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Radial spreading of vertical buoyant jets in shallow water MacLatchy, Michael R 1993-09-15

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RADIAL SPREADING OF VERTICAL BUOYANT JETS IN SHALLOW WATERbyMICHAEL RAY MACLATCHYB.A.Sc, The University of British Columbia, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OF MASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Civil Engineering)We accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Michael Ray MacLatchy, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of Ci\.),^VtC4 ; ef,1121,„\.,3The University of British ColumbiaVancouver, CanadaDate ^[C4V\DE-6 (2/88)AbstractMany wastewaters are discharged from single vertical outlets located at the bottomof shallow water bodies. Often these effluents are buoyant with respect to the receivingenvironment. In many jurisdictions, regulations require that a certain dilution is achievedwithin a specified mixing zone. In shallow water it is likely that a portion of this dilutionwill have to be achieved in the radially spreading surface region of the jet. The degree ofdilution obtained in the radial surface buoyant jet region will depend upon the buoyancyand velocity of the effluent, and the depth of water available. Prior to the installation of anoutfall, the dilution that will be achieved and behaviour of the flow must be modeled indeveloping the design to meet regulatory and other requirements. Unfortunately, radiallyspreading flows are not clearly understood, and their behaviour is the subject of somecontroversy.This study was intended to identify the details of the structure of the radiallyspreading upper layer associated with the discharge of a vertical buoyant jet in shallowwater. Collection and analysis of detailed numerical data was not the focus or intent ofthis study, flow visualization was relied upon to investigate the radially spreading surfaceflow. Experiments to study the mechanisms of the radially spreading surface flow wereconducted with a series of vertical buoyant jets discharged into a shallow tank. Thisexperimental tank was specially designed to simulate an infinite ambient, and avoiddownstream control effects. A range of flow rates and port diameters were utilized todetermine the nature of the flow structure in the surface region. Flow visualization wasused to examine the flow and obtain an understanding of the mechanisms present to aid inmodeling.Four regions of flow are present when a buoyant jet is discharged vertically inshallow water. The first of these is the vertical buoyant jet region, in which the vertical jetis entraining fluid as the flow moves upward to the surface. Next is a surface impingementillregion, where the vertical flow arrives at the surface, and is redirected to an outward,radially spreading surface flow. The first of two surface regions is the radial buoyant jetregion, in which the momentum of the fluid dominates. This study determined that in thisregion the surface flow exists as a series of billows or vortices into which ambient fluid isentrained as they grow and move radially outward. Eventually this flow will have spreadout, and entrained enough fluid, that the interfacial shear will decrease to the point wherebuoyancy will dominate. The large scale billows will collapse, to be replaced by smallerless vigorous interfacial instabilities, as the second surface region, the radial buoyantplume region, is entered. Contrary to the assumptions made by some researchers, noradial internal hydraulic jumps were detected in the radially spreading surface flowregions.ivTable of ContentsAbstract^ iiList of Tables viList of Figures^ viiAcknowledgments ix1. Introduction^ 12. Literature Review 42. 1. Basic Jet and Plume Parameters ^ 52. 1. 1. Pure Jets ^ 62. 1. 2. Pure Plumes 82. 1. 3. Buoyant Jets ^  82.2. Buoyant Jets In Shallow Water ^  102. 2. 1. Water Depth Classification  102. 2. 2. Buoyant Jets In Shallow Water ^  112. 2. 3. Surface Impingement and Disturbance  I52. 2. 4. Additional Effects of Shallow Water ^ 162. 2. 5. Buoyant Jets In Extremely Shallow Water 172. 2. 6. Mechanisms of Near-Field to Far-Field Transition ^ 212. 2. 6. 1. Entrainment Layer Approach ^ 212. 2. 6. 2. Internal Hydraulic Jump Approach 242. 2. 6. 3. Summary of Approaches to TransitionMechanisms ^ 273.^Experiments^ 293. 1. Prediction of Flow Conditions ^ 293. 2. Experimental Design and Procedure 323.2. 1. Experimental Design ^ 323. 2. 2. Experimental Procedure 364.^Results and Comments^ 384. 1. Results ^ 384. 1. 1. Standard Experiments ^ 384. 1. 2. Experiments with Nozzle Extensions ^ 414. 1. 3. Choked Experiments ^ 424. 1. 4. Measured Quantities  434. 2. Comment on Results ^ 464. 3. Comparison To Models 484. 4. Restrictions and Other Problems ^ 515.^Conclusions and Recommendations 53Notation 57References^ 60Appendix A Tables^ 62Appendix B Figures 65List of Tables(Appendix A)3. 1 Experiments Conducted ^ 634. 2 Measured and Predicted Quantities for Experiment Conducted ^ 64VIList of Figures(Appendix B)2.1 Definition sketch for pure round jets^ 662.2 Definition sketch for water depths 662.3 Length of Zone of Flow Establishment as a function of sourceDensimetric Froude number, Lee & Jirka (1981) ^ 672.4 Regions of flow of a vertical buoyant jet in shallow water, accordingto Lee & Jirka (1981) ^ 672.5 Stability Diagram, Lee & Jirka (1981) ^ 682.6 Thickness of surface impingement layer as a function of sourcedensimetric Froude number, Lee & Jirka (1981) ^ 682.7 Stability diagram re-expressed in terms of F, Labridis (1989) ^ 692.8 Flow regions of a vertical buoyant jet in extremely shallow water asdefined by Lawrence & Bratkovitch (1990) ^ 692.9 Internal hydraulic jump in a two layer flow in a channel, AfterWilkinson & Wood (1971) ^  703.10 Definition sketch for flow prediction calculations ^ 703.11A Variation of upper layer Reynold's number with Non-Dimensionalized Radius ^ 713.11B Variation of Composite Froude Number Squared with Non-Dimensionalized Radius ^ 713.12 Plan view schematic of experimental apparatus ^ 723.13 Photograph showing experimental apparatus 724.14 A-D Surface flow of Experiment 15020-2 ^ 734.15 Comparison of experiments to stability criteria of Labridis (1989) ^ 74viiVIII4.16 Plunging Structure at the exit from surface impingement zone inExperiment 7520-4 ^ 744.17 A-D Surface flow under choked conditions, Experiment 15020-C ^ 754.18 Variation of Strouhal number of radial flow with F ^ 764.19 Non-dimensionalized measured upper layer depths vs Fo , includingvalues predicted using solutions of Lee & Jirka (1981) ^ 764.20 Comparison of observed upper layer depths with solutions of Lee &Jirka (1981) ^ 774.21 Variation of upper layer growth rate with Fo ^ 774.22 Comparison of experiments of Lee & Jirka (1981), Wright et al.(1991) and this study ^ 78ixAcknowledgmentsFor his constant encouragement and constructive supervision, I would like tothank my thesis supervisor, Dr. G. A. Lawrence. I would also like to thank KurtNielson, of the Civil Engineering Hydraulics Laboratory, for the construction of theexperimental apparatus used in this study. A special thanks goes to Jason Vine for hisassistance with the scientific laser during experiments, and in the darkroom developingfilm.1Chapter 1Introduction Many wastewater discharges, such as pulp mill effluents and cooling water fromthermal power plants, are buoyant with respect to the receiving water that they areentering. The effluents are frequently released as vertical discharges located at the bottomof shallow water bodies. Such discharges pose a concern from an environmentalstandpoint. Contaminants or heat may be present, with the potential for a negative impacton the receiving environment. Depending upon the location, and regulatory requirements,it is often necessary to achieve a required dilution within a certain distance of the point ofdischarge. In shallow water the degree of dilution will be greatly influenced by the depthavailable in which the jet may entrain fluid. Once surface impingement occurs radialspreading of the jet may account for additional dilution. The extent to which thisadditional dilution occurs will depend on such factors as the buoyancy and velocity of theeffluent, and the relative proportion of the depth occupied by the surface layer.Eventually, as the distance from the discharge point increases, the initial momentum of thedischarge will be of less importance, and the behaviour of the flow will become plume like,buoyancy effects will dominate.While most real situations will involve the presence of ambient currents, the basicsituation in which the ambient is stagnant has considerable relevance, both as afundamental case, and in situations where the ambient currents are relatively small withrespect to the velocities in the radially spreading surface flow. Models for predicting thedilution and behaviour of vertical round buoyant jets in shallow water have been presentedby Lee & Jirka (1981), Wright et. al. (1991) and Lawrence & Bratkovitch (1993), to namea few. These models draw on earlier work, such as Crow & Champagne (1971), Abraham(1965), Wilkinson & Wood (1971), and Chen (1980). Experimental investigation has2largely been limited to a confirmation and calibration of the bulk parameters of thesemodels. Detailed examination of the physical mechanisms of the surface flow has not beenundertaken to any great extent. Certain of the models have been based on assumptions,adopted for mathematical convenience, as to what is occurring in the radial surface flow.In order to better understand the behaviour of the radial surface flow and to formulatebetter models, the present study will undertake to investigate the behaviour andmechanisms of the radial surface flow. The main emphasis of this study will be to examinethe radially spreading surface flow, and identify the structures present by flow visualizationtechniques. Detailed analyisis of hard numerical data, such as the velocities in the surfacespreading layer, or the dilution and entrainment occurring, will be undertaken in futureinvestigations.Most researchers have recognized that there are two basic parameters thatdetermine the behaviour of a vertical buoyant jet surfacing in stagnant shallow water, thesebeing the densimetric Froude number, and the ratio of the depth of ambient water to thejet orifice diameter, HID. Three distinct regions of flow have been identified in the near-field, though the exact behaviour of the surface flow in the near-field is open to debate.Basic relationships used to describe the velocity and concentration profiles, and dilution ofvertical buoyant jets (Fischer et al., 1979, and List, 1982) are presented in Chapter 2, aswell as a review of literature regarding vertical buoyant jets discharged in shallow water.Experiments to investigate the behaviour of the surface flow associated with thedischarge of a vertical buoyant jet in shallow water are discussed in Chapter 3. Athermally buoyant jet was discharged into cold ambient water, which was constantly beingreplenished to replace entrained fluid. Visualization of the mechanisms in the surface flowwas accomplished using a sheet of laser light, and fluorescent dye injected into the jetfluid. Chapter 3 contains a description of the design of the experimental apparatus, anddetails of the experimental procedure. Results and discussion are presented in Chapter 4.3Conclusions and recommendations arising out of this study are presented inChapter 5.The results of this thesis will be of use in increasing the understanding of radialsurface flow. A large number of discharges exist as single vertical buoyant jets located atthe bottom of nearly stagnant shallow water. In these cases, as long as ambient currentsare small relative to the velocities of the radially spreading flow in the near-field, theinformation obtained from this study should assist in predicting the flow conditions andmixing which will occur. Refinement of predictive models used to aid in design of outfallswill hopefully arise out of this study.4Chapter 2Literature ReviewWhile a number of papers have presented models of the dilution which occurs inthe surface regions of a vertical buoyant jet in shallow water, relatively little work hasbeen undertaken to investigate the mixing mechanisms, and the manner in which thetransition from near-field to far-field occurs. This review will start by considering thebasic jet and plume parameters, and the simple relationships which exist for dilution, andvelocity and concentration profiles. Then a number of models regarding the discharge ofvertical buoyant jets into shallow water will be focused on, with emphasis on theassumptions made in developing the models presented. Where experimental results exist,these will be discussed, particularly where hydraulic experiments provide insight into themechanisms involved. Where investigations regarding other situations, such as for planevertical jets or two layer channel flow, contain useful results they will be discussed.The simple case of a vertical buoyant jet discharging into deep stagnant ambientwater has received considerable attention, an introduction to this case, as well as the caseof buoyant plumes, is presented in Fischer et al. (1979). Fischer et al. (1979) also presentsa general approach for the case of a vertical jet or plume in deep flowing water. List(1982) provides an overview of the basic characteristics and parameters for buoyant jetsand plumes. Jirka (1981) presents an approach to the problem of vertical buoyant jetsdischarged into shallow water.This review is divided into two major parts. In the first part the basic parameterswhich describe jets and plumes are presented and discussed. The second part covers thetheory of buoyant jets in shallow water, and the available results of investigations of themixing mechanisms.52. 1. Basic Jet and Plume ParametersThe behaviour of a jet or plume discharged into a fluid is governed by threegeneral categories of parameters. These parameters are environmental, which relates tothe conditions in the ambient fluid; geometrical, which deal with the geometric relationshipbetween the jet and the ambient fluid; and jet parameters which are the specific propertiesof the jet. Examples of environmental conditions are the density stratification, flowvelocities, and degree of turbulence in the ambient fluid. Geometrical parameters ofimportance include the depth of submergence of the jet, angle of the jet to the horizontal,or angle of the jet to the flow in the ambient. Significant jet (or plume) parameters includethe jet velocity, total discharge, and density deficit between the jet and the ambient fluid.Making use of the definitions in Fischer et al. (1979), there are three expressionsthat are useful in describing the conditions in a jet:Volume flux, p=f u dA^ (2.A1)Momentum flux, m = f /42 dA^ (2.A2)Buoyancy flux, /3 =Ag' u dA^ (2. 3)Where: A = cross sectional area of jet.u = time averaged velocity in axial direction.p = specific mass flux or volume flux.m = specific momentum flux.= the specific buoyant or submerged weight of thefluid passing through a cross section per unit time.The specific buoyancy flux.g' = (\p/p)g, effective gravitational acceleration.These quantities will vary with location in the jet as fluid is entrained and velocitiesdecrease (discussed later). For purposes of determining how the jet will behave it is often6more convenient to state these quantities in terms of the initial jet conditions. In terms ofthe initial jet conditions the equivalent expressions to equations 2. 1 to 2. 3 above are:Q = 4—D2 U (2. 4)M= —71. D2 U2 (2. 5)4B. 0) (2. 