Depth of penetration of bubbles entrained by a plunging water jetChristophe Clanet and Juan C. LasherasDepartment of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla,California 92093-0411

~Received 3 January 1997; accepted 11 March 1996!

A model is proposed to predict the depth of penetration of the air bubbles entrained by a round waterjet impacting into a flat, liquid pool. This depth is shown to be determined only by the initial jetmomentum and by the non-monotonic nature of the bubble terminal velocities as a function of theirsize. The model is shown to be in excellent agreement with measurements of the depth and widthof penetration of the bubbles performed over a wide range of jet diameters, velocities, and plungingangles. ©1997 American Institute of Physics.@S1070-6631~97!00907-0#

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Many industrial and environmental processes involveaeration of a liquid by the entrainment of air bubbles pduced when another liquid of the same or different properimpacts on its surface; e.g., a jet plunging into a poolbreaking wave plunging in the ocean, a droplet impinginga liquid surface, etc. As a consequence,plunging jetshavebeen thoroughly investigated over the years, and an exsive bibliography exists on this issue. For a summary ofmost recent studies, see the comprehensive review by B´n.1

Despite this large body of work, several of the mechanisinvolved in this complex two-phase flow are still poorunderstood,2 and the prediction of important parameters suas the volume of entrained air, the bubble size distributfunction, or the depth of penetration of the bubbles still relon semi-empirical correlations valid only over a restrictirange of parameters.

In this Letter, we shall report on some experiments cducted with round plunging jets that reveal interesting nresults concerning the depth of penetration of the bubcloud under a wide range of jet diameters, velocities aplunging angles. We will also propose new scaling lawsrived from a simplified model, and we will show that theare in excellent agreement with our experimental obsetions as well as with previous results. Other issues suchentrainment threshold values, flow rate of air entrained,bubble size distributions throughout the submerged jetbe presented elsewhere.3

The problem concerns the air entrained by a round wjet ejecting from a nozzle of diameter,D, with a mean exitvelocity,V0, plunging at an angle,u, into a pool located at adistance,h, from the nozzle, as shown in Fig. 1. The inertviscous, surface tension, and gravity effects are describemeans of the nondimensional Reynolds, Weber, and Fronumbers defined by Re5 V0D/n, We5 rV0

2D/s, Fr5 V0

2/gh, respectively, whereg is the acceleration due tgravity,r is the liquid density ands is the surface tension owater (rwater51000 kg/m3 andswater50.073 kg/s2).

The experiments reported here were carried out in a t50 cm tall, 50 cm wide, and 200 cm long. The ratioh/D wasalways kept constant and equal to 20, and the ratio ofmaximum penetration depth to the total depth neverceeded 0.35 in order to minimize the effect of the finite sof the pool. De-ionized water was supplied to the nozzle aconstant pressure through a regulator and a flowmeter.

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nozzles consisted of stainless steel needles with lengtdiameter ratios,L0 /D, larger than 50. The mean flow rate othe jets was measured with high accuracy flow meters, wthe exit velocity profile was estimated assuming a fully dveloped pipe flow. The range of variation of the control prameters are reported in Table I, where subindex 1 cosponds to the continuous entrainment threshold, and 2 tomaximum values tested.

The bubble penetration depth was measured experimtally applying anedge findingfilter to the average imageresulting from 125 video frames acquired over a 13 s spThe video images were recorded with a Sony xc-77RR caera ~768 pixels3493 pixels!, and processed with theNIHimage 1.60image processing package.

Let us consider first the case of a jet plunging normala flat liquid surface (u50). For a given nozzle diametewhen the velocity of the jet, at the point of impact, is abothe critical valueV1, the suction generated by the submergjet is strong enough to overcome the capillary forces, anshroud of air is entrained into the receiving liquid~Fig. 1!.The resulting air bubbles are then transported in bothaxial and radial directions by the large scale eddies domiing the evolution of the submerged jet.4 As the bubbles arefurther convected by the flow they grow in size, accordingthe equilibrium established between the break up and coacence processes,5 and reach a maximum depth where tbuoyancy force surpasses the viscous drag exerted bydownward moving jet, and the bubbles begin to rise tosurface. Experimentally, it is observed that this flow reveroccurs sharply at a location we shall define as the maximpenetration depth,H ~see Fig.1!. Although at this point alarge fraction of the entrained bubbles is seen rising tosurface, we observe that a certain portion of the entrainedcomposed of smaller bubbles, is transported to deepertances into the pool~see Fig. 1!.

