A novel ensemble model for predicting the performance of a novel vertical slot fishway

Abstract

We investigate the performance of a novel vertical slot fishway by employing finite volume and surrogate models. Multiple linear regression, multiple log equation regression, gene expression programming, and combinations of these models are employed to predict the maximum turbulence, maximum velocity, resting area, and water depth of the middle pool in the fishway. The statistical parameters and error terms, including the coefficient of determination, root mean square error, normalized square error, maximum positive and negative errors, and mean absolute percentage error were employed to evaluate and compare the accuracy of the models. We also conducted a parametric study. The independent variables include the opening between baffles (OBB), the ratio of the length of the large and small baffles, the volume flow rate, and the angle of the large baffle. The results show that the key parameters of the maximum turbulence and velocity are the volume flow rate and OBB.

Keywords

novel vertical slot fishway / parametric study / finite volume method / ensemble model / gene expression programming

Introduction

For decreasing the negative effect of weirs and dams on migratory fish, fishway structures are employed [¹]. The vertical slot fishway (VSF) is proposed to use in a wide range of biological and hydraulic environments because of its advantages among other kinds of fishways; therefore, this type of fishway is commonly used [²]. To preserve aquatic ecosystems for all kinds of fishes, the development of influence fishways bridge for fish migration is an important topic in environmental and ecological science.

There are various types of fishway shapes, such as VSFs, culverts, denils, and weirs. Several studies focused on these fishways [³–⁷]. VSFs have two important advantages, including the possibility of fish migration at low velocity, and quasi-independent relationship between hydraulic features and depth and discharge [⁸–¹⁰].

Most studies focused on the hydraulic properties of various VSFs because the flow pattern (FP) has the greatest influence on fish migratory behavior; therefore, these studies evaluated the effects of changes in the geometry and dimensions of the VSF on the FP [¹¹–¹⁴]. For example, some studies evaluated the effect of the baffle length and the pool length and width on the FP [³,⁴,¹⁵]. The effect of changes dimensions in the pool [¹⁶–¹⁸] and slope on the FP in pools was evaluated [¹⁹]. Another important parameter, which has a sufficient effect on the FP, is the pool shape [²⁰]. Two previous studies [¹¹,²¹] evaluated the ratio of the width and length of a pool and proposed a relationship between fish passage and flow discharge. Several researchers stated that the FPs differ under various boundary conditions [²²,²³].

Although the fishway is a necessary structure to maintain the ecological conditions based on watercourses, few studies have been focused on fishway design and structure in recent years [²⁴,²⁵]. Some researchers focused on case studies on a specific fishway near a specific dam by experimental and computational approaches [¹⁴,²⁴,²⁶]. There are also important computational studies and design books that presented a relationship between the maximum velocity and water depth of the VSF [⁹,¹⁵,²⁷,²⁸]. As a result, presenting a numerical model for operating and evaluating the performance of each kind of fishway is one of the necessary studies that could decrease the need for future case studies.

The computational fluid dynamics (CFD) model is a useful tool for evaluating and analyzing hydraulic structures. Some studies have employed the CFD model to evaluate the performance of various fishways [²⁹]. Liu et al. [⁹] studied the FP in a fishway located near the Dalles Dam by simulating the 3D hydrodynamic flow using the CFD model. Rodriguez et al. [³⁰] analyzed the flow characteristics of a fishway using ANSYS Fluent. Duguay et al. [³¹] simulated the pool and weir fishway using OpenFOAM and Flow-3D. After using finite volume method (FVM) to simulate the models, the results of the model were compared with the results of experimental test to evaluate the validation of the models. Moreover, they found that the k-\epsilon turbulence model and InterFoam solver in Flow-3D and OpenFOAM predicted the flow characteristics with acceptable accuracy. The improved VSF can provide a better hydraulic and flow performance for the migration of fish; therefore, selecting the most effective VSF for each specific situation is an important task in environmental engineering. In recent years, some studies have focused on designing a VSF for a specific dam and geology [⁹,³²], as discussed below.

