (2007).
[Crossref]

2006 (1)

D. Bodony, “Analysis of sponge zones for computational fluid mechanics,” J. Comput. Phys. 212, 681–702 (2006).
[Crossref]

2004 (1)

A. W. Vreman, “An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications,” Phys. Fluids 16(10), 3670–3681 (2004).
[Crossref]

2003 (2)

Y. Tsuji and T. Ishihara, “Similarity scaling of pressure fluctuation in turbulence,” Phys. Rev. E,  68, 026309 (2003).
[Crossref]

M. Lloyd-Hart, “Taking the twinkle out of starlight,” IEEE Spectrum 40(12), 22–29 (2003).
[Crossref]

2002 (1)

C. Pantano and S. Sarkar, “A study of compressibility effects in the high-speed turbulent shear layer using direct simulation,” J. Fluid Mech. 451, 329–371 (2002).
[Crossref]

2001 (3)

T. Gohto and D. Fukayama, “Pressure spectrum in homogeneous turbulence,” Phys. Rev. Lett. 86(17), 3775–3778 (2001).
[Crossref]

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[Crossref]

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 443, 105–134 (2001).
[Crossref]

2000 (2)

E. J. Fitzgerald and E. J. Jumper, “Two-dimensional optical wavefront measurements using a small-aperture beam technique-derivative instrument,” Opt. Eng. 39, 3285–3293 (2000).
[Crossref]

Z. Warhaft, “Passive scalars in turbulent flows,” Ann. Rev. Fluid Mech. 32, 203–240 (2000).
[Crossref]

1999 (1)

P. Vedula and P. K. Yeung, “Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence,” Phys. Fluids 11(5), 1208–1220, (1999).
[Crossref]

1996 (1)

M. Rogers and R. Moser, “Direct simulation of a self-similar turbulent mixing layer,” Phys. Fluids 6(2), 903–923 (1996).
[Crossref]

1992 (1)

1990 (1)

J. Bell and R. Mehta, “Development of a two-stream mixing layer from tripped and untripped boundary layers,” AIAA J. 28, 2034–2042 (1990).
[Crossref]

1984 (1)

H. Winarto and M. Davis, “Fluctuations of density, pressure, and temperature in a turbulent mixing region,” Proc. R. Soc. A 395, 203–228 (1984).
[Crossref]

1981 (1)

G. A. McBean and J. Elliott, “Pressure and humidity effects on optical refractive-index fluctuations,” Bound.-Layer Meteorol. 20, 101–109 (1981).
[Crossref]

1980 (1)

1975 (1)

1974 (1)

R. Woo and A. Ishimaru, “Effects of turbulence in a planetary atmosphere on radio occultation,” IEEE Trans. Antennas Propag. 22(4), 566–573 (1974).
[Crossref]

1951 (1)

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

1949 (1)

A. M. Obukhov, “The structure of the temperature field in a turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. Geophys. 13, 58–69 (1949).

Batchelor, G.

G. Batchelor, The Theory of Homogeneous Turbulence (Cambridge University, 1982).

Bell, J.

J. Bell and R. Mehta, “Development of a two-stream mixing layer from tripped and untripped boundary layers,” AIAA J. 28, 2034–2042 (1990).
[Crossref]

Bodony, D.

D. Bodony, “Analysis of sponge zones for computational fluid mechanics,” J. Comput. Phys. 212, 681–702 (2006).
[Crossref]

Catrakis, H. J.

A. P. Freeman and H. J. Catrakis, “Direct reduction of aero-optical aberrations by large structure suppression control in turbulence,” AIAA J. 46(10), 2582–2590 (2008).
[Crossref]

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 443, 105–134 (2001).
[Crossref]

Champagne, F. H.

Clifford, S. F.

Corrsin, S.

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

Cress, J. A.

S. Gordeyev, A. E. Smith, J. A. Cress, and E. J. Jumper, “Experimental studies of aero-optical properties of subsonic turbulent boundary layers,” J. Fluid Mech. 740, 214–253 (2014).
[Crossref]

Davis, M.

