A new drainage staircase design for separating pedestrian traffic and surface runoff: a numerical and conceptual approach

Um novo projeto de escadaria drenante para separar o tráfego de pedestres do escoamento superficial: uma abordagem numérica e conceitual

ABSTRACT

Cities with rugged topographies, such as Salvador-Bahia, have neighbourhoods that have developed on hillsides, posing challenges for residents, including mobility and drainage. A solution proposed in 1979 became known as the drainage staircase, a structure designed to channel surface runoff beneath the steps while keeping the upper surface free for pedestrian movement. The original model underwent modifications in the following decades to address issues inherent to local conditions, yet these challenges persist. This study proposes a solution to recurring difficulties associated with classical models by introducing two lowered lateral sections on the steps, relative to the central portion, allowing lateral drainage while enabling pedestrian movement along the central section without competition between flow and users. The results obtained through computational fluid dynamics showed consistency with experimental data regarding dissipated energy and flow regime predictions, confirming the proposed model. The staircase efficiency for the three tested flow conditions—measured by its ability to divert surface runoff to the lateral sections—exceeded 90%, validating the initial hypothesis that the central section remains free for pedestrian movement during rainfall events. Additionally, human body stability was analyzed using existing criteria from the literature alongside the results for the velocity field and flow depths.

Keywords:
Computational fluid dynamics (CFD); Drainage staircase; Open channel flow; Resilient cities; Stepped chutes

RESUMO

Cidades com topografias acidentadas, como Salvador (Bahia), possuem bairros que se desenvolveram em encostas, apresentando desafios para os moradores, incluindo mobilidade e drenagem. Uma solução proposta em 1979 ficou conhecida como escadaria drenante, uma estrutura projetada para canalizar o escoamento superficial sob os degraus, mantendo a parte superior livre para a circulação de pedestres. O modelo original sofreu modificações nas décadas seguintes, em tentativas de solucionar questões inerentes às condições locais, mas esses desafios persistem. Este trabalho propõe uma solução para as dificuldades recorrentes associadas aos modelos clássicos, introduzindo duas seções laterais rebaixadas nos degraus, em relação à porção central, permitindo a drenagem lateral e, ao mesmo tempo, a circulação de pedestres ao longo da seção central, sem competição entre o escoamento e os pedestres. Os resultados obtidos por meio da dinâmica de fluidos computacional mostraram consistência com os dados experimentais quanto às previsões de energia dissipada e regime de escoamento, confirmando o modelo proposto. A eficiência da escadaria drenante para as três condições de escoamento testadas, medida por sua capacidade de desviar o escoamento superficial para as seções laterais, excedeu 90%, validando a hipótese inicial de que a seção central permanece livre para a circulação de pedestres durante os eventos de chuva. Além disso, a estabilidade do corpo humano foi analisada utilizando critérios existentes na literatura, juntamente com os resultados para o campo de velocidades e alturas de escoamento.

Palavras-chave:
Dinâmica dos fluidos computacional; Escadaria drenante; Escoamento em canais; Cidades resilientes; Canais em degraus

INTRODUCTION

The increasing number of dams constructed with roller-compacted concrete and consequent specifications of stepped spillways have significantly contributed to research on flows in stepped chutes and spillways. Previous work by Essery & Horner (1978) identified the existence of two flow regimes: (1) nappe flow and (2) skimming flow. Subsequent studies, such as Sorensen (1985), confirmed the existence of these regimes and further defined sub-regimes, as described by Chanson (2002). Flow characteristics have also been investigated, including aeration, pressures and cavitation, energy dissipation, and interfacial mass transfer. In this context, designs of other hydraulic structures utilizing stepped chutes have benefited from the knowledge derived from stepped spillways. Among these structures are so-called “drainage staircases” used on slopes.

