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International Journal of Multiphase Flow xxx (2011) xxx–xxx

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International Journal of Multiphase Flow

journal homepage: www.elsevier .com/ locate / i jmulflow

Characterization of plunging liquid jets: A combined experimentaland numerical investigation

X.L. Qu a, L. Khezzar a, D. Danciu b, M. Labois c, D. Lakehal c,⇑a Mechanical Engineering Department, The Petroleum Institute, Abu Dhabi, United Arab Emiratesb Forschungszentrum Rossendorf, Institute of Safety Research (FZD) Dresden, Germanyc ASCOMP GmbH, Zurich, Switzerland

a r t i c l e i n f o

Article history:Received 8 September 2010Received in revised form 10 February 2011Accepted 20 February 2011Available online xxxx

Keywords:Plunging jetPenetrationAir entrainmentMixture modelLevel-setCFD

0301-9322/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseflow.2011.02.006

⇑ Corresponding author. Tel.: +41 44 4454070.E-mail address: [email protected] (D. Lakehal).

Please cite this article in press as: Qu, X.L., et aMultiphase Flow (2011), doi:10.1016/j.ijmultiph

a b s t r a c t

This paper presents a combined experimental and numerical study of the flow characteristics of roundvertical liquid jets plunging into a cylindrical liquid bath. The main objective of the experimental workconsists in determining the plunging jet flow patterns, entrained air bubble sizes and the influence ofthe jet velocity and variations of jet falling lengths on the jet penetration depth. The instability of thejet influenced by the jet velocity and falling length is also probed. On the numerical side, two differentapproaches were used, namely the mixture model approach and interface-tracking approach using thelevel-set technique with the standard two-equation turbulence model. The numerical results are con-trasted with the experimental data. Good agreements were found between experiments and the twomodelling approaches on the jet penetration depth and entraining flow characteristics, with interfacetracking rendering better predictions. However, visible differences are observed as to the jet instability,free surface deformation and subsequent air bubble entrainment, where interface tracking is seen to bemore accurate. The CFD results support the notion that the jet with the higher flow rate thus more sus-ceptible to surface instabilities, entrains more bubbles, reflecting in turn a smaller penetration depth as aresult of momentum diffusion due to bubble concentration and generated fluctuations. The liquidaverage velocity field and air concentration under tank water surface were compared to existing semi-analytical correlations. Noticeable differences were revealed as to the maximum velocity at the jetcentreline and associated bubble concentration. The mixture model predicts a higher velocity than thelevel-set and the theory at the early stage of jet penetration, due to a higher concentration of air thatcannot rise to the surface and remain trapped around the jet head. The location of the maximum aircontent and the peak value of air holdup are also predicted differently.

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1. Introduction

In plunging liquid jets the flow consists of an evolving liquidcolumn passing through an ambient atmosphere before impingingon the free surface of a receiving liquid bath. Depending on theflow Reynolds number and height of the falling jet, a bubble swarmmay be entrained beneath the surface by the impacting jet. Thismechanism is frequently encountered in a number of practical sit-uations, such as the aeration of lakes and in waste water treatment,mixing of stagnant water, de-stratification of water reservoirs, andin nuclear and metallurgical industries. Depending on the applica-tion considered, the flow characteristics can play a desired, positiverole, or in contrast an undesired, negative role. Plunging jets arefound to be particularly necessary in the manufacturing of atomiz-ers and emergency cooling techniques in the nuclear industry.

ll rights reserved.

l. Characterization of plungingaseflow.2011.02.006

However, in equipment such as chemical reactors, a plungingliquid jet may give rise to bubbly plumes in the receiving pool withundesirable foaming. In heat-exchanger equipment, the presenceof the air plume can have a significant impact on the heat transfercharacteristics. For all these applications the basic need is to deter-mine the currents induced by the evolving gaseous phase in thesurrounding liquid, and the subsequent mixing and partition ofenergy or species concentration in the flow.

A large amount of experimental work has been devoted to studyplunging jets. Non-invasive techniques, such as photography, lightscattering, laser-Doppler anemometry (Ben-Yosef et al., 1975)were employed to investigate the characteristics of plunging liquidjet including air entrainment, phase velocity fields and size of en-trained air bubbles. For instance, Chirichella et al. (2002) studiedair entrainment due to a translating axisymmetric laminar waterjet plunging into a quiescent pool of water, where several differentair entrainment modes were investigated depending on somecritical values. High-speed visualization was used to successfully

liquid jets: A combined experimental and numerical investigation. Int. J.

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Fig. 1. Sketch of the experimental setup.

