Three-Dimensional Numerical Simulations of Buoyant Jets Discharged from a Rosette-Type Multiport Diffuser

Abstract

In some outfall systems, wastewaters are discharged into ambient water bodies using rosette-type diffusers in the form of multiple buoyant jets, and it is essential to simulate their mixing characteristics for practical applications and optimal design purposes. The mixing processes of a rosette jet group are more complicated than single jets and multiple horizontal or vertical jets, and thus the existing methods cannot be effectively used to simulate their mixing and dilution properties. With the recent advancements in numerical modeling approaches, numerical simulation of wastewater jets as three-dimensional phenomena can be feasible. The present study deals with a fully three-dimensional numerical simulation for buoyant jets discharged from a rosette-type multiport diffuser, with the standard and re-normalization group (RNG) k-ε turbulence models. The simulated results are compared with experimental data, and the results show a good agreement with the experimental data, demonstrating that the numerical model is an efficient and effective tool for simulating rosette jet groups. It was also concluded that the RNG k-ε model leads to better results than the standard k-ε model with a comparable computational cost. The validated model was further utilized to investigate the influences of port inclinations on the mixing behaviors.

1. Introduction

Wastewater jets that have a lower density than the ambient water often emanate from desalination and municipal activities [1,2,3]. These discharges are known as buoyant jets [4], and improper disposal of these discharges may result in significant environmental and ecological impacts [5,6]. Therefore, estimating the mixing processes of these discharges is very important.

The mixing processes of wastewater jets are quite sensitive to the configurations of the diffusers [7,8]. If there are enough spacing and funds, a single discharge is the optimal configuration because the ambient water can be freely entrained into the jets, and thus the effluent is significantly diluted. However, the spacing and funds are often limited in practical engineering projects, so it is becoming a common practice to use diffusers with multiple ports [8,9]. Depending on how the ports are configured, a multiport diffuser can be classified into different types, such as unidirectional [9,10], Tee-shaped [11], and rosette-shaped [12,13]. Rosette-type multiport diffusers are becoming popular nowadays. As shown in Figure 1, a rosette diffuser consists of a cluster of ports that are located around a circle. After discharged from the ports, the jets are typically bent over toward the upward direction because they have a lower density than the ambient water. These jets may interact with each other if the port spacing is not sufficiently large. The buoyant jets discharged from a rosette-type diffuser are hereafter referred to as the rosette buoyant jets. The mixing mechanisms of these jets typically include buoyancy effects, inclined trajectories, and jet interactions from various directions [14], so they are much more complicated than single jets and multiple jets discharged from unidirectional or Tee-shaped multiport diffusers. Therefore, the practice of predicting the mixing characteristics of rosette jets has not yet been adequately handled and needs to be improved.

Physical modeling and experimental approaches are currently the most common ways of studying rosette jets (

Table 1. Summary of relevant previous studies on the mixing properties of multiple wastewater jets.

Study	Method	Type	Outcome
Lai and Lee [14]	Experimental	Rosette buoyant jets in stagnant water	Measured data
Abessi et al. [12]	Experimental	Rosette dense jets in stagnant water	Measured data
Abessi and Roberts [13]	Experimental	Rosette dense jets in flowing currents	Measured data
Wang and Davidson [15]	Theoretical	Unidirectional multiple jets	A theoretical model
Yannopoulos and Noutsopoulos [16]	Entrainment restriction	Unidirectional multiple jets	A theoretical model
Jirka [17]	Jet integral	Unidirectional multiple jets	A jet integral model
Lai and Lee [14]	Semi-analytical	Rosette buoyant jets in stagnant water	A semi-analytical model
Yan and Mohammadian [19]	Artificial intelligence	Multiple inclined dense jets	AI-based models
Yan and Mohammadian [20]	Artificial intelligence	Multiple vertical jets	AI-based models
Yan and Mohammadian [8]	Numerical	Multiple inclined dense jets	A numerical model

