It was shown in recent work that the crests of surface disturbances on a falling jet are a powerful agent for air entrainment at the free surface of a liquid pool. The paper explores the opposite case in which the jet is disturbed so as to form an axisymmetric trough, rather than a crest. It is found that no air is entrained in this case. The paper concludes with some considerations on the validity of a recently proposed model for air entrainment

Oguz, H.N.

Ohl, C.D.

Prosperetti, Andrea

English

Crossref

University of Houston Institutional Repository (UHIR)

Mechanism of air entrainment by a disturbed liquid jetC. D. Ohl,a) H. N. Og˜uz, and A. Prosperettib)Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218~Received 27 April 1999; accepted 7 April 2000!It was shown in recent work that the crests of surface disturbances on a falling jet are a powerfulagent for air entrainment at the free surface of a liquid pool. The paper explores the opposite casein which the jet is disturbed so as to form an axisymmetric trough, rather than a crest. It is found thatno air is entrained in this case. The paper concludes with some considerations on the validity of arecently proposed model for air entrainment. © 2000 American Institute of Physics.@S1070-6631~00!02507-1#I. INTRODUCTIONConsider a liquid jet falling into a pool of an otherwisequiescent liquid. If precautions are taken to condition theflow so that the jet surface is smooth, one can avoid airentrainment up to Reynolds numbers well beyond 10 000.However, surface disturbances on the jet surface are readilyseen to cause entrainment. An example is shown in Fig. 1,where the bulge of liquid falling with the jet is created byrapidly increasing the jet flow rate. In a recent paper in whichwe have studied this process, we argued that the phenom-enon is due to the high stagnation pressure acting on the poolliquid when its surface is struck by the bulge.1 This momen-tary high pressure causes the formation of a cavity that col-lapses from the sides and entraps air as Fig. 1 demonstrates.The role played by the stagnation pressure can be furtherexamined by considering the converse situation in which thedisturbance on the jet surface has the form of an axisymmet-ric trough, rather than a crest ~Fig. 2!. The apparatus we usedin our earlier study can be easily modified to produce thissituation, and we present here some results of this experi-ment. It is found that in this case the phenomenon is muchmilder and no entrainment is observed. At the end of thepaper we discuss a recent model for air entrainment,2,3 thatwas proposed to explain the aeration of the liquid slugs thatseparate the large bubbles ~the so-called Taylor bubbles! en-countered in gas–liquid flows in the slug-flow regime.II. EXPERIMENTThe apparatus is a slight modification of the one de-scribed in Ref. 1. In that study, a rapid increase of the jetflow rate was obtained by simultaneously opening four sole-noid valves connected to a high-pressure liquid reservoir up-stream of the jet exit. For the purposes of the present experi-ment the valves were instead connected to a low-pressurereservoir that caused a decrease of the jet flow when opened.Pictures of the stroboscopically illuminated jet were takenwith a CCD camera ~Kodak DC2101!. Due to the high re-producibility of the process, by varying the delay betweenthe opening of the valves and the time of illumination, it waspossible to capture the temporal evolution of the jet shape.The velocity U0 of the undisturbed jet at the nozzle exitwas obtained from the flow rate Q0 measured with a rotame-ter ~OMEGA FL 4503-V! and the nozzle diameter D j55.4mm. Upon opening of the valves, the velocity decreases fromU0 to U1 according to a relation of the type U(t)5U01(U12U0) f (t). The function f (t) was measured in Ref. 1and found to be accurately represented byf ~ t !5expF210S tT 21 D2G , ~1!in the range 0<t<T , where T513 ms is the response timeof the system. The steady velocity U1 reached after the open-ing of the valves was obtained by measuring the volume Voutof the liquid flowing into the low-pressure reservoir duringthe time Dt the valves remain open:U154pD j2 ~Q02Qout!. ~2!The flow rate Qout into the reservoir is approximated asQout.Vout /Dt , which is sufficiently accurate since Dt. 200ms is much longer than the valve response time T.III. MODELIn Ref. 1 a quasi-one-dimensional model4–6 was used tocalculate the shape of the jet free surface. The same model isappropriate for the present case. The jet is assumed to beaxisymmetric. The z coordinate is vertical and positivedownward; z50 corresponds to the nozzle exit. Upon inte-grating over the jet cross section, the equation of continuitybecomes] A]t1]]z~Au !50, ~3!where A5A(z ,t) is the local cross-sectional