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Article

Vertical Dense Effluent Discharge Modelling in Shallow Waters

Department of Civil Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, ON K1N 6N5, Canada
*
Author to whom correspondence should be addressed.
Water 2022, 14(15), 2312; https://doi.org/10.3390/w14152312
Received: 20 June 2022 / Revised: 18 July 2022 / Accepted: 20 July 2022 / Published: 25 July 2022

Abstract

:
Vertical dense effluent discharges are popular in outfall system designs. Vertical jets provide the opportunity to be efficient for a range of ambient currents, where the jet is pushed away so as not to fall on itself. This study focuses on the worst-case scenario of the dilution and mixing of such jets: vertical dense effluent discharges with no ambient current, in shallow water, where the jet impinges the water surface. This scenario provides conservative design criteria for such outfall systems. The numerical modelling of such jets has not been investigated before and this study provides novel insights into simulations of vertical dense effluent discharges in shallow waters. Turbulent vertical discharges with Froude numbers ranging from 9 to 24 were simulated using OpenFOAM. A Reynolds stress model (RSM) was applied to characterize the geometrical (i.e., maximum discharge rise Zm and lateral spread Rsp) and dilution μmin 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

Dense effluents are residual flows resulting from physical or chemical processes. Brine from desalination plants, wastewater from wastewater treatment plants, and tailings effluents from mining plants are a few examples where the receiving water body has a lower density than that of the effluent. The receiving water body (often called ambient water) can be coastal or inland waters (i.e., lakes and rivers). Dense effluents are discharged back into the ambient waters through outfall systems. The main consideration in designing an efficient outfall is to achieve certain dilution in the initial mixing zone (IMZ), as regulated by environmental authorities. The dilution requirement within the IMZ is to preserve the natural habitats and ecosystems in the receiving water bodies.
Dense effluents are discharged at the water surface or close to the bed. It was shown that submerged outfalls have higher efficiencies due to better mixing and dilution [1]. Different discharge configurations are used in practice, as schematically shown in Figure 1. Depending on the application, ambient condition, and bathymetry of the discharge location, any of these scenarios may be adopted for construction, assuming that they meet the regulatory criteria for concentration and temperature. A case of deep ambient water conditions was considered in Figure 1, which represents an ideal case due to the greater depth and stronger currents available for diluting the effluent [2]. This is why previous numerical and experimental studies have focused on effluent discharges in deep waters. It is worth noting that deep water in this context means that there is enough of a water column that the jet does not interact with the water surface or is not influenced by the water–atmosphere boundary. Although a higher dilution is achieved in deeper waters, construction and maintenance of such outfalls are relatively expensive due to the need for longer pipelines as deeper coastal waters are further offshore. It is therefore necessary to find an optimum location to meet both the environmental and economic criteria for shallower water conditions.
Vertical dense discharges are common in practice as the connections of nozzles to the diffuser pipe are easier to construct and maintain. Vertical jets may not be suitable for small ponds where the wind- or wave-driven currents are constrained, but they are ideal for rivers, large lakes, and coastal waters where ambient water currents exist. Ahmad and Baddour [3] experimentally evaluated vertical dense jets and argued that vertical discharges are preferred over inclined discharges especially in cases where the inclined discharge is opposite to the direction of the ambient current.
Vertical jets (also known as fountains) have been widely researched in recent decades. The research started with simple image analyses of the vertical jets to quantify the geometrical properties of such jets in a calm ambient. Experimental studies advanced into more complex analyses using PIV (particle image velocimetry) and LIF (laser induced fluorescence) systems that were able to reveal kinematic properties of the jets. In the last two decades, numerical studies of vertical jets have also been conducted using CFD (computational fluid dynamics) codes. Vertical discharges in deep waters have been extensively studied in the past, both experimentally and numerically. Table 1 summarizes previous studies with a concise description of their respective findings.
The studies summarized in Table 1 are for vertical discharges in deep waters, without considering the jet impingement into the water surface. There are limited experimental studies on vertical dense discharges in shallow waters impinging into the water surface, as schematically illustrated in Figure 2. Lemckert [10] performed an experimental study to evaluate the spreading radius of the vertical dense discharge in a homogenous calm ambient water with surface attachment. The study tried to quantify the spread radius of the discharge at the water surface before it plunged downstream due to negative buoyancy. It was found that the spreading radius is a function of the nozzle size, discharge Froude number, and ambient water depth (i.e., from the nozzle exit to the water surface). Based on a series of experiments and dimensional analysis, the following empirical equation (Equation (1)) was proposed to estimate the spreading radius [10].
H + R s p r 0 = 4.8 F r 0.74
The coefficient of 4.8 and the exponent of 0.74 of the discharge Froude number on the right-hand side of Equation (1) were derived experimentally. Lemckert [10] discussed the upper limit of the proposed formula for cases where the discharge momentum was large enough to break through the free surface and exit the ambient water.
More recently, an experimental study was conducted to investigate the dilution, maximum height, and spread of vertical dense discharges in shallow waters [11]. The authors included experimental data of deep waters for comparative purposes. A range of Froude numbers from 9 to 24 was examined in their study. They used a shadowgraph technique to identify the maximum discharge rise and its lateral spread. Three regimes were identified in the study: a deep mixing regime (deep, hereafter), intermediate mixing regime (intermediate, hereafter), and impinging mixing regime (impinging, hereafter). The return point dilution was measured and reported to be smallest in the impinging regime. The spreading radius of discharge was measured and then compared with the empirical relationship from [10]. They categorized the deep, intermediate, and impinging regimes as summarized in Table 2 (H, r0, and Zm are water depth, nozzle radius, and maximum rise height, respectively).
The study from [11] was adopted in the current study to verify the CFD model. The main goal of the current study is to numerically investigate the discharge of vertical dense effluents in shallow waters, where the jet impinges onto the water surface, using the OpenFOAM CFD model. To the best of authors knowledge, this aspect has never been investigated in past studies. A RANS turbulence model was used to predict the kinematic and geometric properties of discharges by comparing the current study results with experimental data from [11]. We hope that this study can be a benchmark for future researchers numerically investigating discharges in shallow waters, where jets interact with the water surface.
The structure of this paper is as follows: Section 2 discusses the dimensional analysis used to presenting the Zm and μmin. 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