H. Winarto and M. Davis, “Fluctuations of density, pressure, and temperature in a turbulent mixing region,” Proc. R. Soc. A 395, 203–228 (1984).
[Crossref]

Dimotakis, P. E.

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 443, 105–134 (2001).
[Crossref]

Dreyer, G. F.

Elliott, J.

G. A. McBean and J. Elliott, “Pressure and humidity effects on optical refractive-index fluctuations,” Bound.-Layer Meteorol. 20, 101–109 (1981).
[Crossref]

Fitzgerald, E. J.

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[Crossref]

E. J. Fitzgerald and E. J. Jumper, “Two-dimensional optical wavefront measurements using a small-aperture beam technique-derivative instrument,” Opt. Eng. 39, 3285–3293 (2000).
[Crossref]

Fourguette, D. C.

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 443, 105–134 (2001).
[Crossref]

Freeman, A. P.

A. P. Freeman and H. J. Catrakis, “Direct reduction of aero-optical aberrations by large structure suppression control in turbulence,” AIAA J. 46(10), 2582–2590 (2008).
[Crossref]

Friehe, C.

Fukayama, D.

T. Gohto and D. Fukayama, “Pressure spectrum in homogeneous turbulence,” Phys. Rev. Lett. 86(17), 3775–3778 (2001).
[Crossref]

Gao, Q.

Gibson, C. H.

Gohto, T.

T. Gohto and D. Fukayama, “Pressure spectrum in homogeneous turbulence,” Phys. Rev. Lett. 86(17), 3775–3778 (2001).
[Crossref]

Gordeyev, S.

E. J. Jumper and S. Gordeyev, “Physics and measurement of aero-optical effects: past and present,” Ann. Rev. Fluid Mech. 49, 419–441 (2016).
[Crossref]

S. Gordeyev, A. E. Smith, J. A. Cress, and E. J. Jumper, “Experimental studies of aero-optical properties of subsonic turbulent boundary layers,” J. Fluid Mech. 740, 214–253 (2014).
[Crossref]

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Ann. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

S. Gordeyev and E. J. Jumper, “Aero-optical effects of supersonic boundary layers,” AIAA J. 50(3), 682–690 (2012).
[Crossref]

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[Crossref]

A. E. Smith, S. Gordeyev, T. Saxton-Fox, and B. McKeon, “Subsonic boundary-layer wavefront spectra for a range of Reynolds numbers,” 45th AIAA Plasmadynamics and Lasers Conference (AIAA, 2006), paper 2014-2491.

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting and Exhibit (AIAA, 2008), paper 2008-1076.
[Crossref]

Ham, F.

Y. Khalighi, J. W. Nichols, F. Ham, S. K. Lele, and P. Moin, “Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations,” in 17th AIAA/CEAS Aeroacoustics Conference (AIAA, 2001), paper 2011-2886.

He, L.

Hill, R. J.

Ishihara, T.

Y. Tsuji and T. Ishihara, “Similarity scaling of pressure fluctuation in turbulence,” Phys. Rev. E,  68, 026309 (2003).
[Crossref]

Ishimaru, A.

R. Woo and A. Ishimaru, “Effects of turbulence in a planetary atmosphere on radio occultation,” IEEE Trans. Antennas Propag. 22(4), 566–573 (1974).
[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

Jiang, Z.

Jones, B.

B. Spencer and B. Jones, “Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer,” in 4th Fluid and Plasma Dynamics Conference, Fluid Dynamics and Co-located Conferences (AIAA, 1971), paper 71-613.

Jumper, E.

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[Crossref]

Jumper, E. J.

E. J. Jumper and S. Gordeyev, “Physics and measurement of aero-optical effects: past and present,” Ann. Rev. Fluid Mech. 49, 419–441 (2016).
[Crossref]

S. Gordeyev, A. E. Smith, J. A. Cress, and E. J. Jumper, “Experimental studies of aero-optical properties of subsonic turbulent boundary layers,” J. Fluid Mech. 740, 214–253 (2014).
[Crossref]

S. Gordeyev and E. J. Jumper, “Aero-optical effects of supersonic boundary layers,” AIAA J. 50(3), 682–690 (2012).
[Crossref]

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[Crossref]

E. J. Fitzgerald and E. J. Jumper, “Two-dimensional optical wavefront measurements using a small-aperture beam technique-derivative instrument,” Opt. Eng. 39, 3285–3293 (2000).
[Crossref]

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting and Exhibit (AIAA, 2008), paper 2008-1076.
[Crossref]

Khalighi, Y.