2 X.L. Qu et al. / International Journal of Multiphase Flow xxx (2011) xxx–xxx

capture these entrainment regimes. On the other hand, invasivetechniques such as electrical conductivity microprobes (Serizawaet al., 1975) and optical fibre probes (Yu and Kim, 1991) were alsoused to study the jet penetration depth and the air carried under.These experimental studies were complemented by various analyt-ical contributions which provided useful correlations to predictglobal flow features such as the onset of air entrainment, theamount of entrained air, the characteristics of the resulting bubblyplume and mass transfer, and the radial mean velocity and voidfraction. The reader can refer to the comprehensive review of Bin(1993). On the basis of various experimental campaigns, Chanson(1997) and Chanson et al. (2004) established empirical relationsto predict the entrained air content distribution and plunging jetflow motion. These correlations have been proven to be more pre-cise compared to many theories proposed hitherto. Still, the valid-ity of these empirical laws to predict gas penetration andconcentration are challenged by different measurements with dif-ferent flow properties and configurations; one of which is the pres-ent case study.

Numerical predictions of plunging liquid jets are also abundantin the literature. Sene (1984) proposed a sub-grid air entrainmentmodel to predict the dispersed bubbly flow created by a plungingjet. The model was implemented into an Eulerian–Eulerian, two-fluid model to predict the gas holdup distribution underneath thefree surface at different penetration depths, but turned out to beinaccurate in the area close to the entrainment region, seeSchmidtke and Lucas (2009). The same two-fluid model was em-ployed by Bonetto and Lahey (1993) for a round liquid–liquid jet.The Eulerian non-homogeneous multiple size group (MUSIG) mod-el implemented into CFX was employed by Krepper et al. (2008).The separation phenomenon of small and large bubbles seems tobe well described by the model, although the modelling of bubblefragmentation and coalescence is still debatable in this context.Richards et al. (1994) simulated an axisymmetric jet from its for-mation to the dispersion in liquid–liquid systems using the Vol-ume-of-Fluid method (VOF) without a turbulence model (theapproach cannot be considered as a Direct Numerical Simulation(DNS) though). Liovic and Lakehal (2007a) were the first to useLarge-Eddy Simulation for two-phase jetting flows, and applied itto gas injection in a Boiling Water Reactor (BWR) depressurizationwater pool.

The present study reports on findings of a combined experi-mental and simulation investigation of the behaviour of a plungingwater jet under different flow configurations. The measurementdata include the flow map regime, the jet penetration depth, themean Sauter diameter of entrained air bubbles, and other flowcharacteristics like interface instability and re-entrainment. Thenumerical modelling uses the multiphase mixture model approachand the interface tracking technique (level-set), resorting both tothe standard k–e to model turbulence to predict the jet penetrationbelow the free surface, its deformation and the liquid velocity andair holdup below the surface. A comparison between the two mul-tiphase models is conducted against the data and semi-empiricalresults.

2. Experiments

2.1. Experimental Set up

Fig. 1 shows a schematic of the experimental setup conductedat FZD Rossendorf, Germany. The measurements were performedin a 0.3 m � 0.3 m � 0.5 m water tank equipped with transparentacrylic walls for visualization purposes. The water level was keptconstant at 0.28 m. Water was pumped out of the tank andre-injected through a smooth 6 mm inner diameter and 0.55 m

Please cite this article in press as: Qu, X.L., et al. Characterization of plungingMultiphase Flow (2011), doi:10.1016/j.ijmultiphaseflow.2011.02.006

long steel pipe used as a nozzle to produce a vertical jet. A highspeed camera was used to capture images of the impact of thejet on the water. For each experimental condition, a 10 s long se-quence of images of the region below the surface was taken. Back-lighting was used during the experiments in order to ensure aproper exposure at the required filming rate. Experiments wererealized for different nozzle heights and volumetric flow rates.The water velocity V0 at the nozzle exit ranges between 1 m/sand 3.5 m/s. Based on the nozzle diameter d0, the correspondingReynolds number Re = qld0V0/l0 (ql is the water density and l0 isits viscosity) ranges between 5000 and 20,000. Assuming a criticalReynolds number for pipes of 2300, the range of flow parametersemployed ensures a fully developed pipe flow at the nozzle exit.The jet length before impact was varied between 1 cm and20 cm. Assuming free fall of the water after leaving the nozzle,the jet impact velocity, Vj is defined as

Vj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

0 þ 2gLj

qð1Þ

where Lj is the jet falling length from the nozzle tip to the liquid sur-face. According to Eq. (1), this yields impact velocities Vj between1 m/s and 4 m/s.

