Horizontal round heated jets into calm uniform ambient

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1 Desalination 248 (2009) Horizontal round heated jets into calm uniform ambient Spyros N. Michas, Panos N. Papanicolaou* Hydromechanics and Environmental Engineering Laboratory, Department of Civil Engineering, University of Thessaly, Pedion Areos, Volos, Greece Received 29 October 2008; accepted 1 December 2008 Abstract In this article, we investigate horizontal, round, turbulent buoyant jets that discharge in a homogeneous calm ambient fluid. In a series of experiments performed, the initial jet Richardson numbers ranged from very small (jet-like) to around unity (plume-like), to include the full extent of applications. The mixing characteristics such as trajectories obtained with video imaging, turbulence properties measured with fast response thermistors, and dilution factors are evaluated. The trajectories of buoyancy-conserving horizontal jets bend faster than those observed in heated jets. The flow resembles a jet-like behavior in the horizontal regime, while the mean and turbulent temperature profiles become asymmetrical in the transition and the vertical regime. The maximum intensity of turbulence doubles in the vertical regime if compared to that in the horizontal part of the flow. Minimum dilutions obtained from experiments at the jet axis are compared to the results of a one-dimensional numerical model of a heated horizontal jet, where the loss of buoyancy flux has been accounted for. Keywords: Horizontal; Buoyant jet; Heated; Uniform ambient; Turbulence measurements; Trajectory 1. Introduction Liquid wastes such as hot water from cooling of thermal energy power plants, as well as the effluent from wastewater treatment plants, are usually disposed by means of submerged outfalls in the coast near the plant site. Heavier saltwater from desalination plants can also be discharged from a floating pipeline, at the *Corresponding author. Presented at the Conference on Protection and Restoration of the Environment IX, Kefalonia Greece, June 30 July 3, 2008 surface of the receptor. The regulations regarding disposal of thermal or denser discharges into coastal waters are related to the volumetric discharge rate and the subsequent dilution of the wastewater. Thus, the local seasonal temperature and salinity of coastal waters must not exceed specific values (usually 2 48C or 2 3 g/l). The initial dilution of the wastes obtained when they reach the free surface or the bottom of the receptor is crucial with respect to the environmental effects of the effluent in the vicinity of the discharge area. For a specific receptor depth, use of horizontal jets results in longer /09/$ See front matter 2009 Published by Elsevier B.V. doi: /j.desal

2 804 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) trajectories and subsequently in entrainment and mixing of greater amounts of ambient fluid, if compared to outfalls with vertical risers. Despite the fact that horizontal jets are currently in use worldwide, so far little research has been carried out to study their properties, although a great number of experiments on vertical buoyant jets have been reported [1 5]. Moreover, the majority of research on horizontal buoyant jets [6 8] is basically focused on the trajectory rather than on the turbulence properties of the flow. Modeling of horizontal jets is usually implemented assuming that their cross-section normal to the axis is circular [6, 9 11]. Also, the flow parameters (model constants ) used such as the entrainment coefficient a e and the spreading ratio l of the concentration to the velocity profile are usually those measured in vertical buoyant jets. In this article, a set of experiments performed is employed to investigate round turbulent buoyant jets that discharge horizontally in a motionless, uniform, deep body of water. The scope of the present investigation is to evaluate their turbulence and mixing characteristics, such as their trajectory and distributions of the time-averaged excess temperature and turbulence intensity. Subsequently, the measured dilutions will be compared to the results from a one-dimensional numerical model of a heated horizontal jet Theoretical background A horizontal jet of density r 0, sketched in Fig. 1, comes out of a nozzle with uniform velocity U and mixes with ambient fluid of uniform density r a >r 0. The ambient fluid is assumed to extend to infinity, thus introducing negligible boundary effects to the jet flow field. The initial jet parameters are the specific (per unit mass) mass, horizontal momentum and buoyancy fluxes defined as Q ¼ pd2 4 U; M ¼ QU and B ¼ g0 o Q; go 0 ¼ r a r o gy r a ð1þ respectively. D is the jet nozzle diameter, g the gravitational acceleration and g 0 o the effective gravity that will subsequently produce vertical momentum flux. From the initial jet parameters, one may define two characteristic length scales [12] as s T a, a u z v s T o, o x θ r Fig. 1. Definition sketch.