Y. Khalighi, J. W. Nichols, F. Ham, S. K. Lele, and P. Moin, “Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations,” in 17th AIAA/CEAS Aeroacoustics Conference (AIAA, 2001), paper 2011-2886.

Kincheloe, N.

Lawrence, R. S.

Lele, S. K.

Y. Khalighi, J. W. Nichols, F. Ham, S. K. Lele, and P. Moin, “Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations,” in 17th AIAA/CEAS Aeroacoustics Conference (AIAA, 2001), paper 2011-2886.

Lloyd-Hart, M.

M. Lloyd-Hart, “Taking the twinkle out of starlight,” IEEE Spectrum 40(12), 22–29 (2003).
[Crossref]

Malley, M.

Mani, A.

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Ann. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

Mathews, E.

E. Mathews, Numerical and theoretical analysis of aero-optics with application to an optical turret, Ph.D. Thesis, University of Notre Dame (2017).

McBean, G. A.

G. A. McBean and J. Elliott, “Pressure and humidity effects on optical refractive-index fluctuations,” Bound.-Layer Meteorol. 20, 101–109 (1981).
[Crossref]

McKeon, B.

A. E. Smith, S. Gordeyev, T. Saxton-Fox, and B. McKeon, “Subsonic boundary-layer wavefront spectra for a range of Reynolds numbers,” 45th AIAA Plasmadynamics and Lasers Conference (AIAA, 2006), paper 2014-2491.

Mehta, R.

J. Bell and R. Mehta, “Development of a two-stream mixing layer from tripped and untripped boundary layers,” AIAA J. 28, 2034–2042 (1990).
[Crossref]

Moin, P.

D. You and P. Moin, “A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries,” Phys. Fluids,  19(6), 065110 (2007).
[Crossref]

Y. Khalighi, J. W. Nichols, F. Ham, S. K. Lele, and P. Moin, “Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations,” in 17th AIAA/CEAS Aeroacoustics Conference (AIAA, 2001), paper 2011-2886.

Monin, A. S.

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence (Massachusetts Institute of Technology, 1975).

Moser, R.

M. Rogers and R. Moser, “Direct simulation of a self-similar turbulent mixing layer,” Phys. Fluids 6(2), 903–923 (1996).
[Crossref]

Nichols, J. W.

Y. Khalighi, J. W. Nichols, F. Ham, S. K. Lele, and P. Moin, “Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations,” in 17th AIAA/CEAS Aeroacoustics Conference (AIAA, 2001), paper 2011-2886.

Obukhov, A. M.

A. M. Obukhov, “The structure of the temperature field in a turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. Geophys. 13, 58–69 (1949).

Pantano, C.

C. Pantano and S. Sarkar, “A study of compressibility effects in the high-speed turbulent shear layer using direct simulation,” J. Fluid Mech. 451, 329–371 (2002).
[Crossref]

Prasad, S.

Rogallo, R. S.

R. S. Rogallo, “Numerical experiments in homogeneous turbluence,” NASA Technical Memorandum 81315 (1981).

Rogers, M.

M. Rogers and R. Moser, “Direct simulation of a self-similar turbulent mixing layer,” Phys. Fluids 6(2), 903–923 (1996).
[Crossref]

Rue, J. C. La

Sarkar, S.

C. Pantano and S. Sarkar, “A study of compressibility effects in the high-speed turbulent shear layer using direct simulation,” J. Fluid Mech. 451, 329–371 (2002).
[Crossref]

Saxton-Fox, T.

A. E. Smith, S. Gordeyev, T. Saxton-Fox, and B. McKeon, “Subsonic boundary-layer wavefront spectra for a range of Reynolds numbers,” 45th AIAA Plasmadynamics and Lasers Conference (AIAA, 2006), paper 2014-2491.

Siegenthaler, J. P.