2.2. Measurements of bubble size distribution

Bubble sizes were estimated from single images by means ofimage processing through subsequent background subtraction, cellsegmentation, bubble detection and bubble size calculation using aHough transform (HT) based algorithm. The dynamic generalisedHough transform (DGHT) algorithm (Leavers, 1989) has been pro-ven to be particularly useful when the object to be recognized andreconstructed is symmetric, such as circles and ellipses. For dis-torted images of the bubble, the DGHT algorithm can be used toreconstruct their 2D shape. Nevertheless, it was found that thisalgorithm could only be used confidently for experiments with asmall number of bubbles in the plume. When the plume is denser,bubbles overlap, obstructing the background light and distinguish-ing between them becomes rather difficult and in this particularcase only bubbles outside the plume in the region of rising bubblesare identified and measured.



X.L. Qu et al. / International Journal of Multiphase Flow xxx (2011) xxx–xxx 3

2.3. Measurements of penetration depth

The penetration depth is the lowest point of plume at a certaininstant. Because the shape of the plume is not completely stable, atime averaged penetration depth is used. Fig. 2 shows two repre-sentative plume snapshots for jets with an impact velocity ofVj = 2.2 m/s. The jet with the higher flow rate (higher V0) entrainsmore bubbles, but its penetration depth is smaller in the snap shot.However, due to the fluctuation of the plumes snapshots are notconclusive. Therefore, for the calculation of the penetration depth,two different approaches were used. Both of them were imple-mented using the Image Processing Toolbox of Matlab. The firstone is an averaging algorithm, which involves several processingsteps for each image, e.g. background subtraction, filtering, re-moval of the small air bubbles at the walls and thresholding. Thus2000 thresholded images were recorded over 10 s with a frequencyof 200 frames/s to obtain an average. In view of the fact that tem-poral resolution is quite high, a total of 4 runs were made for thesame experimental conditions. The second algorithm calculatesthe histogram of the penetration depth for each experimental se-quence and in this case the penetration depth with the highest rep-etition rate is chosen. Results obtained from both methods werecompared and seem to be in good agreement with one another.The difference between the results delivered by these two methodswas found not to exceed 5%, in fact between 2% and 5%.

Fig. 3 presents pictures of time averaged concentration mea-surements for the cases mentioned previously. Despite havingthe same impact velocity, it can be observed that the left plumeis less populated with bubbles than the right one with, however,a larger penetration depth. This reflects the general trend observedin this experiment, that higher flow rates produce higher entrain-ment rates. The phenomenon could perhaps be physically ex-plained as follows: increasing the nozzle jet velocity with a fullydeveloped turbulent pipe flow hence with an appreciable turbu-lence content – for a given diameter – may trigger jet-interface

(a) Lj = 0.1 m, V0 = 1.7 m/s (b

Fig. 2. Two different plumes for je


instabilities, which when impacting on the water surface entrainsmore air bubbles beneath the surface. In turn, jet momentum is dif-fused by the presence of the bubbles, causing a reduction of itspenetration depth. The jet-interface instability phenomenon, afunction of initial turbulence content, details of the nozzle designand length of free-falling jet, is therefore a key element affectingits lateral spreading and diffusion and penetration depth. Alterna-tively, larger flow rates in combination with jet-interface instabil-ities could generate larger air bubbles which are more buoyant andpredominantly localised just below the impact point, see Igushiet al. (1998), and hence escape rapidly the gravity effect leadingthereafter to a shorter penetration.

Looking again at the snapshots depicted in Fig. 3, one can easilyobtain a penetration depth by localising the lowest large bubbles.For time-averaged gas plume pictures this is more difficult, sincethe edges of the plumes are smeared out. A slight estimation erroris thus expected and needs to be considered when comparing thedata with the calculations.

2.4. Measurement of flow regime map and bubble size distribution

One of the most interesting aspects of the present experiment isthe qualitative results obtained for air entrainment beneath thepool free surface. Fig. 4a refers to experimental findings and illus-trates the variation of the jet impact velocity against its free falllength before impact and identifies the flow regime taking place.Each data point represents one experiment for a given jet lengthand a nozzle velocity, resulting in a calculated impact velocityaccording to Eq. (1). The observed entrainment regime is reflectedby the symbol type and colour where three regimes wereobserved; see Schmidtke et al. (2009). In general, the amount ofentrained gas increases with V0 if the jet length is kept constant(vertical rows).

The entrainment regime is only a qualitative criterion forcharacterizing bubble entrainment; quantitative criteria can be

) Lj = 0.075 cm, V0 = 1.9 m/s

t impact velocity Vj = 2.2 m/s.



