Abstract

A lateral shearing interferometer was used to measure the slope of perturbed wave fronts after propagating through free turbulent mixing layers. Shearing interferometers provide a two-dimensional flow visualization that is nonintrusive. Slope measurements were used to reconstruct the phase of the turbulence-corrupted wave front. The random phase fluctuations induced by the mixing layer were captured in a large ensemble of wave-front measurements. Experiments were performed on an unbounded, plane shear mixing layer of helium and nitrogen gas at fixed velocities and high Reynolds numbers for six locations in the flow development. Statistical autocorrelation functions and structure functions were computed on the reconstructed phase maps. The autocorrelation function results indicated that the turbulence-induced phase fluctuations were not wide-sense stationary. The structure functions exhibited statistical homogeneity, indicating that the phase fluctuations were stationary in first increments. However, the turbulence-corrupted phase was not isotropic. A five-thirds power law is shown to fit orthogonal slices of the structure function, analogous to the Kolmogorov model for isotropic turbulence. Strehl ratios were computed from the phase structure functions and compared with classical estimates that assume isotropy. The isotropic models are shown to overestimate the optical degradation by nearly 3 orders of magnitude compared with the structure function calculations.

© 1996 Optical Society of America

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References

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1994

1992

1990

L. Chew, W. Christiansen, “Coherent structure effects on the optical performance of plane shear layers,” AIAA J. 29, 76–80 (1990).
[CrossRef]

1985

1983

1980

1977

1974

G. L. Brown, A. Roshko, “On density effects and large structure in turbulent mixing layers,” J. Fluid Mech. 64, 775–816 (1974).
[CrossRef]

1965

Arnold, R.

Barrett, T.

Brown, G. L.

G. L. Brown, A. Roshko, “On density effects and large structure in turbulent mixing layers,” J. Fluid Mech. 64, 775–816 (1974).
[CrossRef]

Chew, L.

L. Chew, W. Christiansen, “Coherent structure effects on the optical performance of plane shear layers,” AIAA J. 29, 76–80 (1990).
[CrossRef]

Christiansen, W.

L. Chew, W. Christiansen, “Coherent structure effects on the optical performance of plane shear layers,” AIAA J. 29, 76–80 (1990).
[CrossRef]

Cuellar, L.

Dimotakis, P. E.

P. E. Dimotakis, “Turbulent free shear mixing layer and combustion,” in High-Speed Flight Propulsion Systems, Vol. 137 of Progress in Astronautics and Aeronautics, S. N. Murthy, E. T. Curran, eds. (American Institute of Aeronautics and Astronautics, 1991), Chap. 5, pp. 265–340.

Fried, D. L.

Gilbert, K. G.

K. G. Gilbert, L. J. Otten, W. C. Rose, “Atmospheric propagation of radiation,” in Infrared and Electro-Optical Systems Handbook, J. S. Accetta, D. L. Shumaker, eds., Vol. PM10 of SPIE Press Monographs Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), Chap. 3, pp. 233–285.

K. G. Gilbert, “Overview of aero-optics,” in Aero-Optical Phenomena, K. Gilbert, L. Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), Vol. 80, pp. 1–9.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, pp. 361–392.

Grechko, G. M.

Gurvich, A. S.

Havener, G.

G. Havener, “Optical wave front variance: a study on analytical models in use today,” presented at the 30th Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA-92-0654.

Hesselink, L.

L. Hesselink, “Optical tomography,” in Handbook of Flow Visualization, W. J. Yang, ed. (Hemisphere, New York, 1989), pp. 307–328.

Ho, C. M.

C. M. Ho, P. Huerre, “Perturbed free shear layers,” in Annual Review of Fluid Mechanics (Annual Reviews, Palo Alto, Calif., 1984), Vol. 16, pp. 365–424.
[CrossRef]

Hudgin, R. H.

Huerre, P.

C. M. Ho, P. Huerre, “Perturbed free shear layers,” in Annual Review of Fluid Mechanics (Annual Reviews, Palo Alto, Calif., 1984), Vol. 16, pp. 365–424.
[CrossRef]

Johnson, P.

Kan, V.

Kay, S. M.

S. M. Kay, Modern Spectral Estimation (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 15, pp. 489–491.

Lefebvre, M.

Magee, E. P.

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[CrossRef]

E. P. Magee, “Characterization of laboratory generated turbulence by optical phase measurements,” M.S. Thesis (U.S. Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1993).

Mahajan, V. N.

