Vertical Round Buoyant Jets and Fountains in a Linearly, Density-Stratiﬁed Fluid

: Turbulent round buoyant jets and fountains issuing vertically into a linearly density-stratiﬁed calm ambient have been investigated in a series of laboratory experiments. The terminal (steady-state) height of rise and the mean elevation of subsequent horizontal spreading have been measured in positively buoyant jets (at source level), including pure momentum jets and plumes, as well in momentum-driven negatively buoyant jets (fountains). The results from experiments conﬁrmed the asymptotic analysis that was based on dimensional arguments. The normalized terminal height and spreading elevation with respect to the elevation of injection of momentum-driven (positively) buoyant jets and fountains attained the same asymptotic values. The numerical results from the solution of entrainment equations, using an improved entrainment coe ﬃ cient function, conﬁrmed the results related to buoyancy dominant ﬂows (plumes), while their predictions in momentum-driven ﬂows were quite low if compared to measurements.


Introduction
A round buoyant jet is the flow out of a pipe or circular nozzle that discharges into a quiescent or moving volume of fluid of different density. A vertical jet is positively buoyant if the direction of flow initially is that of the effective gravity, due to the density difference between jet and ambient fluid at the elevation of discharge; otherwise, it is called a negatively buoyant jet or fountain. A positively buoyant jet rises and mixes up to the surface of a surrounding motionless fluid of higher uniform density. A (momentum-dominated) fountain rises and mixes in an ambient of uniform, lower density to an elevation where the momentum flux of the mixed fluid vanishes. Then, the flow reverses and returns to the elevation of injection.
In a density-stratified, stable, calm ambient, a vertical jet that is positively buoyant at the elevation of injection entrains and mixes with the surrounding fluid up to an elevation where the average jet density is equal to that of the ambient fluid (neutral buoyancy). From that point on, the vertical kinematic momentum flux (inertia) of the jet will force it to a higher elevation, while ambient fluid still entrains and mixes with jet fluid, until the vertical momentum flux vanishes. Then, the jet flow reverses and returns to the elevation of neutral buoyancy, where it spreads horizontally. A vertical fountain (jet negatively buoyant at injection elevation) in a stable, density-stratified ambient will rise and mix with ambient up to the elevation where its kinematic momentum flux vanishes, under the influence of reverse buoyancy. Then, the direction of motion reverses and the fluid returns to an elevation of neutral density where it spreads horizontally. The maximum steady-state elevation attained by a vertical jet that is initially positively buoyant or a fountain in a density-stratified ambient is defined as "the terminal height of rise" (THR), and the mean elevation of neutral density where it spreads horizontally is defined as the "spreading height" (SH).
Atmosphere and oceans are both density-stratified. When studying the initial dilution of turbulent buoyant jets which discharge in the atmosphere or in a body of water, the ambient is assumed to be motionless, since the timescale of the initial evolution of the flow is rather short. Subsequently, it may be considered as a moving body of fluid for the study of the dispersion in the horizontal spreading field which is formed.
Following the pioneering work of Morton, Taylor, and Turner [1], the mechanics of positively buoyant jets and fountains in a calm, linearly density-stratified ambient has been the subject of theoretical and experimental investigations for several decades. Vertical fountains in a uniform ambient fluid have also been studied in experiments [2,3]. Interest has been focused on the terminal rise height reached by the flow, as well as the mean spreading elevation where the mixed fluid subsequently expands horizontally. Direct applications of the results of such studies are related to the design of outfalls for the disposal of treated municipal wastewater and brine from desalination plants into the sea, as well as to the dispersion of smoke from factory stacks in the atmosphere. Further applications are the study of geophysical flows such as the motion of smoke plumes above fires, volcano eruptions, black smokers in the deep ocean, and magma chambers. Note that, after the eruptions of the Eyjafjallajökull volcano in Iceland (April-May 2010), ash and gases spread at an elevation around 7000 m in a very large area, causing great problems in the aviation industry for a long time.
