Re-aeration in skimming flow over stepped chutes: two-dimensional CFD modeling and empirical equations

Reaeração em escoamentos deslizantes sobre turbilhões em vertedouros e canais em degraus: modelagem via CFD e proposição de equações empíricas

ABSTRACT

Flows in stepped chutes and spillways are highly turbulent and have the potential of absorbing and desorbing volatile compounds. The present work aimed to model the interfacial mass transfer in skimming flows using the two-resistance model and the advection-diffusion equation. The simulation was conducted in 2D, with two multiphase eulerian models. An empirical formulation is also proposed for predicting the reaeration efficiency in skimming flows with slopes of 45°, 21.8° and 11.3°. The results obtained with the homogeneous and inhomogeneous multiphase models showed behaviors similar to the experimental data by adjusting the global mass transfer coefficient of the two-resistance model and the turbulent Schmidt number, with values in the ranges of 0.0015 to 0.01 m/s and 1 to 5, respectively. The proposed empirical equation resulted in high correlation coefficients and adherence to the experimental data.

Keywords:
Computational fluid dynamics; Mass transfer; Oxygen transfer; Stepped spillways

RESUMO

Escoamentos em calhas e vertedouros em degraus são altamente turbulentos e apresentam potencial para reoxigenar a água e promover a dessorção de compostos voláteis. O presente trabalho teve como objetivo modelar a transferência de massa interfacial em escoamentos do tipo deslizante sobre turbilhões utilizando o modelo de dois filmes e a equação de advecção-difusão. A simulação foi realizada em 2D, com dois modelos eulerianos multifásicos. Uma formulação empírica também é proposta para prever a eficiência da reaeração em escoamentos deslizantes sobre turbilhões em canais com 45°, 21,8° e 11,3°. Os resultados obtidos com os modelos multifásicos homogêneo e não homogêneo apresentaram comportamentos estatisticamente aderentes aos dados experimentais, ajustando-se o coeficiente global de transferência de massa do modelo de dois filmes e o número de Schmidt turbulento, com valores nas faixas de 0,0015 a 0,01 m/s e 1 a 5, respectivamente. A equação empírica proposta resultou em altos coeficientes de correlação e aderência aos dados experimentais.

Palavras-chave:
Dinâmica de fluidos computacional; Transferência de massa; Oxigênio dissolvido; Vertedouros em degraus

INTRODUCTION

Flows in stepped chutes and spillways under skimming flow conditions have been studied for decades due to the extensive applications of these hydraulic structures as part of dam spillway systems. During these years, special attention has been given to understanding flow characteristics such as energy dissipation, cavitation, aeration, and reoxygenation or re-aeration. Re-aeration, in particular, plays a critical role in environmental and engineering applications. One of its most relevant implications is enhancing river water quality, where stepped chutes can be strategically implemented to increase dissolved oxygen levels in degraded water bodies. Additionally, re-aeration is an essential factor in fish passage systems in dams, ensuring that oxygen concentrations remain adequate to maximize the success rate of fish migration. Beyond these practical applications, studying re-aeration also provides a gateway to numerical investigations of gas transfer across the air-water interface, a topic of increasing relevance in environmental hydraulics. In some cases, excessive gas dissolution downstream of stepped chutes may lead to total dissolved gas supersaturation, potentially causing gas bubble disease in fish. This phenomenon can negatively impact aquatic ecosystems.

Relatively few studies on re-aeration have originated from the cooperation between computational fluid dynamics and experimentation when it comes to stepped chutes and spillways, although this association is showing promise for water processes, as demonstrated in Shah et al. (2024), Hoiberg & Shah (2021), and Kouzbour et al. (2020). As a likely early work on the subject, Essery et al. (1978) investigated re-aeration in channels inclined at 45°, 21.8°, and 11.3°, with steps whose heights, s, in physical models varied from 0.025 m to 0.5 m and with specific discharge, q, ranging from 0.01 m²/s to 0.145 m²/s. The results of Essery et al. (1978), expressed in terms of reaeration efficiency (Equation 1) at 15 °C, were analyzed by Chanson (2002) in comparison to a semi-empirical model based on a simplified form of the advection – diffusion equation with the two-resistance model, resulting in a good agreement when using the 45° angle.