Matson, C. L.

Monin, A. S.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, pp. 337–368.

Murty, M.

M. Murty, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 4, pp. 105–148.

Otten, L. J.

K. G. Gilbert, L. J. Otten, W. C. Rose, “Atmospheric propagation of radiation,” in Infrared and Electro-Optical Systems Handbook, J. S. Accetta, D. L. Shumaker, eds., Vol. PM10 of SPIE Press Monographs Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), Chap. 3, pp. 233–285.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991), Chap. 8, pp. 188–189.

Rego, A.

Roggemann, M. C.

Rose, W. C.

K. G. Gilbert, L. J. Otten, W. C. Rose, “Atmospheric propagation of radiation,” in Infrared and Electro-Optical Systems Handbook, J. S. Accetta, D. L. Shumaker, eds., Vol. PM10 of SPIE Press Monographs Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), Chap. 3, pp. 233–285.

Roshko, A.

G. L. Brown, A. Roshko, “On density effects and large structure in turbulent mixing layers,” J. Fluid Mech. 64, 775–816 (1974).
[CrossRef]

Sandler, D. G.

Savchenko, S. A.

Smith, G.

Sokolovskii, S. V.

Southwell, W. H.

Spivey, B.

Steinmetz, W. J.

W. J. Steinmetz, “Second moments of optical degradation due to a thin turbulent layer,” in Aero-Optical Phenomena, K. Gilbert, L. Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), Vol. 80, pp. 78–100.

Streeter, V. L.

V. L. Streeter, Handbook of Fluid Dynamics, 1st ed. (McGraw-Hill, New York, 1961), Chap. 10, pp. 3–32.

Sutton, G. W.

G. W. Sutton, “Aero-optical foundations and applications,” AIAA J. 23, 1525–1537 (1985).
[CrossRef]

G. W. Sutton, “Optical imaging through aircraft turbulent boundary layers,” in Aero-Optical Phenomena, K. Gilbert, L. Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), Vol. 80, pp. 15–39.

B. T. Vu, G. W. Sutton, “Laser beam degradation through optically turbulent mixing layers,” presented at the AIAA 13th Fluid and Plasmadynamics Conference, Snowmass, Colo., 14–16 July 1980, paper AIAA-80-1414.

Taylor, G.

Truman, C. R.

C. R. Truman, “The influence of turbulent structure on optical phase distortion through turbulent shear flows,” presented at the AIAA SDIO Annual Interceptor Technology Conference, Huntsville, Ala., 19–21 May 1992, paper AIAA-92-2817.

Vu, B. T.

B. T. Vu, G. W. Sutton, “Laser beam degradation through optically turbulent mixing layers,” presented at the AIAA 13th Fluid and Plasmadynamics Conference, Snowmass, Colo., 14–16 July 1980, paper AIAA-80-1414.

Welsh, B. M.

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[CrossRef]

White, F. M.

F. M. White, Viscous Fluid Flow, 2nd ed. (McGraw-Hill, New York, 1991), Chap. 6, p. 470.

Wissler, J. B.

J. B. Wissler, “Transmission of thin light beams through turbulent mixing layers,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1992).

Yaglom, A. M.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, pp. 337–368.

AIAA J.

G. W. Sutton, “Aero-optical foundations and applications,” AIAA J. 23, 1525–1537 (1985).
[CrossRef]

L. Chew, W. Christiansen, “Coherent structure effects on the optical performance of plane shear layers,” AIAA J. 29, 76–80 (1990).
[CrossRef]

J. Fluid Mech.

G. L. Brown, A. Roshko, “On density effects and large structure in turbulent mixing layers,” J. Fluid Mech. 64, 775–816 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

E. P. Magee, B. M. Welsh, “Characterization of laboratory-generated turbulence by optical phase measurements,” Opt. Eng. 33, 3810–3817 (1994).
[CrossRef]

Other

B. T. Vu, G. W. Sutton, “Laser beam degradation through optically turbulent mixing layers,” presented at the AIAA 13th Fluid and Plasmadynamics Conference, Snowmass, Colo., 14–16 July 1980, paper AIAA-80-1414.

J. B. Wissler, “Transmission of thin light beams through turbulent mixing layers,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1992).

M. Murty, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 4, pp. 105–148.

E. P. Magee, “Characterization of laboratory generated turbulence by optical phase measurements,” M.S. Thesis (U.S. Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1993).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991), Chap. 8, pp. 188–189.

