Abstract

A lateral shearing interferometer was used to measure the slope of perturbed wave fronts after they propagated through a He–N2 mixing layer in a rectangular channel. 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. Phase structure functions, computed from the reconstructed phase surfaces, were stationary in first increments. A five-thirds power law is shown to fit streamwise and cross-stream slices of the structure function, analogous to the Kolmogorov model for isotropic turbulence, which describes the structure function with a single parameter. Strehl ratios were computed from the phase structure functions and compared with a measured experiment obtained from simultaneous point-spread function measurements. Two additional Strehl ratios were calculated by using classical estimates that assume statistical isotropy throughout the flow. The isotropic models are a reasonable estimate of the optical degradation only within a few centimeters of the initial mixing, where the Reynolds number is low. At higher Reynolds numbers, Strehl ratios calculated from the structure functions match the experiment much better than Strehl ratio calculations that assume isotropic flow.

© 1997 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

1996 (2)

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
[CrossRef] [PubMed]

1995 (1)

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

1994 (1)

1993 (1)

1990 (1)

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

1985 (1)

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

1983 (1)

1977 (1)

1966 (1)

Arnold, R.

Barrett, T.

Bishop, K.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Bowersox, R. D.

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
[CrossRef] [PubMed]

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

Chen, E.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[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]

L. Chew, W. H. Christiansen, “Experimental investigation of free shear layer optics,” presented at the 22nd AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hi., 24–26 June 1991, paper AIAA 91-1722.

Christiansen, W.

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

Christiansen, W. H.

L. Chew, W. H. Christiansen, “Experimental investigation of free shear layer optics,” presented at the 22nd AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hi., 24–26 June 1991, paper AIAA 91-1722.

Clark, N.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[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, Washington, D.C., 1991), Chap. 5, pp. 265–340.

Fried, D. L.

Gardner, P. J.

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
[CrossRef] [PubMed]

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6, pp. 101–140.

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

Hammoud, A. M.

Hudgin, R. H.

Hugo, R. J.

E. J. Jumper, R. J. Hugo, “Optical phase distortion due to turbulent-fluid density fields: quantification using the small-aperture beam technique,” presented at the 23rd AIAA Plasma Dynamics and Lasers Conference, Nashville, Tenn., 6–8 July 1992, paper AIAA 92-3020.

Jewell, D. W.

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

Johnson, P.

Jumper, E. J.

E. J. Jumper, R. J. Hugo, “Optical phase distortion due to turbulent-fluid density fields: quantification using the small-aperture beam technique,” presented at the 23rd AIAA Plasma Dynamics and Lasers Conference, Nashville, Tenn., 6–8 July 1992, paper AIAA 92-3020.

Lefebvre, M.

Luke, T. E.

Magee, E. P.

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.

Masson, B.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

McMackin, L.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

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.

Pierson, R.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Rego, A.

Roggemann, M. C.

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
[CrossRef] [PubMed]

Sandler, D. G.

Smith, G.

Snyder, D. L.

Spivey, B.

Sutton, G. W.

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

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

Taylor, G.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 1, pp. 5–15.

Vu, B. T.

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

Welsh, B. M.

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, T. E. Luke, “Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers,” Appl. Opt. 35, 4879–4889 (1996).
[CrossRef] [PubMed]

White, R. L.

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. (3)

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

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wave front sensor studies of dynamic organized structure in flowfields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

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

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

Measurement (1)

M. C. Roggemann, P. J. Gardner, B. M. Welsh, R. D. Bowersox, D. W. Jewell, “Gas flow visualization by means of sheared beam interferometry: sensitivity, maximum measurable gradient of the density fluctuation, and integrated density estimation,” Measurement 17, 251–265 (1996).
[CrossRef]

Other (12)

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, Washington, D.C., 1991), Chap. 5, pp. 265–340.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), Chap. 1, pp. 5–15.

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).

L. Chew, W. H. Christiansen, “Experimental investigation of free shear layer optics,” presented at the 22nd AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hi., 24–26 June 1991, paper AIAA 91-1722.

E. J. Jumper, R. J. Hugo, “Optical phase distortion due to turbulent-fluid density fields: quantification using the small-aperture beam technique,” presented at the 23rd AIAA Plasma Dynamics and Lasers Conference, Nashville, Tenn., 6–8 July 1992, paper AIAA 92-3020.

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

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

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968), Chap. 6, pp. 101–140.

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

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

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

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Figures (13)

Fig. 1
Fig. 1

Optical bench used to obtain phase measurements and PSF’s for a laser beam propagating through the turbulent mixing layer.

Fig. 2
Fig. 2

(a) Side view of the orientation of the turbulence chamber and an aperture location. (b) Top view of the laser propagation through the shear layer. The He–N2 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 is 95 µm; sample spacing in y is 105 µm.

Fig. 4
Fig. 4

PSF images for the turbulent mixing layer experiments: (a) reference PSF (no turbulence), (b) single realization, (c) average PSF taken over 160 independent realizations. The aperture is centered at 8.25 cm.

Fig. 5
Fig. 5

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 is 95 µm; sample spacing in y is 105 µm. Spacing between each anchor point is five samples.

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 81 anchor points. The aperture is centered at 8.25 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 ϕx, Δy).

Fig. 9
Fig. 9

Average structure function surface contours for 81 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. The aperture is centered at (a) 1.25, (b) 2.25, (c) 3.75, (d) 5.75 cm from the exit of the turbulence generator.

Fig. 10
Fig. 10

Average structure function surface contours for 81 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. The aperture is centered at (a) 8.25, (b) 10.75, (c) 13.25, (d) 15.75 cm from the exit of the turbulence generator.

Fig. 11
Fig. 11

Plot of α x and α y as a function of the eight locations in the flow. Units are waves2 samples-5/3.

Fig. 12
Fig. 12

Coherence parameters x 0 and y 0 as a function of downstream location. The ordinate is log10 scale.

Fig. 13
Fig. 13

Relative SR calculations as a function of downstream location.

Equations (17)

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Dϕx1, y1; x2, y2=Eϕx1, y1-ϕx2, y22,
¯sνx, νy=exp-12 Dϕλ¯fνx, λ¯fνy,
¯νx, νy=0νx, νy¯sνx, νy,
s¯x, y=-1¯νx, νy,
II0=s¯xa, yas0xa, ya,
SR=- νx, νy|aberrateddνxdνy- νx, νy|unaberrateddνxdνy.
II0=exp-σϕ2.
II0=exp-0.5k2α2Δn2δ2,
Rex=xΔUν,
δxxCδ1-Ur1+ρr21+Urρr×1-1-ρr/1+ρr1+2.91+Ur/1-Ur,
0.25Cδ0.45.
Reδ=δxRexx.
ϕmeasx, y=ϕavgx, y+ϕx, y,
D˜ϕx, y; x+Δx, y+Δy=1Ni=1Nϕix, y-ϕi×x+Δx, y+Δy2,
D˜ϕΔx, 0=6.88Δxx05/3,  D˜ϕ0, Δy=6.88Δyy05/3,
x0=αx6.88-3/5,  y0=αy6.88-3/5,
D˜ϕΔx, Δy6.88Δxx02+Δyy025/6,

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