6)Where:^Q = initial volume flow rate (at jet exit).M = initial mass flux rate.B = initial buoyancy flux rate.go = initial effective gravitational acceleration.Note that these expressions are for round jets, similar expressions are possible forother jet configurations. From research, it has been determined that the three initialparameters, Q, M, and B will govern the behaviour of round buoyant jets as long as thejet is fully turbulent, that is, if the jet Reynolds number is greater than about 4000. If thiscondition is met, other factors will have relatively little significance in determining thebehaviour of the jet. We will now discuss the behaviour, and the relationships used todescribe the behaviour, of pure jets, pure plumes and buoyant jets.2. 1. 1.^Pure JetsUpon entering the ambient fluid, a shear layer will form between the jet and theambient fluid. The behaviour of the jet may be broken into two distinct zones, the Zone ofFlow Establishment (ZFE), and the Zone of Established Flow (ZEF), refer to Figure 2.1.Within the first zone (ZFE), the shear forces generated by the interaction of the jet and theambient fluid have not penetrated into the center of the jet, and there exists a jet core inwhich the velocity remains equal to the jet exit velocity. After a short distance from thejet exit, approximately six to seven port diameters (z,), the Zone of Established Flow(ZEF) begins. Within this zone the velocity and concentration profiles become Gaussian.As a result, the time-averaged profiles of velocity or concentration can be expressed as a7maximum value at the jet centerline, and the distance from the jet centerline. The resultingformula for the velocity distribution takes the form of:u = uc exp[-(z/b) 2 ] .^ (2. 7)Where u, is the centerline (maximum) velocity, x is the distance from thecenterline, and b is the characteristic width of the profile. For a pure jet a characteristiclength may be defined :1 = Q Q M°5 (2. 8)Expressions for the centerline velocity and momentum flux as functions ofdistance from the jet exit, z , can be derived (Fischer et al., 1979), as can expressions forthe characteristic width, b :u —Q = 7.0-Qc A,/ (2. 9)(2. 10)0.107^ (2. 11)z= 0.127 (2. 12)Where^bN = the characteristic width of the velocity profile(Gaussian).b, = the characteristic width of the concentration ortemperature profile.Provided that z » /Q .In a similar fashion, it is also possible to define the centerline concentration as afunction of z, and the concentration with distance from the centerline:C„, /C0 = 5. 640Q /4^(2. 13)8C = Cm exp{—(x/b, ) 2 1 (2. 14)As a consequence of the shear between the jet and ambient fluid, entrainment ofambient fluid into the jet will occur. In formulating an approach to the entrainmentproblem, Morton et al. (1956), proposed that the velocity of the inflowing diluting fluidbe proportional to the local centerline velocity of the jet. The resulting expression for theentrainment flux was Qe =2ffabue , where a is an entrainment coefficient. Furtherresearch ( List and Imberger, 1973) revealed that a was not a constant, but in fact was afunction of the local densimetric Froude number.An alternate approach, and one that is more easily applied, is to assume a constantspreading angle for the jet, as proposed by Abraham (1965). With this approach thediffusion layer is assumed to spread linearly. It has been shown that the spreading angle,dbldz= c , varies by less than 10% between the extreme end cases of pure plumes andjets, while a varies widely with Fo . Jirka (1975) further demonstrated that the twoapproaches were consistent, and c could be related to a for both the case of a jet orplume.2. 1. 2. Pure PlumesFor a pure plume, both the initial momentum and volume fluxes are zero. Anexpression for the center line vertical velocity in a plume is:Ll'i3tic = 4.7  0z (2. 15)As the plume rises the momentum flux increases from its initial value of zero to:p= 0.254mv2z^ (2. 16)2. 1. 3.^Buoyant JetsBuoyant jets are hybrids of jets and plumes, the fluid being discharged is buoyant,but also has significant initial momentum. In addition to the length scale le , used to9characterize pure jets, an additional characteristic length scale is introduced for buoyantjets:1113/411 = ^m^B1/2The ratio of IQ to 1„ is the initial jet Richardson number:BQ 112R = 1Q =i^m5/4M(2. 17)(2. 18)When the initial jet Richardson number, Ro , approaches a critical value, the plumeRichardson number (I?, = 0.557), then the buoyant jet actually behaves as if it were a pureplume right from its origin. In a manner similar to that for pure jets, Fischer et al. (1979),presents expressions for the dilution of a buoyant jet, where:Q (2. 19)RpC= cp ZIQRo j (2. 20)RPWhere cp = 0. 254, and the volume flux then becomes:N= S «1 (2. 21)c»1 (2. 22)It is interesting to note that all buoyant jets will eventually become plumes, oncethey have traveled a great enough distance. The determining parameter for whether abuoyant jet will behave as a plume, or as a jet, is the ratio of z to IM . If z >> 124 then thebuoyant jet will behave as a plume, conversely if z Iv then the buoyant jet will retain itsjet like behaviour. Therefore, if /Q, and lm are of the same order, the buoyant jet willbecome plume like in behaviour very close to the jet exit.102. 2. Buoyant Jets In Shallow Water2. 2. 1.^Water Depth ClassificationThere are a number of different schemes for classifying water depth when dealingwith jets and plumes, with each researcher apparently using a different one. For thepurposes of this review a uniform classification of water depth will be applied. Under thisscheme water depths will be divided into four categories, each category being defined bythe characteristics of a jet discharged into it (Refer to Figure 2.2). The four categories ofdepths that will be employed are: deep water, shallow water, very shallow water, andextremely shallow water. These categories are defined as follows:Deep WaterGenerally speaking, deep water can be defined as water with a depth above theoutlet much greater than 20 port diameters (h, /D » 20). Velocity and density profiles ofthe vertical jet can be treated as self-similar or Gaussian. Should the jet reach the surface,no significant surface disturbance can be expected. If the water is sufficiently deep,H » IQ , then a buoyant jet will develop into a plume and buoyancy effects will bedominant.Shallow WaterWithin shallow water, velocity and density profiles of the jet outside the zone offlow establishment may still be treated as Gaussian. However a surface disturbance andsurface spreading layer may be expected. If the jet does not have sufficient buoyancy orhas high momentum flux then there is a possibility of a recirculating or unstable flow.Based on the definition of Lee & Jirka (1981), shallow water depths are from 6 to 20 portdiameters (6 < h, /D 20).Very Shallow WaterFor jets with Fo > 25 the length of the Zone of Flow Establishment (ZFE), z1 , isapproximately 6D. As Fo decreases below approximately 8, the length of the ZFE11decreases quickly to a minimum of 2D for F0 of approximately 1 (refer to Figure 2.3).Since for most practical cases the densimetric Froude number will be greater than 8, z, canbe assumed to be approximately 6D. When z < 6D, there exists a jet core in which thevelocity remains at the jet exit velocity, and turbulent shear forces have not penetrated intothe jet. As a result, in the ZFE, Gaussian velocity and density profiles are not appropriate.Also, the thickness of the surface impingement region may be a significant proportion ofthe water depth over the discharge. Based on this, the range for very shallow water isdefined as 2 to 6 port diameters (2 < h , /D < 6).Extremely Shallow WaterCrow and Champagne (1971) demonstrated that the entrainment flux for a jet isconstant, and that a "top-hat" velocity profile was appropriate for water depths of lessthan 2 port diameters (k /D 2). From this, extremely shallow water is defined as havinga depth less than 2 port diameters (h, /D 2).2. 2. 2. Buoyant Jets In Shallow WaterThe study by Lee & Jirka (1981) on shallow water jets has formed the basis forlater work by many researchers. As part of their study, Lee & Jirka (1981) examinedshallow water jets over a wide range of flow conditions. Experimental studies werecarried out on semicircular vertical buoyant wall jets with densimetric Froude numbers(F0 = 111(g 'D) 112 ) ranging from 8 to 583. Depths used in the study were between 6 and35 port diameters, and jet Reynolds numbers were sufficiently high for all runs to ensurefully turbulent conditions. In developing their model of shallow water jets, Lee & Jirka(1981) divide the flow into four regions (described below): buoyant vertical jet region,surface impingement regions, radial internal hydraulic jump, and stratified counterflowregion. The first three regions comprise the near field (momentum dominated), and thelast region makes up the far field (buoyancy dominated). See Figure 2.4.12Buoyant Vertical Jet RegionIn this region the vertical buoyant jet entrains ambient fluid as it moves toward thesurface. Where the jet enters the surface impingement region, the velocity and densityprofiles are assumed to have become self similar or Gaussian, since Lee & Jirka (1981)assume depths greater than six port diameters.Surface Impingement Region.Within this region, the vertical flow of the buoyant jet is redirected to a horizontalradially spreading flow. A surface boil, or fountain, is associated with this region. Thereis assumed to be intense turbulent mixing, but no entrainment, within this region. It isassumed that little or no entrainment occurs in this region, since the surface impingementregion is bounded almost completely by the vertical buoyant jet and radial internalhydraulic jump region. The surface impingment region has almost no interfacial area withthe ambient fluid. At the point where the flow exits from the surface impingement region,the velocity profile is assumed to be half-Gaussian.Radial Internal Hydraulic Jump RegionThe flow is transformed from a supercritical surface flow to a subcritical flow bypassing through an internal hydraulic jump. There is an abrupt increase in upper layerthickness, and an energy loss, associated with the radial internal hydraulic jump. Based ontheory for two-layer flows, Lee & Jirka (1981) were able to develop expressions forestimating the conjugate layer depths upstream and downstream of the internal hydraulicjump. This region will exist when the jet discharge is considered stable. That is, when thedischarged fluid has sufficient buoyancy to form a surface layer upon exit from the surfaceimpingment zone, and to prevent being re-circulated and re-entrained into the jet. If themomentum of the jet were high relative to the buoyancy of the jet, an unstable flowconfiguration would exist. As will be discussed later, there is some dispute over the13existence of an internal hydraulic jump in this transition from near-field to far-fieldconditions.Stratified Counterflow RegionThis region is made up of the buoyant upper layer flowing outward from thesource and the denser ambient fluid of the lower layer flowing inward to the verticalbuoyant jet. Velocities are relatively low, and entrainment across the interface of the twolayers is considered negligible.An unstable configuration is one in which the buoyancy of the discharged fluid isnot sufficient to cause the flow to form a distinct, stable upper layer. The downwardmomentum of the fluid exiting the surface impingment zone, combined with the shearbetween the outward flowing upper layer and the inward flowing ambient fluid in thelower layer, causes the upper layer to fill the entire depth of the near-field, and re-circulation and re-entrainment into the jet fluid occurs. In an unstable configuration, adistinct surface layer only forms in the Stratified Counterflow Region. This flowconfiguration is not considered desirable as it has the effect of reducing dilution.Lee & Jirka (1981) developed a criteria for stability of the near-field region. Thiscriteria may be expressed as a function of the densimetric Froude number(Fo = 111(g' , ), and H/D . Refer to Figure 2.5. The flow configuration is consideredstable when:Fo < 4.6(H/D) for H D > 6 (2. 23)Thus, when the densimetric Froude number is greater than a certain valuedetermined by H/D, the flow will be unstable. If the source conditions (F,,D) are fixed,then decreasing the depth H beyond a certain value will result in the flow becomingunstable.Central to Lee & Jirka's approach to the problem of vertical buoyant jets inshallow water, is the assumption that the transition from near-field to far-field conditionsis accomplished by means of an internal hydraulic jump. This assumption is disputed by14other researchers, as will be discussed below. Lee & Jirka (1981) do not present anyvelocity measurements to verify that there are conditions of internally supercritical andinternally subcritical flow upstream and downstream of the supposed location of theinternal hydraulic jump, to support their contention that this is a valid approach. It ispossible that the presence of a downstream control in Lee & Jirka's experimental workmay have created an internal hydraulic jump that would not otherwise have been present.A further, modified development of Lee & Jirka's approach is presented in Wrightet al. (1991). In this model, it was assumed that the radial surface flow results in greaterdilution than that in the vertical buoyant jet alone, it was further assumed that entrainmentdoes occur in the internal hydraulic jump, where as Lee & Jirka (1981) did not. Informulating their model, Wright et al. (1991) assumed that the flow exiting the internalhydraulic jump is internally critical. This contradicts the formal definition of an internalhydraulic jump in which the flow must be internally supercritical upstream of the jump,and internally subcritical downstream and is probably incorrect. However, this assumptionhas the advantage of allowing the equations of Wright et al.'s model to be more easilysolved, since the form and degree of downstream control could be ignored, and wasprobably adopted for this reason. It is not clear whether this is actually valid for the realsituation or is merely an expedient measure. It was noted that the model was mostsensitive to the assumed length of the internal hydraulic jump.Wright et al (1991) also presented a stability criteria, the flow configuration wasunstable, and re-circulation of the upper layer back into the vertical jet would occur when:H > 6 (2.24)By substituting for 44 , and rearranging equation 2.24, the following is obtained:Fo 7.63(Li-^ (2.25)15This result for the stability criterion has the same form as that of Lee & Jirka(1981). Since Wright et al. (1991) noted that the coefficient on the right hand side ofequation 2.23 could vary between 3 and 6, with the result that the coefficient in Equation2.24 could vary between 3.8 and 7.63, this indicates that these results are in relativelyclose agreement with those of Lee & Jirka.2. 2. 3. Surface Impingement and DisturbanceIf the ambient water is sufficiently shallow, then the jet momentum will still beconcentrated into a relatively small jet core, and will not have been spread out byentrainment. As the buoyant jet impinges on the free surface, a surface disturbance or boilis created. This surface boil produces a radial pressure gradient which redirects the flowfrom a vertical jet into a radial horizontal flow. The radius of the region of surfaceimpingement is defined as corresponding to the radius in the vertical jet at which thevelocity, u, is 5% of the centerline (maximum) value, ui., at the point of entry to thesurface impingement region. At the exit from the surface impingement region mostresearchers assume that the radial velocity profiles are half-Gaussian. Lee & Jirka (1981)present diagrams for determining the thickness of the surface impingement region asfunctions of F, and HID (Refer to Figure 2.6).It is possible to estimate the maximum height of the surface disturbance. Makinguse of the results of Murota & Muraoka (1967), Labridis (1989) provided an expressionfor the maximum height of the surface disturbance:110 = 1.61 D714^F312H314(2.26)Many researchers including Murota & Muraoka (1967), and Labridis (1989), havenoted that the surface disturbance does not approach the ambient water surfacemonotonically, but that a small ring around the boil is depressed below the level of theambient water. This result is particularly noted as the ambient water becomes increasingly16shallow. It is believed that this occurs because the water which flows up to the top of thehump then develops downward momentum as it flows outward horizontally from the crestof the hump.Due to the complexity of the flow structure within the surface disturbance, itwould be difficult to analytically describe the flow within the surface impingement region.As a result, most researchers have relied upon a control volume approach, and simplifyingassumptions, such as half-Gaussian velocity profiles at the exit from the region, e.g. Lee &Jirka (1981) and Labridis (1989).2. 