(a) Lj = 0.1 m, V0= 1.7 m/s (b) Lj = 0.075 cm, V0= 1.9 m/s

Fig. 3. Time averaged images for a jet impact velocity Vj = 2.2 m/s (time interval: 10 s).


obtained by measuring the structure of the bubble plume and bydetecting bubble sizes. Both can be obtained by an evaluation ofthe video images. It is observed from the regime map plotted inFig. 4 that air entrainment by a plunging jet takes place whenthe jet impact velocity exceeds a critical value, which dependsamong other things on jet instability and its interaction with tur-bulence, see Cummings and Chanson (1997a,b). The Sauter meandiameter in the region of the rising bubbles is about 3–4 mm andin agreement with previous data from the literature, e.g. Bin(1993). The smallest observed bubbles are spherical with diame-ters between 0.3 mm and 1.5 mm (smaller bubbles are not re-solved by the present measuring technique). The largest bubbleshave horizontal diameters around the value of 7 mm. They appearhowever, only at high jet flow rates (high V0).

3. Numerical models

3.1. The mixture approach

The first approach employed to predict the two-phase flow pat-tern of the plunging liquid jet is the mixture model, see Manninenet al. (1996). The unsteady equations of continuity, momentum forthe mixture and the volume fraction of a single dispersed phase areexpressed as follows:@

@tqm þr � ðqmumÞ ¼ 0 ð2Þ

@

@tqmum þr � ðqmumumÞ ¼ �rpm þ qmg �r � ½qmcp

� ð1� cpÞucp0ucp0� þ r � sGm ð3Þ
@
@tap þr � ðapum � DMprapÞ ¼ �r½apð1� cpÞucp0� ð4Þ

The mixture density qm and the mixture velocity um are definedas

qm ¼Xk¼2

k¼1

akqk ð5Þ


um ¼1qm

Xk¼2

k¼1

akqkuk ð6Þ

where k denotes the phase k, cp is the mass fraction of the dispersedgaseous phase.

In Eq. (3) and Eq. (4), the phase slip terms are expressed explic-itly as functions of the relative velocity ucp0, which is obtained fromthe constitutive equation as follows:

jucp0jucp0 ¼4dp

3CD

ðqp � qmÞqc

g � ðum � rÞum �@um

@t

� �ð7Þ

Note that p represents the dispersed phase and c the continuousphase. DMp is the diffusivity coefficient of the dispersed phase.

sGm ¼ ðlm þ lTmÞ rum þ ðrumÞTh i

� 23qmkmI ð8Þ

where km is the turbulent kinetic energy for the mixture, lm is thedynamic viscosity for the mixture and lTm is the turbulent eddy vis-cosity. I is the unit tensor. The correlation of generalised stress ten-sor sGm is written in the Reynolds Averaged Navier–Stokes (RANS)single-fluid or mixture context, representing both viscous and tur-bulent diffusion mechanisms. For the present work, it is assumedthat the dispersed phase consists of spherical particles of a singleaverage size (dp = 2 mm). This particular value necessary for the cal-culation of interphase drag indeed contrasts with the measuredmean Sauter diameter, which is only limited to certain regions ofthe flow particularly for dense jets. This is obviously a roughapproximation and introduces an uncertainty. This is indeed oneof the weaknesses of such phase averaged models. Inherent in themodel is the assumption that the two phases move at differentvelocities, but do reach local equilibrium over a very short spatiallength scale. Slip velocity function employed the algebraic slipmethod of Manninen et al. (1996). The inter-phase mass transferdoes not apply here, and surface tension is neglected. The dragcoefficient CD appearing in Eq. (7) is determined by the correlationproposed by Schiller and Nauman, see Clift et al. (1978). The stan-dard k–e two-equation turbulence model was used and all turbulent



Fig. 4. Entrainment regime: impact velocity Vj (m/s) versus jet length Lj (m).


quantities are based on the primary phase flow. Turbulence gener-ation in the gaseous phase is not accounted for, and the turbulenceof the primary phase is not directly affected by the presence of sec-ondary phase. The viscous and diffusion stresses can often be omit-ted because the turbulence viscosity of the continuous fluid is muchlarger than the dynamic viscosity. These are two other assumptionsof the model.

Fig. 5. A schematic of computational domain and mesh.

