"Applied Science, Faculty of"@en . "Civil Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Labridis, Christodoulos"@en . "2010-08-30T02:34:04Z"@en . "1989"@en . "Master of Applied Science - MASc"@en . "University of British Columbia"@en . "A common way to dispose of sewage or heated waste water is to discharge it through a submerged outfall located at the bottom of the receiving water. Such outfalls are usually situated in shallow coastal water. The dispersion and dilution of these buoyant jets depends greatly both on the shallowness of the ambient water and the presence of any current. A series of experiments with vertical buoyant jets in shallow water with a crossflow, were conducted in order to understand better the effect of the various parameters. Flow visualization and temperature probes were used. The results were compared to theoretical equations and previous results.\r\nThe basic parameters of the problem are the ratio of the ambient to the jet velocity (less than 0.5), the ratio of the ambient depth to the jet diameter (less than 10) and the jet Froude number. It is shown that for shallow water jets the buoyancy of the jet doesn't have time to affect its dilution. The use of the densimetric Froude number is, therefore, not necessary and often misleading.\r\nThree flow regimes were identified according to these parameters. A crossflow dominated flow (maximum surface dilution), a typical shallow water flow with an upstream recirculation zone and a fountain-like flow (minimum surface dilution). It is, thus, possible to predict the flow regime of an outfall (and, therefore, the minimum surface dilution), for given ambient conditions. It is also possible to study the effect that a change of a basic parameter, will have on the flow regime."@en . "https://circle.library.ubc.ca/rest/handle/2429/27898?expand=metadata"@en . "B U O Y A N T JETS IN S H A L L O W W A T E R W I T H A CROSSFLOW By Christodoulos Labridis Dipl.Eng. (Civil Engineering) University of Athens, Greece A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF AP P L I E D SCIENCE in T H E FACULTY OF GRADUATE STUDIES CIVIL ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA July 1989 \u00C2\u00A9 Christodoulos Labridis, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Cfv/IL iU^tFP-fNQ The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract A common way to dispose of sewage or heated waste water is to discharge it through a submerged outfall located at the bottom of the receiving water. Such outfalls are usually situated in shallow coastal water. The dispersion and dilution of these buoyant jets depends greatly both on the shallowness of the ambient water and the presence of any current. A series of experiments with vertical buoyant jets in shallow water with a crossflow, were conducted in order to understand better the effect of the various parameters. Flow visualization and temperature probes were used. The results were compared to theoretical equations and previous results. The basic parameters of the problem are the ratio of the ambient to the jet velocity (less than 0.5), the ratio of the ambient depth to the jet diameter (less than 10) and the jet Froude number. It is shown that for shallow water jets the buoyancy of the jet doesn't have time to affect its dilution. The use of the densimetric Froude number is, therefore, not necessary and often misleading. Three flow regimes were identified according to these parameters. A crossflow dom-inated flow (maximum surface dilution), a typical shallow water flow with an upstream recirculation zone and a fountain-like flow (minimum surface dilution). It is, thus, possi-ble to predict the flow regime of an outfall (and, therefore, the minimum surface dilution), for given ambient conditions. It is also possible to study the effect that a change of a basic parameter, will have on the flow regime. ii Table of Contents Abstract ii List of Tables v List of Figures v i Acknowledgements vii i 1 Introduction 1 2 Literature Review 4 2.1 Basic Equations And Parameters 4 2.1.1 Zones of Flow and Entrainment 6 2.1.2 Dimensional analysis 8 2.1.3 Effects of Crossflow 9 2.2 Turbulent Buoyant Jets in Shallow Water . 10 2.2.1 Buoyant Jets in Stagnant Shallow Water 10 2.2.2 Length of Zone of Flow Establishment 13 2.2.3 Surface Disturbance 15 2.2.4 Effects of Crossflow 17 2.3 Surface Buoyant Jets 18 3 Theoretical Development 20 3.1 General Considerations 20 iii 3.1.1 Relative Importance of F and F0 20 3.1.2 Surfacing Position of Jet 23 3.1.3 Large Surface Disturbances 24 3.2 Flow Regimes 24 3.2.1 Typical Shallow Water Jets 25 4 Experiments 29 4.1 Experimental Set-Up and Procedure . 