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Buoyant jets in shallow water with a crossflow Labridis, Christodoulos 1989

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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 © 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 Regime Diagram 35 Effect of F on Regime Diagram 38 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 — 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 ' ' ° " 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 — 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„ 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 — ucexp[—(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 €j — 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—at 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 — • 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 ~ ° % * c~ (cj «0.25) (2.2) In a pure plume there is no initial volume or momentum flux. The vertical velocity is given by: um = (h « 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 = £ 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 £ > 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 = § (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 » ( 2 . i i ) for z < ZM-, and by: — = C1(—)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 — 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 . - — ~~ 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) • 3 - d i m . z0/D.= 4 ~ 9 o 2 -d im. y0/bo=4 ~ 2 0 r • •• + + + > ^ + + o, o » o ° 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) • 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 • V t ) . = l 4 . 8 • 10.2 • 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: • The position that the jet will surface (downstream, above, or upstream from the jet exit). • The way it will spread (once it surfaces). • 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 >• 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—i—i—I i — I — i — i — i — i — I — i — i — i — i — i — i — i — i — r 1.5 «\ o 1.0 — o£_ — —^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 /„, have been found using dimensional analysis (Chen, 1980): y{x) ~ M1/4(x/U)1/2 (2.21) L = * % - (2-22) where d — = 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 « 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 <C IM- By substituting IM from equation 2.5 and using M = u2^- and B = u<7/2L^p, the following equation can be derived at z — H: H/D < 0.94Fo (3.23) This is also illustrated by using Lee fo Jirka's diagram for dilution (see fig.2.3). By using data included in Lee fo Jirka (1981), Fo can be reverted to F and a similar diagram is 20 Chapter 3. Theoretical Development 21 obtained in fig.3.9. In Lee & Jirka's experiments the temperature difference, AT, between the ambient and the jet water varied only from 11.9°C to 16.1°C, and the ambient temperature from 21.2°C to 26.2°C. Because of these small variances, an average value of (Ap/pfl2 = 0.069 can be used for the majority of Lee &; Jirka's experiments. Thus, the stability criterion (eq. 2.13): F0 = i.6H/D can be transformed, using F = FoiAp/p)1'2 (3.24) to: F = 0.32(|) (3.25) The dilution contours in fig.3.9 are plotted using Lee & Jirka's equations and the relation F = 0.069i*o- The nature of the diagram doesn't change much. The importance of using F, instead of F0, becomes more evident when data from Pryputniewicz & Bowley (1975), fig.2.4, are plotted in fig.3.8 (in terms of F0) and fig.3.9 (in terms of F). In P k B's experiments the ambient and jet temperatures were kept constant at 70°C and 170°C respectively. Thus (Ap/p)1'2 = 0.155. By using F = 0.155.Fo their results are plotted in fig.3.9. Using Fo, only one point falls in the 'unstable' near field (point R), whereas using F, three points (points R,P,Q) do. The same three points are shown in fig.2.4 to have distinctly lower dilution (or higher maximum surface excess temperature ratio). P & B noted this discrepancy and attributed it to 'the proximity of the free surface acting as a constant pressure boundary'. It would be expected then that these three cases fall in the 'unstable' region of L & J's diagram. It