Numerical modeling and empirical formulation for re-aeration in nappe flow without hydraulic jump in stepped chutes

Modelagem numérica e formulação empírica para a re-aeração de escoamentos em quedas sucessivas sem ressalto hidráulico em canais em degraus

ABSTRACT

Flows in stepped chutes have been widely studied in recent decades due to the numerous dams built with roller-compacted concrete and stepped spillways. Besides this application, there is practical interest in using stepped chutes for re-aeration and desorption of volatile compounds. This work aimed to numerically model the nappe flow without a hydraulic jump, including the advection-diffusion equation for studying re-aeration. The homogeneous multiphase model was used with the advection-diffusion equation, rewritten in terms of average fields and including the turbulent Schmidt number and the k-ε turbulence model. The two-film resistance model was applied at the interface using the global mass transfer coefficient. Numerical results were compared to literature data, allowing model adjustment to experimental observations using the Schmidt number and mass transfer coefficient. Solutions show increasing dissolved oxygen concentrations along the stepped chute, with re-aeration efficiencies up to 77%. An empirical equation fitting the experimental data with a high correlation coefficient was then proposed.

Keywords:  Air-water interfaces; Computational fluid dynamics; Mass transfer; Stepped spillways

RESUMO

Os escoamentos em canais em degraus têm sido amplamente estudados nas últimas décadas devido às numerosas barragens construídas com concreto compactado a rolo e aos vertedouros em degraus incorporados. Além dessa aplicação, há interesse prático no uso de canais em degraus para promover a re-aeração e a dessorção de compostos voláteis. Este trabalho teve como objetivo modelar numericamente o escoamento em quedas sucessivas sem ressalto hidráulico, incluindo a equação de advecção-difusão para estudar a re-aeração. O modelo multifásico homogêneo foi utilizado com a equação de advecção-difusão, reescrita em termos de campos médios e incluindo o número de Schmidt turbulento e o modelo de turbulência k-ε. Para modelar a interface ar-água em relação ao fluxo de massa, empregou-se o modelo de resistência de dois filmes com o coeficiente global de transferência de massa. Os resultados numéricos, quando comparados com dados da literatura, possibilitaram o ajuste do modelo às observações experimentais, utilizando para isso o número de Schmidt turbulento e o coeficiente global de transferência de massa. As soluções mostram concentrações crescentes de oxigênio dissolvido ao longo do canal em degraus, com eficiências de re-aeração de até 77%. Uma equação empírica ajustada a partir dos dados experimentais foi então proposta, tendo apresentado aderência aos mesmos e elevado coeficiente de correlação.

Palavras-chave:  Interface ar-água; Dinâmica de fluidos computacional; Transferência de massa; Vertedouros em degraus

INTRODUCTION

The use of stepped chutes as part of spillway systems in dams, referred to as stepped spillways, and in drainage channel has been common mainly due to their high energy dissipation capacity. The stepped chutes and spillways allow for the occurrence of highly turbulent flows, which may have different patterns depending on the flow rate and geometry of the chutes. For lower discharge and less steep channels, the nappe flow regime is established, which can occur with or without the formation of hydraulic jumps on the step floors. For higher discharges and steeper channels, the formed regime is referred to as skimming flow. Between these two regimes, a transitional pattern takes place, as described by Ohtsu & Yasuda (1997) and graphed in a more direct way by Simões et al. (2012).

The intense turbulence associated with flows in stepped chutes generates “collateral effects” that relate to the energy dissipation, such as the surface breaking and the consequent incorporation of air bubbles into the flow. In this context, mass transfer at the air-water interfaces can occur, providing to the impacted rivers, e.g., those affected by dams, an opportunity to improve the water quality of the upstream reservoirs through re-aeration (or re-oxygenation). Stepped chutes can also be employed as hydraulic transition structures in artificial channels and rivers, enabling the increasing of oxygen levels in the water, and, in some cases, the desorption of volatile compounds dissolved in the water.

