Abstract

Recent laser technology provides accurate measures of the dynamics of fluids and embedded particles. For instance, the laser-extinction measurements (LEM) uses a laser beam passing across the fluid and measures the residual laser light intensity at the fluid output. The particle concentration is estimated from this measurement. However, the particle flow is submitted to random time-varying fluctuations. This study thus proposes to model the received intensity by an appropriate random process. This paper first models the particle flow by a queueing process. Second, the measured intensity power spectrum is derived according to this random model. Finally, the simple case of a constant particle velocity is developped. The proposed model allows to generalize results previously obtained in the litterature with simplified models. Moreover, the particle celerity estimate is provided.

© 2006 Optical Society of America

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References

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  1. R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids,  5, 17–28, (1987).
  2. A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. 25, No. 10, 689–691, (2000).
    [CrossRef]
  3. A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
    [CrossRef]
  4. D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998.
  5. N. Johnson and S. Koltz, Discrete distributions, Houghton mifflin Co.1969.
  6. B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. 232, 83–90, (2004).
    [CrossRef]
  7. K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
    [CrossRef]
  8. E. Lukacs, Characteristic Functions, Griffin, London, 1970.
  9. M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame,  141, 371–391, (2005).
    [CrossRef]
  10. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991.

2005 (2)

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame,  141, 371–391, (2005).
[CrossRef]

2004 (1)

B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. 232, 83–90, (2004).
[CrossRef]

2000 (1)

1999 (1)

A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
[CrossRef]

1987 (1)

R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids,  5, 17–28, (1987).

Adrian, R. J.

R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids,  5, 17–28, (1987).

Chen, A.

A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. 25, No. 10, 689–691, (2000).
[CrossRef]

A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
[CrossRef]

Chung, J.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

Gross, D.

D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998.

Han, Y.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

Hao, J.

A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. 25, No. 10, 689–691, (2000).
[CrossRef]

A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
[CrossRef]

Harris, C. M.

D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998.

He, K.

Johnson, N.

N. Johnson and S. Koltz, Discrete distributions, Houghton mifflin Co.1969.

Koltz, S.

N. Johnson and S. Koltz, Discrete distributions, Houghton mifflin Co.1969.

Lacaze, B.

B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. 232, 83–90, (2004).
[CrossRef]

Lee, C.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

Lee, K.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

Lee, W.

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

Lukacs, E.

E. Lukacs, Characteristic Functions, Griffin, London, 1970.

Musculus, M.

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame,  141, 371–391, (2005).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991.

Pickett, L.

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame,  141, 371–391, (2005).
[CrossRef]

Yao, C. S.

R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids,  5, 17–28, (1987).

Zhou, Z.

A. Chen, J. Hao, Z. Zhou, and K. He, “Particle concentration measured from light fluctuations,” Opt. Lett. 25, No. 10, 689–691, (2000).
[CrossRef]

A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
[CrossRef]

Zu, J.

A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
[CrossRef]

Combustion and Flame (1)

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in highpressure transient combustion environments,” Combustion and Flame,  141, 371–391, (2005).
[CrossRef]

Exp. in Fluids (1)

R. J. Adrian and C. S. Yao, “Power spectra of fluid velocities measured by laser Doppler velocimetry,” Exp. in Fluids,  5, 17–28, (1987).

Meas. Sci. Technol. (1)

K. Lee, Y. Han, W. Lee, J. Chung, and C. Lee, “Quantitative measurements of soot particles in a laminar diffusion flame using LII/LIS technique,” Meas. Sci. Technol. 16, 519–528, (2005).
[CrossRef]

Opt. Commun. (2)

A. Chen, J. Hao, Z. Zhou, and J. Zu, “Theoretical solutions for particular scintillation monitors,” Opt. Commun. 166, 15–20, (1999).
[CrossRef]

B. Lacaze, “Spectral properties of scattered light fluctuations,” Opt. Commun. 232, 83–90, (2004).
[CrossRef]

Opt. Lett. (1)

Other (4)

A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991.

D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Wiley, 1998.

N. Johnson and S. Koltz, Discrete distributions, Houghton mifflin Co.1969.

E. Lukacs, Characteristic Functions, Griffin, London, 1970.

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Figures (4)

Fig. 1.
Fig. 1.

Laser extinction measurement system

Fig. 2.
Fig. 2.

Approached Lorentzian intensity power spectrum

Fig. 3.
Fig. 3.

Constant celerity model notations

Fig. 4.
Fig. 4.