2. 4. Additional Effects of Shallow WaterLee & Jirka (1981) presented a graph that shows that for Fo < 10, the non-dimensionalized length of the ZFE, Z, = (zr ID), varies considerably with I. For Fo >25, z, asymptotically approaches a value of approximately 5.74, (Refer again to Fig 2.3).Thus for many practical situations, when the ambient water depth (H) is less than six portdiameters, the jet flow will not reach a state of established flow, and velocity and densityprofiles will not be Gaussian. Some researchers (Labridis (1989)) define water depthswhere H 6D as very shallow water because of this situation.Crow & Champagne (1971) studied non-fully developed jets, and determined thatfor h, /D <2 and HID < 6 the following equations applied for Q and M, at the boundary(section 1) between the vertical jet and the surface impingement zone:Q, = Q9 (1 + 0.136h, /D) (2.27)M, =M0 (1+0.136h1D) (2.28)It has also been reported (Murota & Muraoka (1967)), that for H <20D, that thespreading angle of the jet within the Zone of Flow Establishment is greater than that for afree jet. This could be due to the formation of helical vortices around the jet. Theunbalanced forces produced by the helical vortiex on alternating sides of the jet would17cause the jet to oscillate about its centerline. This would make the jet appear wider than itactually was as it oscillates through its range of movement.Labridis (1989) demonstrated that for ze « 44 , the jet Froude number(F =U1(gD) 112 ) will be of more significance than Fo, since when .ze « 4,4 the momentumof the jet dominates the flow and the jet's buoyancy is relatively unimportant. This impliesthat for shallow and very shallow water F will be more important than Fo , since z,. « lM .As a result, Labridis re-expressed the stability diagram of Lee & Jirka in terms of Finstead of Fe , refer to Fig 2.7. The stability criteria of Lee & Jirka (1981) re-expressed interms of F by Labridis (1989) is:F = 0.32(H/D) (2. 29)2. 2. 5. Buoyant Jets In Extremely Shallow WaterLawrence & Bratkovitch (1993) developed an alternate approach for the behaviourof a vertical buoyant jet discharged into extremely shallow water (H/D <2 andh, /D < 2 ). Because of the extremely shallow depths, an assumption of Gaussian velocityand density profiles in the vertical jet was inappropriate. Instead, the results of Crow &Champagne (1971) were made use of, and "top-hat" velocity profiles were assumed in thevertical buoyant jet. This also allowed the assumption of constant entrainment flux in thisportion of the jet. The situation involving jets in very shallow water (6 < HID > 2, and6 < h, /D < 2) was not dealt with, as the behaviour of a jet under these conditions is notwell understood, representing as it does a transition from the ZFE to ZEF conditions inthe jet.The Lawrence & Bratkovitch (1993) model is broken up into four regions in amanner similar to that employed by Lee & Jirka (1981), but there are some importantdifferences. The four regions in the Lawrence & Bratkovitch (1993) model are; verticalbuoyant jet; surface impingement zone; radial buoyant jet; and radial buoyant plume (Referto Figure 2. 8). The chief difference between the Lawrence & Bratkovitch (1993) model18and the Lee & Jirka (1981) model lies in the assumption made by Lee & Jirka (1981) thatthe transition from near-field to far-field (radial buoyant jet to radial buoyant plume) isaccomplished by means of an internal hydraulic jump. Lawrence & Bratkovitch (1993)argue that the radial buoyant jet entrains ambient fluid until buoyancy causes entrainmentto cease, and the radial buoyant jet becomes the radial buoyant plume in the far field.Lawrence & Bratkovitch dispute the existence of an internal hydraulic jump as themechanism of transition between near-field and far-field, as a downstream control wouldbe required to induce the internal hydraulic jump, and in the absence of a downstreamcontrol such a jump would not form. The mechanism by which transition from near-fieldto far-field may occur will be discussed further below.In addition to the assumption of "top-hat" velocity profiles in the vertical buoyantjet region, the Lawrence & Bratkovitch (1993) model makes use of the followingassumptions:• Constant momentum flux in the vertical buoyant jet, radial buoyant jet, and radialbuoyant plume. The existence of energy and momentum loss in the surfaceimpingement region was recognized, but was considered negligible based on Lee &Jirka's result that their solution was not sensitive to energy loss.• Constant buoyancy flux in near field, with buoyancy flux decreasing in far-field overtime due to assumed heat loss. In modeling the decrease in buoyancy due to heat lossfrom the radial buoyant plume, it was assumed that density varies linearly withtemperature. For warm discharges, the assumption that density varies linearly withtemperature is reasonable, since the relationship between temperature and density isapproximately linear above 15°C. In the far-field, where dilution and heat loss to theatmosphere will lead to significantly decreased temperatures, and a non-lineartemperature-density relationship, this would not be valid and would be subject togreater error.19Lawrence & Bratkovitch developed the following equations for conditions atsection 1, the entrance of the vertical buoyant jet to the surface impingement region:Q1^r' =^=^= 1 + 0.065AQo^ro^u,(2.30)Provided that hi /r0 5 4 (h, /D 5_ 2 ).In the surface impingement region negligible entrainment was assumed byLawrence & Bratkovitch (1993), and therefore at section 2, the exit from the surfaceimpingement region to the radial buoyant jet region, Q2 = Q1 and g', = g' 1 . Lawrence &Bratkovitch (1993) followed Lee & Jirka (1981), in assuming that section 2 occurs at theradius, r.,, at which the velocity in the vertical buoyant jet falls to 5% of its centerline valueat section 1. As a result of this assumption, and making use of equation (2.30), Lawrence& Bratkovitch obtained:r2 =r o (1 + 0 .125 hd ro ) (2.31)Again provided that hi ro S 4 (hi /D 2).Based on this result, and in applying conservation of mechanical energy in thesurface impingement region, Lawrence & Bratkovitch (1993) then predicted that thethickness of the surface impingement region was nearly constant for the range0 /-1,11.0 5 4 with h2 r, = 0.5.Within the radial buoyant jet region, Lawrence & Bratkovitch (1993) reason thatshear instabilities as a result of the radial flow of the buoyant jet over the counter flowingambient fluid cause further entrainment and the upper layer will grow in thickness. Thegrowth rate of the radial buoyant jet was modeled as:dh = av r f(Ri)drWhere:^a = growth rate coefficient.(2.32)U-U velocity ratio.v r^U+ Urree:Ure, = the depth averaged return flow in the lower layer.20U = the depth averaged flow in the upper layer.Ri— g' h2 the bulk Richardson number.AUAu= U -The bulk Richardson number contained in equation 2.32 is important to Lawrence& Bratkovitch's (1993) model. Initially, the bulk Richardson number will be less than acertain critical Richardson number, Ri c . While this is the case, shear instabilities willentrain ambient fluid into the radial buoyant jet, increasing its thickness until Ri c isreached, and buoyant forces stabilize the flow, and entrainment ceases.If the bulk Richardson number does not approach its critical value, and the velocityratio, v„ instead approaches infinity, then the flow will become unstable. The velocityratio will approach infinity when the velocities in the upper and lower layers are ofapproximately equal magnitudes, though in opposite directions. This will occur when thewater is too shallow and/or the jet has a very high volume flux.If the flow is stable in nature then the transition from near-field to far-field occursat the point where the critical Richardson number, Ri c , is reached. Lawrence &Bratkovitch (1993) made use of the results of Chen (1980) in determining the growth rateof the radial buoyant jet. In working with Chen's data the best fit was found for a = 0.08and f(Ri)=1— Ri/Ri c . In addition, Ri c was found to be approximately 0.3. Others havesuggested that a should be in the range of 0.015 and that a different form of f(11, ) shouldbe employed. However, a = 0.08 is close to the values found for circular plumes.The composite Froude number is defined as follows:62G = ^+ 172^7'1 ^(2.33)g' k g' (H — k)The composite Froude number indicates whether the flow is internallysupercritical, internally critical, or internally subcritical. The significance of the compositeFroude number, is that when the flow is internally supercritical or internally critical, long21internal waves can not propogate upstream against the flow. Thus, under internallysupercritical or internally critical conditions the effects of internal hydraulic jumps, orother similar features, cannot propagate upstream.If constant momentum flux in the far-field is assumed, then sample calculationsusing equation 2.32 (refer to section 3.1) reveal that the composite Froude number, G,approaches 1 asymptotically. This result is essentially the same as the assumption ofinternally critical flow in the distant far-field made by Wright et al. (1991), in developingtheir model. According to Wright et al. (1991), these conditions imply that the internalhydraulic jump is of the maximum entraining type, and shear layer growth is maximized.2. 2. 6. Mechanisms of Near-Field to Far-Field TransitionThere are two contrasting approaches to the treatment of the transition of theradial buoyant surface jet from near-field to far-field conditions. Generally, mostresearchers have followed the approach of Lee & Jirka (1981) in assuming that a radialinternal hydraulic jump exists. Others have argued that this is only valid under certainconditions, namely an imposed downstream control, which would induce an internalhydraulic jump. The most important of these is Wood & Wilkinson (1971).2. 2. 6. 1. Entrainment Layer ApproachWilkinson & Wood (1971) examined the occurrence of an internal hydraulic jump,or density jump, for a dense liquid discharged at the bottom of a channel. Two majorregions were identified in a density jump (Refer to Figure 2.9). In the first of these, theentrainment region, there exists interfacial shear which gives rise to interfacial instabilities,and therefore entrainment. Downstream from the entrainment, region a roller region mayoccur depending on downstream conditions. In the roller region there is a reverse flownear the interface with the ambient fluid, and interfacial shear is considerably lower.Entrainment in the roller region is negligible. It is important to note the definition of adensity jump as employed by Wilkinson & Wood (1971). In their scheme, a density jump22is composed of either an entrainment zone, or internal hydraulic jump, or possibly both,depending upon flow conditions. Since an internal hydraulic jump is not necessarilypresent, the term density jump may be misleading, it may not resemble a "jump" at all.Of importance is Wilkinson & Wood's observations of the behaviour of the internalhydraulic jump as downstream control is varied. In order for a roller region to exist,Wilkinson & Wood (1971) determined that it was necessary for a downstream controlsuch as a weir, or channel constriction, to exist. The greater the control that was exerted,the less the extent of the entrainment region, with the roller region being pushed fartherupstream. If the downstream control, such as weir height, were increased beyond acertain point, it was found that the entrainment zone ceased to exist, any further increasein the control exerted would then result in the density jump becoming flooded. As thedegree of downstream control was reduced, the greater became the extent of theentrainment region, until finally, with little or no downstream control, the entrainmentzone occupied the complete length of the density jump, and the roller region had died out.In short, in the absence of a downstream control, only an extended entrainment regionfeaturing interfacial instabilities due to shear existed.Lawrence (1985) investigated the occurrence of internal hydraulic jumps, andassociated mixing, in two layer flow in a channel. The flume used in these experimentswas 12.8 m long by 0.38 m wide. When the flume was free from obstacles, the flowlacked an internal hydraulic jump, and there was no significant entrainment occurring.Internal hydraulic jumps were only present when a stationary obstacle, in the form of a sill,was introduced to the channel. The internal hydraulic jump would form downstream ofthe obstacle. Often shear instabilities, of the Kelvin-Helmholtz variety, were observed inthe approach to the internal hydraulic jumps. These instabilities resulted in entrainment offluid occurring at the density interface. Of most importance though, is the observationthat internal hydraulic jumps would only occur when an obstacle was placed in the23channel. Even though the channel was very long, friction from the bed and sides of thechannel was insufficient to cause the occurrence of an internal hydraulic jump.Koop & Browand (1979) investigated the entrainment and mixing which occurredat a density interface in a two layer flow in a long channel. In their experiments bothlayers were flowing in the same direction, though at different velocities. As for Lawrence(1985), they did not observe the presence of an internal hydraulic jump in the channel.Instead, interfacial instabilities, in the form of vortices, initiated at the entrance to thechannel. As these vortices moved down the channel they grew in size, and merged. Atsome point in the channel the instabilities would reach a maximum size. When thisoccurred the instabilities would collapse into a stable interface, with only small instabilitiespresent, the turbulence generated by these large scale instabilities having been dissipated.Koop & Browand (1979) reason that the large scale structures are critical in the formationand maintenance of turbulence. The statically stable conditions arising from the densitydifference results in the ultimate destruction of the turbulence generated. Koop &Browand (1979) provide an explanation as to why the collapse of the vortex structuresoccurs. As a vortex grows, it lifts denser fluid, and depresses lighter fluid into the lowerlayer. As vortex size increases, it requires greater energy to accomplish this, eventuallythere will be insufficient energy, and the turbulent structure collapses.In studying the discharge of a horizontal surface buoyant jet, Koh (1971),observed the existence of an entrainment region immediately after the jet exit. In thisregion ambient fluid is entrained into the surface jet, which grows in thickness. The end ofthis region marks the transition from near-field to far-field. In a manner similar toLawrence & Bratkovitch (1993), this transition was modeled as occurring when theRichardson number of the flow reached a critical value, and the turbulent entrainmentcollapsed under the influence of buoyancy.Similar results were reported by Chen (1980). Experiments with a radial surfacebuoyant jet in a circular tank produced a region of flow with increasing thickness, and24entrainment occurring, until at a distance equal to the length scale defined by M and B(4,), the entrainment structure collapsed, and the radial jet became a radial plume. Fromthis point outward, until the spreading radial plume contacted the walls of theexperimental tank, the radial plume was of constant depth. It is interesting to note thatChen (1980) also observed that once the jet had contacted the walls, the surface layerincreased in thickness due to the flow being choked off. This resulted in the entrainmentzone being "drowned" and an internal hydraulic jump forming.2. 