Hydraulic Characteristics of a New Vertical Slot Fishway with Staggered Baffles Configuration

Abstract

The vertical slot fishway (VSF) has proven effective in mitigating the severe fragmentation of rivers caused by artificial hydraulic structures. While fishways with steeper slopes exhibit better economic performance, increased slope can raise the flow velocity and turbulence, which may hinder fish migration. To address this issue, this study investigated the application of a VSF with a staggered baffle configuration. Through numerical modeling, the hydraulic characteristics of the VSF under various slope ratios and chamber length-to-width (L/B) ratios were investigated, with data validated by physical models. An increase in the slope gradient resulted in higher flow velocities, greater maximum attenuation rates of mainstream velocity, and elevated turbulent kinetic energy (TKE) at the corners of the rectifier baffles and the ends of the divider baffles. Additionally, the overall maximum volumetric energy dissipation ($D_{\epsilon }$) increased, although its distribution pattern remained unaffected. Conversely, increasing the chamber L/B ratio significantly altered the distribution patterns of the flow velocity, TKE, and $D_{\epsilon }$, influencing their generation mechanisms. For instance, a higher chamber L/B ratio caused the maximum flow velocity (V_m) to deviate from the vertical slot and raised the maximum attenuation rate of the mainstream velocity. The L/B ratio also caused changes in the TKE distribution; as the ratio increased, the proportion of the chamber’s internal region with $D_{\epsilon }\le 150 \mathrm{W}/{\mathrm{m}}^3$ initially decreased and then increased. Overall, considering the flow velocity, TKE, and $D_{\epsilon }$, it is recommended that the chamber L/B ratio be maintained between 0.9 and 1.1 for slope ratios ranging from 1:20 to 1:50. The research results will offer practical insights for engineering applications, in engineering construction, contribute theoretical guidance for the optimized design of fish passages, promote sustainable hydraulic engineering practices, and aid in the protection of aquatic biodiversity.

1. Introduction

The establishment of hydraulic structures, including sluices and dams, is crucial for flood control, disaster mitigation, agricultural irrigation, alleviating water scarcity, and developing clean energy [1,2]. While these structures provide significant benefits, they also fragment river ecosystems by transforming continuous waterways into isolated sections that obstruct fish migration, adversely affecting fish populations and reproduction [3,4,5]. Consequently, the development of fish passage systems has become increasingly important. Fish ladders, the most commonly implemented solution, assist fish in overcoming barriers and re-establishing their migratory routes. These structures have demonstrated positive ecological effects by conserving fish resources and maintaining aquatic biodiversity. Among various fishway designs, the VSF has been widely utilized owing to its straightforward construction, efficient energy dissipation, operational flexibility, and excellent adaptability to fluctuations in upstream and downstream water levels [6].

In fishway design, the bottom slope gradient significantly impacts the overall length of the structure, thereby influencing the duration of fish ascension, layout, and construction costs [7]. Ahmadi et al. [8] highlighted that optimizing low-slope VSFs offers a more cost-effective solution, as steeper slopes reduce the overall length and construction costs. However, high flow velocities and turbulent flow fields induced by steep slopes pose significant challenges to fish species with limited swimming abilities during upstream migration [9]. To balance ecological restoration and engineering feasibility, researchers have performed numerous studies on the hydraulic optimization of VSFs [10,11,12,13]. Chorda et al. [14] tested three longitudinal slopes and validated the k–$\epsilon$ closure model. They found that the energy dissipation rate substantially affects fish passage efficiency. Tarrade et al. [15] identified two distinct flow topologies with a swirling pattern based on fishway size and slope. Puertas et al. [16] examined the performance of two VSFs at different slopes and concluded that the velocity remains independent of discharge and constant with depth. Ahmadi et al. [17] numerically examined the impact of the slope and cylindrical structure within the fishway on fish swimming performance.

Rajaratnam et al. [18] highlighted the L/B ratio of fishways as a key factor influencing flow variations when examining fishways with different baffle configurations. Wang et al. [19] studied the turbulent characteristics in a scaled VSF model with varying width and slope and revealed that changes in the chamber width directly affected the flow structure and turbulence intensity. Therefore, the L/B ratio can be an important parameter for the optimal design of a fishway.

