Measurements of free-surface profilein transient flow with a simple light-slicing method

Luis P. Thomas, Roberto Gratton, Beatriz M. Marino, and Juan M. Simon

A simple light-slicing technique for the direct measurement of the free-surface shape h(x) in one-directional transient liquid flows is studied. The method is used to determine h(x) with a precision of theorder of 10-2 mm along a line several centimeters long, allowing for height differences of somemillimeters. It is useful for transient flows in rectangular channels in which capillarity and liquidadhesion at walls perturb lateral observations; the method can also be used for nondiffusive fluids in casesin which it is not advisable to add diffusive particles.

Key words: Liquid current, free-surface measurements, light slicing.

1. Introduction

In this paper we are concerned with the measurementof the free-surface height h(x) in one-directionalliquid flows that are too large to make measurementwith interference techniques- 3 practical. Here weanalyze a simple light-slicing technique (light-slicingtechniques have been used for turbulent flows; see,for instance, Refs. 4 and 5) for complete measurementof h(x) along a line several centimeters long within aheight-variation range of some millimeters, with aprecision of 10-2 mm. The method is particularlyuseful for transient flows in rectangular channels, inwhich there are no possible precise sideways observa-tions, and also for nondiffusive fluids when theaddition6 of diffusive particles is not advisable.However, both reflective and diffusive surfaces can beproved; in the former case the height profile can bemeasured only where the slope I lax is below alimiting value, which is approximately 0.20 for ourdevice, whereas in the latter case this limitation doesnot exist.

In the system described here the profile h(x) isdisplayed with high magnification on a semitranspar-ent far screen, together with adjustable reference

L. P. Thomas, R. Gratton, and B. M. Marino are with theInstituto de Fisica Arroyo Seco, Universidad Nacional del Centrode la Provincia de Buenos Aires, Pinto 399, 7000 Tandil, Argen-tina. J. M. Simon is with the Laboratorio de Optica, Universidadde Buenos Aires, Pabellon 1, Ciudad Universitaria, 1428 BuenosAires, Argentina.

Received 19 January 1993; revised manuscript received 13September 1993.

0003-6935/94/132455-04$06.00/0.© 1994 Optical Society of America.

lines. Recordings are made with a standard videocamera with a 2 x 10-2 s time resolution that isfocused on a screen. In Section 2 we describe theoptical system and its performance; some tests onsolid surfaces are included to illustrate the sensitivityand the reliability of the method. Finally, we pre-sent a demonstrative sequence of height profilescorresponding to the impinging of a solitary waterwave on an inclined plane edge.

2. Optical System

A schematic diagram of the system is shown in Fig. 1.The light beam of a 15-mW He-Ne polarized laserwas first slightly expanded and then focused by acombination of a cylindrical lens, L (focal lengthf = 45 mm), and an objective, Lf (f = 400 mm, clearaperture D = 80 mm), in such a way that a parallellight slice that was 80-mm wide in the x direction wasformed. The slice had a slightly decreasing thick-ness, and a well-defined line segment (focal line) wasformed at a focal plane near the surface that was to bemeasured. The lenses were chosen to make the slicethickness small along a depth of some millimeters toallow a precise test of surfaces for which heightdifferences are of this order; we return to this pointbelow.

The incoming light slice was at 450 with respect tothe horizontal plane, which contains the x and y axesand, the focal line (see Fig. 1). A second high-aperture objective, Li (f = 178 mm, D = 70 mm),linked the slice plane ri with the screen plane Tr,.Mirrors M, and M2 may be omitted in the case inwhich the laser and the screen must be placed at 450with respect to the vertical. The intersection of theincoming light-slice plane rri with the test surface I

1 May 1994 / Vol. 33, No. 13 / APPLIED OPTICS 2455

Fig. 1. Schematic view of the optical system, with a cylindricalsolid surface being probed. The x-y plane is horizontal, and it is60 mm below the mirrors.

determined a curve that was contained in the ri planewhose focused image appears on wr. At the sametime, the beam splitter, BS, and mirror M3 (shownonly in Fig. 2) produce a straight line on the screen forreference.

