# CFD Modeling of Effluent Discharges: A Review of Past Numerical Studies

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Jet Studies: Details of Numerical Analysis

#### 2.1. Dimensional Analysis

_{0}, jet density ${\rho}_{0}$, jet nozzle diameter D, angle ϴ to the horizontal, and ambient water density ${\rho}_{a}$ (with ${\rho}_{0}$ > ${\rho}_{a}$). As it is discharged, the jet reaches a maximum or terminal rise height, y

_{t}, and then falls due to the negative buoyancy, mixing with the ambient water. The jet lands on the seabed and then it spreads horizontally as a density current.

_{0}, D, and ϴ, described above, plus the initial density difference $\Delta {\rho}_{0}$ = ${\rho}_{0}$ − ${\rho}_{a}$, the jet discharge concentration, C

_{0}, the turbulence intensity of the jet, I

_{TJ}, and the ambient water depth H

_{a}. The jet densimetric Froude number (Fr

_{d}) is a key parameter for dense jet analysis. Fr

_{d}is the ratio of inertia to buoyancy, and it is calculated with the reduced gravitational acceleration (${g}_{0}^{\prime}$) as follows:

_{0}), the jet discharge volume flux (Q

_{0}), and the kinematic momentum flux (M

_{0}):

_{t}), for a jet may be written with the momentum and source length scales (L

_{M}and L

_{Q}), using dimensional analysis as follows:

_{r}, and centerline peak dilution, S

_{m}, can be expressed as:

#### 2.2. CFD Governing Equations

_{eff}is effective kinematic viscosity (υ

_{eff}= υ

_{t}+ υ); υ

_{t}is turbulent kinematic viscosity; P is fluid pressure; g is gravitational acceleration; $\rho $ is fluid density; and ${\rho}_{0}$ is the reference fluid density.

_{eff}is the heat transfer coefficient, Pr is the Prandtl number, and Pr

_{t}is the turbulent Prandtl number.

## 3. Discharge through Inclined Dense Jets

_{ct}= 0.9. Both of these simulations with the k-ε turbulence model were more accurate than the integral models and the analytical solutions. In the experimental studies, the buoyancy-induced instabilities were observed on the lower (inner) half of the jet. However, in the study of [35], the k-ε simulations (both the standard and the calibrated) underpredicted the spread of the jet and the integrated centerline dilution at the top rise of the jet, because they overestimated the influence of the stabilizing density gradients. This study only achieved qualitative comparisons and did not arrive at any conclusions about which model performed the best.

_{0}, where C is the computed concentration and C

_{0}is the discharge concentration) and the dilution isolines for inclinations of 30° and 45°; realizable k-ε and LRR turbulence models were used for each case.

_{a}is ambient concentration) was used to calculate the dilution values for plotting in Figure 3. The discharge trajectory and flow growth are also illustrated in Figure 3 as the jets travel downstream with clear differences between the two turbulence models. LRR is an anisotropic turbulence model and performs more reliably for calculating the shear forces on jet edges and providing more realistic predictions. Table 2 and Table 3 summarize the discharge trajectory and flow growth compared to other turbulence models.

_{0}− C

_{a}/C − C

_{a}

_{μ}= 0.09, C

_{1ε}= 1.44, C

_{2ε}= 1.92, and a calibrated coefficient, C

_{3ε}for sensitivity tests with C

_{3ε}= 0.9, 0.6, and 0.4.

_{3ε}), the discharge growth rate is larger, and the numerical results align more closely to the data by experiments (not presented in this paper). The SGDH and GGDH methods showed similar results when the computed and experimental discharge trajectories were compared. This could be attributed to the small difference in density between the discharge and the receiving water (smaller than 1%). The buoyancy effect on the effluent discharge mixing characteristics should be investigated further, especially for larger density differences between the discharge and ambient water (larger than 1% in this study) as well as C

_{3ε}calibration for mixing applications.

_{s}) in the eddy viscosity, ${\upsilon}_{t}=\rho {\left({C}_{s}\u25b3\right)}^{2}{S}_{ij}$, where Δ is the LES filter size and S

_{ij}is the strain rate tensor, needs to be specified. The accuracy in choosing this parameter is directly related to the kinetic energy dissipation.

_{s}. Rather than obtaining the minimal coefficient from predetermined expressions, the model analyzed large-scale turbulences during the numerical simulation to deduce them. This dynamic model approach was proved to be more applicable in the cases with complex interactions.

