# Vertical Dense Effluent Discharge Modelling in Shallow Waters

^{*}

## Abstract

**:**

_{m}and lateral spread R

_{sp}) and dilution μ

_{min}properties of such jets. Three flow regimes were reproduced numerically, based on the experimental data: deep, intermediate, and impinging flow regimes.

## 1. Literature Review and Research Needs

_{0}, and Z

_{m}are water depth, nozzle radius, and maximum rise height, respectively).

_{m}and μ

_{min}. Section 3 reviews the numerical details of the CFD model used in this study. Section 4 presents the numerical model results and discusses jet behavior, geometrically and kinematically. Finally, Section 5 summarizes the conclusions and recommendations of our study.

## 2. Dimensional Analysis

_{0}, jet velocity of U

_{0}, jet density of ρ

_{0}, and ambient water density of ρ

_{a}(ρ

_{0}> ρ

_{a}). The jet mixes with ambient water as it is discharging and reaches a terminal (maximum) rise height, Z

_{m}, and then falls because of the negative buoyancy and creates a density current which moves it further horizontally. The mixing of concentrations relies on both discharge and ambient water properties, such as the concentration at the nozzle, C

_{0}; the density difference between the ambient water and discharge, Δρ

_{0}= ρ

_{0}− ρ

_{a}; the jet velocity, U

_{0}; the discharge nozzle, r

_{0}; and the depth of ambient water, H. The jet densimetric Froude number is one of the key properties in vertical dense discharge analysis and is defined as the ratio of inertia to buoyancy forces:

_{0}′ is the reduced acceleration due to gravity. The jet mixing characteristics of interest are Z

_{m}and μ

_{min}.

_{0}), kinematic momentum flux (M

_{0}), and the buoyancy flux (B

_{0}) [11], which are provided by the equations below:

_{M}is the momentum length scale, which can be calculated as:

_{0}and B

_{0}, the length scale below is suitable to nondimensionalize vertical discharge height [5].

_{0}, C

_{a}, and C

_{1}represent the discharge concentration at the nozzle, the ambient water concentration, and the discharge concentration at the return point, respectively.

_{0}and B

_{0}, we derive the following effective gravity scale to nondimensionalize jet buoyancy:

_{sp}.

## 3. Numerical Model

#### 3.1. Governing Equations

#### 3.2. Numerical Solver and Schemes

#### 3.3. Boundary Conditions

_{0}and density of ${\rho}_{0}$. The discharge density (ρ

_{0}) and density of ambient water (ρ

_{a}) ranged from 1012.4 to 1013.6 kg/m

^{3}and 993 to 994.7 kg/m

^{3}, respectively. The density ratio Δ

_{ρ}/ρ

_{a}ranged from 0.015 to 0.019. The inlet values of k and ε were selected based on [16], as k = 0.06U

_{0}

^{2}and ε = 0.06U

_{0}

^{3}/D. The ambient water condition was still and had a homogeneous density. The nozzle was raised 0.14 m above the bed level to eliminate any influence from the lower boundary. The outlet boundary was considered all around the tank (i.e., four side faces), applying a zero-gradient boundary condition perpendicular to those planes. The atmosphere condition was modelled applying a symmetry boundary using a zero-gradient boundary condition. The symmetry plane boundary condition means that flux and the components of the gradient normal to the plane should be zero but free to slide in tangential directions. A Dirichlet condition was utilized on the bottom wall boundary to apply the zero-velocity boundary condition. The OpenFOAM’s standard wall functions (e.g., epsilonWallFunction) were applied for the wall surface to simulate hydraulically smooth walls.

^{−5}and 1 × 10

^{−6}, respectively.

#### 3.4. Turbulence Model

#### 3.5. Numerical Cases

## 4. Results and Discussion

#### 4.1. Discharge Evolution

#### 4.2. Discharge Dilution

_{0}) profiles against the nondimensionalized radial distance from the nozzle (r − r

_{0})/r

_{0}Fr at the horizontal plane at the nozzle level (z = 0). The numerical cases from the B series of experiments have been compared with those from [11]. A concentration profile from a deep regime of the experimental study is shown in Figure 6 for reference. In the two numerical cases (from the B series), the water depth to nozzle radius ratio, H/r

_{0}= 37.2, is kept fixed and the jet exit velocity (i.e., Froude number) was increased from 9.44 to 18.20. As seen in Figure 6, the experimental results of the B series experiments are different from those of the deep regime, with more spikes in the concentration profile due to the shallow water impact. In other words, when the jet Froude number increases and the front of the jet meets the water surface (in both intermediate and fully impinging scenarios), the concentration at the nozzle level will experience ups and downs due to jet pulses at the attachment level. However, the numerical results show a smooth decrease in concentration as the distance from the nozzle is increased, with much smaller fluctuations. This could be due to the nature of the RANS models and time-averaging of the results. It is expected that employing the LES models may improve the capturing of the fluctuations in concentrations at the nozzle level. It is understood that in shallow water vertical discharges with fixed water depth, jet instabilities are increased with an increased Froude number. The root mean square error (RMSE) of 0.02 and mean error (ME) of 0.00 were obtained for C/C

_{0}when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.