Earlier studies of round buoyant jets and fountains in a linearly density-stratified ambient that focused on the terminal height of rise (THR) are summarized in [4]. In cases where the initial momentum of the flow dominates the buoyancy force at the injection point, the flow is characterized as momentum-driven or as jet-like. If the initial buoyancy force dominates the momentum at the injection point, the flow is characterized as buoyancy-driven or as a plume. The fountain flow is jet-like since it is a consequence of the initial jet momentum. Fischer et al. [4] using the experiments in [5,6] proposed that the normalized THR c j (to be defined in Section 2) for positively buoyant, momentum-driven flows (jets) is c j = 3.80, as also presented in [7]. In (positively buoyant) plumes, from the experiments in [1,8,9], [4] proposed the same constant c p ≈ 3.80 (c p is defined in Section 2) for the normalized THR, while [7] recommended a constant equal to 5.
Experiments on positively buoyant, vertical, round jets that discharge into a linearly density-stratified fluid were implemented by [10]. The authors classified the flow into two asymptotic cases, jet-like flows and plumes, on the basis of the dimensionless parameter MN/B, where M and B are the initial specific (per unit mass) momentum and buoyancy fluxes and N is the buoyancy frequency of the density stratification to be defined in Section 2. Experiments in the regime 0.80 < MN/B < 2.40 regarding the THR were also reported [11] from measurements. Similar experiments were reported for a wide range of MN/B by [12,13], the latter [13] including data for the mean spreading elevation of the buoyant jet. Experiments were reported by [14] for pure momentum-driven and negatively buoyant jets (fountains) in a linearly, density-stratified ambient. The authors [14] also solved the steady-state equations of motion numerically and found the predictions to be congruent with their experiments.
Recently, the spreading height of buoyant jets in a linearly density-stratified ambient was measured [15], concluding that the normalized spreading height is 1.53 when MN/B > 7 and 2.775 when MN/B < 7. Measurements of the velocity field of buoyant jets in linearly stratified ambient were performed [16] using particle image velocimetry (PIV). The terminal rise height in plume-like flows was measured, along with the turbulence parameters that were evaluated for modeling.
The model proposed in [1] was extended [17] to compute the THR and SH in geophysical flows on Earth, as well as on other planets. An attempt was made to connect the rise height to a plume length scale (L p defined in next chapter), where both are normalized by the plume diameter, via the coefficient of entrainment that is variable in the range 0.05 to 0.16. This normalization does not comply with dimensional arguments presented in the next section. The scope of the present investigation is to provide a systematic new set of data regarding the THR and SH of buoyant jets and fountains in a linearly density-stratified ambient and compare it to earlier experiments. Following the dimensional arguments (see next section), the experiments extend to a range of the dimensionless parameter MN/B that includes the two asymptotic regimes of initially jet-like flows and plumes, as well as the transition in between. Emphasis is given to the spreading height (SH) that is a very important parameter in the dispersion of buoyant jets in a stratified ambient, where available data are limited. An integral model using an improved entrainment coefficient function derived in Appendix A from the entrainment equations is applied, in an effort to match the results obtained from the experiments. The entrainment function obtains a reduced coefficient if compared to that of a simple jet, when the buoyancy acts against the movement, in accordance with the results in [18,19].