Stepped weirs are large hydraulic structures consisting of an upper part known as an ogee or simply a crest, although the crest specifically refers to the top of the ogee, with a hydrodynamic shape typically of the Creager or Scimemi type. The crest may include a stepped section with varying step heights to prevent jumps and free-surface oscillations at low and high discharges, respectively. Downstream of the dam, the stepped chute itself is constructed with uniform steps, transitioning to a stilling basin for energy dissipation, with or without a counter-curvature. In contrast, drainage staircases do not have a Creager-type crest; instead, they feature landings between steps, as their primary function differs. Their design is intended to facilitate drainage while ensuring safe pedestrian access, allowing people to use the staircase without directly encountering surface runoff. The typical slopes of stepped spillways are chosen based on an analysis that also involves the structural stability of the dam. For drainage staircases, the slope, in many cases, must adapt to the realities of the location where they are to be built, but without losing sight of the essential comfort characteristics of a staircase that pedestrians will use. Stepped spillways, in most cases, operate under the skimming flow regime. Because staircases are generally less steep and operate with much lower flow rates than stepped spillways, they operate under the nappe flow regime in most cases. This general scenario can be modified in specific cases when there is a restriction on width, which is not uncommon in neighbourhoods located on slopes.

Drainage staircases are hydraulic structures employed as pedestrian pathways and channels for water evacuation during rainy periods. According to Mangieri (2012), in the thirty years since the conception of the first drainage staircase in 1979, six different types of staircases have been proposed, and over 370 structures have been built in the city of Salvador, Bahia, Brazil. These staircases must divert surface flow to the auxiliary subsurface drainage system, allowing people to use the staircase without competing with the flow. Up to 2012, literature analysis did not identify any studies on physical models or methodologies proposed to design and analyze drainage staircases. Mangieri (2012) was the first to address the topic using equations related to flow studies in stepped channels.

Simões et al. (2015) employed computational fluid dynamics and simulated flows in a three-dimensional domain of a drainage staircase with holes on the upper landing, providing access to a staircase beneath the main staircase. The authors also included an obstacle on the floor of one of the steps, resembling a human foot, to calculate drag force. By obstructing 5 out of 10 holes, the authors observed increased drag force on the obstacle, highlighting the need for adequate drainage maintenance. This aspect integrated the objectives of Simões et al. (2015) as nearby residents closed or partially closed the holes responsible for drainage. This action aimed to prevent accidents involving children moving around the landing area where the grates of the drains are located, as well as to prevent access to the upper part of the floors and steps by rodents inhabiting the lower part of these structures, as demonstrated by Mangieri (2012) through reports and photographs.

Oliveira et al. (2018), using mass conservation equations, Reynolds-averaged Navier-Stokes equations (RANS) under the form of the inhomogeneous model and the k-ε turbulence model, simulated water flow and a fluid as dense as sand in a stepped channel under nappe flow regime. As a result, the authors identified sedimentation of the denser fluid at the initial part of the floor. Although through numerical means, this finding contributes to the knowledge of flow in successive falls applicable to drainage staircases by demonstrating the possibility of mud accumulation and a potential increase in human fall risk without proper maintenance. In a subsequent study, Simões et al. (2019) simulated 2D flows in low-gradient stepped channels and identified the location of the stagnation point on each step floor. The occurrence of a stagnation point on the floor divides the downstream flow from the recirculating flow over the floor, which is the likely mechanism for sediment capture.

Rodrigues et al. (2020) studied flow behavior in a drainage staircase built in some neighborhoods of Salvador, Bahia, comprising a conventional grate drain on the upper landing that accesses a box directing water to a subsurface pipe. Among the study's conclusions was the demonstration of elevated pressures at the base of the pipe, which may have caused the reported ruptures in Mangieri (2012). Rodrigues et al. (2020) also showed that the grate could only capture 23% of the total flow for the simulated flow rate, indicating low efficiency.

Considering the problems described in the cited works, Simões et al. (2020) developed a staircase with a centrally lowered channel compared to the lateral steps. With this proposal, the authors calculated 3D flows with mass conservation equations, RANS, the inhomogeneous model, and the k-ε model, finding solutions where there is virtually no flow in the higher sections, eliminating subsurface drainage.