area of the jetand u(z ,t) is the jet velocity ~positive downward! averagedover the cross section.The momentum equation isa!Present address: Department of Applied Physics, University of Twente, AE7500 Enschede, The Netherlands.b!Also at: Department of Applied Physics, Twente Institute of Mechanics,and Burgerscentrum, University of Twente, AE 7500 Enschede, The Neth-erlands. Electronic mail: prosper@jhu.eduPHYSICS OF FLUIDS VOLUME 12, NUMBER 7 JULY 200017101070-6631/2000/12(7)/1710/5/$17.00 © 2000 American Institute of Physics]u]t1u]u]z5g21r]p]z, ~4!where the average of u2 has been approximated by thesquare of u. This is a good approximation because, due to thethinning of the viscous boundary layer in the contracting partof the nozzle and to the presence of the free surface, thevelocity distribution is close to uniform over the jet crosssection. Because of the thinness of the jet, the pressure p canbe taken to be uniform over the cross section, which leads tothe relationp 5 p01s C, ~5!where p0 is the ambient pressure outside the jet, s the sur-face tension coefficient, and C the local curvature of the in-terface. We assume that, before opening the valves, the jet issteady and in free fall. Thus, the initial velocity distributionFIG. 1. The sequence of events leading to air entrapment by a disturbanceon a falling water jet. The bulge on the jet is caused by a rapid increase ofthe jet flow rate from 0.98 to 1.35 m/s over a time of about 13 ms. Thenozzle diameter is 5.4 mm and its distance from the undisturbed free surface53 mm. The time interval between successive frames is 5 ms. The field ofview in air is 30.0 by 96.4 mm.FIG. 2. Evolution of the jet surface after opening the valves that decreasethe velocity from 0.98 to 0.66 m/s. Successive frames are separated by 5 ms.The field of view in air is 27.8 by 83.0 mm.1711Phys. Fluids, Vol. 12, No. 7, July 2000 Mechanism of air entrainment by a disturbed liquid jetis u(z ,0) 5 (U0212gz)1/2. As a consequence of the massconservation equation ~3! with ]A/]t50, this relation givesthe initial jet shape A(z ,0).Equations ~3! and ~4! are solved numerically by finitedifferences using a Lax–Wendroff two-step scheme and aLax scheme, respectively, for time advancement.7IV. RESULTSFigure 2 is a sequence of photographs separated by 5 ms.Although, as explained before, each photo corresponds to adifferent event, due to the high reproducibility of the experi-ment the sequence can be considered a faithful depiction ofan individual run. The dark fuzzy horizontal stripe in thelower part of each frame is the meniscus of the pool liquidon the wall of the container. This area separates the air abovefrom the water below. The flow rate at the nozzle exit isU050.98 m/s, and the flow rate after opening the valves isU150.65 m/s, so that U1 /U050.66. Although it would bedesirable to generate a stronger disturbance on the jet sur-face, we were unable to do so as attempts to reduce thevelocity any further led to a break-up of the jet. The poolsurface is at a distance z553 mm ~i.e., 9.8 jet diameters!from the nozzle. It is seen here that, contrary to Fig. 1, the jetdisturbance causes no air entrainment.Figure 3 compares the experimental jet shape with thatcalculated according to the previous model ~dashed line!.The shape of the jet surface is extracted from the photo-graphs by means of an edge-detection algorithm based on thelocal maxima of the gradient of the smoothed image.8 In thecalculations we used r5998.2 kg/m3 for the liquid density,s50.072 75 N/m for the surface tension, and g59.81 m/s2for the acceleration of gravity.Overall, the agreement between the experiment and thequasi-one-dimensional model is very good. Some differencesare encountered only in regions of strong curvature, e.g., thefifth frame of Fig. 3. As in Ref. 1, the model tends to under-estimate the curvature in these conditions.From these theoretical results we can calculate the accel-eration with which the contact line between the jet and thepool liquid moves in the horizontal plane. These results areshown in Fig. 4 for the case of Figs. 2 and 3. Here the solidline is the apparent horizontal acceleration of the contact lineat the level of the free surface of the pool liquid and thedashed line is the jet radius R ~to be read on the right verticalscale!. As the jet diameter decreases, the acceleration fallsbelow 22g while it reaches nearly 2g in the opposite direc-tion as the jet diameter recovers. Since, as noted before, themodel tends to underestimate the jet curvature, the actualacceleration is probably somewhat more negative than theestimated model value.It is also interesting to present a different view of thesurface disturbance of the pool liquid, which is given in Fig.5. Here the individual photographs are taken for the sameexperimental parameters as in Fig. 2 and correspond to thesame instants of time, but the camera is set at an angle so asto capture the pool surface together with the jet shape. As thejet disturbance approaches the pool ~frame 5 of Fig. 5!, adepression forms on the surface. This depression reaches itsmaximum at about the time the thinnest part of the jetFIG. 