Continuous part of the intensity power spectrum-Constant celerity case

Equations (33)

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I = I 0 k = 1 N ( 1 A k ) A k = B k 𝓢 1 < 1
m 1 B = E [ B k ] , m 2 B = E [ B k 2 ] m 1 A = m 1 B 𝓢 1 , m 2 A = m 2 B 𝓢 1 2
I ( t ) = I 0 k J t ( 1 A k )
J t = { n , t n < t , t n t }
E [ t n + 1 t n ] = 1 λ
G ( x ) = Pr [ t n t n < x ]
m 1 I = E [ I ( t ) ] K I ( τ ) = E [ I ( t ) I ( t + τ ) ]
K I ( τ ) = I 0 2 exp [ λ ( m ( 2 m 1 A m 2 A ) + m 2 A 0 τ ( 1 G ( u ) ) du ) ]
lim τ K I ( τ ) = m 1 I 2
{ m 1 I = I 0 exp [ λ m m 1 A ] m 2 I = K I ( 0 ) = I 0 2 exp [ λ m ( m 2 A 2 m 1 A ) ]
K I ( τ λ ) exp [ λ m ( m 2 A 2 m 1 A ) ] I 0 2 exp [ m 2 A τ ]
Δ = var [ I ] m 1 I 2 = m 2 I m 1 I 2 m 1 I 2
Δ = exp [ λ m m 2 A ] 1
Δ λ m m 2 A
s I ( ω ) = m 1 I 2 δ ( ω ) + 1 π 0 [ K I ( τ ) m 1 I 2 ] cos ω τ d τ
s I ( ω ) = I 0 2 e 2 λ m m 1 A [ δ ( ω ) + 1 π 0 ( e λ m 2 A ( τ ( 1 G ( u ) du ) ) 1 ) cos ω τ d τ ] .
s I ( ω ) I 0 2 e λ ( m ' ( 2 m 1 A m 2 A ) ) π λ m 2 A ω 2 + ( λ m 2 A ) 2
λ = ρ vlL
m = E [ E [ t n t n X ] ] = E [ 1 v ( f 2 ( X ) f 1 ( X ) ) ]
m = 1 v 0 l ( f 2 ( x ) f 1 ( x ) ) dx l = 𝓢 1 lv
Δ = m 2 I m 1 I 2 m 1 I 2 = exp [ ρ L 𝓢 1 m 2 B ] 1 ρ L 𝓢 1 m 2 B .
K I ( τ ) = { I 0 2 exp [ ρ L ( 2 m 1 B + 1 𝓢 1 2 m 2 B α ( τ ) ) ] , 0 < τ < h v I 0 2 exp [ 2 ρ L m 1 B ] , τ > h v
s I ( ω ) = I 0 2 e 2 ρ L m 1 B [ δ ( ω ) + 1 π 0 h v ( exp { ρ Ll 𝓢 1 2 m 2 B α ( τ ) } 1 ) cos ω τ d τ ]
K I ( τ ) = { I 0 2 exp [ ρ L { 2 m 1 B + m 2 B ( 1 lv τ ) } ] τ < 1 lv I 0 2 exp [ 2 ρ L m 1 B ] τ > 1 lv
s I ( ω ) = I 0 2 e 2 d [ δ ( ω ) + 1 π 1 lv g ( ω lv ) ] with { g ( ω ) = sin ω ω + d e d d cos ω + ω sin ω ω 2 + d 2 d = ρ L m 2 B
{ J t = { n , t n < t , t n t } B = J t J t + τ = { n , t n < t , t n t + τ } C = J t J ¯ t + τ = { n , t n < t , t t n < t + τ } D = J ¯ t J t + τ = { n , t t n < t + τ , t n t + τ } N ( t , τ ) = { n , t t n < t + τ }
E [ I ( t ) I ( t + τ ) ] = E [ k B ( 1 A k ) 2 j C ( 1 A j ) m D ( 1 A m ) ]
E [ I ( t ) I ( t + τ ) ] = E [ ( m 2 A 2 m 1 A + 1 ) B ( 1 m 1 A ) C ] E [ ( 1 m 1 A ) D ] .
{ b t = 1 t 0 t Pr [ t k t k > t + τ u ] du = 1 t 0 t [ 1 G ( u + τ ) ] du c t = 1 t 0 t Pr [ t u < t k t k < t + τ u ] du = 1 t 0 t [ G ( u + τ ) G ( u ) ] du
d = 1 τ t t + τ Pr [ t k t k > t + τ u ] du = 1 τ 0 τ [ 1 G ( u ) ] du
K I ( τ ) = E [ ( ( m 2 A 2 m 1 A ) b t m 1 A c t + 1 ) N ( 0 , t ) ] E [ ( 1 m 1 A d ) N ( t , τ ) ]
K I ( τ ) = exp [ λ t ( ( m 2 A 2 m 1 A ) b t m 1 A c t ) m 1 A λ τ d ]
{ K I ( τ ) = exp [ α + β τ ( 1 G ( u ) ) du ] m = E [ t k t k ] G ( u ) = Pr [ t k t k < u ] α = 2 λ m m 1 A β = λ m 2 A

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