Essery et al. (1978) experimentally investigated skimming flow re-aeration in stepped chutes with slopes of 11°, 22°, and 45°. Chanson (2002), using the re-aeration efficiency at 15 °C, E₁₅, defined by Equation 1 with T = 15 °C, compared the data from Essery et al. (1978) to data from Rindels & Gulliver (1989) obtained for smooth chutes with slopes of 55° and 69° relative to the horizontal. As a result of this analysis, it was observed that E₁₅ reached a maximum value below 30% for the smooth channel with a slope of 55°. The stepped chute achieved E₁₅ values close to 64% for a slope of 22°, considering approximately the same fall height from the crest to the base of the downstream channel, H_dam, and same critical flow depth (or critical depth), h_c.

$E_T=\frac{C_d-C_u}{C_s-C_u}$ (1)

in which E_T = re-aeration efficiency at a specific temperature T, C_d = concentration of dissolved oxygen downstream, C_u = concentration of dissolved oxygen upstream, and C_s = saturation concentration for oxygen.

Essery et al. (1978) also investigated re-aeration in nappe flows with hydraulic jumps in physical models with slopes of 11.3°, 21.8°, and 45°. Their data demonstrate that E₁₅ increases with H_dam/h_c, reaching a maximum between 75% and 80% for H_dam/h_c ≈ 80. For situations where temperatures are different from the reference value of 15 °C, the authors proposed Equation 2, valid for 10 < T < 30 °C.

$r_T-1=\left(r_{20}-1\right)\left[1+0.0335\left(T-20\right)\right]$ (2)

where r₂₀ = deficit ratio at 20 °C, r_T = deficit ratio at T [°C]; r_T = 1/(1-E_T).

Toombes & Chanson (2000) conducted experiments in a stepped chute with step heights (s) of 0.143 m, step lengths (l) of 2.4 m, ten steps, and a flow rate corresponding to h_c/s = 0.607. The authors measured the concentration of dissolved oxygen at the lowest step and found a re-aeration efficiency at 20 °C equal to 42%.

Chanson (2002) used the data from Essery et al. (1978) to compare the one-dimensional advection-diffusion equation combined with the two resistance model. The calculations were performed for the skimming flow regime and showed good agreement with experimental data for the channel inclined at 21.8° and lower adherence for the channel inclined at 45°. Chanson (2002) employed the equations from Kawase & Moo-Young (1992) to determine the overall mass transfer coefficient, k_L, and additional empirical formulations to determine the average specific interfacial area, as well as the average flow velocity along the stepped chute. At 15 °C, the mentioned equations result in k_L values around 8.32 x 10^-5 m/s for d_ab < 0.25 mm and around 4.11 x 10^-4 m/s for d_ab > 0.25 mm, where d_ab is the reference diameter of the gas bubbles.

Felder & Chanson (2009) conducted experiments on stepped chutes with slopes of 3.4°, 5.7°, 15.9°, 18.8°, and 21.8° for nappe flow, transition flow, and skimming flow regimes. They analyzed the hydrodynamics and re-aeration efficiency using data obtained with a phase detection probe. Results were presented in terms of the dimensional ratio between the re-aeration efficiency and the drop in invert elevation expressed as a function of the dimensionless energy dissipated (head loss) for the maximum specific head. Further, Felder & Chanson (2015) studied re-aeration in stepped chutes with angles ranging from 3.4° to 26.6°, and, similar to the presentation by Felder & Chanson (2009), their results were expressed in terms of the dimensionless head loss.

Dermawan et al. (2017) presented results of an experimental study comparing flat and pooled steps in a laboratory flume. The authors did not present adjusting equations, but tables and graphs of their experimental values, which may be used in further comparative studies. The authors worked with slopes of 30^o and 45^o, specific flow rates varying from 69 to 613 cm²/s, critical depths varying from 1.75 to 7.50 cm, and presented dissolved oxygen concentration values (mg/L) as a function of the Froude number and of the for H_dam/h_c ratio. The authors concluded that, for skimming flow conditions, higher dissolved oxygen levels are attained for higher slopes, notably for Froude numbers less than 2.0. Above this value of the Froude number, the oxygen concentration reduces by further increasing the Froude number. The roughness of the step, characterized by the pooled step, also interferes in the value of the oxygen concentration, being higher for the pooled steps.