The first VSF was built at the Hell’s Gate Dam in Canada [³³], which significantly influenced the migration of fish to North America. Some researchers have focused on the design of the VSF. For example, Rajaratnam et al. [³] experimentally investigated the FP in several pools with various designs, such as a VSF. Some studies focused on various VSF designs for construction in the United States [⁴,³⁴]. The FP was evaluated for various designs and the slopes [¹⁹]. A velocimetry approach was employed to investigate the flow structure in two various slopes [³²]. A linear relationship was found between the flow depth and the discharge based on a previous study [⁹]. Although many studies were evaluated the FP in different fishways based on parametric studies by CFD models and experiments, there is still a lack of research on numerical models that predict the flow properties of different VSFs.

Gene expression programming (GEP) has been widely employed in hydrology and hydraulic problems for predicting flow performance. For example, parameters, such as evapotranspiration [³⁵,³⁶], the depth of pipeline scour in a river [³⁷], Manning’s roughness coefficient for a stream with a high slope [³⁸], the river discharge in stages [³⁹], and the height of scour at downstream of sills [⁴⁰], have been predicted by GEP. The multiple linear regression (MLR) technique has also been used in previous studies [⁴¹,⁴²]. For instance, Chenini and Khemiri [⁴³] used the MLR model to predict the groundwater depth in Tunisia using input parameters obtained by evaluating the relationship between the input parameters and the output. Chen and Liu [⁴⁴], and Nasir et al. [⁴⁵] utilized MLR to predict the water quality of a reservoir and the Klang River, respectively. A developed MLR method was also proposed for predicting air quality, which included several log equations that should be summed up to predict the output [⁴¹].

The novelty of this study is highlighted by three aspects. The first is the proposed fishway structure, in which the locations of large and small baffles were changed along the length of the fishway. As a result, the flow in this fishway should perform more laminar flow in comparison to that of the conventional VSF because of the creation of a zigzag flow. For a better and clearer understanding of this concept, assume a fishway. It is clear that the length of the zigzag flow is more than the length of the direct flow in the assumed fishway; this indicates that the slope of the zigzag flow is lower than that of the direct flow. Thus, the zigzag flow is more laminar than the direct flow because of the lower slope. The zigzag flow has a greater surface contact with concrete because of the difference between the length of its flow and the length of the direct flow. The zigzag flow also changes direction in each pool; therefore, the energy is dissipated. It is expected that laminar flow would be created in this type of fishway in contrast to an ordinary VSF. Another advantage of this type of fishway is related to the resting of fishes. The fish can rest on two sides of each pool, where the flow is more laminar. The second novelty of this study is referred to the evaluation of changes in design parameters, including the opening between baffles (OBB), length of baffles, angle of the large baffle (ALB), and volume flow rate as an operating parameter. A parametric study is performed to evaluate the effect of changing the design parameters and one operating parameter on the FP. Presenting the surrogate models to predict the maximum turbulence, maximum velocity, rest area, and water depth.

Methodology

The main aims of this study are to conduct a parametric study of the plug-flow VSF (PFVSF) and propose a surrogate model to predict its performance. This study is divided into two parts. In the first part, a FVM is developed for implementing the parametric study. Two various experimental tests of the VSF, which were published [¹⁴,⁴⁶], were modeled by Flow-3D software in this study to develop the FV model and evaluate its validity. There are various pool geometries and volume flow rates in these experimental tests; therefore, the validation of the FV model was evaluated for different variables. To simulate the VSF, various hydraulic models were tested and evaluated, and the k-\epsilonmodel was selected for developing the VSF and PFVSF. Mesh sensitivity analysis was carried out for each validation model to select the best mesh size. Then, the validated FV model was employed to simulate 81 PFVSFs with various volume flow rates, ALB, ratio of the length of the large and small baffles (RLB), and OBB. The maximum turbulence, maximum velocity, rest area percentage, and flow depth were considered as outputs of the FV models, which demonstrate the performance of PFVSF. The outputs of the PFVSF were evaluated and compared to determine the most efficient PFVSF for each volume flow rate. The volume flow rate is an operating parameter, while the OBB, ALB, and RLB are considered as design parameters.

In the second part, GEP, MLR, multiple log-equation regression (MLER), and an ensemble model were used to predict the outputs of each FV sample. The error terms and statistical parameters (i.e., maximum positive and negative errors, mean absolute percentage error (MAPE), root mean square error (RMSE), normalized mean square error (NMSE)), and error distribution were used to evaluate and compare the accuracy of the prediction models and determine the best model for predicting the outputs of the FV models. Figure 1 illustrates a flowchart of the present study.