is obvious that both Lee & Jirka's (1981) and Pryputniewicz & Bowley's (1975) experiments were within the range of eq.3.23. The fact that P & B's data plot better Chapter 3. Theoretical Development 22 10 — Stable Near Field 8.3 5.6 5.3 4,4 - A A A A 3.3 4 i 3 J J _ — r r " 4 4 « D A A A P J 2 1.4 U Stability Criterion / ^ * ^ ^ 3 3 - — ~ 2 " ' - 1.5 1 1 1 1 1 1 II Unstable Near Field O a Data from Pryputniewicz &Bowley (1975) 1 1 1 1 1 1 1 1 1 II 1 10 p 100 1000 Figure 3.8: Observed and predicted near field dilution in terms of F0 (data from Lee & Jirka,1981) Figure 3.9: Observed and predicted near field dilution in terms of F (data from Lee & Jirka,1981 and Pryputniewicz & Bowley, 1975) Chapter 3. Theoretical Development 23 with reference to F, is a strong indication that for small H/D, F describes the flow better than Fo. 3.1.2 Surfacing Position of Jet A buoyant jet discharged vertically upward in deep water, follows a trajectory that is influenced by: the jet momentum and buoyancy, the ambient crossflow, and stratification, if present. When there is no significant stratification, there is one characteristic length Z M = is important for the description of the flow. The velocity, concentration and coordinates of the jet are a function of Z/ZM-Following the analysis presented in Fischer et al (ch.9, 1979), jets can be divided to momentum-dominated jets and pure plumes, according to the ratio Z M / Z B - The three relations that define the present problem are: 1. Momentum Dominated Jets: ZM ^ ZB-2. Trajectory equation for jet region: — = C\{—W2 Substituting, zM = M1/2/U and zB = B/U3 and also M = u2^- and B = ug'^-, the 3. Limit of jet region: z < ZM above three relations can be written as: U (3.26) u z O M C ^ Y ' 2 ^ 2 (3.27) D (3.28) Chapter 3. Theoretical Development 24 So, given enough depth most jets will be carried downstream by the current, no matter how weak the current is. If, however, the depth becomes small, the jet may surface above the jet exit. For this to happen, x/D must be small (less than 1-2) at z = H. It is also evident that the jet will be in the first stage of its trajectory (where x ~ z2), since in the next stage x grows faster (x ~ z3) (see eqs.2.11,2.12). It is not easy, at this stage to be more precise about the max. value of x/D that the jet will surface above its source. The free surface boundary affects the spreading of the jet. The above relations, however, provide a good insight on the resulting flow. It can be seen that for constant jet conditions (u, D, g'), a decrease of U means that the jet will surface above its exit at a higher z/D (or H/D). It should be noted that since the z coordinate is proportionate to (u/t/)1/2, there is a max. value of u/U above which the third restriction is not satisfied. Beyond that value the jet begins to 'bend' and z ~ x1/3. 3.1.3 Large Surface Disturbances When a jet is discharged into stagnant ambient water, there is a depth below which, the free surface of the receiving water is greatly disturbed due to the impact of the jet. As the depth decreases further, the surface disturbance increases. The same is also true when there is a current. The presence of a current, however, reduces the size of the surface disturbance that would occur in stagnant water. Thus, there are two parameters that determine the size of the surface disturbance: F and U/u. 3.2 Flow Regimes The above discussion leads to a classification of the possible flow regimes that can occur for a given F (see fig.3.10). Chapter 3. Theoretical Development 25 1. A crossflow-dominated flow where the jet surfaces downstream from the exit. The jet travels a longer distance before it surfaces, thus increasing its entrainment length. Because of this, the dilut ion is high. The width of the jet is also restricted by the strong current. 2. A flow where the jet surfaces above its exit and then spreads downstream. A n upstream recirculation zone is observed. The length of this zone depends on the strength of the current. The dilut ion is greatly reduced compared to regime 1. 