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

in which E_T represents the re-aeration efficiency at a specific temperature T; C_d, C_u, and C_s are, respectively, the downstream, upstream, and saturation concentration of the dissolved oxygen in water.

Felder & Chanson (2009) conducted measurements using a phase detection probe in nappe flows, transition flows, and skimming flows in stepped chutes with slopes of 3.4°, 5.7°, 15.9°, 18.8°, and 21.8°. Among all the obtained data, six data points corresponded to the skimming flow regime. The results were expressed in terms of the ratio between the re-aeration efficiency and the elevation drop as a function of the dimensionless energy dissipated with the maximum specific head (head loss). Further, Felder & Chanson (2015) also studied re-aeration in stepped chutes with angles of 3.4° to 26.6°, and, similar to the presentation by Felder & Chanson (2009), expressed their results in terms of dimensionless head loss.

In studies similar to those mentioned in the previous paragraph, Simões et al. (2012), using dimensional analysis and semi-empirical information also developed a formulation for calculating the re-aeration coefficient in stepped chutes, and fitting the data from Essery et al. (1978). A further step in that formulation would be the search of a more adequate relationship between efficiency and angles.

Bung & Valero (2018) analyzed experimental data obtained for stepped channels with 1V:2H and 1V:3H, corresponding to specific discharges between 0.07 and 0.11 m²/s and considering the skimming flow regime. They determined values of the overall mass transfer coefficient from the two-resistance model, k_L, ranging from 1.08 × 10^-5 to 3.5 × 10^-5 m/s. The authors also presented an equation relating k_L/u to C, where u is the mean flow velocity of the aerated flow, and C is the average volumetric air fraction determined for h₇₅. The parameter h₇₅ corresponds to the flow height at which the volumetric air fraction is equal to 75% perpendicularly to the pseudo-bottom (the alignment of the external step vertices).

Jahad et al. (2022) investigated re-aeration in conventional stepped chutes, stepped chutes with end sills, and channels with end sills shaped like a quarter circle. The slopes of the formed angles of 8.9°, 21.8°, and 26.6° with the horizontal line, and the regimes of skimming flow, transition flow, and nappe flow were studied. Considering all the data presented by the authors, there was an average increase of approximately 19.5% in re-aeration efficiency at 20 °C for the channel with the curvilinear terminal sill compared to the others.

Nina et al. (2022) investigated skimming flows in physical models of stepped chutes with a ratio of 1V:1H, s = 10 cm, using both horizontal steps and downward inclined steps, within the range of 0.8 ≤ h_c/s ≤ 1.6, where h_c = critical depth and s = step height. The authors conducted measurements of aerated flow characteristics at 20 °C and integrated the mass transfer equation to obtain the needed re-aeration information. They concluded for a higher re-aeration efficiency using horizontal steps in relation to inclined steps. Nina et al. (2022) combined their findings with experimental data from other authors and proposed an improved equation for the ratio between the re-aeration efficiency and the elevation drop in relation to the dimensionless dissipated energy, following the procedures of Felder & Chanson (2015).

In a recent study, Minho et al. (2024) numerically simulated flow over stepped chutes under nappe-flow conditions without hydraulic jump formation, employing a homogeneous multiphase model, the k-ε turbulence model, and the 2D advection-diffusion equation to estimate the spatial distribution of dissolved oxygen concentration along the chute. The authors compared numerical predictions with experimental data and determined the turbulent Schmidt number and the overall mass-transfer coefficient as fitting parameters between theory and experiment. Turbulent Schmidt numbers ranging from 0.9 to 6.0 and overall mass-transfer coefficients in the order of 10^-2 to 10^-5 m/s were found.