3. Comparison of the experimental shapes of the jet surface at differentinstants of time ~dotted lines! with the calculated profiles ~shaded areas!.The first frame depicts the calculated jet shape 30.5 ms after the opening ofthe valve; subsequent frames are at 5 ms intervals.FIG. 4. Normalized horizontal acceleration a/g ~negative inward, solidcurve! of the contact line between the jet and the pool liquid and normalizedjet radius R/D j ~dashed line, right vertical scale!.FIG. 5. Oblique view of the same event shown in Fig. 2 looking downtoward the pool surface. Note the depression formed by the thinning of thelocal diameter, followed by the generation of capillary waves when thenormal diameter is restored.1712 Phys. Fluids, Vol. 12, No. 7, July 2000 Ohl, Og˜uz, and Prosperettireaches the surface, which happens between frames 7 and 8.The process is similar to a ‘‘rarefaction wave’’ on the surfaceof the pool liquid. At later times, as the jet diameter recovers,circular capillary waves are seen to radiate away from theimpingement point.V. DISCUSSIONIn recent work on air entrainment in gas–liquid slugflow,2,3 an essentially geometric criterion for air entrainmentwas proposed. The postulated mechanism is shown in Fig. 6and the condition for entrainment is phrased in terms of thetwo slopes S1 ,S2 shown in the figure asS2 . S1 . ~6!The authors assume that a series of surface waves is presenton both the vertical and the horizontal liquid surfaces andfurther elaborate this condition formulating it in terms ofwavelengths and wave amplitudes. Since in our case we onlyhave a single disturbance on the jet, and no waves on thepool surface ~at least initially!, it is best to consider the en-trainment criterion in the original form ~6!, rather than in thederived form based on a conceptual picture not directly ap-plicable to our situation. It may be noted that a similar geo-metric picture of air entrainment has been considered before~see e.g., Ref. 9 cited in Ref. 10 and Refs. 11 and 12! andsuccessfully used, in a slightly different context, in Ref. 13.Let us first consider the case of a ‘‘negative’’ bulgeshown in Figs. 2 and 5. In this case the pool surface pertur-bation is very mild so that S2. 0 while, after the passage ofthe point of minimum diameter, S1 is relatively close to 12 pso that ~6! is not satisfied and, indeed, no entrainment isobserved.It would be hard, however, to explain air entrainment inthe case of a positive bulge on the basis of the criterion ~6!.After striking the water surface, not only Fig. 1, but all ourother pictures ~some of which were published in Ref. 1!clearly show that the bulge loses its integrity and the jetsurface in the developing cavity is smooth, except for theeffect of optical distortions due to capillary waves. It is evi-dent from the photos that, after the initial bulge impact, theprocess of air entrapment is not related in any way to thepresence of disturbances on the jet, but is simply the result ofthe closing of the cavity against the jet surface. The processis, therefore, governed by dynamics rather than geometry asthe criterion ~6! would imply. We reached the same conclu-sion in an earlier work14 devoted to the initial impact of acylindrical liquid mass on an undisturbed water surface. Alsoin that case the evolution of the phenomenon is very similarto the present one, with a cavity produced by the large stag-nation pressure that eventually collapses on itself entrappingan air bubble. In this case as well the criterion ~6! does notapply.These conclusions of course do not imply that a particu-lar spatial arrangements of the free surfaces coming into con-tact cannot entrap bubbles. An example is shown in Fig. 7where a small bubble is caught in the underside of a strongpositive bulge similar to that of Fig. 1. Here the pool surfaceis nearly flat so that S2.0, while the lower surface of thebulge bends downward toward the rim, so that S1, 0 and ~6!is satisfied. In frames 3 and 4 of this sequence the entrappedbubble is seen to split in two, probably due to the shear layerbetween the jet flow and the pool liquid. The early stages offormation of the large cavity produced by the stagnationpressure are visible in the last two frames.A more dramatic example occurs when the jet surface isvery rough as, in these circumstances, the repeated impactsof the surface disturbances give rise ~on average! to an airsheath