). For example, Lai and Lee [14] published the measured trajectories of the jets discharged from rosette-shaped diffusers to stagnant water, Abessi et al. [12] measured the mixing processes of rosette dense jets in stationary receiving water, and Abessi and Roberts [13] reported the measurements of the concentration fields for rosette dense jets in flowing currents. These previous experimental works have significantly advanced the knowledge of rosette jets. However, laboratory methods are typically very time-consuming and expensive, so it is urgent to develop supplementary methods. Complementing physical modeling and experimental approaches, several studies on analytical or simplified numerical models for unidirectional multiple jets have been reported [15,16,17]. Different from unidirectional or Tee-shaped multiple jets, the effects of jet interactions and entrainment restrictions involved in rosette jets occurred in various directions, and thus the well-established theory for unidirectional multiple jets is less applicable for rosette jets. Lai and Lee [14] proposed a general semi-analytical model for multiple buoyant jets that can be used for rosette jets, which provided a very good avenue of predicting the mixing characteristics of rosette jets. However, the uncertain parameters, such as the contraction coefficient, involved in the model may result in unacceptable uncertainties, and utilizing the model for complicated cases may not be feasible. Recently, Yan and Mohammadian have successfully utilized an artificial intelligence approach based on multi-gene genetic programming for buoyant jets subjected to lateral confinement [18], multiple inclined dense jets [19], and multiple vertical buoyant jets [20]. However, it has been stated by the authors that this type of model requires extensive reliable data for further training and generalization.

With the recent advancements in numerical modeling approaches, numerically modeling wastewater jets as three-dimensional (3D) phenomena can be feasible. The majority of the existing studies on numerical modeling of wastewater jets focused on single jets in unbounded conditions [21,22,23]. A small number of works modeled single jets near boundaries [2,24], and Yan and Mohammadian [1] simulated laterally confined single jets. Studies on numerical modeling of multiple jets are very few. Lou et al. [25] modeled two coalescing plumes in stratified conditions, and Yan and Mohammadian [8] simulated multiple inclined dense jets discharged from moderately spaced ports. The dilution mechanisms of rosette buoyant discharges are much more complicated than the reported studies on single or multiple unidirectional jets, so it is necessary to evaluate the applicability of a model for rosette buoyant jets. To the authors’ knowledge, 3D numerical simulations of rosette buoyant jets have rarely been reported.

The key purpose of the present research is to assess if the dilution properties of rosette buoyant jets can be simulated by a 3D model based on the Reynolds-averaged Navier–Stokes equations. Another objective of the present study is to compare the competences of the standard and re-normalization group (RNG) k-ε turbulence models. The standard k-ε model is the most popular closure for turbulence modeling because it has a good balance between efficiency and accuracy, and it is very beneficial to figure out a turbulence closure that is more accurate with a comparable computational cost. The study also aims to carry out some additional computations using the validated model to check the influences of port inclinations on the mixing properties. The present study can improve the practice of numerical modeling of wastewater effluents, so it can also provide useful information for the numerical simulations of other types of wastewater discharges.

2. Methodology

2.1. Dimensional Analysis of Rosette Buoyant Jets

The mixing properties of a jet can be indicated by the concentration of the jet, which can be normalized as C/Co, in which C is the concentration at a certain location, and Co is the initial concentration of the jet. It is well documented that the mixing properties of a single jet are mainly affected by the discharge volume flux Q, kinematic momentum flux M, and buoyancy flux B [26,27], as:

$$\frac{C}{Co}=f\left(Q,M,B\right);Q=\frac{\mathsf{\pi }}{4}{D}^2{U}_j;M=U_{j}Q;B=g^{\prime }Q$$

(1)

where D is the diameter of the discharge port; U_j is the initial velocity of the jet; g’ is the reduced gravitational acceleration.

