Abstract

We describe a novel stereocamera for particle image velocimetry (PIV) applications that incorporates the Scheimpflug condition that the object plane, lens plane, and image plane must be collinear. We examined the governing equations for this system using a computer-based sensitivity analysis to predict the accuracy of the in-plane and out-of-plane measurement. We evaluated the performance of the Scheimpflug PIV system with a three-dimensional uniform translation test. Results indicate that the Scheimpflug PIV stereocamera performs as expected. The larger off-axis angles possible with the Scheimpflug system can provide a higher accuracy in the out-of-plane component when compared with a translation PIV stereocamera.

© 1997 Optical Society of America

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References

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  1. A. K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
    [CrossRef]
  2. A. K. Prasad, K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Opt. 34, 7092–7099 (1995).
    [CrossRef] [PubMed]
  3. R. E. Altenhofen, “Rectification,” in Manual of Photogrammetry (American Society of Photogrammetry, Washington, D.C., 1952), p. 457.
  4. A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
    [CrossRef]
  5. A. Boillot, A. K. Prasad, “Optimization procedure for pulse separation in cross-correlation PIV,” Exp. Fluids 21, 87–93 (1996).
    [CrossRef]
  6. R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Ann. Rev. Fluid Mech. 23, 261–304 (1991).
    [CrossRef]

1996 (1)

A. Boillot, A. K. Prasad, “Optimization procedure for pulse separation in cross-correlation PIV,” Exp. Fluids 21, 87–93 (1996).
[CrossRef]

1995 (1)

1993 (1)

A. K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

1992 (1)

A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
[CrossRef]

1991 (1)

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Ann. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Adrian, R. J.

A. K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
[CrossRef]

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Ann. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Altenhofen, R. E.

R. E. Altenhofen, “Rectification,” in Manual of Photogrammetry (American Society of Photogrammetry, Washington, D.C., 1952), p. 457.

Boillot, A.

A. Boillot, A. K. Prasad, “Optimization procedure for pulse separation in cross-correlation PIV,” Exp. Fluids 21, 87–93 (1996).
[CrossRef]

Jensen, K.

Landreth, C. C.

A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
[CrossRef]

Offutt, P. W.

A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
[CrossRef]

Prasad, A. K.

A. Boillot, A. K. Prasad, “Optimization procedure for pulse separation in cross-correlation PIV,” Exp. Fluids 21, 87–93 (1996).
[CrossRef]

A. K. Prasad, K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Opt. 34, 7092–7099 (1995).
[CrossRef] [PubMed]

A. K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
[CrossRef]

Ann. Rev. Fluid Mech. (1)

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Ann. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Appl. Opt. (1)

Exp. Fluids (3)

A. K. Prasad, R. J. Adrian, C. C. Landreth, P. W. Offutt, “Effect of resolution on the speed and accuracy of particle image velocimetry interrogation,” Exp. Fluids 13, 105–116 (1992).
[CrossRef]

A. Boillot, A. K. Prasad, “Optimization procedure for pulse separation in cross-correlation PIV,” Exp. Fluids 21, 87–93 (1996).
[CrossRef]

A. K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

Other (1)

R. E. Altenhofen, “Rectification,” in Manual of Photogrammetry (American Society of Photogrammetry, Washington, D.C., 1952), p. 457.

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Figures (6)

Fig. 1
Fig. 1

Configurations for the (a) translation and (b) angular-displacement stereoscopic systems.

Fig. 2
Fig. 2

Schematic of the Scheimpflug stereoscopic PIV system.

Fig. 3
Fig. 3

Error analysis for 3-D displacement.

Fig. 4
Fig. 4

Distorted view of the field from two cameras in the Scheimpflug system.

Fig. 5
Fig. 5

RMS error in measured displacements for various Δz.

Fig. 6
Fig. 6

Vector map from (a) camera 1 and (b) camera 2 for the case Δx = −0.25 mm, Δy = 0 mm, Δz = −0.25 mm; average displacements (〈ΔX1〉 = 0.34 mm, 〈ΔY1〉 = 8.48 × 10−4 mm; 〈ΔX2〉 = 0.17 mm, 〈ΔY2〉 = 4.1 × 10−4 mm) have been subtracted from each vector.