separating the jet from the pool liquid.9–12,15,16 Inthese conditions, the crest of a wave on the surface of thefalling liquid often entraps an air mass by ‘‘capping’’ a sur-face depression as postulated in Refs. 2, 3, and 17. ~Thissituation is different from that encountered at low Reynoldsnumbers, where the air sheath is nearly steady and is causedby viscous effects.18 Under these circumstances, air entrain-ment probably occurs through an instability of the air film ashypothesized in Ref. 19.! Thus, while the model ~6! does notfit the present observations, it is certainly applicable in otherconditions. The point we wish to stress, however, is that thedegree of surface disruption necessary for the criterion ~6! tolead to air entrapment is only possible under the action of theFIG. 6. Sketch of the air entrapment mechanism proposed in Refs. 2 and 3.It is postulated that surface disturbances on the jet surface entrap air ~shadedarea! by a ‘‘capping’’ action.FIG. 7. The bulge shown here is generated by increasing the jet velocityfrom 0.69 to 1.06 m/s and is, therefore, bigger than that of Fig. 1. As aconsequence, the outer rim of the bulge bends downward, the criterion ~6! issatisfied, and a small bubble is entrapped ~frame 3!, which subsequentlybreaks up due to shear ~frames 4 and 5!. In the last two frames the onset ofthe large cavity generated by the stagnation pressure is visible. As before,the images in this sequence correspond to different events but the consistentappearance of the small entrapped bubble shows the remarkable detailedreproducibility of the process. The time interval between frames is 5 ms andthe field of view in air 28 by 58 mm.1713Phys. Fluids, Vol. 12, No. 7, July 2000 Mechanism of air entrainment by a disturbed liquid jetpowerful stagnation pressures caused by the repeated impactof positive ~outward! surface disturbances.ACKNOWLEDGMENTSThe authors express their appreciation to the authors ofRefs. 2, 3, and 17 for sharing with them their data prior topublication and for motivating this work. This study hasbeen supported by the Office of Naval Research.1Y. G. Zhu, H. N. Og˜uz, and A. Prosperetti, ‘‘Air entrainment by imping-ing liquid jets with surface disturbances,’’ J. Fluid Mech. 404, 151 ~2000!;see also Phys. Fluids 10, S3 ~1998!.2J. Kockx, ‘‘Experiments on the gas exchange between a Taylor bubble andits liquid slug in a vertical tube,’’ Ph.D. thesis, Delft University of Tech-nology, Delft, The Netherlands, 1999.3J. P. Kockx, F. T. M. Niewstadt, R. V. A. Oliemans, and R. Delfos,‘‘Instantaneous film thickness measurements on a Taylor bubble in a ver-tical tube’’ ~in preparation!.4D. B. Bogy, ‘‘Wave propagation and instability in a circular semi-infiniteliquid jet harmonically forced at the nozzle,’’ J. Appl. Mech. 45, 469~1978!.5G. Meier, A. Klo¨pper, and G. Grabitz, ‘‘The influence of kinematic waveson jet breakdown,’’ Exp. Fluids 12, 173 ~1992!.6G. Meier, S. Loose, and B. Stasicki, ‘‘Unsteady liquid jets,’’ Appl. Sci.Res. 58, 207 ~1998!.7W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. Flannery,Numerical Recipes in C, 2nd ed. ~Cambridge U.P., Cambridge, 1995!.8J. Canny, ‘‘A computational approach to edge detection,’’ IEEE Trans.Pattern Anal. Mach. Intell. 8, 679 ~1986!.9M. J. McCarthy, J. Henderson, and N. A. Molloy, ‘‘Entrainment by plung-ing jets,’’ in Proc. Chemeca 1970 Conf. ~Butterworths, London, 1970!, pp.86–100.10J. M. Burgess, N. A. Molloy, and M. J. McCarthy, ‘‘A note on the plung-ing liquid jet reactor,’’ Chem. Eng. Sci. 27, 442 ~1972!.11D. A. Ervine, E. McKeogh, and E. M. Elsawy, ‘‘Effect of turbulenceintensity on the rate of air entrainment by plunging water jets,’’ Proc. Inst.Chem. Engrs. 69, 425 ~1980!.12A. K. Bin´, ‘‘Gas entrainment by plunging liquid jets,’’ Chem. Eng. Sci.48, 3585 ~1993!.13H. N. Og˜uz, ‘‘The role of surface disturbances in the entrainment ofbubbles by a liquid jet,’’ J. Fluid Mech. 372, 189 ~1998!.14H. N. Og˜uz, A. Prosperetti, and A. R. Kolaini, ‘‘Air entrapment by afalling water mass,’’ J. Fluid Mech. 294, 181 ~1995!.15E. Van de Sande and J. M. Smith, ‘‘Surface entrainment by high-velocitywater jets,’’ Chem. Eng. Sci. 28, 1161 ~1973!.16E. J. McKeogh and D. A. Ervine, ‘‘Air entrainment rate and diffusionpattern of plunging liquid jets,’’ Chem. Eng. Sci. 39, 1161 ~1981!.17R. Delfos, ‘‘Experiments on air entrainment from a stationary slug bubblein a vertical tube,’’ Ph.D. thesis, Delft University of Technology, Delft,The Netherlands, 1996.18T. J. Lin and H. G. Donnelly, ‘‘Gas bubble entrainment by plunginglaminar liquid jets,’’ AIChE. J. 12, 563 ~1966!.19A. M. Lezzi and A. Prosperetti, ‘‘The stability of an air film in a liquidflow,’’ J. Fluid Mech. 226, 319 ~1991!.1714 Phys. Fluids, Vol. 12, No. 7, July 2000 Ohl, Og˜uz, and Prosperetti