A dimensional analysis indicates that the normalized jet concentration is a function of the densimetric Froude number, Fr, and the initial angle of the jet, θ, as [28,29]:

$$\frac{C}{Co}=f\left(Fr,\theta \right)$$

(2)

with

$$Fr=\frac{U_j}{\sqrt{g^{\prime }D}}$$

(3)

For uni-directional multiple jets, the jet dilution is also affected by the other jets. Therefore,

$$\frac{C}{Co}=f\left(Fr,\theta ,s\right)$$

(4)

where s denotes the port spacing.

For buoyant jets discharged from a rosette-type multiport diffuser, the actual port spacing is difficult to define because the jet interactions and entrainment restrictions may occur in various directions and the mutual effects cannot be simply represented by the port spacing. Therefore, the mechanisms of rosette buoyant jets are more complicated than single or unidirectional multiple jets, and are thus difficult to describe or predict based on simple dimensional or theoretical analysis.

2.2. Governing Equations

The 3D Navier–Stokes equations for mixing two fluids can be expressed as [30,31]:

$$\nabla \cdot \mathbf{U}=0$$

(5)

$$\frac{\partial \rho \mathbf{U}}{\partial t}+\nabla \cdot \left(\rho \mathbf{UU}\right)=-\nabla \cdot \left(p_{rgh}\right)-gh\nabla \rho +\nabla \cdot \left(\rho \mathbf{T}\right)$$

(6)

with:

$$\rho ={\alpha }_1{\rho }_1+{\alpha }_2{\rho }_2={\alpha }_1{\rho }_1+\left(1-{\alpha }_1\right){\rho }_2$$

(7)

$$\mathbf{T}=-\frac{2}{3}{\overline{\mu }}_{eff}\nabla \cdot \mathbf{UI}+{\overline{\mu }}_{eff}\nabla \mathbf{U}+{\overline{\mu }}_{eff}{\left(\nabla \mathbf{U}\right)}^T$$

(8)

$${\overline{\mu }}_{eff}={\alpha }_1{\left({\mu }_{eff}\right)}_1+{\alpha }_2{\left({\mu }_{eff}\right)}_2$$

(9)

$${\left({\mu }_{eff}\right)}_i={\left(\mu -{\mu }_t\right)}_i$$

(10)

where t indicates time, U indicates velocity, and ρ represents density. The variable p_rgh is defined as the static pressure minus hydraulic pressure, and h denotes the height of the fluid column. The variable α represents the volume fraction of the fluids, μ denotes the dynamic viscosity, and μ_t is turbulent viscosity. The subscript i denotes either the clean water or wastewater jet.

A transport equation is utilized to calculate the field of α, as:

$$\frac{\partial {\alpha }_1}{\partial t}+\nabla \cdot \left(\mathbf{U}{\alpha }_1\right)=\nabla \cdot \left(\left(D_{ab}+\frac{{\nu }_t}{S_C}\right)\nabla {\alpha }_1\right)$$

(11)

where D_ab represents the molecular diffusivity, ν_t denotes the turbulent eddy viscosity, and S_C indicates the turbulent Schmidt number. The typical and default values for D_ab and S_c, 1×10⁻⁶ m²/s and 1, were adopted in this study.

2.3. Turbulence Models

The standard k-ε turbulence model can be written as:

$$\frac{\partial k}{\partial t}+\frac{\partial k{u}_i}{\partial x_i}-\frac{\partial }{\partial x_i}\left(D_{keff}\frac{\partial k}{\partial x_i}\right)=G-\epsilon$$

(12)

$$\frac{\partial \epsilon }{\partial t}+\frac{\partial \epsilon u_i}{\partial x_i}-\frac{\partial }{\partial x_i}\left(D_{\epsilon eff}\frac{\partial \epsilon }{\partial x_i}\right)=c_{1\epsilon }\frac{\epsilon }{k}G-c_{2\epsilon }\frac{{\epsilon }^2}{k}$$

