Abstract

Fluid flow measurement capabilities of a traveling wave laser oscillator (or ring laser) are assessed theoretically and experimentally. The feasibility of this alternative optical technique for measuring gas flow profiles without insertion of a mechanical probe is examined. Measurements are reported for a range of Reynolds numbers including both laminar and turbulent flows and for a variety of tube diameters. The experimental results are compared with the theory of conditions preceding the establishment of the equilibrium laminar flow profile in round tubes.

© 1968 Optical Society of America

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References

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  1. H. Fizeau, Compt. Rend. 33, 349 (1851) and Ann. Inst. Phys. Chem. Eng. 3, 457 (1853).
  2. W. M. Macek, J. R. Schneider, R. M. Salamon, J. Appl. Phys. 35, 2556 (1964).
    [CrossRef]
  3. J. C. Owens, Appl. Opt. 6, 51 (1967).
    [CrossRef] [PubMed]
  4. O. Reynolds, Trans. Roy. Soc. London 174(1883).
  5. V. L. StreeterFluid Mechanics (McGraw-Hill Book Company, Inc., New York, 1958), p. 137.
  6. See Ref. 4.
  7. Ref. 5, pp. 141–154.
  8. Ref. 5, p. 149.
  9. H. L. Langhaar, J. Appl. Mech. 9, A55 (1942).
  10. L. Prandtl, Essentials of Fluid Mechanics (Hafner, New York, 1952), pp. 105–145.
  11. Ref. 5, p. 159.
  12. H. Chu, V. L. Streeter, “Fluid Flow in Artificially Roughened Pipes,” Project 4918, Unpublished Rept., Illinois Institute of Technology, Chicago (August, 1949).
  13. J. Nikuradse, V. D. I. Forsch, 356 (1932).
  14. W. M. Macek, E. J. McCartney, Sperry Engr. Rev. 19, No. 1, 8 (1966).
  15. W. Tollmien, Z. Angew, Math. Mech 6, 468 (1926).
  16. See Ref. 10.

1967 (1)

1966 (1)

W. M. Macek, E. J. McCartney, Sperry Engr. Rev. 19, No. 1, 8 (1966).

1964 (1)

W. M. Macek, J. R. Schneider, R. M. Salamon, J. Appl. Phys. 35, 2556 (1964).
[CrossRef]

1942 (1)

H. L. Langhaar, J. Appl. Mech. 9, A55 (1942).

1932 (1)

J. Nikuradse, V. D. I. Forsch, 356 (1932).

1926 (1)

W. Tollmien, Z. Angew, Math. Mech 6, 468 (1926).

1883 (1)

O. Reynolds, Trans. Roy. Soc. London 174(1883).

1851 (1)

H. Fizeau, Compt. Rend. 33, 349 (1851) and Ann. Inst. Phys. Chem. Eng. 3, 457 (1853).

Angew, Z.

W. Tollmien, Z. Angew, Math. Mech 6, 468 (1926).

Chu, H.

H. Chu, V. L. Streeter, “Fluid Flow in Artificially Roughened Pipes,” Project 4918, Unpublished Rept., Illinois Institute of Technology, Chicago (August, 1949).

Fizeau, H.

H. Fizeau, Compt. Rend. 33, 349 (1851) and Ann. Inst. Phys. Chem. Eng. 3, 457 (1853).

Langhaar, H. L.

H. L. Langhaar, J. Appl. Mech. 9, A55 (1942).

Macek, W. M.

W. M. Macek, E. J. McCartney, Sperry Engr. Rev. 19, No. 1, 8 (1966).

W. M. Macek, J. R. Schneider, R. M. Salamon, J. Appl. Phys. 35, 2556 (1964).
[CrossRef]

McCartney, E. J.

W. M. Macek, E. J. McCartney, Sperry Engr. Rev. 19, No. 1, 8 (1966).

Nikuradse, J.

J. Nikuradse, V. D. I. Forsch, 356 (1932).

Owens, J. C.

Prandtl, L.

L. Prandtl, Essentials of Fluid Mechanics (Hafner, New York, 1952), pp. 105–145.

Reynolds, O.

O. Reynolds, Trans. Roy. Soc. London 174(1883).

Salamon, R. M.

W. M. Macek, J. R. Schneider, R. M. Salamon, J. Appl. Phys. 35, 2556 (1964).
[CrossRef]

Schneider, J. R.

W. M. Macek, J. R. Schneider, R. M. Salamon, J. Appl. Phys. 35, 2556 (1964).
[CrossRef]

Streeter, V. L.

H. Chu, V. L. Streeter, “Fluid Flow in Artificially Roughened Pipes,” Project 4918, Unpublished Rept., Illinois Institute of Technology, Chicago (August, 1949).

V. L. StreeterFluid Mechanics (McGraw-Hill Book Company, Inc., New York, 1958), p. 137.

Tollmien, W.

