# Revisiting Mean Flow and Mixing Properties of Negatively Round Buoyant Jets Using the Escaping Mass Approach (EMA)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}= A

_{0}w

_{0}, the initial flux of momentum m

_{0}= A

_{0}w

_{0}

^{2}and the initial flux of buoyancy β

_{0}= g

_{0}′ A

_{0}w

_{0}c

_{0}, where w

_{0}is the initial discharge velocity, A

_{0}= πd

_{0}

^{2}/4 is the area of the exit nozzle, d

_{0}is the diameter of the exit nozzle, c

_{0}is the initial concentration of tracer, ${{g}^{\prime}}_{0}=g\left({\rho}_{a}-{\rho}_{0}\right)/{\rho}_{0}$ is the apparent gravitational acceleration, g is the gravitational acceleration, ρ

_{0}and ρ

_{a}are the densities of the fluid at the jet exit and the ambient, respectively. Yannopoulos and Bloutsos [74], following Shao and Law [50], stated that negative buoyancy occurs when the vertical component of the initial momentum of the buoyant jet is zero, or the jet discharges with purely horizontal momentum and ρ

_{0}> ρ

_{a}so ${{g}^{\prime}}_{0}<0$ or alternatively when the vertical component of the initial momentum is non-zero, and the resulting buoyancy force acts in the opposite direction as the initial motion of the buoyant jet. Additionally, dimensional analysis shows that the behavior of such a flow field is described by three independent non-dimensional variables: the initial densimetric Froude number ${F}_{0}={w}_{0}\sqrt{\left|{{g}^{\prime}}_{0}\right|{d}_{0}}$, the initial inclination angle θ

_{0}measured with respect to the horizontal plane and the dimensionless axial distance Ξ = ξ/l

_{m}= (ξ/d

_{0}) F

_{0}

^{−1}, where ξ is the axial geometrical distance and l

_{m}is the characteristic length scale l

_{m}= d

_{0}F

_{0}that is usually employed to make geometrical distances non-dimensional [74]. All other parameters are either constants or variables depending upon independent ones. The constant parameters have the same values both for vertical or inclined buoyant jets of either positive or negative buoyancy. The only extra parameter included in inclined buoyant jets with negative buoyancy is the parameter Λ, which is defined in Chapter 2. More details about the description of the flow field of an inclined buoyant jet discharged into a stationary ambient fluid of uniform density can be found at the relevant papers of Yannopoulos [9] and Yannopoulos and Bloutsos [74]. As the present paper revisits the previously published work regarding the negatively round buoyant jet using the EMA model [74], the authors effort is to incorporate in the current paper an overall review of all the experimental data and numerical results of such problems.

_{0}≤ 75° and initial Froude number range 4 < F

_{0}< 100 in a stationary and homogenous ambient environment. EMA’s performance was checked by comparing geometric (centerline and external edge terminal height, horizontal location of centerline terminal height and return point, centerline length to terminal height and centerline and external edge length to return point) and mixing (minimum dilution at centerline terminal height and return point) characteristics to available experimental data, numerical and integral model predictions. The good agreement of the results of the EMA model with experimental data obtained from the international literature confirms the reliability of the model. In addition, the better performance in comparison to other integral or numerical model results makes the EMA model appropriate in the design of systems for the disposal of liquid, thermal and gas waste under these conditions. It could also be used for the verification of complex numerical models, but also for laboratory studies of related models.

_{0}is discharged upwards with initial velocity w

_{0}from a round nozzle of size d

_{0}with an initial inclination θ

_{0}to the horizontal plane into a quiescent ambient fluid of uniform density ρ

_{a}(ρ

_{0}> ρ

_{a}). At the first stage, the buoyant jet moves upwards, due to its initial momentum and the mixing with ambient fluid starts. Meanwhile, the negative buoyancy decreases the momentum flux causing its deceleration. The jet rises to a terminal (maximum) height, where the vertical component of momentum becomes zero, and then it falls as a positively buoyant jet to the initial discharge elevation. The general configuration of such a flow is shown in Figure 1, related to a Cartesian coordinate system (O, x, y, z). The flow is assumed to be steady state. The flow centerline (trajectory) is defined as the location of maximum velocity or concentration at each cross-section of the flow. In Figure 1, y and z are the horizontal and vertical distances and subscripts c, e, t and r denote the jet centerline, the external boundary, the terminal rise height and the return point to the source elevation, respectively. Following these definitions z

