Abstract

Transformation equations are derived that may be used to determine a third orthogonal velocity component from measurements made at a common point by two rotationally displaced, two orthogonal component laser Doppler velocimeter (LDV) systems. These equations also may be used to relate velocity measurements made in a particular coordinate system to any other coordinate system. It is shown that the set of smallest angles that may be used to separate the two, two component, LDV systems is very sensitive to the relative magnitudes and directions of the individual velocity components. When the magnitudes of all velocity components are about the same, it is found that the minimum angle of separation is of the same order as the instrumental error of the LDV systems.

© 1972 Optical Society of America

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References

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  1. D. B. Brayton, W. H. Goethert, Trans. Inst. Soc. Am.10, in press (1971).
  2. A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).
  3. F. H. Smith, J. A. Parsons, AEDC-TR-70-119 (1970).
  4. W. J. Yanta, D. F. Gates, F. W. Brown, “The Use of a Laser Doppler Velocimeter in Supersonic Flow,” presented at AIAA 6th Aerodynamics Testing Conference, Albuquerque, New Mexico (10–12 March 1971).
  5. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [CrossRef]
  6. M. J. Rudd, J. of Phys. E 2,(1969).
  7. Actually the planes lie parallel to the bisector of angle ABO. Since ABOis usually a very small angle, we can simplify the transformation considerably by assuming parallelism to AO.
  8. Maintaining θp,θq≲ 0.1 allows the possibility of transmitting both sets of LDV beams through the same lens. Ordinarily, optimum use of the lens is made when the component beams are separated at the lens by about 0.8-lens diam, and the input beam diameters are about 0.1-lens diam. The interference fringe spacing is then about the Rayleigh resolution limit of the lens. The beam spacing for three beams set orthogonally to each other for a two-component LDV system reduces the optimum beam separation somewhat, the loss being power density in the probe volume. For two, two-component beam sets transmitting through the same lens, the component beam spacing becomes even smaller so that θp,θqmust be small for nearly all practical transmitting lens Fnumbers.
  9. J. D. Trimmer, Response of Physical Systems (Wiley, New York, 1956), p. 150.

1970 (1)

A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).

1969 (1)

M. J. Rudd, J. of Phys. E 2,(1969).

1967 (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Brayton, D. B.

A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).

D. B. Brayton, W. H. Goethert, Trans. Inst. Soc. Am.10, in press (1971).

Brown, F. W.

W. J. Yanta, D. F. Gates, F. W. Brown, “The Use of a Laser Doppler Velocimeter in Supersonic Flow,” presented at AIAA 6th Aerodynamics Testing Conference, Albuquerque, New Mexico (10–12 March 1971).

Fried, D. L.

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Gates, D. F.

W. J. Yanta, D. F. Gates, F. W. Brown, “The Use of a Laser Doppler Velocimeter in Supersonic Flow,” presented at AIAA 6th Aerodynamics Testing Conference, Albuquerque, New Mexico (10–12 March 1971).

Goethert, W. H.

A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).

D. B. Brayton, W. H. Goethert, Trans. Inst. Soc. Am.10, in press (1971).

Lennert, A. E.

A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).

Parsons, J. A.

F. H. Smith, J. A. Parsons, AEDC-TR-70-119 (1970).

Rudd, M. J.

M. J. Rudd, J. of Phys. E 2,(1969).

Smith, F. H.

A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).

F. H. Smith, J. A. Parsons, AEDC-TR-70-119 (1970).

Trimmer, J. D.

J. D. Trimmer, Response of Physical Systems (Wiley, New York, 1956), p. 150.

Yanta, W. J.

W. J. Yanta, D. F. Gates, F. W. Brown, “The Use of a Laser Doppler Velocimeter in Supersonic Flow,” presented at AIAA 6th Aerodynamics Testing Conference, Albuquerque, New Mexico (10–12 March 1971).

J. of Phys. E (1)

M. J. Rudd, J. of Phys. E 2,(1969).

Laser J. (1)

A. E. Lennert, D. B. Brayton, W. H. Goethert, F. H. Smith, Laser J. 2, 19 (1970).

Proc. IEEE (1)

D. L. Fried, Proc. IEEE 55, 57 (1967).
[CrossRef]

Other (6)

D. B. Brayton, W. H. Goethert, Trans. Inst. Soc. Am.10, in press (1971).

F. H. Smith, J. A. Parsons, AEDC-TR-70-119 (1970).

W. J. Yanta, D. F. Gates, F. W. Brown, “The Use of a Laser Doppler Velocimeter in Supersonic Flow,” presented at AIAA 6th Aerodynamics Testing Conference, Albuquerque, New Mexico (10–12 March 1971).