Dimensional Analysis-Review of Earlier Experiments
A round vertical (positively) buoyant jet of density ρ o and uniform velocity W discharges vertically into an infinite volume of calm, linearly density-stratified fluid. The initial jet volume flux Q, specific (per unit mass) momentum M, and buoyancy B fluxes with dimensions L 3 /T, L 4 /T 2 , and L 4 /T 3 , respectively, are calculated as follows: where ρ a is the density of the ambient fluid at the jet nozzle elevation. The buoyancy frequency N of the linearly density-stratified ambient with dimensions T −1 is computed from Let us assume fully developed turbulent flow and adopt the Boussinesq approximation (density differences do not exceed 5%). The terminal (steady-state) height of rise Z is related to the initial flow parameters with a functional relationship Z = Z(Q,M,B,N). Far from the ejection point, the jet volumetric flow rate is much greater than the initial one, such that Q may be neglected [4]; thus, Z = Z(M,B,N). Two length scales L j and L p may be defined, one using the initial jet momentum flux and N and the other using the initial buoyancy flux and N [1,4,9,20], as follows: The jet is characterized initially as momentum (inertia)-driven or as buoyancy-driven, from the ratio of the inertia to buoyancy forces that is the initial densimetric Froude number [4]  (i) A jet-like discharge (B ≈ 0, ρ o ≤ ρ a ) will rise up to an elevation Z = Z(M,N) that is only a function of the initial specific momentum flux M and the buoyancy frequency N, thus leading to a dimensionless elevation, which must be constant. (ii) A plume (M ≈ 0, ρ o ≤ ρ a ), will rise up to a dimensionless elevation, which must also be constant. (iii) A buoyant jet with initial kinematic momentum and buoyancy fluxes M and B, respectively, of similar strength that discharges into an ambient fluid with linear density stratification of buoyancy frequency N will rise to an elevation Z that is a function of M, B, and N. Then, the normalized rise height will attain a functional form, where MN/B is a characteristic dimensionless parameter. In asymptotically jet-like flows (M >> B), MN/B >> 1 and Equation (4) is valid, while, in plumes, MN/B < 1 and Equation (5) meaning that the dimensionless jet rise Z/L p is proportional to ( The analysis presented above is also valid if we substitute the terminal rise height Z with spreading elevation Z S , giving Z S /L j = constant for jet-like flows and Z S /L p = constant for plumes. (iv) A fountain is the flow when the initial momentum flux is large but the buoyancy flux B is negative (opposite to the direction of motion). In a uniform, calm ambient, the normalized terminal height of rise of a turbulent fountain is constant [21], if the length scale l M = M 3/4 /|B| 1/2 [2,4] is large, or alternatively when the initial jet Richardson number . For fountains (large l M ) in a linearly density-stratified ambient, the asymptotic normalized terminal height of rise Z is a function of M and N only and, therefore, Z/L j takes a limiting value as shown from Equation (4). Thus, for large values of the parameter MN/|B|, Z/L j = constant, regardless of the sign of B. Rearranging the terms in Equation (4) gives Fluids 2020, 5, 232 5 of 20 When the stratification parameter N is small (N→0, MN/B << 1) the ambient fluid asymptotically can be considered as homogeneous, and the turbulent fountain will reach a terminal rise height so that Equation (9) is fulfilled, i.e., Note that, in the case of a fountain, the spreading height (SH) elevation obeys the dimensional arguments made for the terminal rise height (THR).
An average dilution at elevation z is the ratio S = µ(z)/Q of the local jet volume flux µ(z) to the initial one Q. Following the same procedure as in the previous paragraphs, if the momentum is initially dominant (MN/B >> 1, jet-like flow), the normalized volume flux may be obtained as If the buoyancy is dominant (MN/B < 1, plume-like flow), where d j and d p are constants. The results from the asymptotic analysis are summarized in Table 1. Table 1. Summary of the asymptotic relationships resulting from dimensional analysis.

MN/B >> 1
Jets and fountains Data from earlier experiments regarding the THR and SH were normalized and plotted together to justify the findings from dimensional analysis and evaluate the constants. The constants of proportionality from different experiments are shown in Table 2. The parameters presented in this section are used for comparison with present data. The descriptions "ortho" and "round" for the data by Wong and Wright [10] refer to the shape of the horizontal cross-section of the dispersion tank they employed, which was orthogonal and circular, respectively, for the two different sets of experiments they implemented. The normalized THR Z/L j and Z/L p are plotted versus the dimensionless parameter MN/B in Figure 1. From Figure 1a, it is evident that, when MN/B > 10, the normalized THR Z/L j is constant, in accordance with dimensional analysis, with an average value around c j ≈ 3.50 according to all authors but [14]. From Figure 1b, it is clear that, when MN/B < 1, the normalized THR Z/L p is a constant, congruent with dimensional analysis regarding initially plume-like flows, with an average constant from the data presented of c p ≈ 4.35, greater than those measured in [1,11,16]. Note that [1,14] added the distance z o of virtual origin to the THR, estimated to be located 5 cm and 1 cm upstream from the pipe outlet, respectively. Data from other authors did not involve z o , including Fan's [5] data, from which we subtracted 6.2 jet diameters.

