tailwater effects on the characteristics of a square jet near a free-surface

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Tailwater effects on the characteristics of a squarejet near a free-surfaceGirish Sankar a , Ram Balachandar b & Rupp Carriveau ca Department of Civil and Environmental Engineering , University of Windsor , N9B 3P4 ,Windsor, Ontario , Canada E-mail:b Department of Civil and Environmental Engineering , University of Windsor , N9B 3P4 ,Windsor, Ontario , Canada Phone: 519 253 3000 Fax: 519 253 3000 E-mail:c Department of Civil and Environmental Engineering , University of Windsor , N9B 3P4 ,Windsor, Ontario , CanadaPublished online: 26 Apr 2010.

To cite this article: Girish Sankar , Ram Balachandar & Rupp Carriveau (2008) Tailwater effects on the characteristics of asquare jet near a free-surface, Journal of Hydraulic Research, 46:4, 504-515, DOI: 10.3826/jhr.2008.3043

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Journal of Hydraulic Research Vol. 46, No. 4 (2008), pp. 504–515

doi:10.3826/jhr.2008.3043

© 2008 International Association of Hydraulic Engineering and Research

Tailwater effects on the characteristics of a square jet near a free-surface

Les effets de l’aval sur les caractéristiques d’un jet carré près d’unesurface libreGIRISH SANKAR, Graduate Student, Department of Civil and Environmental Engineering, University of Windsor, N9B 3P4,Windsor, Ontario, Canada. E-mail: [email protected]

RAM BALACHANDAR, (IAHR Member), Professor, Department of Civil and Environmental Engineering, University of Windsor,N9B 3P4, Windsor, Ontario, Canada. Tel.: 519 253 3000; fax: 519 971 3686; e-mail: [email protected]

RUPP CARRIVEAU, Assistant Professor, Department of Civil and Environmental Engineering, University of Windsor, N9B 3P4,Windsor, Ontario, Canada. Tel.: 519 253 3000; fax: 519 971 3686; e-mail: [email protected] (author for correspondence)

ABSTRACTCharacteristics of a square jet near a free surface were experimentally studied at four different tailwater depths. Velocity measurements were madewith a two-component laser Doppler anemometer at five stations along the jet streamwise axis. A qualitative assessment was made of the location ofjet impingement on the free surface. The results reveal that the tailwater depth has little influence on the mean velocity distribution beyond a certaindepth. Turbulence intensities and shear stress profiles indicate jet anisotropy and higher spread rates than an axi-symmetric free jet. The analysis ofthird-order moments divulged a marked increase in the turbulent ejection and entrainment events for the jet as the tailwater depth increased and theeffect of free surface confinement was reduced. A quadrant analysis confirmed that the confinement effect due to the proximity of the free surfaceacted to curtail the ability of the jet to eject or entrain fluid.

RÉSUMÉLes caractéristiques d’un jet près d’une surface libre ont été expérimentalement étudiées pour quatre profondeurs différentes en aval. Des mesures devitesse ont été faites avec un anémomètre laser Doppler à deux-composantes, en cinq stations le long de l’axe de l’écoulement du jet. Une évaluationqualitative a été faite du lieu d’impact du jet sur la surface libre. Les résultats indiquent que la profondeur aval influence peu la distribution moyennede vitesse au delà d’une certaine profondeur. Les intensités de turbulence et les profils d’effort de cisaillement indiquent une anisotropie du jet et destaux de dispersion supérieurs à ceux d’un jet libre axisymétrique. L’analyse des moments de troisième ordre a montré une augmentation marquée desphénomènes turbulents d’éjection et d’entraînement pour le jet à mesure que la profondeur aval augmentait et que l’effet de confinement de la surfacelibre était réduit. Une analyse par quadrant a confirmé que l’effet de confinement dû à la proximité de la surface libre diminuait la capacité du jet àéjecter ou entraîner du fluide.