To estimate the mean velocity of the submerged jet afunction of the downstream location, we model the jet asequivalent water jet with an initial momentum flux given bprVj0

2 Dj02 /4, whereVj0 andDj0 are the diameter and veloc

ity of the jet at the point of impact with the flat interfaceSince most of the reported data correspond to turbulent(Re.2300), the assumptionVj05V0, and Dj05D holds.Applying conservation of linear momentum between the ipact section and the section of the submerged jet locate

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z5H, we estimate the mean velocity of the jetUj (z) as:

Uj~z!5V0

D

Dj~z!. ~1!

Consistent with our experimental observation, and pviously reported measurements for high Reynonumbers,1,6 we assume that the submerged biphasicspreads linearly with a constant half angle,a'12.5°, inde-pendent of Re~a result well known in turbulent single phasjets7,8!. The penetration depthH, can then be estimated athe distance downstream where the mean downward veloof the jet equals the terminal velocity of the bubbl(Uj (H)5UT).

6 Thus,

H

D5

1

2 tana

V0

UT. ~2!

The remaining problem is to calculate the terminal vlocity UT of the bubbles that reach the maximum penetratdepth,H. First, recall that beyond the point defined as tmaximum penetration, we observed a sharp drop inbubble size whereby bubbles much smaller than those riwere seen being transported deeper into the pool by themerged jet. The explanation for this sharp transitionbubble size observed atz5H rests on the dependency of th

TABLE I. Range of variation of the different relevant parameters.

D~mm! V1 ~m/s! V2~m/s! Re1 Re2 We1 We2 Fr1 Fr2

2.16 1.9 6.3 4102 13601 106 1173 8 91.60 2.1 7.3 3360 11680 96 1168 14 161.19 2.5 12 2978 14292 101 2349 26 610.838 2.8 15 2346 12570 89 2582 47 1360.584 3.9 13 2278 7591 121 1351 132 140.394 5.2 13 2049 5122 145 912 349 2180.241 8.0 18 1928 4338 211 1069 1353 68

FIG. 1. Plunging jet (V052.4 m/s,D52.159 mm,h/D57 andu50°).

Phys. Fluids, Vol. 9, No. 7, July 1997

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bubble terminal velocity on its size. If this velocity increasmonotonically with the size of the bubbles, the bubbwould have been sorted out by the pull of gravity risingthe surface from depths which would increase as their sizdecreased. The actual behavior of the bubbles in the smerged jet is drastically different since the terminal velocfollows a non-monotonic law and undergoes a markchange of behavior between the gravity and viscosity donated regime~Stokes regime!, and the purely buoyancydominated regime~Taylor regime! as shown in Fig. 2~seeMendelson,9 Clift et al.,10 and Maxworthyet al.11!. In ourexperiments, Mo'2.53 10211. Thus, as themean jet velocitdecreases, it eventually reaches the velocity correspondinthe minimum in Fig. 2~note that the terminal velocity to thiminimum isUT'22 cm/s independent of the Morton numbMo!, at which point all bubbles with sizes greater than 1 mwill suddenly rise to the surface~see Fig. 2!, thereby, pro-ducing a sharp drop in the bubble size from 6 mm to 1 mOur preliminary measurements of the bubble size at

FIG. 2. Raw data curves of the terminal velocity ofUT vs the diameterd ofthe bubble for selected values of the Morton number Mo5 gn4r3/s3 ~Fig. 4from reference 11!. Note that the terminal velocity corresponding to thminimum isUT ' 22 cm/s independent of the Morton number Mo!.

FIG. 3. Penetration depthH/D and widthW/D as a function ofV0/UT fordifferent diameters.