29 4.1.1 Equipment and Instrumentation 29 4.1.2 Flow Visualization 31 4.1.3 Procedure of Experiments 32 4.2 Results and Comments 32 4.2.1 Restrictions and Other Problems 35 4.2.2 Comments on the Results 35 4.2.2.1 Regime Diagram 35 4.2.2.2 Effect of F on Regime Diagram 38 4.2.2.3 Dilution Diagram 41 4.3 Comparisons with theory 47 4.3.1 Limits of Shallow Water Effect 49 5 Conclusions and Recommendations 53 Appendix : Notation 55 Bibliography 57 iv / List of Tables 4.1 Experimental Results 33 4.2 Experimental Results (Surface Temperatures) 34 v List of Figures 2.1 Initial buoyant jet regions 7 2.2 Near field flow regions for buoyant jets in stagnant water (according to Lee& Jirka,1981) 11 2.3 Observed and predicted near field dilution (Lee & Jirka, 1981) 12 2.4 Dependence of maximum surface excess temperature on discharge Froude number (from Pryputniewicz h Bowley, 1975) 14 2.5 Length of region of flow establishment as a function of source densimetric Froude number (from Lee k, Jirka, 1981) 14 2.6 Dimensionless expression and shape of the maximum height of surface disturbance for stagnant ambient water (from Murota & Muraoka, 1967) 16 2.7 Relation of starting jet coefficient to jet Froude number (from Chen, 1980) 19 3.8 Observed and predicted near field dilution in terms of F0 (data from Lee &Jirka,1981) 22 3.9 Observed and predicted near field dilution in terms of F (data from Lee & Jirka,1981 and Pryputniewicz h Bowley, 1975) 22 3.10 Flow regimes of buoyant jets in shallow water with a crossflow 26 4.11 Set up of experiments 30 4.12 lu/D for F = 1.17 36 4.13 Characteristic lengths for F \u00E2\u0080\u0094 1.17 37 4.14 Flow patterns 39 vi 4.15 Characteristic lengths for F = 2.53 40 4.16 Effect of F on the flow regime 41 4.17 Effect of Tt - Ta on Dilution S = ( % ~ T j a ' ' \u00C2\u00B0 \" r c e 43 4.18 Effect of flow regime on Dilution 44 4.19 Dilution diagram for F = 1.17 and F = 2.53 45 4.20 Effect of H/D on Dilution for F = 1.17 and F = 2.53 (values of S for U/u \u00E2\u0080\u0094 0 are derived from Lee & Jirka,1981) 46 4.21 Diagram for prediction of flow regimes, according to F, H/D, U/u . . . . 48 4.22 Photographs 1-4 : Effect of U/u (as the current weakens, the upstream recirculation zone increases)-experiments 1/1,2,4,5 50 4.23 Photographs 5-6 : Transition from regime 3 to 3a (as the depth decreases, part of the downward deflected jet reaches the bottom and is deflected again before it is carried downstream)-experiments 8/13,14 51 4.24 Photographs 7-8 : Effect of F on upstream edge (when u increases, h\u00E2\u0080\u009E increases and that results in a smaller upstream edge)-experiments 8/3,18 52 vii Acknowledgements I would like to thank my thesis supervisor Dr. G.A. Lawrence, for the constant support and advice he gave me throughout this work. I am also grateful to Dr. M. Quick for his comments and suggestions and K. Nielson of the Hydraulics Lab at UBC, for his technical assistance in the lab. viii Chapter 1 Introduction Waste heat from thermal plants is often disposed by discharging the heated water into a large body of receiving water. One of the simplest and most economic ways to do this is through a circular outfall at the bottom of the receiving water. Such outfalls are usually located in relatively shallow coastal waters. The major concern in such cases is the environmental impact that the discharged waste heat has on the receiving water. The available ambient depth determines to a great extent the dispersion and dilution of the heated water. Restrictions are often imposed on the maximum allowable surface concen-tration of the effluent. At the same time there are certain minimum quantities of effluent that have to be disposed within a given time period. These conflicting requirements have to be satisfied by the design of the outfall. Since the basic parameters may often vary within a certain range (i.e. changing currents, tides, exit velocity), an understanding of their effect on the resulting flows is necessary. The dilution of round buoyant jets in deep water has been studied by many researchers (e.g. List, 1982). One case, however, that has received little attention in previous studies of round vertical jets, is the case of such a jet discharging into shallow water with a crossflow. The object of the present study is to acquire a better understanding of this flow by developing a theoretical background, and laboratory experimentation. 1 Chapter 1. Introduction 2 Most of the previous experiments conducted in stagnant shallow water used the den-simetric Froude number as a basic parameter for the description of the jet flow. Three near field flow regions were defined, and a stability criterion, according to FQ and H/D, has been formulated (see Lee & Jirka, 1981). Once the jet surfaces, its behavior is very similar to surface buoyant jets (Chen, 1980). A review of relevant studies is included in chapter 2, along with a brief overview of the basic characteristics of a buoyant jet. In chapter 3, results from previous experiments are combined in an attempt to derive equations that can describe the flow of round buoyant jets discharged in shallow water with crossflow. It is shown that, in shallow water, the use of the jet Froude number is more appropriate than the densimetric Froude number. There are three possible flow regimes of a buoyant jet in shallow water with a crossflow: a crossflow dominated flow, a flow with an upstream recirculation zone and a fountain-like flow. A classification scheme for these possible flow regimes (in terms of F, H/D, U/u), is also presented in chapter 3. The experiments of the present study were performed in a flume through which the ambient water flowed. The warmer, buoyant jet was discharged into the flume through a circular opening at the bottom of the flume. The visualization of the flow was achieved by adding dye in the warmer water of the jet. The experimental set-up, details of the flow visualization and measurement techniques, are described in chapter 4 along with a discussion on the results. Conclusions