Currently, conventional vertical slot fishways (VSFs) position both the divider and rectifier baffles on the same side. This linear alignment of vertical slots generates a shorter mainstream flow path compared to staggered configurations. To enhance energy dissipation, this study proposes a staggered configuration for both baffle types, which elongates the mainstream flow path. This modification reduces the maximum velocity (V_m) in the pool chamber to below 1.2 m/s—a value lower than the burst swimming speed (1.2 m/s) for the four major Chinese carps. The staggered design also improves flow uniformity, while maintaining a recirculation zone with resting velocities of 0.3–0.5 m/s. These velocities align with the sustained swimming capacity of cyprinid species, thereby facilitating fish migration and resting. Numerical simulations carried out employed the RNG k-ε turbulence model, while physical modeling was performed to validate the results. Various structural configurations, including the bottom slope ratios and length-to-width ratios of the pool chamber, were examined for their impact on the hydraulic performance of this type of fishway. The findings of this research offer theoretical insights for the optimized design of fish passages and hold practical significance for engineering applications.

2. Materials and Methods

2.1. Research Object

This study involved a fishway located at a water conservancy project on the Han River in Hubei Province, China, which is a critical protection area for the Four major Chinese carps. The V_m within the fishway must not exceed the burst swimming speed of the fish [20], which was identified as less than 1.2 m/s for the four major Chinese carps [21,22,23]. The fishway, designed with a water depth of 2 m, a design discharge of 3.2 m³/s, and a flow velocity of 0.8 m/s, was developed considering the hydrological conditions and the specifications of the hydraulic structure.

The cross-section of the fishway pool chamber is rectangular, measuring 2.0 m in width (B) and featuring a divider baffle length of 1.26 m. The rectifier baffles extend 0.5 m. Both the divider and rectifier baffles have a thickness of 0.20 m. The vertical slot width (b) is 0.33 m. In this study, the L/B ratio of the fishway was controlled by varying the length of the fishway pool chamber (L) from 1.20 to 3.00 m in increments of 0.20 m, so that the L/B ratios varied between 0.6 and 1.5. The selected slope ratios (J_i) for this study are 1:20, 1:30, 1:40, and 1:50. The structural design of the fishway pool chamber is depicted in Figure 1.

2.2. Physical Model

The validation data in this study were obtained from local model tests of the fishway associated with the aforementioned hydraulic engineering project. The experimental setup includes an inlet, an outlet, and ten conventional pools, each measuring L = 2.4 m, B = 2 m, and J = 1:20. The experimental water depth was maintained at 2 m. The model was constructed based on the Froude number similarity. With reference to existing research [14,24], the experimental ratio was set at 1:5. Tests were conducted in a multifunctional test tank, which features plexiglass panels for the sides, bottom, and baffles, with installation errors maintained blow 0.2 mm. The test tank consists of a water reservoir, an energy dissipation mesh, an electromagnetic flowmeter, a manual valve, a downstream sluice gate, a submersible pump, a reflux corridor, and other components. Its inlet section is 4 m long and equipped with an energy dissipation mesh to ensure uniform and smooth flow upstream to the first level of the fishway, while the 4 m long tailgate exit eliminates water turbulence. The experimental setup is illustrated in Figure 2. During testing, the inlet flow was regulated using the manual valve, and the flow rates were measured by the electromagnetic flowmeter, which has an accuracy of 0.4% and a sampling frequency of 10 Hz. To ensure a consistent water depth across all pools, the water depth was regulated by downstream sluice gates, with a water level difference of 0.12 m between adjacent pools.

Due to the vertical slot design and varying side configurations of the fishway, the flow structure can differ between adjacent pools. For validation purposes, the pools from slots “5” to “7” were chosen. In each pool, a cross-section labeled S_i was selected at 0.4 m intervals, where i = 1 to 10. This operation resulted in a total of 10 validation cross-sections, each of which contained 10 measurement points; the spacing between measurement points is 0.18 m. To validate the vertical flow velocity, two measurement lines were established at the center of each slot, each of which had 9 measurement points that were vertically spaced at 0.2 m. To assess grid convergence, the velocities at the midpoints (V_S2 and V_S9) of cross-sections S2 and S9 were utilized to compute the GCI value. The layout of the measurement points for the flow velocities within the pool chamber is illustrated in Figure 3.

2.3. Numerical Simulation

2.3.1. Model Establishment

The numerical model was constructed using FLOW-3D^® v11.2, a widely utilized computational fluid dynamics (CFD) software in fishway studies [25,26]. This general purpose program is able to provide transient 3D solutions for complex multiphysics flow problems by solving fluid motion equations [27]. FLOW-3D^® has found extensive applications in studies related to hydraulic systems and engineering (e.g., [28,29,30,31,32,33]). A detailed description of FLOW-3D^® is available in the Flow Science documentation [27].