_{min}) in discharges of vertical dense jets. The minimum dilution at the return point increases with a reduction in water depth and when the flow regime changes from deep to intermediate (Figure 7). However, after impinging the discharge into the water surface, the return point dilution decreases. The numerical results were able to capture the same mechanism as that observed in the experimental tests. Even though the discharge impingement onto the water surface resulted in jet expansion on the water surface (i.e., longer jet trajectory) and a delay in the fallback of the jet to the return point, the dilution was reduced in the case of the impinging flow regime under steady-state conditions. This suggests that surface attachment does not really contribute to the jet dilution, because water entrainment is really limited (i.e., not much water entrainment from bottom and top). It is also clear that an optimal design of vertical discharges in shallow water will be within an intermediate flow regime. Therefore, given the shallow bathymetry of a discharge location, the exit discharge should be designed such that the system reaches an intermediate flow regime. In the impinging regime, dilution decreased rapidly, which is a design point of concern. Fully impinging regimes can be recognized by monitoring the water surface disturbances visually or with instruments. It is therefore advised to reduce the discharge flow rate (or velocity) when surface disturbances are observed. A RMSE of 0.01 and ME of 0.01 were obtained for μ

_{min}/Fr when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.

#### 4.3. Discharge Maximum Rise

_{m}was nondimensionalized by the length scale r

_{0}Fr (Equation (9)) and was plotted against the nondimensionalized water depth (H/r

_{0}Fr) in Figure 8. As shown in this plot, the Z

_{m}/r

_{0}Fr reaches an almost constant level for deep water conditions. Note that both numerical and experimental cases are for the C series experiments in which the Froude number was kept similar and only the water depth was varied. As the water depth decreased, the discharge height increased until the jet front starts impinging the water surface. As expected, beyond this point, the discharge height decreased with water depth reduction. In the intermediate flow regime, the smaller water depth over the discharge generated less pressure on the jet front and caused the jet to expand in height and laterally. When the water depth increased and was sufficiently deep (i.e., H/r

_{0}> 1.5 Z

_{m}/r

_{0}, based on [11]), the influence of the water column pressure on the discharge remained constant and the increase in water column pressure on the discharge became negligible. A RMSE of 0.02 and ME of 0.01 were obtained for Z

_{m}/r

_{0}Fr when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.

_{m}/r

_{0}versus Fr). As expected, the discharge height increases with an increase in the Froude number until the jet’s attachment to the water surface, at which point the discharge height reaches an asymptotic limit by further increasing the Froude number. Numerical results from this study are in good agreement with the experimental results of [11] for H/r

_{0}= 37.2. Among the three cases modelled in the E series, the one with the lowest Fr is closest to the results of [6], which represents the deep flow regime. A RMSE of 2.94 and ME of −2.00 were obtained for Z

_{m}/r

_{0}when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.

#### 4.4. Spreading Radius

_{sp}as the radial distance of the spread where lateral flow separates the rigid plate.

_{sp}, discharge Fr number, and water depth H. Holstein and Lemckert [22] concluded that the total trajectory travelled by the buoyant fluid, prior to its separation from the plate (R

_{sp}+ H), is related to the discharge characteristics. Lemckert [10] discussed that the jet in shallow ambient water travels a distance of (R

_{sp}+ H) in the same way Z

_{m}does in deep water with no surface attachment.

_{sp})/r

_{0}when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.

#### 4.5. Quantitative Comparison

## 5. Conclusions and Recommendations

- The vertical discharge under an intermediate regime results in higher discharge maximum rise compared with that observed in a deep regime. This is because there is less pressure of the water column above the discharge in the intermediate regime. Another contributing factor may be the stress conditions between two boundaries (i.e., discharge front boundary and water surface boundary) and viscous forces between two fluids with different densities.
- The return point minimum dilution is higher for the vertical jet in an intermediate regime when compared with a deep flow regime. The return point minimum dilution significantly reduces in the fully impinging regime. The surface attachment will make the overall trajectory of the jet longer, but the mixing reduces the area of surface attachment, due to the reduced water entrainment (e.g., from atmosphere).