3.2. The level-set approach

The level-set technique (Sussman et al., 1994; Lakehal et al.,2002) is an Interface Tracking Method (ITM), where the interfaceseparating the two phases is tracked directly in an Eulerian way,without phase averaging. In ITMs the flow system involves onesingle velocity–pressure field, with material properties includingdensity, viscosity, and conductivity changing in time based onthe phase marker scalar function. In the level-set technique theinterface between immiscible fluids is represented by a continuousfunction (/), representing the distance to the interface, which is setto zero on the interface and is positive on one side and negative onthe other. The fluid motion equations under isothermal


incompressible flow conditions without phase change or masstransfer expressed within this context take the form:

r � u ¼ 0 ð9Þ

@

@tuþr � ðquuÞ ¼ �rpþ qg þr � s� crndð/Þ ð10Þ

The last term on the right hand side of Eq. (10) denotes the sur-face tension force, with n standing for the normal vector to theinterface, r for the interface curvature, c for the surface tensioncoefficient of the fluid, and d for a smoothed Dirac delta functioncentred at the interface. The evolution equation for the level-setfunction reads:

@

@t/þ u � r/ ¼ 0 ð11Þ

Material properties are updated locally based on / and distrib-uted at the interface using a smooth Heaviside function denoted byH(/):

q;l; . . . ¼ q;l; . . . jL � HðUÞ þ q;l; . . . jG � ð1� HðUÞÞ ð12Þ

where the subscripts L and G refer to liquid and gas, respectively.Depending on the flow, the level-set function ceases to be a signeddistance from the interface after an advection step of Eq. (11). Var-ious methods have been proposed to restore the level-set functioncorrect distribution near the interface. In the code TransAT of AS-COMP used here, a re-distancing equation is solved, using thenon-oscillatory 3rd-order WENO scheme, enhanced by a globalmass-conservation scheme (Lakehal et al., 2002).

3.3. Computational parameters

Simulations have been carried out based on the experimentalsetup; the flow configuration is one in which a cylindrical tank isthe receiving bath with the water level 280 mm above the tankbottom. The employed grid is generated to represent the injection




pipe installed vertically above the free water surface and alignedwith the axis of the tank (see Fig. 5). The diameter of the tank is150 mm and the height from the nozzle exit to the water levelwas varied in the simulations to study the influence of jet lengthon the penetration and entrained-air concentration. In both simu-lations, a 2D axisymmetric domain was set covering the pool, thejet and the surrounding air. The structured mesh generated forthe mixture-model simulations covered between 75,775 and223,260 elements. The grids in the jet falling area (from the tubeexit to the free surface of water) and the liquid part of the compu-tational domain where jet penetration occurs were refined to trackthe accurate jet behaviour. For the level-set simulation, structuredsingle-block meshes of 31,750–47,500 cells were used, consistingof one block only, albeit refined around the jet impact region andthe free surface. The boundary conditions included an axis of sym-metry, wall type, velocity inlet, and pressure and velocity.

The jet is injected through the nozzle with a constant bulkvelocity. A pressure outlet condition was adopted at the top ofthe domain fixing the pressure to atmospheric conditions. Tomaintain a constant water level, an outlet condition was used inthe bottom-right corner of the tank for the mixture-model simula-tions. The mass flow rate of water leaving the tank was set equal tothe water mass flow rate introduced from the nozzle. In the level-set simulations, a pressure outlet was also imposed at the rightboundary above the tank, so that water in excess could leave itand compensates the excess. It was assumed that the flow in thecore of the nozzle in the flow upstream is fully developed, so thatthe turbulence intensity at inlet can be estimated from the follow-ing empirical correlation for pipe flows: I = 0.16(Re)�1/8 where Re isthe Reynolds number based on the nozzle diameter. However, thebackflow turbulence intensity for the pressure outlet was esti-mated according to previous experiences and it was set to be0.05%.

The equations in the homogeneous model simulations arediscretized using the 2nd-Order Upwind scheme within a finite-volume formulation. The PISO algorithm was used for pressure–velocity coupling. An adaptive time-marching strategy wasadopted during the calculation. Computations were continued tillthe global residuals reached the absolute criterions set in thatcode: 10�4 for continuity, velocity and volume fraction, and 10�3

for k and e. The time for calculation convergence depends on thecase studied and ranges from 0.5 s to 3.5 s. The code TransAT usedfor level-set simulations is finite-volume based too, using thoughthe SIMPLEC algorithm for pressure–velocity coupling. Time advec-tion is achieved using the 3rd-order Runge–Kutta explicit scheme.The convective fluxes are discretized using the 3rd-order quickscheme. The k–e model is employed for turbulence, with specificnear-interface treatment (Liovic and Lakehal, 2007b). The surfacetension coefficient is set equal to 0.072 N/m. Transient simulationswere averaged over time between 1 and 1.7 s which is probablynot long enough to obtain statistically averaged results.

Fig. 6. Jet penetration depth hp (cm) against impacting velocity Vj (m/s). Compar-ison between the models, the experiments and the correlation data.