As can be inferred, there are few studies on re-aeration directly related to the skimming flow regime, although the existing experimental data already allow to test different conceptual approaches for this phenomenon. The present work was conducted with the aim to model and simulate skimming flows and the induced re-aeration using computational fluid dynamics (CFD). Specific objectives were: (1) to determine the overall mass transfer coefficient of the two-resistance model and the turbulent Schmidt number by fitting the mathematical physical model to experimental data of the literature for slopes of 21.8° and 45°; (2) analyze how the Eulerian-Eulerian formulation using homogeneous and non-homogeneous multiphase models impacts the solution; (3) to provide an improved empirical formulation for the re-aeration efficiency in stepped chutes with slopes of 11°, 21.8°, and 45°.

MATERIALS AND METHODS

The two-dimensional numerical study conducted in the present work considered flows and geometry that allowed comparisons with the experimental data of Essery et al. (1978). Those authors conducted measurements of dissolved oxygen in skimming flows in chutes with 11.3°, 21.8°, and 45° at a temperature of 15 °C. The present simulations considered the cases of 21.8° and 45°, having a maximum nondimensional ratio H_dam/h_c of 32, a restricting condition that approaches well the mentioned cases, whose maxima were 28.8 for 45° and approximately 32.5 for 21.8°. In the nondimensional ratio H_dam is the height of the crest of the chute in relation to its downstream base; and h_c is the critical depth of the flow in the rectangular channel. The needed information for the adjusting of the model (geometric, numerical, boundary conditions, and velocities at the inlet, V_e) are summarized in Table 1 and Figure 1.

Table 1
Information about the simulations conducted.

Simulation	Multiphase model	Angle	Mesh		Iterations	RMS	Ve
			Elements	Nodes		residual *	[m/s]
1	Homogeneous	45°	4.1E+05	8.2E+05	6712	2.40E-07	0.096
2	Inhomogeneous		4.1E+05	8.2E+05	2590	3.30E-07
3	Homogeneous	21.8°	6.6E+05	1.3E+06	4835	1.20E-07	0.098
4	Inhomogeneous		6.6E+05	1.3E+06	3934	2.00E-08

• ^*
  For oxygen mass fraction.
• RMS = root mean square.

Figure 1
Computational domain characteristics: (a) 45° and (b) 21.8°.

Tetrahedral meshes were employed with the number of elements and nodes specified in Table 1, chosen after conducting preliminary tests to assess mesh convergence. The adopted numerical schemes are of high resolution for the advective terms of the equations and using the finite volume method, as described in CFX (Ansys, 2021). For the calculations, the academic version of the Ansys CFX^® software (ANSYS, Inc. Products 2021 R2) was used on a computer running the Windows 10 64-bit operating system, with an 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80GHz/1.69 GHz processor and 16 GB of installed RAM, taking approximately a real time of thirty days to complete each simulation.

Multiphase flow – homogeneous model

The mass conservation and Navier-Stokes equations were originally formulated for a one-phase fluid. 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, both used in this study. The homogeneous model assumes that there is a sharing of fields between phases, except for the volumetric fraction field. The inhomogeneous model solves the fields for each phase, except for the pressure, which is shared between phases to close the system of equations. The description of these models is presented below, and is based on the references of Manninen & Taivassalo (1996) and CFX (Ansys, 2021).

Let U_α be the velocity field of phase α, and N_p the total number of phases. For the homogeneous model, Equation 2 represents the existence of a single velocity field for all phases. With r_α being the volumetric fraction of phase α and ρ_α the density of the fluid composing phase α, Equation 3 defines the mass density of the mixture of the N_p phases present in a control volume.

$U_{\alpha }=U$ for $$1\le a\le N_p$$(2)

$\rho ={\displaystyle \sum }_{\alpha =1}^{N_p}{r}_{\alpha }{\rho }_{\alpha }$ (3)

Equation 4 corresponds to the mass conservation for the N_p phases (Ansys, 2021). This equation is valid for both homogeneous and inhomogeneous models. For the homogeneous model, the velocity field is simplified with Equation 2.