2. 6. 2. Internal Hydraulic Jump ApproachIn contrast to the evidence presented above, which argues the existence of anentrainment zone in the absence of a downstream internal control, rather than an internalhydraulic jump, is the conflicting position taken by some researchers which promotes theinternal hydraulic jump argument. Andreopoulos et al. (1986) and Wright et al. (1991)both state that an internal hydraulic jump should be present, though the results of theirown experiments contradict them.In experiments designed to investigate the effect of an imposed downstreamcontrol on a vertical plane buoyant jet discharged in shallow water, Andreopoulos et al.(1986) (inadvertently) largely confirmed the results of Wood and Wilkinson (1971). Inthe absence of an imposed downstream control, in this case a submerged weir, no rollerregion was observed, and only an entrainment region was present. Within the testchannel, entrainment was observed to cease as buoyancy stabilized the flow downstream.In spite of the apparent absence of the roller region, Andreopoulos et al. (1986) still termthis an internal hydraulic jump. Andreopoulos et al. (1986) further argue that thedownstream control would have been provided by channel and/or interfacial friction, hadtheir experimental channel been long enough.The reasoning employed by Andreopoulos et al. (1986), that channel frictionwould be sufficient to force an internal hydraulic jump, does not appear to be supportable.25Wood and Wilkinson (1971) were explicit in their results from experiments in a longchannel. An internal hydraulic jump with roller region was only present when a stronginternal control, such as a hump or contraction, was placed in the channel. If channel andinterfacial friction were likely to produce an internal hydraulic jump, it would have beenmost evident in Wood & Wilkinson's experiments. In their experimental configuration, thehigher speed flow was in the dense layer located at the bottom, against the solid floor ofthe channel, friction would have been greatest in this situation. Yet an internal hydraulicjump was not created without a channel control. In the experiments conducted byAndreopoulos et al. (1986) the fast flowing layer was the less dense one at the top,farthest from the channel bottom, and hence channel friction was likely less than that ofWood & Wilkinson (1971). No internal hydraulic jump was observed until Andreopoulosat al. (1986) introduced a sill into the upper layer.Andreopoulos et al. (1986), provide further theory into the behaviour of the jetdischarge in the absence of the downstream control. Andreopoulos et al. (1986) states:"The surface jet flow is supercritical, i. e. the densimetric Froude numberbased on the local layer thickness, velocity, and density difference to theambient is larger than 1"This is not a valid concept, since whether the surface flow itself is supercritical or notwould depend on the regular Froude number, not on the densimetric Froude number. Ifthe two layer flow itself is being referred to, then the quantity of importance fordetermining whether the flow is internally supercritical or not is the composite Froudenumber, G, discussed previously. Andreopoulos et al. (1986) then go on to state that asentrainment occurs the Froude number is reduced until a locally critical flow condition isachieved ( F=1). This is in contrast to Wright et al (1991), who state that for a radialflow, the downstream flow is internally critical, e. g. F,2 + F22 = 1 as opposed to F1 2 = 1.Andreopoulos et al. (1986) also note that when a roller is present the upstream flow islocally (externally) supercritical, and the downstream flow is locally (externally)subcritical, as would be expected for a "genuine" hydraulic jump.26As stated previously, Wright et al. (1991) assumed that the downstream (far-field)flow would be internally critical, since this would allow a steady state condition to existwithout the need to specify downstream conditions. If the flow is internally critical, thiswould have the effect of blocking internal long waves, and therefore negate the effect ofany downstream control. However, it is not stated what conditions would be necessary toensure internally critical flow in the far-field, or whether this would actually occur inpractice. This condition was also implicitly assumed by Lawrence & Bratkovitch (1993)in the structure of their model, as the flow moves outward radially, G2 approaches 1asymptotically. Thus, the flow approaches the internally critical condition as it spreadsfrom the surface impingement zone.Wright et al. (1991) formulate their model based on the presence of an internalhydraulic jump in at least a portion of the near-field region. Yet, further evidencesupporting the existence of an entrainment region and interfacial instabilities, as opposedto an internal hydraulic jump, is provided by Wright et al. (1991) themselves. Wright et al.(1991) conducted laboratory experiments to validate their model. Two differentexperiments, one with a negatively buoyant jet, and the other with a positively buoyant jetdischarged upward, were conducted. Only written descriptions are provided of theresulting flows, with the exception of two photographs. Wright et al. (1991) refer to theoccurrence of ring vortices propagating outward from the jet location. These vorticesappeared to extend through the whole depth of the upper layer, and may be a form ofshear instability. At the same time, no roller region was observed, indicating that theprocess does not involve an internal hydraulic jump. Closer to the jet, smaller scaleturbulent fluctuations were observed, these tended to die out with increasing distance fromthe point where the jet was discharged. Finally only the large scale fluctuations, the ringvortices, remained, at the transition to far-field conditions even the large scale fluctuationswere observed to die out. This is analagous to the results provided by Koop & Browand(1979).27Vortex ring structures were reported by Garvine (1984) in a radially spreadingsurface buoyant jet where a river flowed into ocean water. Observations of vortex ringstructures in other surface buoyant plumes undergoing radial spreading were also made byScarpace & Green (1973), and McClimans (1978). Laboratory investigations by Alavian& Hoopes (1982) have also revealed vortex ring structures. Rottman & Simpson (1983)also observed vortex rings in the radially spreading regions of a negatively buoyant jetdischarged vertically downward to a surface.2. 2. 6. 3. Summary of Approaches to Transition MechanismsTo summarize, many researchers have reported results supporting the existence ofan entrainment region in the near-field, leading into the stable, stratified far-field. Theexistence of large scale instabilities, in the form of vortices, has been reported in both thecases of two layer flow in a channel, and for radial spreading flow from a vertical jet inshallow water. Internal hydraulic jumps have only been reported when an obstacle ofsufficient size to act as an internal control has been placed in the flow. In spite of thisevidence, some researchers continue to maintain that the transition from near-field to far-field conditions will occur by way of an internal hydraulic jump. This appears to be basedon an intuitive belief that a radial internal hydraulic jump will be produced by interfacialfriction, though experimental results do not support this. The convenience of solvingmathematical equations formulated using the assumption of an internal hydraulic jump mayalso help to perpetuate this position.Interfacial or channel friction has been cited as justifying the use of an internalhydraulic jump in models. However, as discussed previously, investigations of two layerflow in channels has shown that friction effects do not produce internal hydraulic jumps.Friction is even less likely to produce an internal hydraulic jump in stratified radial flows.As the radial distance increases, the velocities in the flows, and hence the friction, whetherinterfacial or boundary, decreases. Also, radial flows are not subject to sidewall friction.28This is in contrast to channel flow, where constant width, and therefore constantvelocities, are maintained throughout the channel, and friction against the sidewalls ispresent.29Chapter 3Experiments3. 1. Prediction of Flow ConditionsIn order to design the experiments undertaken in this study, it was considereddesirable to attempt to predict the behaviour of the radially spreading flow in the proposedexperimental apparatus. In order to make these predictions, it was necessary to make useof one of the models mentioned in the previous chapter. The best available model forpredicting the radially spreading flow was that of Lawrence and Bratkovitch (1993). Thismodel has not been confirmed experimentally and is likely somewhat coarse, but doesprovide estimates necessary in designing the experiments. Basic jet relationships, asdescribed in Fischer et al. (1979), were used to estimate the flow conditions in the verticaljet. These predictions were necessary in order to estimate the variation of the compositeFroude number, G, of the two layer flow, and the Reynolds' number of the upper layer.These quantities are important for determining, respectively, whether the flow is internallysupercritical, internally critical, or internally subcritical, and whether the upper layer flowis turbulent.First the flow conditions and the resulting entrainment into the vertical jet had tobe estimated. In order to determine the entrainment length, the thickness of the surfaceblocking layer was estimated using Figure 2.6, from Lee & Jirka (1981). This providesthe depth of the surface blocking layer as a function of the depth ratio, H/D, and thedensimetric Froude number of the discharge, F o. Subtracting the surface blocking layerthickness from the available depth gave the length of vertical jet available for entrainment(z ). Using this length, the flow entrained was calculated by using equation (2.10). Thisequation is not entirely applicable for the depth range over which it was used, equation30(2.10) being intended for z/D >>6, but it should provide a reasonable estimate of the flowentering the surface blocking region.The radius of the boundary of the surface blocking region, ri, was estimated byassuming that it would occur at the point where the vertical time-averaged velocity of theentering vertical jet had dropped to 5% of its centerline value (Refer to Figure 3.10 fordefinition sketch). This assumption has also been used by Lee & Jirka (1981), Wright etal. (1991) and Lawrence & Bratkovitch (1993), and it is employed for convenience. Mostof the volume flux of the vertical jet is contained within this radius (95%) and verticalvelocoties have diminished to relatively low values beyond it. This radius was calculatedby equation (2.7). The flow exiting the surface blocking region was assumed to becompletely mixed, due to the highly turbulent nature of the surface impingement zone, andit was assumed that no entrainment occurs within the surface blocking region.The radially spreading flow was treated as having uniform ("top hat") velocity anddensity profiles to simplify computations. Stepwise calculations were made to determinethe flow, upper and lower layer depths, and upper layer growth rate, at a series of discretesections as a crude numerical solution. The growth rate of the surface layer wascalculated using equation (2.31). The resulting flow at each section was calculated usingconservation of momentum flux. As a result at each section, i:11, = Q = (2gr,h,,,M) 112^(3. 34)Where:^r; = Radius at section i.hu = Depth of upper layer at section i.The resulting flow in the lower layer at each section, i, due to entrainment, is:^Qe = Qi - Q0^ (3. 35)The flow velocities at each section in the upper and lower layers are, respectively:U = QI(221ki r,)^ (3. 36)U„ = 1(224H — hjr,) (3. 37)31Due to dilution the modified gravitational constant decreases as the flowprogresses outward. This variation was assumed to be linear with temperature orconcentration of dissolved solids, with the result that:g', = g'0 Q/Q^ (3. 38)Using the above relationships, the bulk Richardson number could be calculated:R` ^— Urer )2^(3.39)Then, using Equation 2.32, the rate of change of depth with radius (dh/dr) couldbe calculated. The depth of the upper layer could then be calculated for the next sectionby:^^,=I dh lOrdr ), (3.40)Based on the results of Koop & Browand (1979) and Chen (1980), as discussed byLawrence & Bratkovitch (1993), growth of the upper layer was assumed to cease, and astable interface form, when the bulk Richardson number (A) reached 0.3. Theentrainment into the upper layer in the far-field region was then calculated assuming aconstant upper layer depth, and conservation of momentum flux. This has the result thatin the far field the upper layer Reynolds' number varies inversely with the square root ofthe radius.At any section in the radial flow, the Composite Froude number, and the Reynolds'number for flow in the upper layer, could be calculated:U2^Ur2erG= ^ + ^g' - — ki )R = /(271r; ) v(3.41)(3.42)Variation in Composite Froude number for the two layer flow, and Reynolds'number for the upper layer flow, was then predicted for the range of proposed flowconditions for the experiment. Sample plots of Composite Froude and Reynolds' numbers32against the radius, made non-dimensional by dividing by ri , are presented in Figure 3.11 A& B. The Composite Froude number was found to start relatively high, and then decreaserapidly with radius, asymptotically approaching 1 for very large radii. Thus, the flow forall proposed experiments appears to remain internally supercritical for the entire range ofthe experimental tank (from 0 to 82 cm radius), if effects of the free surface control at theweir are neglected, discussed later.Similarly, for the range of flows proposed for the experiment (refer to Section3.2.2.), the Reynolds' number of the upper layer remains above the critical Reynolds'number for open channel flow of approximately 550 (Daugherty et al. 1954), well beyond(>30 cm) where the boundary of the experimental tank would occur (This should not beconfused with the critical Reynold's number for a jet of approximately 4000). While thevalue of 550 represents the critical Reynold's number for parallel flow in a straight openchannel, and is not directly applicable to a developing radial flow, it does provide a pointof comparison for an order of magnitude analysis. Considering that the calculated upperlayer Reynold's numbers in Figure 3.11B are in excess of 2000, and hence well above 550,this indicates that the flow will remain turbulent well into the far-field region in the radiallyspreading surface layer.3. 2. Experimental Design and Procedure3. 2. 