Mechanism of air entrainment by a disturbed liquid jet

Oğuz, H.N.

Prosperetti, A.

University of Twente Research Information

PHYSICS OF FLUIDS VOLUME 12, NUMBER 7 JULY 2000Mechanism of air entrainment by a disturbed liquid jetC. D. Ohl,a) H. N. Og̃uz, and A. Prosperettib)Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218~Received 27 April 1999; accepted 7 April 2000!It was shown in recent work that the crests of surface disturbances on a falling jet are a powerfulagent for air entrainment at the free surface of a liquid pool. The paper explores the opposite casein which the jet is disturbed so as to form an axisymmetric trough, rather than a crest. It is found thatno air is entrained in this case. The paper concludes with some considerations on the validity of arecently proposed model for air entrainment. ©2000 American Institute of Physics.@S1070-6631~00!02507-1#etha0d.yicopocoatththedhieucththejeleuesuneenase.it-ome-gsel isbeee-ityjet, AicsetI. INTRODUCTIONConsider a liquid jet falling into a pool of an otherwisquiescent liquid. If precautions are taken to conditionflow so that the jet surface is smooth, one can avoidentrainment up to Reynolds numbers well beyond 10 0However, surface disturbances on the jet surface are reaseen to cause entrainment. An example is shown in Figwhere the bulge of liquid falling with the jet is created brapidly increasing the jet flow rate. In a recent paper in whwe have studied this process, we argued that the phenenon is due to the high stagnation pressure acting on theliquid when its surface is struck by the bulge.1 This momen-tary high pressure causes the formation of a cavity thatlapses from the sides and entraps air as Fig. 1 demonstrThe role played by the stagnation pressure can be furexamined by considering the converse situation in whichdisturbance on the jet surface has the form of an axisymmric trough, rather than a crest~Fig. 2!. The apparatus we usein our earlier study can be easily modified to produce tsituation, and we present here some results of this expment. It is found that in this case the phenomenon is mmilder and no entrainment is observed. At the end ofpaper we discuss a recent model for air entrainment,2,3 thatwas proposed to explain the aeration of the liquid slugsseparate the large bubbles~the so-called Taylor bubbles! en-countered in gas–liquid flows in the slug-flow regime.II. EXPERIMENTThe apparatus is a slight modification of the one dscribed in Ref. 1. In that study, a rapid increase of theflow rate was obtained by simultaneously opening four sonoid valves connected to a high-pressure liquid reservoirstream of the jet exit. For the purposes of the present expment the valves were instead connected to a low-presreservoir that caused a decrease of the jet flow when opePictures of the stroboscopically illuminated jet were taka!Present address: Department of Applied Physics, University of Twente7500 Enschede, The Netherlands.b!Also at: Department of Applied Physics, Twente Institute of Mechanand Burgerscentrum, University of Twente, AE 7500 Enschede, The Nerlands. Electronic mail: prosper@jhu.edu1711070-6631/2000/12(7)/1710/5/$17.00Downloaded 02 May 2006 to 130.89.85.19. Redistribution subject to AIPeir0.ily1,hm-oll-es.eret-sri-heat-t-p-ri-reed.nwith a CCD camera~Kodak DC2101!. Due to the high re-producibility of the process, by varying the delay betwethe opening of the valves and the time of illumination, it wpossible to capture the temporal evolution of the jet shapThe velocityU0 of the undisturbed jet at the nozzle exwas obtained from the flow rateQ0 measured with a rotameter ~OMEGA FL 4503-V! and the nozzle diameterD j55.4mm. Upon opening of the valves, the velocity decreases frU0 to U1 according to a relation of the typeU(t)5U01(U12U0) f (t). The functionf (t) was measured in Ref. 1and found to be accurately represented byf ~ t !5expF210S tT 21D2G , ~1!in the range 0<t<T, whereT513 ms is the response timof the system. The steady velocityU1 reached after the opening of the valves was obtained by measuring the volumeVoutof the liquid flowing into the low-pressure reservoir durinthe timeDt the valves remain open:U154pD j2 ~Q02Qout!. ~2!The flow rateQout into the reservoir is approximated aQout.Vout/Dt, which is sufficiently accurate sinceDt. 200ms is much longer than the valve response timeT.III. MODELIn Ref. 1 a quasi-one-dimensional model4–6 was used tocalculate the shape of the jet free surface. The same modappropriate for the present case. The jet is assumed toaxisymmetric. Thez coordinate is vertical and positivdownward;z50 corresponds to the nozzle exit. Upon intgrating over the jet cross section, the equation of continubecomes] A]t1]]z~Au!50, ~3!whereA5A(z,t) is the local cross-sectional area of theandu(z,t) is the jet velocity~positive downward! averagedover the cross section.The momentum equation isE,h-0 © 2000 American Institute of Physics license or copyright, see http://pof.aip.org/pof/copyright.jspetheartthesss toet isnanehrfaldase5 ms.1711Phys. Fluids, Vol. 12, No. 7, July 2000 Mechanism of air entrainment by a disturbed liquid jetFIG. 