Abstract

We describe and test a refractive method for determining the height profile of an unsteady liquid-free surface, which is adequate for regions where the slope of the surface changes strongly, as for example the head of a spreading current. The method is developed for situations in which the height depends on a single Cartesian coordinate (plane flows); however, the underlying idea could be applied to more general cases. As a test, we obtained the profiles of a transparent solid object (a cylindrical lens) and of an actual liquid flow. These profiles are determined with high accuracy even if the direction of the normal to the free surface changes approximately 150° within the probed region.

© 1997 Optical Society of America

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  1. B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
    [CrossRef] [PubMed]
  2. B. Marino, “Efectos de las fuerzas de superficie en derrames viscogravitatorios,” Ph.D. dissertation (Universidad Nacional del Centro de la Provicia de Buenos Aires, Argentina, 1994).
  3. S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).
  4. L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, J. Simon, “Measurements of free-surface profile in transient flow with a simple light-slicing method,” Appl. Opt. 33, 2455–2458 (1994).
    [CrossRef] [PubMed]
  5. L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, “Droplet profiles obtained from the intensity distribution of refraction patterns,” Appl. Opt. 34, 5840–5848 (1995).
    [CrossRef] [PubMed]
  6. L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
    [CrossRef]
  7. L. A. Vasil’ev, Schlieren Methods (Israel Program for Scientific Translations, New York, 1971), Chaps. 2–4.
  8. W. Merzkirch, Flow Visualisation, 2nd. ed. (Academic, London, 1987), Chap. 3.
  9. D. W. McAdam, E. G. Hautmann, “Optical measurements of liquid film profile and dynamic contact angle,” Appl. Opt. 14, 1764–1765 (1976).
  10. L. H. Tanner, “The measurement of viscosity by optical techniques applied to a falling liquid film,” J. Phys. E 9, 967–973 (1976).
    [CrossRef]
  11. L. H. Tanner, “A study of the optics and motion of oil droplets,” Opt. Laser Technol. 10, 125–128 (1978).
    [CrossRef]
  12. K. Guo, T. Uemura, W. Yang, “Reflection-interference method to determine droplet profiles,” Appl. Opt. 24, 2655–2659 (1985).
    [CrossRef] [PubMed]
  13. T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
    [CrossRef]
  14. C. Chan, N. Liang, W. Liu, “Measurement of the shape of liquid-liquid interface by the method of light reflection,” Rev. Sci. Instrum. 64, 632–637 (1993).
    [CrossRef]
  15. G. Da Costa, J. Calatroni, “Self-holograms of laser-induced surface depressions in heavy hydrocarbons,” Appl. Opt. 17, 2381–2385 (1978).
    [CrossRef]
  16. C. Allain, D. Ausserre, F. Rondelez, “A new method for contact-angle measurement of sessile drops,” J. Colloid Interface Sci. 107, 5–13 (1985).
    [CrossRef]

1997

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

1996

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
[CrossRef] [PubMed]

1995

1994

1993

C. Chan, N. Liang, W. Liu, “Measurement of the shape of liquid-liquid interface by the method of light reflection,” Rev. Sci. Instrum. 64, 632–637 (1993).
[CrossRef]

1988

T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
[CrossRef]

1985

K. Guo, T. Uemura, W. Yang, “Reflection-interference method to determine droplet profiles,” Appl. Opt. 24, 2655–2659 (1985).
[CrossRef] [PubMed]

C. Allain, D. Ausserre, F. Rondelez, “A new method for contact-angle measurement of sessile drops,” J. Colloid Interface Sci. 107, 5–13 (1985).
[CrossRef]

1978

G. Da Costa, J. Calatroni, “Self-holograms of laser-induced surface depressions in heavy hydrocarbons,” Appl. Opt. 17, 2381–2385 (1978).
[CrossRef]

L. H. Tanner, “A study of the optics and motion of oil droplets,” Opt. Laser Technol. 10, 125–128 (1978).
[CrossRef]

1976

L. H. Tanner, “The measurement of viscosity by optical techniques applied to a falling liquid film,” J. Phys. E 9, 967–973 (1976).
[CrossRef]

D. W. McAdam, E. G. Hautmann, “Optical measurements of liquid film profile and dynamic contact angle,” Appl. Opt. 14, 1764–1765 (1976).

Allain, C.