3. A flow where the jet causes a large disturbance on the free surface changing the character of the flow. The upstream zone remains more or less constant. The upward momentum of the jet is deflected (due to the big surface disturbance) downwards and then (due to the current) downstream. As the depth decreases, a variation of this flow (regime 3a) is observed. The downwards deflected jet reaches the bot tom and is deflected again before it is, finally, carried downstream. 3.2.1 Typical Shallow Water Jets The second flow Regime is the most interesting and the one that is usually connected wi th shallow water jets. The geometry of this flow is defined by: • The upstream length /„. • The thickness of the upstream zone l2. • The size of the surface disturbance hs. • The downstream thickness of the spreading jet h&. • The surface spreading width y(x). Chapter 3. Theoretical Development 26 Flow Regime 1 Surfacing position I Flow Regime 2 j(D,u) Flow Regime 3 Figure 3.10: Flow regimes of buoyant jets in shallow water with a crossflow Chapter 3. Theoretical Development 27 Equations for the dimensionless form of these lengths (i.e.lu/D) are derived below, as a function of H/D, U/u, F. Upstream length. As the jet reaches the free surface it spreads radially (on a hor-izontal plane). Thus some of its particles move upstream with a horizontal velocity that is a fraction of u. This velocity is negated, at some point upstream, by the velocity of the ambient crossflow. The jet particles are then driven downstream creating a recircu-lation zone. The length of this upstream recirculation zone is expected to increase as the crossflow decreases, or as U/u decreases. Using Crow & Champagne's (1971) results for non-fully developed jets (eq 2.15) the following relation is derived: M2 = 0.8M (1 + 0.136fci/D) (3.29) (where M2 is the jet momentum at the exit of the impringement zone and h\ = H — l2) Substituting this result into eq. 2.22 an expression for lu can be derived: ^ = 0.437(^)(1+ 0.136^)1/2 (3.30) The thickness of the upstream layer l2, can be assumed equivalent to the upper layer thickness of stagnant shallow water jets. Then, from fig.2.3 it is seen that l2/D 0.1H/D for H/D < 10. If such a relation is true for shallow water jets in a crossflow, then : h/D « (H - l2)/D « 0.9H/D (3.31) Equation 3.29 would then become: k « 0.46(^ ) A 1 / 2 (3.32) Chapter 3. Theoretical Development 28 This implies that lu is more sensitive to changes of u/U than changes of H/D. Ex-perimental results will examine the validity of eqs. 3.29 and 3.30. Surface Disturbance. The height of the surface disturbance hs will be studied experimentally. Equations 2.17 and 2.18 (for stagnant ambient water) will be compared with the experimental results. The influence of the crossflow is expected to reduce the height of the disturbance. All other parameters being the same, the surface disturbance will be maximum when the crossflow is zero. D o w n s t r e a m thickness. A definition of a critical Hc (as in eq.2.20) could be made for the present case. An unstable flow would then be a flow where = H. The possibility and nature of such a case will be investigated. Finally, the surface spreading width can be given by eq.2.21. Substituting M with the expression of eq.3.29: Ku' KD' (3.33) D Chapter 4 Experiments 4 .1 Experimental Set-Up and Procedure 4.1.1 Equipment and Instrumentation The experiments were conducted in a 20m long flume (0.50m wide, 1.00m high), with glass walls. The velocity and depth of flow were controlled by a valve for the discharge, and a gate at the end of the flume. The temperature of the water in the flume varied from 12°C - 16°C. The jet water came from a tap with hot and cold water valves. It was collected in an overflow tank situated above the flume (see fig.4.11). A system of pipes connected the tank with a circular outlet at the bottom of the flume. The jet velocity was controlled by a valve at the exit of the tank. The diameter of this outlet was adjustable up to 0.025m. The temperature of the jet varied up to ~ 50°C. The velocity of the 'ambient' water of the flume was measured using a propeller meter (for velocities greater than 0.05m/s and depths greater than 3cm). In some runs that the conditions didn't allow the use of a propeller meter, the