Bung & Valero (2018) analyzed experimental data obtained for stepped chutes with 1V:2H and 1V:3H configurations, corresponding to specific flow rates between 0.07 and 0.11 m²/s and subjected to the skimming flow regime. They determined k_L values ranging from 1.08 x 10^-5 to 3.5 x 10^-5 m/s. The authors also presented an equation that relates k_L/u to C, where u is the average velocity of the aerated flow and C is the average volumetric air fraction determined for h₇₅, corresponding to the flow height at which the volumetric air fraction is equal to 75%.

Felder et al. (2019) conducted experiments in a stepped chute with s = 0.50 m, l = 1.87 m in a nappe flow without hydraulic jump, measuring the concentration of dissolved oxygen with a probe positioned in the cavities between steps at various locations of the unaerated portion of the flow below the jet. Considering the performed description of the runs together with a figure provided by the authors, it is likely that the probe was positioned below the jet and upstream of the shear layer. Felder et al. (2019) mentioned that measurements at different positions in the cavity did not result in significant variations in the concentration of dissolved oxygen.

Jahad et al. (2022) conducted experiments on physical models of conventional stepped chutes and channels with the insertion of end sills with different shapes. The angles studied were 8.9°, 21.8°, and 26.6°, and among the experimental results, the authors presented E₂₀ for all three flow regimes, where E₂₀ is the efficiency of re-aeration at 20 °C. The highest efficiency was achieved for the nappe flow regime and an angle of 8.9°, reaching E₂₀ ≈ 0.557.

Nina et al. (2022) examined the skimming flow in physical models featuring a 1V:1H configuration, a step height of 10 cm, horizontal steps, sloping steps, and 0.8 ≤ h_c/s ≤ 1.6. Employing the data acquired for re-aeration in conjunction with the one-dimensional mass transfer equation, the researchers computed k_L values within the range of 0.0015 m/s to 0.0020 m/s for Reynolds numbers ranging between 3x10⁵ and 4x10⁵. This data is imperative for the execution of numerical simulations with computational fluid dynamics.

Modeling turbulent multicomponent flows with computational fluid dynamics (CFD) typically involves using the advection-diffusion equation. As a strategy for turbulence modeling, a constant turbulent Schmidt number and turbulent viscosity are often employed along with an adequate turbulence model. Gibson & Launder (1978) made predictions for the Schmidt number under various flow conditions, including buoyancy, and found values ranging from 0.5 to 2.0. Donzis et al. (2014) analyzed 46 direct numerical simulations (DNS) from different authors modeling multicomponent flow. As a result of their analysis, they calculated turbulent Schmidt number values ranging from 1.038 to 6.069. In a literature review, Gualtieri et al. (2017) presented experimental and numerical values from 23 different studies, with turbulent Schmidt number values ranging from 0.1 to 2.11.

In this context, it is essential to highlight that in mild slope stepped chutes without end sills on the stepped floors, nappe flow without the hydraulic jumps is the most likely flow regime. Establishing hydraulic jumps requires sufficiently long floors to develop supercritical flow, which naturally transitions into a hydraulic jump downstream of an M3 (Chow, 1959) backwater curve in this condition. Consequently, unless end sills are employed on the floors, empirical evidence suggests that stepped chutes designed to enhance the water quality of rivers and lakes, typically positioned upstream of sewage treatment plants, in stepped transition chutes, as well as low-slope stepped spillways subjected to low flow conditions, will predominantly operate under a nappe flow without hydraulic jumps.

The provided review suggests a limited number of studies on re-aeration for nappe flow regime and a probable absence of research on the modeling of re-aeration in nappe flow without hydraulic jumps via Computational Fluid Dynamics (CFD). The present study proposes a methodology for the CFD simulation of nappe flow without hydraulic jumps, including the modeling of re-aeration, that is, the mass transfer at the air-water interface. This approach involves the two following specific objectives: (1) determination of the turbulent Schmidt number and the overall mass transfer coefficient of the two-film resistance model; (2) proposal of an empirical formulation for calculating the re-aeration efficiency, developed based on the observed CFD trends and on experimental data available in the literature.

