Abstract

The finite-domain direct inversion method, which was developed for use with sparse data sets, assumes smooth distributions, uses a priori information, and is well suited to the study of fluid mechanical and combustion phenomena. We successfully applied the inversion method, together with shifting functions that improve the reconstruction of distributions with nonzero values at the boundaries of their domain, to a real experimental situation and reconstructed the density distribution of methane in a nonuniform, nonreacting flow of methane and argon from projections measured optically. A point-by-point probe measurement of the methane concentration through the use of a hydrocarbon analyzer was performed to confirm the quality of the reconstruction of the optical measurement data with the inversion method.

© 1995 Optical Society of America

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References

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  1. T. D. McCay, J. A. Roux, eds., Combustion Diagnostics by Nonintrusive Methods, Vol. 92 of Progress in Astronautics and Aeronautics (American Institute of Aeronautics and Astronautics, New York, 1984).
  2. R. Goulard, ed., Combustion Measurements: Modern Techniques and Instrumentation (Academic, New York, 1976).
  3. P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
    [CrossRef]
  4. H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).
  5. S. R. Ray, H. G. Semerjian, “Laser tomography for simultaneous concentration and temperature measurements in reacting flows,” in Combustion Diagnostics by Nonintrusive Methods, T. D. McCay, J. A. Roux, eds. (American Institute of Aeronautics and Astronautics, 1984), pp. 301–324.
  6. R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet diffusion flame,” presented at The Winter Annual Meeting of The American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.
  7. T. C. Liu, W. Merzhirch, K. Oberste-Lehn, “Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density,” Exp. Fluids 7, 157– 163 (1989).
    [CrossRef]
  8. M. Ravichandran, F. C. Gouldin, “Reconstruction of smooth distributions from a limited number of projections,” Appl. Opt. 27, 4084–4097 (1988).
    [CrossRef] [PubMed]
  9. M. Ravichandran, F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption measurements,” Combust. Sci. Technol. 60, 231–248 (1988).
    [CrossRef]
  10. K. B. Chung, “Laser tomographic reconstruction of the spatial density distribution of a nonreacting flow with a small number of absorption measurements,” Ph.D. dissertation (Cornell University, Ithaca, New York, 1992).
  11. B. N. Edwards, D. E. Burch, “Absorption of 3.39 micron helium-neon laser emission by methane in the atmosphere,” J. Opt. Soc. Am. 55, 174–177 (1965).
    [CrossRef]
  12. W. G. Mallard, W. C. Gardiner, “Absorption of the 3.39 μm He-Ne laser line by methane from 300 to 2400 K,” J. Quantum Spectrosc. Radiat. Transfer 20, 135–149 (1978).
    [CrossRef]
  13. M. Ravichandran, F. C. Gouldin, “Efficient retrieval of temperature and concentration profiles from absorption measurements with error,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R.K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), paper HTD-106.

1989

T. C. Liu, W. Merzhirch, K. Oberste-Lehn, “Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density,” Exp. Fluids 7, 157– 163 (1989).
[CrossRef]

1988

M. Ravichandran, F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption measurements,” Combust. Sci. Technol. 60, 231–248 (1988).
[CrossRef]

M. Ravichandran, F. C. Gouldin, “Reconstruction of smooth distributions from a limited number of projections,” Appl. Opt. 27, 4084–4097 (1988).
[CrossRef] [PubMed]

1982

H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).

1980

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
[CrossRef]

1978

W. G. Mallard, W. C. Gardiner, “Absorption of the 3.39 μm He-Ne laser line by methane from 300 to 2400 K,” J. Quantum Spectrosc. Radiat. Transfer 20, 135–149 (1978).
[CrossRef]

1965

Burch, D. E.

Chung, K. B.

K. B. Chung, “Laser tomographic reconstruction of the spatial density distribution of a nonreacting flow with a small number of absorption measurements,” Ph.D. dissertation (Cornell University, Ithaca, New York, 1992).