Ribeiro et al. (2021) conducted experiments on a physical model of a drainage staircase with capture through a grate drain on the upper landing. They included obstacles on the step floors to calculate the flows resulting in obstacle instability. In the same work, a mathematical model based on fundamental physical laws was developed to determine the unstable conditions of the obstacles, which approximately acted as representatives of potential staircase users.

Li et al. (2022) emphasize the importance of studies on flow dynamics in staircases leading people to underground facilities such as subway stations. The aforementioned authors numerically investigated flow patterns in a staircase using large-eddy simulations with the Smagorinsky subgrid-scale model. Among the obtained results, drag coefficients were calculated for a human body model positioned at different locations on the staircase, and low- and high-risk zones were delineated.

The cited studies underscore the necessity of improving existing drainage staircase designs to ensure they fulfill their intended functions without introducing new challenges for residents, such as accidents involving children's feet caught in grates or the proliferation of rodents near homes. Additionally, the low efficiency of grate drains and the vulnerability of subsurface drainage systems—prone to pipe ruptures and blockages caused by solid waste—further emphasize the urgency for new solutions. Beyond regional concerns, recurrent flooding events affecting staircases used to connect surface areas with underground transportation systems highlight a broader, global relevance.

In this context, this study builds upon the concept originally introduced by Menezes et al. (2022) and proposes a drainage staircase design featuring side sections of the steps lowered relative to the central portion. This configuration directs most of the surface flow to the side channels, keeping the central section free of pedestrian traffic. The specific objectives are: (1) to compare the dissipated energy with data from the literature, considering different possible approaches for calculating the dimensionless values, given that the total flow is divided into two or three parts; (2) to evaluate the efficiency of the side channels in draining the total flow while keeping the central section free for pedestrian movement; (3) to assess the flow regimes across the sections of the proposed staircase; (4) to determine the maximum flow heights near the entrance under the influence of the highest step; and (5) to apply human body stability criteria to the flow velocity and height data to analyze the risk of falls.

MATERIALS AND METHODS

Definition of the drainage staircase shape

The conceived staircase, illustrated in Figure 1, consists of steps with a height s = 0.16 m, a step length l = 0.32 m, a total width of 2.40 m, with each of the three sections having a width of 0.80 m and a total height of 3.2 m between the upper and lower landings. The width and height of the staircase depend on the local implementation conditions, with the present dimensions considered feasible for observed staircases. It is worth mentioning that in neighborhoods situated on slopes, there usually is limited space to meet the widths specified in technical standards, such as NBR 9050 (ABNT, 2015), both in the lowered sections and in the central section, which proposes a minimum width of 1.80 m. Hence, narrower widths of 0.80 m per section were employed. As depicted in Figure 1a, there is no subsurface drainage, and the geometry indicates that flow will be diverted to the lateral channels due to an additional step at the upper landing, which should be equipped with an access gate and a signaling plate to prevent accidents.

Figure 1
Staircase developed in the present work: (a) isometric view; (b) longitudinal section.

Physical-mathematical modeling

For high Reynolds numbers and problems with large dimensions, conducting direct numerical simulation (DNS) remains infeasible, as it requires the use of meshes with a high number of elements to capture the minor Kolmogorov scales, in addition to memory limitations, since turbulence is variable, demanding simulations in unsteady state. Therefore, turbulence modeling is necessary to solve typical engineering problems. In the present study, the mass conservation equation, Reynolds-averaged Navier-Stokes equations in the form of the multiphase inhomogeneous model, and the k-ε turbulence model were employed using Ansys CFX^® software. A description of the equations of the inhomogeneous model, based on CFX (Ansys, 2021), is provided below. It is worth noting that the simulations were conducted in a steady state. Equation 1 corresponds to the mass conservation equation for the inhomogeneous model.