Finite volume model

Although the finite element method (FEM) and finite difference method can be used for CFD problems, the FVM is a natural choice for solving these problems. Each of these methods represents a systematic numerical approach to solve partial differential equations; however, one important difference among them is the ease of implementation. The FVM is an effective method because it only needs to carry out flux evaluation for the cell boundaries. This evaluation can be used for nonlinear problems, which make FVM robust in handling nonlinear conservation laws that appear in transport problems. The FVM is based on the fact that many physical laws are conservation laws that go into one cell on one side and need to leave the same cell from another side. Thus, in this method, there is a formulation that includes flux conservation equations, which were defined in an averaged sense over the cells. As a result, this approach is very effective in solving fluid flow problems. The local accuracy of the FVM, such as the accuracy close to a corner of interest, is increased by refining the mesh around that corner, similar to the FEM [⁴⁷–⁴⁹]. In this study, FV samples were developed with the FLOW-3D software (FlowScience, Inc.), which is based on the volume-of-fluid (VOF) approach [⁴⁷]. This software permits the use of the one-fluid method for free surface flows [⁴⁸,⁴⁹].

Geometry and size of the considered meshes

Previous studies have shown that structured meshes generally have more accurate performance than unstructured meshes [⁵⁰,⁵¹]. Structured meshes are also faster than unstructured meshes because of the decreasing simulation latency [⁵²]. In this study, all specimens were modeled by 3D structured cubic cells. The orthogonal mesh reduced the accuracy of the numerical problems in multiphase flows; thus, a structured rectangular hexahedral mesh was employed in the models, which is more suitable for simulating the VSF and PFVSF samples. A uniform cubic mesh with \mathrm{\Delta }X lengths was utilized for the mesh selectivity analysis. The mesh sizes, which are considered for mesh sensitivity analysis, are 20, 15, 10, 8, 6, 4, 3, and 2 cm. After completing the mesh sensitivity analysis, the specific mesh size was selected for developing the PFVSF samples.

Flow equations

The Navier-Stokes equations were employed to provide the full description of flow specifications defined as the fluid motion. Although the VOF approaches can be applied to variable-density flows, the Navier-Stokes approach was employed because of the incompressible feature of flow [⁵³]. The FVM was used to employ the flow equations:

(1)\nabla \overline{u}=\mathrm{0},

(2){\displaystyle \frac{\partial \overline{u}}{\partial t}}+\overline{u}·\nabla \overline{u}=-{\displaystyle \frac{\mathrm{1}}{\rho }}\nabla P+\upsilon {\nabla }^{\mathrm{2}}\overline{u}+\overline{f_{\mathrm{b}}},

where u, p, \rho, \upsilon, and f_b are defined as the velocity, pressure, density, kinematic viscosity, and body forces, respectively. The time-step length is adjusted automatically for ensuring the threshold of the Courant number that is less than 0.75.

Free surface modeling

The algorithm, which is utilized to define the interface between fluid phases, is important for applying the effect of the coexistence of two fluids on numerical flow models [⁵⁴]. In this study, Eulerian-Eulerian approaches were employed to apply this effect. These approaches can become more accurate and efficient, which use a single variable value in each mesh [⁵⁵]. \alpha is defined as the scalar variable that explains the fluid fraction contained in each mesh element. If only two fluids are simulated, then their fraction is complementary. Thus, a transport equation should be determined for the fluid fraction value in the computational domain [⁵¹].

(3){\displaystyle \frac{\partial \alpha }{\partial t}}+\nabla ·\left(\overline{u}\alpha \right)=\mathrm{0},

where \alpha, u, and t are the fluid fraction, velocity, and time, respectively. According to the above-mentioned, this method considers two fluids as a single multiphase fluid. If two fluids are named as fluids A and B, the weighted average treats the transport of other properties are defined as:

(4)\xi ={\xi }_{\mathrm{A}}\alpha +{\xi }_{\mathrm{B}}(\mathrm{1}-\alpha ),

where \xiis the transport of other properties. According to Eq. (4), its output includes values between 0 and 1. However, no neat fluid interface is explicitly defined. Donor-acceptor approaches, such as VOF, which were conducted in FLOW-3D, have been widely utilized [⁵⁶].