1. Experimental DesignIn order to experimentally investigate the entrainment mechanisms, and manner inwhich the near-field to far-field transition takes place, it was necessary to produce aradially spreading surface jet. The main emphasis of this investigation was on flowvisualization of the radial regions of the jet, to determine the mixing and transitionmechanisms in the near- and far-fields. There were four important factors whichinfluenced the design, these are discussed more fully below, but briefly they were:1. simulation of an infinite ambient to avoid internal control effects.332. ensuring symmetry of the flow.3. allowing undistorted viewing for flow visualization.4. allowing for replenishment of the ambient to replace entrained fluid.The most serious consideration was to ensure that the experiment was capable ofsimulating an infinite ambient, when in fact the apparatus was very limited in extent. Thisrequired that the design allow for the outflow of the jet flow and the entrained fluid in amanner which would not choke the flow, and cause the onset of a radial internal hydraulicjump, as discussed in Chapter 2. This suggested a form of weir which would allow theupper layer of fluid to spill out of the experimental tank, while still containing the ambientfluid. A free surface local control of the upper layer flow over the weir would exist, butan internal control would not be imposed.The effect of the circular weir upon the upper layer would be in the form of a freesurface control where the flow would accelerate over the weir, in the same manner as overa spillway or similar structure. The existence of the underlying layer would have no effectupon the flow over this weir, since it would be below the level of the weir. The circularwall would not act as an internal control, as ambient fluid supplied from the ring diffuser(discussed below) would be flowing inward, away from the wall. Thus, the circular wall,which represents an artificial constraint not found in prototype situations, does not act asan internal control since the flow of ambient fluid effectively originates "downstream" ofthis wall and does not encounter it.Since the intent of the experiment was to study the radially flowing surface regionsof the jet, the boundary of the tank had to be configured so that it would allow the flow tobe symmetrical. This condition dictated that the tank be circular with the jet discharge inthe center. In order to ensure the symmetry of the flow, the weir had to be carefullyleveled, any significant deviation in the level of the weir would tend to bias the flow in onedirection or another, destroying the desired symmetry.34The circular weir or wall leads to a difficulty, in that the curved surface of the wallwill tend to distort any view through the wall of the flow. Also the fluid spilling over thetop of the weir, and flowing down the sides, would have interfered with photography. Toovercome these problems, a square tank was constructed to contain the circularexperimental tank. This tank was filled with water to within 1 to 2 cm of the top of thecircular weir. This had the effect of minimizing the distortion of the curved wall of thecircular tank, by allowing photography to occur through the flat surface of the squaretank, through water, and then into the circular tank. The square tank also served tocontain the outflow from the circular tank, the water level in the square tank beingcontrolled by valves on the outlet lines.Because the circular tank was relatively small in volume, the available ambient fluidin the tank would have been quickly depleted, particularly at the higher discharge rates ofthe jet. Thus a scheme whereby the ambient would be continually recharged to replacelost fluid, and maintain steady state conditions, was designed. Using a cold water supplyin the lab, a supply line was run through a control valve and flow meter, and then split intotwo lines, which were connected to two diffuser like discharge tubes in the bottom outsideedge of the circular tank. These diffuser tubes were installed such that they dischargedstraight inwards toward the vertical jet, to approximate the gross flow of the ambient fluidtoward the center to replace entrained fluid. In order to minimize mixing created by jetsfrom the ring diffuser, synthetic air filter material was wrapped around the ring diffuser tobaffle the flow, and diffuse the jets. Refer to Figure 3.12 for a schematic of theexperimental apparatus, Figure 3.13 provides a photograph of the apparatus.It was necessary to balance a number of considerations in developing the finaldesign of the experimental tank and apparatus. The maximum flow rates were limited bythe capacity of the water supplies in the lab. Most critical was the available hot watersupply, which, from a 2.5 cm domestic line, was capable of supplying a maximum flow offrom 2 to 2.5 Vs (30 to 37.5 USGPM) depending on other demands. Steady state water35temperature at full demand was approximately 50°C. Temperature of the cold watersupply varied between 8°C and 10°C, with a maximum flow rate of approximately 6 1/s.Minimum flow rates used in experiments were dictated by the requirement toensure that the flow remained turbulent well into the far-field region, so that themechanisms of interest were present. For the vertical jet region, this required that the JetReynolds' Number be greater than 4000. Prediction of the Reynolds' number of theradially spreading flow was discussed in the preceding section.To investigate the effect of the ratio H/D, a series of jet nozzles of differentdiameters were fabricated. These nozzles were equipped with standard fittings forattachment to the circular tank and connection to the hot water line. Extensions to allowthe effect of different h, /D ratios to be investigated were also manufactured.The dimensions of the circular tank were 30 cm deep with a diameter of 183 cm.The square containment tank was 213 cm by 213 cm by 34 cm deep. Both tanks wereconstructed from clear Plexiglas GM. The flow meter used on the jet discharge was aKing Instrument Company variable area flow meter, model number K72-05-0465, for flowranges from 0.1 Vs to 2.5 1/s (1.5 to 37.5 USGPM). The flow meter used on the ambientrecharge was a King Instrument Company variable area flow meter, model number K72-05-0251, for flow ranges from 1.3 1/s to 6.6 Us (20 to 100 USGPM). Ball type valveswere used throughout for control of the ambient recharge, jet discharge, and water levelsin the square containment tank.Flow visualization was accomplished with fluorescene dye injected into the jetdischarge line by a small peristaltic dosing pump with a flow range of 0. 01 to 1 1/s. Flowillumination was provided by a 4 Watt argon ion laser, with the laser sheet produced by anresonant scanning mirror controlled by a function generator. Image recording was donewith both a 35 mm still camera, and video camera, under dark room conditions. Prior tothe commencement of an experiment a scale was placed in the same plane that the lasersheet would be illuminating, and photographed to allow later measurement from the36recorded images. Once the scale had been photographed, the position and focal length ofthe cameras were not changed.3. 2. 2. Experimental ProcedureFor each H/D ratio, a series of three different densimetric Froude numbers, F.,were run to investigate the behaviour of the flow. These different F. were accomplishedprimarily by varying the jet discharge, since there was no practical means to control thetemperatures of either the ambient or the jet. Temperatures of both the ambient and jetwere measured before and after experiments, and were used to determine the appropriatedischarge for the Fo of that experiment.Details of the experiments conducted are provided in Table 3.1. The experimentalparameters of the diameter (D), total depth to nozzle diameter ratio (H/D), depth overnozzle to nozzle diameter ratio (h1 /D) and densimetric Froude number (F.) are provided.The recorded temperatures of the ambient water (TA ) and the jet (Ti ) were essentiallyconstant at 9°C and 49°C, respectively. The following calculated quantities of the jet arealso included; jet flow rate (Q), predicted entrained flow rate (Q,), modified gravitationalconstant (g'0 ), Froude number (F), and Reynolds' number (Re ). The ambient rechargerate (QR ) used is also included in Table 3.1.The entrained flow (Q.), was estimated by using the method described in Section3.1. The jet discharge was subtracted from the total flow occurring at the radiuscoinciding with the wall of the circular tank, to obtain the flow entrained by the jet. Theambient recharge (QR ) was then set to this flow. As this ambient flow was only anestimate, with the possibility of considerable error due to the crude nature of the model,the sensitivity of the flow behaviour to the rate of ambient recharge was investigated byusing ambient recharge flows both greater than and less than that estimated.At the larger nozzle diameters, the estimated ambient entrainment rate for some jetdischarges was well above the capacity of the cold water supply. Experiments for which a37sufficiently high flow rate to match predicted entrainment could not be provided are soindicated in Table 3.1. For these cases, a range of ambient recharge rates from 4 to 6 Uswere utilized.For all experiments, the jet was started some time before dye injection began,usually 2-3 min. This allowed the flow structure to achieve a near steady state before flowvisualization began. Increasing dye concentration in the ambient would have precludedobservation of the steady state flow structure if dye injection had commencedsimultaneously with the jet flow.The scientific laser was set up to provide a sheet of laser light along the tankcenterline. With both the ambient recharge flow and jet flow adjusted to the desired rates,dye injection would commence. Using the variable speed dosing pump, the concentrationof dye in the jet could be adjusted to allow the best visualization of the details of the flowstructures, see Figures 4.14 etc. Both video and still images of the flow were recorded forlater analysis.Further experiments were conducted with the ambient recharge flow completelycutoff, and the water level in the tank set slightly below that of the circular walls. Theseexperiments were intended to investigate the occurrence of a radial internal hydraulic jumpunder choked conditions. The experimental procedure in these cases was essentially thesame as that discussed above, with the exception of the ambient recharge being shut off.Also, experiments were conducted for each F., in which 15 cm extensions were attachedto the nozzles, with the result that h, was approximately 15 cm while H remained at 30cm. This allowed the investigation of the effect of reduced entrainment length in thevertical jet on the radial flow behaviour. These experiments were run with threedensimetric Froude numbers for each jet diameter.38Chapter 4Results and Comments4. 1.^Results4. 1. 1. Standard ExperimentsNo radial internal hydraulic jumps were observed in any experiments. Typically,there existed a region immediately outside the surface impingement zone in whichinstabilities or vortices caused violent, high energy mixing and entrainment, whichpenetrated the entire depth of the upper layer. At some distance from the discharge point,this violent mixing would collapse, and only smaller, less vigorous, interfacial instabilitieswere observed. This area was assumed to represent the transition from near-field to far-field conditions. These instabilities would become less pronounced with increasing radius.Figures 4.14 A-D, from experiment 15020-2 (FG, = 20, HID = 15) presentphotographs of the surface flow. Large scale interfacial instabilities, vortex cells, visible asthe brightest areas, are apparent starting from the exit from the surface impingement zoneat the left. Ambient (darker) fluid can be seen intruding almost to the free surface betweenthe vortex cells. As the cells move outward (to the right), they grow in size, until, justbeyond the extreme right of the photographs, the vortex structure breaks down. Thetranslation of vortex cells can be seen in the movement of features a and b in thesequential photographs. Feature c is a wisp of upper layer fluid caught between the surfaceflow and the underlying inward flow, and as a result, it does not move significantly overthe sequence of photos.The dynamic nature of the flow, the motion of the vortices, and entrainment ofambient fluid, is most apparent when the flow can be seen moving, either viewed first handor on video recording. The point of transition, not visible on the still photographs, as itwas just outside the field of view of the still camere, is easily discernable when the video39recordings are viewed. The rotating, or swirling, motion of the vortices can be clearly seento collapse into a more uniform and continuous upper layer. Though referred to as a pointof transition, the transition from near-field to far-field conditions more correctly occursover some distance, varying from approximately 5 to 10 cm, depending on the magnitudeof the flow.At any given instant, a distinct, uniform surface layer did not exist in the near-field.The thickness of the upper layer varied constantly as vortices would form, grow whilemoving outward, and collapse. In addition, examination of the video recordings revealedthat the vortices or billows tended to slow as the transition from near-field to far-fieldconditions was approached.Observations of the video recordings revealed that the ring vortices in the radiallyspreading flow originated from the vortices or billows formed in the interfacial regions ofthe vertical buoyant jet. Billows in the vertical jet could be clearly seen to travel upwardinto the surface impingment region, to then emerge in the radial flow. The billows werecontinuous in nature from the vertical to the radial flow. Again, this phenomena is notapparent in the still photographs, and is only apparent when the actual motion can beobserved.With the larger nozzle diameters and small jet momentum in experiments 5002,and 5004, a noticeable oscillation of the vertical jet occurred. The vertical jet tended tooscillate back and forth in a sinusoidal shape about the jet centerline, referred to as avaricose mode. This was likely due to the presence of an asymmetrical helical instabilityaround the jet, this asymmetrical structure would produce unbalanced forces which wouldaccount for this behaviour.As discussed in Chapter 3, a range of ambient recharge flows from the ring diffuserwere used to allow for the possibility of error in the predicted entrainment flow rates. Theflow structure in the upper layer did not change appreciably with the variation in ambientflow rate. The only significant effect of the lower ambient flow rates was that the dye40concentration in the ambient increased more quickly than with the other experiments, withthe result that these experiments were of shorter duration than when the ambient flowswere high enough to meet predicted entrainment. At the walls of the circular tank aportion of the flow was drawn down and back toward the vertical jet by the need tocompensate for the deficit in ambient flow for entrainment. This resulted in the increase indye concentration in the ambient fluid observed in these experiments. Since the portiondeflected back by the circular wall was still slightly buoyant with respect to the ambientfluid, it tended to exist as a partial third layer between the radially spreading surfacebuoyant jet and the ambient. As this intermediate flow approached the center of the tankit tended to disappear due to entrainment into the upper layer, and mixing into the ambientfluid.At no time during these experiments was a radial internal hydraulic jump observed.The behaviour of the upper layer was observed to be as with the other experiments wherere-entrainment was not occurring, the only difference being that the surface flow becamedifficult to distinguish more quickly due to the build up of dye in the intermediate layerand ambient. A minor degree of recirculation at the walls, resulting in a slow dye buildup, was observed with all other experiments. While the majority of entrainment was intothe upper layer, a small quantity of the upper layer flow was entrained into the lower layer.This minor entrainment into the lower layer resulted in a gradual build up of dye in thelower layer.As was discussed in Chapter 3, the circular weir did not act as an internal control.The weir's only effect was as a free surface control on the upper layer, which spilled overit. Since the weir could not influence the lower layer, which flowed away from it, the weircould not have acted as an internal control. The minor degree of recirculation observed atthe weir was due largely to the exchange of fluid between the two layers, which resulted ina small quantity of upper layer fluid being entrained and mixed into the lower layer. The41entrainment and mixing occurring was heavily biased into the upper layer, but a smallproportion of upper layer fluid did end up in the lower layer of fluid.When large scale re-circulation did occur at the wall, it was due to the inadequacyof the ambient fluid recharge flows from the ring diffuser. The entrainment demands ofthe radially spreading upper layer were satisified by drawing upper layer fluid down intothe lower layer at the circular weir, to eventually be re-entrained back into the upper layerflow. This effect was not one of an internal control imposed by the circular weir, but onethat arose because there was not an infinite ambient from which large entrainment flowscould be drawn.4. 