Equations (17)

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Δ x = Δ x 2 [ ( p 1 + q 1 ) · x ] - Δ x 1 [ ( p 2 + q 2 ) · x ] ( p 1 + q 1 ) · x - ( p 2 + q 2 ) · x ,
Δ y = Δ y 1 + Δ y 2 2 - Δ z d 1 ( p 1 + q 1 ) · y + ( p 2 + q 2 ) · y 2 ,
Δ z = d 1 ( Δ x 1 - Δ x 2 ) ( p 1 + q 1 ) · x - ( p 2 + q 2 ) · x ,
( p 1 + q 1 ) · x - ( p 2 + q 2 ) · x = S + Δ x 1 - Δ x 2 .
Δ x i = x f i cos ( θ b - θ ) z 3 cos ( θ ) [ x f i cos ( θ b - θ ) tan ( θ ) + x f i sin ( θ b - θ ) + c 2 ] + ( x f i + Δ X i ) cos ( θ b - θ ) z 3 cos ( θ ) [ ( x f i + Δ X i ) cos ( θ b - θ ) tan ( θ ) + ( x f i + Δ X i ) sin ( θ b - θ ) + c 2 ] ,
Δ y i = - z 3 Δ Y i x f i cos ( θ b - θ ) tan ( θ ) + x f i sin ( θ b - θ ) + c 2 ,
Δ z i = 0 ,
Δ x = S / 2 [ - Δ X 1 z 3 2 Δ X 1 sin ( θ ) + c 2 - Δ X 2 z 3 2 Δ X 2 sin ( θ ) + c 2 ] S + Δ X 2 z 3 2 Δ X 2 sin ( θ ) + c 2 - Δ X 1 z 3 2 Δ X 1 sin ( θ ) + c 2 ,
Δ y = - z 3 ( Δ Y 1 + Δ Y 2 ) 2 c 2 ( Δ z d 1 - 1 ) ,
Δ z = d 1 [ Δ X 2 z 3 2 Δ X 2 sin ( θ ) + c 2 - Δ X 1 z 3 2 Δ X 1 sin ( θ ) + c 2 ] S + Δ X 2 z 3 2 Δ X 2 sin ( θ ) + c 2 - Δ X 1 z 3 2 Δ X 1 sin ( θ ) + c 2 ,
σ Δ z / σ Δ X 1 / [ 2 tan ( θ ) ] ,
σ Δ x / σ Δ X 1 / 2 ,
σ Δ y / σ Δ X 1 / 2 .
σ Δ x 2 σ Δ X 2 = { Δ X 2 d 1 Θ 21 c o s ( θ ) A 1 + Θ 21 A 5 A 3 - [ Δ X 2 d 1 A 4 cos ( θ ) A 1 - Δ X 1 d 1 A 5 cos ( θ ) A 2 ] Θ 21 A 3 2 } 2 + { Δ X 1 d 1 Θ 12 cos ( θ ) A 2 + Θ 12 A 4 A 3 - [ Δ X 1 d 1 A 5 cos ( θ ) A 2 - Δ X 2 d 1 A 4 cos ( θ ) A 1 ] Θ 12 A 3 2 } 2 ,
A 1 = 2 Δ X 2 sin ( θ ) + d 1 cos ( θ ) , A 2 = 2 Δ X 1 sin ( θ ) + d 1 cos ( θ ) , A 3 = 2 tan ( θ ) d 1 - Δ X 1 d 1 cos ( θ ) A 2 + Δ X 2 d 1 cos ( θ ) A 1 , A 4 = tan ( θ ) d 1 - Δ X 1 d 1 cos ( θ ) A 2 , A 5 = - tan ( θ ) d 1 - Δ X 2 d 1 cos ( θ ) A 1 , Θ i j = 2 Δ X j d 1 sin ( θ ) cos ( θ ) A i 2 - d 1 cos ( θ ) A i . σ Δ y 2 σ Δ X 2 = ( Φ 12 B 5 ) 2 + ( Φ 21 B 5 ) 2 + B 4 2 B 3 + [ 2 tan ( θ ) d 1 B 3 2 ] 2 ,
B 1 = 2 Δ X 2 sin ( θ ) + d 1 cos ( θ ) , B 2 = 2 Δ X 1 sin ( θ ) + d 1 cos ( θ ) , B 3 = 2 tan ( θ ) d 1 - Δ X 1 d 1 cos ( θ ) B 2 + Δ X 2 d 1 cos ( θ ) B 1 , B 4 = - Δ X 1 d 1 cos ( θ ) B 2 + Δ X 2 d 1 cos ( θ ) B 1 , B 5 = 2 tan ( θ ) d 1 B 3 ( Δ Y 1 + Δ Y 2 B 3 ) , Θ i j = 1 2 [ 2 Δ X i d 1 tan ( θ ) B i 2 - d 1 cos ( θ ) B j ] . σ Δ z 2 σ Δ X 2 = ( 12 C 3 - C 4 12 C 3 2 ) 2 + ( 21 C 3 - C 4 21 C 3 2 ) 2 ,
C 1 = 2 Δ X 2 sin ( θ ) + d 1 cos ( θ ) , C 2 = 2 Δ X 1 sin ( θ ) + d 1 cos ( θ ) , C 3 = 2 tan ( θ ) d 1 - Δ X 1 d 1 cos ( θ ) C 2 + Δ X 2 d 1 cos ( θ ) C 1 , C 4 = - Δ X 1 d 1 cos ( θ ) C 2 + Δ X 2 d 1 cos ( θ ) C 1 , i j = d 1 [ 2 Δ X i d 1 tan ( θ ) C j 2 - d 1 cos ( θ ) C j ] .

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