S. M. Kay, Modern Spectral Estimation (Prentice-Hall, Englewood Cliffs, N.J., 1988), Chap. 15, pp. 489–491.

C. M. Ho, P. Huerre, “Perturbed free shear layers,” in Annual Review of Fluid Mechanics (Annual Reviews, Palo Alto, Calif., 1984), Vol. 16, pp. 365–424.
[CrossRef]

G. W. Sutton, “Optical imaging through aircraft turbulent boundary layers,” in Aero-Optical Phenomena, K. Gilbert, L. Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), Vol. 80, pp. 15–39.

G. Havener, “Optical wave front variance: a study on analytical models in use today,” presented at the 30th Aerospace Sciences Meeting and Exhibit, Reno, Nev., 6–9 January 1992, paper AIAA-92-0654.

F. M. White, Viscous Fluid Flow, 2nd ed. (McGraw-Hill, New York, 1991), Chap. 6, p. 470.

V. L. Streeter, Handbook of Fluid Dynamics, 1st ed. (McGraw-Hill, New York, 1961), Chap. 10, pp. 3–32.

L. Hesselink, “Optical tomography,” in Handbook of Flow Visualization, W. J. Yang, ed. (Hemisphere, New York, 1989), pp. 307–328.

K. G. Gilbert, L. J. Otten, W. C. Rose, “Atmospheric propagation of radiation,” in Infrared and Electro-Optical Systems Handbook, J. S. Accetta, D. L. Shumaker, eds., Vol. PM10 of SPIE Press Monographs Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), Chap. 3, pp. 233–285.

K. G. Gilbert, “Overview of aero-optics,” in Aero-Optical Phenomena, K. Gilbert, L. Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), Vol. 80, pp. 1–9.
[CrossRef]

W. J. Steinmetz, “Second moments of optical degradation due to a thin turbulent layer,” in Aero-Optical Phenomena, K. Gilbert, L. Otten, eds. (American Institute of Aeronautics and Astronautics, 1982), Vol. 80, pp. 78–100.

C. R. Truman, “The influence of turbulent structure on optical phase distortion through turbulent shear flows,” presented at the AIAA SDIO Annual Interceptor Technology Conference, Huntsville, Ala., 19–21 May 1992, paper AIAA-92-2817.

P. E. Dimotakis, “Turbulent free shear mixing layer and combustion,” in High-Speed Flight Propulsion Systems, Vol. 137 of Progress in Astronautics and Aeronautics, S. N. Murthy, E. T. Curran, eds. (American Institute of Aeronautics and Astronautics, 1991), Chap. 5, pp. 265–340.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8, pp. 361–392.

A. S. Monin, A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (MIT, Cambridge, Mass., 1975), Vol. 2, Chap. 8, pp. 337–368.

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Figures (13)

Fig. 1
Fig. 1

Shear layer turbulence generator: (a) side view, (b) top view.

Fig. 2
Fig. 2

(a) Orientation of the turbulence chamber and an aperture location as viewed from the side. (b) Laser propagation through the shear layer as viewed from the top. The N2–He mixing layer is in the center of the flow.

Fig. 3
Fig. 3

Reconstructed wave-front phase. Deterministic optical system effects have been removed. Flow is in the +x direction, propagation is in the +z direction. Sample spacing in x and y is 95 and 105 μm, respectively.

Fig. 4
Fig. 4

Surface contour of one phase realization, ϕ′(x, y), with an overlay of 81 anchor points. Each cross indicates an anchor point. Sample spacing in x and y is 95 and 105 μm, respectively. Spacing between each anchor point is eight samples.

Fig. 5
Fig. 5

Surface contour of the ACF for a single anchor point. Each unit of Δx and Δy is 95 and 105 μm, respectively.

Fig. 6
Fig. 6

Surface contour of the structure function for a single anchor point. Each unit of Δx and Δy is 95 and 105 μm, respectively.

Fig. 7
Fig. 7

Average structure function surface contour for 59 anchor points. Aperture is centered at 5.75 cm in the flow. Contours are in units of waves2. Each unit of Δx and Δy is 95 and 105 μm, respectively.

Fig. 8
Fig. 8

Orthogonal slices of D ˜ ϕ ( Δ x , Δ y ). Upper curve: D ˜ ϕ ( Δ x , 0 ), overlaid (×) with αxx|5/3 for αx = 0.0078 waves2 samples−5/3. Lower curve: D ˜ ϕ ( 0 , Δ y ), overlaid (○) with αyy|5/3 for αy = 0.0027 waves2 samples −5/3.