ambient velocity was measured by timing beads carried on the free surface. The velocity of the jet was measured by closing the inlet tap and timing the withdrawal of the tank water. The volume of the withdrawn water, multiplied by the withdrawal 29 Chapter 4. Experiments Figure 4.11: Set up of experiments Chapter 4. Experiments 31 time, was equated to the product of the outlet area multiplied by the exit velocity. This procedure was repeated 3 times at the beginning of each run and was verified at the end of the run. The temperature of both ambient and jet water was measured by using an AD594 semi-conductor probe. Readings were taken from a volt meter that was connected to the probe. (10mV=lC ) . The probe was dipped in the water at the desired location. Because of the often varying temperature, a time of about 3-5sec. was required for an averaged temperature reading to be made. An absolute temperature measurement was accurate within 0.5°C. The relative temperature differences were accurate within 0.1°C. 4.1.2 Flow Visualization The flow was visualized using red food dye which was mixed in the overflow tank. The density of the added dye was negligible and did not affect the flow (especially since the buoyancy of the jet was not a determining parameter in these experiments). A 1000W reflector was positioned on one side of the flume; transparent paper was placed on the outside of the glass wall between the reflector and the flume. This way, by placing a camera on the opposite side of the flume a clear photograph of the jet could be taken. In most of the runs, the camera was placed on a level with the free surface and opposite the jet exit at x=0. The more dye was added in the tank, the clearer the outline of the jet became. Less amount of dye, however, gave more details of the flow structure. The measurement of the various lengths (from the photographs) was done by mea-suring the length / and the H on the centerline of the flume (which was also the jet center line). The ratio l/H was then multiplied by the known H/D. The result gave l/D. Measuring the centerline lengths, no corrections for perspective or diffraction had to be made. The accuracy of the measurements taken from the photographs was within Chapter 4. Experiments 32 l-5mm. 4.1.3 Procedure of Experiments Most of the runs were conducted by setting and then keeping constant the jet parameters (D,u,g'). Then, a series of ambient conditions (H,U) was set and photographs and temperature readings were taken. For each change of H or U a certain time of 10-20min was required for the flow to stabilize. Other runs were done in the opposite way. A set of H and U were kept constant and either the jet temperature or jet velocity were changed. This way the effect of a certain parameter was easier to observe. The above two procedures reduced the amount of time required for each run and minimized the relative error. It was also possible to include a run in two different series of runs, thus double checking the results of some runs. The temperature measurements were taken at fixed positions on the water surface. The initial jet temperature was verified, before and after each run, by placing the end of the probe in the jet outlet (enough time was allowed after that for the flow to re-stabilize). 4.2 Results and Comments The results are shown in table 4.1. They are organized according to F,H/D,U/u. The jet D and u are also shown. The jet and the ambient temperatures are given for each run. Surface temperature measurements are given in table 4.2 along with the relevant information of each run. Chapter 4. Experiments 33 Run No Tj Ta D (m) u (m/s) F Fo U/u H/D lu/D I2/H hs/H hd/H 1 /5 42 1 6 0.015 0.4 1 .04 1 2 0.4 6 0 0.5 1 /4 42 1 6 0.015 0.4 1 .04 1 2 0.3 6 0 1 /3 42 1 6 0.01 5 0.4 1 .04 1 2 0.25 6 4.7 0.2 0 0.3 1 /1 42 1 6 0.015 0.4 1 .04 1 2 0.2 6 7.6 0.35 0.05 0.4 112 42 1 6 0.015 0.4 1 .04 1 2 0.15 6 10.5 0.7 0 1 1 /6 42 1 6 0.015 0.4 1 .04 1 2 0.38 * 8 0 0 0 0.55 1 /9 42 1 6 0.015 0.4 1 .04 1 2 0.25 8 0 0 0 0.3 1 n 42 1 6 0.015 0.4 1 .04 1 2 0.23 8 5.8 0.4 0 0.4 1 /8 42 1 6 0.015 0.4 1 .04 1 2 0.18 8 6.4 0.4 0 0.4 6/6 46 1 4 0.015 0.45 1 .17 1 2 0.2 2 1.7 1 0.5 1 8/1 46 1 4 0.015 0.45 1 .17 1 2 0.59 2 0 0 0 1 8/1 0 46 1 4 0.015 0.45 1 . 