The sketch of a vertical dense discharge in shallow waters is presented in Figure 2. The jet is vertically discharged through a nozzle with a radius of r0, jet velocity of U0, jet density of ρ0, and ambient water density of ρa (ρ0 > ρa). The jet mixes with ambient water as it is discharging and reaches a terminal (maximum) rise height, Zm, 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, C0; the density difference between the ambient water and discharge, Δρ0 = ρ0ρa; the jet velocity, U0; the discharge nozzle, r0; 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:
F r = U 0 g 0 r 0
g 0 = ( Δ ρ 0 ρ a ) g
where g is the acceleration due to gravity and g0′ is the reduced acceleration due to gravity. The jet mixing characteristics of interest are Zm and μmin.
The vertical dense discharges are characterized by the jet discharge volume flux (Q0), kinematic momentum flux (M0), and the buoyancy flux (B0) [11], which are provided by the equations below:
Q 0 = U 0 π D 2 4
M 0 = U 0 2 π D 2 4
B 0 = Q 0 g 0
In dimensional analysis, one may define a length scale for the discharge (e.g., maximum discharge rise), as shown in [11]:
Z m = c o n s t a n t   ( L M )
where LM is the momentum length scale, which can be calculated as:
L M = M 0 3 / 4 B 0 1 / 2
By replacing M0 and B0, the length scale below is suitable to nondimensionalize vertical discharge height [5].
L s = r 0 F r
For dilution, a simple definition is given by [11], as formulated below:
μ m i n = Δ C 0 Δ C 1
where Δ C 0 = C 0 C a and Δ C 1 = C 1 C a . C0, Ca, and C1 represent the discharge concentration at the nozzle, the ambient water concentration, and the discharge concentration at the return point, respectively.
Using a similar approach to that used to derive Equation (9), by substituting for M0 and B0, we derive the following effective gravity scale to nondimensionalize jet buoyancy:
g s = g 0 F r
The effective gravity scale is proportional to the concentration scale when the water equation of state is linear [11]. Thus:
C s = Δ C 0 g 0 g s
Combining Equations (11) and (12) will result in:
C s = Δ C 0 F r
Using Equation (10), the dilution scale for the dense jet can be written as:
μ s = g 0 g s = Δ C 0 C s
Substituting Equation (13) into Equation (14) results in the following nondimensional dilution scale [11]:
μ s = F r
Finally, for the lateral spreading of the discharge, ref. [12] found that ambient water depth above the nozzle, H, is a proper scale for nondimensionalizing Rsp.