_{m}, to optimize the design to meet the depth criteria.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

B_{0} | buoyancy flux [m^{4}/s^{3}] |

C | concentration at each cell [ppm/ppt] |

C_{0} | discharge concentration [ppm/ppt] |

C_{a} | ambient concentration [ppm/ppt] |

C_{1} | return point concentration [ppm/ppt] |

Fr | densimetric Froude number [-] |

G | gravitational acceleration [m/s^{2}] |

g_{0}′ | reduced gravitational acceleration [m/s^{2}] |

g_{s}′ | effective gravity scale [m/s^{2}] |

H | water depth above nozzle level [m] |

K | turbulent kinetic energy [m^{2}/s^{2}] |

L_{s} | length scale to normalize jet height [m] |

M_{0} | momentum flux [m^{4}/s^{2}] |

Pr_{t} | turbulent Prandtl number [-] |

Q_{0} | discharge volume flux [m^{3}/s] |

${q}_{j}$ | turbulent scalar flux [-] |

r | radial distance [m] |

r_{0} | nozzle radius [m] |

R_{sp} | jet lateral spread [m] |

u | fluid velocity [m/s] |

U_{0} | discharge initial velocity [m/s] |

Z_{m} | discharge maximum rise [m] |

$\Gamma $ | calar diffusivity [kg/ms] |

${\Gamma}_{t}$ | turbulent dispersity |

${\delta}_{ij}$ | Kronecker delta [-] |

$\mu $ | fluid viscosity [Ns/m^{2}] |

${\mu}_{min}$ | minimum return point dilution [-] |

${\mu}_{t}$ | turbulent eddy viscosity [m^{2}/s] |

$\rho $ | fluid density at each cell [kg/m^{3}] |

${\rho}_{0}$ | discharge density [kg/m^{3}] |

${\rho}_{a}$ | ambient density [kg/m^{3}] |

${\tau}_{ij}$ | Reynolds stresses [-] |

## References

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**Figure 3.**Model geometry and computational mesh: (

**a**) full mesh, (

**b**) x-z plain, (

**c**) zoomed-in top face of the nozzle, and (

**d**) x-y view of the nozzle.

**Figure 4.**Vertical jet evolution in shallow water with surface interaction (C

_{0}is the discharge concentration and C is discharge computed at each cell): (

**a**) lateral cross section along the central plane, (

**b**) top view at the water surface at t = 60 s, and (

**c**) top view at the water surface at t = 20 s.

**Figure 5.**Top view of water surface disturbance observed in the experimental study for H/r

_{0}= 37.2 and (

**a**) Fr = 10.9, (

**b**) Fr = 18.2, and (

**c**) Fr = 24.2 ([11]; Reprinted from Water Science and Technology, volume 73, issue number 12, pages 2986–2997, with permission from the copyright holders, IWA Publishing).

**Figure 7.**Minimum dilution at the return point of vertical dense discharges in shallow waters [3].

**Figure 8.**Discharge maximum rise under fixed Froude number and variable ambient water depths [6].

**Figure 9.**Discharge maximum rise under fixed ambient water depths and variable Froude number [6].

**Figure 10.**Lateral spread of the vertical dense discharges at the surface in the impinging flow regime [10].

**Figure 11.**Numerical results (from current study) against experimental data [11] for different parameters.

Study | Findings |
---|---|

Yannopoulos and Noutsopoulos [4] | Studied the plane vertical turbulent buoyant jets to find the discharge flow spreading coefficients (Kc and Kw for velocity and concentration, respectively). They included a large range of discharge Froude numbers and showed that γ (spreading parameter) is constant (γ = 0.6). |

Zhang and Baddour [5] | Investigated the maximum height of vertical dense fountains with a large range of Froude numbers from small to large. They found out that Z_{m}/L_{m} reaches an asymptotic value for high Froude numbers (i.e., Fr > 7) and argued that the mass flux at discharge point has a negligible influence on the maximum discharge penetration. |

Baddour and Zhang [6] | Conducted an experimental investigation to find out the effect of density on round turbulent fountains. Their results revealed that the maximum penetration height of the discharge (Z_{m}/L_{m}) was a function of relative density difference (Δ_{ρ}/ρ_{a}). They concluded that the Z_{m}/L_{m} decreases from 3.06 to 2.59 at Δ_{ρ}/ρ_{a} = 0.001 and Δ_{ρ}/ρ_{a} = 0.1, respectively. |

Elhaggag et al. [7] | Studied the vertical dense discharges numerically using a computational fluid dynamics (CFD) model. They used FLUENT to model the vertical jets, however, this study did not verify the model using experimental data. |