and recommendations for future research can be found in chapter 5. The results of this thesis may be useful in the design of thermal or sewage outfalls. Many such disposers consist of a single port and are situated in shallow coastal or river waters. The mixing characteristics of these discharges are similar to the ones of the Chapter 1. Introduction 3 experiments of this study. The ambient conditions in reality, usually vary within a certain range. The ability, therefore, to have a rough estimate of the minimum surface dilution for any set of jet and ambient conditions, will undoubtedly assist the design of shallow water outfalls. As the demand for a better control of our environmental pollution increases, the necessity of better design of our waste disposal systems becomes more urgent. Hopefully, the present thesis will contribute towards that goal. Chapter 2 Literature Review The emphasis of the theory reviewed in this chapter lies on round buoyant jets discharged into shallow water with a crossflow. A detailed review of the basic parameters and characteristics of turbulent buoyant jets and plumes is given by List (1982). For a review of buoyant jets discharging into shallow water see Jirka (1982). The effect of a crossflow on a buoyant jet has also been studied, but only for deep ambient water. There have been suggestions for the development of equations to describe the effect of a crossflow on shallow water jets (Jirka, 1981). However, no such expressions have been derived or relevant experiments performed. Studies of surface buoyant jets (with or without crossflow) can be used in combination with studies of vertical shallow water jets, to describe the flow of the latter in the presence of a crossflow. This chapter consists of three parts. The first reviews briefly the basic parameters and equations of a round buoyant jet in deep water. The second deals with vertical round buoyant jets discharged in shallow water. Finally experiments concerning surface buoyant jets are reviewed. 2.1 Basic Equations A n d Parameters The behavior of turbulent jets depends on three sets of parameters: jet, environmental and geometrical. Jet parameters include the initial jet velocity and turbulence level, 4 Chapter 2. Literature Review 5 the jet mass and momentum flux, and the flux of any tracer material. Environmental parameters include the ambient turbulence level and existence of currents and density stratification. Finally, geometrical parameters are the shape of the jet, its orientation, altitude and its proximity to other jets or physical boundaries. The five primary variables affecting the behaviour of a buoyant jet are: 1. The jet diameter D. 2. The jet exit velocity u. 3. The ambient velocity U. 4. The ambient depth H. 5. The density difference between the jet and the ambient water, expressed by g' = These can be combined to form three dimensionless parameters: 1. H/D 2. U/u 3. F = u/igDfl2 or FQ = u/(g'D)^2 It is also useful to define a jet in terms of three fluxes: 1. Volume flux Q = u^f-2. Momentum flux M = u2^-3. Buoyancy flux B = ug11^-Chapter 2. Literature Review 6 2.1.1 Zones of flow and Entrainment The flow of any round or axisymmetric jet, passes through two characteristic phases immediately after its exit. First is the Zone of Flow Establishment (ZFE) very close to the exit. In this zone the original momentum controls the flow and there is a jet core in which the velocity is constant and equal to the exit velocity. Then we have the Zone of Established Flow (ZEF) which begins at a distance of 6-10D (where D is the diameter of the exit) from the exit. In this zone the velocity and concentration profiles are self-similar or Gaussian. That means that the time-averaged velocity or concentration at any section,can be expressed in terms of a length measure (radius) and a maximum value (at centerline). So in the ZEF we have u \u00E2\u0080\u0094 ucexp[\u00E2\u0080\u0094(z/b)2] (see fig.2.1). The entrainment of ambient fluid in the jet (usually defined by the entrainment flux Qe ) has been the focus of many theoretical and experimental studies. Morton et al (1956) proposed the entrainment hypothesis method, whose basic assumption is that the velocity of the inflowing diluting fluid is proportional to the locally maximum (centerline) velocity of the jet. Thus Qe = 2Tcctbuc where a is an entrainment coefficient. However, Priestley & Ball (1955) using a conservation of energy equation, and List & Imberger (1973,1975) using dimensional analysis, concluded that a depends on the local densimetric Froude number. The value of a depends on the profile chosen; for Gaussian profiles a, = 0.057 and ap = 0.085. Another approach is the one, first proposed by Abraham (1965), in which an as-sumption of a constant spreading angle is used. The diffusion layer is assumed to spread linearly; & = e. Jirka (1975) showed that the two approaches are consistent if \u00E2\u0082\u00ACj \u00E2\u0080\u0094 2ctj = 0.114 and ep = 6/5ap = 0.106. The advantage of using Abraham's method is that ct varies considerably with FQ, whereas e varies by less than 10% between the two Chapter 2. Literature Review 7 Figure 2.1: Initial buoyant jet regions Chapter 2. Literature Review 8 extreme cases of a jet and a plume. 