Experiments-The Integral Model
Experiments were performed in a tank made of 1.25 cm thick Lucite with horizontal dimensions 0.90 m × 0.60 m and 0.80 m deep. An overflow around the top was used for excess fluid removal. The jet fluid was dyed with food dye, and the flow was illuminated with two 1000 W flood lights through a 5 mm thick panel of milky-white plexiglass, placed behind the rear window to diffuse the light uniformly. In the front and back (opposite) Lucite windows, square grids of 5 cm × 5 cm were drawn. For a vertical jet placed at the mid-plane of the tank, the terminal height of rise (THR) and mean spreading height (SH) were the average values read in the front and rear grid [22]. The linear density gradient was obtained using the "two-tank technique" [23]. Two similar tanks connected with a pipe were used, the first filled with fresh water and the other with an aqueous salt (NaCl) solution. Heavy fluid withdrawn from the continuously stirred second tank at small flow rates was pumped toward two floating surface spreaders positioned inside the dispersion tank. In this manner, lighter fluid was continuously added on the surface of the underlying heavier one, until the dispersion tank was filled up to the elevation of the overflow.
Four nozzles with diameters 0.5, 0.75, 1, and 1.5 cm were used to create the buoyant jets. They were mounted at the lower end of a jet plenum made of a 200 mm long piece of polyvinyl chloride (PVC) pipe with an inner diameter of 40 mm. A 2 cm long cylinder of sponge and a subsequent 6 cm long honeycomb section were set inside the jet plenum, to destroy any large eddies launched at the upstream end of it from the supply pipe and reduce the jet turbulence intensity. The inner side of all nozzles (inside the jet plenum) was rounded in order to provide smooth transition from the 40 mm pipe diameter to the size of the nozzle diameter, thus producing a uniform velocity profile at the exit [24,25].
When the stratification tank was filled up to the overflow, the surface spreaders were removed. The density profile was measured 20 min later waiting for the tank fluid to settle, using a calibrated microscale conductivity probe [26] that was mounted on a vertical, 60 cm long, linear positioning device, the accuracy of which was 0.1 mm. Calibration of the conductivity probe was performed before each experiment using samples of salt water, the density of which was measured with a hydrometer of 2% accuracy. The conductivity probe was lowered abruptly, while the density and position were recorded simultaneously at a rate of 200 Hz with a 12 bit data acquisition card and Labview ® software. From the density profile (a typical one is shown in Figure 2), the density gradient dρ/dz was evaluated and the buoyancy frequency N was computed. The jet plenum was placed at the top of the stratification tank pointing downward. The nozzle elevation was located at least 15 cm below the free surface, to ensure that the density gradient was linear throughout the whole depth where the jet flow developed. For positively buoyant jets at nozzle elevation, the jet fluid was heavier than the ambient, while, for negatively buoyant ones, the jet fluid was lighter than the ambient.
Fluids 2020, 5, x FOR PEER REVIEW 8 of upstream end of it from the supply pipe and reduce the jet turbulence intensity. The inner side of nozzles (inside the jet plenum) was rounded in order to provide smooth transition from the 40 m pipe diameter to the size of the nozzle diameter, thus producing a uniform velocity profile at the ex [24,25]. When the stratification tank was filled up to the overflow, the surface spreaders were remove The density profile was measured 20 min later waiting for the tank fluid to settle, using a calibrat microscale conductivity probe [26] that was mounted on a vertical, 60 cm long, linear positionin device, the accuracy of which was 0.1 mm. Calibration of the conductivity probe was perform before each experiment using samples of salt water, the density of which was measured with hydrometer of 2% accuracy. The conductivity probe was lowered abruptly, while the density an position were recorded simultaneously at a rate of 200 Hz with a 12 bit data acquisition card an Labview ® software. From the density profile (a typical one is shown in Figure 2), the densi gradient dρ/dz was evaluated and the buoyancy frequency N was computed. The jet plenum w placed at the top of the stratification tank pointing downward. The nozzle elevation was located least 15 cm below the free surface, to ensure that the density gradient was linear throughout t whole depth where the jet flow developed. For positively buoyant jets at nozzle elevation, the fluid was heavier than the ambient, while, for negatively buoyant ones, the jet fluid was lighter th the ambient.