Keywords: Confined square jet, free surface, jet flow, tailwater effect, turbulence characteristics

1 Introduction

A considerable effort was devoted to the study of circular jetsas they are considered as one of the benchmarks for researchinto the physics of turbulent fluid flow (e.g. Noutsopoulos andYannopoulos, 1987; Hussein et al., 1994; Shinneeb et al., 2006).Most studies were conducted under unconfined conditions forwhich the jet characteristics are not affected by physical con-straints. Non-circular jets such as square jets are also of interestdue to their enhanced mixing characteristics. Miller et al., (1995)performed a simulation of spatially developing three-dimensionaljets issued from circular and non-circular jets. They found thatnon-circular jets are more efficient mixers than circular jets. In acomputational and experimental investigation by Grinstein et al.(1995), it was found that the non-circular jets have enhanced

Revision received September 27, 2007/Open for discussion until February 28, 2009.

504

entrainment properties relative to circular jets. Ai et al. (2005)in a study on the vortex dynamics of starting square jets foundthat due to the enhanced entrainment ratio and mixing efficiency,square jets penetrate slower than circular jets having the sameexit conditions. In the past and most recently, systematic testswere conducted to obtain the characteristics of such jets (Sforzaet al., 1966; Grinstein et al., 1995; Sankar, 2005). Sforza et al.(1966) noticed that the axial velocity distributions of non-circularjets are slightly different to those of axi-symmetric jets.

In many applications, jets occur under confined situations.The confinement can occur either in the form of a solid wall ora free surface. Madina and Bernal (1994) investigated the char-acteristics of a round jet interacting with the free-surface. Theynoticed the presence of large-scale disturbances near the freesurface which propagated away from the jet axis. Liepmann and

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Journal of Hydraulic Research Vol. 46, No. 4 (2008) Tailwater effects on the characteristics of a square jet near a free-surface 505

Gharib (1992) also showed that the free surface can cause insta-bilities in the near field region. Anthony and Willmarth (1992)identified surface currents near the free surface. These currentswere mainly comprised of vortical structures ejected from thejet. Gaskin et al. (2004) considered how the entrainment of aconfined plane jet was impacted by background turbulence.

A survey of the literature indicates that while the study ofaxi-symmetric, free jets is reasonably well established, squarejets have not been as thoroughly explored. Such a configura-tion sees many applications in the field where enhanced mixingis desired in jet discharging to open water bodies. The presentstudy examines the characteristics of a jet generated using a three-dimensional square cross-sectional nozzle at various tailwaterconditions. The proximity to the free-surface is varied from 1.5to 5 times the nozzle width measured from the bottom of the noz-zle. Velocity measurements were conducted at five axial stationsdownstream of the nozzle exit.

2 Experimental setup and procedure

A schematic of the flow field is shown in Fig. 1. Experimentswere conducted in a rectangular cross-section flume 9.5 m long,1.2 m wide and 0.6 m high. The side walls of the flume are madeof Plexiglas to facilitate unobstructed transmission of the laserbeams into the flow. A tailgate was used to control the depthof the downstream water level. The supply system consisted ofa pump, a head tank, and a flow control valve. The upstreamflow conditions were adjusted to deliver a constant velocity of1 m/s at the nozzle exit. A well-designed smooth nozzle having acontraction ratio of 17:1 and an exit square cross-section of widthB = 40 mm was designed using a third degree polynomial. Thenozzle was positioned at the upstream end of the flume withthe horizontal centerline positioned at 0.3 m from the base ofthe flume. The flume width to nozzle width ratio was 30 and islarge enough to consider that the confinement resulting from theflume side walls is low. This effectively provided a jet which wasvertically confined by a solid wall on one side and a free surfaceon the other.

Measurements were conducted using a two-component fiberoptic LDA system (BSA® F60 Dantec Inc.). It was powered

Figure 1 Schematic of flow field

by a 300 mW argon-ion laser. The measuring volume for thepresent configuration was 3.54×10−10 m 3. Due to the restrictionimposed by the geometry of the transmitting optics, no measure-ments were possible at locations closer than 0.5B downstream ofthe nozzle exit. The measurement probe was mounted on a two-dimensional traversing system capable of traversing to the samelocation with an accuracy of ±0.01 mm. The jet exit Reynoldsnumber was R = UjB/ν = 40,000 at all tailwater depths. Thewater temperature was ensured to be constant throughout the testperiod and the proper temperature was used to evaluate R. Nodifference prevailed between the temperature of the jet and theambient fluid.