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting and Exhibit (AIAA, 2008), paper 2008-1076.
[Crossref]

Smith, A. E.

S. Gordeyev, A. E. Smith, J. A. Cress, and E. J. Jumper, “Experimental studies of aero-optical properties of subsonic turbulent boundary layers,” J. Fluid Mech. 740, 214–253 (2014).
[Crossref]

A. E. Smith, S. Gordeyev, T. Saxton-Fox, and B. McKeon, “Subsonic boundary-layer wavefront spectra for a range of Reynolds numbers,” 45th AIAA Plasmadynamics and Lasers Conference (AIAA, 2006), paper 2014-2491.

Spencer, B.

B. Spencer and B. Jones, “Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer,” in 4th Fluid and Plasma Dynamics Conference, Fluid Dynamics and Co-located Conferences (AIAA, 1971), paper 71-613.

Sutton, G. W.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Media (McGraw-Hill, 1961).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations; Reproduced by National Technical Information Service, U.S. Dept. of Commerce, 1971).

Tsuji, Y.

Y. Tsuji and T. Ishihara, “Similarity scaling of pressure fluctuation in turbulence,” Phys. Rev. E,  68, 026309 (2003).
[Crossref]

Vedula, P.

P. Vedula and P. K. Yeung, “Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence,” Phys. Fluids 11(5), 1208–1220, (1999).
[Crossref]

Vreman, A. W.

A. W. Vreman, “An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications,” Phys. Fluids 16(10), 3670–3681 (2004).
[Crossref]

Wang, K.

K. Wang and M. Wang, “Computational analysis of aero-optical distortions by flow over a cylindrical turret,” AIAA J. 54(5), 1461–1471 (2016).
[Crossref]

K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech. 696, 122–151 (2012).
[Crossref]

Wang, M.

K. Wang and M. Wang, “Computational analysis of aero-optical distortions by flow over a cylindrical turret,” AIAA J. 54(5), 1461–1471 (2016).
[Crossref]

K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech. 696, 122–151 (2012).
[Crossref]

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Ann. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

Warhaft, Z.

Z. Warhaft, “Passive scalars in turbulent flows,” Ann. Rev. Fluid Mech. 32, 203–240 (2000).
[Crossref]

Wilson, D.

D. Wilson, “Turbulence models and the synthesis of random fields for acoustic wave propagation calculations,” Army Research Lab. Tech. Rep. ARL-TR-1677 (1988).

Winarto, H.

H. Winarto and M. Davis, “Fluctuations of density, pressure, and temperature in a turbulent mixing region,” Proc. R. Soc. A 395, 203–228 (1984).
[Crossref]

Woo, R.

R. Woo and A. Ishimaru, “Effects of turbulence in a planetary atmosphere on radio occultation,” IEEE Trans. Antennas Propag. 22(4), 566–573 (1974).
[Crossref]

Yaglom, A. M.

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence (Massachusetts Institute of Technology, 1975).

Yeung, P. K.

P. Vedula and P. K. Yeung, “Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence,” Phys. Fluids 11(5), 1208–1220, (1999).
[Crossref]

Yi, S.

You, D.

D. You and P. Moin, “A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries,” Phys. Fluids,  19(6), 065110 (2007).
[Crossref]

Zhao, Y.

AIAA J. (4)

S. Gordeyev and E. J. Jumper, “Aero-optical effects of supersonic boundary layers,” AIAA J. 50(3), 682–690 (2012).
[Crossref]

A. P. Freeman and H. J. Catrakis, “Direct reduction of aero-optical aberrations by large structure suppression control in turbulence,” AIAA J. 46(10), 2582–2590 (2008).
[Crossref]

K. Wang and M. Wang, “Computational analysis of aero-optical distortions by flow over a cylindrical turret,” AIAA J. 54(5), 1461–1471 (2016).
[Crossref]

J. Bell and R. Mehta, “Development of a two-stream mixing layer from tripped and untripped boundary layers,” AIAA J. 28, 2034–2042 (1990).
[Crossref]

Ann. Rev. Fluid Mech. (3)