3 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) l Q ¼ Q and l M 1=2 M ¼ M 3=4 B ; 1=2 ð2þ the ratio of which is defined to be the initial jet Richardson number R o R o ¼ l Q ¼ p pffiffiffiffiffiffiffiffi 1=4 g 0 o D l M 4 U ¼ p 1=4 Fo 1 4 : ð3þ F o is the initial densimetric Froude number of the jet. The density difference between jet and ambient fluid may be due to the relative temperature or salinity of the two fluids. In a heated jet, temperature differences produce the density deficiency that is responsible for the initial jet specific buoyancy flux. The initial jet heat flux Y=Q(T o T a ), where T o is the jet temperature and T a the uniform ambient fluid temperature, is conserved. The dilution S at a point of the jet flow field is defined to be the ratio of the initial temperature difference between jet and ambient fluid, to the local time-averaged excess (above ambient) temperature S ¼ T o T a T T a : ð4þ The normalized jet trajectory [11] can be plotted as z/l M versus x/l M, x and z being the horizontal and vertical distances from the nozzle, of a point located on the jet axis. The normalized dilution S c along the axis S c F o ¼ 1 F o T o T a T c T a ð5þ is a function of the normalized elevation z/l M, T c being the (maximum) time-averaged excess temperature. The local, over a jet cross-section A normal to the axis as shown in Fig. 1, volume flux m(s), momentum m(s) and buoyancy b(s) specific fluxes due to the mean flow are Z bðsþ ¼ Z mðsþ ¼ A Z mðsþ ¼ A A uðr; sþda u 2 ðr; sþda ð6þ ð7þ g r a rðr; sþ r a uðr; sþda ð8þ respectively. Coordinates s and r are the distance along the axis from the nozzle and the radial distance from the jet axis, respectively. Assuming that distributions of the time-averaged velocity u(r,s) that is parallel to the axis and the density deficiency Dr(r,s) are Gaussian, and also that the jet cross-section is circular, the integrated over the cross-section time-averaged equations of motion can be written as [6, 10, 13] dm ds ¼ 2 p ffiffiffiffiffi 2p ae m 1=2 ð9þ dm ds ¼ 1 þ l2 2 mb m sin y dy ds ¼ 1 þ l2 mb 2 m 2 cos y db ds ¼ 0 ð10þ ð11þ ð12þ where a e is the local entrainment coefficient proposed by Morton et al. [1], l =b e /b u is the ratio of density deficiency width over velocity width, and y is the angle between the jet axis and horizontal. Two more equations needed to determine the jet trajectory are dx ds ¼ cos y ð13þ

4 806 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) and dz ds ¼ sin y ð14þ The system of six nonlinear differential equations is an initial value problem, and it can be solved numerically (using for example a Runge- Kutta routine), once the entrainment coefficient a e and the jet width ratio l are estimated properly. 2. Experiments Experiments were performed in a transparent orthogonal tank with dimensions 1.00 m 0.80 m and 0.70 m deep, equipped with a peripheral overflow to remove excess water. A set of three circular jet nozzles of diameters 5, 10 and 15 mm could be mounted to the properly designed end of a 20 cm long, hollow PVC cylinder (plenum) of internal diameter 4 cm, with a 7 cm long honeycomb section positioned at midlength. The insulated against heat loss water supply pipe was mounted on the other end of the PVC cylinder, in line with a PT100 temperature sensor for recording of the jet fluid temperature. The horizontal jet plenum was supported on the short side transparent tank wall with a special mount that did not allow any leaks from the motionless tank fluid. It was positioned 10 cm below the free surface for the heavier saltwater jets, or 10 cm above the tank bottom for the hot water jets, ensuring that the tank boundaries had minimal effects on the flow. A water heater served as constant head tank since it was pressurized with air at 2 atm, and a pressurereducing valve installed in line maintained constant head conditions throughout the experiment. For the saltwater jet experiments, the water heater has only served as a pressurized constant head tank. The jet flow rate at the nozzle was measured using a simple, ball-type flowmeters that allowed the precise setting and control of the desired discharge. The experimental procedure was similar for both thermal and saltwater jets. For the saltwater jets, the solution was initially prepared in a separate tank and when ready it was transferred to the pressurized tank. For the heated buoyant jets, a bypass pipe located upstream of the inlet to the jet plenum was used to dispose some amount of hot water before each experiment until the temperature sensor PT100 displayed constant water temperature. Then the hot water was let to fill the jet plenum and measurements were taken after about 1 min, the time that was necessary for establishment of steady-state. During the experiment, a series of still photographs and video were taken simultaneously to be used for the subsequent trajectory analysis. An adequate number of experiments [14] in the full range of initial jet Richardson numbers from 0.02 to about 1 (corresponding to asymptotically pure jets and plumes respectively) was executed, for Reynolds numbers from 660 to 13,000. The range of the initial conditions of the flow is shown in Table 1. Jet trajectories Table 1 Range of the initial conditions for the saltwater and heated jet experiments Type # Runs r a (g/l) r o (g/l) R e R o saltwater trajectory ~ ~ ~ ~0.799 Type # Runs T a (8C) T o (8C) R e R o Hot water trajectory Hot water temperature ~ ~ ~ ~0.842