Flow aeration

Based on the concept of air-water flow, aeration modifies the macroscopic flow density, adding compressibility properties to the flow and increases the flow depth; therefore, changes the momentum distribution [⁵⁷–⁵⁹]. There is no approach to describe this phenomenon because the length of the scale is smaller than the mesh size of bubbles and droplets [⁶⁰–⁶²]. To overcome this problem, a subscale air-entrainment model should be developed and used. According to previous studies, Eulerian-Lagrangian methods are sufficient to resolve the issue. These approaches include the estimation of the Navier-Stokes Equations [⁶³,⁶⁴]. In this study, an entirely Eulerian method was used to apply the effect of two fluids.

Turbulence modeling

CFD is one of the methods, which can be used to apply and determine turbulence [⁵⁰]. Large eddy simulation (LES), re-normalization group (RNG), and k-\epsilonapproaches were proposed for simulating the multiphase flow in previous studies [⁴⁶,⁴⁸,⁶⁵–⁶⁷]. Here, LES, RNG, and k-\epsilon were selected as the turbulence models to simulate the samples, which were tested by Bombač et al. [¹⁴] and An et al. [⁴⁶]. The results of these turbulence models are shown in the validation section. Based on the results in validation sections, the accuracy of the k-\epsilonmodel was more than those of other turbulence models; hence, it was selected for developing the PFVSF samples. The general form of the k-\epsilonequations is as follows:

(5){\displaystyle \frac{\partial }{\partial t}}\left(\rho K\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho K{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[(\mu +\frac{{\mu }_{\mathrm{t}}}{{\sigma }_{\mathrm{k}}}){\displaystyle \frac{\partial k}{\partial x_j}}\right]+P_{\mathrm{k}}-\rho \epsilon ,

(6){\displaystyle \frac{\partial }{\partial t}}\left(\rho \epsilon \right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho \epsilon u_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[(\mu +\frac{{\mu }_{\mathrm{t}}}{{\sigma }_{\epsilon }}){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+C_{1\epsilon }\frac{\epsilon }{K}{P}_{\mathrm{K}}-C_{2\epsilon }\rho \frac{{\epsilon }^2}{K},

where k, \epsilon, \rho, t, and {\mu }_{\mathrm{t}}are the turbulence kinetic energy (TKE), dissipation rate, density, time, and dynamic viscosity, respectively. X_i is considered as coordinate in the i axis, and P_k is defined as the production of TKE. C_{\mathrm{1}\epsilon }, C_{\mathrm{2}\epsilon }, {\sigma }_{\mathrm{k}}, and {\sigma }_{\epsilon } are model parameters that should be calculated. The turbulence viscosity is calculated as

(7){\mu }_{\mathrm{t}}=\rho C_{\mu }{\displaystyle \frac{k^{\mathrm{2}}}{\epsilon }},

where the value of C_{\mu }is considered as 0.085.

Boundary conditions

As shown in Fig. 2, four different boundary conditions were considered for various surfaces of the samples. The flow depth and volume flow rate were selected as the inlet (Q) for the boundary condition of the channel upstream. The outlet boundary condition (O) was selected as the outlet downstream of the VSF. The rigid boundary condition (wall roughness) was used for the walls and bottom of the VSF and PFVSF. The symmetry boundary condition was chosen as the Z_max level. The upper of the fluid level was selected as infinitely the atmospheric condition when the fluid level of the PFVSF and VSF does not reach the maximum level of the PFVSF and VSF. Furthermore, the roughness coefficient of rigid objects, which was considered as concrete, was assumed to be 0.014. The explicit approach is considered as the advection and diffusion scheme for analyzing samples. All analyses were continued until a steady-state was reached.

Figure 2 shows the geometry of the PFVSF. There are two kinds of baffles: small and large, as shown in Fig. 2. There are four independent variables: volume flow rate, length of small and large baffles, baffle angle, and OBB. The small and large baffles are denoted as L_s and L_l, respectively. The ALB is denoted as Θ, and the volume flow rate is denoted as Q. The surface slope of the VSF is 0.0167. Water at ambient temperature was considered in the FV model, and its properties are listed in Table 1.

In the following section, the validity of the developed FV models was evaluated by comparison with two various experimental results of the VSFs.