(13)

with

$$D_{keff}={\nu }_t+\nu$$

(14)

$$D_{\epsilon eff}=\frac{{\nu }_t}{{\sigma }_{\epsilon }}+\nu$$

(15)

$${\nu }_t=c_{\mu }\frac{k^2}{\epsilon }$$

(16)

$$G=2{\nu }_t{S}_{ij}{S}_{ij}$$

(17)

$$S_{ij}=\frac{1}{2}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)$$

(18)

where k represents the turbulent kinetic energy, ε indicates the turbulent energy dissipation rate, and G is the production of turbulence due to shear. The values of the model constants σ_ε, c_1ε, c_2ε, and c_μ are 1.3, 1.44, 1.92, and 0.09, respectively.

The RNG k-ε turbulence model can be expressed as:

$$\frac{\partial \epsilon }{\partial t}+\frac{\partial \epsilon u_i}{\partial x_i}-\frac{\partial }{\partial x_i}\left(D_{\epsilon eff}\frac{\partial \epsilon }{\partial x_i}\right)=\left(c_{1\epsilon }-R_{\epsilon }\right)\frac{\epsilon }{k}G-c_{2\epsilon }\frac{{\epsilon }^2}{k}$$

(19)

with

$$R_{\epsilon }=\frac{\eta \left(1-\eta /{\eta }_0\right)}{1+\beta {\eta }^3}$$

(20)

$$\eta =\sqrt{S_2}\frac{k}{\epsilon }$$

(21)

where model constants σ_k, σ_ε, c_1ε, c_2ε, c_μ, η₀, and β are 0.71942, 0.71942, 1.42, 1.68, 0.0845, and 0.012, respectively.

2.4. Mesh Configurations and Model Setup

The present study utilized the multi-fluid solver “twoLiquidMixingFoam” within the framework of OpenFOAM to solve the governing equations. This solver has been validated for single jets by Zhang et al. [32] and Yan and Mohammadian [8] for multiple dense jets. However, to the best of the authors’ knowledge, it has not been demonstrated to be capable of simulating rosette jets. The present study first numerically simulated the experiment of rosette buoyant jets reported by Lai and Lee [14], and then used the validated model to carry out six additional computations. The relevant parameters used in the numerical simulations are summarized in

Table 2. Relevant parameters used in the numerical study.

D (m)	Uj (m/s)	Fr (—)	Dc (m)	Hc (m)	Lmin (m)	Lmax (m)
0.0044	0.365	2.5	0.6	0.65	0.001	0.005

Note: D = jet diameter; U_j = initial jet velocity; Fr = densimetry Froude number; D_c = diameter of the cylinder for the computational domain; H_c = height of the cylinder; L_min = minimum grid size; and L_max = maximum grid size.

.

In the considered cases, the number of ports was six (Figure 2); the jet diameter, D, was 0.0044 m; the initial jet velocity, U_j, was 0.365 m/s; and the densimetric Froude number, Fr, was 2.5. As indicated by Figure 2a, the entire domain can be subdivided into 12 segments by the planes of symmetry. To save computational costs, the present simulations only considered the first segment S1, utilized the symmetric boundary condition [8,33] for the planes of symmetry, and employed the “reflect” and “mirror” functions available in ParaView and Tecplot to mirror the results. Preparatory tests indicated that the differences between the present results and those obtained by simulations with the entire domain covered were negligible. The no-slip boundary condition with the standard wall function was assigned to the bottom patch. The fixed-value velocity was employed for the jet inlets, and the “inlet-outlet” boundary condition [1,8] was assigned to the top and farther patches.

The diameter of the cylinder for the computational domain was 0.6 m, and the differences in the results obtained by the present simulations and those with a larger computational domain were negligible. The height of the cylinder was 0.65 m, which was consistent with the reported water level in the experiment [14]. The initial angles of the jets in different cases, θ, are summarized in

Table 3. Geometrical parameters and number of computational cells of the considered cases.