C. Allain, D. Ausserre, F. Rondelez, “A new method for contact-angle measurement of sessile drops,” J. Colloid Interface Sci. 107, 5–13 (1985).
[CrossRef]

Ausserre, D.

C. Allain, D. Ausserre, F. Rondelez, “A new method for contact-angle measurement of sessile drops,” J. Colloid Interface Sci. 107, 5–13 (1985).
[CrossRef]

Betelú, S.

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

Calatroni, J.

Chan, C.

C. Chan, N. Liang, W. Liu, “Measurement of the shape of liquid-liquid interface by the method of light reflection,” Rev. Sci. Instrum. 64, 632–637 (1993).
[CrossRef]

Da Costa, G.

Diez, J. A.

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, “Droplet profiles obtained from the intensity distribution of refraction patterns,” Appl. Opt. 34, 5840–5848 (1995).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, J. Simon, “Measurements of free-surface profile in transient flow with a simple light-slicing method,” Appl. Opt. 33, 2455–2458 (1994).
[CrossRef] [PubMed]

Endoh, K.

T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
[CrossRef]

Gratton, R.

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, “Droplet profiles obtained from the intensity distribution of refraction patterns,” Appl. Opt. 34, 5840–5848 (1995).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, J. Simon, “Measurements of free-surface profile in transient flow with a simple light-slicing method,” Appl. Opt. 33, 2455–2458 (1994).
[CrossRef] [PubMed]

Guo, K.

Hautmann, E. G.

Liang, N.

C. Chan, N. Liang, W. Liu, “Measurement of the shape of liquid-liquid interface by the method of light reflection,” Rev. Sci. Instrum. 64, 632–637 (1993).
[CrossRef]

Liu, W.

C. Chan, N. Liang, W. Liu, “Measurement of the shape of liquid-liquid interface by the method of light reflection,” Rev. Sci. Instrum. 64, 632–637 (1993).
[CrossRef]

Marino, B.

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, “Droplet profiles obtained from the intensity distribution of refraction patterns,” Appl. Opt. 34, 5840–5848 (1995).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, J. Simon, “Measurements of free-surface profile in transient flow with a simple light-slicing method,” Appl. Opt. 33, 2455–2458 (1994).
[CrossRef] [PubMed]

B. Marino, “Efectos de las fuerzas de superficie en derrames viscogravitatorios,” Ph.D. dissertation (Universidad Nacional del Centro de la Provicia de Buenos Aires, Argentina, 1994).

McAdam, D. W.

Merzkirch, W.

W. Merzkirch, Flow Visualisation, 2nd. ed. (Academic, London, 1987), Chap. 3.

Mikami, A.

T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
[CrossRef]

Mori, Y.

T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
[CrossRef]

Ohyama, T.

T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
[CrossRef]

Rondelez, F.

C. Allain, D. Ausserre, F. Rondelez, “A new method for contact-angle measurement of sessile drops,” J. Colloid Interface Sci. 107, 5–13 (1985).
[CrossRef]

Simon, J.

Tanner, L. H.

L. H. Tanner, “A study of the optics and motion of oil droplets,” Opt. Laser Technol. 10, 125–128 (1978).
[CrossRef]

L. H. Tanner, “The measurement of viscosity by optical techniques applied to a falling liquid film,” J. Phys. E 9, 967–973 (1976).
[CrossRef]

Thomas, L. P.

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, “Droplet profiles obtained from the intensity distribution of refraction patterns,” Appl. Opt. 34, 5840–5848 (1995).
[CrossRef] [PubMed]

L. P. Thomas, R. Gratton, B. Marino, J. A. Diez, J. Simon, “Measurements of free-surface profile in transient flow with a simple light-slicing method,” Appl. Opt. 33, 2455–2458 (1994).
[CrossRef] [PubMed]

Uemura, T.

Vasil’ev, L. A.

L. A. Vasil’ev, Schlieren Methods (Israel Program for Scientific Translations, New York, 1971), Chaps. 2–4.

Yang, W.

Appl. Opt.

Int. J. Numer. Methods Fluids

S. Betelú, J. A. Diez, L. P. Thomas, R. Gratton, B. Marino, “A boundary-elements method for viscous gravity currents,” Int. J. Numer. Methods Fluids (1997).