4. Results and discussions

Both experimental and numerical results are presented. Exper-imental findings will serve to validate and complement the calcu-lation results. The present computations target the continuousregime of Fig. 4. Fig. 4b represents the corresponding calculatedtarget regime for both multiphase models used which exhibit anexcellent agreement with experiments. The results are discussedin terms of flow regimes, characteristics of entrained bubbles, jetpenetration depth and falling water jet deformation for whichexperimental results and visualisations are available. The air–water flow properties such as the liquid velocity and the air holdup under the free surface are also discussed. The discussion will


also contrast the performance of the mixture and level-setapproaches.

4.1. Jet penetration depth

Fig. 6 compares the jet penetration depth for nozzle exit veloc-ities of 1.8 m/s and 2 m/s, respectively. The penetration depth isplotted against the jet velocities at impingement given by Eq. (1).Fig. 6 includes the results from the empirical relationship of Bin(1984), where the penetration depth hp is determined by:

hp ¼ 2:1V0:775j d0:67

0 ð13Þ

The results of penetration depth from the experiments are ob-tained using the averaging algorithm mentioned in Section 2. Inthe mixture model calculations, the penetration depth is definedas the location where air volume fraction reaches zero at the bub-ble plume edge. In the level-set simulations, this criteria is some-what correlated to the vertical speed of the jet, which shouldconverge to zero.

From Fig. 6, it can be seen that the simulation results and theexperimental values are relatively overall in good agreement. Thebehaviour is well captured and the results reveal that both themixture model and the Interface tracking variant are able to cap-ture the plunging liquid jet penetration depth. Bin’s expression isalso plotted in the figure and as expected, it displays a fundamen-tally different behaviour. This can be explained by the fact thatBin’s correlation is based on global parameters like jet impactvelocity and diameter of injection nozzle, but neglects jet falling/breaking length and jet-free surface interaction, which are impor-tant parameters in this phenomenon and have considerable influ-ence on the jet penetration depth. It also ignores viscous shear atthe interface and surface tension. Fig. 6 shows, however, differentresults between the two modelling approaches: while the predic-tions are equally good for low impact velocities, the level-set is rel-atively better than the mixture model for high impact velocities,for nozzle exit velocities of 1.8 m/s and 2 m/s. Several reasonscould be invoked to explain the discrepancies, among which thefact that the level-set predicts a wide range of frequency scalesor unsteadiness of the flow than the average model. The mixturemodel tends naturally to smear the interfaces, which could leadto wrong definition of the topology of the flow, and thus



Fig. 7. Time averaged pictures of air volume fraction delivered by the mixture and level-set models at V0 = 1.8 m/s, for different jet lengths.


under-estimates the shear-induced turbulence. Be it as it may, it istoo premature to attribute these drawbacks to turbulence model-ling alone.

Fig. 7 compares the volume fraction contours predicted by bothmodels, for jet nozzle velocity of 1.8 m/s. The penetration locationis also indicated on the figures. The simulation results for the im-pact velocities plotted in Fig. 6 are shown here (three values onlyfor level-set simulations). It is obvious that both simulations re-quire more samples to achieve complete statistical convergence.The right panels depict a smeared interface, which is an artefactof time averaging only; the interface is actually sharper in the le-vel-set simulations as indicated further below in Fig. 8. TheLj = 12 cm and 16 cm calculations in the left panels show incom-plete statistical convergence, which should have predicted a com-plete ascension of the gas swarm to the free surface. Differencesbetween the homogeneous model and level-set are clearly high-lighted when looking at the Lj = 2.5 cm results.

4.2. Jet instability and free surface deformation

As the water jet penetrates ambient air, a gaseous film developsalong its trajectory as an envelope surrounding its surface. The for-mation of a gas film adjacent to the surface of the jet is associatedwith the relative velocity between gas and liquid phases, resultingin the generation of surface jet instabilities and deformations.According to previous studies, both axisymmetric and asymmetri-cal deformations can occur on the jet surface depending on itslength, nozzle exit velocity – and perhaps the pipe diameter andsurface tension as well, as illustrated by the images from theexperiment of plunging liquid jets provided by Schmidtke et al.(2009). A detailed theoretical analysis of jet instability is beyondthe scope of this paper; we restrict ourselves to reporting the sim-ulations results and how they compare with the data.