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maximum penetration depth (z5H) agree qualitatively withthe above values. In our experiments, we measure a tration in the bubble size from approximately 4 mm to 0.5 mThe reasons for the observed differences could be due tomain effects: an error in our estimation ofs due to surfactantcontamination, and bubble-bubble interaction or clustereffects that are not considered here. The data in Fig. 2respond to isolated single bubble experiments, however,could modify the expression for the rise velocity of thbubble in a swarm asUT5UTbubble

A12aG, whereaG is thegas volume fraction. Measurements of the void fraction aits influence on the prediction of the maximum penetratdepth will be presented elsewhere.3

The above discussion implies that the terminal velocof the bubbles located at the maximum depth is determiexclusively by the value of the minimum in Fig. 2. Therfore, it depends only on the density and surface tensionthe liquid and is independent of the jet’s parameters. TakUT50.22 m/s in Eq.~2!, leads toH/D'10.25V0 ~which isvery close to the well known result of Suciu and Smigelsc6

H/D'10V0). The penetration reported by these authors athe one evaluated by the above method corresponds todistanceZeq in Fig. 1. However, the way we measured tmaximumpenetrationH also involves the turbulent fluctuations at the centerline of the jet. The largest scales of thfluctuations at the locationz5Zeq, are of the order of magnitude ofDeq52 tanaZeq ~see Fig. 1!. Equation~2! can thenbe further corrected to give:

H

D511tana

2 tana

V0

UT. ~3!

The maximum width of the bubble cloud,W, can then beevaluated as the width of the jet at theZeq location~Fig. 1!:

FIG. 4. Non dimensional penetration depthH/D. UT/V0 as a function of thepenetration angleu, for different diameters and velocities.

1866 Phys. Fluids, Vol. 9, No. 7, July 1997

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In Fig. 3, we present the measurements of the penetion depth,H/D, and of the widthW/D for the whole rangeof diameters and velocities~Table I! and compare them withthe predictions given by Eqs.~3! and ~4!. Noteworthy inthese results is the excellent agreement of all the datathe predictions, where the bubble terminal velocity has bassumed to be in all cases that of the minimum in Fig(UT522 cm/s), and the spreading angle of the submerjet a512.5°.

Finally, the above argument can be extended toplunging into the free liquid surface at an arbitrary angu. The maximum penetration of the cloud is then calculafrom simple geometrical relations as:

H

D5

~11tana!cosu1tana sinu

2 tana

cos~u2a!

cosa

V0

UT. ~5!

The function (H/D)(UT /V0), that only depends onu, ispresented in Fig. 4. The comparison of the predicted val@Eq. ~5!# with the measurements reveals an agreement wi15% for u.30°, while the ‘‘overshoot’’ observed for0,u,30 is underestimated by our model.

ACKNOWLEDGMENTS

We would like to thank Ken Kiger and E.Villermaux fomany fruitful discussions and G. Pre´aux for making some ofthe measurements presented in this letter. Finally, this wwas partially supported by ONR Grant No. N0014-96-0213.

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2A. Prosperetti, H. N. Oguz, and A. M. Lezzi, ‘‘Examples of air-entraininflows,’’ Phys. Fluids A4, 649 ~1992!.

3C. Clanet and J. C. Lasheras, ‘‘Air entrainment by a plunging water j~unpublished!.

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6G. D. Suciu and O. Smigelschi, ‘‘Size of the submerged biphasic regioplunging jet systems,’’ Chem. Eng. Sci.31, 1217~1976!.

7L. Landau and E. Lifchitz,Physique Theorique: Mecanique des Fluide~Mir, Moscow, 1971!.

8G. Horn and M. W. Thiring, ‘‘Angle of spread of free jets,’’ Nature178,205 ~1956!.

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10J. R. Grace, R. Clift, and M. E. Weber,Bubbles, Drops, and Particles~Academic, New York, 1978!.

11T. Maxworthy, C. Gnann, M. Kurten, and F. Durst, ‘‘Experiments on trise of air bubbles in clean viscous liquids,’’ J. Fluid Mech.321, 421~1996!.

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