Edwards, B. N.

Emmerman, P. J.

H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
[CrossRef]

Gardiner, W. C.

W. G. Mallard, W. C. Gardiner, “Absorption of the 3.39 μm He-Ne laser line by methane from 300 to 2400 K,” J. Quantum Spectrosc. Radiat. Transfer 20, 135–149 (1978).
[CrossRef]

Goulard, R.

H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
[CrossRef]

Gouldin, F. C.

M. Ravichandran, F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption measurements,” Combust. Sci. Technol. 60, 231–248 (1988).
[CrossRef]

M. Ravichandran, F. C. Gouldin, “Reconstruction of smooth distributions from a limited number of projections,” Appl. Opt. 27, 4084–4097 (1988).
[CrossRef] [PubMed]

M. Ravichandran, F. C. Gouldin, “Efficient retrieval of temperature and concentration profiles from absorption measurements with error,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R.K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), paper HTD-106.

Joklik, R. G.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet diffusion flame,” presented at The Winter Annual Meeting of The American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

Liu, T. C.

T. C. Liu, W. Merzhirch, K. Oberste-Lehn, “Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density,” Exp. Fluids 7, 157– 163 (1989).
[CrossRef]

Mallard, W. G.

W. G. Mallard, W. C. Gardiner, “Absorption of the 3.39 μm He-Ne laser line by methane from 300 to 2400 K,” J. Quantum Spectrosc. Radiat. Transfer 20, 135–149 (1978).
[CrossRef]

Merzhirch, W.

T. C. Liu, W. Merzhirch, K. Oberste-Lehn, “Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density,” Exp. Fluids 7, 157– 163 (1989).
[CrossRef]

Oberste-Lehn, K.

T. C. Liu, W. Merzhirch, K. Oberste-Lehn, “Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density,” Exp. Fluids 7, 157– 163 (1989).
[CrossRef]

Ravichandran, M.

M. Ravichandran, F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption measurements,” Combust. Sci. Technol. 60, 231–248 (1988).
[CrossRef]

M. Ravichandran, F. C. Gouldin, “Reconstruction of smooth distributions from a limited number of projections,” Appl. Opt. 27, 4084–4097 (1988).
[CrossRef] [PubMed]

M. Ravichandran, F. C. Gouldin, “Efficient retrieval of temperature and concentration profiles from absorption measurements with error,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R.K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), paper HTD-106.

Ray, S. R.

S. R. Ray, H. G. Semerjian, “Laser tomography for simultaneous concentration and temperature measurements in reacting flows,” in Combustion Diagnostics by Nonintrusive Methods, T. D. McCay, J. A. Roux, eds. (American Institute of Aeronautics and Astronautics, 1984), pp. 301–324.

Santoro, R. J.

H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
[CrossRef]

Semerjian, H. G.

H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
[CrossRef]

S. R. Ray, H. G. Semerjian, “Laser tomography for simultaneous concentration and temperature measurements in reacting flows,” in Combustion Diagnostics by Nonintrusive Methods, T. D. McCay, J. A. Roux, eds. (American Institute of Aeronautics and Astronautics, 1984), pp. 301–324.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet diffusion flame,” presented at The Winter Annual Meeting of The American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

Snyder, R. E.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet diffusion flame,” presented at The Winter Annual Meeting of The American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

Appl. Opt.

Combust. Sci. Technol.