METHODS

Computational domain and boundary conditions

The two-dimensional numerical study conducted in this work shares similarity with the experimental study by Felder et al. (2019), who measured dissolved oxygen in a nappe flow without hydraulic jumps. Geometric information and boundary conditions are summarized in Table 1 and Figure 1. The s/l and s/h_c values presented in Table 1 correspond to the occurrence of nappe flow, as per the methodology outlined by Ohtsu et al. (2001) and graphically simplified by Simões et al. (2012) (see Figure 3b of cited study). This method predicts a minimum s/h_c value of 1.31. Additionally, in accordance with the predictions by Chanson (2001), a minimum s/h_c of 1.28 is expected.

Table 1
Geometric information and boundary conditions.

Simulations	s [m]	l [m]	he [m]	q [m2/s]	Ve [m/s]	hc [m]	s/hc	s/l	Fr	Re	We
1	0.5	1.87	0.5	0.637	1.27	0.346	1.45	0.27	1.89	7.1x105	24881
2				0.478	0.96	0.286	1.75		1.82	5.4x105	16513
3				0.319	0.64	0.218	2.29		1.86	3.6x105	9753
4				0.159	0.32	0.137	3.65		1.84	1.8x105	3823

Figure 1
Geometric characteristics of the studied step channel.

Figure 3
Efficiency variability upstream of the stagnation point: details of the velocity field in the vicinity of the stagnation point for simulation 1; (b) variation in efficiency along the line indicated in (a). Notation: x* = x/x_máx, with x_max = 14 cm and 0 ≤ x ≤ 14 cm.

Table 1 shows the values of the Froude, Reynolds, and Weber numbers, where Fr = V_b/(gh_b)^0.5 is the Froude number calculated at the downstream brink for z = 0 (see Figure 1); Re = q/ν, where ν is the kinematic viscosity of water, equal to 8.92x10^-7 m²/s at 20 °C, and We = ρV_b²h_b/σ is the Weber number calculated at the same position. The Re and We values presented in Table 1 indicate the suitability of the dimensions for the present work about similarity, as suggested by Boes & Hager (2003) for stepped spillways, who recommend Re ≥ 10⁵ and We ≥ 100 when studying aeration.

The designed computational domain contains an inlet reservoir with a total height of 3.0 m from the base to the highest step and a width of 1.0 m; the length of the highest step level is 1.0 m; the further lower steps have s = 0.50 m and l = 1.87 m. The four simulations performed correspond to four specific discharges, q, between 0.159 and 0.637 m²/s, as shown in Table 1. For each q, considering a single inlet height, h_e, the average inlet velocities, V_e, was calculated.

Conceptually, the model used is of the Eulerian-Eulerian type and the fluids are considered continuous. In order to simulate steady-state water re-aeration, the flow that enters the computational domain was set with both the oxygen volume fraction and the oxygen mass fraction equal to zero; while the fraction of water was set as the unity at the same position. To compose the atmospheric air in the vicinity of the flow, a mixture of oxygen and nitrogen needed to considered. The advection–diffusion equation of the multicomponent modeling of oxygen transport calculates the nitrogen concentration in a way that the mass fraction of the mixture equals 100%. This implies in setting a mass fraction equal to unity at the input of the computational domain.

The open-type boundary condition for the outlet considers the possible occurrence of supercritical flow at this position, a conclusion resulting from the joint analysis of the hyperbolic part of the Navier-Stokes equations and the mass conservation equation. The upper part of the domain, named “top” in Figure 1, remained open only to gases; In this position, a mass fraction for oxygen with value of 0.0077 was imposed, which, for the used meshes and the dimensions of the domain, guarantees a saturation concentration for oxygen around 9.0 mg/L at 20 °C (McCutcheon, 1989), temperature that corresponds to the experiments by Felder et al. (2019). For the exit, a mass fraction for oxygen equal to zero was adopted to avoid convergence problems related to the exit of water through the domain. This way, gas exchange through the boundary occurs only at the open top.