2.1.2 Dimensional analysis Jets and plumes represent extreme cases of the same problem. In plumes, the buoyancy is much greater than the momentum, and in jets, the momentum is dominant\u00E2\u0080\u0094at least in the first stages of the flow. It is, therefore, easier to find solutions for these two cases and then proceed to the in-between case of a buoyant jet. Dimensional analysis provides order of magnitude estimates for the trajectories, ve-locities and dilutions. These are given for the cases of pure jets, pure plumes and buoyant jets. The characteristic length for a round pure jet is lq \u00E2\u0080\u0094 \u00E2\u0080\u00A2 For z > lg (that is, some distance away from the jet orifice) we have: uMQ ^IQ ( F L I ? A 7 0 ) ( 2 J ) M z EL - 1 Q ~ \u00C2\u00B0 % * c~ (cj \u00C2\u00AB0.25) (2.2) In a pure plume there is no initial volume or momentum flux. The vertical velocity is given by: um = (h \u00C2\u00AB 4.7) The momentum flux, however, increases along the axis of the plume (in a pure jet it's practically constant). An expression for p can be derived, similar to the one for pure jets: p = cpm1/2z w 0.254) (2.3) * Chapter 2. Literature Review 9 A buoyant jet has both jetlike and plumelike properties. It has two characteristic length scales: M 3/4 Their ratio is the initial jet Richardson number 1M = \u00C2\u00A3 1 / 7 (2- 5) ^ \" W - M ^ \" ( 2 - 6 ) When Ro = Rp the buoyant jet is initiated as a pure plume. The dilution of a buoyant jet is given by: ' - 5 ( !> ( 2 ' 7 ) C = c , f f (2.8) with p = C for ( < 1 and p, = C 5 / 3 for \u00C2\u00A3 > 1 2.1.3 Effect of Crossflow When a buoyant jet is ejected into an ambient fluid that has a crossflow, the basic parameter for the description of the flow is the ratio of jet to plume length scales: M 1 / 2 ZM = - j j - (2.9) *B = \u00C2\u00A7 (2.10) If ZM > ZB the momentum is dominant and there are three stages, best described as: 'vertical' jet,'bent' jet and 'bent' plume. The trajectory of the jet is given by: Chapter 2. Literature Review 10 j- = c ^ y \u00C2\u00BB ( 2 . i i ) for z < ZM-, and by: \u00E2\u0080\u0094 = C1(\u00E2\u0080\u0094)1/3 (2.12) for z > ZM-If ZM < ZB the buoyancy is dominant and there are three stages: 'vert ical ' jet, 'vert ical ' plume and 'bent' plume. Solutions for these cases are given by the use of dimensional analysis. These solutions are verified by different versions of Morton's integral analysis (Schatzmann (1978,1979) and Hofer, Hutter (1981) ) 2.2 Turbulent Buoyant Jets in Shallow Water If a vertical jet reaches the free surface of the receiving water and spreads horizontally, the ambient water isrefered to as shallow. Generally, the depth of the ambient water in such cases is not greater than 10 or 20 times the exit diameter. The ini t ia l jet conditions, as well as the ambient ones, determine the exact nature of the flow. 2.2.1 Buoyant Jets in Stagnant Shallow Water LEE & JlRKA (1981) examined the case of shallow water jets for H > 6D (H=6-35D) , F0 from 8 to 583, and high Reynolds number (6800-62500). They defined three main regions of flow (see fig.2.2). The Buoyant Jet Region. In this region, while the momentum flux remains con-stant, the volume flux increases because of the entrainment of ambient fluid. The velocity Chapter 2. Literature Review 11 Figure 2.2: Neax field flow regions for buoyant jets in stagnant water (according to Lee & Jirka,1981) profile at the end of this region has become self-similar (since H > 6D ). Lee &; Jirka assume in this region, a constant spreading angle e \u00E2\u0080\u0094 0.109 . Surface Impringement Region. When the round buoyant jet reaches the free surface it spreads radially. A control volume is used to describe the flow characteristics. It is assumed that there is no entrainment in this region. The velocity profile of this radially spreading surface jet is assumed to be half-Gaussian. In this region there is a loss of momentum, and since there is no entrainment the volume flux remains the same. Lee & Jirka (1981) present diagrams that give the thickness of the resulting upper layer as a function of F0 and H/D. Radial Flow Region. The flow is similar to that of a radial surface jet. The thickness of the upper layer (moving outward) is gradually increasing while the jet keeps entraining ambient fluid. Lee &; Jirka treat the resulting flow as an internal jump. Using results from the two-layer theory they develop equations for the conjugate jump heights. Defining the limits within which, these equations have a solution, we can have a criterion for the stability of flow: Chapter 2. Literature Review 12 10 - Stable Near Field 4 . - \u00E2\u0080\u0094 ~~ Stability Criterion \F=4.6(S) / 3 ^^^^^^^^^ 2 ~ 2 1.5 1 1 1 1 1 1 II 1 1 1 1 1 III Unstable Near Field 1 1 1 1 1 1 1 1 1 10 pQ 100 looo Figure 2.3: Observed and predicted near field dilution (Lee & Jirka, 1981) F0 = 4.6H/D (H > 6D) (2.13) This criterion implies that for a given H/D there is a FQ below which the flow is unstable. And, similarly, that for given source conditions (FQ, D), there is a certain depth H below which the flow becomes unstable. In an unstable flow the outward spreading jet does not have enough buoyancy to stay close to the surface. Recirculation cells are created near the jet exit re-entraining the effluent and thus reducing its dilution. In most practical applications such an unstable flow is undesirable, because it prohibits the entrainment of ambient fluid and