$\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }\right)+\nabla ·\left(r_{\alpha }{\rho }_{\alpha }{U}_{\alpha }\right)=S_{\alpha }+{\displaystyle \sum }_{\beta =1}^{N_p}{\Gamma }_{\alpha \beta }$ (4)

in which Sα is a source term for mass, and Γαβ corresponds to the temporal rate of change of mass per unit volume, resulting from the mass flow between phases. Equation 5 establishes the relationship between Γαβ, the mass flow rate ${\dot{m}}_{\alpha \beta }$, and the interfacial area density, Aαβ, with units in m⁻¹, according to the International System of Units (SI).

${\Gamma }_{\alpha \beta }={\dot{m}}_{\alpha \beta }{A}_{\alpha \beta }$ (5)

The calculation of interfacial area density for flow in channels with stepped bottoms can be performed using the mixture model, defined by Equation 6, or the free surface model, defined by Equation 7, as studied by Simões (2012). In this study, the mixture model was adopted after conducting preliminary tests.

$A_{\alpha \beta }=\frac{r_{\alpha }{r}_{\beta }}{d_{\alpha \beta }}$ (6)

$A_{\alpha \beta }=\left|\nabla r_{\alpha }\right|$ (7)

in which d_αβ is the interfacial length scale, assumed to be 1 mm in the Ansys CFX^® software, a value also employed in the present study

Equation 8, derived from the second law of Newton, is the Navier-Stokes equation for a single phase, but with variable density and dynamic viscosity, μ, calculated with Equations 3 and 9, respectively.

$\frac{\partial }{\partial t}\left(\rho U_{\alpha }\right)+\nabla ·\left(\rho U_{\alpha }{U}_{\alpha }-\mu \left(\nabla U_{\alpha }+{\left(\nabla U_{\alpha }\right)}^T\right)\right)=S_M-\nabla p$ (8)

$\mu ={\displaystyle \sum }_{\alpha =1}^{N_p}{r}_{\alpha }{\mu }_{\alpha }$ (9)

In addition to the equations described earlier, the employed modeling resolves a transport equation for volume. Considering that the sum of the volumetric fractions of the phases equals unity, the use of Equation 4 results in Equation 10.

${\displaystyle \sum }_{\alpha }\frac{1}{{\rho }_{\alpha }}\left(\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }\right)+\nabla ·\left(r_{\alpha }{\rho }_{\alpha }{U}_{\alpha }\right)\right)={\displaystyle \sum }_{\alpha }\frac{1}{{\rho }_{\alpha }}\left(S_{\alpha }+{\displaystyle \sum }_{\beta =1}^{N_p}{\Gamma }_{\alpha \beta }\right)$ (10)

It is worth noting that in all presented cases, the variables correspond to mean values as per the use of Reynolds averaging, enabling the use of turbulence models. In this study, turbulence was modeled using the k-ε model (Jones & Launder, 1972), for both multiphase approaches.

Multiphase flow – inhomogeneous model

The flows were also modeled with the aforementioned equations written in the form of the inhomogeneous model (Ansys, 2021; Ishii & Hibiki, 2011). The conservation equations for mass and volume are the same as indicated earlier for the homogeneous model. The Navier-Stokes equation, however, takes the form presented in Equation 11, under a condition where only the pressure field is shared between the phases. The modeling of the interfacial area was carried out using the mixture model, as adopted for the homogeneous model.