Fig. 9
Fig. 9

Average structure function surface contours for 59 anchor points at four aperture locations in the turbulent mixing layer. Contours are in units of waves2. Each unit of Δx and Δy is 95 and 105 μm, respectively. Aperture is centered at (a) 8.25 cm, (b) 10.75 cm, (c) 18.75 cm, (d) 28.75 cm from the exit of the turbulence generator.

Fig. 10
Fig. 10

Plot of coefficients αx and αy as a function of the six locations in the flow. Units are waves2 samples−5/3.

Fig. 11
Fig. 11

Coherence parameters x0 and y0 as a function of downstream location. The ordinate is on a log10 scale.

Fig. 12
Fig. 12

Structure function surface approximation, using coherence dimensions x0 and y0, for the aperture centered at 5.75 cm: (a) surface contour, (b) difference between actual and approximation. Each unit of Δx and Δy is 95 and 105 μm, respectively.

Fig. 13
Fig. 13

SR calculations as a function of downstream location. The ordinate is on a log10 scale.

Tables (1)

Tables Icon

Table 1 Summary of Fluid Parameters for the Free Mixing Layers

Equations (25)

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

σ ϕ 2 = 2 G 2 k 2 0 δ σ ρ 2 ( z ) l z ( z ) d z ,
R ϕ ( x 1 , y 1 ; x 2 , y 2 ) = E [ ϕ ( x 1 , y 1 ) ϕ ( x 2 , y 2 ) ] ,
Φ n ( κ ) = 0 . 033 C n 2 κ 11 / 3 ,
D ϕ ( x 1 , y 1 ; x 2 , y 2 ) = E { [ ϕ ( x 1 , y 1 ) ϕ ( x 2 , y 2 ) ] 2 } .
H ¯ s ( ν x , ν y ) = exp [ 1 2 D ϕ ( λ ¯ f ν x , λ ¯ f ν y ) ] ,
H ¯ ( ν x , ν y ) = H 0 ( ν x , ν y ) H ¯ s ( ν x , ν y ) ,
D ϕ ( r ) = 2 . 91 ( 2 π λ ¯ ) 2 C n 2 z r 5 / 3 ,
s ¯ ( x , y ) = F 1 [ H ¯ ( ν x , ν y ) ] ,
I I 0 = s ¯ ( x a , y a ) s 0 ( x a , y a ) ,
I I 0 = exp ( σ ϕ 2 ) .
I I 0 = exp ( 0 . 5 k 2 α 2 Δ n 2 δ 2 ) ,
Re x = x Δ U ν ,
δ ( x ) x C δ ( 1 U r ) ( 1 + ρ r ) 2 ( 1 + U r ρ r ) × [ 1 ( 1 ρ r ) / ( 1 + ρ r ) 1 + 2 . 9 ( 1 + U r ) / ( 1 U r ) ] ,
0 . 25 C δ 0 . 45 .
Re δ = δ ( x ) Re x x .
ϕ meas ( x , y ) = ϕ avg ( x , y ) + ϕ ( x , y ) ,
SNR n = n [ f ¯ ( v ¯ ) 1 / 2 ] ,
f ¯ = 1 n i = 1 n f i ,
v ¯ = 1 n 1 i = 1 n ( f i f ¯ ) 2
R ϕ ( x , y ; x + Δ x , y + Δ y ) = 1 N i = 1 N ϕ i ( x , y ) ϕ i ( x + Δ x , y + Δ y ) ,
D ˆ ϕ ( x , y ; x + Δ x , y + Δ y ) = 1 N i = 1 N [ ϕ i ( x , y ) ϕ i ( x + Δ x , y + Δ y ) ] 2 .
D ϕ ( r ) = 6 . 88 ( r r 0 ) 5 / 3 ,
D ˜ ϕ ( Δ x , 0 ) = 6 . 88 ( | Δ x | x 0 ) 5 / 3 , D ˜ ϕ ( 0 , Δ y ) = 6 . 88 ( | Δ y | y 0 ) 5 / 3 ,
x 0 = ( α x 6 . 88 ) 3 / 5 , y 0 = ( α y 6 . 88 ) 3 / 5 ,
D ˜ ϕ ( Δ x , Δ y ) 6 . 88 [ ( Δ x x 0 ) 2 + ( Δ y y 0 ) 2 ] 5 / 6 .

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