The law of the wall was used to model the solid contours, with an equivalent sand roughness of 1.0 mm, which is similar to a concrete with a normal finishing according to Porto (2006). The use of the symmetry-type boundary condition imposed on the lateral surfaces of the domain in conjunction with the adoption of a domain thickness equal to the width of the smallest mesh element guaranteed the adequate performance of the two-dimensional simulations.

Modeling multiphase and multicomponent flow

Multiphase flow – homogeneous model

The Navier-Stokes and the mass conservation equations were originally written for a single phase. Considering the existence of two or more fluids separated by an interface, it becomes necessary to rewrite the equations following a specific approach, such as the homogeneous model or the inhomogeneous model. The homogeneous model, used in the present work, assumes that fields of relevant variables are shared between the phases, with exception of the volumetric fraction field (ANSYS, Inc., 2021, p. 183). Alternatively, the inhomogeneous model solves the fields for each phase, with exception of the pressure field, which is shared between the phases to close the system of equations (ANSYS, Inc., 2021, p. 185). A description of the homogeneous model is presented in the sequence, using as reference the texts of Manninen & Tavassalo (1996) and Ansys CFX (ANSYS, Inc., 2021, p. 179-186), a manual of the software used in the present work.

Let U_α be the velocity field of phase α and N_p be the total number of phases. For the homogeneous model, Equation 3 represents the existence of a single velocity field for all phases. With r_α equal to the volumetric fraction of the α phase and ρ_α the density of the fluid that makes up the α phase, Equation 4 defines the density of the mixture of N_p phases present in a control volume.

$U_{\alpha }=Ufor1\le \alpha \le N_p$ (3)

$\rho ={\displaystyle \sum }_{\alpha =1}^{N_p}{r}_{\alpha }{\rho }_{\alpha }$ (4)

Equation 5 corresponds to the mass conservation of the N_p phases in the flow (ANSYS, Inc., 2021). This equation is valid for both the homogeneous and inhomogeneous models. For the homogeneous model, the velocity field is simplified with Equation 3.

$$\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }\right)+\nabla \cdot \left(r_{\alpha }{\rho }_{\alpha }{U}_{\alpha }\right)=S_{\alpha }+{\displaystyle \sum _{\beta =1}^{N_p}{\Gamma }_{\alpha \beta }}$$ (5)

where S_α is a source term for the mass and Γ_αβ corresponds to the rate of temporal variation of the mass per unit volume, resulting from the mass flow between the phases. Equation 6 establishes the relationship between Γ_αβ, the mass flow ${\dot{m}}_{\alpha \beta }$ and the interfacial area density, A_αβ, which has the unit of m^-1 when using the International System (SI).

${\Gamma }_{\alpha \beta }={\dot{m}}_{\alpha \beta }{A}_{\alpha \beta }$ (6)

The calculation of the interfacial area density for flows in channels with a stepped bottom can be carried out using the mixing model, defined by Equation 7 or with the free surface model, defined by Equation 8 (Simões, 2012). In the present work, the mixture model was adopted after performing preliminary tests that demonstrated better practical results.

$A_{\alpha \beta }=\frac{r_{\alpha }{r}_{\beta }}{d_{\alpha \beta }}$ (7)

$A_{\alpha \beta }=\left|\nabla r_{\alpha }\right|$ (8)

where d_αβ is the interfacial length scale, assumed equal to 1 mm in the Ansys CFX^® software, a value also used in the present work.

Equation 9, originated from the Newton's second law, is essentially the Navier-Stokes equation for one phase, but with varied specific mass and dynamic viscosity, μ, calculated with Equations 4 and 10, respectively.