_{ct}and z

_{et}are the corresponding terminal heights of the centerline and the external boundary of the jet, which occur at a horizontal distance y

_{ct}from the source. The horizontal location of the return point of the jet axis is y

_{cr}and the corresponding location for the external boundary is y

_{er}. Additionally, S

_{ct}and S

_{cr}are the axial dilutions at terminal rise height and the return point, correspondingly. Previous studies [32,38,40,45,49,50,53,54,56,57,58,59,60,61,62,81] have verified that the geometrical quantities non-dimensionalized by d

_{0}, and dilutions are proportional to F

_{0}for a specific angle θ

_{0}.

## 2. Materials and Methods

_{B}of the escaping masses from the buoyant jet field, employing the general definition of the local Richardson number for plume flows [9]. The concentration of these masses c

_{B}is estimated as a portion Λ of its centerline value c

_{m}. The value of Λ coefficient was adopted equal to 0.34 [74], where Λ includes the value π/4 for round buoyant jets) for all cases in the inclination range −75° ≤ θ

_{0}≤ −15° (or 15° ≤ θ

_{0}≤ 75° of negatively buoyant jets). Additionally, EMA is equipped with the second order approach (SOA), to include the variable second-order effect of turbulence to the mean flow properties [9].

_{l}(r, φ, ξ), the governing equations of the model are:

^{2}+ ψ

^{2})

^{1/2}is the transverse (radial) distance; $h=1+r\mathrm{sin}\phi d\theta /d\xi $ is the scale factor of the coordinate system, θ is the local inclination angle of the ξ axis, $c=\left({\rho}_{a}-\rho \right)/\left({\rho}_{a}-{\rho}_{0}\right)$ is the local mean concentration and c′ the corresponding turbulent fluctuation; ρ is the local mean fluid density; w′

^{2}, w′c′, u′c′ are the local fluxes due to turbulent fluctuations of w, u, c; τ

_{rξ}is the mean turbulent shear stress; p

_{d}is the dynamic pressure; ${S}_{1}=\partial \left(h{q}_{r}\right)/\partial r$ is a source term for the escaping masses from the flow field of the buoyant jet; and q

_{r}is the escaping load of masses per unit area in the r direction.

_{m}, c = c

_{m}, τ

_{rξ}= u′c′ = q

_{r}= 0, p

_{d}= p

_{dm}

_{w})

_{B}, w = c = τ

_{rξ}= u′c′ = 0, q

_{r}= q

_{B}sinφ, p

_{d}= p

_{de}

_{B}= −u

_{B}− w

_{B}cosθ sinφ. Based on experimental measurements, the region between the nominal half-width of a buoyant jet, b

_{0.5}, and the total half-width, B

_{w}, is characterized by intermittencies of the velocity field (see Yannopoulos & Bloutsos 2012 [74]). Therefore, our assumption is that all this region could contribute to escaping masses. The velocity w

_{B}becomes zero at r = b

_{0.5}, while has a positive value at the boundary of the buoyant jet (r = B

_{w}). The escaping masses from the convex boundary enter the buoyant jet and are thus swept by the main flow. However, only masses from the concave boundary manage to escape when the entrainment of the buoyant jet becomes very weak or negative. Consequently, in addition to Equation (6), on the jet boundary (r = B

_{w}) further boundary conditions are applied:

_{B}and c

_{B}are the vertical velocity and concentration of escaping masses; u

_{B}is the entrainment velocity of the inclined buoyant jet at the actual boundary B

_{w}= d

_{0}/2 + n

_{w}b

_{w}and n

_{w}= 2

^{−1/2}[7,8]).

^{−1}of the maximum value ${\varphi}_{m}$ and ${K}_{\varphi}$ is the spreading rate coefficient. Thus, $\varphi $ stands for either the mean axial velocity w or the mean concentration c.