E. J. Jumper and S. Gordeyev, “Physics and measurement of aero-optical effects: past and present,” Ann. Rev. Fluid Mech. 49, 419–441 (2016).
[Crossref]

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Ann. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

Z. Warhaft, “Passive scalars in turbulent flows,” Ann. Rev. Fluid Mech. 32, 203–240 (2000).
[Crossref]

Appl. Opt. (1)

Bound.-Layer Meteorol. (1)

G. A. McBean and J. Elliott, “Pressure and humidity effects on optical refractive-index fluctuations,” Bound.-Layer Meteorol. 20, 101–109 (1981).
[Crossref]

IEEE Spectrum (1)

M. Lloyd-Hart, “Taking the twinkle out of starlight,” IEEE Spectrum 40(12), 22–29 (2003).
[Crossref]

IEEE Trans. Antennas Propag. (1)

R. Woo and A. Ishimaru, “Effects of turbulence in a planetary atmosphere on radio occultation,” IEEE Trans. Antennas Propag. 22(4), 566–573 (1974).
[Crossref]

Izv. Akad. Nauk. SSSR, Ser. Geogr. Geophys. (1)

A. M. Obukhov, “The structure of the temperature field in a turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. Geophys. 13, 58–69 (1949).

J. Appl. Phys. (1)

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

J. Comput. Phys. (1)

D. Bodony, “Analysis of sponge zones for computational fluid mechanics,” J. Comput. Phys. 212, 681–702 (2006).
[Crossref]

J. Fluid Mech. (4)

C. Pantano and S. Sarkar, “A study of compressibility effects in the high-speed turbulent shear layer using direct simulation,” J. Fluid Mech. 451, 329–371 (2002).
[Crossref]

K. Wang and M. Wang, “Aero-optics of subsonic turbulent boundary layers,” J. Fluid Mech. 696, 122–151 (2012).
[Crossref]

S. Gordeyev, A. E. Smith, J. A. Cress, and E. J. Jumper, “Experimental studies of aero-optical properties of subsonic turbulent boundary layers,” J. Fluid Mech. 740, 214–253 (2014).
[Crossref]

P. E. Dimotakis, H. J. Catrakis, and D. C. Fourguette, “Flow structure and optical beam propagation in high-Reynolds-number gas-phase shear layers and jets,” J. Fluid Mech. 443, 105–134 (2001).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

E. J. Fitzgerald and E. J. Jumper, “Two-dimensional optical wavefront measurements using a small-aperture beam technique-derivative instrument,” Opt. Eng. 39, 3285–3293 (2000).
[Crossref]

Opt. Express (1)

Phys. Fluids (4)

A. W. Vreman, “An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications,” Phys. Fluids 16(10), 3670–3681 (2004).
[Crossref]

D. You and P. Moin, “A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries,” Phys. Fluids,  19(6), 065110 (2007).
[Crossref]

P. Vedula and P. K. Yeung, “Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence,” Phys. Fluids 11(5), 1208–1220, (1999).
[Crossref]

M. Rogers and R. Moser, “Direct simulation of a self-similar turbulent mixing layer,” Phys. Fluids 6(2), 903–923 (1996).
[Crossref]

Phys. Rev. E (1)

Y. Tsuji and T. Ishihara, “Similarity scaling of pressure fluctuation in turbulence,” Phys. Rev. E,  68, 026309 (2003).
[Crossref]

Phys. Rev. Lett. (1)

T. Gohto and D. Fukayama, “Pressure spectrum in homogeneous turbulence,” Phys. Rev. Lett. 86(17), 3775–3778 (2001).
[Crossref]

Proc. R. Soc. A (1)

H. Winarto and M. Davis, “Fluctuations of density, pressure, and temperature in a turbulent mixing region,” Proc. R. Soc. A 395, 203–228 (1984).
[Crossref]

Prog. Aerosp. Sci. (2)

E. J. Jumper and E. J. Fitzgerald, “Recent advances in aero-optics,” Prog. Aerosp. Sci. 37, 299–339 (2001).
[Crossref]

S. Gordeyev and E. Jumper, “Fluid dynamics and aero-optics of turrets,” Prog. Aerosp. Sci. 46, 388–400 (2010).
[Crossref]

Other (12)

B. Spencer and B. Jones, “Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer,” in 4th Fluid and Plasma Dynamics Conference, Fluid Dynamics and Co-located Conferences (AIAA, 1971), paper 71-613.