Fig. 8 depicts the simulation (using both models) and experi-mental topologies of the jet interface around the water-bath freesurface, right at impingement on the water surface. It can be seenthat the mixture-model simulations fail to capture the shape of thejet interface with the present 2D modelling assumptions and com-putational parameters, in particular the Rayleigh type of instabilitydepicted in the left experimental panel and the bubble entrain-ment at the free surface. The level-set simulations reveal the exis-tence of the above mentioned instability in the left panel, depictingalso a stronger entrainment and more violent free surface deforma-


tion in the right panel. The reasons for these discrepancies are thesame as evoked previously. A 3D modelling strategy is perhapsnecessary to capture such details, using a finer mesh. The videosproduced during the measurement campaign suggest that the bub-bles trapped at the surface partly originate in the core liquid sub-sequent to jetting effects and mixing and then rise by buoyancyeffects, and partly induced by jet breaking at the impingemententraining air. The videos show that these bubbles trapped at thesurface could remain ‘floating’ for a very long time (seconds). Thisis obviously computationally unaffordable.

4.3. Velocity and air hold up below the free surface

Below the impingement point, the flow structure basically com-prises two distinct regions: (1) a diffusion cone with a downwardflow motion induced by the plunging liquid jet, and (2) a swarmof rising bubbles which surrounds the former one (Chanson,1997). In the present study no measurements of liquid velocitiesand air concentrations were performed. However, in this sectionthe computed distribution of liquid velocity and void fraction usingthe mixture and level-set methods are discussed and comparedwith semi-analytical and empirical results taken from the litera-ture. The velocity distribution in the fully developed region validbeyond a distance of 10 jet diameters from the origin of the jetfor a round turbulent single phase jet is given by the followingsemi-analytical correlation, see Chanson (2004):

VVj¼ 5:745

x=d0

1

1þ 0:125 18:5 rx

� �2

!ð14Þ

where x is measured from the plane of impingement of the plungingjet. The profile predicted by Eq. (14) together with the computa-tional results from the mixture model and the level-set variantare plotted in Fig. 9. The general shape of the velocity profile isGaussian as found by McKeogh and Ervine (1981), Bonetto andLahey (1993), Hammad (2010), and Ma et al. (2010). It can be seenthat the velocity of the entraining liquid column decays laterallyfrom the nozzle centreline to reach zero away from the jet centre-line. The source of the discrepancies between the simulations andtheoretical profiles can be traced to the nature of Eq. (14) which,strictly speaking, is valid for liquid–liquid jet flow, while the veloc-ity distributions of impinging jet flow will be affected by the drag/buoyancy of the entraining gas. Nevertheless, the profiles generated



Fig. 8. Jet instability and surface deformations at impact (snapshots). Comparisons between the models and the data for case (a) V0 = 1.2 m/s, Lj = 25 cm, and case(b) V0 = 3 m/s, Lj = 25 cm.


by the two models are consistent in trend with the profile ofEq. (14), although the two models predicted lower velocities awayfrom the centreline than Eq. (14). A further observation indicatesthat the level-set model predicts marginally higher diffusion, byits radial spreading, in contrast to the mixture model. The maxi-mum velocity value of the jet on the centreline is somewhat overpredicted by both models. The local minimum predicted by themixture model and its subsequent shape is another deficiency, inthat they reflect a higher concentration of air around the jet head.The bubbles are not allowed to rise up to the free surface, a draw-back rooted in the slip model used, as already suggested by Fig. 7.A consistent explanation of the phenomenon in support of this anal-ysis is given below when discussing void fraction distributions.

The distributions of mean void fraction obtained by the mixtureand level-set methods are displayed in Fig. 10. For the discussedcase, the penetration depth predicted by the mixture model is14.5 cm while for level-set it is 15.5 cm. Thus in order to probethe air concentration distribution around this area, the planes with


h = 12 cm, 13 cm and 14 cm are considered for the purpose. Inaddition to the numerical results, air concentration values obtainedfrom the empirical correlation proposed by Chanson et al. (2004)are also shown:

C ¼Q air

Q w

14D#ðx� x1Þ=YCmax

exp�1

4D#

1þ rYCmax

� �2

x�x1YCmax

0B@

1CA� I0

12D#

rYCmaxx�x1YCmax

!

ð15Þ

where Qair is the air flux, Qw is the water jet flow rate, x is the lon-gitudinal coordinate, D# = 2Dt/(Vjd0) is the dimensionless air bubblediffusivity, Dt is the advection diffusion coefficient, which averagesthe effects of turbulent dispersion and streamwise velocity gradi-ent, and I0 is the modified Bessel function of the first kind of orderzero. To keep the empirical nature of Eq. (15), the values of Qair/Qw

and D# were determined from the best fit of data from the bank ofexperiments performed by Chanson et al. (2004). They are assumed



Fig. 9. Empirical versus computational radial average velocity profiles at h = 5 cm,6 cm and 7 cm (Vj = 2.12 m/s, Lj = 2.5 cm).