_{m}that is produced by integrating the equation of continuity (1) with respect to distance r [9,74]:

_{B}of the escaping masses at the inner boundary is estimated as:

_{w}= 0.11, λ

_{Bp}= 1.15, λ

_{p}= 1.04, R

_{p}= 0.3521 and Y

_{p}= 1.001 [74]. Additionally, the concentration c

_{B}of the escaping fluid by the inner boundary is assumed proportional to the corresponding centerline concentration as:

## 3. Variation of Basic Parameters

_{0}until θ

_{0}$\cong $ 40°–45° and then decreases up to 75°. Thus, as seen in Section 4, the distance y

_{cr}increases until θ

_{0}$\cong $ 40°–45° and then decreases up to a minimum value approaching zero for vertical discharges.

_{ct}occurs, the buoyancy flux becomes positive. The major part of this region is beyond Ξ $\cong $ 10 (Figure 2). Therefore, the stability of the buoyancy flux means that the assumption made regarding the escaping mass has no effect in the region of positive buoyancy flux.

_{0}F

_{0}) for various initial inclination angles θ

_{0}(θ

_{0}= 15°, 30°, 45°, 60°, 75°) and initial Froude numbers F

_{0}(F

_{0}= 5, 20) is shown in Figure 3. It is apparent that the effect of F

_{0}to θ is rather insignificant for practical applications.

## 4. Results and Discussion

_{0}≤ 75° and initial densimetric Froude numbers 4 ≤ F

_{0}≤ 100. For each angle, in the range 15° ≤ θ

_{0}≤ 75°, the variation of the non-dimensional geometric quantities of z

_{ct}/d

_{0}, z

_{et}/d

_{0}, y

_{ct}/d

_{0}, y

_{cr}/d

_{0}and y

_{er}/d

_{0}and mixing characteristics S

_{ct}and S

_{cr}are calculated for every initial Froude numbers 4 ≤ F

_{0}≤ 100. For every θ

_{0}, these quantities vary proportionally to the Froude number by a constant k

_{i}(among others: [49,53,54]) as:

_{i}(i = 1, …, 7) is the proportionallity constant. The results are compared to experimental data available in the literature published by several researchers, as well as to other numerical models predictions such as CorJet, VisJet, the reduced buoyancy flux (RBF) model by Oliver et al. [58], modified RBF by Crowe et al. [61] and analytical solutions proposed by Kikkert et al. [49]. Available data from CFD analysis [76,78,81,82] are also used for comparisons.

_{ct}to the initial inclination angle. EMA’s prediction is compared to available experimental data. The terminal rise height EMA passes through the scatter of experimental data following their trend. The latter is more obvious if a polynomial fit to experimental data [45,47,49,50,51,54,57,58,59,61,73,81,87,88] is made. For this purpose, a third degree polynomial is fitted to experimental data using the least square method. The equation of the polynomial is presented in Table 1. In the range 15° ≤ θ

_{0}≤ 75° EMA’s prediction is very close to the polynomial fit. It must be noted that EMA predicts the centerline as the midpoint of the round cross-section. The same procedure is followed by Bosanquet et al. [87] and Bashitialshaaer et al. [88]. Instead, Cipollina et al. [45], Kikkert et al. [49], Ferrari and Querzoli [47], Shao and Law [50], Lai and Lee [51], Oliver et al. [58], Nikiforakis et al. [89], Crowe et al. [61], Ramakanth [73], and Zhang et al. [81] determine the position of centerline as the point where the transverse profile of velocity or concentration is maximized. In particular, this point is shifted towards the external edge, moving away from the midpoint of cross-section. As the initial inclination angle is rising, the deviation of actual transverse profile is deviates intensively from the Gaussian presenting this difference, while, when the initial inclination reaches the vertical, this deviation is reduced, and EMA’s prediction coincides with the experimental data. Thus, the deviation between EMA’s prediction and the experimental data in the range 20° < θ

_{0}< 70° is reasonably expected. In Figure 4b, EMA’s prediction of centerline terminal rise height is compared to other models results. For θ

_{0}< 30° all models almost coincide. For θ

_{0}> 30°, Kikkert’s analytical solution and the RBF model are diverging upwards calculating higher values than EMA’s and modified RBF’s predictions that approximately coincide up to 75°. The commercial packages CorJet and VisJet are underestimating the centerline terminal rise height for θ

_{0}> 30°. In Figure 4b, the results of CFD analysis from Gildeh et al. [78,79] Oliver et al. [76] and Zhang et al. [81,82] for a variety of simulations are included. CFD’s results are for the most usual initial inclination angles of 15°, 30°, 45° and 60°. For 15°, all CFD’s results coincide, showing that the option of RANS or the most sophisticated LES approach does not have major effect on the results. There are significant differences among CFD simulations for θ

_{0}= 30°, 45° and 60°, where the scatter of the results is maximized for θ

_{0}= 45°. In any case, EMA predictions are within experimental data.