3. Numerical Model

A RANS turbulence model (Launder-Reece-Rodi, LRR) was used in this study. While LES may enhance the simulation of dense jets in quiescent ambient waters, their benefits are minimal, and they suffer from high computational expenses [13].

3.1. Governing Equations

Full Navier-Stokes equations are the governing equations for the vertical dense effluent discharges of incompressible fluids. As described by [14,15], the time-averaged Navier-Stokes equations for continuity, scalar, and momentum transport, the derived equations may be written as [13]:
ρ ¯ t + x i ( ρ ¯ u i ¯ ) = 0
( ρ ¯ u i ¯ ) t + x j ( ρ ¯ u i ¯ u j ¯ ) = P ¯ x i + ρ ¯ g i + x j ( μ u i ¯ x j ) τ i j x j
( ρ ¯ u i ¯ ) t + x j ( ρ ¯ c ¯ u j ¯ ) = x j ( Γ c ¯ x j ) q j x j
where subscripts i, j, and k represent the axis of system of coordinates; ρ is the fluid density; u is the fluid velocity; p is the pressure; g is acceleration due to the gravity; μ is the fluid viscosity; Γ is the scalar diffusivity; c is the scalar concentration; and the overbar denotes time-averaged variables. The Reynolds stresses, τ i j and turbulent scalar flux q j , can be written as:
τ i j = μ t ( u i ¯ x j + u j ¯ x i ) 2 / 3 ( ρ k + μ t u k ¯ x k ) δ i j
q j = Γ t c ¯ x j
where μ t is the turbulent eddy viscosity, k is the turbulent kinetic energy, δ i j is the Kronecker delta, and Γ t is the turbulent dispersion. The Reynolds Stress Models (RSM), directly calculate the Reynolds stresses.

3.2. Numerical Solver and Schemes

A FVM (finite volume method) was utilized to discretize the RANS equations noted earlier. Simulations were completed using OpenFOAM (an open-source CFD model), applying the TwoLiquidMixingFoam solver.
The transient models in this study were run long enough to reach a steady-state condition for the discharge flows. A computational time-step of 0.001 s and a total time duration of 80 s were chosen for the simulations.

3.3. Boundary Conditions

The OpenFOAM model domain and geometry are shown in Figure 3. The tank dimensions were chosen to be 1.15 m long, 1.15 m wide and a variable depth depending on the model scenario, based on a physical model from [11]. These dimensions ensured no recirculation of fluids due to the presence of boundary walls. The effluent discharges from a circular nozzle (of D = 9.45 mm) with an exit velocity of U0 and density of ρ 0 . The discharge density (ρ0) and density of ambient water (ρa) ranged from 1012.4 to 1013.6 kg/m3 and 993 to 994.7 kg/m3, respectively. The density ratio Δρ/ρa ranged from 0.015 to 0.019. The inlet values of k and ε were selected based on [16], as k = 0.06U02 and ε = 0.06U03/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.
A hexahedral mesh was used to spatially discretize the model domain with a minimum cell size of 0.001 m around the nozzle and maximum cell size of 0.020 m at the farthest end of tank. A refined mesh system was designed during the initial simulations to better capture the regions of jet trajectory for a more accurate computation of flow characteristics. A mesh sensitivity analysis was performed in this study to make sure that the velocity and concentration results were reliable and independent of mesh sizes. The grid refinement criterion was set to reach a difference of less than 2% between the predicted results when using the last two grids. When this goal was achieved, it was assumed that no further refinement was needed, and the mesh was satisfactory. The final mesh configuration used in this study included ~1.5 million cells.
For the entire simulation performed in the current study, a computational time step of 0.001 s was used, which results in a Courant-Friedrichs-Lewy (CFL) number smaller than 0.5 for stability considerations. Smaller time steps were also tested but the results remained unchanged. The convergence criteria for each time step were set such that the residuals for the velocity components and pressure were 1 × 10−5 and 1 × 10−6, respectively.

3.4. Turbulence Model

The LRR turbulence model was used in this study based on its proven performance (e.g., [16,17,18,19,20]). This model was developed by [21]. For deriving equations of the LRR model and discussion about its performance for dense effluent discharges in shallow water, please refer to [19].

