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

Abstract

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 coefficient function, confirmed the results related to buoyancy dominant flows (plumes), while their predictions in momentum-driven flows were quite low if compared to measurements.

1. 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].

2. 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 L³/T, L⁴/T², and L⁴/T³, respectively, are calculated as follows:

$$Q=W=\frac{\mathsf{\pi }{D}^2}{4}W, M=QW, B=g_o^{\prime }Q;\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{g}_o^{\prime }=\frac{{\rho }_a-{\rho }_o}{{\rho }_o}\mathrm{g},$$

(1)

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

$$N^2=-\frac{g}{{\rho }_o}\frac{\mathrm{d}\rho }{\mathrm{d}z}.$$

(2)

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:

$$L_j=\frac{M^{1/4}}{N^{1/2}} \mathrm{and} L_p=\frac{B^{1/4}}{N^{3/4}}.$$

(3)

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] Fr_o = W/(g’_oD)^1/2, g’_o = (ρ_a − ρ_o)/ρ_o. Thus, the flow can be classified into four categories on the basis of the initial momentum and buoyancy fluxes B and M, respectively (or on Fr_o), as well as the (initial) density difference between jet ρ_ο and ambient fluid ρ_a at nozzle elevation: (i) jet-like flow when the momentum flux is dominant and ρ_ο ≤ ρ_a or Fr_o is very large, (ii) plume-like flow, when the buoyancy flux is dominant and ρ_ο ≤ ρ_a or Fr_o→1, (iii) buoyant jet (forced plume) flow when momentum and buoyancy fluxes affect the flow equally and ρ_ο ≤ ρ_a, and (iv) fountain flow when the momentum flux is dominant and ρ_ο ≥ ρ_a (B < 0).

(i)

A jet-like discharge (B ≈ 0, ρ_ο ≤ 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,

$$Z{\left(\frac{N^2}{M}\right)}^{1/4}=\frac{Z}{L_j}=c_j,$$

(4)

which must be constant.

(ii)

A plume (M ≈ 0, ρ_ο ≤ ρ_a), will rise up to a dimensionless elevation,

$$Z\frac{N^{3/4}}{B^{1/4}}=\frac{Z}{L_p}=c_p,$$

(5)

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,

$$\frac{Z}{L_p}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\frac{Z}{L_j}\hspace{0.17em}=f\left(\frac{M}{B}N\right),$$

(6)

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) is valid. If MN/B >> 1, dividing both sides of Equation (4) by L_p obtains

$$\frac{Z}{L_p}=c_j\frac{L_j}{L_p}=c_j{\left(\frac{M}{B}N\right)}^{1/4},$$

(7)

meaning that the dimensionless jet rise Z/L_p is proportional to (MN/B) ^1/4. Similarly, in plumes (MN/B < 1),

$$\frac{Z}{L_j}=c_p\frac{L_p}{L_j}=c_p{\left(\frac{M}{B}N\right)}^{-1/4}.$$

(8)

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],

$$\frac{Z}{l_M}=C,$$

(9)

if the length scale l_M = M^3/4/|B|^1/2 [2,4] is large, or alternatively when the initial jet Richardson number Ri_o = Q|B^1/2|/M^5/4 <<1 [21]. 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

$$\frac{Z}{l_M}=c_j\frac{L_j}{l_M}=c_j{\left(\frac{MN}{B}\right)}^{-1/2}.$$

(10)

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.,

$$\frac{Z}{l_M}=C\Rightarrow \frac{Z}{L_j}=C\frac{l_M}{L_j}=C{\left(\frac{MN}{B}\right)}^{1/2} \mathrm{or} \mathrm{alternatively} \frac{Z}{L_p}=C{\left(\frac{MN}{B}\right)}^{3/4}.$$

(11)

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

$$\frac{\mu (Z)N^{1/2}}{M^{3/4}}=d_j \mathrm{or} \frac{\mu (Z)N^{5/4}}{B^{3/4}}=d_j{\left(\frac{MN}{B}\right)}^{3/4}.$$

(12)

If the buoyancy is dominant (MN/B < 1, plume-like flow),

$$\frac{\mu (Z)N^{5/4}}{B^{3/4}}=d_p \mathrm{or} \frac{\mu (Z)N^{1/2}}{M^{3/4}}=d_p{\left(\frac{MN}{B}\right)}^{-3/4},$$

(13)

where d_j and d_p are constants. The results from the asymptotic analysis are summarized in

Table 1. Summary of the asymptotic relationships resulting from dimensional analysis.