_{et}of the dense jets is of great environmental importance, because it indicates whether the mixing processes are taking place under the water surface or not. The dimensionless values of z

_{et}that are calculated by the EMA model through the initial inclination angles are shown on Figure 5a,b. In Figure 5a, the EMA’s results are compared to previous published experimental data, and in Figure 5b, the performance of EMA is compared to other models. Again EMA, without intense deviation, follows the trend of a third degree polynomial fitted to experimental data [38,40,41,42,43,44,45,48,49,50,51,53,56,58,60,69,73,81,88,89,90,91,92,93] (Table 1). In Figure 5a, the experimental data of Nemlioglou and Roberts [48] and Bloomfield and Kerr [44] are the upper and lower data of the scatter, respectively. The wide scatter of experimental results is due to the different definitions of the external edge of the buoyant jet among the investigators. Indicatively, Kikert et al. [49] and Oliver et al. [58,78] define the external boundary at a distance twice the concentration spread, while Papakonstantis et al. [54] define this length as 1.5 times the concentration spread. Lai and Lee [51] and Abessi and Roberts [60] define the boundary at a locus where concentration values c are 25% or 10% the maximum concentration (c

_{max}) at centerline, respectively. Similarly, Jiang et al. [52] and Zhang et al. [81] define the external boundary at c/c

_{max}= 5% and Shao and Law [50] and Gildeh et al. [78] define the external boundary at c/c

_{max}= 3%. According to EMA, the outer boundary of the jet is defined at a distance B

_{c}= d

_{0}/2 + n

_{c}λ b

_{w}, where b

_{w}is the nominal spread width of the velocity field of the buoyant jet, λ is the concentration-to-velocity spreading rate coefficient ratio and n

_{c}is the non-dimensional total spread of concentrations of a buoyant that according to profile measurements by Ramaprian and Chandrasekhara [94] and Shao and Law [50] is approximately 1.9. More details can be found at the relative paper of Yannopoulos and Bloutsos [74]. Among EMA’s prediction and Kikkerts’s solution, the RBF and modified RBF models there are slight differences up to approximately 70°. A little deflection occurs between EMA’s and CorJet’s prediction for the cutoff level of c/c

_{max}= 3%. As VisJet’s boundary is defined at 0.25 c

_{max}, its results are like CorJet’s cutoff level of 25%, and obviously differ to the EMA relative values.

_{ct}predicted by EMA is shown. The distance y

_{ct}is increasing gradually up to initial angle of 45°, approximately, and then decreases smoothly up to 75°. Similar behavior is observed at experimental data of various previous works demonstrating the good performance of EMA. Excluding the experiments of Bosanquet et al. [87] and Lindberg [43], the experimental scatter is quite narrow, and EMA predictions are within the experimental data for the range 15° ≤ θ

_{0}≤ 75°. CFD analyses provide data only for initial angles of 15°, 30°, 45° and 60°. The scatter of CFD data seems to be wider than experiments. Both CorJet and VisJet diverge from experimental data, underestimating the horizontal distance of centerline peak. EMA’s predictions underestimate the corresponding experimental data, following the trend of a third degree polynomial fitted to experimental data [43,45,49,50,51,53,56,58,61,73,81,87,89] (Table 1).