$\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }\right)+\nabla .\left(r_{\alpha }{\rho }_{\alpha }\overrightarrow{V_{\alpha }}\right)=S_{MS\alpha }+{\displaystyle \sum }_{\beta =1}^{N_p}{\Gamma }_{\alpha \beta }$ (1)

where rα represents the volumetric fraction of phase alpha, such that the summation of rα for all phases equals unity; ρ_α is the mass density of phase alpha, $\overrightarrow{V_{\alpha }}$ is the velocity field of phase alpha, $S_{MS\alpha }$ is a source term, and ${\Gamma }_{\alpha \beta }$ is the mass transfer rate from phase beta to phase alpha per unit volume, where N_p is the total number of phases.

Equation 2 corresponds to the Navier-Stokes equation in the form of the inhomogeneous model.

$\begin{array}{l}\frac{\partial }{\partial t}\left(r_{\alpha }{\rho }_{\alpha }\overrightarrow{V_{\alpha }}\right)+\nabla .\left(r_{\alpha }{\rho }_{\alpha }\overrightarrow{V_{\alpha }}\overrightarrow{V_{\alpha }}\right)=-r_{\alpha }\nabla p_{\alpha }+\nabla .\left(r_{\alpha }{\mu }_{\alpha }\left(\nabla \overrightarrow{V_{\alpha }}+{\left(\nabla \overrightarrow{V_{\alpha }}\right)}^T\right)\right)+\dots \\ \dots +{\displaystyle \sum }_{\beta =1}^{N_p}\left({\Gamma }_{\alpha \beta }^{+}{\overrightarrow{V}}_{\beta }-{\Gamma }_{\beta \alpha }^{+}{\overrightarrow{V}}_{\alpha }\right)+S_{M\alpha }+M_{\alpha }\end{array}$ (2)

where p_α is the pressure field of the alpha phase, which is the same for all N_p phases, a necessary condition to close the system of equations; μα is the viscosity of the alpha phase; $\left({\Gamma }_{\alpha \beta }^{+}{\overrightarrow{V}}_{\beta }-{\Gamma }_{\beta \alpha }^{+}{\overrightarrow{V}}_{\alpha }\right)$ represents the transfer of momentum between phases induced by mass transfer between phases. The term S_Mα is a source term for the transfer of linear momentum caused by field forces, such as weight force. Mα corresponds to the interfacial forces acting on the α phase due to the presence of the other phases.

Boundary conditions, meshes, and numerical methods

The boundary conditions include the inlet, outlet, top of the computational domain, bottom, and side walls. The inlet's average velocities were set within a rectangular area measuring 0.15 m in height and 2.4 m in width. The outlet was modeled with zero gradients for the variables, an appropriate condition for outlets with supercritical flows, from the viewpoint of the hyperbolic part of the equations. The top was considered open to the air and closed to water, while the walls and bottom were modeled with wall laws, with an equivalent absolute roughness of 1.0 mm, a representative value for concrete with a normal finish (Porto, 2006).

The chosen flow rates for the simulations, as presented in Table 1, correspond to values of s/h_c, where h_c = critical depth, resulting in the occurrence of nappe flow for simulations 1 and 2 and in the transition flow that exists between nappe flows and skimming flow for simulation 3. The predictions of these flow regimes were made based on the graph in Figure 2, developed based on equations proposed by different authors, which shows the existence of regions in the s/l - s/h_c plane where there is agreement between the equations (continuous colors) and disagreement between the equations or only one equation (dotted colors).

Table 1
Flow rates and related quantities used in the simulations.

Simulation	Q [m3/s]	B [m]	q [m2/s]	hc [m]	s/hc	s/l	he [m]	Ve [m/s]
1	0.0871	2.4	0.036	0.051	3.1	0.5	0.15	0.242
2	0.1991		0.083	0.089	1.8			0.553
3	0.4168		0.174	0.145	1.1			1.158

• Symbols: Q = discharge; B = channel width; q = Q/B; h_c = critical depth; s = step height; l = length of the step tread; h_e = height of domain entry; V_e = q/h_e, average velocity at the entrance to the domain.