2.3.2. Governing Equations

The Navier–Stokes equations were discretized using the finite difference method, while the free surface was modeled using the SOLution Algorithm-based Volume of Fluid method [34]. This method focuses solely on the fluid phase, disregards the air phase, and treats gas elements as voids. Previous studies have demonstrated that this method reduces computational time and effectively visualizes the free surface shape [34,35,36,37]. A Cartesian staggered grid was employed to solve the Reynolds-Averaged Navier–Stokes equations, which consist of continuity and momentum equations. The Cartesian coordinates (x, y, z) employed are defined as follows:

$$V_{\mathrm{f}}{\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial (\rho u{A}_{\mathrm{x}})}{\partial x}}+{\displaystyle \frac{\partial (\rho v{A}_{\mathrm{y}})}{\partial y}}+{\displaystyle \frac{\partial (\rho w{A}_{\mathrm{z}})}{\partial z}}=R_{\mathrm{SOR}}$$

(1)

$${\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{1}{V_{\mathrm{f}}}}(u{A}_{\mathrm{x}}{\displaystyle \frac{\partial u}{\partial x}}+v{A}_{\mathrm{y}}{\displaystyle \frac{\partial u}{\partial y}}+w{A}_{\mathrm{z}}{\displaystyle \frac{\partial u}{\partial z}})=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial P}{\partial x}}+G_{\mathrm{x}}+f_{\mathrm{x}}$$

(2)

$${\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{1}{V_{\mathrm{f}}}}\left(u{A}_{\mathrm{x}}{\displaystyle \frac{\partial v}{\partial x}}+v{A}_{\mathrm{y}}{\displaystyle \frac{\partial v}{\partial y}}+w{A}_{\mathrm{z}}{\displaystyle \frac{\partial v}{\partial z}}\right)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial P}{\partial y}}+G_{\mathrm{y}}+f_{\mathrm{y}}$$

(3)

$${\displaystyle \frac{\partial w}{\partial t}}+{\displaystyle \frac{1}{V_{\mathrm{f}}}}\left(u{A}_{\mathrm{x}}{\displaystyle \frac{\partial w}{\partial x}}+v{A}_{\mathrm{y}}{\displaystyle \frac{\partial w}{\partial y}}+w{A}_{\mathrm{z}}{\displaystyle \frac{\partial w}{\partial z}}\right)=-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial P}{\partial w}}+G_{\mathrm{z}}+f_{\mathrm{z}}$$

(4)

where u, v, and w represent the velocity components in the x-, y-, and z-directions; V_f is the volume fraction of fluid in each element; R_SOR $R_{\mathrm{SOR}}$ is the spring term; A_x, A_y, and A_z are the flow area components; $\rho$ is the fluid density; P is the hydrostatic pressure; G_x, G_y, and G_z are the gravitational accelerations in the respective directions; and f represents the Reynolds stress. The viscous accelerations in the x-, y-, and z-directions are denoted by f_x, f_y, and f_z, respectively. The VOF transport equations are expressed as follows:

$${\displaystyle \frac{\partial F}{\partial t}}+{\displaystyle \frac{1}{V_F}}\left[{\displaystyle \frac{\partial (F{A}_{x}u)}{\partial x}}+{\displaystyle \frac{\partial (F{A}_{y}v)}{\partial y}}+{\displaystyle \frac{\partial (F{A}_{z}w)}{\partial z}}\right]=0$$

(5)

where F denotes the fluid fraction, with F = 1 in fluid-filled elements and F = 0 in fluid-free elements (empty regions) [38,39]. The free surface region corresponds to intermediate values of F (typically F = 0.5, with customizable intermediate values).

2.3.3. Turbulence Model

Studies have demonstrated that the RNG k-ε model offers enhanced capabilities for handling high strain rates and large curvature streamlines [40,41,42,43] and provides more accurate reproductions of the velocity fields [44]. Additionally, it has also demonstrated its effectiveness in addressing complex issues in the simulation of hydraulic structures [45,46,47,48]. Consequently, this model was selected for simulating the water flow within the fishway pool chamber.

The model comprises two essential equations: Equation (6) defines the TKE, denoted k, while Equation (7) describes the turbulent dissipation rate (ε).