5 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) were determined from shadowgraph videos using a double grid based technique to minimize the error in the measurement of the linear jet characteristics. Two series of jet images were obtained, the first using saltwater jets into calm homogeneous freshwater (buoyancy preserving) and the second using hot water jets into uniform calm cold water ambient (buoyancy is a function of the dilution). The photographs were corrected for optical errors and subsequently the jet centerline coordinates were evaluated, from which the normalized trajectories were determined. Apart from the trajectories, detailed temperature measurements were carried out using an array of seven, equally spaced, fast response NTC-type thermistors. The array was used to measure either the vertical temperature distribution in the horizontal regime of the jet, or the horizontal temperature distribution in the vertical regime of it, as shown in Fig. 2. The ambient fluid temperature ranged between 198C and 268C while the jet temperature varied between 508C and 658C. The analog (voltage) temperature data were amplified then converted into digital form by means of a 16-bit data acquisition A/D converter and stored in the computer. The array of thermistors was mounted on a threedimensional traverse that enabled movement in any direction, while its position relative to the jet nozzle coordinates could be read from calibrated rulers. The sampling rate at each location was 80 Hz (temperature samples per second) for each one of the thermistors, while sampling was 40 s long. Then the array of thermistors was moved to a different location of the crosssection, and the sampling procedure was repeated until measurements were taken over the whole jet vertical or horizontal section on the plane of symmetry. Sampling time was limited to 40 s to ensure that the mixed fluid interface where the temperature was greater than that of the ambient T a did not reach the elevation of measurement. After a set of measurements was completed, the ambient, tank water was stirred to obtain homogeneous temperature and a new experiment was run once it became motionless. From the temperature time-series records, we computed the distributions of the local turbulence characteristics such as the mean temperature, turbulent intensity as well as higher moments, intermittency factor, probability density distribution functions, and turbulence spectra. The dilution was computed in characteristic jet locations along the axis. 3. Results 3.1. Trajectories The data collected from each experiment (optical material) were processed in a standard manner to derive the trajectories. The procedure interface z x Fig. 2. Temperature measurement in a heated horizontal jet using an array of fast response thermistors for horizontal temperature profile (left), and verical temperature profile (right).