1. 2. Experiments with Nozzle ExtensionsThe shorter entrainment length in the vertical jet, resulting from the use of thenozzle extensions, produced differences in the structure of the radially spreading surfaceflow. With less ambient fluid entrained prior to exiting the surface impingement region,the momentum remained concentrated in a smaller volume of fluid than had been the casewhere the nozzle extensions were not utilized. When nozzle extensions were used theradially spreading surface flow was thinner, and the billows had higher velocities, than wasthe case for comparable experiments. The point of transition from near-field to far-fieldconditions tended to be moved outward when nozzle extensions were used.A modified form of behaviour was noted in the experiments for which the nozzleextension was utilized, and the jet momentum was relatively high. In these cases, themomentum of the fluid moving outward and downward from the surface impingementzone was great enough to cause a "plunge ring", where the buoyant fluid penetrated fairlyfar down into the lower layer before buoyancy caused it to return upwards, and enter intoa normal near-field layer as described above. A photograph of this phenomena is providedin Figure 4.15. This phenomena was observed in experiments 7510-4, 7520-4 and 15020-4, where high jet momentum was combined with the smaller I; produced by the nozzle42extensions. The Froude numbers (F), and HID ratios for these experiments are plottedagainst the stability criteria of Labridis (1989) in Figure 4.15. From Figure 4.15 it appearsthat these experiments were ones in which instability and the occurrence of a recirculatingnear-field was approached. Application of Labridis's stability criteria to these threeexperiments is approximate, since this criteria does not account for the presence of thenozzle extensions (H # 1;) used in the experiments in question, which would explain whyexperiment 7520-4 plots in the unstable region when in fact it was a stable flow. All otherseries of experiments were within the stable region, and are plotted on Figure 4.15 forcomparison purposes.The high jet momentum and small h, of experiments 7520-4, and 7510-4, alsoproduced an extremely large surface disturbance, with resulting large surface wavesspreading outward from the center, and then being reflected back by the circular walls.These surface waves are apparent in Figure 4.16. The flow structure in the surfacespreading layer was somewhat disrupted by these waves, tending to break up into a seriesof large billows which only moved slowly outward from the center. The thickness of theupper layer also remained relatively constant in these cases.4.1.3. Choked ExperimentsChen (1980) reported that a radial internal hydraulic jump would form after sometime when the surface flow was completely blocked. Experiments conducted in this study,in which the ambient recharge flow was shut off, and starting water level was below thecircular tank wall, failed to produce any internal hydraulic jumps. These experiments werenot of as long a duration as Chen's, since there was only a limited volume before the tankwould fill and overflow the sides, and the blocking effect would be removed. There was agradual build up of dye in the experimental tank, but even after some time, internalhydraulic jumps were not observed to have formed for any of the choked experiments.Experiments 15020-C, 7520-C, and 5008-C, were choked experiments.43'Figure 4.17 A-D presents photographs of the flow under choked conditions forthe same Fo and HID as that for Figure 4.14 A-D. The overall structure of the upperlayer flow remains the same as for the experiment featured in Figure 4.14 A-D, however,the intrusion of the intermediate layer described above is apparent in the lower right regionof the photographs.4. 1.4. Measured QuantitiesThough the main emphasis of this study was on investigation of the flow structureby flow visualization techniques, it was possible to obtain some preliminary data from therecorded images. Using the still photographs and video recordings it was possible tomeasure certain quantities of the radially spreading surface flow. These quantities are;average billow spacing (S), depth of flow at exit from surface impingement zone (km ),approximate speed of movement of billows (U„ ), radius at which transition from near-fieldto far-field occurs (Re ), and average growth rate of the surface layer in the near-field(dhl dr). The frequency at which billows exited the surface impingement zone (B1 ), wasalso determined from the video recordings. The Strouhal Number (Sr = BID/U) of theradial flow, which expresses the billow frequency relative to the jet parameters U and D,was calculated for each experiment. Data obtained for each of these quantities areprovided in Table 4.2. Figure 2.6 was used to predict the thickness of the upper layer(hop,) at the exit from the surface impingement zone, for comparison to measured values.With experiments in the 5002, 5004, and 5008 series, encompassing the largest jetdiameter and highest jet flow rates, consistent results were not apparent in the measuredquantities for the radially spreading surface region. The flow structure does not appear tobe different in form from the other experiments, it is possible that shallow water effects arecausing a change in the trends in upper layer growth rate and upper layer depth from thatobserved in the other experiments. However, the preliminary nature of this study preventsany further analysis of this phenomena, as sufficient data is not available to allow a more44complete consideration of these experiments. As a result, data regarding upper layerdepth (km ), and upper layer growth rate (dhldr), are considered unreliable, and are notdiscussed further.At the larger jet diameters and higher jet flow rates, covering experiments in series7510, 7520, and 5008, the radius at which the apparent transition from near-field to far-field conditions occurs is nearly constant at 65-70 cm. It appears then, that the tank wallsare forcing a change in the flow conditions at smaller radii than would occur in an infiniteambient for these jet parameters.In Figure 4.18, the Strouhal number of the radial flow was plotted against thedensimetric Froude number. As can be seen by examining Figure 4.18, the frequency ofthe billows in the radial flow decreases relative to the jet parameter U and D (D beingfixed for each HID ratio) with increasing densimetric Froude number, for all depth tonozzle diameter ratios. As the densimetric Froude number is increased significantly above10, the Strouhal numbers for all HID ratios appear to converge. Nozzle extensions resultin higher Strouhal numbers than for the corresponding experiments without nozzleextensions. A possible cause for the decrease in S„ with increasing Fo , is that the higherflow rates are accomodated by larger billows. While Table 4.2 indicates that the actualfrequency of billows, Bf , does increase with F, it does not increase as quickly as U,resulting in a decease in S,. At lower Froude numbers, an increase in the HID ratio resultsin a lower Sr . In this case, this can be assumed to translate directly into a decrease in thebillow frequency. This may occur, since a greater depth allows pairing of instabilities tooccur in the vertical jet, with the result that there are fewer instabilities which are fartherapart. This is then reflected in the frequency of billows exiting the surface impingementregion. The convergence of the various curves as F, is increased indicates that the depthratio is of decreasing importance as F„ increases above 10. The slope of the curves alsodecreases with increasing Fo , indicating that S, may be constant for very large Fo .45The upper layer depth at the exit from the surface impingement zone was plottedagainst Fo (Figure 4.19), after non-dimensionaliimg with respect to D (h. = k /D).Both measured values, and predicted values, from Figure 2.6 (Lee & Jirka, 1981), areincluded. When F. is increasing, he increases. The increase in the upper layer depth withincreased Fo , is due to the increased volume flux, the depth of the radial flow and the sizeof the associated instabilities are increased to accomodate the increased flow. DecreasingHID results in a decrease in h, with fixed F . The decrease in entrainment length reducesthe entrained volume in the vertical jet, with the result that the upper layer thickness at theexit from the surface impingment zone is reduced. There is little correlation between thepredicted curves and the measured curves.Figure 4.20 is a plot of the upper layer depth data, from each series of experimentsin this study, on to the predicted solutions by Lee & Jirka (1981) for upper layer depth atthe exit from the surface impingement zone. For this plot the upper layer depth has beennon-dimensionalized by dividing by the total water depth, H. In comparing the observedvalues to the solutions of Lee & Jirka (1981), it is apparent that there is little correlation.Figure 4.21 presents a plot of the average rate of upper layer growth against thedensimetric Froude number. For the two series of experiments with the larger depth ratios,HID = 15 and HID= 7.5, the rate of upper layer growth decreases with increasing Fa .One might expect that the upper layer growth rate should increase with increasing Fa .However, there is a possible explanation for the decrease in upper layer growth rate withincreasing Fa . As the volumetric flow rate is increased (as a result of increasing F.), thevelocities in both the upper and lower layer are increased. The lower layer velocityincreases to meet the increased entrainment demands of both the vertical jet region, andthe radially spreading regions. In the upper layer, the velocity increase is much greaterthan the lower layer, since the flow in the upper layer is composed of both the jet flow andthe entrainment flow, both of which are increased by an increase in Fa . There is a slightincrease in upper layer thickness with F, but not enough to negate the higher velocities46due to the higher volumetric flow rates. Since the increase in velocity in the upper layer isgreater than the increase in velocity in the lower layer, there is a net increase in theconvection speed of the instabilities. Thus, while the instabilites or vortices are increasingin size at a faster rate (dh/dt increases), the change in size with distance (dh/dr) isreduced by the increase in convection speed. The billows are being swept along. Thiseffect is apparent when one considers the variation in billow speed (U„) with increasing F oin Table 4.2.The use of nozzle extensions is reflected in a decrease in the upper layer growthrate from that for the experiments where the nozzle extensions were not utilized. With thenozzle extensions, the entrainment in the vertical jet is reduced, and hence the flow in theradial region is less than it would be without the nozzle extensions. The lesser flow in theupper layer reduces interfacial shear, and therefore entrainment, resulting in a lower upperlayer growth rate. The effect of a smaller HID ratio for a given Fo is similar, entrainmentin the vertical region is reduced, which reduces the upper layer growth rate.4.2. Comment on ResultsA large range of flow conditions has been examined and under none of theseconditions were radial internal hydraulic jumps observed. This result is in agreement withthat of Wood & Wilkinson (1971) for a long channel, where it was found that an internalhydraulic jump would only form when a downstream sill or hump was placed in thechannel. In the absence of a strong internal control only an entrainment zone will bepresent. The results of Chen (1980), and Lawrence (1985), and the predictions ofLawrence & Bratkovitch (1993), also support this result.In all experiments, except 7510-4, 7520-4 and 15020-4, a radial form ofentraining shear layer was observed. In the near-field region, violent mixing andentrainment occurred in the form of instabilities or vortices which penetrated the full depthof the upper layer. These vortices were continuations of the instabilities formed in the47vertical buoyant jet region. At the transition to far-field conditions this violent mixing wasobserved to collapse, to be replaced by a stable upper layer with smaller instabilities at theinterface between the upper layer and lower ambient layer. Even when the experimentwas contained within the circular walls, and no replenishment flow was provided, theabove described flow structure was still observed. However, dye concentrations increasedvery quickly, precluding the observation of flow structure as unstable conditions wereapproached due to warming of the ambient.When ambient recharge flows were too low to meet demand, the deficit was madeup for by recirculation of upper layer fluid at the walls. This resulted in the gradualincrease in dye concentration in the ambient. Even when this recirculation occurred noradial internal hydraulic jumps were observed. The existence of the recirculation andintermediate layer was entirely due to the presence of the circular wall forcing flow backto meet entrainment demands, since ambient flow was not sufficient, and hence wasentirely artificial. If the circular wall had not been present, as in a genuine infinite ambient,this structure would not have been observed.Because the densities of the ambient and jet fluid could not be controlled, theeffects of varying the buoyancy and momentum/volume fluxes independently of each othercould not be investigated. Therefore, the results obtained with varying Fo morerealistically represent the effects of varying volumetric flow rates with fixed densitydifferences, and not the behaviour of the flow under varying momentum and buoyancyconditions.It should be noted that there was not a distinct interface in the two layer flow inthe near-field. Instead the upper layer flow took the form of a series of instabilities whichpropogated outward from the surface impingement region. Only once the far-field wasreached, did the flow settle down and a distinct stable interface form. The nature of theflow in the near-field means that conventional internal hydraulic principles do not apply tothe flow. A continuous upper layer does not exist, which implies that internal long waves48are not a possibility, since there is not an interface for them to act along. Since internallong waves cannot exist in the near-field, and cannot govern the flow, and the concepts ofinternally supercritical and internally subcritical flow are hard to apply, this effectivelyprecludes the existence of internal hydraulic jumps. While it would be possible todetermine a composite Froude number (G), based on the time averaged velocities, depths,and effective gravitational accelerations of the flow, this quantity would have littlerelevance. The application of standard hydraulic equations, such as continuity andmomentum, while possible on a time-averaged basis, do little to indicate the flowconditions in the upper layer due to its discontinuous, intermittent, nature. For this reason,and since internal hydraulic theory as discussed above is not really applicable, the upperlayer flow could be considered to be not truly hydraulic in the traditional sense. Thepredictions of the composite Froude number carried out in Chapter 3 are irrelevant in lightof the above discussion.4. 