R. S. Rogallo, “Numerical experiments in homogeneous turbluence,” NASA Technical Memorandum 81315 (1981).

E. Mathews, Numerical and theoretical analysis of aero-optics with application to an optical turret, Ph.D. Thesis, University of Notre Dame (2017).

V. I. Tatarski, Wave Propagation in a Turbulent Media (McGraw-Hill, 1961).

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations; Reproduced by National Technical Information Service, U.S. Dept. of Commerce, 1971).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

J. P. Siegenthaler, E. J. Jumper, and S. Gordeyev, “Atmospheric propagation vs. aero-optics,” in 46th AIAA Aerospace Sciences Meeting and Exhibit (AIAA, 2008), paper 2008-1076.
[Crossref]

A. E. Smith, S. Gordeyev, T. Saxton-Fox, and B. McKeon, “Subsonic boundary-layer wavefront spectra for a range of Reynolds numbers,” 45th AIAA Plasmadynamics and Lasers Conference (AIAA, 2006), paper 2014-2491.

G. Batchelor, The Theory of Homogeneous Turbulence (Cambridge University, 1982).

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence (Massachusetts Institute of Technology, 1975).

D. Wilson, “Turbulence models and the synthesis of random fields for acoustic wave propagation calculations,” Army Research Lab. Tech. Rep. ARL-TR-1677 (1988).

Y. Khalighi, J. W. Nichols, F. Ham, S. K. Lele, and P. Moin, “Unstructured large eddy simulation for prediction of noise issued from turbulent jets in various configurations,” in 17th AIAA/CEAS Aeroacoustics Conference (AIAA, 2001), paper 2011-2886.

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Figures (5)

Fig. 1
Fig. 1 Spanwise cross-section of the computational domain for the temporally evolving mixing layer simulations.
Fig. 2
Fig. 2 Contours of instantaneous density, ρ/ρ0, in a spanwise plane for three different lower-to-upper stream density ratios: (a) s = 1.0, (b) s = 1.05, and (c) s = 1.35.
Fig. 3
Fig. 3 Self-similar mean streamwise velocity profiles: ——, s = 1.0 simulation; oe-27-4-5670-i001, A3 Pantano & Sarkar [38]; – ○ –, Spencer & Jones [44]; oe-27-4-5670-i002, Bell & Mehta [43].
Fig. 4
Fig. 4 Root-mean-square values of (a) streamwise, (b) transverse, and (c) spanwise velocity fluctuations, and (d) the square-root of Reynolds shear stress across the s = 1.0 shear layer: ——, s = 1.0 simulation; oe-27-4-5670-i003, A3 Pantano & Sarkar [38]; oe-27-4-5670-i004, Rogers & Moser [42]; – ○ –, Spencer & Jones [44]; oe-27-4-5670-i005, Bell & Mehta [43].
Fig. 5
Fig. 5 Streamwise wavenumber spectra of (a) density at z/δθ(0) = 0 and (b) normalized optical path difference for mixing layers with (i) s = 1.0, (ii) s = 1.05 and (iii) s = 1.35. For clarity, in (a), the s = 1.05 data has been multiplied by 100.5; in (b), the s = 1.05 data has been multiplied by 100.75 and s = 1.35 data has been multiplied by 100.5. Dotted lines in each figure reflect the specified power-law slopes.