$\begin{array}{l}\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }{U}_{\alpha }\right)+\nabla ·\left(r_{\alpha }{\rho }_{\alpha }{U}_{\alpha }{U}_{\alpha }\right)=-r_{\alpha }\nabla p+\nabla ·\left(r_{\alpha }{\mu }_{\alpha }\left(\nabla U_{\alpha }+{\left(\nabla U_{\alpha }\right)}^T\right)\right)+\dots \\ \dots +{\displaystyle \sum }_{\beta =1}^{N_p}\left({\text{Γ}}_{\alpha \beta }^{+}{U}_{\beta }-{\text{Γ}}_{\beta \alpha }^{+}{U}_{\alpha }\right)+S_{M\alpha }+M_{\alpha }\end{array}$ (11)

in which rα = volumetric fraction of phase α; ρα = mass density of the fluid composing phase α; Uα = velocity field of phase α; Sα is a mass source term; Γαβ corresponds to the temporal rate of change of mass per unit volume, resulting from mass flow between phases; Np = total number of phases; p = pressure field; μα is the dynamic viscosity (when interpreted without turbulence modeling, or the sum of this with turbulent viscosity for turbulence modeling condition); SMα is a source term representing external field forces, and sources and sinks defined for specific problems; Mα corresponds to interfacial forces acting on phase α due to the presence of other phases; the term ${\Gamma }_{\alpha \beta }^{+}{U}_{\beta }-{\Gamma }_{\beta \alpha }^{+}{U}_{\alpha }$ models the transfer of momentum between phases due to mass transfer between phases.

Multicomponent flow

Modeling the multicomponent behavior of multiphase flow employs Fick's law in conjunction with the mass conservation equation for a species, a condition that results in the advection-diffusion equation (Equation 12), written for the mass fraction Y_Aα. The subscript A represents a particular species present in phase α.

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

in which D_Aα [m²/s] is the kinematic diffusivity of component A in phase α, μ_t [kg/(ms)] is the turbulent viscosity, Sc_t is the turbulent Schmidt number, defined as Sc_t = μ_t/D_tAα, with D_tAα [kg/(ms)] 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 employed code uses Henry's law, specifying the Henry's coefficient of the mole fraction with pressure units, or the Henry's coefficient for molar concentration with pressure units per molar concentration unit. For the temperature of 15 °C used in the simulations, the mentioned coefficient is equal to 36959.2 bar, employing quadratic interpolation applied to the data from Spalding (1963).

Additionally, the model originated from the two-resistance theory was adopted. The use of this model requires specifying a mass transfer coefficient for the air phase and another for the water phase, in relation to the air-water interface. Considering the low solubility of oxygen in water, only a single overall mass transfer coefficient was necessary, and it was applied to the water side. As there is limited information available on this coefficient for flow in stepped channels, a trial-and-error process was followed in this study to approximate the numerical results to the experimental data used for comparisons. It is also noteworthy that the saturation concentration was calculated as an average of values obtained with the equations of Genet et al. (1974), Johnson & Duke Junior (1976), Baca & Arnett (1976), Popel (1979), and Roesner et al. (1981), resulting in C_s ≈ 10.10 mg/L, at 15 °C.

RESULTS AND DISCUSSION

The calculations performed led to the adjustment of the overall mass transfer coefficient, k_L, and the turbulent Schmidt number, Sc_t, with the results outlined in Table 2. It was observed that the same k_L value was necessary for simulations 1 to 3, while a relatively higher value was required for simulation 4. The trial-and-error process began with a value on the order of 10^-5 m/s, similar to that suggested by Chanson (2002), but it did not converge, yielding efficiencies close to 90%. Only the value of 0.0015 m/s allowed for a viable adjustment, leading to maximum efficiencies close to those found by Essery et al. (1978).

Table 2
Information on simulations and adjustments conducted.

Simulation	Multiphase model	Angle	Ve [m/s]	kL [m/s]	Sct [-]	Henry [bar]	σ [N/m]
1	Homogeneous	45°	0.096	0.0015	5.0	36959.2	0.0739
2	Inhomogeneous			0.0015	4.5
3	Homogeneous	21.8°	0.098	0.0015	3.8
4	Inhomogeneous			0.0100	1.0

• Notation: Sc_t = turbulent Schmidt number; Henry = Henry's coefficient; σ = surface tension coefficient for air-water interface at 15 °C (Incropera et al., 2007).