3. Comparison to Existing ApproachesThe assumption of the presence of a radial internal hydraulic jump by Lee & Jirka(1981) is clearly not appropriate in representing the actual mechanisms and structurepresent in the radially spreading flow. Lee & Jirka (1981) conducted experiments to aid inadjusting their model. They relied on an arrangement where the flow out of theexperimental tank was partially or completely choked by matting at the outflow weir.Thus, their model is probably inaccurate, due to having been based on the assumption ofan internal hydraulic jump, and also having been calibrated using a choked flow in whichatypical conditions existed. Choked conditions similar to those of Lee & Jirka (1981)were not possible to reproduce in this study due to the configuration of the apparatus andthe inability to produce steady state conditions in the choked experiments of this study.Steady state conditions could not be established in the choked experiments undertaken inthis study as the choked conditions were produced by starting the experiment with the49ambient water level below the level of the circular weir. After a short period of time hadpassed, the tank would fill, and flow over the weir would occur, eliminating the chokedconditions.Wright et al. (1991) was more accurate in recognizing the possibility ofentrainment into the radially spreading surface flow, but their model was still based on theexistence of an internal hydraulic jump in a portion of the near-field. Wright's other mainassumption, that the flow becomes internally critical in the far-field, could not be verifiedat this point, since velocity measurements are not available from this study, or from Wrightet al. (1991). Wright's observations of ring vortices, which propagated outward from thesurface impingement region before collapse, were confirmed.Both Lee & Jirka (1981), and Wright et al. (1991) may assume the presence of aninternal hydraulic jump because of the manner in which they conducted their experiments.In neither of these investigations was a thin cross-section of the flow examined using flow-visualization techniques similar to those used in this study, these techniques having onlyrecently become practical to implement with the availability of lasers and advanced videorecording equipment. Both were viewing the flow in bulk, and were not able todistinguish the fine details of the flow structure. From the side, or above, the radialbuoyant jet region would appear to be a near-continuous layer which increased inthickness with distance until the far-field was reached, and as such could be mistaken foran internal hydraulic jump.Lawrence & Bratkovitch (1993) correctly identified the existence of theentrainment zone comprising the near field, and the existence of entrainment to a lesserdegree in the far-field. Although the assumption of the entrainment zone was correct, theaccuracy of the model, as for the other models discussed above, in predicting the dilutionand flows was not investigated, and no comment can be made on its effectiveness.Lawrence & Bratkovitch (1993) assume the existence of a distinct upper layer in the near-50field, as do the other models. This may be a useful approximation of the time-averagedbehaviour of the flow, but it does not explicitly recognize the actual mechanisms present.In comparing the thickness of the radial surface jet at the exit from the surfaceimpingement zone (h.) to the predicted thickness of the surface impingement zone (h,,),(refer to Figure 4.19 and Figure 4.20) it appears that there is little correlation between thesolutions of Lee & Jirka (1981) and the measured values.A graphical comparison of the range of experiments undertaken in this study withthose undertaken by Lee & Jirka (1981), and Wright et al. (1991), is provided in Figure4.22. Wright et al. (1991) used a range of Froude numbers from 1.5 to greater than 1000,in combination with small nozzle diameters (2.2 and 4.7 mm), and large flow rates anddensity deficits. Most of Wright et al.'s experiments utilized depth ratios greater than 80,with none less than 15. Lee & Jirka's (1981) experiments had a similar range ofparameters, but did use depth ratios as low as 2. The experiments of this study utilized arange of densimetric Froude numbers from 2 to 20, and depth ratios from 5 to 15, andoccupy an envelope essentially missed by the two other studies mentioned here. In manycases, Wright et al.'s experiments utilized parameters (F0 and HID) well in excess ofthose in this study, however the mechanisms investigated in this study are alluded to byWright et al.'s reference to ring vortices, indicating that the results of this study areapplicable beyond the range of this study.Since this study did not collect numerical data regarding the dilution or velocityprofiles in the radially spreading surface flow, it is not possible to comment on theaccuracy of the models discussed above in modeling the flow. The results of this study areof a preliminary nature and are focussed mainly on identifying the details of the flowstructure in the radially spreading surface layer. Future investigations will be of greaterscope and will include evaluation and refinement of the numerical output of the modelsdiscussed above.514. 4. Restrictions and Other ProblemsIn spite of the apparatus being designed in an effort to achieve a steady stateexperiment that could be operated indefinitely, any given run was limited to 5 minutes orless before dye build up in the ambient would make flow visualization impossible. At thewalls of the circular tank, it was observed that some of the surface layer would be carrieddownward and back into the ambient, instead of over the sides to waste. Thisrecirculation effect would occur to compensate for any deficit between the volume beingentrained and the ambient recharge flow being provided. A minor degree of recirculationoccurred in all experiments, whether a deficit in available ambient fluid occurred or not,due to the entrainment of upper layer fluid into the lower ambient layer. This fluid wouldbe diverted down and back into the tank at the wall, to become mixed with the ambientfluid. This resulted in a gradual dye build up, limiting the life of experiments. This effectmight be eliminated by attaching matting or filter material to the inside of the circular wall,just below the crest, to stifle downward currents at the wall.The range of experimental flows that could be utilized was smaller than expected.This was due to the cold water supply not being capable of providing enough flow to meetthe entrainment demands of the jet discharge at its higher flow rates, and hencerecirculation effects set in during these experiments, necessitating the use of lowerdensimetric Froude numbers with the larger jet orifices, in order to avoid it. Also theretended to be large scale surface disturbances when high jet flow rates and large jetdiameters were utilized.The range of conditions which could be investigated was also limited by thetemperatures of the cold and hot water supplies. Since these temperatures wereessentially beyond the control of the researcher, and remained constant during the periodin which experiments were being conducted, it was not possible to change the ratio of Fto F.. This had the result that the buoyancy of the jet could not be varied independently52of momentum. As a result, the behaviour of extremely buoyant jets or of unstable ornearly unstable flows could not be investigated. Modification of the experimentalapparatus to allow salt injection in the jet, or control of the jet temperature, wouldovercome this limitation.53Chapter 5Conclusions and RecommendationsWhen a vertical buoyant jet enters shallow water there are four distinct regions:the vertical buoyant jet region, the surface impingement region, the momentum dominatednear-field and the buoyancy dominated far-field. As the flow exits the surfaceimpingement region there will be entrainment of ambient fluid, and an increase in thethickness of the radial surface flow as it spreads outward. An internal hydraulic jump doesnot occur as part of this flow. The radially spreading surface layer exists as a series of ringvortices or instabilities which propagate outward, until the far-field is reached, and theycollapse into a continuous, more stable layer. The far-field is characterized by a sharperand more stable interface between the two layers, with smaller, less vigorous instabilities,which are remnants of the more active interface in the near-field region. The upper layerflow in the near-field is an intermittent dynamic flow, lacking a continuous distinctinterface, which makes the occurrence of internal hydraulic jumps impossible.The present study represents a preliminary investigation to identify, and explain,the mechanisms of the radially spreading surface flow associated with the discharge of abuoyant jet in shallow water. As such, collection of detailed data regarding velocitydistributions and dilutions in the radially spreading surface layer was not undertaken.Nevertheless, the results of the present study serve to improve the understanding of themechanisms, and behaviour, of radially spreading surface flows resulting from thedischarge of a vertical jet in shallow water. Experimental investigation of these flows inthe past had been limited. With the increased knowledge of the structure of radial flowsarising out of this investigation, existing models used for predictive purposes can beimproved or refined. This in turn should result in more efficient design of outfalls, andbetter protection of receiving water environments.54The transition to far-field conditions from the near-field is accomplished by gradualentrainment and mixing into the upper layer, as described above. Even during extremeflow conditions, in which the upper layer is completely blocked off, a radial internalhydraulic jump did not occur. While preliminary in nature, the results of this study raisequestions as to the accuracy of some models, such as Lee & Jirka (1981), Wright et al.(1991), and their derivatives, which were formulated based on the assumption of theoccurrence of a radial internal hydraulic jump in the transition from momentum dominatedto buoyancy dominated conditions. The general predictions by Lawrence & Bratkovitch(1993) as to the nature of the flow have been confirmed.Due to decreased entrainment length in the vertical jet region at lower depth tonozzle diameter ratios, the surface layer would be thinner, and velocities of thebillows/vortices would be higher, than would occur at larger depth to nozzle diameterratios. The depth of the surface layer at the exit from the surface impingement region andvelocity of the billows was shown to increase with increasing densimetric Froude number.Flow structures were not significantly affected by variation in the ambient recharge rates inthe experimental tank.High densimetric Froude numbers (>10), in combination with short entrainmentlengths in the vertical jet due to the use of nozzle extensions, were found to produce adifferent structure in the radially spreading surface layer. The surface boil was morepronounced, and there was a ring of buoyant fluid immediately outside the surfaceimpingement zone, which protruded downwards into the ambient fluid as far as half thewater depth. The flow exited this "plunge ring" into a radially spreading surface layercharacterized by billows larger than occurred in the unmodified experiments, and a thickersurface layer which had nearly constant depth. These flow conditions represent theapproach to a recirculating flow regime.As the jet densimetric Froude number is increased, the Strouhal number decreases.This indicates that the billow frequency of the upper layer decreases with respect to the jet55velocity as the Froude number increases. The Strouhal number will increase, if, for agiven densimetric Froude number, the ratio of the water depth to the jet nozzle diameter isdecreased. This effect arises because in the shorter entrainment length in the vertical jet,less pairing or merging of billows occurs. The billows in the radial flow at the exit fromthe surface impingement zone are directly linked to those in the vertical jet. The effect ofthe depth ratio was found to decrease sharply as the densimetric Froude number wasincreased above approximately 10.Lee & Jirka (1981) provided a graph (Figure 2.6) for estimating the thickness ofthe surface blocking layer as a function of the densimetric Froude number (Fo ), and thedepth to nozzle diameter ratio (HID). When compared to the experimentally measuredthickness of the surface layer at the exit from the surface impingement region, the graphwas found to be inaccurate for most flows, particularly those with small HID (5 or 7. 5)where the error was as much as 50%.As the densimetric Froude number is increased for a given depth ratio, the growthrate of the upper layer is decreased, due to an increase in the convection speed of theinstabilities. A change in the depth ratio for a given densimetric Froude number results inan opposite change in the upper layer growth rate. The change in the entrainment length inthe vertical jet region accounts for this effect.Further experimental work involving measurement of dilution and velocities in theradially spreading layer can be undertaken. Direct comparison to the predictions providedby existing models would then make possible evaluation and refinement of these models indetail. The experimental apparatus should be modified to enable changing the buoyancyof the jet, allowing the effects of variations in buoyancy and momentum to be fully studiedover a wider range of flow conditions. Dilution and surface layer behaviour in arecirculating regime, or with a very stable jet, may then be investigated. The effect ofbuoyancy and momentum variation on the frequency of billow generation, and the growth56rate of the upper layer, should be investigated. Further refinement of the circular tank toeliminate or minimize the recirculation effect at the tank wall should be considered.Notation a : entrainment coefficienta : growth rate coefficient (in Equation 2. 31)a . : jet entrainment coefficienta : plume entrainment coefficientB = Ug'^14: initial buoyancy flux of jet (Equation 2. 6)• = 0.127z : characteristic width of tracer profile (Equation 2. 11)k = 0.107z : characteristic width of velocity profile (Equation 2. 10)B1 : billow frequency at exit from surface impingement zone= f g' udA: specific buoyancy flux in jet (Equation 2. 3)AC tracer concentrationCo : initial tracer concentrationCo, = 5.64Co (ldz) : tracer concentration on jet centerline (Equation 2. 13)c = 0.254: plume coefficientD: diameter of jet nozzled3 : depth of upper layer at exit from surface impingement region (measured)Ap : density difference between ambient and jet-dh : average growth rate of radially spreading upper layer (measured)dr—dh = av 1– —Ri  upper layer growth rate (Equation 2. 