Actually the planes lie parallel to the bisector of angle ABO. Since ABOis usually a very small angle, we can simplify the transformation considerably by assuming parallelism to AO.

Maintaining θp,θq≲ 0.1 allows the possibility of transmitting both sets of LDV beams through the same lens. Ordinarily, optimum use of the lens is made when the component beams are separated at the lens by about 0.8-lens diam, and the input beam diameters are about 0.1-lens diam. The interference fringe spacing is then about the Rayleigh resolution limit of the lens. The beam spacing for three beams set orthogonally to each other for a two-component LDV system reduces the optimum beam separation somewhat, the loss being power density in the probe volume. For two, two-component beam sets transmitting through the same lens, the component beam spacing becomes even smaller so that θp,θqmust be small for nearly all practical transmitting lens Fnumbers.

J. D. Trimmer, Response of Physical Systems (Wiley, New York, 1956), p. 150.

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Figures (6)

Fig. 1
Fig. 1

A One-Velocity-component, long range, dual scatter, backscatter LDV system.

Fig. 2
Fig. 2

Photomicrograph of one-velocity-component interference fringes in the probe volume.

Fig. 3
Fig. 3

Geometry for two rotationally separated two-velocity-component LDV systems observing a common point.

Fig. 4
Fig. 4

End view of xyz coordinate system showing θ octant end faces that ABC would occupy after the indicated rotation directions for θq,θp.

Fig. 5
Fig. 5

Relaxation of angular separation requirement for case one rotations.

Fig. 6
Fig. 6

Relaxation of angular separation requirement for case two rotations.

Equations (23)

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( x y ) = [ cos ( θ ) ± sin ( θ ) sin ( θ ) cos ( θ ) ] × ( x y ) ,
( e 1 e 2 e 3 ) = [ cos ( θ ) r sin ( θ ) r 0 sin ( θ ) r cos ( θ ) r 0 0 0 1 ] ( e 1 e 2 e 3 ) .
( e 1 e 2 e 3 ) = [ cos ( θ ) p 0 sin ( θ ) p 0 1 0 sin ( θ ) p 0 cos ( θ ) p ] ( e 1 e 2 e 3 ) .
( e 1 e 2 e 3 ) = [ 1 0 0 0 cos ( θ ) q sin ( θ ) q 0 sin ( θ ) q cos ( θ ) q ] ( e 1 e 2 e 3 ) .
e = R ˆ e ,
R ˆ = { [ cos ( θ p ) cos ( θ r ) ] [ sin ( θ r ) cos ( θ p ) ] [ sin ( θ p ) ] [ sin ( θ r ) cos ( θ q ) sin ( θ q ) sin ( θ p ) cos ( θ r ) ] [ cos ( θ q ) cos ( θ r ) sin ( θ q ) sin ( θ p ) sin ( θ r ) ] [ sin ( θ q ) cos ( θ p ) ] [ sin ( θ q ) sin ( θ r ) sin ( θ p ) cos ( θ q ) cos ( θ r ) ] [ sin ( θ q ) cos ( θ r ) cos ( θ q ) sin ( θ p ) sin ( θ r ) ] [ cos ( θ q ) cos ( θ p ) ] } .
e = R ˆ ˜ e .
sin θ i θ i i = p , q , cos θ i 1 θ i 1.
( e e 2 e 3 ) = [ 1 θ p θ q θ p 0 1 θ q θ p θ q 1 ] ( e 1 e 2 e 3 ) .
V x = V x θ p θ q V y θ p V z ,
V y = V y θ q V z ,
V z = θ p V x + θ q V y + V z .
V z 1 = ( V y V y ) / θ q ,
V z 2 = ( V x V y V x V y + θ p θ q V y V y ) / ( θ q V x θ p V y ) ,
( Δ V i j / V i j ) = ( Δ V i + Δ V j ) / ( V i V j ) ,
V i V j ( 1 / C ) ( Δ V i + Δ V j ) .
V i V j a ( V i + V j ) ,
θ q 1 [ a ( V y + V y ) / V z ] .
θ q 1 2 a ( V y / V z ) ,
θ q 2 2 a V y V z [ 1 ( ± ) q 1 ( ± ) p ( V y / V x ) f ] ,
θ q 2 ± 2 a ( V y / V z ) { 1 / [ f ( V y / V x ) + 1 ] } .
θ q 2 θ q 1 { 1 / [ f ( V y / V x ) + 1 ] } .
θ q 2 θ q 1 { 1 / [ f ( V y / V x ) 1 ] } .

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