M. Ravichandran, F. C. Gouldin, “Retrieval of asymmetric temperature and concentration profiles from a limited number of absorption measurements,” Combust. Sci. Technol. 60, 231–248 (1988).
[CrossRef]

Exp. Fluids

T. C. Liu, W. Merzhirch, K. Oberste-Lehn, “Optical tomography applied to speckle photographic measurement of asymmetric flows with variable density,” Exp. Fluids 7, 157– 163 (1989).
[CrossRef]

J. Energy

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular absorption diagnostics of a turbulent argon-methane jet,” J. Energy 4, 70–77 (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Quantum Spectrosc. Radiat. Transfer

W. G. Mallard, W. C. Gardiner, “Absorption of the 3.39 μm He-Ne laser line by methane from 300 to 2400 K,” J. Quantum Spectrosc. Radiat. Transfer 20, 135–149 (1978).
[CrossRef]

Temperature

H. G. Semerjian, R. J. Santoro, P. J. Emmerman, R. Goulard, “Laser tomography for temperature measurements in flames,” Temperature 5, 649–660 (1982).

Other

S. R. Ray, H. G. Semerjian, “Laser tomography for simultaneous concentration and temperature measurements in reacting flows,” in Combustion Diagnostics by Nonintrusive Methods, T. D. McCay, J. A. Roux, eds. (American Institute of Aeronautics and Astronautics, 1984), pp. 301–324.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet diffusion flame,” presented at The Winter Annual Meeting of The American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

K. B. Chung, “Laser tomographic reconstruction of the spatial density distribution of a nonreacting flow with a small number of absorption measurements,” Ph.D. dissertation (Cornell University, Ithaca, New York, 1992).

T. D. McCay, J. A. Roux, eds., Combustion Diagnostics by Nonintrusive Methods, Vol. 92 of Progress in Astronautics and Aeronautics (American Institute of Aeronautics and Astronautics, New York, 1984).

R. Goulard, ed., Combustion Measurements: Modern Techniques and Instrumentation (Academic, New York, 1976).

M. Ravichandran, F. C. Gouldin, “Efficient retrieval of temperature and concentration profiles from absorption measurements with error,” in Heat Transfer Phenomena in Radiation, Combustion, and Fires, R.K. Shah, ed. (American Society of Mechanical Engineers, New York, 1989), paper HTD-106.

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Figures (12)

Fig. 1
Fig. 1

Coordinate system and a projection at (s, θ). The position of a point in the plane is specified by (x, y) or (s, t). A projection is specified by its perpendicular distance (s) from the origin and its orientation (θ).

Fig. 2
Fig. 2

Two outermost strips for one view used in the generation of shifting functions; values of the distribution inside the strips are raised to correct peripheral values.

Fig. 3
Fig. 3

Optical experiment setup. See text for details.

Fig. 4
Fig. 4

Electronic and data path setup. See text for details.

Fig. 5
Fig. 5

Gas-flow and control system. In the gas-flow controller the two circles represent pressure gauges, and the eight rectangles represent rotameters.

Fig. 6
Fig. 6

Distribution measured with the hydrocarbon analyzer.

Fig. 7
Fig. 7

Residual norm versus solution norm for the experimental distribution with projections measured optically.

Fig. 8
Fig. 8

First reconstructed distribution (λ = 11) from experimental projections to be used as a starting distribution for a shifting function.

Fig. 9
Fig. 9

Shifting function generated from the first reconstructed distribution from Fig. 8.

Fig. 10
Fig. 10

Reconstructed result from experimental projections after three iterations with relaxation parameters.

Fig. 11
Fig. 11

Error plot of the reconstructed distribution in Fig. 10

Fig. 12
Fig. 12

Contour plot of the percentage error of the reconstructed distribution shown in Fig. 10

Tables (1)

Tables Icon

Table 1 Reconstructed Results Obtained with the Use of Shifting Functions and Three Iterations of Different Distributions from Optically Measured Projections

Equations (5)

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p ( s , θ ) = T T f ( x , y ) d t = T T f ( s cos θ t sin θ , s sin θ + t cos θ ) d t ,
Given p ( s i , θ j ) for i = 1 , , M , j = 1 , , N , determine f ( k Δ x , l Δ y ) subject to f Q ,
ln I ν ( S , θ ) I ν 0 ( S , θ ) = p ( S , θ ) = T T α ν ( x , y ) d t ,
C f = p ,
minimize D f p subject to f Q ,

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