Recently, Nina et al. (2022) reported k_L values between 0.0015 m/s and 0.0020 m/s, for Reynolds numbers between 3.10⁵ and 4.10⁵ for stepped cascades inclined at 26.6°. The mentioned values are close to most of the results found in the present work for numbers Reynolds also of the order of 10⁵. The turbulent Schmidt number initially employed for each simulation was set to unity, following the Reynolds analogy. Only simulation 4 converged using this condition, while an increase in Sc_t values for the other simulations was required, converging to the values indicated in Table 2.

The numerical solutions for the dissolved oxygen concentration along the stepped chutes were utilized to calculate the corresponding efficiency fields at 15 °C, as depicted in Figure 2 (in blue). The same figure also displays the volumetric water fraction fields, enabling the identification of the free surface position (in black). It is noteworthy that both homogeneous and inhomogeneous models resulted in similar free surfaces. However, the homogeneous model still exhibited the occurrence of some air pockets between steps, while the inhomogeneous model completely filled the step cavities with water.

Figure 2
Water volume fraction (greyscale) and re-aeration efficiency at 15 °C for the four simulations (blue): (a) 45°, homogeneous model; (b) 45°, inhomogeneous model; (c) 21.8°, homogeneous; (d) 21.8°, inhomogeneous.

Regarding the special distribution of re-aeration efficiency at 15 °C (E₁₅) presented in Figure 2 (blue), the numerical solutions exhibited some distinct characteristics, particularly in the interface predicted by the inhomogeneous model in the approach channel region and the distribution of E₁₅ between the steps. It is observed that, for the 45° angle, the homogeneous model resulted in a more uniform E₁₅ distribution compared to the inhomogeneous model in the flow region between the steps, with the inhomogeneous model predicting less oxygenated zones localized in the inner corners of the steps. For the 21.8° angle, the same pattern was not observed, which may be attributed to modeling limitations or the difference in slope. The latter induces a different skimming flow sub-regime due to the displacement of the stagnation point on the step and variations in the wake development and its interaction with the step floor, as described by Chanson (2002).

The velocity fields obtained in all simulations demonstrate the occurrence of the skimming flow regime, as indicated in the longitudinal sections of two simulations shown in Figures 3a and 3b, for the angles of 45° and 21.8°. This provides further evidence of the suitability of the proposed methodology concerning its adequacy to represent experimental data, specifically in predicting the occurrence of the mentioned skimming flow regime. The direct incorporation of air by the water flow after the flow boundary layer reaches the surface was not reflected in the calculations as local air pockets (below the interface). However, the persistence of air pockets on the steps for the homogeneous model results may suggest this type of effect. The velocity fields presented in Figures 3a and 3b also demonstrate the existence of a stagnation point between the large vortex confined between steps and the downward flow. The location of the stagnation point was used as a reference for determining the dissolved oxygen concentration and calculating the oxygenation efficiency for comparisons with experimental data.

Figure 3
Hydrodynamic results: (a) 45° and (b) 21.8°, illustrating the occurrence of skimming flow; (c) CFD and analytical solution for the free surface profile upstream of the inception point.

The free surface profiles computed with CFD for angles of 21.8° and 45° were used to determine the Darcy-Weisbach friction factor, f. The numerical results were those upstream of the surface aeration onset point, determined using the equation proposed by Chanson (2002). The free surface profile was calculated following the methodology described by Simões et al. (2010), who analytically solved the differential equation for gradually varied steady flow using the Darcy-Weisbach resistance equation, leaving the Darcy-Weisbach friction factor as the sole fitting parameter.