_{cr}, is shown in Figure 7. EMA’s results are compared to several experimental data for various initial inclinations in Figure 7a. Comparing EMA’s prediction to experimental fit line [41,45,47,48,49,50,51,52,54,57,58,60,61,73,81,89,91,92] (Table 1), it is obvious that EMA follows the fit line, underestimating the return point location in the whole range of 15° ≤ θ

_{0}≤ 75°. The difference is maximized for θ

_{0}= 15°, decreases approaching approximately θ

_{0}= 60° and slightly increases at θ

_{0}= 75°. Comparing to other models and CFD predictions (Figure 7b), the EMA’s predictions seem to be better and rather satisfactory. Additionally, the corresponding variation of the external edge to initial inclination angle is presented in Figure 8. EMA’s predictions are compared to the experimental data of Bashitialshaaer et al. [88], Crowe [69], Nikiforakis et al. [89], Oliver [68], Papakonstantis et al. [53], Papakonstatis and Tsatsara [56], Ramakanth [73] and Zeitoun et al. [38], and the modified RBF Model [61]. Again, a third-degree polynomial is fitted to experimental data [38,53,56,68,69,73,88,89] (Table 1). The two models predict quite similar values. The divergence of the EMA model to the experimental fit line is less than 10% for initial angles in the range 15° ≤ θ

_{0}≤ 75°.

_{ct}). The prediction is compared to available experimental data. EMA’s dilution values are overestimated, compared to a third degree polynomial fit [38,40,42,50,51,54,57,58,73,81,91] (Table 1) to experimental data, in the range of initial inclination angles 15° ≤ θ

_{0}≤ 65°. In the range 65° < θ

_{0}≤ 75°, the maximum underestimation is 24.0%, which occurred for θ

_{0}= 75°.

_{cr}) for initial inclination angles in the range 15° ≤ θ

_{0}≤ 75° is presented in Figure 10. Figure 10a shows the predicted EMA’s dilutions compared to available experimental data. A third degree polynomial is fitted to experimental data [40,41,48,50,51,52,54,57,59,60,73,81,91,92] (Table 1). EMA’s predictions perform a continuous increment, slightly underestimating dilution values in the range of 15° ≤ θ

_{0}≤ 65°, while the maximum overestimation is 12.6% for initial inclination angles greater than 65°. Comparing to other model predictions (Figure 10b), it is obvious that EMA has the better overall performance. CFD analysis by Gildesh et al. [78,79] for θ0 = 30° gave similar values approximating experimental data.

_{ct}(Figure 11) almost coincides to corresponding experimental data of Crowe [69], Oliver [68] and Ramakanth [73], where the scatter of experimental data is negligible.

## 5. Conclusions

_{0}from 35° to 40°. This difference is attributed to the way that centerline is defined by each researcher. EMA’s predictions of horizontal distance of centerline terminal height are lower than the experimental fit polynomial, but within the range of experimental values. Regarding the horizontal location of return point of centerline and external edge, EMA’s predictions follow the corresponding polynomial fits presenting a slight difference for the case of centerline variation. As an overall observation, EMA’s predictions agree well to the experimental data, which assures that EMA performs better than other models. The definition of buoyant jet’s centerline affects the estimated variation of axial dilution at centerline terminal height regarding the available experimental data, leading to overestimations for initial inclinations less than 45°. Additionally, EMA’s prediction of the axial dilution values at centerline return point is rather conservative compared to the corresponding experimental fit, but much closer than other models’ predictions. Finally, EMA’s of centerline length to terminal height and external edge length to return point coincides excellently to available experimental data. Overall, it seems that the application of EMA model gives reliable predictions without heavy computational cost or adopting controversial assumptions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Variation of the normalized buoyancy flux β/β

_{0}with dimensionless distance Ξ = ξ/(d

_{0}F

_{0}) and initial inclination θ

_{0}.

**Figure 3.**Variation of local centerline angle θ to horizontal with dimensionless distance Ξ = ξ/(d

_{0}F

_{0}) and initial inclination θ

_{0}for F

_{0}= 5 and F

_{0}= 20.

**Figure 4.**Escaping mass approach’s (EMA’)s prediction of dimensionless centerline terminal height and comparison with (

**a**) experimental data, and (

**b**) other models’ predictions.

**Figure 5.**EMA’s prediction of dimensionless terminal height of external edge and comparison with (

**a**) experimental data, and (

**b**) other models’ predictions.

**Figure 6.**EMA’s prediction of dimensionless horizontal location of centerline terminal height and comparison with (

**a**) experimental data, and (

**b**) other models’ predictions.

**Figure 7.**EMA’s prediction of dimensionless horizontal location of centerline terminal height and comparison with (

**a**) experimental data, and (

**b**) other models’ predictions.

**Figure 8.**EMA’s prediction of dimensionless horizontal location of external edge return point and comparison with experimental data.