#### Discussion on Differences in RANS and LES Models for Effluent Mixing Problems

_{t}) instead of defining the turbulent diffusivity (${\kappa}_{t}$) explicitly. The turbulent Prandtl number is defined as:

## 4. Vertical Jets

_{µ}, as a function of mean flow and turbulence properties, instead of assuming that it was a constant as in the standard model. This expression (i.e., realizability) satisfies certain constrains on the Reynolds stresses, which makes this model stronger compared to the other k-ε models for mixing problems.

- An assumption was made that improves the determinacy and aids in the calibration of CFD modeling, which is: the Prandtl number Pr and turbulent Prandtl number Pt
_{r}would be related to the densimetric Froude number Fr_{d}:Pt_{r}= Pr = (0.032Fr_{d}+ 0.89) – 1 - The influence of water surface is smaller than the influence of confinement, although the distribution of concentration along a cross-section in a confined discharge could be impacted by boundaries.
- A conclusion was made to enable engineers and researchers to quickly estimate the evolution of a laterally confined vertical buoyant discharge, which is that the rate of discharge concentration growth is almost equal to b
_{gc}/s = 0.0938, where b_{gc}is the concentration 1/e width (e is the Napier’s constant) and that is where the jet concentration reaches to 1/e of the centerline maximum concentration.

_{p}is the port diameter, p

_{s}is the port spacing, and α, β, and γ are regression coefficients.

## 5. Horizontal Jets

_{d}. An exponential relationship was found for the jet centerline velocity.

_{d}. The temperature dilution in the wall jet region decays according to Equation (32). (ii) The central surface velocity profiles resemble that of a turbulent wall jet and generally agree with the classical curve of wall jets. (iii) The velocity profile shows strong similarity in the vertical plane beyond an x > 5D distance from the nozzle, fitting to a Gaussian shape. (iv) The centerline velocity decay is related to the densimetric Froude number Fr

_{d}and the nozzle diameter D and fits Equation (33), where U

_{0}is the velocity at the source and U

_{m0}is the centerline maximum velocity.

_{0}/U

_{m0}= 0.65x/(DFr

_{d}

^{0.5}) + 0.3

_{0}) are distributed in the wall normal direction parabolically, while the maximum value decreases along the stream-wise direction linearly. The concentration (C/C

_{m}) profiles were similar in the region with a steady C

_{0}value.

_{d}= 11.61 to 42.33). The linear (standard k-ε, RNG k-ε, realizable k-ε and SST k-ω) and Reynolds stress (Launder-Gibson and LRR) turbulence models were tested in their study. This was the first study for wall jet modeling that compared several turbulence models.

_{a})/(T

_{0}− T

_{a}) = 3% [58]. Figure 7 shows the numerical cling length results from [23] compared to the experimental data and numerical results of [58].

_{d}for each turbulence model evaluated in [23].

_{m}is the x-direction velocity (along y at the central plane), U

_{m0}is the maximum of U

_{m}, with the ordinate of y, and y

_{m/2}is the velocity-half-height, which is the height where U

_{m}=U

_{m0}/2. All profiles for stream-wise velocity are self-similar and in good agreement with the data by [63], as plotted in Figure 9. Ref. [60]’s equation, which is suitable for 2D wall jets, is also plotted along with other data for comparison. Ref. [60]’s equation reads:

_{m/2}> 1) where the momentum forces dissipate, and buoyancy forces get stronger. The literature often reports the velocity self-similarity profiles at the central plane for both experimental and numerical studies, although usually without presenting results for the offset measurement from the centerline.

_{m/2}is the local length scale and U

_{ms}is the maximum velocity for the offset sections. As seen in the figure, the numerical results (current study) for z/D = 3.636 do not agree well with Verhoff’s curve in the area close to the nozzle (x/D = 5 and x/D = 10). This is primarily due to the development of the jet in the tank width (Figure 11). As z/D increases, the jet may not develop at the values along the width of the tank yet, and thus the scatters show less self-similarity.

_{rd}, and ambient water depth H

_{a}, etc.). It was observed that the effect of distance (the path that the jet travels through) on the dilution factor was larger than the effect of the Froude number at the nozzle.

_{rd}, ranging from 9.9 to 29.8) and density differences ($\Delta \rho $, ranging from 5.1 to 17.41). Three RANS turbulence models were adopted for their numerical study: standard k-ε, realizable k-ε, and buoyancy-modified k-ε. They concluded that the realizable k-ε model was more successful in predicting the discharge trajectories. The main finding of this study was that while using different combinations of parameters in discharge (salinity versus temperature) for keeping the same properties of the jet (the same values of F

_{rd}and same $\Delta \rho $), the trajectory and mixing characteristics of the jets would be different in the same ambient water. Therefore, it is important not to only look at the relative buoyancy between discharge and ambient water, but also the properties of discharge such as salinity and temperature, which could be very important in the overall mixing efficiency of the jets.