J. Colloid Interface Sci.

B. Marino, L. P. Thomas, J. A. Diez, R. Gratton, “Capillarity effects on viscous gravity spreadings of wetting liquids,” J. Colloid Interface Sci. 177, 14–30 (1996).
[CrossRef] [PubMed]

C. Allain, D. Ausserre, F. Rondelez, “A new method for contact-angle measurement of sessile drops,” J. Colloid Interface Sci. 107, 5–13 (1985).
[CrossRef]

J. Phys. E

L. H. Tanner, “The measurement of viscosity by optical techniques applied to a falling liquid film,” J. Phys. E 9, 967–973 (1976).
[CrossRef]

Meas. Sci. Technol.

L. P. Thomas, R. Gratton, B. Marino, S. Betelú, J. A. Diez, “Measurement of the slope of an unsteady liquid surface along a line by an anamorphic schlieren system,” Meas. Sci. Technol. 7, 1134–1139 (1996).
[CrossRef]

Opt. Laser Technol.

L. H. Tanner, “A study of the optics and motion of oil droplets,” Opt. Laser Technol. 10, 125–128 (1978).
[CrossRef]

Rev. Sci. Instrum.

T. Ohyama, K. Endoh, A. Mikami, Y. Mori, “Optical interferometry for measuring instantaneous thickness of transparent solid and liquid films,” Rev. Sci. Instrum. 59, 2018–2022 (1988).
[CrossRef]

C. Chan, N. Liang, W. Liu, “Measurement of the shape of liquid-liquid interface by the method of light reflection,” Rev. Sci. Instrum. 64, 632–637 (1993).
[CrossRef]

Other

L. A. Vasil’ev, Schlieren Methods (Israel Program for Scientific Translations, New York, 1971), Chaps. 2–4.

W. Merzkirch, Flow Visualisation, 2nd. ed. (Academic, London, 1987), Chap. 3.

B. Marino, “Efectos de las fuerzas de superficie en derrames viscogravitatorios,” Ph.D. dissertation (Universidad Nacional del Centro de la Provicia de Buenos Aires, Argentina, 1994).

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Figures (7)

Fig. 1
Fig. 1

Basic idea of the method.

Fig. 2
Fig. 2

Schematic diagram of the experimental arrangement.

Fig. 3
Fig. 3

Three-dimensional diagram of the optical system.

Fig. 4
Fig. 4

Amplification of a small δs given by the reconstruction process for decreasing s; the curves correspond to the values of θ i and Δh/s in which the amplification ratio χ = δs(h + Δh)/δs(h) is 1.2, 1.5, and 2.

Fig. 5
Fig. 5

Ray deviations at the interfaces between walls and the liquid for a plane free surface (initial configuration of the test experiment).

Fig. 6
Fig. 6

Reconstruction of the profile of a cylindrical lens. (a) Central points of the image curves (arbitrary units) are obtained by using standard software from the recorded images. The reference straight line corresponds to the image in the absence of a refractive medium. (b) Height profiles obtained from the data shown in (a). The solid curve is the ideal contour of the cylindrical lens calculated from independent measurements (the convex face is circular). (c) Distribution of ds/dh as a function of height h. The solid curve is the derivative of the curvex face of the ideal contour shown in (b). The symbols correspond to s f = 0.58 cm (■) (i.e., the diffuser is just on the plane surface of the lens), 0.965 cm (●), and 1.35 cm (▲).

Fig. 7
Fig. 7

Reconstruction of the profiles of a viscous current. (a) Image curves and the reference straight curve (diamonds). At t = 0 the profile is wedge shape; ■, t = 0.2 s; ●, t = 27.5 s; ▲, t = 51.8 s. The horizontal line marks the zero height. (b) Parts of the height profiles obtained by the reconstruction starting from x = s f and x = 0; the solid curve shows the intial wedgelike profile determined by independent measurements of s f the wedge angele. (c) Derivative distributions given by the reconstruction process.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

tan θd=h-hds,
η sinθi-θd=sinθi
cot gθi=η cos ϑd-1η sin ϑd.
dhds=-cot gθi=fh, hd, s,
dhds=gh, s.
δsh+Δhδshexpη-cos θdη cos θd sin θdη cos θd-12Δhs.
d=dw1-tan θs tan θd,
θs=sin-1ηηw cosθd.
hd=-s-h/tan θdcot gθs.

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