**Figure 9.**EMA’s prediction of axial dilution at centerline terminal height and comparison with experimental data.

**Figure 10.**EMA’s prediction of axial dilution at centerline return point and comparison with (

**a**) experimental data, and (

**b**) other models’ predictions.

**Figure 11.**EMA’s prediction of dimensionless centerline length to terminal height and comparison with experimental data.

**Figure 12.**EMA’s prediction of dimensionless external edge length to return point and comparison with experimental data.

Quantity | Figure | Polynomial Fit |
---|---|---|

Centerline terminal height | 2 | ${z}_{ct}/\left({d}_{0}{F}_{0}\right)=1.6275\times {10}^{-1}-1.2691\times {10}^{-2}{\theta}_{0}+1.2672\times {10}^{-3}{\theta}_{0}^{2}-1.0583\times {10}^{-5}{\theta}_{0}^{3}$ |

External edge terminal height | 3 | ${z}_{et}/\left({d}_{0}{F}_{0}\right)=4.4066\times {10}^{-1}-2.1578\times {10}^{-3}{\theta}_{0}+1.0565\times {10}^{-3}{\theta}_{0}^{2}-9.1929\times {10}^{-6}{\theta}_{0}^{3}$ |

Horizontal distance to terminal height | 4 | ${y}_{ct}/\left({d}_{0}{F}_{0}\right)=6.0332\times {10}^{-1}+6.7018\times {10}^{-2}{\theta}_{0}-8.4160\times {10}^{-4}{\theta}_{0}^{2}+4.4949\times {10}^{-7}{\theta}_{0}^{3}$ |

Horizontal distance to centerline return point | 5 | ${y}_{cr}/\left({d}_{0}{F}_{0}\right)=1.3833\times {10}^{0}+1.0256\times {10}^{-1}{\theta}_{0}-1.3447\times {10}^{-3}{\theta}_{0}^{2}+7.3392\times {10}^{-7}{\theta}_{0}^{3}$ |

Horizontal distance to external edge’s return point | 6 | ${y}_{er}/\left({d}_{0}{F}_{0}\right)=3.6327\times {10}^{0}-1.0627\times {10}^{-2}{\theta}_{0}+9.8744\times {10}^{-4}{\theta}_{0}^{2}-1.3289\times {10}^{-5}{\theta}_{0}^{3}$ |

Minimum dilution at terminal height | 7 | ${S}_{ct}/{F}_{0}=-3.5982\times {10}^{-3}+2.3515\times {10}^{-2}{\theta}_{0}-3.9088\times {10}^{-4}{\theta}_{0}^{2}+2.1287\times {10}^{-6}{\theta}_{0}^{3}$ |

Minimum dilution at return point | 8 | ${S}_{cr}/{F}_{0}=4.9351\times {10}^{-1}+1.2654\times {10}^{-2}{\theta}_{0}+5.0248\times {10}^{-4}{\theta}_{0}^{2}-6.5504\times {10}^{-6}{\theta}_{0}^{3}$ |

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**MDPI and ACS Style**

Bloutsos, A.A.; Yannopoulos, P.C.
Revisiting Mean Flow and Mixing Properties of Negatively Round Buoyant Jets Using the Escaping Mass Approach (EMA). *Fluids* **2020**, *5*, 131.
https://doi.org/10.3390/fluids5030131

**AMA Style**

Bloutsos AA, Yannopoulos PC.
Revisiting Mean Flow and Mixing Properties of Negatively Round Buoyant Jets Using the Escaping Mass Approach (EMA). *Fluids*. 2020; 5(3):131.
https://doi.org/10.3390/fluids5030131

**Chicago/Turabian Style**

Bloutsos, Aristeidis A., and Panayotis C. Yannopoulos.
2020. "Revisiting Mean Flow and Mixing Properties of Negatively Round Buoyant Jets Using the Escaping Mass Approach (EMA)" *Fluids* 5, no. 3: 131.
https://doi.org/10.3390/fluids5030131