Validation

To evaluate the validity of the FV models, two various VSFs with different geometries and hydraulic parameters were modeled. In subsection 4.1, the results of the proposed FV models were compared with the results of an experimental test, which was published by Bombač et al. [¹⁴]. To evaluate the effect of variations in geometry and hydraulic parameters on the results of the FV model, the results of the developed FV model were compared with the results of another experimental test, which was published by Bombardelli et al. [⁶⁶]. The results of the second validation are presented in subsection 4.2.

Validation of FV model for developing VSF samples

To carry out the parametric study, the developed FV model should be validated. Thus, the FP in a VSF, which was tested by Bombač et al. [¹⁴], was simulated by the FV model, and the results were compared with experimental results to validate the FV model. Based on Fig. 3, there are nine pools in this VSF sample, and the velocity was measured at the two points in pool 5 in the experiment and FV model. The velocity results of pool 5 were used to validate the FV model. Figures 3(a) and 3(b) show the boundary condition considered in the FV model, the geometry of this VSF, and the geometry of all pools in this VSF, respectively. The locations of the measured velocities are shown in Fig. 3(b).

The same boundary condition, depth, and volume flow rate at the upstream as in the experiment were considered as inputs for simulating the FV model. To verify the FV model, the velocity in all directions obtained at two points (V₁ and V₂) from the FV model were compared with the velocity measured at the same points in the experiment, as shown in Fig. 3(b).

In Section 3, the mesh size was selected based on the mesh sensitivity analyses, which are provided in the validation subsections. In the present study, RMSE was used to evaluate the accuracy of the FV models with various mesh sizes for analyzing mesh sensitivity, as in previous studies [¹⁶,¹⁷,⁵¹]. The mesh sizes for analyzing mesh sensitivity were considered as 20, 15, 10, 8, 6, 4, 3, and 2 cm. To perform the mesh sensitivity analysis, the velocities obtained at two points from the FV model and experiment were compared. Figure 4 shows the results of the mesh sensitivity analysis, in which the vertical and horizontal axes indicate the error and mesh sizes, respectively. Three turbulence models were employed with various mesh sizes. The results of the k-\epsilon turbulence model showed good agreement with experimental results. The RMSE of this model was less than that of the other models; therefore, the k-\epsilon turbulence model was considered as the best one for developing the FV model to simulate the VSF. The best mesh size for the k-\epsilonmodel is 3 cm (Δx = 3 cm).

Figure 5 shows the velocities obtained at the two points from the FV model and experiment. The velocities in the x-, y-, and z-directions at 11 levels were measured in the experiment. Then, the measured velocities in the experiment were compared with the obtained velocities from the FV model at the same point, direction, and level. The results show that the velocities obtained from the FV model correspond to the values measured at the same point, direction, and level.

Figures 6 shows the obtained velocity distribution from the FV model in points 1 and 2 of the VSF. According to Fig. 6, the obtained velocity distribution from the FV model is close to the measured velocity from the experiment [¹⁴]. The results in Figs. 5 and 6 indicate that the accuracy of the developed FV model is acceptable.

Validation of FV model for various geometries and hydraulic parameters

To find the validated FV model for simulating the PFVSF, another VSF with various geometric dimensions was modeled, which was tested by An et al. [⁴⁶]. Their experimental results were compared with the obtained results from the developed FV model. Figure 7(a) demonstrates the geometry of the VSF. In this figure, there are seven pools in this VSF, and the velocity was measured in four pools. According to Fig. 7(b), the velocity was measured at seven points on the surface of the water in pools 1–4, where the velocity at the same points and level were determined by the FV model. Figure 7(c) shows the dimensions of each pool of this VSF.

To find the suitable mesh size, mesh sensitivity analysis was conducted as in the previous subsection. The same mesh size must be verified for both samples to develop the FV model. Figure 8 shows the RMSE values obtained from the experimental results and the results of the FV models, which were solved by RNG, LES, and k-\epsilon methods. As shown in Fig. 7(b), the velocity at each point was measured at the surface level. The considered mesh sizes for analyzing the mesh sensitivity were 20, 15, 10, 8, 6, 4, 3, and 2 cm. For implementing mesh sensitivity analysis, the obtained velocity at all points from the FV model and experiment were compared using the RMSE. Figure 8 shows the results of the mesh sensitivity analysis, where the vertical and horizontal axes indicate the error and the considered mesh sizes, respectively. The results of this figure indicate that the k-\epsilon turbulence model and experimental results have a good agreement. The RMSE of this model was less than that of the other models; therefore, k-\epsilon turbulence was considered as the best model for developing the FV model to simulate the VSF. The best mesh size for the k-\epsilon model was considered to be 3 cm (Δx= 3 cm).