1. The sequence of events leading to air entrapment by a disturbon a falling water jet. The bulge on the jet is caused by a rapid increasthe jet flow rate from 0.98 to 1.35 m/s over a time of about 13 ms. Tnozzle diameter is 5.4 mm and its distance from the undisturbed free su53 mm. The time interval between successive frames is 5 ms. The fieview in air is 30.0 by 96.4 mm.Downloaded 02 May 2006 to 130.89.85.19. Redistribution subject to AIP]u]t1u]u]z5g21r]p]z, ~4!where the average ofu2 has been approximated by thsquare ofu. This is a good approximation because, due tothinning of the viscous boundary layer in the contracting pof the nozzle and to the presence of the free surface,velocity distribution is close to uniform over the jet crosection. Because of the thinness of the jet, the pressurep canbe taken to be uniform over the cross section, which leadthe relationp 5 p01s C, ~5!wherep0 is the ambient pressure outside the jet,s the sur-face tension coefficient, andC the local curvature of the in-terface. We assume that, before opening the valves, the jsteady and in free fall. Thus, the initial velocity distributioceofeceofFIG. 2. Evolution of the jet surface after opening the valves that decrethe velocity from 0.98 to 0.66 m/s. Successive frames are separated byThe field of view in air is 27.8 by 83.0 mm. license or copyright, see http://pof.aip.org/pof/copyright.jspssdmtorineuidoisistojethtontcthr-cel-thearelidlinehelalls, thetualtheeig.metheo asthes itsjetrengzedwnthethe1712 Phys. Fluids, Vol. 12, No. 7, July 2000 Ohl, Og̃uz, and Prosperettiis u(z,0) 5 (U0212gz)1/2. As a consequence of the maconservation equation~3! with ]A/]t50, this relation givesthe initial jet shapeA(z,0).Equations~3! and ~4! are solved numerically by finitedifferences using a Lax–Wendroff two-step scheme anLax scheme, respectively, for time advancement.7IV. RESULTSFigure 2 is a sequence of photographs separated by 5Although, as explained before, each photo correspondsdifferent event, due to the high reproducibility of the expement the sequence can be considered a faithful depictioan individual run. The dark fuzzy horizontal stripe in thlower part of each frame is the meniscus of the pool liqon the wall of the container. This area separates the air abfrom the water below. The flow rate at the nozzle exitU050.98 m/s, and the flow rate after opening the valvesU150.65 m/s, so thatU1 /U050.66. Although it would bedesirable to generate a stronger disturbance on the jetface, we were unable to do so as attempts to reducevelocity any further led to a break-up of the jet. The posurface is at a distancez553 mm ~i.e., 9.8 jet diameters!from the nozzle. It is seen here that, contrary to Fig. 1, thedisturbance causes no air entrainment.Figure 3 compares the experimental jet shape withcalculated according to the previous model~dashed line!.The shape of the jet surface is extracted from the phographs by means of an edge-detection algorithm based olocal maxima of the gradient of the smoothed image.8 In thecalculations we usedr5998.2 kg/m3 for the liquid density,s50.072 75 N/m for the surface tension, andg59.81 m/s2for the acceleration of gravity.Overall, the agreement between the experiment andquasi-one-dimensional model is very good. Some differenare encountered only in regions of strong curvature, e.g.,FIG. 3. Comparison of the experimental shapes of the jet surface at diffeinstants of time~dotted lines! with the calculated profiles~shaded areas!.The first frame depicts the calculated jet shape 30.5 ms after the openithe valve; subsequent frames are at 5 ms intervals.Downloaded 02 May 2006 to 130.89.85.19. Redistribution subject to AIPas.a-ofvesur-heltat-theheesefifth frame of Fig. 3. As in Ref. 1, the model tends to undeestimate the curvature in these conditions.From these theoretical results we can calculate the aceration with which the contact line between the jet andpool liquid moves in the horizontal plane. These resultsshown in Fig. 4 for the case of Figs. 2 and 3. Here the soline is the apparent horizontal acceleration of the contactat the level of the free surface of the pool liquid and tdashed line is the jet radiusR ~to be read on the right verticascale!. As the jet diameter decreases, the acceleration fbelow 22g while it reaches nearly 2g in the opposite direc-tion as the jet diameter recovers. Since, as noted beforemodel tends to underestimate the jet curvature, the acacceleration is probably somewhat more negative thanestimated model value.It is also interesting to present a different view of thsurface disturbance of the pool liquid, which is given in F5. Here the individual photographs are taken for the saexperimental parameters as in Fig. 2 and correspond toame instants of time, but the camera is set at an angle sto capture the pool surface together with the jet shape. Asjet disturbance approaches the pool~frame 5 of Fig. 5!, adepression forms on the surface. This depression reachemaximum at about the time the thinnest part of thentofFIG. 