thus reduces the dilution of the jet. Lee & Jirka (1981) also give the near-field dilution under steady (and also unsteady) discharge conditions. It is measured experimentally by the ratio of the discharge excess temperature to the maximum excess temperature in the upper layer (see fig.2.3). Previous to this work by Lee & Jirka, which dealt with round jets, another paper by Jirka & Harleman (1979) had studied a similar case with plane jets. In the buoyant jet region they followed Mortons' (1956) analysis. They had to modify the coefficients (which Chapter 2. Literature Review 13 are constant only in the cases of pure jets or plumes). They also used an alternative expression of the basic equations by using the jet width instead of the volume flux. Their solutions agreed with those of Schmidt (1957), Abraham (1965) and Kotsovinos & List (1977). In the surface impringement region they derived a relation for hi/H (hi is the thick-ness of the spreading layer), in terms of Fo and H/B. They showed that buoyancy has little effect on spreading conditions. Andreopoulos, Praturi & Rodi (1986) verified the above results performing experi-ments with H/B = 100 and FQ = 9.9 and 21. Due to the small length of their channel, however, they couldn't compare the results for the dilution. Pryputniewicz & Bowley (1975) measured surface excess temperatures for H/D = 10,20,40, 80 and F0 = 1,2,4,5,8,16,25,50. Their results show that the influence of the Froude number on the maximum surface excess temperature is almost negligible for F0 greater than 20 . P&B's results will be compared to L &; J's results in chapter 3. 2.2.2 Length of Zone of Flow Establishment The determination of the length of the ZFE (ze), is important in the description of the jet flow, before it reaches the free surface. Albertson et al (1950), determined that Ze = ze/D = 6.2 for three-dimensional momentum jets. Lee & Jirka (1981), however, show that Ze is a function of F0. For F0 > 25, Ze reaches asymptotically a value of 5.74, (see fig.2.5). So it becomes evident that when a jet is discharged vertically upward in water whose depth is H < 6D, the velocity profile is not Gaussian. This ambient water is defined as 'very shallow'. Crow & Champagne (1971) studied non-fully developed jets and give the following Chapter 2. Literature Review (VERTICAL JET) 14 O - H/D.- 10.0 D E N S I M E T R I C F R O U D E N U M B E R F0 Figure 2.4: Dependence of maximum surface excess temperature on discharge Froude number (from Pryputniewicz & Bowley, 1975) A = 1.14 e = 0.109 Ze = zJD A S Y M P T O T I C V A L U E O F A P L U M E 10 ZO SO 40 Figure 2.5: Length of region of flow establishment as a function of source densimetric Froude number (from Lee & Jirka, 1981) Chapter 2. Literature Review 15 equations for the Q and M of the jet at the entrance of the impringement zone: Qi = Qo(l+ 0.136^) (2.14) Mi = M 0(l + 0.136^) (2.15) for hx/D < 2 and H/D < 6 (see fig.2.2). The shallowness of the ambient water, also affects the spreading angle of the buoyant jet. Murota & Muraoka (1967) observed that when H < 20D, the spreading angle of the ZEF is greater than that of a free jet. This implies more entrainment of ambient water. 2.2.3 Surface Disturbance When the buoyant jet reaches the free surface it causes a surface hump (accompanied with small surface fluctuations), which results in a radial pressure gradient and horizontal spreading of the discharge. The maximum height of this hump is estimated by using dimensional analysis and experimental results (Murota Sz Muraoka, 1967). Thus, the ratio of this maximum height to the depth of the ambient water is given as: H=f{D'Tg& )= 6 7 W * (2'16) From the above result, the following two expressions can be derived: ^ = 1.61 A 3 / 4 F 3 / 2 (2.17) D H ^ = 1.61(^)7/4F3/2 (2.18) H tl Chapter 2. Literature Review 16 ho/y . h./z. 10 -1 10 - 2 10 - 3 + 3 - d i m . Zo / D o=4~25 (by Hunt) \u00E2\u0080\u00A2 3 - d i m . z0/D.= 4 ~ 9 o 2 -d im. y0/bo=4 ~ 2 0 r \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 + + + > ^ + + o, o \u00C2\u00BB o \u00C2\u00B0 10 - 5 10 - 4 I O \" 3 I O \" 2 10\" 0.5 r / rV + 3-d imens iona l z . /D.= l.O (by Hunt) \u00E2\u0080\u00A2 3 -d imens iona l Zo/Do= 4 ~ 4 . 7 o 2-dimensional y . / b . = 4 . 4 ~ 5 . 5 0.5 0 V . ii 2-d imens iona l \u00E2\u0080\u00A2 V t ) . = l 4 . 8 \u00E2\u0080\u00A2 10.2 \u00E2\u0080\u00A2 5.5 Figure 2.6: Dimensionless expression and shape of the maximum height of surface dis-turbance for stagnant ambient water (from Murota & Muraoka, 1967) Chapter 2. Literature Review 17 Another interesting observation is that for shallower water, the profile of the hump does not approach the still water surface monotonically. Instead, there is a small de-pression around the hump (see fig.2.7). It is obvious from the above, that an analytical description of the flow in the impringement area is very complicated. For that reason, it is more practical to use a control volume analysis (see Ch.3). 