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho k{u}_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}[{\alpha }_k{\mu }_{eff}{\displaystyle \frac{\partial k}{\partial x_j}}]+G_k-G_B-\rho \epsilon -Y_M+S_k$$

(6)

$${\displaystyle \frac{\partial (\rho \epsilon )}{\partial t}}+{\displaystyle \frac{\partial (\rho \epsilon u_i)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_j}}[{\alpha }_{\epsilon }{\mu }_{eff}{\displaystyle \frac{\partial \epsilon }{\partial x_j}}]+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}(G_k+C_{3\epsilon }{G}_b)-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}-R_{\epsilon }+S_{\epsilon }$$

(7)

where $G_k$ denotes the TKE arising from the velocity gradient; $G_B$ indicates the buoyancy-induced TKE; $S_k$ and $S_{\epsilon }$ represent the source terms; ${\alpha }_k$ and ${\alpha }_{\epsilon }$ are the inverse Prandtl numbers for k and ε, respectively; ${\mu }_{eff}$ is the effective viscosity, defined as ${\mu }_{eff}=\mu +{\mu }_t$, where ${\mu }_t$ is the eddy viscosity.

$$R_{\epsilon }={\displaystyle \frac{C_{\mu }\rho {\eta }^3(1-\eta /{\eta }_0){\epsilon }^2}{k(1+\beta {\eta }^3)}}$$

(8)

$${\mu }_t={\displaystyle \frac{\rho C_{\mu }{k}^2}{\epsilon }}$$

(9)

The constants of the model are as follows [43]:

$C_{\mu }$ = 0.0845, $C_{1\epsilon }$ = 1.42, $C_{2\epsilon }$ = 1.68, $C_{3\epsilon }$ = 1.0, ${\sigma }_k$ = 0.7194, ${\sigma }_{\epsilon }$ = 0.7194, ${\eta }_0$ = 4.38, and $\beta$ = 0.012.

2.3.4. Boundary Conditions

The boundary conditions were defined based on site-specific hydraulic calculations and biological requirements. The upstream velocity (0.8 m/s) was derived from the fishway’s design discharge (3.2 m³/s). This value aligns with the burst swimming speed limit of the four major Chinese carps (≤$1.2 \mathrm{m}/\mathrm{s}$) and ensures hydrodynamic compatibility with the fishway’s slope (1:20) and roughness (n = 0.015). The downstream boundary was set as a pressure outlet with a fixed water depth of 2 m, consistent with engineering practice. The top boundary was specified as a pressure boundary with zero relative pressure. No-slip boundaries were applied to the side walls and bottom surface of the fishway, with a surface roughness of 0.015 m. An initial water depth of 2 m was established in the fishway model to facilitate convergence and reduce computational time. The model was solved using the Generalized Minimal Residual (GMRes) implicit solver, with an initial time step of 0.001 s and a minimum time step of 10⁻⁷ s.

2.4. Grid Independence Verification

A full-scale physical model was utilized to construct a numerical fishway model. A structured hexahedral mesh that demonstrates good convergence properties was utilized. A containing block was applied to the entire model, while a nested block with refined elements was employed for the target study area. The mesh was automatically aligned with the geometry, and grid processing was performed using the FAVOR function. To enhance numerical accuracy and solution convergence, the model’s geometry was refined by selecting an appropriate mesh size for grid division. Based on related research [49], a recommended grid refinement ratio (r = Gcoarse/Gfine) of 1.3 was adopted. The mesh size and total number of elements are presented in

Table 1. Details of the constructed meshes.

Mesh	Nested Block Element Size	Containing Block Element Size	Number of Element	Mesh Type
1	3.8 cm	5.0 cm	2,689,588	Fine
2	4.2 cm	5.5 cm	2,008,890	Medium
3	4.6 cm	6.0 cm	1,533,248	Coarse

, and Figure 3 shows the mesh configuration.

The GCI values were used to evaluate the convergence of the mesh. This analysis was conducted based on the velocities at the midpoints of cross-sections S2 and S9 (V_S2 and V_S9), with the results presented in

Table 2. Summary of grid convergence index (GCI).

Quantity	${\mathit{\epsilon }}_{32}$	${\mathit{\epsilon }}_{21}$	p	$\mathbf{GC}{\mathbf{I}}_{\mathbf{fine}}^{21}$	GCI32	$\mathbf{GC}{\mathbf{I}}_{\mathbf{fine}}^{21}$/GCI32
$V_{S2}$	−0.15	−0.12	0.77	0.49	0.55	0.90
$V_{S9}$	−0.12	−0.09	0.99	0.38	0.42	0.92

.

The approximate relative error, denoted by “ε”, was calculated using the following expressions: ${\epsilon }_{32}={\varphi }_3-{\varphi }_2$, ${\epsilon }_{21}={\varphi }_2-{\varphi }_1$, where ε₃₂ and ε₂₁ represent the approximate relative error between the coarse and medium meshes, as well as between the medium and fine meshes, respectively; ${\varphi }_3$, ${\varphi }_2,$ and ${\varphi }_1$ denote the velocity values derived from the FLOW-3D^® simulations for the fine, medium, and coarse meshes, respectively.