## 6. Surface Discharges

- Surface discharges using different channel geometries in calm ambient water
- Surface discharges using different channel geometries in co-flow ambient water
- Surface discharges using different channel geometries in cross-flow ambient water

## 7. Discharge Port Configuration

- The RNG k-ε model accounts for the influence of the Reynolds number on the effective turbulence transport.
- The RNG k-ε model calculates the inverse effective Prandtl numbers, using a more advanced equation.
- The RNG k-ε model includes a new term in the ε transport equation that improves the calculation of the turbulent viscosity.

## 8. Critical Review and Future Research Needs

## 9. Conclusions

- Numerically, the most studied effluent discharge configuration has been in inclined dense jets, due to their applicability in industry. Most studies focused on lab-size experiments to calibrate their models. Details on jet trajectories and dilution and velocity characteristics have been investigated and compared to experimental data. RANS and LES turbulence models are popular for such studies.
- Vertical jets are also popular in CFD studies, and the new trend in studying these jets involves considering the ambient conditions that may affect these jets such as lateral confinement and water shallowness, where the jet is attached to the top boundary. Cross-flow in vertical jets could have a significant influence in terms of the trajectory and dilution, both of which are getting more attention from researchers using CFD models.
- Horizontal jets could be either positively buoyant or negatively buoyant with attachment to the bed (i.e., wall jets) or elevated (i.e., offset jets). Single jets have been studied experimentally and numerically during the past years, and more attention is now given to interactions of multiple horizontal jets when they merge after a certain distance from the discharge point.
- Surface discharges have not been studied yet using a CFD approach, even though they are used in the industry and experimental data are available on such jets. This literature gap also exists for more complex jet configurations such as multiport and rosette diffusers.
- Previous studies have mainly focused on the stagnant ambient water due to the simplicity of internal and boundary conditions. However, to replicate real-life conditions more precisely, there is a need to move toward more complex ambient conditions to study the effects of wave, wind, co-flow and crossflow, density stratification, etc. on jet mixing and dispersion in CFD models.
- A wide range of turbulence models are available and have already been implemented in different CFD platforms that could be used for discharge mixing studies. Modifications of turbulence models, such as implementing the buoyancy terms, has been shown to be effective in improving the prediction of jet characteristics.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Mathematical Symbols

B_{gc} | Jet concentration 1/e width |

B_{0} | Jet buoyancy flux |

C_{a} | Ambient water concentration |

C_{m} | Jet centerline maximum concentration |

C_{0} | Jet exit concentration |

D or d | Nozzle diameter or diffusion coefficient |

Fr_{d} | Densimetric Froude number |

G | Swirl number |

g | Gravitational acceleration |

${g}_{0}^{\prime}$ | Reduced gravitational acceleration |

H_{a} | Ambient water depth |

k | Turbulent kinetic energy |

k_{eff} | Heat transfer coefficient |

L | Cling length |

L_{M} | Jet momentum-length scale |

L_{Q} | Jet source-length scale |

M_{0} | Jet kinematic momentum flux |

P | Fluid pressure |

Pr | Prandtl number |

Pr_{t} | Turbulent Prandtl number |

Q_{0} | Jet discharge volume flux |

Re | Reynolds number |

Ri | Richardson number |

S | Dilution |

s | Jet stream-wise distance |

S_{m} | Jet centerline peak dilution |

S_{r} | Jet dilution at return point |

t | Time |

T | Jet temperature at any location |

T_{a} | Ambient water temperature |

T_{0} | Jet exit temperature |

U_{m} | Jet velocity component in the x-direction |

U_{m0} | Jet centerline maximum velocity |

U_{0} | Jet exit velocity |

u | Mean velocity component in the x-direction |

u_{o} | Inlet velocities of the offset jet |

u_{w} | Inlet velocities of the wall jet |

v | Mean velocity component in the y-direction |

V_{r} | Velocity ratio |

w | Mean velocity component in the z-direction |

x_{i} | Horizontal location of impact |

x_{m} | Jet centerline peak horizontal location |

x_{r} | Jet return point |

y_{m} | jet centerline peak vertical location |

y_{m/2} | Jet velocity-half-height (the height of U_{m}=U_{m0}/2) |

y_{t} | Terminal rise height |

## Greek Symbols

α | Jet entrainment constant |

ε | Turbulent dissipation rate |

ϴ | Angle of discharge |

µ_{t} | Turbulent viscosity |

$\rho a$ | Ambient water density |

$\rho 0$ | Jet exit density |

υ | Kinematic viscosity |

υ_{t} | Turbulent kinematic viscosity |

υ_{eff} | Effective kinematic viscosity |

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**Figure 1.**Surface discharge of the Al-Ghubrah desalination plant, the largest such facility in Oman (Source: H.H. Al-Barwani).