Ahmad and Baddour [8] | Performed an experimental study of the terminal rise height and dilution of vertical dense jets. They quantified vertical and horizontal penetrations of the jet using thermometers and comparing with findings of previous studies. They concluded that the vertical jet penetration matched those from previous studies but the horizontal spread was smaller with a value of δ_{m} = 1.4r_{0}Fr, where r_{0} is the radius of nozzle and Fr is the discharge densimetric Froude number. |

Yan and Mohammadian [9] | Studied the lateral confinement impact on the vertical buoyant jets numerically using the OpenFOAM CFD model. They stated that the buoyancy-modified k-ɛ turbulence model was able to produce reasonable results for such jets. They also varied the Prandtl number (Pr) and turbulent Prandtl number (Pr_{t}) for various Froude numbers and claimed this affects the jet predictions. |

**Table 2.**Flow regime in shallow vertical dense jet discharges [11].

Deep | Intermediate | Impinging |
---|---|---|

H/r_{0} > 1.5 Z_{m}/r_{0} | Z_{m}/r_{0} < H/r_{0} < 1.5 Z_{m}/r_{0} | H < Z_{m} |

Experiment Series | Experiment # | D (2r_{0}) (mm) | U_{0} (m/s) | Δρ/ρ_{a} | ρ_{0} (kg/m^{3}) | Fr | H/r_{0} | Fr/(H/r_{0}) |
---|---|---|---|---|---|---|---|---|

A (Initial run) | 1 | 9.45 | 0.54 | 0.018 | 1011.89 | 18.69 | 34.0 | 0.55 |

B (Fixed H and variable Fr) | 2 | 9.45 | 0.28 | 0.019 | 1012.89 | 9.44 | 37.2 | 0.25 |

3 | 9.45 | 0.54 | 0.019 | 1012.89 | 18.20 | 37.2 | 0.49 | |

C (Fixed Fr and variable H) | 4 | 9.45 | 0.25 | 0.015 | 1008.91 | 9.48 | 25.4 | 0.37 |

5 | 9.45 | 0.25 | 0.015 | 1008.91 | 9.48 | 37.2 | 0.25 | |

6 | 9.45 | 0.25 | 0.015 | 1008.91 | 9.48 | 166.0 | 0.06 | |

D (Variable Fr and variable H) | 7 | 9.45 | 0.56 | 0.019 | 1012.89 | 18.87 | 37.2 | 0.51 |

8 | 9.45 | 0.45 | 0.019 | 1012.89 | 15.16 | 44.4 | 0.34 | |

9 | 9.45 | 0.54 | 0.019 | 1012.89 | 18.20 | 52.8 | 0.34 | |

E (Fixed H and variable Fr, intermediate Fr range) | 10 | 9.45 | 0.32 | 0.019 | 1012.89 | 10.80 | 37.2 | 0.29 |

11 | 9.45 | 0.36 | 0.019 | 1012.89 | 12.13 | 37.2 | 0.33 | |

12 | 9.45 | 0.42 | 0.019 | 1012.89 | 14.15 | 37.2 | 0.38 | |

F (Fixed H and variable Fr, high Fr range) | 13 | 9.45 | 0.43 | 0.019 | 1012.89 | 14.50 | 37.2 | 0.39 |

14 | 9.45 | 0.60 | 0.019 | 1012.89 | 20.22 | 37.2 | 0.54 | |

15 | 9.45 | 0.72 | 0.019 | 1012.89 | 24.26 | 37.2 | 0.65 |

Parameter/Error | C/C_{0} | Z_{m}/r_{0}Fr | Z_{m}/r_{0} | μ_{min}/Fr | (H + R_{sp})/r_{0} |
---|---|---|---|---|---|

RMSE | 0.02 | 0.02 | 2.94 | 0.01 | 2.38 |

ME | 0.00 | 0.01 | −2.00 | 0.01 | −0.03 |

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**MDPI and ACS Style**

Gildeh, H.K.; Mohammadian, A.; Nistor, I.
Vertical Dense Effluent Discharge Modelling in Shallow Waters. *Water* **2022**, *14*, 2312.
https://doi.org/10.3390/w14152312

**AMA Style**

Gildeh HK, Mohammadian A, Nistor I.
Vertical Dense Effluent Discharge Modelling in Shallow Waters. *Water*. 2022; 14(15):2312.
https://doi.org/10.3390/w14152312

**Chicago/Turabian Style**

Gildeh, Hossein Kheirkhah, Abdolmajid Mohammadian, and Ioan Nistor.
2022. "Vertical Dense Effluent Discharge Modelling in Shallow Waters" *Water* 14, no. 15: 2312.
https://doi.org/10.3390/w14152312