The apparent order of the computational method was determined by the following equations.

$$p={\displaystyle \frac{1}{\ln (r_{21})}}\left|\ln \right|{\epsilon }_{32}/{\epsilon }_{21}|+q(p)|$$

(10)

$$q(p)=\ln \left({\displaystyle \frac{r_{21}^p-s}{r_{32}^p-s}}\right)$$

(11)

$$s=1\cdot \mathrm{sgn}({\epsilon }_{32}/{\epsilon }_{21})$$

(12)

Given a constant value of r = 1.3, q(P) is determined to be 0. The extrapolated values of the method were computed using Equation (13).

$${\varphi }_{\mathrm{ext}}^{21}=(r_{21}^p{\varphi }_1-{\varphi }_2)/(r_{21}^p-1)$$

(13)

The calculation method for the fine-grid convergence index ${\mathrm{GCI}}^{32}$ was consistent with that employed for ${\mathrm{GCI}}_{\mathrm{fine}}^{21}$.

$${\mathrm{GCI}}_{\mathrm{fine}}^{21}={\displaystyle \frac{1.25e_a^{21}}{r_{21}^p-1}}$$

(14)

$$e_a^{21}=\left|{\displaystyle \frac{{\varphi }_1-{\varphi }_2}{{\varphi }_1}}\right|$$

(15)

$$e_{\mathrm{ext}}^{21}=\left|{\displaystyle \frac{{\varphi }_{\mathrm{ext}}^{12}-{\varphi }_1}{{\varphi }_{\mathrm{ext}}^{12}}}\right|$$

(16)

where $e_a^{21}$ and $e_{\mathrm{ext}}^{21}$ denote the approximate and extrapolated relative errors, respectively.

The results indicated that the value of ${\mathrm{GCI}}_{\mathrm{fine}}^{21}$ was smaller than that of GCI³². It can be concluded that the lattice is independent. The computed value of the ${\mathrm{GCI}}_{\mathrm{fine}}^{21}$/GCI³² ratio approaching 1 indicated that the numerical solution fell within the asymptotic convergence region. Figure 4 illustrates the mesh configuration, where the dimensions of the containing and nested blocks are 5.0 cm and 3.8 cm, respectively.

3. Results and Discussion

3.1. Model Validation

As shown in Figure 5, the velocity vectors were extracted at different depths of z = 0.75 h, 0.50 h, and 0.25 h (where z is the depth of the cross-section and h is the water depth), referred to as the surface, middle, and bottom layers, respectively. The flow patterns at these layers for the VSF were fundamentally similar, and the flow field distribution patterns at different water depths were also similar. Due to the resemblance in the flow field distribution patterns at different water depths, a representative mid-level flow field was selected for analysis.

To validate the numerical model, the absolute velocities were calculated using the formula $V=\sqrt{u^2+v^2}$ and compared with the experimental results, as depicted in Figure 6. The numerical results derived from the medium and fine meshes showed good agreement with the experimental results.

Table 3. Comparison of velocities from numerical modeling and experimental measurements at different water layers.

Location	Vl1		Vl2		VS2		VS9
	Num	Exp	Num	Exp	Num	Exp	Num	Exp
z = 0.4 m	0.018	0.064	0.080	0.100	0.014	0.080	0.014	0.075
z = 0.8 m	0.432	0.422	0.232	0.216	0.186	0.173	0.174	0.162
z = 1.2 m	0.446	0.495	0.446	0.455	0.357	0.364	0.335	0.341
z = 1.6 m	0.821	0.885	0.949	0.750	0.960	0.600	0.900	0.563
z = 2 m	0.081	0.036	0.060	0.098	0.050	0.078	0.045	0.090
Mean Error (%)	4.28		5.63		5.5		4.7

presents a comparison between the numerical and experimental results at different water layers.

Table 3 demonstrated that the largest discrepancies occurred at Vl₁ and Vl₂, which correspond to the vertical slot region where the flow structure is highly complex. The maximum error was 5.63%, which indicated that the simulation effectively captured the flow characteristics of the VSF.

3.2. Flow Field

Figure 7 illustrates the velocity distribution in adjacent pools under varying slope ratios, revealing two distinct flow patterns. The first pattern features a mainstream zone at the center of the pool chamber, primarily formed by the jet flow through the slots. The second pattern consists of recirculation zones on either side of the mainstream zone, with one side exhibiting a larger recirculation zone. These recirculation zones were symmetrically distributed along the central axis of the pool chamber. The flow velocities in the recirculation zones were opposite to those in the mainstream zone, generating vertical vortices around the Z-axis perpendicular to the riverbed, which dissipated energy.