3.5. Numerical Cases

The numerical experiments were selected based on the experimental data in [11], as summarized in Table 3. As claimed in that study, the experiments were designed to obtain a set of specific objectives. These objectives included investigating the influence of varying water depths on dense jet heights, the relationship between the minimum return dilution and discharge Froude number under varying water depths, and the impact of the Froude number on the maximum discharge rise and discharge lateral spread. The basic goal was to cover deep, intermediate, and impinging flow regimes in the proposed experiments.

4. Results and Discussion

The LRR turbulence model used in the current study has been successfully employed to simulate inclined dense effluent discharge modelling in shallow waters in previous experiments [19]. This section summarizes the results of the current numerical simulations and their comparison to experimental cases, where applicable, both qualitatively and quantitatively. The overall discharge behavior of vertical jets in shallow waters is evaluated first. The discharge dilution rates at return point are characterized for the cases studied herein, followed by geometrical properties for such discharges. Finally, the spreading radius of vertical dense discharges studied in this paper are evaluated.

4.1. Discharge Evolution

The first investigation of our study was on general discharge behavior. The objective was to observe overall vertical discharge behavior in shallow waters. Figure 4 shows a vertical dense effluent discharge into shallow water with surface attachment, while the jet is developing laterally at the surface. The initial momentum of the jet at the discharge point pushes the jet upward until the jet impinges on the water surface. Vertical momentum forces then convert to lateral momentum forces and the jet develops in a symmetrical pattern (Figure 4b,c). At a certain distance from the jet center, the horizontal momentum dissipates and buoyancy forces dominate the flow, the jet sinking toward the bottom of the tank. The lateral expansion of vertical dense discharges with surface attachment is larger than that of the jets without surface attachment, because the strong vertical momentum leads to the generation of horizontal forces due to jet attachment.
When an intermediate or impinging flow regime is compared with a deep flow regime (i.e., in the case of a same Froude number but variable H), the impact of the water shallowness on jet height is obvious. The jet height is larger in the intermediate and impinging flow regimes, due to the stress conditions between two boundaries and the viscous forces between two fluids with different densities. The other physical reason behind the difference in discharge heights is the smaller pressure field in the intermediate and impinging regimes compared with that in the deep regime. In a deep regime, the larger water column will increase the pressure on the jets, which dissipates the vertical momentum forces faster.
Experimental modelling of vertical jets in shallow waters showed the water surface disturbances in cases of intermediate and impinging regimes [11]. Figure 5 (from the experimental study of [11]) is comparable to the top views shown in Figure 4b,c.

4.2. Discharge Dilution

Figure 6 illustrates the mean nondimensionalized concentration (C/C0) profiles against the nondimensionalized radial distance from the nozzle (rr0)/r0Fr 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/r0 = 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/C0 when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.
The dilution rate is one of the key parameters in the design of outfalls in inland and coastal waters. An optimal design is one that achieves the highest dilution rate. It is therefore important to understand the minimum dilution achieved at the return point (μmin) 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 μmin/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

To quantify and better understand the impact of water depth on the discharge maximum rise, Zm was nondimensionalized by the length scale r0Fr (Equation (9)) and was plotted against the nondimensionalized water depth (H/r0Fr) in Figure 8. As shown in this plot, the Zm/r0Fr 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/r0 > 1.5 Zm/r0, 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 Zm/r0Fr when numerical results were compared with experimental data. Table 2 in Section 4.5 provides a summary of error measures for quantitative comparison purposes.
To analyze the impact of constant water depth and variable Froude numbers, results from the B, E, and F series experiments are shown in Figure 9 (Zm/r0 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/r0 = 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 Zm/r0 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

There have been several studies on the lateral spread of impacting jets along a rigid and free surface [10,12,22].
Cooper and Hunt [12] focused on air dynamics and fluid mechanics applications, such as gas metal welding, air curtains, and aircraft vertical takeoff. They defined Rsp as the radial distance of the spread where lateral flow separates the rigid plate.
Lemckert [10] experimented with saline jets impinging into rigid and free surfaces, respectively. Both of these studies aimed to find a relationship between the Rsp, 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 (Rsp + H), is related to the discharge characteristics. Lemckert [10] discussed that the jet in shallow ambient water travels a distance of (Rsp + H) in the same way Zm does in deep water with no surface attachment.
The surface attachment and size of the plume appearing on the surface are of particular regulatory concerns. Therefore, it is very important to understand the spreading radius of vertical dense jets impinging into the water surface. Figure 10 shows the numerical results of the B, E, and F series experiments compared with the experimental data of [11] and the powerline of [10]. The numerical results are in good agreement with both studies, especially with the results from [10]. The powerline from the numerical results suggests the following equation for the discharge spread at the surface in an impinging regime, which is very close to that of [10]. It is noted that the Froude numbers of the numerical study did not range as high as those in [10].
H + R s p r 0 = 5.04 F r 0.73
A RMSE of 2.38 and ME of −0.03 were obtained for (H + Rsp)/r0 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