1 7 1 2 0.26 3 1.5 1.5 0.3 1 8/2 46 1 4 0.015 0.45 1 .17 1 2 0.33 4 1.3 0.5 0 0.4 6/1 46 1 4 0.015 0.45 1 .17 1 2 0.24 4 1.4 0.3 0.15 0.3 6/3 46 1 4 0.015 0.45 1 .17 1 2 0.18 4 4.8 0.35 0.15 0.35 6/2 46 1 4 0.015 0.45 1 .17 1 2 0.13 4 10.5 0.5 0.2 0.5 6/10 30 1 4 0.01 5 0.45 1 .17 20 0.26 6 0 0 0 0.3 8/3 46 1 4 0.015 0.45 1 .17 1 2 0.22 6 2.4 0.15 0 0.35 6/4 46 1 4 0.015 0.45 1 .17 1 2 0.18 6 5 0.2 0 0.25 6/1 1 30 1 4 0.015 0.45 1 .17 20 0.18 6 6 0.25 0.1 0.35 8/5 46 1 4 0.015 0.45 1 .17 1 2 0.2 8 2.1 0.15 0 0.3 6/8 46 1 4 0.015 0.45 1 .17 1 2 0.18 8 3.5 0.1 0 0.2 6/9 30 1 4 0.015 0.45 1 .17 20 0.18 8 2 0.15 0 0.3 8/4 46 1 4 0.015 0.45 1 .17 1 2 0.16 8 3.5 0.2 0 0.4 6/7 46 1 4 0.015 0.45 1 .17 1 2 0.16 9 2.2 0.1 0 0.3 8/6 46 1 4 0.015 0.45 1 .17 1 2 0.16 1 0 3 0.2 0 0.35 8/7 46 1 4 0.015 0.45 1 .17 1 2 0.13 1 0 5 0.3 0 0.35 1 /1 7 42 1 6 0.015 0.75 1 .96 20 0.16 4 2.2 0.6 0.4 1 1 /1 3 42 1 6 0.01 5 0.75 1 .96 20 0.2 6 1.5 0.15 0.1 0.25 1 /1 4 42 1 6 0.015 0.75 1 .96 20 0.15 6 2.4 0.4 0.1 0.4 1 /1 5 42 1 6 0.015 0.75 1 .96 20 0.11 6 3.6 0.4 0.2 0.5 1 /1 6 42 1 6 0.015 0.75 1 .96 20 0.1 6 9 0.6 0.25 0.7 1 /1 2 42 1 6 0.015 0.75 1 .96 20 0.2 8 0.5 0 0 0.25 1/1 1 42 1 6 0.015 0.75 1 .96 20 0.16 8 4.8 0.2 0.1 0.25 1/10 42 1 6 0.015 0.75 1 .96 20 0.12 8 7.5 0.25 0.1 0.3 8/19 46 1 4 0.015 0.97 2 .53 27 0.33 4 0 1 0.5 1 8/13 46 1 4 0.015 0.97 2 .53 27 0.08 5 0.6 0.35 1 1 8/14 46 1 4 0.015 0.97 2 .53 27 0.07 5 0.3 0.15 0.4 8/1 7 46 1 4 0.015 0.97 2 .53 27 0.15 6 1 0 0.15 0.35 8/1 8 46 1 4 0.015 0.97 2 .53 27 0.2 6 0.5 0 0.2 0.35 8/1 5 46 1 4 0.015 0.97 2 .53 27 0.12 8 1.4 0 0.25 0.5 8/1 5 46 1 4 0.015 0.97 2 .53 27 0.18 8 0.5 0 0.15 0.4 Table 4.1: Experimental Results er 4. Experiments Horizontal distance trom jet exit -10 -5 -2.5 0 5 10 20 30 Run No F U/u H/D S |T|- Ta Surface Temperature 8/1 1 1 7 0.59 2 3.2 46-14C 1 4 2 4 21 1 9 17.5 8/1 0 1 1 7 0.26 3 1.6 44-15C 3 3 24.5 23 22 8/2 1 1 7 0.33 4 3.75 44-14C 1 5 1 6 2 2 1 8 1 7 6/1 1 1 7 0.24 4 2.2 46-15C 18.5 2 9 23 20 19.5 6/3 1 1 7 0.18 4 1.7 46-15C 1 7 22.5 3 3 24 21 6/5 1 1 7 0.17 4 1.7 46-15C 21 24.5 3 3 22 20 6/2 1 1 7 0.13 4 1.9 46-15C 20 23 31.5 24 20 1 9 6/10 1 1 7 0.26 6 5.3 30-14C 1 6 1 7 1 5 8/3 1 1 7 0.22 6 4.3 44-14C 1 5 2 1 1 8 1 7 6/4 1 1 7 0.18 6 3.1 46-15C 1 5 18.5 2 5 18.5 1 8 6/1 1 1 1 7 0.18 6 3.2 30-14C 1 5 1 6 1 8 1 9 17.5 1 6 6/1 2 1 1 7 0.1 8 6 2.9 24-14C 14.5 15.5 17.5 16.5 1 5 8/5 1 1 7 0.2 8 5 44-14C 1 7 1 9 2 0 1 9 1 7 6/8 1 1 7 0.18 8 4.6 46-1 4C 1 5 1 8 2 1 20 1 6 6/9 1 1 7 0.18 8 4.6 30-14C 1 5 1 6 1 6 8/7 1 1 7 0.1 3 1 0 4.3 44-14C 1 9 2 1 19.5 1 8 1 7 1 75 0.19 5 2 49-15C 1 9 3 2 23 21 19.5 1 75 0.19 5 2.3 31-15C 1 7 1 8 2 2 1 9 1 8 1 7 1 75 0.06 1 2 3.8 49-15C 20 22 2 4 21 20 1 9 1 75 0.06 1 2 3.8 30-1 5C 17.5 1 8 1 9 18.5 1 8 1 8 1 7 2 53 0.19 2 2.2 34-15C 2 7 23.5 2 53 0.13 4 1.7 34-16C 17.5 2 7 21 .5 21 20 2 53 0.42 4 4.75 34-15C 2 3 1 9 1 8 2 53 0.15 4 2.5 34-15C 25.5 22.5 2 53 0.3 6 7.6 34-15C 17.5 16.5 2 53 0.26 6 4.7 34-15C 15.5 1 9 1 8 1 6 1 5.5 2 53 0.16 6 3.3 34-16C 21.5 1 9 1 8 16.5 2 53 0.23 8 7.9 34-15C 1 8 1 6 8/16 2 53 0.18 8 6 44-14C 1 8 1 9 1 6 2 53 0.1 2 8 3.5 34-1 5C 20.5 1 9 1 8 2 53 0.1 8 2.8 34-15C 1 6 1 8 21.8 1 9 1 8 2 53 0.06 8 2.7 34-1 5C 1 9 20 2 2 1 9 1 8 1 7 2 53 0.05 1 0 3.2 34-1 5C 1 8 2 1 1 9 1 8 Table 4.2: Experimental Results (Surface Temperatures) Chapter 4. Experiments 35 4.2.1 Restrictions and Other Problems The length of the upstream recirculation zone was in some cases oscillating up to 3-4D . However, in the majority of the runs this oscillation is negligible (less than ID). In other runs, where the oscillation is 2-3D, the average value of lu is used. The above problem, however, is indicative of the difficulties of clearly defining lu for each run. Although some basic trends were observed and measured, it is hard to give precise coefficients. Thus, lu/D is given with an accuracy of ±1. Another restricting factor was the width of the flume. Although it was narrow enough for good photographs and observations to be made, it wasn't wide enough to allow the unrestricted spreading of the flow. All the runs were designed so that the presence of the flume-walls did not affect the surface spreading width for about 20-30cm down-stream from the jet exit. For this reason the upstream length could not be greater than 10-15cm, otherwise the spreading flow would be affected by the walls. For the same rea-son, the downstream thickness of the spreading jet is not studied further than 20-30cm downstream from the jet exit. Within the uninfluenced zone, the thickness of the jet is practically constant, so that an average value for it can be estimated. The cases where this thickness is equal to the ambient depth are also noted. 