31)dr^RidFo = U Ag' c, DY12 : jet densimetric Froude numberF = U AgDY12 : jet Froude number172^U2 G'2 = ^ + , : composite Froude number squared (Equation 2. 32)g' k g' (1-1 – kg' 0 = g Apo p : initial effective gravitational accelerationg' g Apl p : effective gravitational acceleration5758g' = g'0 QI : effective gravitational acceleration at section i (Equation 3. 37)el : effective gravitational acceleration at entrance to surface impingement regionH: total depth of ambient waterIts : height of surface humpk : depth of upper layerh. = h /D: non-dimensionalized upper layer depth, at exit from surface impingment zoneh,: depth of water above jet nozzleIQ = QIM'12 : jet characteristic length scale (Equation 2. 8)= M3/4 /B1/2 : characteristic length scale for buoyant jets (Equation 2. 17)M = U2 IV /4 :initial momentum flux of jet (Equation 2. 5)M, : momentum flux at entrance to surface impingement regionm= f u2dA : specific momentum flux in jet (Equation 2. 2)AQ = U zD2 14 : initial volume flux of jet (Equation 2. 4): estimated entrained flowQR : ambient recharge flow rateQ, : volume flux at entrance to surface impingement region• = Q — Q: entrained flow at section i (Equation 3. 34)• : total flow at section ip.= fudA: specific volume flux in jet (Equation 2. 1)Ap = 0 .254m112z: specific volume flux in plume (Equation 2. 16)Re: jet Reynolds' number at exit from nozzleA _ g'uh„A : bulk Richardson numberzRo =1Q 11,„ : initial jet Richardson number (Equation 2. 18)Rp : plume Richardson numberA: radius at which transition from near-field to far-field conditions occurs (measured)R„ = Q1(2 71; ) : Reynolds' number of upper layer flow (Equation 3. 40)ri : radius at section i: radius of jet at entrance to surface impingement region59r, : radius at exit from surface impingement regionU : average velocity at jet exitS: average billow spacing (measured)Sr = BfDIU: Strouhal number of radially spreading flowTi : temperature of jet fluid at nozzle exit7:4 : temperature of ambient fluidUb : average velocity of billows (measured)U : averaged velocity in upper layer of radially spreading flowtire, :averaged return velocity in layer below radially spreading flowu : time averaged jet velocityu,=7.01QMIzQ: time averaged velocity at jet center line (Equation 2. 9)u, : velocity in jet at entrance to surface impingement region, assuming top hat velocityprofile (Lawrence & Bratkovitch, 1993)—F1v =.e,1v:^velocity ratio for two layer flow+ Ur )v: kinematic viscosityz : distance along jet centerlinez, : length of Zone of Flow Establishment4 z, /D: non-dimensionalized length of Zone of Flow Establishmentze : entrainment lengthZe = zr /D: non-dimensionalized entrainment length60References Abraham, G. 1965. "Entrainment Principle and its Restriction to Solve JetProblems", Journal of Hydraulic Research, Vol. 3, No. 2, pp. 1-23.Alavian„ V. , Hoopes, J. A. 1982. "Thermal Fronts in Heated Water Discharges",Journal of the Hydraulics Division, ASCE, Vol. 108, pp. 707-725.Andreopoulos, J. , Praturi, A. , Rodi, W. 1986. "Experiments on Vertical PlaneBuoyant Jets in Shallow Water", Journal of Fluid Mechanics, Vol. 168, pp 305-336.Chen, J. C. 1980. "Studies on Gravitational Spreading Currents", W. H. KeckLaboratory Report No. KH-R-40, California Institute of Technology, Pasadena,California, 436 pp. .Crow, S. C. , Champagne, F. H. 1971. , "Orderly Structure in Jet Turbulence",Journal of Fluid Mechanics, Vol. 48, pp 547-591.Daugherty, R. L. , Franzini, J. B. , Finnemore, E. J. 1985. "Fluid Mechanics withEngineering Applications", 8th edition, McGraw-Hill, New York, 583 pp. .Fischer, H. B. , List, E. J. , Imberger, J. , Koh, R. C. Y, Brooks, N. H. 1979."Mixing in Inland and Coastal Waters", Academic Press, New York, 483 pp. .Garvine, R. W. 1984. "Radial Spreading of Buoyant, Surface Plumes in CoastalWaters", Journal of Geophysics Research, Vol. 89, No. C2, pp. 1989-1996.Jirka, G. H. , Harleman, D. R. F. 1979. "Stability and Mixing of a Vertical PlaneBuoyant Jet in Confined Depth", Journal of Fluid Mechanics, Vol. 94, part 2, pp.275-304.Koh, R. C. Y. 1971. "Two-dimensional surface warm jets", Journal of theHydraulics Division, ASCE, Vol. 97, HY6, pp. 819-836.Koop, C. G. and Browand, F. K. 1979. "Instability and Turbulence in a StratifiedFluid with Shear", Journal of Fluid Mechanics. , Vol. 93, 135-159.Labridis, C. 1989. "Buoyant Jet in Shallow Water With a Crossflow", Thesissubmitted in partial fulfillment of the degree Master of Applied Science, Universityof British Columbia, Vancouver, Canada, 57 pp.Lawrence, G. A. 1985. "The Hydraulics and Mixing of Two-Layer Flow Over anObstacle", Thesis submitted in partial fulfillment of the degree Doctor ofPhilosophy, University of California, Berkeley, California. 130 pp.61Lawrence, G. A. , Bratkovitch, A. 1993. "Vertical buoyant jet into very shallowwater with current", submitted to: Journal of Hydraulic Engineering, ASCE.Lee, J. H. W, Jirka, G. H. 1981. "Vertical round buoyant jet in shallow water",Journal of The Hydraulics Division, ASCE, Vol. 107, pp. 1651 - 1675.List, E. J. , Imberger, J. 1981. "Turbulent entrainment in buoyant jets and plumes",Journal of the Hydraulics Division, ASCE, Vol. 99, pp 1651 - 1675.List, E. J. 1982. "Turbulent jets and plumes", Annual Review of Fluid Mechanics,Vol. 14, pp. 189-212.McClimans, T. A. 1978. "Fronts in Fjords", Geophysics, Astrophysics and FluidDynamics, Vol. 11, pp. 23-34.Murota, A. , Muraoka, K. 1967. "Turbulent diffusion of the vertically upward jet",Proceedings of the 12th I. A. H. R. Congress, Vol. 4, pp. 60 -70.Morton, B. R. , Taylor, G. , Turner, J. S. 1956. "Turbulent gravitationalconvection from maintained and instantaneous sources", Proceedings of the RoyalSociel), of London, Vol. 234(A), pp. 1-23.Rottman, J. W. , Simpson, J. E. 1983. "The initial development of gravity currentsfrom fixed volume releases of heavy fluids", Proceedings of the IUTAM Symposiumon Atmospheric Dispersion of Heavy Gases and Small Particles", Delft, TheNetherlands, August 1983.Scarpace, F. L. , Green, T. 1973. "Dynamic surface temperature structure ofthermal Plumes", Water Resources Research, Vol. 9, pp. 138-153.Wilkinson, D. L. , Wood, I. R. 1971. "A rapidly varied flow phenomena in a twolayer flow", Journal of Fluid Mechanics, Vol. 47, part 2, pp 241-256.Wright, S. J. , Roberts, P. J. W. , Zhongmin, Y. , Bradley, N. E. 1991. "Surfacedilution of round submerged buoyant jets", Journal of Hydraulic Research, Vol.29, No. 1, pp. 67-89.Appendix A62Tables63EXP.No.Run No. DcmHID- hl/D QWsQe1/sOr1/sg'cm/s"2Fo F Mcm^4/sA2Re'^1 15005-1 -^2 15 15 0.08 0. 20 0. 15 10.9 . 5 0. 53 2000 8.4002 16005-2 2 15 15 0.08 0. 20 0.20 10.9 5 0.53 2000 84003 150054 2 15 15 0.08 0. 20 0. 25 10.9 5 0.53 2000 84004 150054 2 15 7. 5 0. 08 0. 20 0. 20 10. 9 5 0. 53 2000 84005 15010-1 2 15 15 O. 15 2. 00 1.50 10.9 10 1.05 7100 168006 16010-2 2 15 15 O. 15 2. 00 2. 00 10.9 10 1.05 7100 168007 15010-3 2 15 15 O. 15 2.00 2.50 10.9 10 1.05 7100 168008 150104 2 15 7. 5 O. 15 2. 00 2. 00 10.9 10 1.05 7100 168009 16020-1 2 15 15 0. 30 6.00 4.00 10.9 20 2. 11 28600 3370010 15020-2 2 15 15 0.30 6.00 5.00 10.9 20 2. 11 28600 3370011 150204 2 15 15 0.30 6.00 6.00 10.9 20 2.11 28600 3370012 150204 2 15 7.5 0.30 6.00 6.00 10.9 20 2.11 26600 3370013 7605-1 4 7. 5 7. 5 0. 43 2. 90 2.20 10. 9 5 0. 53 14700 2380014 7505-2 4 7. 5 7. 5 0. 43 2. 90 2. 90 10. 9 5 0. 53 14700 2380015 75054 4 7. 5 7. 5 0. 43 2. 90 3. 60 10. 9 5 0. 53 14700 2380016 75054 4 7. 5 3. 7 0. 43 2. 90 2. 90 10. 9 5 0. 53 14700 2380017 7510-1* 4 7. 5 7. 5 0. 86 8. 60 4. 00 10.9 10 1.05 58800 4770018 7610-2' 4 7. 5 7.5 0.86 8.60 5.00 10.9 10 1.05 58800 4770019 75104* 4 7. 5 7.5 0.86 8.60 6.00 10.9 10 1.05 58800 4770075104* 4 7.5 3.7 0.86 8.60 6.00 10.9 10 1.05 58800 4770021 7520-1* 4 7.5 7.5 1. 70 26.20 4.00 10.9 20 2. 11 229900 9550022 7520-r 4 7. 5 7. 5 1. 70 26. 20 5. 00 10. 9 20 2. 11 229900 9550023 75204* 4 7.5 7.5 1.70 26.20 6.00 10.9 20 2. 11 229900 9550024 76204* 4 7. 5 3. 7 1. 70 26. 20 6. 00 10. 9 20 2. 11 229900 9550025 5002-1 6 5 5 0.47 2.00 1.50 10.9 2 0.21 7800 1750026 5002-2 6 5 5 0. 47 2. 00 2. 00 10. 9 2 0. 21 7800 1750027 60024 6 5 5 0. 47 2. 00 2. 50 10. 9 2 0. 21 7800 1750028 6002-4 6 5 2. 5 0. 47 2. 00 2. 00 10. 9 2 0. 21 7800 1750029 5004-1 6 5 5 0. 94 5. 10 4. 00 10. 9 4 0. 42 31200 3500030 5004-2 6 5 5 0. 94 5. 10 5. 00 10. 9 4 0. 42 31200 3500031 60044 6 5 5 0. 94 5. 10 6. 00 10.9 4 0. 42 31200 3500032 60044 6 5 2.5 0.94 5. 10 5.00 10.9 4 0.42 31200 3500033 5008-1* 6 5 5 1.80 15.60 4.00 10.9 8 0. 84 114500 7010034 50084* 6 5 5 1.80 15.60 5.00 10.9 8 0.84 114500 7010035 60084* 6 5 5 1. 80 15. 60 6. 00 10. 9 8 0. 84 114500 7010036 60084* 6 5 2. 5 1. 80 15. 60 6. 00 10. 9 8 O. 84 114500 7010037 15020-C 2 15 15 0. 30 N/A 0.00 10. 9 20 2. 11 28600 3370038 7520-C 4 7. 5 7. 5 1. 70 N/A 0.00 10. 9 20 2. 11 229900 9550039 5008-C 6 5 5 1.80 N/A 0.00 10.9 8 0.84 114500 70100Note: * denotes experimental runs where cold water supply was insufficient to meet predictedentrainment.Table 3.1 Experiments Conducted64EXP.No.Run No. D(cm)H/D h1/D Fo F Rt(cm)hup(cm)hum(cm) (cm)S Ub(cm/s)Bf(hz)Sr dh/dr115005-1 2 15 15 5 0.53  35.00 2.40 2.4 3 6.50 0.8 0.063 0.122 15005-2 2 15 15 5 0.53 35.00 2.40 2.4 3 6.50 0.80 0.063 0.123 15005-3 2 15 15 5 0.53 35.00 2.40 2.4 3.2 6.50 0.73 0.058 0.11415006-4 2 15 7.5 5 0.53 35.00 2.40 1.5 2.5 7.00 1.00 0.079 0.105 15010-1 2 15 15 10 1.05 40.00 2.40 2.7 3 7.50 0.93 0.039 0.11616010-2 2 15 15 10 1.05 40.00 2.40 2.7 3 7.50 0.87 0.036 0.107 150104 2 15 15 10 1.05 40.00 2.40 2.6 3 7.50 0.93 0.039 0.098 150104 2 15 7.5 10 1.05 43.00 2.40 2 2.5 8.00 1.00 0.042 0.079 15020-1 2 15 15 20 2.11 45.00 2.40 3.2 2.5 8.00 1.00 0.021 0.0910 15020-2 2 15 15 20 2.11 45.00 2.40 3.2 2.5 8.00 0.93 0.02 0.0911 15020-3 2 15 15 20 2.11 47.00 2.40 3.1 2.5 8.00 1.00 0.021 0.0912 15020-4 2 15 7.5 20 2.11 55.00 2.40 N/A N/A 10.00 N/A N/A N/A13 7505-1 4 7.5 7.5 5 0.53 54.00 2.40 2.7 5.4 7.40 0.87 0.101 0.0514 7505-2 4 7.5 7.5 5 0.53 56.00 2.40 2.7 5.4 7.40 0.93 0.109 0.0516 76054 4 7.5 7.5 5 0.53 56.00 2.40 2.7 5.4 7.40 0.87 0.101 0.0516 75054 4 7.5 3.7 5 0.53 67.00 2.40 2.2 6 8.00 1.13 0.132 0.0417 7510-1 4 7.5 7.5 10 1.05 64.00 2.40 3 4 10.00 1.00 0.058 0.0618 7510-2 4 7.5 7.5 10 1.05 65.00 2.40 3 4 10.00 1.00 0.058 0.0619 75104 4 7.5 7.5 10 1.05 65.00 2.40 3 4 10.00 1.00 0.058 0.0620 7510-4 4 7.5 3.7 10 1.05 70.00 2.40 N/A N/A 11.00 1.07 0.062 0.0021 7520-1 4 7.5 7.5 20 2.11 65.00 2.40 3.2 4 13.00 1.20 0.035 0.0222 7520-2 4 7.5 7.5 20 2.11 70.00 2.40 3.2 4 13.00 1.13 0.034 0.0223 7620-3 4 7.5 7.5 20 2.11 70.00 2.40 3.2 4 13.00 1.20 0.035 0.0224 7520-4 4 7.5 3.7 20 2.11 N/A 2.40 N/A N/A N/A N/A N/A N/A26 5002-1 6 5 5 2 0.21 50.00 2.50 3 8 5.50 0.47 0.168 0.0426 6002-2 6 5 5 2 0.21 50.00 2.50 3 8.5 5.50 0.53 0.193 0.0427 6002-3 6 5 5 2 0.21 50.00 2.50 3 8.5 5.50 0.47 0.168 0.0428 50023 6 5 2.5 2 0.21 50.00 2.50 2 7 6.00 0.80 0.289 0.0429 5004-1 6 5 5 4 0.42 60.00 2.50 4 5 6.50 0.73 0.132 0.0530 6004-2 6 5 5 4 0.42 60.00 2.50 4 5 6.50 0.73 0.132 0.0531 50044 6 5 5 4 0.42 60.00 2.50 4 5 6.50 0.73 0.132 0.0432 6004^4 6 5 2.5 4 0.42 65.00 2.50 2.5 4 7.00 1.00 0.18 0.0433 6008-1 6 5 5 8 0.84 65.00 2.50 4 4 8.00 0.80 0.075 0.0934 6008-2 6 5 5 8 0.84 60.00 2.50 4 4 8.00 0.87 0.082 0.1035 5008-3 6 5 5 8 0.84 60.00 2.50 4 4 8.00 0.80 0.075 0.0936 5008-4 6 5 2.5 8 0.84 65.00 2.50 2.5 3 9.00 1.07 0.101 0.1037 15020-C 2 15 15 20 2.11 N/A N/A 3 3 8.00 0.93 0.02 0.1038 7620-C 4 7.5 7.5 20 2.11 N/A N/A 3.2 4 13.00 1.13 0.034 0.0238 5008-C 6 5 5 8 0.84 N/A N/A 4 4 8.00 0.87 0.082 0.10Note: due to large surface disturbance in Experiments 7520-4, 7510-4, and 15020-4, somequantities could not be measured, indicated N/A.Table 4.2 Measured and Predicted Quantities for Experiments ConductedAppendix B65Figures Figure 2.1 Definition sketch for pure round jets66—..Figure 2.2 Defintition sketch for water depths67Z,Z, = z, /DASYMPTOTIC VALUE OF A PLUME10^20^30^40Figure 2.3 Length of Zone of Flow Establishment as a function of Source DensimetricFroude NumberFAR-FIELD^NEAR-FIELD1111"71111 ^No,STRATIFiED COUNERFLOW REGIONSURFACE IMPINGEMENT REGIONV HRADIAL INTERNAL HYDRAULICJUMP REGIONQ, wtFigure 2.4 Regions of flow of a vertical buoyant jet in shallow water, according to Lee &Jirka (1981)F. 100 1000I MitI10168I...P■  1.=gol■Mu..nue4321.5Stability CriterionHI:=4.6( 1-5)Unstable Near FieldStable Near FieldII.3^167.3^4Aa41. A tt3.3^4 130 •' 0 —321.5 ^0 A Data from Ptyputniewiez & Bowley (1975)1i^I^i i !lit!^I^i^I Hill100H/D10Figure 2.5 Stability Diagram, Lee & Jirka (1981)F0Figure 2.6 Thickness of surface impingement zone as a function of Source DensimetricFroude Number, Lee & Jirka (1981)411FAR-FIELDRADIAL BUOYANT PLUME REGIONSURFACE IMPINGEMENT REGIONRADIAL BUOYANTJET REGIONHQ, WNEAR-FIELDVERTICAL BUOYANT JET REGION.",...—Stability CriterionFu032(1:1—)Stable Near Field D4— 323 4.4 4P4 2—^ A AA A A A^ 1.5.-----4-0"---------H--",_—,3 .......---^23^23022,40011111 . ■ ^■sR,...^2— Unstable Near Field—^1.5so^A Data from Pryputniewiez & Bowley (1975)•t^t^t^11111 I L 111111 4 1^I^I^111110.1^ F^10^ 100Figure 2.7 Stability diagram re-expressed in terms of F, Labridis (1989)69100101Figure 2.8 Flow regions of a vertical buoyant jet in shallow water, as defined byLawrence and Bratkovitch (1993)70Figure 2.9 Internal hydraulic jump in a two layer flow in a channel, after Wilkinson &Wood (1971)FAR-FIELD^NEAR-FIELD.41----111074Figure 3.10 Definition sketch for flow prediction calculations1 0 0 . 0 0 ^ Fo= 4, H/D=5^ Fo=5, HID=5Fo=10, H/D=15Fo=20, H/D=15LriO1 0 .00-r --F --+ -+1.00COI Fo=4, H/D=5 Fo=5, HID=5Fo=10, H1D=15Fo=20, H/D=151 000 0 01 000 0s+---f- --+ --+^--+1 0 00NON-DIM ENSIONA L IZ ED DISTANCEFigure 3.11 A Variation of Composite Froude Number Squared with Non-Dimensionalized RadiusNON-DIM ENSIONALIZED DISTANCEFigure 3.1 lB Variation of Upper Layer Reynold's number with Non-DimensionalizedRadius71CIRCULAR WEIR0 - CONTAINMENT TANK DRAIN (2)-.RING DIFFUSERERTICAL JET NOZZLELASER SHEET ONCENTERLINECOLD WATER SUPPLYTO RING DIFFUSER (2).41-.CONTAINMENT TANKPHOTOGRAPHY INTO TANKFigure 3.12 Plan view schematic of experimental apparatusFigure 3.13 Photograph showing experimental apparatus7273C.L.^FLOWFigure 4.14 A-D^Surface flow of Experiment 15020-2.Time interval 0.7s, approximate scale 1cm 5 cm.7416 —• • •14 — 15005 15010 1502012 —-±- 1 008 7505 7510 7520^15020-4♦.c •6 5002 5008Stability Criteria F < 0.32(H/D)A A4 — 50047520-42 —7510-400^0.5^1^1.5^2^2.5^3Froude Number (F)Figure 4.15 Comparison of experiments to stability criteria of Labridis (1989)Experiments where plunge ring structure occurred plotted seperately, andindicated in bold face.Figure 4.16 Plunging structure at the exit from surface impingement zone in Experiment7520-4 Approximate scale lcm 3 cm.Figure 4.17 A-D^Surface flow under choked conditions, Experiment 15020-C.Time interval 0.7s, approximate scale lcm — 5cm750.3(3 0.25LL 0.20.152z 0.1g 0.05014/1:15, h1/D=15- h/D=15, h1/12w7.6—0-- H/13=7.5, h1/11:7.5^ H/1:=7.5, h1/D=3.7- 0^ ti/D=5, h1/13=5H/D=5, h15:=2.5760^5^10^15^20Densimetric Froude NumberFigure 4.18 Variation of Strouhal number of radial flow with FoFigure 4.19 Non-dimensionalized measured upper layer depths vs Fo ,including values predicted using solutions of Lee & Jirka (1981)0 1 0I 000•15010, hu/H 0.0915005, hu/H = 0.08 • Z• •15020. hu/H = 0.107505, hu/H = 0.09^7520, hu/H is 0.11•^•^•0 :7510, hu/H =iti • 0.300 :0.200.100.06237710^100^l000F0Figure 4.20 Comparison of observed upper layer depths with solutions of Lee & Jirka,(1981). Upper layer depth non-dimensionalized by dividing by H. Solutionsof Lee & Jirka in solid lines. Data this study indicated: • expt. series, hu/H =upper layer depth divided by H, e.g • 7505, hu/H=0.09Figure 4.21 Variation of upper layer growth rate with F.78Figure 4.22 Comparison of range of experiments of Lee & Jirka (1981), Wright et al.(1991), and this study.

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