Previous sections presented qualitative comparisons between the numerical and experimental data in graphs. Table 4 summarizes the root mean square error (RMSE) and mean error (ME) calculated for the parameters discussed above. As shown in Table 4 and Figure 11, the performance of the numerical model is promising for such applications (i.e., the vertical dense jet discharges in shallow water).

5. Conclusions and Recommendations

In this study, for the first time, we performed a comprehensive numerical modelling experiment to investigate the vertical effluent discharges into shallow waters. The LRR turbulence model (known as the Reynolds stress models—six equations to model anisotropic behavior) was used in this study based on previous studies (e.g., [19]). The following conclusions can be drawn based on our results:
  • 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).
The authors recommend a control system (such as flange valves) for effluent discharge in areas with shallow water or a large tidal range to reduce discharge, if the flow regime changes from intermediate to impinging. More detailed experimental and numerical studies will be needed to quantify the flow regime changes.
For next research steps, properties such as current, wave, and ambient water stratification need to be added to the current model to account for environmental forcing. The turbulence and vorticity of the jet, especially in the attachment zone, will also need to be studied in more depth using LES turbulence models.
We also intend on investigating the effect of nozzle geometry on the discharge maximum rise Zm, to optimize the design to meet the depth criteria.

Author Contributions

Conceptualization, H.K.G., A.M. and I.N.; methodology, H.K.G., A.M. and I.N.; software, H.K.G.; validation, H.K.G.; formal analysis, H.K.G.; investigation, H.K.G., A.M. and I.N.; resources, H.K.G., A.M. and I.N.; data curation, H.K.G.; writing—original draft preparation, H.K.G., A.M. and I.N.; writing—review and editing, H.K.G., A.M. and I.N.; visualization, H.K.G.; supervision, A.M. and I.N.; project administration, H.K.G., A.M. and I.N.; funding acquisition, H.K.G., A.M. and I.N. All authors have read and agreed to the published version of the manuscript.

Funding

The financial support of the NSERC Postgraduate Scholarship—Doctoral (PGS-D), Ontario Graduate Scholarship (OGS), and Queen Elizabeth II Graduate Scholarship in Science and Technology (QEII-GSST) held by the first author, as well as of the NSERC Discovery Grants held by the second and third authors are kindly acknowledged.

Acknowledgments

This research was conducted during the graduate studies of the first author at the University of Ottawa. The financial support of the NSERC Postgraduate Scholarship—Doctoral (PGS-D), Ontario Graduate Scholarship (OGS), and Queen Elizabeth II Graduate Scholarship in Science and Technology (QEII-GSST) held by the first author, as well as of the NSERC Discovery Grants held by the second and third authors are kindly acknowledged.

Conflicts of Interest

There is no conflict of interest.

Abbreviations

The following symbols are used in this study:
B0buoyancy flux [m4/s3]
Cconcentration at each cell [ppm/ppt]
C0discharge concentration [ppm/ppt]
Caambient concentration [ppm/ppt]
C1return point concentration [ppm/ppt]
Frdensimetric Froude number [-]
Ggravitational acceleration [m/s2]
g0reduced gravitational acceleration [m/s2]
gseffective gravity scale [m/s2]
Hwater depth above nozzle level [m]
Kturbulent kinetic energy [m2/s2]
Lslength scale to normalize jet height [m]
M0momentum flux [m4/s2]
Prtturbulent Prandtl number [-]
Q0discharge volume flux [m3/s]
q j turbulent scalar flux [-]
rradial distance [m]
r0nozzle radius [m]
Rspjet lateral spread [m]
ufluid velocity [m/s]
U0discharge initial velocity [m/s]
Zmdischarge maximum rise [m]
Γ calar diffusivity [kg/ms]
Γ t turbulent dispersity
δ i j Kronecker delta [-]
μ fluid viscosity [Ns/m2]
μ m i n minimum return point dilution [-]
μ t turbulent eddy viscosity [m2/s]
ρ fluid density at each cell [kg/m3]
ρ 0 discharge density [kg/m3]
ρ a ambient density [kg/m3]
τ i j Reynolds stresses [-]