The water in the flume was filtered for several days usinga 5 µm filter. Then, 5 µm spherical seed particles were mixedinto the flow to enable LDA measurements. Further details ofthe LDA and the associated uncertainty in measurements arereported in Sankar (2005). For the measurements reported here,the origin of the coordinate system is located on the nozzlecenterline at the exit. The streamwise coordinate x is positivedownstream, the vertical coordinate y is positive towards thefree surface and the lateral coordinate is z. Five streamwise sta-tions in the range 0.5 ≤ x/B ≤ 15 were chosen to conduct theLDA measurements. As indicated in Fig. 1, the tailwater depthTW was varied from 1.5B to 5B and Uj denotes the jet exitvelocity.

3 Results and discussion

Prior to presenting details of the results, it is useful to mentionthat for the lowest tailwater depth TW = 1.5B the jet was visu-ally seen to attach to the free-surface soon after it exited thenozzle. For the other tailwater depths (TW = 2.5B, 3.5B and5B) the jet was seen to impinge on the free surface at a stream-wise distance of about 5B, 8B and 18B, respectively. Figure 2shows the mean streamwise velocity (U) measured 10 mm belowthe free surface in the streamwise direction. With increasing x/B

one would expect negative velocities close to the nozzle and posi-tive velocities following jet impingement. However, at the lowesttailwater one notes that the mean velocity is positive right fromx/B = 0.5. The velocity histogram at this location indicates theoccurrence of both positive and negative instantaneous veloci-ties. In the remaining three tests, a changeover from negative to

0−0.05

0.05

0.15

0.25UUj

0.35 1.5B

2.5B

3.5B

5B

0.45

5 10

X / B

15 20 25

Figure 2 Jet streamwise mean velocity measured 10 mm below the freesurface

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506 G. Sankar et al. Journal of Hydraulic Research Vol. 46, No. 4 (2008)

Figure 3 Streamwise mean velocity profiles from jet central vertical plane

positive mean velocity is noted indicating that the flow near thefree surface occurs in either direction following impingement.This figure should be viewed in a qualitative sense as the wavynature of the free surface tends to complicate the measurements.The cross-over point from negative to positive velocities confirmsthe visual impingement noted earlier.

Figures 3 and 4 provide a comprehensive description of thevelocity characteristics in the vertical central plane of the nozzle

from x/B = 0.5 and extending to the last streamwise stationx/B = 15. The corresponding streamwise locations are identi-fied in the first column of the graphs. In these plots, the nozzlewidth B and the jet exit velocity Uj are used as the normalizingscales. Figure 3 shows the streamwise mean velocity plots at thefour tailwater depths. It is clear that at x/B = 0.5 in the first col-umn that the exit conditions are identical at all tailwater depthswith a typical top hat type velocity profile. At this station in the

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Figure 4 Profiles of turbulence intensity in the streamwise and vertical direction, and shear stress

middle 70% of the flow, the velocity is constant at 1 m/s. Eventhough the jet at TW = 1.5B was visually seen to behave differentfrom that at a deeper recovery water condition, the measurementsat z = 0 are identical to that at other tailwater depths. In any given

column of the graphs in Fig. 3, proceeding on to x/B = 2 andx/B = 5, the top-hat profile changes slowly to a Gaussian typeprofile. An evaluation of the first three graphs in any column indi-cates that the region of constant velocity in the central portion of