Thus, Figure 3c presents the analytical solution according to Simões et al. (2010) alongside the numerical results obtained via CFD. The fitting parameter used to adjust the analytical solution to match the numerical points computed with CFD was the Darcy-Weisbach friction factor. For the 45° angle, both the homogeneous and inhomogeneous models resulted in f = 0.196; for the 21.8° angle, the same models yielded f = 0.170, demonstrating excellent agreement between the numerical modeling and the analytical solution for the S₂ profile (using Ven Te Chow’s terminology), as indicated in Figure 3c. These values for the Darcy-Weisbach friction factor are consistent with experimental data for slopes greater than 20°. Chanson (2002) reported f = 0.16 as the most probable value based on experimental results from multiple authors, while Simões et al. (2010) calculated f values ranging from 0.08 to 0.20. In this context, H_dam is also interpreted as the vertical coordinate axis originating at the crest of the stepped spillway or at the upper platform of the stepped chute, positive downwards.

The data from Essery et al. (1978), compared to the numerical solutions, can be seen in Figures 4a and 4b, along with Equation 13 developed by Simões et al. (2012). Upon initial observation, it appears that the homogeneous model predicted slightly lower efficiencies compared to the inhomogeneous model for some points. The trial-and-error process employed here, due to its high computational cost, did not allow for the further refinement of the fit, which is seen as a challenge for this field, given the considerable number of possible combinations between k_L and Sc_t. Considering the overall behavior, there is agreement between Equation 13 and the numerical solutions, suggesting an extension of its validity range to H_dam/h_c = 0. The results presented in Figure 4 also demonstrate that, for the 21.8° angle, there was less dispersion of the results around the experimental data points and Equation 13 than for the 45° angle.

$E_{15}=1-\text{exp}\left[-0.011{\left(\frac{H_{dam}}{h_c}\right)}^{1.3}\right]$ (13)

Equation 13 is valid for 11°, 21.8°, 45°, and 7.8 < H_dam/h_c < 32.5.

Figure 4
Comparisons between homogeneous and inhomogeneous models with experimental data from Essery et al. (1978) for angles of 45° (a) and 21.8° (b).

The analysis of Essery et al. (1978) data in this study led to the formulation of Equations 14, 15, and 16, with forms similar to Equation 13 but with coefficients restricted to the angle corresponding to each experiment. As shown in Figure 5a, the proposed equations follow the same trend as the experimental data, demonstrating good agreement, which is more precisely observed in Figure 5b, where the experimental data, the values calculated using Equations 14 to 16, and the exact fit line are compared. It is also noteworthy that the data points are well distributed around the proposed curves, indicating the absence of systematic deviations. For Equations 14, 15, and 16, the correlation coefficients further reinforce the strong agreement between the empirical models proposed in this work and the experimental data, with values of 0.986, 0.995, and 1.0, respectively.

$E_{15}=1-\text{exp}\left[-0.019{\left(\frac{H_{dam}}{h_c}\right)}^{1.107}\right]$ (14)

$E_{15}=1-\text{exp}\left[-0.0148{\left(\frac{H_{dam}}{h_c}\right)}^{1.194}\right]$ (15)

$E_{15}=1-\text{exp}\left[-0.0078{\left(\frac{H_{dam}}{h_c}\right)}^{1.44}\right]$ (16)

Equations 14 to 16 have the following validity intervals: Equation 14: 45°, 15.53 < H_dam/h_c < 28.57; Equation 15: 21.8°, 11.59 < H_dam/h_c < 46.38; Equation 16: 11.3°, 5.29 < H_dam/h_c < 10.58.

Figure 5
Comparison between Equations 14, 15, and 16 and the data from Essery et al. (1978).