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Figure 1. Dense effluent discharge configurations in deep waters.
Figure 1. Dense effluent discharge configurations in deep waters.
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Figure 2. Schematic representation of dense effluent discharge with surface attachment.
Figure 2. Schematic representation of dense effluent discharge with surface attachment.
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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 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.
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Figure 4. Vertical jet evolution in shallow water with surface interaction (C0 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 4. Vertical jet evolution in shallow water with surface interaction (C0 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.
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Figure 5. Top view of water surface disturbance observed in the experimental study for H/r0 = 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 5. Top view of water surface disturbance observed in the experimental study for H/r0 = 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).
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Figure 6. Concentration profiles at the nozzle level.
Figure 6. Concentration profiles at the nozzle level.
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Figure 7. Minimum dilution at the return point of vertical dense discharges in shallow waters [3].
Figure 7. Minimum dilution at the return point of vertical dense discharges in shallow waters [3].
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Figure 8. Discharge maximum rise under fixed Froude number and variable ambient water depths [6].
Figure 8. Discharge maximum rise under fixed Froude number and variable ambient water depths [6].
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Figure 9. Discharge maximum rise under fixed ambient water depths and variable Froude number [6].
Figure 9. Discharge maximum rise under fixed ambient water depths and variable Froude number [6].
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Figure 10. Lateral spread of the vertical dense discharges at the surface in the impinging flow regime [10].
Figure 10. Lateral spread of the vertical dense discharges at the surface in the impinging flow regime [10].
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Figure 11. Numerical results (from current study) against experimental data [11] for different parameters.
Figure 11. Numerical results (from current study) against experimental data [11] for different parameters.
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Table 1. Previous studies of vertical dense discharges in deep waters.
Table 1. Previous studies of vertical dense discharges in deep waters.
StudyFindings
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 Zm/Lm 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 (Zm/Lm) was a function of relative density difference (Δρ/ρa). They concluded that the Zm/Lm decreases from 3.06 to 2.59 at Δρ/ρa = 0.001 and Δρ/ρa = 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 δm = 1.4r0Fr, where r0 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 (Prt) for various Froude numbers and claimed this affects the jet predictions.
Table 2. Flow regime in shallow vertical dense jet discharges [11].
Table 2. Flow regime in shallow vertical dense jet discharges [11].
DeepIntermediateImpinging
H/r0 > 1.5 Zm/r0Zm/r0 < H/r0 < 1.5 Zm/r0H < Zm
Table 3. Numerical experiment parameters.
Table 3. Numerical experiment parameters.
Experiment SeriesExperiment #D (2r0) (mm)U0 (m/s)Δρ/ρaρ0 (kg/m3)FrH/r0Fr/(H/r0)
A
(Initial run)
19.450.540.0181011.8918.6934.00.55
B
(Fixed H and variable Fr)
29.450.280.0191012.899.4437.20.25
39.450.540.0191012.8918.2037.20.49
C
(Fixed Fr and variable H)
49.450.250.0151008.919.4825.40.37
59.450.250.0151008.919.4837.20.25
69.450.250.0151008.919.48166.00.06
D
(Variable Fr and variable H)
79.450.560.0191012.8918.8737.20.51
89.450.450.0191012.8915.1644.40.34
99.450.540.0191012.8918.2052.80.34
E
(Fixed H and variable Fr, intermediate Fr range)
109.450.320.0191012.8910.8037.20.29
119.450.360.0191012.8912.1337.20.33
129.450.420.0191012.8914.1537.20.38
F
(Fixed H and variable Fr, high Fr range)
139.450.430.0191012.8914.5037.20.39
149.450.600.0191012.8920.2237.20.54
159.450.720.0191012.8924.2637.20.65
Table 4. Root mean square error (RMSE) and mean error (ME) for different parameters.
Table 4. Root mean square error (RMSE) and mean error (ME) for different parameters.
Parameter/ErrorC/C0Zm/r0FrZm/r0μmin/Fr(H + Rsp)/r0
RMSE0.020.022.940.012.38
ME0.000.01−2.000.01−0.03
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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

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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

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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

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