4.2.2 Comments on the Results Regime Diagram The most characteristic parameter is the upstream recirculation zone, /„. In figure 4.12 the length of this zone is plotted (in its dimensionless form lu/D), for constant F = 1.17, in terms of H/D and U/u. Similar diagrams for I2/H, hs/H, hi/H are shown in fig.4.13. The three flow regimes are shown and are separated by two lines. Line OA separates regime 1 from regime 2 and line OB separates regime 2 from 3. Line OA is given by the Chapter 4. Experiments 36 H/D « 15.5 - 30U/u 0.05 < U/u < 0.5 (4.34) All points to the right of line OA have lu/D = 0. To the left of line OA, lu/D increases as U/u decreases. When there is no crossflow (U/u —* 0), lu/D —• oo. It is important to note that, since F is constant in this diagram, an decrease of U/u means that the crossflow becomes weaker; similarly, a decrease of H/D means that the ambient depth becomes smaller. Visually, the distinction between flow Regimes 1 and 2 is relatively easy to make (see fig.4.22). When the crossflow is very strong (i.e. U/u is big) there is no upstream length /„. As U/u decreases (for constant H/D) lu increases. The free surface in both regimes acts as a boundary that deflects the surfacing jet. The / u is determined by the upstream momentum of the reflected jet and the strength of the crossflow. Due to the (often rapidly) changing surfacing position of the jet, the distinction between the two Chapter 4. Experiments 10 H/D 6 10 0 10 0.3 8-1- A A A H/D A 0.1 A A A 0.2 0.1 0.15 Thickness of Upstream Edge A A A lj/H 0.2 0.15 0 A A A A 0.5 0.35 0.3 A 0.5 A 1 —i— 1.5 0.1 0.2 U/u 0.3 0.4 0.5 A 0 0.05 A A A 8 H/D 6 £0.09 A A A 0 0 o 4 • r l 0 1 8 A A A A 0.2 0.15 0.15^ 0 2 -fJO.61 A 0.3 0.5 n Size of Surface 0 0 Disturbance tu/H -+-0 0.1 0.2 u / u 0.3 0.4 0.5 " predicted by equation 2.18 (no crossflow) A Thickness of downstream 0.4 0.2 0.3 spreading jet 6 } A A A h H / r l 0.25 0.35 0.3 4 -j- A A A A 0.5 0.15 0.3 0.4 A 2 + A 1 1 0 0 0.1 0.2 0.3 0.4 0.5 U/u Figure 4.13: Characteristic lengths for F = 1.17 Chapter 4. Experiments 38 Regimes is more difficult to make when lu w 0 — ID. The transition from flow Regime 2 to 3 is described below. For a given F, U/u and H/D, a jet in a flow Regime 2 has an upstream length /„. As H/D decreases this length increases slightly. At the same time the surface disturbance increases too. When the height of the surface disturbance becomes larger than a certain value (hs/H greater than ~ 0.4), the free surface disturbance (induced by the jet) causes the deflected jet to spread more in a vertical direction (see fig.4.14). Part of the jet is still deflected upstream, but the horizontal momentum of the deflected jet, is much smaller than before. Thus, the upstream length is reduced and doesn't vary much in flow Regime 3. As the depth (or the U) decreases more, another variation of Regime 3 flows occurs (regime 3a). The deflected jet has so much downward momentum that it reaches the bottom of the flume and then spreads downwards (see fig.4.23). At the same time there is still a relatively small upstream zone which doesn't vary much. In the extreme case where the ambient depth becomes very small, the flow is 'fountain-like'. Effect of F on Regime Diagram Results for F = 2.53 are shown in fig.4.15. It is important to note that the limits of the three Regimes of flow depend on F. When u increases, the F increases too. Therefore, the flow Regime can be predicted by using a different diagram. This is illustrated with the following example. A jet with F = 1.17 (u = 0.45m/s, D = 0.015m) is