The jet water supply was through a constant head tank with an overflow, positioned 2 m abo the free surface of the dispersion tank. It was filled with water of known density from a supply tan on the laboratory floor, via a submersible pump at flow rates higher than that of the jet, in order ensure overflow. The jet flow rate was measured via a calibrated rotameter in the range 7-55 cm with 2.5% accuracy. All experiments were planned according to the dimensional analysis discuss earlier. Experiments were performed in the regime 0.10 < MN/B < 1000. The buoyancy frequen parameter N 2 varied from 0.058 to 0.16 s −2 , the Reynolds number varied between 1870 and 8710 (t transition to turbulence occurred within five jet diameters from the nozzle, [24,27]), and the initial Richardson number varied between 0.003 and 0.406. The jet water supply was through a constant head tank with an overflow, positioned 2 m above the free surface of the dispersion tank. It was filled with water of known density from a supply tank on the laboratory floor, via a submersible pump at flow rates higher than that of the jet, in order to ensure overflow. The jet flow rate was measured via a calibrated rotameter in the range 7-55 cm 3 /s with 2.5% accuracy. All experiments were planned according to the dimensional analysis discussed earlier. Experiments were performed in the regime 0.10 < MN/B < 1000. The buoyancy frequency parameter N 2 varied from 0.058 to 0.16 s −2 , the Reynolds number varied between 1870 and 8710 (the transition to turbulence occurred within five jet diameters from the nozzle, [24,27]), and the initial jet Richardson number varied between 0.003 and 0.406.
The experiments were video-taped at 25 fps with a three-CCD (charge-coupled device) camera, positioned 4 m far from the dispersion tank. The terminal rise height and spreading elevation were evaluated from the time series of the movie frames obtained, using Autocad ® software and the two grids in the front and back transparent Lucite panels of the dispersion tank. The THR fluctuated around an average value, while the SH did not vary essentially throughout the experiment. Approximately 60 frames (an example is shown in Figure 3) sampled every 1 s were used to evaluate an average of THR and SH, before the laterally expanding "wastefield" reached the tank walls. Initial and measured flow parameters in buoyant jets and fountains are shown in Table 3. Table 3. Initial flow parameters and measured terminal height of the rise (THR) and spreading height (SH) from positively buoyant jet and fountain experiments [28].  Note that N was computed as an "average over depth" least-square line from density profiles such as in Figure 2. This means that deviations from the "linear density" due to possible flow rate changes of the pump supplying the surface spreaders would produce local density gradients that affect both THR and SH; hence, repeatability of the experiment was impossible. Therefore, our experiments were planned and executed by varying only the dimensionless parameter MN/B.
The equations of motion of the mean flow in round vertical turbulent buoyant jets that discharge in a density-stratified fluid with buoyancy frequency N, assuming top-hat distributions for the time-averaged velocity and density excess or deficiency, are written [4] as follows: where µ, m, and β are the local volume, specific momentum, and buoyancy fluxes, and α is the local entrainment coefficient. The initial conditions at the jet origin z o are An estimate of the virtual origin distance z o from the nozzle can be obtained from the theory by List and Imberger [29]. In a buoyant jet, the ratio C p = µ/zm 1/2 is defined to be the constant jet width parameter [30] found to be equal to 0.27. Substituting the local volume and momentum fluxes in the equation above with the initial values, the jet virtual origin can be estimated as z o = Q/(C p M 1/2 ) = 3.28D. The system of Equation (14) can be integrated starting at z o with the initial conditions in Equation (15).