6 808 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) included removal of color, adjustment of the grayscale-level distribution to enhance the jet contrast, rotation, layer alignment, perspective correction, and barrel distortion removal (only when necessary). Subsequently, the corrected photographs were inserted in vector CAD software, where they were properly scaled to the dimensions of the jet midplane (of symmetry). Then the jet visual boundaries defined in every set of photographs for each experiment were superimposed, and an average boundary was drawn on each side. Using a simple in-house built algorithm, the jet-axis trajectory was determined as the mid-distance from the average visual boundaries. Additionally, the video recordings helped to refine the process, by improving the selection of the visual boundaries. The jet trajectories obtained were normalized using the characteristic length scale l M. The procedure described above was applied to describe the trajectory in both thermal and saltwater jets. Normalized trajectories of round horizontal jets are shown in Fig. 3 for saltwater and hot water jets at different initial Richardson numbers. The saltwater jet trajectories have been plotted upside-down for comparison with those of hot water ones. In Fig. 3, one may note that the normalized trajectories of saltwater jets are quite similar to the hot water ones, in the sense that they become vertical faster for high initial jet Richardson numbers. Trajectories of jets with low initial Richardson numbers persist traveling longer distances, before they bend over to follow vertical motion. Normalized trajectories from earlier experiments [6 8] performed in buoyancy-conserving jets seem to collapse in a narrow regime, which corresponds to the lower values of R o of the present investigation. This occurs because the initial jet Richardson numbers used in earlier investigations were quite low (Fan [6], R o = ; Anwar [7], R o = ; and Davidson [8], R o = ) D=0.50, Ro=0.039 D=0.50, Ro=0.054 D=1.00, Ro=0.084 D=1.00, Ro=0.159 D=1.00, Ro=0.198 D=1.50, Ro=0.311 D=1.50, Ro=0.448 D=1.00, Ro= D=0.50, Ro=0.040 D=0.50, Ro=0.056 D=1.00, Ro=0.099 D=1.00, Ro=0.238 D=1.50, Ro=0.271 D=1.50, Ro=0.430 D=1.50, Ro=0.495 D=1.50, Ro=0.625 D=1.50, Ro=0.846 z/l M 8 z/l M x/l M x/l M Fig. 3. Normalized trajectories of horizontal round saltwater (left) and thermal (right) jets into fresh water, at various initial Richardson numbers.

7 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) To our knowledge, there are no data available regarding trajectories of horizontal heated jets. One major difference in the trajectories of heated jets if compared to those of saltwater jets (with similar kinematic and buoyancy characteristics) is that the former are longer. This means that a heated jet travels longer horizontal distances x/l M than a saltwater one, to reach a certain elevation z/l M. This was expected because hot water jets do not maintain their initial buoyancy flux due to the variation of the thermal expansion coefficient of water with temperature. Once the warmer jet fluid mixes with cold ambient water its temperature is reduced, and so is the volume expansion coefficient. Therefore the initial jet specific buoyancy flux is reduced, resulting gradually in lower vertical momentum flux along the jet trajectory, if compared to that of the buoyancy-conserving saltwater jets. Reduction in vertical momentum results in trajectories with longer horizontal coordinates for the same vertical elevation of the buoyant jet axis. In both saltwater and hot water jets, one may clearly note that some trajectories start at an angle rather than initially being horizontal. This occurred at relatively high initial Richardson numbers (buoyancy flux is dominant), regardless of the fact that the initial Reynolds numbers some times were high (exceeded 2000). There was a long laminar region before the transition to turbulence, beyond which the turbulent jet appeared to be initially at an angle. This long initial laminar region was generally not observed in the heated jets of comparable initial Richardson numbers as it is shown in Fig. 4, since the reduced hot jet fluid kinematic viscosity resulted in higher Reynolds numbers. Thus, hot water jets became turbulent immediately downstream from the nozzle. In Fig. 4, one may clearly observe that for similar jet exit velocities, transition to turbulence of a 5 mm hot water jet has occurred at around four jet diameters from the nozzle, while in a saltwater jet at twice this distance, although their initial inertial and buoyancy characteristics were similar Turbulence properties and dilution The turbulence properties have been analyzed from the time records of temperature measurements taken during the experiment at each point of measurement, as described in the Fig. 4. Breakdown to turbulence of a 5 mm round jet downstream from the nozzle; heated jet at R o = (left), and saltwater jet at R o = after a significant length of laminar flow (right).