In the staggered-side slot-type fishway, the planform of the mainstream zone in adjacent pools exhibited an S-shape, contrasting significantly with the same-side slot-type fishway, where the mainstream zone within a single pool chamber consistently exhibited an S-shape pattern [50,51]. In both configurations, the location of maximum velocity at different cross-sections consistently aligned with the mainstream centerline, and the planform remained unchanged under varying slope ratios. Additionally, the flow velocity increased as the slope ratio increased. The maximum velocity ($U_i$) values for each cross-section of the fishway were extracted from the numerical results. These values, along with their corresponding coordinates, were processed using a dimensionless method. By connecting the $U_i$ values from each cross-section, the $U_i$ curve of the main flow path was constructed [51], where $U_{max}$ represents the maximum $U_i$ and $U_{min}$ denotes the minimum $U_i$. The dimensionless maximum mainstream velocity was determined using the formula $U_i/U_{max}$, and the maximum attenuation rate of the mainstream velocity was calculated using the formula $1-U_{min}{/U}_{max}$. The resulting variation patterns are illustrated in Figure 8 and

Table 4. The maximum attenuation rate of the mainstream velocity.

Slope Ratio	$1-{\mathit{U}}_{\mathbf{min}}/{\mathit{U}}_{\mathbf{max}}$
J1 = 1:20	0.51
J2 = 1:30	0.48
J3 = 1:40	0.42
J4 = 1:50	0.43

Note: U_min is the trough value of the mainstream maximum velocity curve.

.

The results indicated that for a slope ratio of J₁ = 1:20, the maximum attenuation rate of the mainstream velocity reached 0.51, indicating significant energy dissipation. When the slope ratio further decreased to J₃ = 1:40, the maximum attenuation rate of the mainstream velocity dropped to 0.42 before increasing slightly. This behavior can be attributed to the higher flow velocity at steeper slope ratios, which creates a larger difference between the slot jet velocity and the pool chamber water velocity, thereby enhancing diffusion and energy dissipation. As the slope ratio decreased, the slot jet velocity decreased, weakening both the diffusion and energy dissipation effects. With further reduction in the slope ratio, the slot jet velocity continued to decrease, which increased the time required for the mainstream to traverse the pool chamber. This extended interaction allowed the slot jet to diffuse more completely and merge with the pool chamber water, leading to a slight recovery in energy dissipation efficiency.

The mainstream region serves as the primary passage for fish during upstream migration, and the L/B ratio of the pool chamber significantly influences the curvature of the mainstream flow. Consequently, the degree of curvature affects fish migration behavior. Figure 9 illustrates the velocity distribution within pools at different L/B ratios. When the L/B ratio was 0.6, the curvature of the mainstream flow between adjacent pools was pronounced, characterized by a small radius of curvature and significant distortion. At the vertical slot positions, the mainstream direction underwent abrupt changes, resulting in a linear flow distribution within each pool chamber, while the flow between adjacent chambers formed a complex Z-shaped pattern. At this stage, the smaller recirculation zones on either side of the mainstream were not fully developed, suggesting that when the pool chamber length was too short, the jet flow could not dissipate sufficient energy before entering the next chamber. With increase in the L/B ratio, the curvature of the mainstream flow gradually decreased, and the flow transitioned from a Z-shaped to an S-shaped pattern. During this transition, the recirculation zones in the smaller areas began to develop, although they remained relatively small, with limited effects on energy dissipation. These flow conditions remained suboptimal for fish upstream migration. When the L/B ratio exceeded 0.8, the flow pattern in the mainstream region of a single pool chamber exhibited a semi-parabolic shape, while adjacent chambers displayed an S-shaped flow pattern. As the L/B ratio increased further, the S-shaped flow pattern became more pronounced, and the recirculation zones beneath the rectifier and divider baffles expanded. When the L/B ratio was between 1.0 and 1.2, the mainstream region clearly exhibited an S-shaped flow pattern, with particularly prominent recirculation zones on both sides. This flow structure enhanced energy dissipation within the pool chamber and provided optimal flow conditions and suitable resting areas for fish. However, when the L/B ratio exceeded 1.2, a wall-attached flow phenomenon may occur in the mainstream region, potentially hindering fish migration.