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the jet gradually decreases and the existence of a conventionalpotential core region is seen to extend up to x/B = 2. The dataat x/B = 5 for all the tailwater depths are more or less identicalexcept for the lowest tailwater condition where the jet has had amore significant interaction with the free surface. At TW = 1.5B,obviously due to the close proximity of the free surface, the jet hasreduced entrainment from the top thereby restricting its expan-sion. This is discussed when dealing with higher-order momentsand quadrant decomposition of the velocity data. Column 2 ofFig. 3 shows also selected data from off-centre planes locatedat z/B = 0.25 and z/B = 0.5. It is apparent from any of thegraphs where these data appear, that mean streamwise velocitiesdrop as one moves away from the jet centre plane in a span-wise direction. Where available, the streamwise mean velocityprofiles for a typical axi-symmetric jet (Shinneeb et al., 2006)are also shown in Fig. 3. One should recall that the exit profilehas an influence on the jet evolution and the similarity of theprofiles at x/B = 0.5 render the comparisons more appropri-ate. It is clear from these graphs that the evolution of the meanprofiles is quite different in the present study as compared toround jets.

At the lowest tailwater depth, as the flow is highly restrictedon the top, there is less possibility of fluid entrainment into thejet. This results in a jet having higher capacity for penetrationindicated by higher velocities along the jet centerline at fartherdistances from the nozzle. At x/B = 10, the centerline veloc-ity is of the order of 0.6Uj compared to a centerline velocity ofabout 0.4Uj at a tailwater of 2.5B. At tailwater depths of 3.5B and5B, the centerline velocities at various downstream sections areslightly higher than at 2.5B. One would expect higher velocitiesalong the jet centerline at TW = 2.5B as compared to TW = 3.5B

and TW = 5B. However, for TW = 2.5B, there is some degreeof entrainment in the vertical direction above the jet centerline,more so that could occur at the shallower tailwater depth of 1.5B.Further, the momentum loss due to the interaction with the freesurface is still substantial. These two effects are coupled to pro-vide a mechanism for lower velocities at TW = 2.5B. For deepertailwater depths, there are higher velocities noted along the cen-terline compared to a tailwater depth of 2.5B due to the reducedinteraction with the free surface. The dominant mechanism forreducing the velocity is fluid entrainment from the ambient intothe jet.

It is also clear from the plots that the tailwater has no sig-nificant effect on the velocity distribution beyond TW = 2.5B.Progressing to x/B = 15, as the jet expands into shallow tailwa-ter depths at x/B = 1.5B and 2.5B, it gradually loses its formand one notes higher velocities in the upper regions of the jetas compared to equivalent distances along the lower half of thejet. For example, at TW = 2.5B corresponding to x/B = 10, thevelocity magnitude at y/B = 1 is 0.30 m/s as opposed to 0.22 m/sat y/B = −1. At the lower tailwater depths, upon impingement,the free surface begins to move and this effect migrates into thejet causing velocities above the jet centerline to be higher thanthose below the jet centerline. For the deeper tailwater depths of3.5B and 5B, the jet retains its near symmetrical distribution evenat x/B = 15.

It is quite clear from the evolution of the velocity profiles forall the TW depths that the characteristics are clearly differentfrom that of an axi-symmetric free jet. Further, for the range ofTW conditions considered, the mean velocity characteristics arenot significantly different. This is due to the strong confinementeffects present in the flow and also due to the nature of the jetemanating from the square nozzle. For the range of TW of thisstudy the effects cannot be decoupled. The jet expansion in squarejets is expected to occur at a faster rate (Sankar, 2005).

The first column of the plots in Fig. 4 represents the variation ofthe turbulent fluctuations in the streamwise direction. At x/B =0.5, as expected, distinct double peaks are observed.