Figure 2
Regions for predicting flow regimes in stepped chutes and spillways based on experimental studies by several authors. Source: Simões et al. (2011).

Finite volume numerical schemes were adopted, using a high-resolution method for the advective part and turbulence. Residuals lower than 10^-4 were adopted as a stopping criterion, considering this value as RMS (root mean square). For each simulation, unstructured tetrahedral meshes with 3,334,775 elements were adopted, requiring approximately 24 hours of simulation for each flow on a computer with an i7-1165G7 processor and 16GB of RAM installed. Tests with other degrees of refinement for the meshes indicated convergence towards the adopted mesh.

RESULTS

Three-dimensional numerical solutions obtained via Computational Fluid Dynamics (CFD) can be visualized through various graphs. For this study, determining the position of the water-free surface and the flow location is of practical interest. Thus, isosurfaces of the volumetric air fraction equal to 0.90 were generated, a value commonly employed in studies on stepped spillways, as seen in Chanson (2002). Nappe flow was observed for the lowest simulated discharge for the recessed section. Using a longitudinal section located at the center of the recessed section, flow heights at the ends of each step, denoted as h₁, were determined. With these heights and Equation 3, as presented in Chanson (2002), dimensionless head losses ΔH/H_max were calculated to compare the results with experimental data available in the literature.

$\frac{\Delta H}{H_{max}}=1-\left(h_1+\frac{h_c^3}{2h_1^2}\right)/\left(\text{z}+\frac{3}{2}{h}_c\right)$ (3)

where z = positive axis downwards, originating at the upper level; H_max = z + 1.5h_c.

Figure 3 allows one to compare the numerical results obtained in the present work with the experimental data for s/l = 0.42, from Horner (1969) and s/l = 0.47 from Jahromi et al. (2008). Although there is a difference between the s/l values and the s/l = 0.5 of this work, a relative approximation between the numerical points and the experimental data is noted, indicating the consistency of the solution obtained. The analysis was also conducted using the critical depth determined from the specific discharge calculated for the lateral section, denoted as ℎ_𝑐*. It is observed that the results for simulations 2 and 3 followed the same trends as those obtained using ℎ_𝑐, remaining close to them. For simulation 3, the dimensionless dissipated energy was lower than that observed in simulations 1 and 2 for both approaches, i.e., using ℎ_𝑐 and ℎ_𝑐*. This outcome is attributed to the occurrence of skimming flow in the lateral part for simulation 3.

Figure 3
Numerical results from simulations 1, 2, and 3, and experimental data from Horner (1969) and Jahromi et al. (2008).

Figure 4a, corresponding to the lowest discharge, indicates that the flow was diverted to the lowered lateral sections, leaving the central part of the channel practically free. Figure 4b presents a similar pattern but with a larger volume of water on the steps compared to the lower flow rate simulation. The results of the higher flow rate, as illustrated in Figure 4c, show that the central channel starts to operate with a significant volume on the first steps, followed by subsequent lateral spreading.

Figure 4
Representation of the water free-surface for simulations: (a) 1, (b) 2, and (c) 3.

The calculation of the flow rates for the cross sections at the inlets of the lowered lateral channels, located at the upper level, was performed by integrating the velocity distributions, resulting in the values indicated in Table 2. For simulations 1 and 2, the efficiency results are defined as the lateral flow rate divided by the total flow rate, which corresponds to approximately 100%, consistent with the images of the isosurfaces in Figures 4a and 4b. Simulation 3 showed an efficiency of 93.5% for the upper level and increasing efficiencies along the channel due to the lateral propagation of the flow, as shown in Figure 5.

Table 2
Results for the lateral sections.