The longitudinal maximum velocity curves for different pool chamber L/B ratios under varying slope ratios were derived from numerical simulations. As an example, the longitudinal maximum velocity curves for pools with different L/B ratios at a bottom slope ratio of J₁ = 1:20 were analyzed, as shown in Figure 10. The distributions of these curves under varying L/B ratio conditions were similar. After the first vertical slot jet, the flow decelerated, reached a trough before the second vertical slot, and then accelerated as it passed through the slot. As the L/B ratio increased, the location of the peak velocity shifted towards the vertical slot. This was due to the fact that when the L/B ratio was between 0.6 and 0.8, the mainstream water did not dissipate sufficient energy during its forward movement in the pool chamber. The flow was further dissipated by the disturbance effects of the rectifier baffle, causing the water to decelerate and then accelerate upon entering the vertical slot. In contrast, when the L/B ratio was greater than or equal to 0.9, the mainstream water dissipated sufficient energy before reaching the next vertical slot, and the flow velocity increased immediately as it passed through the slot. To ensure appropriate curvature at the junctions of adjacent pools and to prevent the occurrence of wall-attached flow, it is recommended that the L/B ratio be maintained within the range of 0.9 to 1.1.

The maximum attenuation rate of the mainstream velocity for different pool chamber L/B ratios at various bottom slope ratios is shown in Figure 11. Generally, a larger change in the maximum mainstream velocity along the flow path corresponded to a more significant decay rate, indicating higher head loss and enhanced energy dissipation, which is beneficial for fish migration [52,53]. The results showed that, for a fixed L/B ratio, the distribution patterns of the mainstream maximum velocity decay rate along the flow path were similar across different slope ratios. Specifically, as the L/B ratio increased, the decay rate along the flow path also increased. When the L/B ratio exceeded 1, the velocity decay rate stabilized along the flow path.

3.3. TKE

TKE is a critical hydraulic parameter that characterizes the turbulence within the pool chamber and plays a vital role in assessing the hydraulic performance of fishways. Generally, lower TKE levels are more conducive to fish migration, as higher TKE can hinder fish movement by increasing swimming resistance and reducing their ability to navigate the migration paths. TKE levels are typically classified into two categories: “low” (TKE < 0.05 J/kg) and “high” (TKE > 0.05 J/kg) [54,55]. In fishways, fish tend to prefer areas with “low” TKE levels [56,57]. Figure 12 illustrates the distribution of TKE in pools under varying slope ratios. It was evident that higher TKE values were concentrated around the rectifier baffle angles and the divider baffle heads. This observation indicated that these baffle angles and ends induce flow infusion, which promoted energy dissipation and enhanced turbulence near these structures. The TKE in the recirculation zones flanking the mainstream varied between 0.01 and 0.05 J/kg, suggesting that these zones are suitable for fish to rest during their upstream migration. When the slope ratio was altered while maintaining the configuration and dimensions of the rectifier and divider baffles constant, the TKE distribution pattern remained unchanged, although the TKE decreased as the slope ratio decreased. Notably, the reduction in TKE was most pronounced near the rectifier baffle angles and the ends of the divider baffles. This indicated that a larger slope ratio led to higher energy dissipation efficiency within the fishway.

Figure 13 illustrates the distribution of TKE in pools with varying L/B ratios under a slope ratio of J₁ = 1:20. As the L/B ratio increased, the area of the turbulent flow region expanded. As the L/B ratio increased from 0.6 to 1.2, the increase in TKE was primarily due to the combined effects of higher flow velocity, flow guidance, and energy dissipation from the rectifier and divider baffles. When the L/B ratio is greater than or equal to 1.2, this increase was attributed to mainstream flow impacting the chamber walls in the form of a jet, which led to the onset of wall-attached flow. This phenomenon intensified turbulence, thereby increasing TKE and affecting the upstream migration of fish.

3.4. Volumetric Energy Dissipation

The design of VSFs needs to account for the dissipation rate $D_{\epsilon }$, which represents the volumetric energy dissipation within the pool. Insufficient dissipation can result in excessive energy expenditure by the fish during their upstream migration, potentially hindering successful passage. To simplify fishway design, the global volumetric dissipated power D_V has commonly been employed as a key discriminant parameter. It is calculated using the formula D_V = P/V_P, where $P=\rho gQ\delta h$ ($\rho$ is the water density, g is the gravity, Q is the flow rate, and $\delta h$ is the head difference). The D_V encapsulates power and friction losses in addition to turbulence losses, making it a global pool parameter [14]. In contrast, $D_{\epsilon }$ reflects the dissipation distribution within the pool. The determination of $D_{\epsilon }$ within the pool chamber is theoretically significant for refining the design and optimizing the internal structure of the pool chamber. Existing research indicated that $D_{\epsilon }$ values between 150 and 200 W/m³ are suitable for fish species with strong swimming abilities. For smaller fishways or species with weaker swimming abilities, such as many cyprinids, $D_{\epsilon }$ should be maintained below 150 W/m³ [58]. Accurately determining the spatial distribution of $D_{\epsilon }$ is essential for evaluating migratory opportunities for fish passing through VSFs [7].