discharged into a crossflow with U = 0.10ra/s and H/D = 6. The flow is represented by point A at fig.4.16. If u increases to u = 0.97m/s and U = 0.20m/s then F = 2.53 and U/u = 0.20. The new flow Regime is shown by point B on the same figure. Since the reference frame (i.e. F) has changed, a smaller upstream length is to be expected. This is, indeed, the case as fig.4.24 shows. Chapter 4. Experiments 39 Flow Regime 1 Row Regime 2 ' (D,u) Flow Regime 3 Flow Regime 3 a Figure 4.14: Flow patterns Chapter 4. Experiments 10 H/D 2 •• A 0 A 0 03** 0.6 0 A 0 A 1 0 10 T 8 - 3 H/D 6 a 0.17 0.1 A 0.25 0.2 u/u 0.3 ,0.28 0.15 A 0.15 A A 0.15 0.2 0.57 0.35 10 j 8 •• H/D 6 + A 0.5 0.1 0.2 u/u 0.3 A 0.4 A 0.5 0.4 A* A 0.35 A 0.35 A 1 0.1 0.2 u/u 0.3 0.4 0.4 Thickness of Upstream Edge l y H — i 0.5 Size of Surface disturbance h s/H 0.4 0.5 o predicted by equation 2.18 (no crossflow) Thickness of downstream spreading jet h/H d 40 0.5 10 j 8 •• H/D 6 4 2 0 A 1.4 A 0.5 A 0.5 0.1 0.2 A 0 U/u 0.3 0.4 Upstream Length 1 /D u — I 0.5 Figure 4.15: Characteristic lengths for F = 2.53 Chapter 4. Experiments 41 Figure 4.16: Effect of F on the flow regime An increase of u (and therefore F) could result in a change of the flow Regime. Such a case is shown by points C and D on fig.4.16. Point C (F = 1.17) represents a flow Regime 2, but an increase of u would result in a Regime 3 flow, shown by point D (F = 2.53). A main characteristic of the above diagrams, is that F has a more pronounced effect on the limit between flow Regime 2 and 3 than on the limit between flow Regime 1 and 2. So, depending on H/D, an increase in u could result in either a flow Regime 2 or 3. Dilut ion Diagram The near field dilution was estimated by dividing the excess temperature at the jet exit by the max. excess temperature at the surface. Obviously, due to the symmetry of the flow, maximum excess surface temperatures were measured along the jet center line. Measurements of surface temperatures along the width of the flume were also taken in order to study the spreading of the jet. The dilution of the jet is independent of the difference between Tj and Ta. This is shown in figure 4.17, where for two different flow regimes, the initial jet temperature was Chapter 4. Experiments 42 first set at 49° C and then at 30° C. The dilution in each regime was not affected by the change. On the same figure, a comparison of the two flow regimes can also be made. It is obvious that the dilution in regime 2 is greater than in regime 3. A similar comparison is made in figure 4.18. Three cases are shown. In the first two (Runs No. 6/10 and 6/11) everything is the same except u/U. A decrease of U reduces the dilution S. Comparing runs no. 6/11 and 6/9, it is evident that an increase of H (everything else being the same) increases the dilution S. Figure 4.19, shows the dilution S in terms of H/D and U/u, for two different F. The dilution is greatly reduced in Regime 2 flows. Once the crossflow becomes dominant, the dilution increases rapidly. This is also shown in fig.4.20, where for a given H/D the dilution is shown to increase as U increases. Naturally, for higher H/D, the dilution is greater. The ratio U/u, however, affects the dilution more than H/D. This can be seen by the drawn dilution contours. The above results can also be interpreted by observing that in Regime 2 flows there is an upstream recirculation zone. The recirculation results in higher surface temperatures, and therefore lower dilution. The existence of a recirculation zone and a downstream spreading jet, create a situation where the vertical jet discharges in a stratified ambient crossflow, the top layer of which is formed by the jet itself. As a result, the vertical entrainment length of the jet is reduced. Since the temperature difference between the jet and the ambient water doesn't affect the dilution, the reduction in dilution is due mainly to the available entrainment length, and not to the temperature difference. Chapter 4. Experiments H/D u/U S Tj Ta F o 4.7 5.2 2 49 15 1.75 • 4.7 5.2 2.3 31 15 1.75 • 12 16.8 3.8 49 15 1.75 12 16.8 3.8 30 15 1.75 Figure 4.17: Effect of T, - T* on Dilution S = (^Tr°j<V'<>ur" °  