8 810 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) previous chapter. The first four moments of the temperature time series were computed. From the time-averaged (mean) excess (above ambient) temperature profiles, we computed the jet width b e, that is the radial distance from the axis (point with the highest mean temperature) where the mean excess temperature is 1/e (e being the base of Neperian logarithms) times that at the axis by fitting a Gaussian curve to the data. The Gaussian was primarily fit to the data corresponding to the outer jet boundary. The secondary plume of hot chunks of slowly rising to the surface fluid, observed in the inner jet boundary, affected the symmetry of time-averaged temperature distribution. The normalized time-averaged excess temperature and turbulence intensity profiles can be plotted versus the normalized distance from the jet axis with width b e. Alternatively, the statistical values can be normalized by the distance s of the measurement cross-section from the jet origin. Apart from the abovementioned moments, other statistical values can be computed, namely minimum and maximum values, hot/cold interface frequencies and intermittency factors. In a horizontal turbulent buoyant jet, we have noted three separate flow regimes. The one close to the jet nozzle, characterized as the horizontal part of the trajectory, is limited to the normalized distance x/l M <1.50, where the flow is essentially driven by the initial horizontal jet momentum (jet-like). The flow regime where z/l M >1.50 can be characterized as the bent over regime of the flow (mainly for buoyant jets with low R o ), that is the result of the buoyancy force acting on the jet (plume-like). The regime in between (1.50<x/l M <5) is a transition from jet-like to plume-like flow. In Fig. 5 the normalized time-averaged excess (above ambient) temperature T=T c and pffiffiffiffiffiffi intensity of turbulence T 02 =T c in round heated jets are plotted versus r/b e for x/l M <1.50. One x/lm:0.64 x/lm:0.98 x/lm:1.06 x/lm:1.09 x/lm: x/lm:0.64 x/lm:0.98 x/lm:1.06 x/lm:1.09 x/lm: r/b e 0.00 r/b e T/T C T 2 /T C Fig. 5. Normalized vertical mean excess temperature (left) and turbulent intensity (right) profiles in horizontal round jets for x/l M <1.50.

9 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) may clearly note that both mean and turbulent temperature profiles are symmetrical with respect to r/b e, while a double peak is apparent in the second one around r/b e = 1. The profiles are very similar to those reported earlier [3] for round vertical heated jets. Therefore, the flow can be characterized as jet-like when x/l M <1.50, as in the vertical heated jets. When x/l M >1.50 (Fig. 6), the vertical timeaveraged normalized excess temperature profile does not remain symmetric any longer, since the temperature distribution extends to r/b e»5 in the inner side of the jet. The normalized turbulent intensity profile is skewed, while the peak close to the outer side of the jet is enhanced and the one toward the inner side seems to disappear. The asymmetries observed for x/l M >1.5 are maintained in the horizontal profiles of mean excess temperature and turbulent intensity of round jets are plotted versus r/b e, as shown in Fig. 7. One may clearly note that they are not symmetric with respect to the axis, while the normalized intensity of turbulence shows a maximum of 0.45 around r/b e = 0.50 that is twice as high as the one measured in the horizontal jet-like regime. Papanicolaou and List [3] have also reported higher turbulence intensity (0.40) around the axis in vertical heated plumes. In both Figs 6 and 7, we observe the asymmetry in the time-averaged normalized excess temperature and in turbulence intensity. The peak of the latter though is not coincident with the peak mean temperature. While modeling horizontal jets, we assume that the axis is the tangent to the maximum mean velocity. From Figs 6 and 7, one may wonder whether the coordinate of maximum mean velocity coincides with the maximum temperature at each cross-section, or whether the peak mean velocity and turbulent velocity appear on the jet axis. Since there is lack of experimental data regarding the velocity distribution, one may question what may be defined as jet axis on the plane of symmetry. It x/lm:1.39 x/lm:1.82 x/lm:2.06 x/lm:2.18 x/lm: x/lm:1.39 x/lm:1.82 x/lm:2.06 x/lm:2.18 x/lm: r/b e 0.00 r/b e T/T C T 2 /T C Fig. 6. Normalized vertical mean excess temperature (left) and turbulent intensity (right) profiles in horizontal round jets for x/l M >1.50.