Positively Buoyant Jet Experiments
The terminal height of rise and spreading height normalized with L j are plotted in Figure 4a,b, respectively, versus the dimensionless parameter MN/B. For large values of MN/B, i.e., for momentum-driven flows initially, both normalized THR Z/L j and SH Z S /L j take constant values around 3.58 ± 0.12 and 1.94 ± 0.11, respectively. These results are congruent with earlier experiments [12,13], while the data spread of Z S /L j is lower if compared to that of the data by [13]. When the dimensionless parameter MN/B is small, i.e., the flow is initially buoyancy-driven, the normalized THR Z/L p and SH Z S /L p are proportional to (MN/B) −1/4 as predicted from dimensional analysis. The normalized THR and SH take the values Z/L p = 4.52 ± 0.34 and SH Z S /L p = 2.95 ± 0.34, respectively, in agreement with results reported by the latter authors.

Negatively Buoyant Jet (Fountain) Experiments
The terminal height of rise and spreading height normalized with L j are also plotted in Figure 4a,b, respectively, versus MN/B. For large values of MN/B, i.e., for initially momentum-driven flows, both THR and SH take asymptotically constant values Z/L j ≈ 3.60 and Z S /L j ≈ 2.0 as in buoyant jets. When MN/B is small, i.e., the flow is initially buoyancy-driven, Z/L j ≈ 2(MN/B) 1/2 , while Z S /L j →−∞. In fact, from Figure 4b, the present data indicate that Z S /L j →0 when MN/B→1. Our negatively buoyant jet data are limited to MN/B > 1 because, at lower values, the spreading elevation coincides with the receptor boundary that is located upstream from the nozzle at the top layers of the dispersion tank.

Numerical Results: THR and SH, Average Dilution
The system of Equation (14) with initial conditions in Equation (15) is an initial value problem. The solution proposed is similar to those in [10,14]. Nevertheless, there are differences regarding the variable entrainment coefficient used in the present investigation and the virtual origin. In our computations, we made the assumption that the density gradient in the spreading (waste) field does not vary significantly if compared to the original one. The system of entrainment equations is an initial value problem and was solved using a fourth-order Runge-Kutta routine, starting from the jet virtual origin at elevation z o . The entrainment coefficient at distance z from the source was computed as a function of the local Richardson number Ri(z) = µ(z)β(z) 1/2 /m(z) 5/4 (see Appendix A) from the equation below.
In positively buoyant jets (B ≥ 0), the jet entrainment coefficient varies from α j = 0.072 to a p = 0.119 (considering a corrected C p = 0.254 and z o = 3.49D), while, when the buoyancy reverses, Ri 2 (z) is negative; therefore, a < 0.072. The spreading height can be roughly computed as assuming no further mixing during the descending flow from THR to SH [19], where ∆ρ, µ(Z) and β(Z) are the average density difference, the specific mass, and buoyancy fluxes, respectively, at the THR. Note that the findings from computations using top-hat, time-averaged velocity and density difference profiles are not any different from those using Gaussian distributions.
The results from computations regarding the normalized THR and SH are plotted in Figure 4 for both buoyant jets and fountains. For jet-like flows and fountains (MN/B >> 1), the numerical predictions are quite lower than our measurements, since the computed Z/L j = 3.0 and Z S /L j = 1.07 are lower than measured values of 3.50 and 2.0, respectively. For initially buoyancy-driven jets (MN/B < 1), the numerical predictions give Z/L p = 4.25 and Z S /L p = 3.1. The normalized THR is near the measured one (4.50), while the normalized SH is nearly the same as the predicted SH.
Measurements regarding the dilution of buoyant jets in a linearly stratified uniform ambient were reported by Wong and Wright [10], who measured the dilution of fluorescein dye in the spreading layer. They reported the normalized minimum dilution (that corresponds to maximum time-averaged concentration c m ) S m Q/(B 3/4 /N 5/4 ) versus MN/B. If the initial dye concentration of a jet is C o , then S m = C o /c m . From tracer conservation equation QC o = µ(z)c (where c is the average dilution), the dimensionless minimum dilution can be expressed as where the ratio c/c m < 1, as the average over a cross-section dye concentration, is only a portion of the maximum time-averaged concentration. In Figure 5, the normalized minimum dilution from experiments in [10] is plotted along with the computed normalized average dilution µN 1/2 /M 3/4 . From this figure, one may note several findings. In plumes (MN/B < 1), the normalized average dilution takes the value d p = 0.87, which is near the 0.84 suggested in [4] and higher than 0.80, an average value of the minimum dilution proposed by [10], as expected. From Equation (16), c m /c = 1.09, a value that is different from 1.40 proposed by [4] for round plumes in uniform ambient. From experiments in [10], the vertical distribution of concentration in the spreading field seems to be uniform. Therefore, an average concentration should be near the maximum, resulting in similar values of minimum and average dilutions. In jets (MN/B > 10), the computed dimensionless average dilution is d j = 0.52, a value that is lower than the minimum dilution 0.67 measured in [4], following the trend of the normalized THR and SH in this regime.