Cases	Inclination (—)	θ (°)	Lc (m)	Yc (m)	nC (—)
C1	Horizontal	0	0.028	0.140	952,073
C2	Upward	30	0.032	0.108	953,578
C3	Upward	45	0.040	0.084	958,582
C4	Upward	60	0.056	0.043	962,545
C5	Downward	−30	0.032	0.172	961,120
C6	Downward	−45	0.040	0.196	961,177
C7	Downward	−60	0.056	0.237	968,761

Note: θ = initial jet angle; Lc = the port length; Yc = vertical location of the port ends; and nC = number of computational cells.

. The coordinates of the jet inlets (the port surfaces) were kept the same in different cases. The horizontal distances between the jet inlets and the centerline of the computational domain was kept as 0.028 m, and the vertical distances between the jet inlets and the bottom patch was kept as 0.140 m. To achieve this consistency, the port lengths and the vertical locations of the port ends at the centerline of the study domain were varied (Table 3).

The computational geometry and mesh were created using the open-source platform Salome. An unstructured computational mesh with local refinements near the port (Figure 2) was employed for domain discretization. Mesh sensitivity analyses were performed following the approach reported by Yan and Mohammadian [1,8], and final grid size was between 0.001 m and 0.005 m. The number of cells was less than 1 million, so the simulations can even be easily run on a personal PC. The numerical time step was dynamically determined by the “adjustTimeStep” codes available in OpenFOAM based on the pre-defined maximum Courant number [8], which was set as 1. The solver utilized in the present study was a transient solver, so the criterion of computational termination was defined by the user-defined end time. For the purpose of validating the model [34,35], the jets along the entire trajectories should be simulated, so the simulation was run up to 120 s for the case C1. The simulations showed that the mixing properties in the locations farther from the ports were similar in different cases, so the objective of simulating the other cases was to compare the mixing characteristics in the initial dilution region, and a simulation time of 30 s was found adequate. All the simulations were conducted employing the parallel simulation technique in the Linux operating system.

3. Results

3.1. General Observations

Figure 3 presents the dimensionless concentration distribution at a sample plane that went through the centerline of the computational domain and the centers of two ports in case C1, provided by the RNG k-ε model. This plane is hereafter referred to as the central plane. The results at the other central planes were identical to the current plane because of the nature of symmetry. As shown in the figure, the jets emanated horizontally into the receiving water body. The shear stress between the jets and ambient water resulted in water entrainment and jet dilution. The jets continuously spread throughout and mix with the ambient water, resulting in a continuous dilution when the jets moved. Overall, the jets moved along a parabolic curve. The initial vertical velocity of the jets at the inlet was zero, but the jets were bent over toward the upward direction because they had a lower density than the ambient water. The buoyancy effects became weaker along the jet trajectories because of the mixing processes, so the vertical velocity of the jets decreased as they moved. The shear stress between the fluid layers led to momentum transfer, so the initial horizontal velocity of the jets also continuously decreased. The horizontal velocity gradually approached zero, and then the jet advections in the horizontal direction became insignificant, and the jets behaved like vertical buoyant jets. These general observations about the dilution properties of rosette buoyant jets at the central plane were consistent with the experimental observations reported by Lai and Lee [14] from a qualitative viewpoint.