10 812 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) T/TC z/lm:1.34 z/lm:1.91 z/lm:2.26 z/lm:3.93 z/lm:4.38 z/lm:5.84 z/lm:6.47 z/lm:9.02 z/lm:12.66 z/lm:14.69 T 2 /TC z/lm:1.34 z/lm:1.91 z/lm:2.26 z/lm:3.93 z/lm:4.38 z/lm:5.84 z/lm:6.47 z/lm:9.02 z/lm:12.66 z/lm: r/b e r/b e Fig. 7. Normalized horizontal mean excess temperature (left) and turbulent intensity (right) profiles in horizontal round jets for z/l M >2. could be the set of points that correspond to the maximum mean excess temperature, as well as the set of points of the maximum turbulent temperature One-dimensional modeling normalized dilution The set of six nonlinear equations (9) to (14) is an initial value problem. It was solved for a heated jet using a fourth order Runge-Kutta routine in the following manner. The initial conditions used werethespecificvolumeq, momentumm, and buoyancy B fluxes at the source, while the coordinate system was assumed to be located at the center of the nozzle (x o =0,z o =0) and the jet was assumed to be horizontal (y o = 0). The initial jet temperature was considered to be T o =608C and the uniform calm ambient temperature T a = 208C. The entrainment coefficient was assumed to vary [4] between a j =0.0545anda p = according to equation a e ¼ a j a j a p RðzÞ ¼ mb1=2 ð1:10mþ 5=4 RðsÞ 2 ; R p ð15þ proposed by List [12], where R(s) is the local buoyant jet Richardson number computed from the local jet parameters m(z), b(s), and m(s), and R p = 0.56 is the plume Richardson number evaluated in [4] and adjusted according to [5, 13]. The jet width ratio l = b e /b u = 1.20 is assumed to be a constant [4]. To account for the buoyancy flux loss with temperature, we have used a procedure similar to the one proposed by [15]. The equation of heat flux conservation (neglecting the heat transported by turbulence) may be written as QðT o T a Þ&mðsÞðT T a Þ ð19þ

11 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) where T is the average, over the jet crosssection temperature. Since an average over the cross-section density can be expressed as a function of the average temperature rðsþ ¼fðTðsÞÞ, the local specific buoyancy flux at s+ds can be computed approximately from equation bðs þ dsþ&mðsþ rðsþ r a r o g: ð20þ In Fig. 8, the normalized average dilution computed by the procedure described above is plotted versus the normalized elevation z/l M from the nozzle for initial Richardson numbers in the range In the same figure, we have plotted the normalized average dilution computed using Visjet software (Computer Ocean Outfall Modeling System) by Lee and Wang from the University of Hong Kong for fresh water jets into salt water (buoyancy is conserved) and of initial Richardson numbers and One may note that the average dilutions computed by both schemes are not any different. In the same figure, we have plotted the normalized centerline (minimum) dilution from the temperature measurements. The computed normalized dilution is higher that the centerline (minimum) dilution measured in heated horizontal jets, as expected, since the average over the jet cross-section temperature is lower than the maximum time-averaged temperature at the jet axis. The experimental observations though follow the asymptotic behavior predicted by the model for the lower values of z/l M (horizontal part of the flow) as well as for the higher values of z/l M (vertical part of the flow). 4. Conclusions The experimental investigation of horizontal turbulent buoyant jets coming out of round nozzles has led to the following conclusions: 1. The trajectories of round, buoyancy-conserving horizontal jets bend faster (at shorter horizontal distances) than those observed in heated jets, because of the buoyancy flux decay in the latter ones. Data from earlier S/F o Visjet Ro = Ro = Ro = Ro = Ro = Ro = Vertical Horizontal z/l M Fig. 8. Normalized centerline dilution (open squares from vertical and solid squares from horizontal temperature profiles). Computed average dilution using Visjet software (solid circles) at high F o, and the model proposed for heated jets (lines) at different R o.