Discussion
From Figure 4, it is evident that positively buoyant jets and fountains converge asymptotically to the same THR Z/L j ≈ 3.50 and SH Z S /L j ≈ 2 when MN/B > 10. The results regarding THR are congruent with earlier measurements [5,10,12,13,16] but higher than the 3.0 reported in [14], as shown in Table 2. The results regarding SH are also higher than the 1.50 reported in [14,16] and lower than the 2.38 in [13], as shown in Table 2 as well. Analyzing the data in [16] using dimensional arguments from Section 2, the normalized SH elevation must be corrected to 1.89, a value that is near the measurements of the present study.
In the same figure, one may note that the data by [14] regarding fountains in a linearly, density-stratified fluid for MN/B >> 1 give asymptotic dimensionless constants of 2.88 for THR and 1.52 for SH, which are also lower from those in the present investigation. There are also differences in the dimensionless THR and SH between the present measurements and those in [14] when the parameter MN/B is small, the values of the latter being systematically lower than those measured in the present experiment. According to dimensional analysis presented in Section 2, as MN/B→0 when N→0 (uniform ambient density) and the jet momentum is driving the flow, the normalized THR Z/l M →C [21] or Z/L j = C(MN/B) 1/2 . Recent experiments [21,31] indicated that C ≈ 2, while [32] reported a constant C = 1.70. In Figure 4a, the line Z/L j = 2(MN/B) 1/2 is plotted for comparison, which is closer to the present data than those reported in [14]. In the same graph, we show four points from the unpublished data of [12] that are near the present findings.
When the flow is buoyancy-driven (B > 0) using dimensional arguments in Section 2, analysis of the present data in Table 3 shows that the normalized THR Z/L p ≈ 4.50 and SH Z S /L p ≈ 3.0 when MN/B < 1. The results regarding the normalized THR are congruent to those in [10,12] but higher than those in [1,13,16]. The results regarding the normalized SH are very close to 2.80 [15] and 3.0 [13], thus confirming their validity.
Numerical predictions using the entrainment coefficient derived from mass and momentum conservation equations led to accurate predictions regarding the spreading elevation and average dilution if compared to our measurements, as seen in Figures 4 and 5, when the flow is buoyancy-driven (MN/B < 1). In this regime the predicted THR is a little lower than our measurements. In jet-like flow and fountain regimes (MN/B >> 1), however, the predictions are lower than our measurements (3.06 vs. 3.50) for THR and much lower (1.06 vs. 2) for SH. The normalized average dilution in this regime turned out to be lower than the measured minimum dilution [10], a result that is against the physics of flow.
Nevertheless, in this regime, the results from integral models employed by others provided even lower values of normalized parameters. For example, the computed normalized THR in plumes [10] was 3.55 vs. measured 4.45 and that in jets was 3.06 vs. 3.60. The computed normalized dilutions in [10] were 0.42 vs. measured 0.68 in jets and 0.70 vs. measured 0.80 in plumes, results that are lower than minimum dilutions measured and contradictory to physics. Jirka [33] computed the normalized THR and SH using CorJet software and found values of 3.30 and 1.50, respectively, values that are lower than those measured in jets. To estimate THR, in addition to the elevation Z of the THR where momentum flux vanishes, √ 2b (where b is the local jet width) was added in [34]; thus, (Z + √ 2b)/L j = 3.30.