This is due to the enhanced turbulence activity as the jet pene-trates into the ambient flow. For the region −1 < y/B < +1, nooverwhelming tailwater effects can be noticed from x/B = 0.5to x/B = 15. Column 2 of Fig. 4 shows the vertical compo-nent of the turbulence intensity at the four tailwater conditionswith increasing distance from the nozzle. Comparing columns1 and 2, one can note the prevalence of anisotropic conditions.The last column in Fig. 4 shows the shear stress profiles at thefour tailwater conditions. It is clear that the magnitude of thepeaks is smaller at the lower tailwater conditions at the very firstlocation. Qualitatively though there is a similarity in the shearstress distribution, because the effect of the proximity of the freesurface is distinguishable at the lowest tailwater condition. Forthe highest tailwater condition the evolution of the shear stressprofile resembles a typical jet. However, the jet spread rate asdistinguished from the change in the shape of the profile occursmuch faster than a typical axi-symmetric free jet. In the deepertailwater conditions, the shear stress profiles are anti-symmetricabout the jet centerline. Similar to the mean streamwise profilesconsidered in Fig. 3, the profiles of Fig. 4 show that the confine-ment effects are severe for x/B = 5 and no distinct TW effectscan be noticed for TW ≥ 2.5B.

The turbulent diffusion of u2 in the y direction can be thoughtof as Dv = u2v. Figure 5 shows the variation of third-ordermoments Dv = u2v for the four tailwater conditions. Attentionis drawn to the first row of graphs (x/B = 0.5). The value Dv

is zero in the mid 60% of the flow at all tailwater depths. Thepotential core region has characteristics that are consistent withthat observed above in the mean and turbulence intensity profiles.At the edges of the jet, four distinct peaks are observed, onepositive pair denoted as A and B, and another negative pair C andD. A positive value of Dv = u2v indicates an upward transport ofthe u-momentum. In the region above the jet centerline, a positiveDv represents flow out of the jet while below the jet axis; a positivepeak represents entrainment into the jet. At x/B = 0.5, whencomparing the shallowest tailwater TW = 1.5B with the deepestTW = 5B condition, the magnitudes of all the peaks are seen tobe larger for TW = 5B. This corresponds to increased turbulenceactivity with increasing tailwater depth. The peaks denoted as A

and D represent ejection of fluid. At any given tailwater condition,these peaks are similar in magnitude. Consequently, in terms ofejection activities at the jet exit, there is no significant differencebetween the behaviors in the upper portions of the jet as comparedto the region below the jet axis (i.e. free surface versus solid

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Figure 5 Variation of the third order moment, u2v, for each tailwater at all five streamwise axial stations

surface) though the distances to the boundary (y/B = 1 versus 7at TW = 1.5B) are different.

This observation is consistent with that of a free round jet inthat the diffusion characteristics are similar along the perimeterof the jet. The peaks denoted as B and C in the Dv = u2v plotat y/B ≈ 0.4 are a consequence of the fluid parcels entrainedinto the jet. Near the jet exit, at any given tailwater condition, thetwo peaks are of similar magnitude and are also lower for lowertailwater conditions. Once again, in terms of entrainment, there

are no major differences in the characteristics between the upperand lower portions of the jet.

At x/B = 2 (row 2 in Fig. 5), the peaks are larger in mag-nitude as compared to x/B = 0.5. Further, for TW ≥ 2.5B,the entrainment peaks B and C there have a larger magnitude ascompared with the ejection peaks at A and D. However, withincreasing axial distance downstream, at x/B = 5, the peaks B

and C tend to become smaller while the peaks A and D tend to bemore spread out over a larger range of y/B. This can be clearly

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noticed in the graphs for TW = 3.5B at x/B = 2 and 5. Withincreasing distance downstream of x/B = 10, the entire curvesrepresenting the distribution of Dv collapse onto single profilesas the jet begins to diffuse and no coherent turbulent diffusioneffects are visible. Comparing Figs 3 to 5, one can note that the

Figure 6 Influence of tailwater on third order moments u3, v3 and u2v

TW effects are hardly visible in the mean and turbulence velocitymoments. They are distinctly visible in the third order momentsfor x/B ≤ 5.

Figure 6 shows the variation of additional third-order momentsu3, v3 and Dv = u2v. All moments were normalized by U3

j .