**Figure 2.**A schematic view of an inclined dense jet with negative buoyancy in stagnant ambient water.

**Figure 3.**Mixing regimes for 30° and 45° inclined dense jets. (

**a**) 30°, realizable k-ε; (

**b**) 30°, LRR; (

**c**) 45°, realizable k-ε; (

**d**) 45°, LRR (Source: [25]).

**Figure 4.**Overall discharge trajectory for 45° inclined dense jets (Source: [25]). (

**a**) Centerline comparison (modeled vs. experiments); (

**b**) standard k-ε; (

**c**) modified k-ε with SGDH, C

_{3ε}= 0.9; (

**d**) modified k-ε with GGDH, C

_{3ε}= 0.9; (

**e**) modified k-ε with standard Boussinesq gradient diffusion hypothesis (SGDH), C

_{3ε}= 0.6; (

**f**) modified k-ε with general gradient diffusion hypothesis (GGDH), C

_{3ε}= 0.6; (

**g**) modified k-ε with SGDH, C

_{3ε}= 0.4; (

**h**) modified k-ε with GGDH, C

_{3ε}= 0.4.

**Figure 5.**Discharge growth width at various cross-sections: the contour lines represent S = C

_{0}/C = 1 (Source: [25]). (

**a**) 30°, realizable k-ε; (

**b**) 30°, LRR; (

**c**) 45°, realizable k-ε; (

**d**) 45°, LRR.

**Figure 6.**Schematic view of the model by [23].

**Figure 7.**Cling length comparison of numerical vs. experimental results (Source: [23]).

**Figure 8.**Centerline trajectory. (

**a**) Fr

_{d}: approximately 12; (

**b**) Fr

_{d}: approximately 20 (Source: [23]).

**Figure 9.**Self-similarity of stream-wise velocity profiles for various turbulence models (Source: [23]).

**Figure 10.**Velocity at offset sections z/D = 1.818 and 3.636. Solid fill scatters are for z/D = 1.818 and the no-fill scatters are for z/D = 3.636 at the x/D values on the plot.

**Figure 12.**Temperature dilution contours at the plane of symmetry. Dilution rates are 12, 15, 20, 30, and 60 (realizable k-ε turbulence model).

**Figure 13.**Stream-wise self-similarity temperature profiles for three cases at various cross-sections.

**Figure 14.**Multiport diffusers. (

**a**) unidirectional diffusers with cross-flow, (

**b**) alternating diffuser.

Models | Mathematical Approaches for Jet/Plume Mixing | Availability | Major Functionalities and Capabilities |
---|---|---|---|

CORMIX [11] | Empirical solutions; Eulerian jet integral method | Commercial model | Prediction of jet and (or) plume geometry and dilution in the near field; single or multiple jets |

VISJET | Lagrangian jet integral method | Commercial model | |

Visual PLUMES | Empirical solutions; Eulerian and Lagrangian jet integral methods | Free package | |

NRFIELD | Empirical solutions | Free package | Prediction of jet and (or) plume geometry and dilution in the near field of multiport diffusers |

Sophisticated Multidisciplinary Models | |||

OpenFOAM | FVM; RWPT method | Free package | Predictions of ocean hydrodynamics; pollutant fate and transport in the near and far fields; water quality; sediment processes |

MIKE21/3 | FVM; RWPT method | Commercial package | |

Delft3D | FDM; RWPT method | Free package | |

ANSYS CFX | FVM; RWPT method | Commercial package | |

ANSYS Fluent | FVM; RWPT method | Commercial package | |

FLOW-3D | FDM; RWPT method | Commercial package | |

TELEMAC-2D/3D | FEM; RWPT method | Free package | |

EFDC–Hydro | FDM; RWPT method | Free package | Predictions of ocean hydrodynamics; Pollutant dispersion in the far field; Near-field processes using the embedded jet model JETLAG; Suspended sediment transport |

HydroQual–ECOMSED | FDM; RWPT method | Free package | Predictions of ocean hydrodynamics; Pollutant fate and transport in the far field; Sediment processes |

**Table 2.**Comparison of numerical and experimental coefficients for the 30° inclined jets (Source: [24]). LIF: Laser-Induced Fluorescence, LRR: Launder-Reece-Rodi.