Works like those of Felder & Chanson (2015) and Nina et al. (2022) present results for the efficiency per meter drop in the invert elevation, Δz_o, as a function of the dimensionless dissipated energy. Figure 6 presents the comparison of the results obtained in the present study with data from different authors, for angles close to or similar to those of the simulated stepped chutes, but at a temperature of 20 °C. The same graph presents the curves of the empirical models proposed by Felder & Chanson (2015) and Nina et al. (2022). For the 45° angle, the solutions obtained with computational fluid dynamics are close to the experimental data of Nina et al. (2022) mainly for lower values of ΔH/H_max. Deviations are observed between the two sets of data for higher values of head losses. This behavior may have occurred due to the use of a dimensional variable on the ordinate, i.e., E_T/Δz_o (E_T = re-aeration efficiency at a temperature T), which may be more sensitive to higher losses. In this sense, despite the dimensional variable, a greater adherence between numerical solutions and experimental data was observed for the angle of 21.8°, including data from Schlenkhoff and Bung (2009), for an angle of 26.6º and a step height of 6 cm.

Figure 6
Comparison with data and models from different authors.

CONCLUSIONS

Through the research presented in this work, it was possible to model and simulate skimming flows in stepped chutes with angles of 45° and 21.8° by employing homogeneous and inhomogeneous multiphase models, which were implemented in conjunction with the k-ε turbulence model and the advection-diffusion equation. The numerical solutions obtained exhibit the general pattern observed for skimming flows, with the formation of large eddies between steps, below the main flow. The incorporation of air downstream of the point where the boundary layer reaches the free surface (inception point) could not be observed in the numerical solutions as an expansion of the depth, or the presence of air pockets close to the air-water interface. Nevertheless, a persistence of air pockets was observed between the steps for some of the tested flow conditions, which may be reflecting the input of air through the interface. The results for reaeration expressed through efficiency fields demonstrate increasing values towards the toe of the stepped channel, which is consistent with experimental data. The adjustments made in this study allowed for the determination of the overall mass transfer coefficient of the two-resistance model and the turbulent Schmidt number, using homogeneous and inhomogeneous multiphase models. The comparison between numerical solutions and experimental data available in the literature showed adherence between the proposed models and the mentioned data. Through the analysis of solutions and experimental data, a set of equations was proposed for angles of 45°, 21.8°, and 11.3°, whose behaviors were in agreement with experimental data and exhibited high correlation coefficients.

The results obtained in this work can be related to different practical applications related to hydraulic engineering. Stepped channels with low slopes, such as those investigated, can be adapted to canalized river sections to enhance re-aeration and improve water quality in polluted rivers, such as the Ganges River, the Tietê River, and others. More steep channels can serve shorter transitions, such as lateral connections with tributaries in similar river systems. In these configurations, stepped channels act as energy dissipators and provide localized re-aeration zones. Furthermore, two-dimensional modelling, and with further research, three-dimensional modelling, can provide information for the design of fish passage channels by identifying dissolved oxygen distributions at different flow positions, including low-velocity zones where fish rest during migration. These applications highlight the broader potential of stepped chute hydrodynamics beyond their conventional use in spillways, extending their role to environmental engineering.

ACKNOWLEDGEMENTS

The first author acknowledges the support of CNPq for the Master's scholarship. This research was funded by the Coordination for the Improvement of Higher Education Personnel – Brazil (CAPES) under Financing Code 001. It was supported through process number 88881.708215/2022-01, as part of the Emergency PDPG for the Strategic Consolidation of stricto sensu Graduate Programs (PPGs) with CAPES grades 3 and 4, and process number 88881.691452/2022-01, under the PDPGPOSDOC/Strategic Postdoctoral Program of the Graduate Development Program (PDPG). Both grants were awarded within the scope of the Master’s Program in Environment, Water, and Sanitation (MAASA) at UFBA.

DATA AVAILABILITY STATEMENT

All data generated or analyzed during this study are included in this published manuscript.

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Edited by

• Editor-in-Chief:
  Adilson Pinheiro

• Associated Editor:
  Iran Eduardo Lima Neto

Publication Dates

• Publication in this collection
  09 June 2025
• Date of issue
  2025

History

• Received
  07 Feb 2025
• Reviewed
  16 Mar 2025
• Accepted
  14 Apr 2025