Figure 4 shows the concentration contours at various vertical cross sections for the case C1 provided by the RNG k-ε model. These contours clearly showed the processes of jet dilution and interactions. In the initial dilution region, which was near the inlets (e.g., y = 5D), the jets had clear individual features, and the concentrations at the jet centerlines was much higher than those at the ambient positions. As the jets moved upward, the jets continuously spread and mixed with the ambient water. Therefore, the peaks of concentrations at a farther location (e.g., y = 25D) were generally lower, the difference between the centerline values and ambient values was generally smaller, and the area that was impacted by the discharge was generally larger. The jets expanded in the span-wise directions, and the jet widths gradually exceeded at the spacing between two adjacent ports, so mutual interactions occurred. As the jets moved farther, the jets continuously spread and merged (such as at the cross section y = 45D), and the jet interactions were very obvious at even farther locations (e.g., y = 65D). These general observations about the dilution properties of rosette buoyant jets at various vertical cross sections were also qualitatively consistent with the experimental observations reported by Lai and Lee [14]. The results obtained by the standard k-ε turbulence model were similar from a qualitative viewpoint. Therefore, these observations confirmed that the model can reasonably predict the general mixing characteristics of rosette buoyant jets, and a quantitative evaluation will be provided in the subsequent section.

3.2. Model Evaluations

The measured and simulated trajectories of the rosette buoyant jets in case C1 were presented in Figure 5, in which the vertical and horizontal coordinates of the trajectories were normalized by the jet diameter. The trajectory of the corresponding single jet provided in the experiment conducted by Lai and Lee [14] was also marked. Being consistent with the experiment, the horizontal coordinates for the rosette buoyant jets were generally smaller than the single buoyant jet at the same vertical levels; i.e., the trajectories of the rosette buoyant jets were closer to the centerline of the study domain. This can be explained by the Coanda effect: the jet interactions restricted the jet spreading and ambient water entrainment, and the resultant low pressure near the centerline tended to attract the jets moving towards the low-pressure region.

The data comparisons exhibited a good match between the experimental and numerical results. The line for the RNG k-ε turbulence closure matched the symbols for the measurements very well, demonstrating the satisfactory capability of the RNG k-ε model. The line for the standard k-ε turbulence closure model was also close to the symbols for the experimental measurements, but the match was not as good as that corresponding to the RNG k-ε model.

To obtain a quantitative measure of the model performances, the root-mean-squared error (RMSE), normalized root-mean-squared error (NRMSE), and coefficient of determination (R²), were computed and indicated in Figure 5. The RMSE value corresponding to the results obtained by the RNG k-ε model was much smaller than those corresponding to the standard k-ε model, indicating that the RNG k-ε model performed better than the standard k-ε model. The R² value confirmed the superiority of the RNG k-ε model over the standard k-ε model. The NRMSE values can be utilized to indicate the level of relative errors, and the values showed that the relative errors for the RNG k-ε model were below 5%, while the relative errors for the standard k-ε model were below 10%. The levels of errors corresponding to the two models were both satisfactory for practical applications, but the RNG k-ε model outperformed the standard k-ε model.

It is necessary to consider both the model accuracy and efficiency for the evaluation of a model’s performance, but the accuracy and efficiency are often in conflict. The standard k-ε model is currently the most popular turbulence model because it has an excellent balance between accuracy and efficiency. A model cannot be regarded as being better than the standard k-ε model if it significantly increases computational costs because the model efficiency is an important consideration for practical engineering applications. The computational costs of the simulation for case C1 were tested, and the results showed that the computational costs for the standard k-ε and RNG k-ε models with 32 parallel computations being performed were 45.14 and 45.92 h, respectively. Therefore, it is reasonable to conclude that the RNG k-ε model led to better results than the standard k-ε model, and the computational costs of the two models were comparable.

3.3. Application of the Model to Rosette Buoyant Jets from Inclined Diffusers

In practical applications, wastewater effluents are often discharged from inclined rosette-type diffusers. It is well known that the mixing and dispersion processes of wastewater jets are very sensitive to the jet angles, so it is important to take the influences of jet inclinations into account. It is difficult to use empirical or theoretical methods to investigate inclined rosette jets because of the complicated mechanisms, but the validated model with the RNG k-ε turbulence closure can be a useful tool for investigating this. In the present study, the validated model with the RNG k-ε turbulence closure was utilized to carry out six additional computations. In the additional cases C2-C4, the multiport jets were discharged from upward-inclined ports, while those jets in cases C5-C7 were discharged from downward-inclined ports. Sample results of the simulated normalized concentration fields at the central planes are presented in Figure 6. As can be observed in the figure, the impact areas of the jets in various cases were quite different.