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The data trends at all stations indicate that there is consistencyin behaviour at all tailwater depths. In all columns, the tailwa-ter effect can be noted to diminish with increasing axial distancefrom the nozzle. The first column illustrates the streamwise con-vection of streamwise momentum, u2. At x/B = 0.5, around theedges of the jet, there are large positive fluctuations towards theouter periphery of the jet, denoted by peaks P and Q, accom-panied by the negative fluctuations in the inner periphery of thejet, denoted by peaks R and S. The peaks P and Q representa sweeping type motion in the streamwise direction. The nega-tive peaks R and S have a smaller magnitude compared with thepeaks P and Q. When the same peaks are viewed in the graphspresented in the second column (u3), the peaks R and S repre-sent entrainment into the jet, while the peaks P and Q representejection activities. Close to the nozzle exit, in the inner regionsof the jet periphery, the entrainment events into the jet are smallwhile the ejecting eddies with a streamwise sweeping motion arethe dominant turbulent motion in the outer periphery of the jet.At x/B = 2, the penetration by entrainment events into the jetincreases though the ejection-type sweeping motion continues toprevail in the outer periphery of the jet. In the second column,the negative peaks R and S tend to merge with increasing axialdistance as the entrainment eddies fully penetrate the jet. Withfull penetration of turbulence into the jet, its ability to entrainasymptotically decreases. Following this merger for x/B > 5,no significant tailwater effect can be noticed.

Column 3 in Fig. 6 represents the variation of Du = u2v

for different tailwater depths at varying distances downstream.A positive value of Dv = v2u indicates a streamwise transportof v-momentum. At x/B = 0.5 and y/B ∼= ±0.60, the meanstreamwise velocity is low (as shown in Fig. 2), and periodicvortices in this region result in high fluctuations causing positivepeaks. This can also be considered as a sweeping action of theentrained eddies traveling in the streamwise direction. Consistentwith columns 1 and 2, at x/B = 0.5 and y/B = ±0.40, negativepeaks occur which tend to merge with increasing x/B. At x/B =15, the three third-order moments are negligible indicating nosignificant turbulence activity.

4 Quadrant analysis

Though the third-order moments plots are useful, some infor-mation is lost in the averaging process associated with theseparameters. Thus, to further enhance the illustration of theprocesses of jet mixing, a quadrant analysis was carried out. Fol-lowing the work of Lu and Willmarth (1973), the present quadrantanalysis breaks the mixing process down into 4 event types inturbulent boundary layers. The jet merges with the ambient fluideither by entraining fluid from its surroundings; or by ejectingfluid from the jet flow into the ambient fluid. These two mix-ing mechanisms can be further sub-classified to occur as fast orslow events. Therefore, each of these 4 events, namely fast ejec-tions, fast entrainments, slow ejections, and slow entrainmentsare assigned an individual quadrant in this analysis. To detectextreme events with large contributions to the turbulent shear

stress, a detector indicator function λi(t) was defined using themethod suggested by Lu and Willmarth (1973) for wall boundedflows as

λi(t) ={

1 when |uv|i ≥ H(urms)(vrms)

0 otherwise(1)

where i denotes the quadrant of interest, and urms and vrms are theroot-mean-square values. Then, only the large contributions to〈uv〉 from each quadrant are extracted leaving the smaller fluctu-ations in the hole region corresponding to more quiescent periods.The contribution to 〈uv〉 from a particular quadrant may then bewritten as

〈uv〉i = limT→∞

1

T

∫ T

o

uv λi(t) dt (2)

Data is available at many thresholds at all locations consideredin the study. Here, results are presented for H = 0 and H = 2.A value of H = 2 corresponds to stronger events that have shearstress values greater than five times the average shear stress. Fig-ures 7 and 8 show the contributions to the shear stress normalizedby U2

j from each of the quadrants at x/B = 2 for threshold levelsof H = 0 and 2, respectively. A legend on each of Figs 7 and8 depicts the quadrants and illustrates the corresponding domi-nant velocity components. For example, fluid parcels have strongupward and streamwise velocity components in quadrant 1, thusif the upper half of the jet is considered, an event in this quadrantwould represent fast ejection of a fluid parcels from the jet into theambient fluid. Similarly, quadrant 3 represents parcels that havestrong counter-streamwise and downward velocity components,thus if the top half of the jet is considered, events in this quadrantwould be classified as a slow entrainment of a fluid parcels fromthe ambient fluid into the jet.