Tables (1)

Tables Icon

Table 1 Initial freestream values for shear layer simulations a

Equations (21)

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𝒰 ( x , y , L ) = 𝒰 ( x , y , 0 ) exp [ i k 0 L n ( x , y , z ) d z ] ,
n = 1 + K GD p RT .
n = K GD ρ K GD ρ ¯ ( p p ¯ T T ¯ ) .
n n ( x 0 , r ) = n ( x 0 , t ) n ( x 0 + r , t ) ¯ ,
n n ( r ) = K GD 2 ρ ρ ( r ) ( K GD ρ ¯ ) 2 ( p p ( r ) p ¯ 2 + T T ( r ) T ¯ 2 T p ( r ) p ¯ T ¯ p T ( r ) p ¯ T ¯ ) .
g h ( x 0 , r ) = g ( x 0 , t ) h ( x 0 + r , t ) ¯ .
Φ n n ( K ) = K GD 2 Φ ρ ρ ( K ) ( K GD ρ ¯ ) 2 ( Φ p p ( K ) p ¯ 2 + Φ T T ( K ) T ¯ 2 2 Re [ Φ p T ( K ) ] p ¯ T ¯ ) ,
E p p ( κ ) p ¯ 2 = B p ρ ¯ 2 ε 4 / 3 κ 7 / 3 p ¯ 2 = γ 2 a ¯ 4 B p ε 4 / 3 κ 7 / 3 ,
E T T ( κ ) T ¯ 2 = β χ ε 1 / 3 κ 5 / 3 T ¯ 2 = γ 2 a ¯ 4 R 2 β χ ε 1 / 3 κ 5 / 3 ,
Φ n n ( κ ) ( γ K GD ρ ¯ ) 2 4 π a ¯ 4 ( B p ε 4 / 3 κ 13 / 3 + R 2 β χ ε 1 / 3 κ 11 / 3 2 R ρ ¯ κ 2 Re [ E p T ( κ ) ] ) .
Φ n n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
ϕ ϕ ( x 0 , r , z ) = ϕ ( x 0 , z ) ϕ * ( x 0 + r , z ) ¯ ,
ϕ ϕ ( r , L ) = 2 π 2 k 2 L 0 κ J 0 ( κ r ) f ϕ ( κ ) Φ n n ( κ ) d κ ,
f ϕ ( κ ) = 1 + sin ( κ 2 L / k ) κ 2 L / k .
ϕ ϕ ( r , L ) = ( 2 π k ) 2 0 L Φ 1 n ( z ) [ 0 κ J 0 ( κ r ) cos 2 ( L z 2 k κ 2 ) Φ 2 n ( κ ) d κ ] d z .
E ϕ ϕ ( κ x ) = ϕ ϕ ( r x , L ) e i r x κ x d r x = 2 0 ϕ ϕ ( r x , L ) cos ( r x κ x ) d r x .
E ϕ ϕ ( κ x ) = 4 π 2 k 2 L 0 [ 0 κ x J 0 ( κ x r x ) Φ n n ( κ x ) d κ x ] cos ( r x κ x ) d r x .
Φ g g ( κ ) = C g g g 2 ¯ 3 ( 1 + κ 2 2 ) m g 3 D / 2 ,
E ϕ ϕ ( κ x ) = 2 π 5 / 2 k 2 K GD 2 C ρ ρ ρ 2 ¯ L 2 ( 1 + κ x 2 2 ) 1 m ρ 3 D 2 Γ ( ( m ρ 3 D 1 ) / 2 ) Γ ( m ρ 3 D / 2 ) .
Φ n n ( κ ) ( γ K GD ρ ¯ ) 2 4 π a ¯ 4 ( B p ε 4 / 3 13 / 3 [ 1 + κ 2 2 ] 13 / 6 + R 2 β χ ε 1 / 3 11 / 3 [ 1 + κ 2 2 ] 11 / 6 2 R ρ ¯ κ 2 Re [ E p T ( κ ) ] ) .
E ϕ ϕ ( κ x ) π ( γ K GD ρ ¯ k L 1 / 2 a ¯ 2 ) 2 ( π 2 B p ε 4 / 3 10 / 3 [ 1 + κ x 2 2 ] 5 / 3 Γ ( 5 / 3 ) Γ ( 13 / 6 ) + π 2 R 2 β χ ε 1 / 3 8 / 3 [ 1 + κ x 2 2 ] 4 / 3 Γ ( 4 / 3 ) Γ ( 11 / 6 ) 8 π R ρ ¯ 0 [ 0 J 0 ( κ x r x ) Re [ Φ p T ( κ x ) ] d κ x ] cos ( r x κ x ) d r x ) .

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