For upward-inclined jets, the horizontal extends that were impacted by the jets decreased with increasing jet inclinations, primarily because the horizontal component of the initial horizontal velocity decreased with increasing jet inclinations. In the initial dilution region (the region close to the ports), the jet concentration at a certain vertical location was typically lower in the case with a smaller angle, as compared to the case with a greater angle. The differences in the concentration distribution were partially induced by the jet trajectories; for example, in the cases with a smaller jet angle, the jets traveled farther in the horizontal direction, so there was more ambient water entrained into the jets when it reached a certain vertical direction, and thus the jets were more diluted. In the farther region (the region farther from the ports), the jets in different cases were quite similar, because they all behaved like vertical buoyant jets.

The downward-inclined buoyant jets behaved like reversed upward-inclined dense jets. In the initial dilution region where the jets were falling, the vertical velocities of the jets decreased because of the buoyancy effects and momentum transfer between the fluids. At the positions where the jets arrived at the lowest locations, the vertical velocities of the jets became zero. The vertical velocities then became positive due to buoyancy effects, so the discharges started traveling back upwards. Similar to the other cases, the shear stress between the fluid layers led to momentum transfer, so the initial horizontal velocity of the jets also continuously decreased. The horizontal velocity gradually approached zero, and then the jet advections in the horizontal direction became insignificant, and the jets behaved like vertical buoyant jets. Similar to upward-inclined jets, the horizontal extends that were impacted by the jets decreased with increasing jet inclinations, primarily because the horizontal component of the initial horizontal velocity decreased with increasing jet inclinations. It is quite obvious in Figure 6 that the jets were less diluted in the cases with greater angles, which can be explained by the Coanda effect. When the jet angle increased, the horizontal travel distance decreased, and so the fluids moving back upward interacted with the initial jets, and the actual spacing between multiple jets decreased. These jet interactions restricted the jet spreading and ambient water entrainment, and thus reduced the jet dilutions.

4. Discussion

The major contribution of this study is that it demonstrated the good capability of the 3D numerical model in simulating the dilution properties of rosette buoyant jets. The levels of errors corresponding to the standard and RNG k-ε turbulence models were both satisfactory for practical applications. Thus, the validated model with either turbulence closure can be further applied to investigate different scenarios for rosette buoyant jets in research or practical projects. This conclusion is very meaningful, although several other approaches have been proposed for wastewater jets. First, most of the existing theoretical or analytical approaches, such as the jet integral, entrainment restriction, and superposition methods [15,16,17] are not very suitable for rosette buoyant jets due to the complicated mechanisms. Second, the user-defined parameters in the previous models may lead to unacceptable uncertainties. Third, applying the previous models to complicated practical cases is not feasible. Although Yan and Mohammadian [18,19,20] have demonstrated the capability of the multi-gene genetic programming approach for wastewater jets, numerical simulations are still necessary to provide reliable data for model training and validation. It should be noted that the AI-based models may provide comparable or even better predictions than numerical models within the range of the training data, but their predictions out of the training range should be used with caution. Therefore, numerical models, which are based on realistic physics, are still important.

It is understood that 3D numerical simulations are more computationally expensive than the previous methods. However, it should be noted that studies on wastewater discharges are not typically conducted for emergency events. The present study showed that the computational cost was only about 45 h, and this is absolutely satisfactory for most practical applications. If needed, the computational costs can be further reduced because OpenFOAM allows for parallel simulations. With the advancement in computing techniques and resources, the computational costs of numerical models can be less of a concern.