For height profiles that vary in space only along theslice width (x axis), i.e., ah/dx • 0 and dh/dy = 0, theimage curve on the screen reproduces h(x) with theheight scale magnified by a factor of f2M, where M isthe amplification factor of the x scale, as given by lensLi. For the system described here M = 9.40, so thatthe height amplification was 13.3.

In Fig. 3 we show d, the thickness of the light lineon the screen, as a function of the height h of ahorizontal water surface. The thickness was mea-sured on the screen with the aid of an eyepiece. Thevalues of d have been reduced by the magnification, sothey give the equivalent light-slice thickness on thewater surface. The sensitivity of the technique canbe estimated as a small fraction of d, i.e., of the orderof 0.01 mm.

Only the rays captured by lens Li contributed to theimage formation, indicating that high-slope regionsmay be lost if the surface is reflective. In this caseeach point of the image is formed by rays passingthrough a small region near a diameter of lens Li.On the other hand, when the surface is diffusive, theLi lens collects part of the light that is diffused by eachpoint of the surface, and then even an image ofhigh-slope regions is formed. The highest slope thatwe could observe for reflective surfaces was 0.15-0.20.

Thicknessslice

M,Lc LI L d t2 r

M, i ~ ~ M

Laser CylndicVideo

lens y as Screen camera

Channel

Fig. 2. Sectional view of the optical arrangement. The beamsplitter, BS, and mirror M3 yield straight reference lines on thescreen.

Height h (mm)

Fig. 3. Light-slice thickness d as a function of the height h withrespect to the focal-line position (zero height). The asterisksrepresent the experimental data, and the solid curve is the best-fitparabola.

This limit depends slightly on the position of thehigh-slope region along the x axis.

In Fig. 4 we show the image corresponding to apolished Lucite cylindrical pipe (75-mm diameter,3.5-mm wall thickness), whose axis is parallel to theyaxis. Diffusive powder has been scattered on boththe internal and the external surfaces. The stronglyilluminated part of the curves corresponds to re-flected rays coming from the internal and the externalpipe regions, where the absolute value of the slope isless than 0.18; the remaining weak part of the curveis due to the diffusive contribution, which forms animage from regions with higher slopes, quite near±'rr/2. The eventual distortions of the reflective

curve (resulting from, for instance, optical aberra-tions of the Li lens) with respect to the diffusive curve

Fig. 4. Cross-section profiles of a transparent cylindrical pipe.The upper curve corresponds to the external surface of the pipe,which measured 75 mm in diameter, whereas the lower curvecorresponds to the internal surface, which measured 68 mm indiameter. On each curve there are two regions, a central highlyilluminated region resulting from the specular reflection on thelucite-air interface and another region resulting from the diffusivescattering of the powder deposited on the surfaces.

2456 APPLIED OPTICS / Vol. 33, No. 13 / 1 May 1994

0.14

0.12

E 0.10

-0 0.08U)

a)-' 0.060

0.04

0.02

0.00

IEjE

(a) I

(b)- -

0.00 S1mm

0.08 s

0.16s

0.24 s

0.32s

Fig. 5. (a) Image corresponding to the height of the upper surfaceof a coin. (b) Frontal image of the coin; the dashed-dotted line isthe line along which the coin height is obtained. The coin has a23.9-mm diameter and a 2-mm thickness. (c) Enlarged view of a0.07-mm jump height.

are less than the line thickness, i.e., less than 0.25mm on the screen. We observed strong distortionswhen we used simple lenses or lower-quality objec-tives.

A pictorial view of the technique sensitivity ispresented in Fig. 5, in which we show the crosssection of a coin [Fig. 5(a)] and its corresponding

0.00 S 1m

0.08 s

0.12 s

0.20s

0.24s

0.28s

0.36 sFig. 6. Weak solitary wave impinging on an inclined plane. Theheight and the length scales as well as the time corresponding toeach frame are reported. The free surface of the water is at theleft of the figure, and the emerging edge is at the right of the figure;a horizontal dashed line is included for reference. The 0.00-sframe gives the initial condition just before the solitary wave comesfrom the left.