MN/B >> 1
Jets and fountains	$\frac{Z}{L_j}=c_j$	$\frac{Z}{L_p}=c_j{\left(\frac{MN}{B}\right)}^{1/4}$	$\frac{Z}{l_M}=c_j{\left(\frac{MN}{B}\right)}^{-1/2}$	$\frac{\mu (Z)N^{1/2}}{M^{3/4}}=d_j$
MN/B < 1
Plumes	$\frac{Z}{L_j}=c_p{\left(\frac{MN}{B}\right)}^{-1/4}$	$\frac{Z}{L_p}=c_p$	$\frac{Z}{l_M}=c_p{\left(\frac{MN}{B}\right)}^{-3/4}$	$\frac{\mu (Z)N^{5/4}}{B^{3/4}}=d_p$

.

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. Evaluation of the dimensionless parameters from earlier experiments.

Author	Z/Lj	ZS/Lj	Z/Lp	ZS/Lp	dj	dp	Comments
Morton, Taylor, and Turner [1] 1			3.81 ± 0.25				Plume
Fan [5]	3.41 ± 0.13						MN/B > 10
Abraham and Eysink [11]			3.63 ± 0.20				MN/B = 0.80–2.40
Wong and Wright [10] for rectangular tank	3.72 ± 0.17		4.50 ± 0.46		0.77 ± 0.14	0.78 ± 0.11	MN/B > 10 MN/B < 1
Wong and Wright [10] for circular tank	3.23 ± 0.09		4.28 ± 0.37		0.68 ± 0.03	0.84 ± 0.12	MN/B > 10 MN/B < 1
Wong and Wright [10] for all data	3.60 ± 0.27		4.46 ± 0.44		0.74 ± 0.12	0.81 ± 0.11	MN/B > 10 MN/B < 1
Papanicolaou et al. [12]	3.46 ± 0.16		4.60 ± 0.39				MN/B > 10 MN/B < 1
Bloomfield and Kerr [14] 1,2	3 ± 0.23 (3.15) *	1.53 ± 0.10 (1.52) *					B = 0
Bloomfield and Kerr [14] 1,3	2.88 ± 0.10	1.35 ± 0.11					B < 0
Konstantinidou and Papanicolaou [13]	3.55 ± 0.16	2.38 ± 0.39	3.97 ± 0.29	2.99 ± 0.29			MN/B > 10 MN/B < 1
Richards et al. [15]		1.50 (1.89) *		2.7 (2.78) *			MN/B = 0.4 ÷ 46
Zhang et al. [16]			3.47 ± 0.26				MN/B < 2.80
Fischer et al. [4]	3.80		3.80				Proposed values
Chen and Rodi [7]	3.80		5.00				Proposed values

¹ Including virtual origin distance; ² zero buoyancy flux at source; ³ fountain; * linear fit.

. 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.

3. 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.

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 cm³/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 N² 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. Initial flow parameters and measured terminal height of the rise (THR) and spreading height (SH) from positively buoyant jet and fountain experiments [28].