$\frac{\partial }{\partial t}\left(\rho U_{\alpha }\right)+\nabla \cdot \left(\rho U_{\alpha }{U}_{\alpha }-\mu \left(\nabla U_{\alpha }+{\left(\nabla U_{\alpha }\right)}^T\right)\right)=S_M-\nabla p$ (9)

$\mu ={\displaystyle \sum }_{\alpha =1}^{N_p}{r}_{\alpha }{\mu }_{\alpha }$ (10)

In addition to the equations described previously, the proposed model also solves a transport equation for the volume. Considering that the sum of the volumetric fractions of the phases must be equal to unity, the use of Equation 5 results in Equation 11.

$${\displaystyle \sum _{\alpha }\frac{1}{{\rho }_{\alpha }}}\left(\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }\right)+\nabla \cdot \left(r_{\alpha }{\rho }_{\alpha }{U}_{\alpha }\right)\right)={\displaystyle \sum _{\alpha }\frac{1}{{\rho }_{\alpha }}}\left(S_{\alpha }+{\displaystyle \sum _{\beta =1}^{N_p}{\Gamma }_{\alpha \beta }}\right)$$ (11)

It is worth noting that in all presented cases, the variables correspond to the average values, obtained following the procedures of the Reynolds averages, in such a way that the use of turbulence models is possible. In the present work, the k-ε (Jones & Launder, 1972) turbulence model was adopted. The review of the literature on the topic did not reveal works focused on calculating re-aeration in successive drops type flows using Reynolds averages and turbulence models. Therefore, the choice of the turbulence model was based on the fact that the k-ε model does not present significant divergences in relation to the others when intending to simulate the average position of the free surface, as demonstrated by Simões et al. (2012).

Multicomponent flow

In the modeling of the multicomponent behavior of a multiphase flow, the Fick's law is applied in conjunction with the mass conservation equation for a species, a condition that produces the advection-diffusion equation (Equation 12) of the mass fraction Y_Aα. The subscript A represents a specific species present in the α phase.

$\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }{Y}_{A\alpha }\right)+\nabla \cdot \left(r_{\alpha }\left({\rho }_{\alpha }{U}_{\alpha }{Y}_{A\alpha }-{\rho }_{\alpha }\left(D_{A\alpha }+{\mu }_t/S{c}_t\right)\left(\nabla Y_{A\alpha }\right)\right)\right)=S_{A\alpha }$ (12)

where D_Aα is the kinematic diffusivity of component A in phase α, μ_t is the turbulent viscosity, Sc_t is the turbulent Schmidt number, defined as Sc_t = μ_t/D_tAα, with D_tAα being the turbulent diffusivity; S_Aα is the source term for species A present in phase α, which may include, for example, chemical reactions.

To model the dissolution of oxygen in water, the code uses the Henry's law, which may be specified for the Henry coefficient of the mole fraction (unit of pressure), or for the Henry coefficient of the molar concentration (unit of pressure per molar concentration). For the temperature of 20 °C used in the simulations, the first of the aforementioned coefficients has the value of 40771.2 bar, obtained through quadratic interpolation applied to the data of Spalding (1963).

Additionally, the model originated from the theory of two-films resistance, proposed initially by Nernst at the end of the 19th century (see Schulz, 2003), was adopted here. This model requires one mass transfer coefficient for the air phase and another for the water phase at the air-water interface. Considering the low solubility of oxygen in water, the water phase coefficient dominates the mass transfer, being then taken as the global mass transfer coefficient. As there is little information available about this coefficient for stepped chutes and spillways, a trial and error process was followed in the present work, with the aim of bringing the results closer to the experimental data used in the comparisons.

Mesh and numerical solution of the equations

Seven tests were performed for the analysis of convergence taking the flow rate of simulation 1 (Table 1), using 1.13x10⁵ to 1.75x10⁶ nodes, and meshes predominantly formed by hexahedrons. The results showed that meshes with approximately 2.00x10⁵ nodes (maximum element size equal to 1.5 cm) presented relative deviations of the order of 0.3% for the maximum efficiency, located at the lower step. To preserve degrees of refinement similar to the aforementioned test and suitable for the simulations, for smaller flows (simulations 3 and 4), it was necessary to use a greater number of nodes due to the reduced flow heights (with a maximum element size equal to 0.4 cm). Figure 2 illustrates the mesh used to perform simulation 4.

Figure 2
Mesh for simulation 4.