Simulation	Qlaterals [m3/s]	Ef [%]	qlateral [m2/s]	hc* [m]	s/hc*
1	0.087	100	0.05	0.07	2.38
2	0.199	99.7	0.12	0.12	1.38
3*	0.390	93.5	0.24	0.18	0.88

• Symbols: Q_laterals = discharge in the lateral sections; Ef = efficiency = Q_laterals/Q, Q = total discharge; q_lateral = specific flow in a lateral section; h_c* critical depth calculated with q_lateral.
• ^*
  For simulation 3, Ef varies along the channel in steps (see Figure 5).

Figure 5
Efficiency variation along the stepped chute for simulation 3. H = 3.2 m (total height).

It is worth highlighting the occurrence of the nappe flow pattern for simulation 1, as indicated in Figure 6a, showing jets successively impacting the steps of subsequent steps and the formation of air cavities under the jets. The images in Figure 6 consider a vertical longitudinal plane located in one of the lateral channels. The value of s/h_c* from Table 2 and Figure 2 demonstrates the coherence between the simulation and the graph obtained through experimentation. The pattern of simulation 2 is not as regular as that of simulation 1, consistent with the prediction of Figure 2, which indicates a zone of divergence between the methodologies, likely due to the undefined nature of the flow regime. In Figure 6c, the steps are entirely submerged in a condition similar to the skimming flow regime.

Figure 6
Volumetric fraction of air and detail of the velocity vector field for simulations (a) 1, (b) 2 and (c) 3, in side views.

This result also agrees with Figure 2 and is confirmed by large eddies between steps, as indicated by the velocity field presented in Figure 6c. The results in Table 2 show that most of the flow through the sides of the stairs. The efficiency for simulation 3 increases along the ladder, reaching 99.8% on the 8th step, from top to bottom. Skimming flow (higher flow) results from rainfall with a longer recurrence period. However, even this condition shows that the human being will not be subject to the entire drag force of the flow but to around 7% of the flow that produces this force. The lower flows tested here (precipitations with shorter recurrence times) indicate that the central part of the ladder will be viable for safe use.

The inlet condition of a system like the one proposed in this study may be influenced by the highest step located at the central portion of the upper landing. Since the simulations were conducted for ℎ_𝑒 > ℎ_𝑐, the flows were necessarily imposed as subcritical at the inlet. Consequently, in the first simulation, there was a reduction in flow depth, which quickly adjusted to a uniform value of approximately 0.10 m. For subcritical flows, the hyperbolic part of the system of equations governing the problem (mass conservation and Navier-Stokes), which has real eigenvalues with opposite signs in the convective matrix, requires the imposition of one variable—such as flow depth—while velocity or discharge is calculated. In the present case, as the model is mixed-type partial differential equation, the adopted strategy was to impose both depth and discharge, with the only drawback being a slight curvature near the inlet, ensuring agreement with the equilibrium depth.

As shown in Figure 7a, three longitudinal planes centered on the three sections of the channel were selected to analyze the influence of the elevated step on flow depth. Figure 7b, corresponding to the lowest discharge, illustrates the curvature near the inlet, as previously mentioned, and also shows that in the central portion, the flow depth was lower than the step height. Figure 7c, corresponding to Simulation 2, displays a slight elevation caused by the presence of the elevated step, while Figure 7d shows a more pronounced increase upstream of the same step. For Simulations 1, 2, and 3, the maximum upstream depths were 0.118 m, 0.206 m, and 0.278 m, respectively. These results are relevant for design considerations, particularly if the inlet configuration is similar to the one adopted in this study.

Figure 7
Flow profiles at the upper landing and analysis of the inlet condition: (a) three analyzed planes, (b) Simulation 1, (c) Simulation 2, (d) Simulation 3.