The formula for calculating the $D_{\epsilon }$ is given by (17):

$$D_{\epsilon }={\displaystyle \frac{1}{V_p}}{{\displaystyle \int }}_{V_p}\rho \epsilon d{V}_P$$

(17)

where $\rho$ is the fluid density (kg/m³); $V_P$ is the unit volume of the pool chamber; $d{V}_P$ is the infinitesimally small volume of the pool chamber; and $\epsilon$ represents the turbulent dissipation rate.

Figure 14 illustrates the distribution of $D_{\epsilon }$ under varying slope ratios. The highest values of $D_{\epsilon }$ were located at the heads of the divider baffles and at the angles of the rectifier baffles. Although these regions exhibited higher $D_{\epsilon }$ values, they were relatively small and had minimal impact on fish upstream migration. In contrast, $D_{\epsilon }$ was lower in the recirculation zones, which occupied a larger area. As the slope ratio decreased, $D_{\epsilon }$ also decreased; however, the overall distribution pattern remained unchanged.

Figure 15 and Figure 16 illustrate the distribution of $D_{\epsilon }$ under different L/B ratios for slope ratios of J₁ = 1:20 and J₂ = 1:30, and the area proportion of the pool chamber with $D_{\epsilon }\le 150 \mathrm{w}/{\mathrm{m}}^3$ (Ap) for different L/B ratios, respectively. Figure 15 indicates that $D_{\epsilon }\le 150 \mathrm{w}/{\mathrm{m}}^3$ was predominantly distributed in the recirculation zones. According to Figure 16, for a slope ratio of J₁ = 1:20, the minimum Ap occurred when the L/B ratio equaled 1, whereas for J₂ = 1:30, the minimum Ap appeared when the L/B ratio equaled 1.1. This phenomenon was attributed to the increase in flow velocity with rising slope ratios, which led to the earlier onset of wall-attached flow and a quicker reduction in Ap. For a given slope ratio, as the L/B ratio increased, Ap initially decreased and then increased. For instance, when J₁ = 1:20 and the L/B ratio was less than 1, an increase in the pool chamber length resulted in higher $D_{\epsilon }$. Furthermore, the increase in maximum flow velocity was relatively high, thereby leading to better energy dissipation and a reduced Ap. However, when the L/B ratio exceeded 1, as the pool chamber length continued to increase, some of the flow from the vertical-slot jet began to interact with the pool chamber walls, further enhancing energy dissipation. At this point, the chamber length became sufficiently long, causing a slight increase in Ap.

4. Conclusions

Based on three-dimensional numerical simulations, this study investigated the hydraulic characteristics of a novel VSF with staggered baffles configuration, using the slope and the length-to-width (L/B) ratio of the pool chamber as variables. The main findings are as follows:

(1)

Steeper slopes enhance the vertical-slot velocities and the chamber velocity differentials, thereby boosting the energy dissipation efficiency that is critical for fish migration. L/B ratios of 0.9–1.1 optimize the flow patterns, though increased slope steepness necessitates proportional adjustments to the L/B ratios.

(2)

Slope elevation amplifies the turbulent kinetic energy (TKE) at baffle junctions, whereas recirculation zones maintain stable turbulence levels, which are crucial for fish resting. When L/B exceeds or equals 1.2, TKE variations become predominantly governed by wall-attached flow.

(3)

Peak $D_{\epsilon }$ zones at baffle corners exhibit slope and L/B-dependent sensitivity; however, these regions are positioned outside the primary fish migration pathways, thus exerting minimal impact on upstream fish passage. The threshold of $D_{\epsilon }$ ≤ 150 W/m³—derived from fish endurance studies [58]—predominantly occurs in recirculation zones. Non-linear L/B relationships suggest optimal configurations that balance hydraulic performance with ecological requirements.

There is potential for further improvement in fishway configurations for steep slopes. Future studies should consider different flow rates and velocities for a comprehensive analysis. To optimize the VSF design, experimental efforts should emphasize hydraulic characteristics while simultaneously accounting for the swimming behaviors and performance of fish across different fish species and life stages. Future research can concentrate on these two aspects.