The discretization of the equations was performed with the finite volume method and high-resolution schemes for the advective term of the equations. After having performed the convergence tests, mean residues lower than 2.2 x 10^-6 were adopted as a stopping criterion for the mass fraction of oxygen, a condition that implies non-significant variations for this and other quantities, having been necessary between 6000 and 7000 iterations to attain this condition, as shown in Table 2.

Table 2
Information about meshes and convergence criteria.

Simulations	q [m2/s]	e [m]	Residual O2	Iterations	Nodes	Elements
1	0.637	0.015	2.2E-06	6837	2.0E+05	9.9E+04
2	0.478	0.015	6.2E-07	6994	2.0E+05	9.9E+04
3	0.319	0.009	2.2E-06	6398	5.3E+05	2.6E+05
4	0.159	0.005	1.2E-06	6622	1.6E+06	7.8E+05

• e = thickness of the computational domain; q = specific discharge.

RESULTS AND DISCUSSION

With the aim of comparing the results obtained in the present work with experimental data presented by Felder et al. (2019), the dissolved oxygen concentrations located in the stagnation points that form on the floors of the steps were used as references for simulations 1 to 3. For simulation 4, due to the reduced flow height in relation to the former simulations, the location chosen for the calculation of the dissolved oxygen concentration and the re-aeration efficiency was the bottom of the free overfall jet. The present positions are specified clearly here because the aforementioned authors did not describe them precisely in their experimental measurements, only stating that they were positioned in the cavities formed by the steps.

Figure 3a illustrates part of the velocity field of simulation 1 on the floor of the lowest step (arrows), superimposed on the field from E₂₀, in blue. In this image, a horizontal line of approximately 14 cm is shown in black, extending from the vicinity of the stagnation point (downstream) to an upstream position located below the downward jet, along which the efficiencies indicated in Figure 3b were determined. The obtained efficiency deviations along the mentioned line, for the four simulations, were 6.4% (Simulation 1), 5.5% (Simulation 2), 5.9% (Simulation 3) and 1.7% (Simulation 4). The low deviation observed with Simulation 4 can be explained by the reduced flow height in relation to the others and greater homogeneity of the dissolved oxygen mixture in the adopted region. Figure 3b also demonstrates that there is a small deviation in efficiency along the line in the step cavity, with a maximum value of 6.4%.

It is also worth noting that, for simulations 1, 2, 3, and 4, the specific flow rates calculated at the outlet were 0.626 m²/s, 0.417 m²/s, 0.281 m²/s, and 0.135 m²/s. These values present an average relative deviation of 10% about the specific flow rates imposed at the domain inlet due to errors resulting from the discretization of the original equations and their solution for problems with abrupt variations, as occurs in multiphase flows.

The numerical results demonstrate that the efficiency decreases when increasing the flow rate, as can be seen in Figure 4, in the E₂₀ fields in blue. The images presented in Figure 4a-d, corresponding to the four numerical simulations, demonstrate the occurrence of flow in successive drops without the formation of projections on the floors (which would lead to hydraulic jumps). They also demonstrate that the re-aeration efficiency increases along the steps of the channel. These findings correspond to new information attained with the model proposed in the present work, which qualitatively indicate the adequacy of the used formulation in relation to the behavior of the data obtained experimentally by Felder et al. (2019).

Figure 4
Water volume fraction and spatial distributions for efficiency, E₂₀: (a) Simulation 1, (b) Simulation 2, (c) Simulation 3, (d) Simulation 4.

To perform calculations with the model of the present work requires to choose values for the turbulent Schmidt number and the mass transfer coefficient of the two-film resistance model. Initially, Sc_t = 1.0 was adopted based on the Reynolds analogy and the k_L coefficient was determined through a trial and error process, starting from 9.44.10^-5 m/s, an experimental value proposed by Chanson (2002). Using data from Felder et al. (2019) as the main reference to carry out the adjustment between the physical-mathematical model and the experimental observation, the subsequent Sc_t values varied from 0.9 to 6.0 and the k_L coefficient varied from 1.00x10^-2 to 9.44x10^-5 m/s, as indicated in Table 3.

Table 3
Results of the adjustments made for Sc_t, k_L.