Discussion on risk zones associated with flow on the staircase

The velocity field analysis for simulation 3 demonstrates that it is predominantly composed of velocities below 0.45 m/s (Figure 8a and 8b). However, there are regions with elevated velocities, reaching up to 4.5 m/s in very small regions and 4.0 m/s in non-negligible volumes, as in Figure 4c,, located near the lower part of the channel where the flow height resulted in h = 0.23 m, as illustrated in Figure 8d. At the same position, the average velocity resulted in 1.44 m/s. By employing the risk index f(V²h) = 1.5 m³/s (high risk of accident) and f(V²h) = 1.2 m³/s (low risk of accident) proposed by Inoue et al. (2003) and utilized by Li et al. (2022) for staircases, it is observed through numerical solutions that V²h = 0.48 m³/s² < 1.5 m³/s², indicating a low risk of accident for staircase users. In this context, it is highlighted that the roughness of the stair surface may vary due to the presence of sediments captured by the vortex located between steps (Oliveira et al., 2018); it is also noted that the age, physical and psychological conditions of staircase users influence their ability to balance, as well as the presence or absence of handrails (Simões et al., 2016). The lack of knowledge about the dynamics of such random variables points to the convenience of using a staircase without flow during rainy events, as proposed in the present study.

Figure 8
Velocity fields of simulation 3: (a) isometric view, (b) isovolume for 0.2 < v < 4.0 m/s, v = velocity field; (c) isovolume region for 4.0 m/s, (d) location of h = 0.23 m.

CONCLUSIONS

The three-dimensional results obtained for the proposed stepped chute in this study indicate that its design has the potential to enable safe pedestrian traffic along the central section of the staircase. The numerical results show adherence to experimental data available in the literature for dimensionless dissipated energy, indicating the coherence of the adopted modeling approach. Defining efficiency based on the channel's ability to divert flow to the lateral sections resulted in efficiencies of 100% and 99.7% for simulations with lower flow rates and an efficiency of 93.5% for simulation 3, with a higher flow rate than the others. The flow regimes observed through isosurfaces with a volumetric air fraction of 0.90 were consistent with predictions based on methodologies developed from experimental data, confirming the suitability of the methodology adopted in this study. The results suggest the development of technical standards that consider steps with recessed parts for safe pedestrian pathways in densely populated hillside cities with rugged topography. The proposed drainage staircase design aims to separate pedestrian traffic from surface runoff by directing most of the flow to the lateral channels, leaving the central portion available for safe pedestrian movement under different flow conditions. The numerical results confirm the design's ability to maintain high efficiency in flow diversion while aligning with experimental data for energy dissipation and flow regime characterization. These findings demonstrate the design's potential as a practical solution to urban drainage and pedestrian safety in areas with complex topography.

ACKNOWLEDGEMENTS

This research was funded by CNPq and 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; and, FAPESB, Bahia, Brazil, through agreement PIE0021/2016.

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• Simões, A. L. A., Menezes, P., Lima, L., Schulz, H. E., & Porto, R. M. (2020). Concepção de uma escadaria drenante com canal central rebaixado empregando simulações numéricas. In Anais do XIII Encontro Nacional de Águas Urbanas (pp. 1-12). Porto Alegre: ABRHidro.
• Simões, A. L. A., Schulz, H. E., Lobosco, R. J., & Porto, R. M. (2011). Stepped spillways: theoretical, experimental and numerical studies. In H. E. Schulz, A. L. A. Simões, & R. J. Lobosco (Eds.), Hydrodynamics: natural water bodies (pp. 237-262). London: InTechOpen.
• Simões, A. L. A., Schulz, H. E., Porto, R. M., Luz, L. D., Mangieri, L. S. G., & Matos, J. E. R. (2015). Contribuição numérica para o estudo de escoamentos em quedas sucessivas e aplicações a drenagem. In Anais do XXI Simpósio Brasileiro de Recursos Hídricos (pp. 1-8). Porto Alegre: ABRHidro.
• Sorensen, R. M. (1985). Stepped spillways hydraulic model investigation. Journal of Hydraulic Engineering, 111(12), 1461-1472. http://doi.org/10.1061/(ASCE)0733-9429(1985)111:12(1461)
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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
  12 Dec 2024
• Reviewed
  20 Mar 2025
• Accepted
  17 Apr 2025