12 814 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) experiments on buoyancy conserving jets are limited to R o <0.235, and congruent with the findings of the present investigation. 2. The flow behaves as a simple jet (jet-like) in the horizontal part of the trajectory, namely for x/l M <1.50, where the horizontal momentum flux is dominant. The distributions of the mean excess and turbulent temperature are symmetric with respect to the axis, similar to those reported for round vertical heated jets. 3. The flow is in transition from jets to plumes when 1<x/l M <5. It is practically considered to be plume-like (vertical momentum flux is dominant) when x/l M >5 or z/l M >2. The distributions of the mean excess and turbulent temperature are not symmetric any longer, and the two peaks characteristic of the turbulent temperature in the jet-like regime coalesce into a single one in the plume-like regime. The intensity of turbulence is doubled if compared to that in jets. The maximum intensity of turbulence appears off the jet axis, around r/b e = This can raise a question regarding the definition of the jet axis, which is usually considered to be the point of the jet section on the plane of symmetry, where the time-averaged excess temperature is the highest. 4. The normalized measured (minimum) dilution along the jet axis seems to follow the same asymptotic power laws for low and high dimensionless elevations from the nozzle, with the ones computed using commercial software and the one-dimensional model of a heated jet presented in this article. As expected, the computed, average over the cross-section dilution is higher if compared to that measured at the jet axis. Acknowledgment The present work was supported by the program EPEAEK II HERAKLEITOS under Grant UTH , jointly funded by the European Social Fund and National Resources. The assistance of Mr. Demetris Karaberopoulos, Mr. Elias Pappas, and Mr. Manolis Lasithiotakis is gratefully appreciated. Nomenclature List of symbols A jet cross-sectional area b u 1/e velocity width, lateral distance from axis where u = 0.37u c b e 1/e temperature width, lateral distance from axis where T ¼ 0:37T c B specific buoyancy flux at the source D jet diameter F o densimetric Froude number at the source g gravitational acceleration g 0 effective gravitational acceleration l Q ; l M characteristic length scales M jet specific momentum flux at the source m local specific momentum flux at distance s along jet axis Q specific mass (volume) flux at the source R o jet Richardson number at the source R(s) local jet Richardson number R p asymptotic plume Richardson number r coordinate perpendicular to the jet axis Re Reynolds number at the source s distance along the jet axis from the origin S dilution T average temperature T time-averaged temperature U jet exit velocity u time-averaged axial velocity v time-averaged radial velocity x, z coordinates Y heat flux at the source Greek Symbols a e entrainment coefficient a j jet entrainment coefficient a p plume entrainment coefficient b local specific buoyancy flux at distance s y angle of jet axis at distance s with respect to horizontal l (= b e /b u ) temperature to velocity 1/e width ratio

13 S.N. Michas, P.N. Papanicolaou / Desalination 248 (2009) m local specific mass flux or volume flux at distance s along jet axis r density Superscripts and Subscripts ðþ 0 deviation from a time averaged mean ðþ time-averaged values c centerline value o initial values at jet origin a values of the ambient fluid variables References [1] B.R Morton, G.I. Taylor, and J.S. Turner. Proc. Roy. Soc. London, A 234 (1956) [2] N.E. Kotsovinos. A study of the entrainment and turbulence in a plane buoyant jet, Report No. KH-R-32, California Institute of Technology, Pasadena, California, [3] P.N. Papanicolaou and E.J. List. Int. J. Heat Mass Transfer 30 (1987) [4] P.N. Papanicolaou and E.J. List. J. Fluid Mech. 195 (1988) [5] H. Wang and A.W.-K. Law. J. Fluid Mech. 459 (2002) [6] L.-N. Fan. Turbulent buoyant jets into stratified or flowing ambient fluids, Report No. KH-R-15, W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California, [7] H.O. Anwar. Proc. ASCE, J. Hyd. Div. 95(4) (1969) [8] I.R. Wood, R.G. Bell, and D.L. Wilkinson. Ocean disposal of wastewater, Advanced Series on Ocean Engineering, Vol. 8, 1993, p. 44, Figure 4.12, from M.J. Davidson, [9] G. Abraham. Proc. ASCE, J. Hyd. Div. 91(4) (1965) [10] L.-N. Fan and N.H. Brooks. Proc. ASCE, J. Hyd. Div. 92(2) (1966) [11] G.H. Jirka. Env. Fluid Mech. 4 (2004) [12] H.B., Fischer, E.J. List, R.C.Y. Koh, J. Imberger, and N.H. Brooks. Mixing in inland and coastal waters, Academic Press, London, [13] P.N. Papanicolaou, I.G. Papakonstantis, and G.C. Christodoulou. J. Fluid Mech. 614 (2008) [14] S.N. Michas, Experimental investigation of horizontal round and non-axisymmetric buoyant jets in a uniform calm ambient, (In Greek), Ph.D. Thesis, Hydromechanics and Environmental Engineering Laboratory, Department of Civil Engineering, University of Thessaly, [15] P.N. Papanicolaou and T.J. Kokkalis. Int. J. Heat Mass Trans. 51 (2008)