4. Normalized horizontal accelerationa/g ~negative inward, solidcurve! of the contact line between the jet and the pool liquid and normalijet radiusR/D j ~dashed line, right vertical scale!.FIG. 5. Oblique view of the same event shown in Fig. 2 looking dotoward the pool surface. Note the depression formed by the thinning oflocal diameter, followed by the generation of capillary waves whennormal diameter is restored. license or copyright, see http://pof.aip.org/pof/copyright.jspndcersthgnigthesnofont-apofo3etufiinurjethi-ththelt ofessy asu-lsoilartag-ingcu-on-. 7ongcepedyerofionisactsther-ldsusedin-asherthethendcitys affore,istenttailedand1713Phys. Fluids, Vol. 12, No. 7, July 2000 Mechanism of air entrainment by a disturbed liquid jetreaches the surface, which happens between frames 7 aThe process is similar to a ‘‘rarefaction wave’’ on the surfaof the pool liquid. At later times, as the jet diameter recovecircular capillary waves are seen to radiate away fromimpingement point.V. DISCUSSIONIn recent work on air entrainment in gas–liquid sluflow,2,3 an essentially geometric criterion for air entrainmewas proposed. The postulated mechanism is shown in Fand the condition for entrainment is phrased in terms oftwo slopesS1 ,S2 shown in the figure asS2 . S1 . ~6!The authors assume that a series of surface waves is pron both the vertical and the horizontal liquid surfaces afurther elaborate this condition formulating it in termswavelengths and wave amplitudes. Since in our case wehave a single disturbance on the jet, and no waves onpool surface~at least initially!, it is best to consider the entrainment criterion in the original form~6!, rather than in thederived form based on a conceptual picture not directlyplicable to our situation. It may be noted that a similar gemetric picture of air entrainment has been considered be~see e.g., Ref. 9 cited in Ref. 10 and Refs. 11 and 12! andsuccessfully used, in a slightly different context, in Ref. 1Let us first consider the case of a ‘‘negative’’ bulgshown in Figs. 2 and 5. In this case the pool surface perbation is very mild so thatS2. 0 while, after the passage othe point of minimum diameter,S1 is relatively close to12 pso that ~6! is not satisfied and, indeed, no entrainmentobserved.It would be hard, however, to explain air entrainmentthe case of a positive bulge on the basis of the criterion~6!.After striking the water surface, not only Fig. 1, but all oother pictures~some of which were published in Ref. 1!clearly show that the bulge loses its integrity and thesurface in the developing cavity is smooth, except foreffect of optical distortions due to capillary waves. It is evdent from the photos that, after the initial bulge impact,FIG. 6. Sketch of the air entrapment mechanism proposed in Refs. 2 aIt is postulated that surface disturbances on the jet surface entrap air~sh dedarea! by a ‘‘capping’’ action.Downloaded 02 May 2006 to 130.89.85.19. Redistribution subject to AIP8.,et. 6eentdlyhe--re.r-steeprocess of air entrapment is not related in any way topresence of disturbances on the jet, but is simply the resuthe closing of the cavity against the jet surface. The procis, therefore, governed by dynamics rather than geometrthe criterion~6! would imply. We reached the same conclsion in an earlier work14 devoted to the initial impact of acylindrical liquid mass on an undisturbed water surface. Ain that case the evolution of the phenomenon is very simto the present one, with a cavity produced by the large snation pressure that eventually collapses on itself entrappan air bubble. In this case as well the criterion~6! does notapply.These conclusions of course do not imply that a partilar spatial arrangements of the free surfaces coming into ctact cannot entrap bubbles. An example is shown in Figwhere a small bubble is caught in the underside of a strpositive bulge similar to that of Fig. 1. Here the pool surfais nearly flat so thatS2.0, while the lower surface of thebulge bends downward toward the rim, so thatS1, 0 and~6!is satisfied. In frames 3 and 4 of this sequence the entrapbubble is seen to split in two, probably due to the shear labetween the jet flow and the pool liquid. The early stagesformation of the large cavity produced by the stagnatpressure are visible in the last two frames.A more dramatic example occurs when the jet surfacevery rough as, in these circumstances, the repeated impof the surface disturbances give rise~on average! to an airsheath separating the jet from the pool liquid.9–12,15,16 Inthese conditions, the crest of a wave on the surface offalling liquid often entraps an air mass by ‘‘capping’’ a suface depression as postulated in Refs. 2, 3, and 17.