The frames in row A of Fig. 7 depict the shear stress contribu-tions for the H = 0 in quadrants 1 and 4. Above the jet centerline(y/B > 0), quadrant 1 represents a measure of fast ejections offluid parcels from the jet as the dominant velocity components arestreamwise and upwards. An inspection of row A indicates thatfast ejections in the streamwise direction tend to increase withincreasing tailwater depth. Further, the location of the peak inquadrant 1, above the jet centerline occurs at slightly larger val-ues of y/B with increasing tailwater depth. These are likely owedto the surface constraint reduction associated with an increasedtailwater depth. Below the jet axis, quadrant 1 represents fastentrainment events. The tailwater depth appears to have a signifi-cantly reduced influence on entrainment events below the jet. Thisis largely due to the secondary influence of the increased waterdepth on the fixed distance from jet centerline to flume floor. RowA also portrays the behavior of quadrant 4 above and below the jetcenterline. Above the centerline, the events in quadrant 4 can becategorized as fast entrainment events and an increasing tailwaterdepth appears to have little influence on these events. Below thecenterline, quadrant 4 describes fast ejections from the jet. Theseincrease with increased tailwater depth. Though the depth fromjet centerline to flume floor is fixed, continuity considerationsrequire a balance between ejection and entrainment events. Thismay influence the behaviour described in quadrant 4.

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Figure 7 Contributions of shear stress in of quadrant analysis for H > 0 threshold at axial station x/B = 2

The frames in row B of Fig. 7 depict the shear stress contri-butions for the H = 0 threshold in quadrants 2 and 3. Above thecenterline, quadrant 2 represents slow ejection events. An inspec-tion indicates that there is little effect of increased tailwater depthon slow ejections. Conversely, slow entrainment events are rep-resented by quadrant 2 below the centerline and quadrant 3 abovethe centerline. These slow entrainment events increase markedlyuntil a tailwater of 2.5B, beyond which changes are marginal.The sudden increase in slow entrainment from TW = 1.5B

to TW = 2.5B could be a response to the increased supplyof fluid that is associated with the increased depth. BeyondTW = 2.5B, increases in tailwater depth have a small effect onthe slow entrainment events. Below the jet centerline, in quadrant3, slow ejections are observed to remain relatively constant with

increasing tailwater depth. Of particular interest in Fig. 7 is thenearly identical image similarity of the behaviour description inquadrants 1 and 3, and quadrants 2 and 4. This suggests that thereexists a correlation between fast ejections and slow entrainment,and fast entrainment and slow ejections. Is there in regions of thejet development a balance of ejection and entrainment mecha-nisms? This balance is more apparent in Fig. 7 than in Fig. 8; asFig. 7 likely represents a better total picture than Fig. 8, whichexclusively maps a higher turbulence threshold. Further, the peakvalues of the shear stress occur at y/B = ±0.5.

Figure 8 illustrates shear stress contributions for more extremeevents occurring for a threshold of H = 2. Compared to Fig. 7,the stronger events are less influenced by differences in tailwaterdepth. Compared to Fig. 7, a careful inspection of row C in

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Figure 8 Contributions of shear stress in of quadrant analysis for H > 2 threshold at axial station x/B = 2

Fig. 8 reveals a vertical shift where the curve peaks occur. Inrow C of quadrants 1 and 4, the peaks shift outwards to abouty/B = ±0.75. This indicates that the most energetic ejectionsoccur at a distance just greater than the nozzle half width for thisparticular station of x/B = 2. An examination of the plots inrows C and D of Fig. 8 indicate that changes in the tailwaterdepth appear to have a greater influence on events in quadrants2 and 3. Both quadrant 2 below the jet centerline and quadrant 3above jet centerline represent slow entrainment events.