2.2.4 Effect of Crossflow When a buoyant jet is discharged into shallow water, the presence and intensity of an ambient crossflow determines: \u00E2\u0080\u00A2 The position that the jet will surface (downstream, above, or upstream from the jet exit). \u00E2\u0080\u00A2 The way it will spread (once it surfaces). \u00E2\u0080\u00A2 Its dilution. If the crossflow is strong, the jet will surface further downstream, and most of the mixing and dilution will take place before it surfaces. If the momentum flux of the jet is more dominant than the crossflow, the jet will surface almost above its source. It is the latter case that will be studied to a greater extent in this thesis. The intensity of the crossflow is usually described by the ratio of the exit momentum of the jet, M 0 , and the velocity of the ambient crossflow, U. The characteristic length is ZM = M1/2/U. When z ZM the crossflow is dominant (at the same time we should have ZM ^ D, for a self similar flow to develop). This last restriction can be written as M 1 / 2 / t V >\u00E2\u0080\u00A2 D and by further substituting M = w 2 i f - , the following condition is derived (for self similar flow to develop): Chapter 2. Literature Review 18 U/u < 0.88 (2.19) 2.3 Surface Buoyant Jets After a vertical buoyant jet reaches the surface, it starts to spread in the direction of the crossflow. Following Lee & Jirka's (1981) classification, the third region of the flow (immediately after the surface impringement region), is very similar to the flow of a surface buoyant jet. A review of the most relevant studies of surface jets is included in this section. There are two basic types of surface buoyant jets, depending on the discharge ge-ometry: the radial and the plane ones. Many theoretical models have been proposed for both these cases. Jirka et al (1981) have summarized surface jet data from various experiments. The present study is interested in radial 3-D horizontal surface buoyant jets. Experiments by Rajaratnam & Subramanyan (1985) for shallow water conditions, show that there is a critical depth if c , below which the flow becomes unstable. For high FQ this is given by: However, in most problems of waste-heat discharges smaller F0 are usually desired by the design. Another parameter of importance is the width of the spreading surface layer. Larsen & Sorensen (1968) presented a model of a surface buoyant jet in a crossflow. They assumed a uniform thickness h of surface jet at a location x downstream from the source of the jet. Rawn &; Palmer (1929) used the concept of a one-dimensional transverse spreading = 0.67 (2.20) Chapter 2. Literature Review 19 \"i I i i I\u00E2\u0080\u0094i\u00E2\u0080\u0094i\u00E2\u0080\u0094I i \u00E2\u0080\u0094 I \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 I \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 r 1.5 \u00C2\u00AB\ o 1.0 \u00E2\u0080\u0094 o\u00C2\u00A3_ \u00E2\u0080\u0094 \u00E2\u0080\u0094^2- \" -J I I I I I I I I I I I I -I I I I I I I L. 0.0 10 15 Fr 2 0 2 5 Figure 2.7: Relation of starting jet coefficient to jet Froude number (from Chen, 1980) field superimposed on a uniform current. This model gives good results as long as the transverse spreading velocity is negligible compared to the current velocity. Relations for the spreading surface width y(z), and the distance of the upstream edge from the center of the jet /\u00E2\u0080\u009E, have been found using dimensional analysis (Chen, 1980): y{x) ~ M1/4(x/U)1/2 (2.21) L = * % - (2-22) where d \u00E2\u0080\u0094 = f(F, Re) is the starting surface jet coefficient. The value of this coefficient decreases with increasing Froude number (see fig.2.7), and approaches asymptotically a value of 1.05 when F > 10. This implies that the buoyancy affects the initial surface jet flow when F < 10. The above relations were verified by Chen (1980) for deep water conditions, Froude numbers between approx. 7 and 33, and u/U between approx. 15 and 24. Chapter 3 Theoretical Development 3.1 General Considerations In this chapter the equations that were presented in the last chapter will be combined and modified to suit shallow water conditions with a crossflow. The possible regimes of flow will be classified on the basis of the values of : H/D, U/u, F. These three dimensionless parameters will be used to describe the various characteristic lengths and quantities of the flow. 3.1.1 Relative Importance of F and FQ At this point a distinction should be made between the Froude number of the jet, F = uZ(gD)1/2, and the densimetric Froude number, Fo = u/(g'D)1/2. Near the jet exit the momentum of the jet dominates the flow. At a distance z \u00C2\u00AB IM, the buoyancy of the jet starts to dominate (List, 1982). Thus the region that F describes the flow better than FQ is given by z ur\" \u00C2\u00B0 3 max(Tj-Ta)attUTfa Chapter 4. Experiments 44 Temperature C 20 downstream -< 10 ambient temperature 10 upstream Run No H/D u/U S 6/10 \u00E2\u0080\u00A2 6 3.8 5.3 6/11 o 6 5.6 3.2 6/9 \u00E2\u0080\u00A2 8 5.6 4.6 F=1.17 Tj-Ta:30-14 Q Figure 4.18: Effect of flow regime on Dilution Chapter 4. Experiments 45 Chapter 4. Experiments 46 8x 7--6--5-S \u00E2\u0080\u00A2\u00E2\u0080\u00A2 4--3--2.5 - -2--1.8 -1.5 \" 1 0.00 H/D=8 /\u00C2\u00AB ^H/D=6 / / v / / 7 / / F=2.53 0.10 0.20 0.30 0-40 0.50 U/u Figure 4.20: Effect of H/D on Dilution for F = U/u = 0 are derived from Lee & Jirka, 1981) 1.17 and F = 2.53 (values of S for Chapter 4. Experiments 47 4.3 Comparisons with theory All the experiments satisfied eq.3.26 for momentum dominated jets. For z \u00E2\u0080\u0094 H equations 3.26 and 3.28 can be combined (at z = H = ZM) to give H/D ~ U/u. This relation represents the line that separates regime 1 from regime 2 and it agrees with