0.40 s

0.52 s

Fig. 7. Same as Fig. 6 but for a stronger solitary wave.

frontal view [Fig. 5(b)], indicated by a dashed-dottedline. In Fig. 5(c) an enlarged view of a part of Fig.5(a) can be seen; the jump height is 0.07 mm, asmeasured with a micrometer.

3. Demonstrative Transient Flow Experiment

As a test problem we observed the impinging of asolitary water wave on an edge formed by a slightlyinclined plane. We used a Lucite channel 30 cm longand 10 cm wide; the liquid used was water with somesurfactant added to reduce the surface tension andthe contact angle. With the static water edge nearthe center of the test region, we produced waves atthe deepest zone simply by moving a small plate.The image formed on the screen was recorded by astandard video camera, and sequential single-frameimages were obtained by photographing the TV screen.In Figs. 6 and 7 we reproduce two brief sequencescorresponding to waves of different amplitudes. Theemerging part of the edge is clearly visible in the firstframes, which correspond to the unperturbed condi-tion. Small parts of the profiles are not shown inFig. 7 because of the high value of the slope, but thegeneral evolution of the waves is well described, andquantitative determinations are possible.

4. Concluding Remarks

The simple method described here provides a valu-able tool for the observation of free-surface profiles inone-directional liquid flows on a laboratory scale.We show that a precision of 10-2 mm is possiblewithin typical full height ranges of some millimeters.Therefore the method is useful for flows that are solarge that interferometry is not suitable for measur-ing them. When compared with point-to-point scan-ning-height measurements, 7 the method has the ad-vantage (especially for fast transient flows) that theprofile along a line of considerable length (several

1 May 1994 / Vol. 33, No. 13 / APPLIED OPTICS 2457

EE

I-mm

centimeters) is instantaneously displayed on thescreen. On the other hand, flows in channels whoseobservations through the walls are usually of lowquality, may be observed from the top.

It is possible to widen the dynamic range of theinstrument by the use of an objective with a highernumerical aperture F, where F = f/D, with f as thefocal distance and D as the aperture. The numericalaperture of the objective used in this paper was F =2.54; this value may be reduced to approximately 1.4(with the consequent improvement in the slope limita-tion) by the use of inexpensive commercial devices.A further reduction of F would require specific de-signed objectives.

Moreover, the method is susceptible to some rel-evant improvements. For instance, objective Li maybe replaced by an anamorphic optical system with amuch greater magnification on the height scale thanon the horizontal scale to achieve higher sensitivities.Finally, because the information is condensed in acurve, the method is suitable for the same digitalimage processing that is used for oscilloscope traces.

L. P. Thomas and R. Gratton were supported by theConsejo Nacional de Investigaciones Cientificas y

T6cnicas (CONICET) as members of its researchstaff. B. Marino was supported by a fellowship of theComisi6n de Investigaciones Cientificas of the Bue-nos Aires Province (CIC). The work was partiallysupported by CONICET and CIC grants.

References

1. K. D. Hinsch, "Holographic interferometry of surface deforma-tions of transparent fluids," Appl. Opt. 17, 3101-3107 (1978).

2. K. Matsuda, S. Watanabe, and T. Eiju, "Real-time measure-ment of large liquid surface deformation using a holographicshearing interferometer, " Appl. Opt. 24, 4443-4447 (1985).

3. C. J. van der Laan and H. J. Frankera, "Thickness measure-ment using Young's interferometric experiment," Appl. Opt.27, 3869-3874 (1988).

4. A. J. Saber, "Light slicing for observation of fluid flows," Appl.Opt. 25, 4605-4608 (1986).

5. P. E. Dimotakis, R. C. Miake-Lye, and D. A. Papantoniou,"Structure and dynamics of round turbulent jets," Phys. Fluids26, 3185-3192 (1983).

6. R. Grousson and S. Mallick, "Study of flow pattern in a fluid byscattered laser light," Appl. Opt. 16, 2334-2336 (1977).

7. F. M. Smolka and T. P. Caudell, "Surface profile measurementand angular deflection monitoring using a scanning laser beam:a noncontact method," Appl. Opt. 17, 3284-3289 (1978).

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