~Thissituation is different from that encountered at low Reynonumbers, where the air sheath is nearly steady and is caby viscous effects.18 Under these circumstances, air entrament probably occurs through an instability of the air filmhypothesized in Ref. 19.! Thus, while the model~6! does notfit the present observations, it is certainly applicable in otconditions. The point we wish to stress, however, is thatdegree of surface disruption necessary for the criterion~6! tolead to air entrapment is only possible under the action of3.FIG. 7. The bulge shown here is generated by increasing the jet velofrom 0.69 to 1.06 m/s and is, therefore, bigger than that of Fig. 1. Aconsequence, the outer rim of the bulge bends downward, the criterion~6! issatisfied, and a small bubble is entrapped~frame 3!, which subsequentlybreaks up due to shear~f ames 4 and 5!. In the last two frames the onset othe large cavity generated by the stagnation pressure is visible. As bethe images in this sequence correspond to different events but the consappearance of the small entrapped bubble shows the remarkable dereproducibility of the process. The time interval between frames is 5 msthe field of view in air 28 by 58 mm. license or copyright, see http://pof.aip.org/pof/copyright.jsppstos-ah-s,veteesi.ryns.g--est.i.ofacitynblet,g1714 Phys. Fluids, Vol. 12, No. 7, July 2000 Ohl, Og̃uz, and Prosperettipowerful stagnation pressures caused by the repeated imof positive ~outward! surface disturbances.ACKNOWLEDGMENTSThe authors express their appreciation to the authorRefs. 2, 3, and 17 for sharing with them their data priorpublication and for motivating this work. This study habeen supported by the Office of Naval Research.1Y. G. Zhu, H. N. Og̃uz, and A. Prosperetti, ‘‘Air entrainment by impinging liquid jets with surface disturbances,’’ J. Fluid Mech.404, 151~2000!;see also Phys. Fluids10, S3 ~1998!.2J. Kockx, ‘‘Experiments on the gas exchange between a Taylor bubbleits liquid slug in a vertical tube,’’ Ph.D. thesis, Delft University of Tecnology, Delft, The Netherlands, 1999.3J. P. Kockx, F. T. M. Niewstadt, R. V. A. Oliemans, and R. Delfo‘‘Instantaneous film thickness measurements on a Taylor bubble in atical tube’’ ~in preparation!.4D. B. Bogy, ‘‘Wave propagation and instability in a circular semi-infiniliquid jet harmonically forced at the nozzle,’’ J. Appl. Mech.45, 469~1978!.5G. Meier, A. Klöpper, and G. Grabitz, ‘‘The influence of kinematic wavon jet breakdown,’’ Exp. Fluids12, 173 ~1992!.6G. Meier, S. Loose, and B. Stasicki, ‘‘Unsteady liquid jets,’’ Appl. ScRes.58, 207 ~1998!.7W. H. Press, W. T. Vetterling, S. A. Teukolsky, and B. P. FlanneDownloaded 02 May 2006 to 130.89.85.19. Redistribution subject to AIPactofndr-,Numerical Recipes in C, 2nd ed.~Cambridge U.P., Cambridge, 1995!.8J. Canny, ‘‘A computational approach to edge detection,’’ IEEE TraPattern Anal. Mach. Intell.8, 679 ~1986!.9M. J. McCarthy, J. Henderson, and N. A. Molloy, ‘‘Entrainment by pluning jets,’’ in Proc. Chemeca 1970 Conf.~Butterworths, London, 1970!, pp.86–100.10J. M. Burgess, N. A. Molloy, and M. J. McCarthy, ‘‘A note on the plunging liquid jet reactor,’’ Chem. Eng. Sci.27, 442 ~1972!.11D. A. Ervine, E. McKeogh, and E. M. Elsawy, ‘‘Effect of turbulencintensity on the rate of air entrainment by plunging water jets,’’ Proc. InChem. Engrs.69, 425 ~1980!.12A. K. Biń, ‘‘Gas entrainment by plunging liquid jets,’’ Chem. Eng. Sc48, 3585~1993!.13H. N. Og̃uz, ‘‘The role of surface disturbances in the entrainmentbubbles by a liquid jet,’’ J. Fluid Mech.372, 189 ~1998!.14H. N. Og̃uz, A. Prosperetti, and A. R. Kolaini, ‘‘Air entrapment byfalling water mass,’’ J. Fluid Mech.294, 181 ~1995!.15E. Van de Sande and J. M. Smith, ‘‘Surface entrainment by high-velowater jets,’’ Chem. Eng. Sci.28, 1161~1973!.16E. J. McKeogh and D. A. Ervine, ‘‘Air entrainment rate and diffusiopattern of plunging liquid jets,’’ Chem. Eng. Sci.39, 1161~1981!.17R. Delfos, ‘‘Experiments on air entrainment from a stationary slug bubin a vertical tube,’’ Ph.D. thesis, Delft University of Technology, DelfThe Netherlands, 1996.18T. J. Lin and H. G. Donnelly, ‘‘Gas bubble entrainment by plunginlaminar liquid jets,’’ AIChE. J.12, 563 ~1966!.19A. M. Lezzi and A. Prosperetti, ‘‘The stability of an air film in a liquidflow,’’ J. Fluid Mech.226, 319 ~1991!. license or copyright, see http://pof.aip.org/pof/copyright.jsp

Mechanism of air entrainment by a distrubed liquid jet

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Mechanism of air entrainment by a disturbed liquid jet

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