Simulations	q [m2/s]	Sct	kL [m/s]
1	0.637	0.9	1.00E-02
2	0.478	0.9	5.00E-03
3	0.319	1.1	1.50E-03
4	0.159	6.0	9.44E-05

The comparison between the experimental data from Felder et al. (2019) and the numerical solutions obtained in the present study indicates adherence between theory and experimentation, as shown in Figure 5. The relative deviations between the numerical solutions and the experimental data ranged from 0.16% to 29.8%, with 62.5% of the relative deviations being less than 10% and 87.5% less than 15%, and having only one point that presented a relative deviation of 29.8%.

Figure 5
Comparison between the numerical results obtained in the present study and the experimental data from Felder et al. (2019): (a) comparison with the exact fit line and (b) histogram of relative error frequencies between experimental data and values calculated with CFD.

The behavior of E₂₀ as a function of z/h_c and z/s, as indicated in Figure 6a,b, revealed trends that could be described by Equation 13. In Figure 6a, for simulation 4, a noticeable shift of the points to the right along the z/h_c axis is observed, attributed to the significantly lower specific flow rate than the other simulations. The coefficients of Equation 13 were determined through nonlinear fitting, using only the experimental data from Felder et al. (2019) as a reference. Figure 6c presents the comparison between the experimental data and the values calculated with Equation 13, indicating agreement between the proposed equation and experimentation. The correlation coefficient between measured and calculated data was 0.984, and the maximum relative error was 11.7%. The equation is valid for 4.1 ≤ z/h_c ≤ 21.9 and 3 ≤ z/s ≤ 6.

Figure 6
(a) CFD-obtained solutions: (a) E₂₀ as a function of z/h_c and (b) E₂₀ as a function of z/s; (c) proposed empirical model compared to the experimental data from Felder et al. (2019); (d) histogram of relative error frequencies between experimental data and values calculated with Equation 13.

$E_{20}=0.0666{\left(\frac{z}{h_c}\right)}^{0.175}{\left(\frac{z}{s}+1\right)}^{0.963}$ (13)

CONCLUSIONS

The conducted research evidenced the feasibility of simulating multiphase and multicomponent flows in the nappe flow regime without hydraulic jumps using the homogeneous multiphase model, the k-ε turbulence model, and the advection-diffusion equation with the turbulent Schmidt number. The inclusion of the two-film model for interfacial transfer was necessary to quantify the interfacial mass transfer from the atmosphere to the water. The overall behavior of the solutions is consistent with experimental observations, obtained for the mentioned regime of nappe flow, specifically without hydraulic jumps. Regarding re-aeration, it was shown to be more efficient for lower flow rates and flow depths. The turbulent Schmidt number varied between 0.90 and 6.0, with the highest value attained for the simulation with the lowest flow rate. The mass transfer coefficient of the two-film resistance model ranged from 10^-2 to 10^-5 m/s with the lowest value corresponding to the lowest flow rate. With these adjustments, the proposed modeling, when compared to experimental data available in the literature, resulted in relative errors below 15% for most of the data, indicating a high agreement between the model and experimental observations. The CFD calculated numerical solutions also suggested correlations between re-aeration efficiency and the dimensionless parameters z/h_c and z/s, leading to the proposal of an empirical model with a maximum relative error of 11.7% and a correlation coefficient of 0.984.

ACKNOWLEDGEMENTS

The first author acknowledges the support of CNPq for the Master's scholarship. This work was conducted with the support of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Funding Code 001; processes 88881.708215/2022-01, PDPG Emergencial de Consolidação Estratégica dos Programas de Pós-graduação (PPGs) stricto sensu acadêmicos com notas 3 e 4, and 88881.691452/2022-01 PDPG-POSDOC/Programa de Desenvolvimento da Pós-Graduação (PDPG) Pós-Doutorado Estratégico, from the Master's Program in Environment, Water, and Sanitation – MAASA – UFBA.

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Edited by

• Editor-in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Iran Eduardo Lima Neto

Publication Dates

• Publication in this collection
  22 Nov 2024
• Date of issue
  2024

History

• Received
  25 July 2024
• Reviewed
  22 Aug 2024
• Accepted
  25 Aug 2024