the empirical equation 4.34. A semi-empirical equation for the line that separates regime 2 from 3, can be derived using equation 2.18 and introducing a factor to account for the effect of U. Equation 2.18 would then become: ^ = 1.61 A 7 ' 4 F3'2 (1 - - ) 0 4 = 0.4 (4.35) H H u The exponent 0.4 and the relation = 0.4,are based on experimental observations. Both these values are subject to changes based on more complete and more accurate experiments. The crosspoint of eqs.4.34 and 4.35 is a characteristic point for each F. By assuming (for practical purposes) that eq.4.34 is true for any F, it is possible to draw a diagram that includes all three basic parameters of the flow: F,U/u and H/D. The diagram, shown in figure 4.21, has a vertical axis for H/D and a horizontal axis for U/u. For each F there is a line, given by equation 4.35, that separates regime 2 from 3. Thus, it is possible, from just one diagram to determine the flow regime that is to be expected according to a set of F,H/D,U/u. The flow regime gives a good indication of the expected dilution and the nature of the flow. It is also possible to understand the effect that a change of one parameter will have on the flow. By using equation 2.18, the size of the surface disturbance can be plotted in terms of F and H/D for stagnant water conditions. The results are compared to the experimental Chapter 4. Experiments 48 H/D F=l Flow Regime 3 (& 3a) 0 -r 0.2 0 0.1 0.3 0.4 0.5 Figure 4.21: Diagram for prediction of flow regimes, according to F, H/D, U/u values of hs/H in fig.4.13,15. The stagnant water disturbance height is always bigger than the experimental ones, as expected. Figure 4.20 shows that as the crossflow decreases the dilution approaches a minimum value (for each H/D). This minimum value can be predicted by using fig.3.9 (derived from Lee & Jirka's results). For F = 1.17 and H/D = 4,6,8 the Smin \u00C2\u00AB 1.5,1.8,2.3 respectively. For F = 2.53 and H/D = 4,6,8 the Smin \u00C2\u00AB 1.5,1.8,2.5. These values are shown in fig.4.20 together with the experimental results of the present study. As expected, the predicted dilution for the stagnant ambient water case, is always less than the observed dilution in the presence of a current. The downstream thickness of the jet is shown to become equal to the ambient depth as flow regime 3 is approached (see figs.4.13,15). There is no significant downstream recirculation. Regime 3 flows are the most unstable of the three regimes. Most of the turbulence observed in these flows is due to the shallowness of the receiving water in the immediate area of the jet exit. Chapter 4. Experiments 49 4.3.1 Limits of Shallow Water Effect In chapter 3 it was shown that the shallowness of the ambient water doesn't allow the buoyancy to affect the jet dispersion and dilution. For that reason, the jet Froude number has been used instead of the densimetric Froude number. The limits, however, of the use of F were not defined. It is logical that, as the ambient depth increases and the jet exit velocity remains the same, the buoyancy will start influencing the jet. If, as the H increases, the u increases too, then the jet momentum could still be more dominant than the buoyancy. So, the limits of the use of F, should be described in terms of H, D, u,g', (for stagnant ambient water). When there is a current, the above limit should be given in terms of H/D, U/u and g'. Equation 3.23 represented the limit that F can be used in stagnant shallow water. When an ambient current, U, is present, there are two cases that the buoyancy can start affecting the flow. First, when the current becomes strong and dominates the jet flow, and second, when the depth becomes becomes big enough to allow the buoyancy of the jet to control the flow. In the first case, the limit between regime 1 and 2 (H/D = 1.6u/U) can give one limiting equation for the range that F should be used: The second case is covered by the relation for momentum dominated jets (see chapter 3) ZM ZB, which can be written also as (4.36) u/U < 1.06F0 (4.37) Chapter 4. Experiments 50 U/u=0.40 Flow Regime 1 TJ/u=0.25 Flow Regime 2 U/u=0.20 Flow Regime 2 U/u=0.15 Flow Regime 2 Figure 4.22: Phot.1-4 Effect of U/u (as the current weakens, the upstream recirculation zone increases)-exp. 1/1,2,4,5 Chapter 4. Experiments 51 U/u=0.07 H/D=5 How Regime 3 U/u=0.08 H/D =4.7 Flow Regime 3a Figure 4.23: Phot.5,6 Transition from regime 3 to 3a (as the depth decreases, part of the downward deflected jet reaches the bottom and is deflected again before it is carried downstream)-exp. 8/13,14 Chapter 4. Experiments 52 F=1.17 U/u=0.22 u=0.45m/s H/D=6 F=2.53 U/u=0.20 u=0.97m/s H/D=6 Figure 4.24: Phot.7-8 Effect of F on upstream edge (when u increases, h, increases and that results in a smaller upstream edge)-exp.8/3,18 Chapter 5 Conclusions and Recommendations The dispersion and dilution of a buoyant jet discharged into very shallow water are determined by three parameters: H/D, U/u and F. Due to the proximity of the free surface, the buoyancy of the jet does not influence the near field characteristics of the jet. The use of F is, therefore, shown to be more appropriate than F0 for the description of such jets. The limits, within which F should be used, are given by H/D E^- : buoyancy flux b : jet width D : jet diameter Ap : density difference F = uKgD)1!2 : jet Froude number -Fo = u/(g'Dy/2 : jet Densimetric Froude number g : gravity acceleration g' = gAp/p : relative gravity acceleration hi : depth of upper layer hs : height of surface disturbance (Equation 2.16) hd : downstream thickness of jet H : depth of ambient water IQ = jpj2 '\u00E2\u0080\u00A2 characteristic length for pure plumes (Equation 2. IM \u00E2\u0080\u0094 ^1/2 : characteristic length for pure jets (Equation 2.5) lu : length of upstream recirculation zone (Equation 2.22) 55 Appendix A. Notation I2 : thickness of upstream recirculation zone M = u 2 \u00C2\u00A3 j - : momentum flux M i : momentum at entrance of impringement region M 0 : momentum at jet exit M2 : momentum at exit of impringement region Q = : volume flux Qe : entrainment flux Qi : volume flux at entrance of impringement region Qo : volume flux at jet exit Ro : Richardson's number p : density S : dilution of jet u : jet velocity uc : centerline jet velocity U : velocity of ambient water x : downstream distance y(x) : surface spreading width (Equation 2.21) z : vertical distance ZM = ^JJ\u00E2\u0080\u0094 '\u00E2\u0080\u00A2 pure jet length scale (Equation 2.9) ZB = xj3 '\u00E2\u0080\u00A2 pure plume length scale (Equation 2.10) ze : entrainment length Ze : dimensionless entrainment length Bibliography Abraham, G., Entrainment principle and its restriction to solve jet problems, J. of Hydraulic Research, vol.3, no.2, 1965, pp.1-23 Ackerman, N.L., Apostol, R.T., Siriyong, S., Surface disturbance from submerged jet, J. of Hydr. Div., vol.97, 1971, pp.937-948 Andreopoulos, J., Praturi, A., Rodi, W., Experiments on vertical plane buoyant jets in shallow water, J. of Fluid Mechanics, vol.108, 1986, pp.305-336 Anwar, H.O., The radial spreading as a free surface layer of a vertical buoyant jet, J. of Eng. Mathematics, vol.6(3), 1972, pp.257-272 Anwar, H.O., Appearance of unstable buoyant jet, J. of Hydr. Div., vol.98, 1972, pp. 1143-1156 Anwar, H.O., Flow of a surface buoyant jet in a crossflow, J. of Hydr. Eng., vol.113, 1987, pp.892-904 Buhler, J., On buoyant surface layers generated by wastewater discharged from submerged diffusers, 17th Congress I.A.H.R., vol.1, 1977, pp.325-332 Buhler, J., Axisymmetric surface layers due to a submerged source of buoyancy, 22nd Congress I.A.H.R., 1987 Chen, J.C., Studies on Gravitational Spreading Currents, W.M.Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Report KH-R-40, March 1980 Chu, V.H., Goldberg, M.B., Buoyant forced plumes in crossflow, J. of Hydr. Div., vol.100, 1974, pp.1203-1214 Engelund, F., Hydraulics of surface buoyant jet, J. of Hydr. Eng., vol.102, 1976, Fisher, H.B. et al, Mixing in Inland and Coastal Waters, Academic Press, New York, 1979 Fox, D.G., Forced plume in a stratified fluid, J. of Geophysical Research, vol.75, 1970, pp.6181-6835 Hart, W.W., Jet discharge into a fluid with a density gradient, J. of Hydr. Div., vol.87, 1961, pp.171-200 Hayashi, T., Ito, M., Diffusion of effluent discharging vertically into stagnant sea water, Coastal Engineering in Japan, vol.17, 1974, pp.199-213 57 Bibliography 58 Hwang, R.R., Chiang, T.P., The flow of round buoyant jets discharging into a crossflow of stratified fluid, Turbulence measurements and flow modeling, by Chen, Chen, Holly, 1985, pp.485-494 Jirka, G.H., Harleman, D.R.F., Stability and mixing of a vertical plane buoyant jet in confined depth, J. of Fluid Mechanics, vol.94, 1979, pp.275-304 Jirka, G.H., Adams, E.E., Stolzenbach, K.D., Buoyant surface jet, J. of Hydr. Eng., vol.107, 1981, pp. 1467-1487 Koh, R.C.Y., Two dim. surface warm jet, J. of Hydr. Div., vol.97, 1971, pp.819-836 Lee, J.H.W., Jirka, G.H., Vertical round buoyant jet in shallow water, J. of Hydr. Eng., vol.107, 1981, pp.1651-1675 List, E.J., Imberger, J., Turbulent entrainment in buoyant jets and plumes, J. of Hydr. Div., vol.99, 1973, pp.1461-1474 List, E.J., Turbulent jets and plumes, Annual Reviews of Fluid Mechanics, vol.14, 1982, pp.189-212 Murota, A., Muraoka, K., Turbulent diffusion of the vertically upward jet, Proc. 12th I.A.H.R. Congress,4 ,1967, pp.60-70 Morton, B.R., Taylor, G., Turner, J.S., Turbulent gravitational convection from maintained and instantaneous sources, Proc. of the Royal Soc. of London, 234(A), 1956, pp. 1-23 Pryputniewicz, R.J., Bowley, W.W., An experimental study of vertical buoyant jets discharged into water of finite depth, J. of Heat Transfer, May 1975, pp.274-281 Rajaratnam, N., Subramanyan, S., Plane turbulent buoyant surface jets and jumps, J. of Hydr. Res., vol.23, 1985, pp.131-146 Wallace, R.B., Sheff, B.B., Two dim. buoyant jets in a two layer ambient fluid, J. of Hydr. Eng., vol.113, 1987, pp.992-1005 Wallace, R.B., Wright, S.J., Spreading layer of a two dim. buoyant jet, J. of Hydr. Eng., vol.110, 1984, pp.813-828 Wright, S.J., Mean behavior of buoyant jets in a crossflow, J. of Hydr. Div., vol.103, 1977, pp.499-513 Wright, S.J., Buoyant jets in density stratified crossflow, J. of Hydr. Eng., vol.110, 1984, pp.643-656 Yoon, T.H., Cha, Y.K., Kim, C.W., Vertical plane buoyant jets in crossflows, 22nd I.A.H.R. Congress, 1987, pp.148-154 "@en . "Thesis/Dissertation"@en . "10.14288/1.0062884"@en . "eng"@en . "Civil Engineering"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Buoyant jets in shallow water with a crossflow"@en . "Text"@en . "http://hdl.handle.net/2429/27898"@en .