Parameter | Proportionality Coefficient | [24] | [21] | [20] | [37] | [38] | [39] | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

Realizable k-ε | LRR | 0.10 ≤ y_{0}/L_{m} ≤ 0.15 | y_{0}/L_{m} > 0.15 | LA data | LIF data | Theory | |||||

Terminal rise height | $\frac{{Y}_{t}}{{L}_{M}}$ | 1.13 | 1.13 | 1.13 | ‒ | 1.14 | 1.34 | 1.08 | 1.49 | 1.15 | 1.22 |

Horizontal location of return point | $\frac{{X}_{r}}{{L}_{M}}$ | 3.40 | 3.34 | 3.06 | 3.19 | 3.34 | 3.66 | 3.14 | 3.51 | 3.22 | 3.70 |

Return point dilution | $\frac{{S}_{r}}{Fr}$ | 1.27 | 1.31 | 1.18 | 1.45 | ‒ | ‒ | ‒ | 1.90 | ‒ | ‒ |

Vertical location of centerline peak | $\frac{{Y}_{m}}{{L}_{M}}$ | 0.71 | 0.69 | 0.70 | ‒ | 0.59 | 0.70 | 0.66 | ‒ | 0.84 | ‒ |

Horizontal location of centerline peak | $\frac{{X}_{m}}{{L}_{M}}$ | 2.05 | 1.97 | 1.81 | 1.64 | 1.86 | 1.97 | 1.81 | ‒ | 2.07 | ‒ |

Centerline peak dilution | $\frac{{S}_{m}}{Fr}$ | 0.65 | 0.63 | 0.62 | 0.66 | ‒ | ‒ | ‒ | ‒ | ‒ | 0.36 |

**Table 3.**Evaluation of performance of two turbulence models (realizable k-ε and LRR models) for 30° inclined jet (Source: [24]).

Parameter | Proportionality Coefficient | [24] | Average of Experiments | Absolute Difference (%) | ||
---|---|---|---|---|---|---|

Realizable k-ε | LRR | Realizable k-ε | LRR | |||

Terminal rise height | $\frac{{Y}_{t}}{{L}_{M}}$ | 1.13 | 1.13 | 1.22 | 8.09 | 8.09 |

Horizontal location of return point | $\frac{{X}_{r}}{{L}_{M}}$ | 3.40 | 3.34 | 3.37 | 1.04 | 0.75 |

Return point dilution | $\frac{{S}_{r}}{Fr}$ | 1.27 | 1.31 | 1.51 | 18.90 | 15.27 |

Vertical location of centerline peak | $\frac{{Y}_{m}}{{L}_{M}}$ | 0.71 | 0.69 | 0.70 | 1.91 | 0.97 |

Horizontal location of centerline peak | $\frac{{X}_{m}}{{L}_{M}}$ | 2.05 | 1.97 | 1.86 | 10.22 | 5.91 |

Centerline peak dilution | $\frac{{S}_{m}}{Fr}$. | 0.65 | 0.63 | 0.55 | 18.90 | 15.24 |

Turbulence Model | Standard k-ε | RNG k-ε | Realizable k-ε | SST k-ω | Launder-Gibson | LRR | Experiment [62] |
---|---|---|---|---|---|---|---|

Cling Length | L/D = 2.68Fr_{d} | L/D = 2.76Fr_{d} | L/D = 2.65Fr_{d} | L/D = 2.50Fr_{d} | L/D = 2.79Fr_{d} | L/D = 2.70Fr_{d} | L/D = 3.2Fr_{d} |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mohammadian, A.; Kheirkhah Gildeh, H.; Nistor, I. CFD Modeling of Effluent Discharges: A Review of Past Numerical Studies. *Water* **2020**, *12*, 856.
https://doi.org/10.3390/w12030856

**AMA Style**

Mohammadian A, Kheirkhah Gildeh H, Nistor I. CFD Modeling of Effluent Discharges: A Review of Past Numerical Studies. *Water*. 2020; 12(3):856.
https://doi.org/10.3390/w12030856

**Chicago/Turabian Style**

Mohammadian, Abdolmajid, Hossein Kheirkhah Gildeh, and Ioan Nistor. 2020. "CFD Modeling of Effluent Discharges: A Review of Past Numerical Studies" *Water* 12, no. 3: 856.
https://doi.org/10.3390/w12030856