Conclusions
Turbulent vertical buoyant jets and fountains in a linearly density-stratified fluid were investigated experimentally and numerically. Data from earlier investigations were collected, analyzed, and put together according to dimensional analysis, leading to asymptotic constants. A positively buoyant jet (B ≥ 0) is momentum-driven if the dimensionless parameter MN/B > 10. In this regime, the normalized terminal height of rise (THR) Z/L j and mean spreading elevation (SH) Z S /L j take constant values around 3.5 and 2, respectively. The flow is buoyancy-driven if MN/B < 1, and, in this regime, the normalized THR Z/L p and SH Z S /L p take values around 4.50 and 3, respectively. The regime 1 < MN/B < 10 is considered to be the transition from plume-like to jet-like flows, where the initial specific momentum and buoyancy fluxes M and B are of equal importance for the development of the flow.
A negatively buoyant jet (B < 0) or fountain is momentum-driven if MN/B > 10. The normalized THR Z/L j and SH Z S /L j take asymptotically the same values as momentum-driven positively buoyant flows, which are around 3.50 and 2, respectively. For MN/B < 10, the flow is buoyancy-driven, and the normalized THR Z/l M takes an asymptotic value around 2 as MN/B→1, which is compatible with dimensional analysis and experiments of vertical fountains in a uniform ambient [21,31], while the SH is that of the nozzle elevation (Z S = 0) as MN/B→1.5.
Both THR and SH were obtained numerically from the equations of motion, using variable entrainment coefficient α that is a function of the local Richardson number of the flow. In buoyancy = driven flows (MN/B < 1), the computed THR and SH were found to be congruent to the measured ones. In momentum-driven flows (MN/B > 10) and fountains (B < 0), both THR and SH obtained numerically were not compatible with measurements.
The average dilution obtained numerically in buoyancy-driven jets (MN/B < 1) was in accordance with earlier dilution measurements [10]. However, the computed average dilution in the jet-like flow regime (MN/B > 10) was far from the measured minimum dilution [10], although the experimental data showed substantial scatter. We believe that velocity and concentration measurements using laser light-based tomography of the flow have to be made in order to understand the physics of dispersion of buoyant jets in a density-stratified ambient.
The entrainment coefficient function that was derived from the equations of motion of vertical buoyant jets in a uniform ambient seems to work quite well, especially in regimes where the discharges are plume-like initially or if MN/B < 1. Since it is related to the squared local Richardson number of the flow, when the buoyancy acts against the movement, the computed entrainment coefficient is lower than that measured in jets, a result that were previously thoroughly discussed [18,19,34].
The values compare well to those proposed by [4] and to those measured by [30,38], as shown in Table A1. Note that the above values are for top-hat velocity and concentration distributions and must be divided by √ 2 in order to obtain those for Gaussian distributions. The analysis presented in this Appendix regards a direct derivation of the entrainment coefficient for buoyant jets using the system of entrainment equations, i.e., the mass and momentum conservation. The entrainment coefficient is a function of the jet growth parameter (jet width) only, which is near the value 0.25z, where z is the elevation from the source. It is evaluated using the local Richardson number squared, such that its sign may be positive or negative if the flow is either positively buoyant or a fountain, respectively. This is congruent with derivations [18,34], as well as the assumption [19] that it should be reduced below the jet entrainment coefficient in flows where buoyancy has the opposite sign with respect to the direction of motion.
The local buoyant jet entrainment coefficient computed directly from the local Richardson number using Equation (A11) is analogous to the linear Equation (A1), since the entrainment coefficient is a linear function of Ri 2 (z), and quite different from the exponential one. The entrainment coefficient computed from Equation (A11) is independent of the sign of buoyancy with respect to the direction of motion. If the buoyancy has opposite sign (negative) from that of the direction of motion, then a is reduced and it takes positive values up to Richardson numbers greater than or equal to −2C p . For local Richardson numbers lower than −2C p , the entrainment coefficient becomes negative; therefore, it is meaningless and, in the computation of flows with negative or reversing buoyancy, should be considered zero. Equation (A11) derived here is better to be applied to flows with negative buoyancy instead of Equation (A1) on the left, since, in the latter, the asymptotic plume Richardson number of positively buoyant jets has to be employed, which does not appear in negative or reversing buoyancy. Note that, in negatively buoyant jets, the local Richardson number Ri 2 (z) has a limit at −∞.