The results also showed that the RNG k-ε turbulence model outperformed the standard turbulence model. This conclusion is in accordance with that reported by Yan and Mohammadian for multiple inclined dense jets [8]. There are two transport equations in the standard and RNG k-ε turbulence models, one is for k and the other one is for ε. Most of the terms in the two closures are the same, but the RNG k-ε contained several improvements, which may contribute to the better performance of predictions for rosette buoyant jets. First, the RNG k-ε model considers the influence of the Reynolds number on the effective turbulence transport. Second, the RNG k-ε model determines the inverse effective Prandtl numbers, employing a more advanced equation. Third, the RNG k-ε model incorporates a new term in the ε that can improve the calculation of the turbulent viscosity. In addition to model accuracy, another merit of the RNG k-ε model is that it did not significantly increase the computational cost. Some more advanced turbulence modeling techniques, such as large-eddy simulation, detached-eddy simulation, and direct numerical simulation techniques would significantly increase the computational cost, so they are not considered in the present study.

The validated model with the RNG k-ε turbulence closure was further applied to investigate the influences of port inclinations on the mixing properties, which are more complicated and cannot be easily predicted using the previous simple models. The results provided some new observations that have not been adequately handled in the literature and an example of how to apply the validated model to conduct further investigations. Although additional validations may be useful in future works to further demonstrate the performances of the model in modeling rosette inclined buoyant jets, the results are in good accordance with the engineering sense, existing knowledge, and theoretical analysis. Considering that Zhang et al. [32] have previously demonstrated the capability of the solver for single jets and Yan and Mohammadian [8] have demonstrated the capability of the solver for multiple inclined dense jets, the present study further indicated that the solver is generally valid for different conditions, so its predictions are believed reliable. However, its performance in modeling other cases, such as rosette buoyant jets in flowing currents, rosette dense jets in stagnant water or flowing currents, jets in stratified conditions, and unequal discharges in stagnant water or flowing currents, requires further validation.

It will also be interesting to further extend the model to study particle plumes [36], sedimentation from buoyant jets [37], and sediment-laden jets [38] in future studies. The present used a simple model for the turbulent Schmidt number (which yield a S_c number of 1), while its values may affect the scalars transport [6,39]. Therefore, it is worthwhile to check if the numerical predictions can be further improved by tuning the S_c number in future studies. The model can also provide some information for practical designs with the consideration of sedimentation and scour. The jets ejected from downward inclined ports may erode the bed surface [40,41,42], and thus the simulated flow field and jet strength could help set the minimum required distance of the downward inclined diffusers from the erodible bed surface.

5. Summary and Conclusions

Three-dimensional numerical modeling of buoyant jets discharged from a rosette-type multiport diffuser was conducted. The simulated results obtained from the standard and RNG k-ε turbulence closures were compared to measurements. The general observations and quantitative comparisons demonstrated that the 3D model can reasonably predict the mixing properties of rosette buoyant jets. Although the performances of the standard and RNG k-ε models were both acceptable for most practical engineering applications, the RNG k-ε model (RMSE = 0.25; R² = 0.98; and NRMSE = 0.04) outperformed the standard k-ε model (RMSE = 0.56; R² = 0.92; and NRMSE = 0.09). The computational costs of the RNG k-ε model (45.92 h) were comparable with the standard k-ε model (45.14 h), so it is reasonable to conclude that the RNG k-ε model led to better results than the standard k-ε model with a comparable computational cost. The validated model was further applied to investigate the influences of port inclinations on the mixing properties, which provided some new observations that have not been adequately handled in the literature, and an example of how to apply the validated model to conduct further investigations. This study focused on rosette buoyant jets, and subsequent studies further utilized the model to investigate the effects of other parameters, such as the number of ports, discharge and flow conditions, and ambient stratification, on the dilution properties. The model will also be evaluated in future studies for modeling other cases, such as rosette dense jets in stagnant water or flowing currents, jets in stratified conditions, and unequal discharges in stagnant water or flowing currents.