Run	D	Q	ρa	ρο	N2	Re	Ro	Z	ZS
	(cm)	(cm3/s)	(gr/cm3)	(gr/cm3)	(s−2)			(cm)	(cm)
Positively buoyant jets
Exp-1	1.00	51.26	1.0073	1.0092	0.1303	7120	0.020	47.3	24.2
Exp-3	1.00	30.25	1.0066	1.0066	0.1350	4304	0.003	33.9	17.7
Exp-4	1.00	21.25	1.0069	1.0100	0.1302	3023	0.061	33.3	18.8
Exp-5	0.50	15.24	1.0057	1.0062	0.1541	4338	0.005	33.3	17.4
Exp-10	1.50	25.75	1.0083	1.0140	0.1074	2055	0.185	33.7	20.7
Exp-14	0.75	27.25	1.0070	1.0091	0.1584	4240	0.019	37.8	20.8
Exp-15	0.50	28.75	1.0075	1.0100	0.1276	7238	0.007	47.8	25.1
Exp-16	0.50	28.75	1.0073	1.0087	0.1269	6711	0.005	48.2	27.7
Exp-20	1.50	21.25	1.0045	1.0142	0.1454	2063	0.294	36.0	23.9
Exp-21	1.50	19.75	1.0031	1.0150	0.0809	1918	0.350	48.6	31.4
Exp-22	1.50	19.75	1.0028	1.0188	0.0705	1873	0.405	55.1	36.6
Exp-23	1.00	24.25	1.0032	1.0101	0.0745	3450	0.079	51.5	33.0
Exp-24	1.00	24.25	1.0070	1.0100	0.1505	3450	0.052	35.1	20.7
Exp-25	1.50	28.75	1.0076	1.0119	0.1520	2727	0.145	35.2	21.5
Exp-26	1.00	33.25	1.0059	1.0095	0.1176	4731	0.042	38.2	22.2
Exp-27	1.00	25.75	1.0065	1.0084	0.1473	3406	0.039	32.8	18.7
Exp-28	1.50	27.25	1.0063	1.0151	0.1196	2403	0.218	34.4	22.0
Exp-29	1.00	36.25	1.0049	1.0060	0.1545	4679	0.021	37.3	19.4
Fountains
Exp-2	1.00	36.25	1.0066	1.0060	0.1492	5036	0.015	38.9	20.7
Exp-6	0.50	21.25	1.0071	1.0032	0.0978	6046	0.012	43.3	24.8
Exp-7	0.50	30.25	1.0065	0.9969	0.1226	8710	0.013	47.6	24.8
Exp-8	0.75	30.25	1.0081	0.9969	0.0584	5807	0.040	34.0	10.0
Exp-9	1.00	36.25	1.0089	0.9986	0.1604	4340	0.065	28.4	9.8
Exp-11	1.00	36.25	1.0074	1.0054	0.1377	4340	0.028	34.3	16.7
Exp-12	0.75	28.15	1.0095	0.9986	0.1634	4493	0.042	28.5	12.5
Exp-13	0.50	27.25	1.0098	0.9988	0.1352	6361	0.016	41.4	20.0
Exp-17	1.00	27.25	1.0105	0.9988	0.1297	3180	0.092	19.4	0.3
Exp-18	1.00	26.65	1.0098	1.0015	0.1565	3526	0.079	22.3	4.0
Exp-19	1.00	21.25	1.0058	0.9975	0.1662	2881	0.099	17.9	1.0
Exp-30	1.00	30.25	1.0049	0.9978	0.1041	4002	0.065	25.3	7.6

.

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:

$$\frac{d\mu }{dz}=2\sqrt{\pi }\hspace{0.17em}\alpha m^{1/2}, \frac{dm}{dz}=\frac{\mu \beta }{m}, \frac{\mathrm{d}\beta }{\mathrm{d}z}=-N^2\mu ,$$

(14)

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

$$\mu (z_o)=Q=\frac{\pi D^2}{4}W, m(z_o)=M=\frac{\pi D^2}{4}{W}^2, \beta (z_o)=B=\frac{{\rho }_a-{\rho }_o}{{\rho }_o}gQ.$$

(15)

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_pM^1/2) = 3.28D. The system of Equation (14) can be integrated starting at z_o with the initial conditions in Equation (15).

4. Results

4.1. 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.

4.2. 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.

4.3. 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.

$$\alpha =\frac{C_p}{2\sqrt{\pi }}\left(1+\frac{1}{2}\frac{R{i}^2(z)}{C_p}\right).$$

(16)

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, Ri²(z) is negative; therefore, a < 0.072. The spreading height can be roughly computed as

$$N^2=-\frac{g}{{\rho }_o}\frac{\overline{\Delta \rho }}{\Delta z}\Rightarrow \Delta z=Z_s-Z=-\frac{g}{{\rho }_o}\frac{\overline{\Delta \rho }}{N^2}=\frac{\beta (Z)}{N^2\mu (Z)},$$

(17)

assuming no further mixing during the descending flow from THR to SH [19], where Δρ, 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_mQ/(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

$$\frac{Q{S}_m{N}^{1/2}}{M^{3/4}}=\frac{Q(C_o/c_m)N^{1/2}}{M^{3/4}}=\frac{(c/c_m)\mu (z)N^{5/4}}{B^{3/4}},$$

(18)

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.

5. 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 Figure 4 and Figure 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.

6. 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 (SΗ) 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 SΗ 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 SΗ 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 SΗ 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 SΗ 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].