W. Tollmien, Z. Angew, Math. Mech 6, 468 (1926).

Appl. Opt. (1)

Compt. Rend. (1)

H. Fizeau, Compt. Rend. 33, 349 (1851) and Ann. Inst. Phys. Chem. Eng. 3, 457 (1853).

J. Appl. Mech. (1)

H. L. Langhaar, J. Appl. Mech. 9, A55 (1942).

J. Appl. Phys. (1)

W. M. Macek, J. R. Schneider, R. M. Salamon, J. Appl. Phys. 35, 2556 (1964).
[CrossRef]

Math. Mech (1)

W. Tollmien, Z. Angew, Math. Mech 6, 468 (1926).

Sperry Engr. Rev. (1)

W. M. Macek, E. J. McCartney, Sperry Engr. Rev. 19, No. 1, 8 (1966).

Trans. Roy. Soc. London (1)

O. Reynolds, Trans. Roy. Soc. London 174(1883).

V. D. I. Forsch (1)

J. Nikuradse, V. D. I. Forsch, 356 (1932).

Other (8)

See Ref. 10.

V. L. StreeterFluid Mechanics (McGraw-Hill Book Company, Inc., New York, 1958), p. 137.

See Ref. 4.

Ref. 5, pp. 141–154.

Ref. 5, p. 149.

L. Prandtl, Essentials of Fluid Mechanics (Hafner, New York, 1952), pp. 105–145.

Ref. 5, p. 159.

H. Chu, V. L. Streeter, “Fluid Flow in Artificially Roughened Pipes,” Project 4918, Unpublished Rept., Illinois Institute of Technology, Chicago (August, 1949).

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Figures (17)

Fig. 1
Fig. 1

TWLO cavity containing a moving medium.

Fig. 2
Fig. 2

γ as a function of σ.

Fig. 3
Fig. 3

Normalized velocity as a function of normalized distance from inlet for several radii.

Fig. 4
Fig. 4

Normalized velocity as a function of radial coordinate for several values of normalized distance from the inlet.

Fig. 5
Fig. 5

Normalized velocity averaged from inlet as a function of distance from inlet for several radii.

Fig. 6
Fig. 6

Normalized velocity averaged from inlet as a function of radical position for several values of axial distance from inlet.

Fig. 7
Fig. 7

Prandtl’s formula for turbulent flow.

Fig. 8
Fig. 8

Schematic of Faraday cell. Note: Optic axis at 45° angle to E field as shown. Output has same polarization as input and paths are parallel.

Fig. 9
Fig. 9

Schematic of flow measuring equipment.

Fig. 10
Fig. 10

Cross section of the flow system.

Fig. 11
Fig. 11

Detection electronics.

Fig. 12
Fig. 12

Normalized averaged velocity for laminar flow development.

Fig. 13
Fig. 13

Normalized averaged velocity for laminar flow development.

Fig. 14
Fig. 14

Normalized averaged velocity for laminar flow development.

Fig. 15
Fig. 15

Normalized averaged velocity for laminar flow development.

Fig. 16
Fig. 16

Normalized averaged velocity for turbulent flow.

Fig. 17
Fig. 17

Normalized averaged velocity for turbulent flow.

Tables (1)

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Table I Data for Experimental Curves

Equations (24)

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Δ V 12 = ± V m ( 1 - 1 / n 2 ) ,
Δ t = Δ t 2 - Δ t 1 = 2 b Δ V / c = ( 2 b V 0 / c 2 ) ( 1 - 1 / n 2 ) ,
Δ f = ( 2 b V 0 / λ l ) ( 1 - 1 / n 2 ) ,
Δ f 4 b V 0 ( n - 1 ) / λ l .
( n - 1 ) 2.9 × 10 - 4 ,
Δ f 460 Hz per m / sec
R = V av ( a ρ / μ ) ,
V ( r / a ) = 2 V av [ 1 - ( r / a ) 2 ] ,
β ( r / a , λ ) = [ I 0 ( λ ) - I 0 ( r λ / a ) ] I 2 ( λ ) ,
σ = Z / s R ,
V = V 0 + Δ V = c + V m ( 1 - 1 / n 2 ) ,
V c + 2 V m ( n - 1 ) .
Δ t - δ l Δ V / c 2
Δ t = - 2 δ l V m ( n - 1 ) / c 2 .
Δ l = 2 δ l V m ( n - 1 ) / c .
λ = P / m ,
Δ λ = Δ P / m = Δ l / m = 2 δ l V m ( n - 1 ) c m = 2 δ l V m λ 0 ( n - 1 ) / c P
Δ f = - f 0 2 δ l V m ( n - 1 ) / c P = 2 δ l V m ( n - 1 ) / λ P .
f d ( r / a , b ) = 2 lim δ l 0 Δ f = 4 λ P ( n - 1 ) 0 b V m ( r / a , l ) d l .
f d ( r / a , Z ) = 4 ( n - 1 ) V av ( Z / λ P ) [ ( 1 / Z ) 0 Z β ( r / a , l ) d l ] ,
β av ( r / a , Z ) = ( 1 / Z ) 0 Z β ( r / a , l ) d l
f d ( r / a , Z ) = 4 ( n - 1 ) Z V av β av ( r / a , Z ) / λ P .
V max / V = ( 1 - r / a ) 1 7 ,
( V max - V ) / V * = ( 1 / x ) ln [ a / ( a - r ) ] , r - a > δ > 0 ,

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