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 • 6 3.8 5.3 6/11 o 6 5.6 3.2 6/9 • 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 •• 4--3--2.5 - -2--1.8 -1.5 " 1 0.00 H/D=8 /« ^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 — 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 « 1.5,1.8,2.3 respectively. For F = 2.53 and H/D = 4,6,8 the Smin « 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 <C 0.94Fo (Equation 3.23) for stagnant water and <f« 1.6 (Equation 4.36) and u/U < 1.06-Fo (Equation 4.37) when an ambient crossflow is present. There are three possible flow regimes for shallow water jets in a crossflow: a crossflow dominated flow; a typical shallow water flow with an upstream recirculation zone; and a 'fountain-like' flow with big surface disturbance. By conducting experiments over a range of parameters, these regimes are defined in terms of H/D and U/u and F. The existence, or not, of an upstream edge distinguishes regime 1 from 2. The size of the surface disturbance distinguishes regime 2 from 3. Figure 4.21 shows graphically the influence of all three parameters in the determination of the flow regime. It is, thus, possible to predict the flow regime, and from it the dilution, of a jet discharged in shallow water with a crossflow. The dilution depends greatly on the flow regime. For a given F the minimum surface dilution is minimum in regime 3 and maximum in regime 1. The entrainment (and therefore the dilution) is independent of the initial temperature difference between the 53 jet and the ambient water. It depends mostly on the available entrainment length, which in flow regimes 2 and 3 is greatly reduced by the recirculation zone. The results of the present study show the extent of the influence of the ambient depth and the crossflow on the dilution of a buoyant jet. When designing an outlet in shallow water, certain conditions are imposed. There are minimum (or maximum) limits for the desired discharge, the concentration at a certain distance from the outlet etc. These must be satisfied under the existing ambient conditions of the area. The results of this study help understand the relative importance of the basic parameters. The smaller the diameter of the outfall is, the higher the jet exit velocity is (the discharge being constant). The jet velocity (and therefore the jet Froude number), however, should be small enough so that a flow regime 1 or 2 is achieved. Thus, a bigger jet diameter is preferable. The diameter, at the same time, cannot be too big because the decrease of H/D would lead to a regime 3 flow. Following these basic concepts and using figure 4.21, a set of D and u, satisfying the designed discharge and the imposed surface concentration levels, can be determined. If, given the ambient conditions, a single outfall cannot satisfy the imposed concentation levels, then a solution involving two (or more) ports should be investigated. The range and accuracy of the above results can be improved by using a bigger flume and relying more on temperature probes for measuring the various lengths (along with visual observations). It would then be possible to study more closely the surface spreading width, and also to extend the range of H/D. By using more detailed and accurate temperature measurements, it will be possible to determine dilution contours with greater accuracy. A study of the effect of a time-varying crossflow is of great practical interest. Such a case would reflect the reality of ocean currents. Similar experiments in shallow water with a current can be conducted for non-vertical buoyant jets, where the angle of the jet and the current can vary. 54 Appendix A Notat ion a : entrainment coefficient ap : plume entrainment coefficient ctj : jet entrainment coefficient B — ug>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 '• characteristic length for pure plumes (Equation 2. IM — ^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 £ 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— '• pure jet length scale (Equation 2.9) ZB = xj3 '• 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 


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