Fig. 10. Empirical versus computational radial gas profiles at h = 12 cm, 13 cm and14 cm (Vj = 2.12 m/s, Lj = 2.5 cm).


constant in the radial direction. Further, Ycmax denotes the lateral dis-tance from the centreline when the air content reaches the maxi-mum, which is probably the weakest point in the above expression.

The choice of Ycmax is important in the calculation as it decidesthe radial location of the air content peak value. The predicted peakvalue of air concentration differs when either the mixture ap-proach or the level-set is used. Therefore two different Ycmax valuesare assigned for the mixture and level-set models respectively

Please cite this article in press as: Qu, X.L., et al. Characterization of plunging liquid jets: A combined experimental and numerical investigation. Int. J.Multiphase Flow (2011), doi:10.1016/j.ijmultiphaseflow.2011.02.006



when comparing to the theory. The air content peak location in thelevel-set simulations is 0.3 cm for all three positions, while for themixture model; this location is predicted at 0.9 cm, 1 cm and1.1 cm, i.e. at around 1 ± 0.1 cm, which is around three times largerthan the level-set results, reflecting an over estimation of the rateof lateral spreading. Comparing the mixture model and the level-set results, it is found that there is an over-prediction of the max-imum concentration values by both the level-set and mixture mod-els. Overall, the prediction of the mixture model is better than thelevel-set model at h = 14 cm, whereas the reverse is true ath = 12 cm, with slightly better prediction by the level-set thanthe mixture model of the maximum concentration at h = 13 cm.The lateral diffusion of the jet is considerably higher with the mix-ture model than with level-set and the rate of decay of air concen-tration predicted by the mixture model is higher than the onepredicted by the level-set. The level-set results seem to be moreconsistent with the empirical relation, in the light of the previouscomparison of velocity profiles. Fig. 9 has indeed shown that atthe centre of the jet within the pool, the maximum average veloc-ities are higher when predicted by the mixture model than boththe level-set and the theory. This is synonymous to a higher airconcentration in the jet centre zone beneath the free surface,which is clearly seen in the context of Fig. 10 comparing air voidfractions. This confirms our observations made earlier that themixture model is not capable of predicting the rise of air bubblesto the free surface. It is seen that the trend of the distribution ofair content is globally similar between CFD and the empirical rela-tion. This gives rather additional credibility to Chanson’s correla-tion, albeit it leaves the air content peak value as an unknown ofthe model.

5. Conclusions

The present work reports findings from a combined experimen-tal and simulation campaign centring around a plunging turbulentwater jet on a free surface of a water pool. The CFD approaches em-ployed for the purpose are deliberately different; the mixture mod-el and level-set approach, (which belongs to the interface trackingframework). The methods were applied under two-dimensionalaxisymmetric conditions to predict several of the flow characteris-tics. Attention was paid to jet penetration depth and jet liquid col-umn instability and free surface deformations. The jet penetrationdepths obtained from the two computational approaches are ingood quantitative agreement with the data. Jet instability and sur-face deformations are predicted, but not with the same accuracy:Interface Tracking Methods are better suited to mimic the detailsof this flow and somehow show the trends occurring in the exper-iments, but are computationally more expensive. 2D axisymmetricsimulations are not sufficiently precise to replicate the flow detailswhich are 3D in nature, but are clearly capable of returning themacroscopic features of the flow.

The liquid average velocity field and air concentration of theentraining two phases under the water pool free surface were alsonumerically investigated and compared to existing analytical cor-relations. The results show reasonable agreement between simula-tions and empirical correlations, in the case of the liquid velocitydecay from the centreline, but noticeable differences were revealedfor maximum velocity at the jet centreline and associated bubbleconcentration. The mixture model predicts a higher velocity thanthe level-set and the theory at the early stage of jet penetration,due to a higher concentration of air that cannot rise to the surface;a model drawback buried in the way the slip velocity is algebrai-cally modelled. As for the air concentration distributions, the twomethods display a similar distribution map. However, the locationof the maximum air content and the peak value of air holdup aredifferent in quantity. This makes the comparison with the theory


rather difficult since the correlation considered here includes pre-cisely the air content peak value as a model unknown. In conclu-sion, the two numerical techniques can generally grasp theglobal characteristics of plunging liquid jets and the differencesin results lie in the different algorithms and computational timefor the simulations.

Acknowledgements

The work was conducted within; the EC-funded projects NUR-ISP and THINS, in the framework of the 7th EURATOM Frame workProgram.

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