It is worth noting, from Figs 5 through 8, the value of the addi-tional information provided by the quadrant analysis of Figs 7and 8. An initial inspection of the third-order moments shown inFigs 5 and 6 may lead to some conflicting interpretation regardingwhether ejection or entrainment events dominate at a particular

section. For example, in Fig. 5, at x/B = 2 and TW = 2.5,the large peaks B and C suggest that entrainment dominates inthis region. Conversely, in the first row of Fig. 6, for the sameflow conditions at x/B = 2 and TW = 2.5, the small peaks R

and S and the relatively larger peaks of P and Q suggest thatthis is a region where ejections prevail. Information is lost inthe averaging process that delivers the third-order moments. Thequadrant analysis provides an additional distinction that furtherclassifies ejection and entrainment events as either fast or slow.Such a distinction may resolve the seemingly conflicting infor-mation provided by the third-order moments that both ejectionand entrainment events could be prevalent at the same location.An examination of Figs 7 and 8 suggests that at x/B = 2 andTW = 2.5 both fast ejection and slow entrainment events are

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prevalent. Consequently, the quadrant analysis reveals how thetwo mechanisms are actually balanced at a location. This is a layerof information not available through exclusive consideration ofthe third-order moments.

5 Conclusions

The characteristics of a square jet near a free surface were studiedexperimentally for four different tailwater depths. Measurementswere conducted at five different streamwise stations in the con-ventional near-jet region. An examination of the results revealed anumber of features typical of submerged jets and some unique tosquare jets, confined in the vicinity of a free surface. The specificconclusions include:

— Jet impingement on the free surface was nearly immediatefor the lowest tailwater, and the free surface impingementwas observed to move further downstream with increasingTW depth.

— Exit velocity profiles at all tailwater depths were top-hatin shape, downstream the potential core region developed,corresponding to a Gaussian profile.

— Free surface had little effect on the measured free streamvelocity profiles above and below jet centerline when the TWdepth exceeded 2.5B. The velocity characteristics of this jetare different from that of an axi-symmetric free jet.

— Turbulence intensities and shear stress profiles indicatedjet anisotropy and higher spread rates than occur in anaxi-symmetric free jet.

— Analysis of third-order moments demonstrated a markedincrease in the turbulent ejection and entrainment events forthe jet as the tailwater depth increased and the effect of freesurface confinement was reduced.

— Turbulent diffusion was found to be relatively constant foreach tailwater at distant downstream locations as the jet beganto disseminate beyond a streamwise axial distance of 10B.

— Quadrant analysis confirmed that the confinement effect ofthe free surface acted to curtail the ability of the jet toeject or entrain fluid. The highest energy events provedto be less affected by changes in the tailwater depth. Thequadrant analysis also provided a more detailed descrip-tion, than third-order moment considerations alone; exclusiveconsideration of third-order moments may lead to a conflict-ing interpretation regarding whether ejection or entrainmentevents dominate at a particular section. The quadrant anal-ysis utilized here has revealed that at particular sections thetwo mechanisms co-exist in a complementary balance of fastejections and slow entrainments or vice-versa.

Notation

B =Width of nozzleg = Gravitational acceleration

Du = u2v

Dv = v2u

TW = Tailwater depthu = Instantaneous mean velocity in streamwise

directionuv = Turbulent shear stress

u2v = Turbulent diffusion in normal directionu2 = Streamwise turbulence intensityu3 = Streamwise flux of streamwise momentumU Mean streamwise velocityUj = Jet exit velocity

Ucl, UO = Centerline velocity along x-directionv2 = Normal turbulence intensity

v2u = Turbulent diffusion in streamwise directionv3 = Normal flux of v-momentumx = Longitudinal distance from nozzley = Lateral distance from nozzle

y1/2 =Value of y where velocity is Um/2z = Lateral distance from centerlineν = Kinematic viscosityρ = Mass density of waterδ =Velocity half-width

References

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tailwater effects on the characteristics of a square jet near a free-surface

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