Figure 9 shows the measured velocities in the experiment and the obtained velocities from the FV model at all points. The horizontal axis shows the point at which the velocity was measured, while the vertical axis shows the determined velocity. The obtained velocity at all points from the FV model is close to the measured velocity by An et al. [⁴⁶]; therefore, the developed FV model was validated for simulating the VSF. The velocity obtained from the FV model is close to the velocity measured by the experiment.

Figure 10 shows the velocity distribution in both the experiment and FV model in pool 3 of this VSF. The same velocity distribution and FP were obtained from both the experiment and FV model; therefore, the validation of the FV model for simulating this VSF is acceptable.

The obtained results from the FV model are in good agreement with the experimental results; therefore, the FV model was developed for simulating VSF in the parametric study section. The FV model can be employed to simulate VSF with various geometrical and hydraulic variables.

Parametric study and thermomechanical analysis

In this section, the PFVSF for the first time was simulated using the developed FV model to evaluate the performance of this type of VSF. According to the hydraulic concept of plug flow, the flow has a tendency to change into laminar flow; therefore, the PFVSF is considered to improve the rest area for fishes. There are nine pools in the simulated PFVSF, as shown in Fig. 11(a). The dimensions of the designed pools are shown in Fig. 11(b). The OBB, RLB, and ALB are considered as design variables in this study. This figure also shows the points where the velocity was measured. To create the zigzag flow and the plug flow in the pool, the location of the large and small baffles in each pool were changed along length of the PFVSF, as shown in Figs. 11(a) and 11(b).

Table 2 shows the levels of the variables, including lengths of the small and large baffles, ALB, OBB, and volume flow rate. Eighty-one PFVSF samples were simulated to analyze the maximum turbulence, maximum velocity, rest area, and flow depth to evaluate the performance of the PFVSF. The outputs were measured at a depth of Z/H=0.5_. Several studies have reported the outputs at this depth because the velocity is constant from the water surface to this level, and decreases with increasing depth [¹⁴,²³]. The volume flow rate is considered as an operating parameter. The lengths of the small and large baffles, ALB, and OBB are the design parameters of the PFVSF.

As mentioned above, the maximum turbulence and velocity were measured at a depth of Z/H=0.5 in pool 5 of the PFVSF. The regions of pool 5, with velocity less than 0.3 m/s, are considered as rest areas. On the other hand, the volume of the pool in which fishes can rest is considered as the rest area that Quaranta et al. [⁶⁸] proposed this concept for the rest area. The water depth was measured at the center of pool 5, as shown in Fig. 11(b).

Figure 12 shows the maximum turbulence of each sample in pool 5 of the PFVSF. The maximum turbulence decreased when the length of the OBB increased, and increased when the volume flow rate increased. The ALB has a slight effect on the maximum turbulence. According to previous studies, when the maximum TKE in the VSF is greater or less than 0.05 m²/s², the turbulence is considered high or low, respectively, although different values for other types of fishways have been proposed [⁹,⁶⁹–⁷¹]. The maximum turbulence decreased from 0.133 to 0.036 m²/s² in the PFVSF, as shown in Fig. 12. The maximum turbulence of the sample, which its volume flow rate, OBB, RLB, and ALB were considered as 1000 L/s, 30 cm, 3.875, and 18°, is more than other samples and its value is 0.133 m²/s². In contrast, for the sample with volume flow rate, OBB, RLB, and ALB of 400 L/s, 60 cm, 2.9, and 0°, respectively, its maximum turbulence is 0.036 m²/s² and is less than that of the other samples.

Figure 13 shows the maximum velocity in pool 5 of the PFVSF. According to the figure, the maximum velocity increased when the volume flow rate increased. The maximum velocity decreased when the OBB increased. When the volume flow rate is 1000 or 800 L/s, and OBB is 30 or 45 cm, the ALB and RLB have sufficient effects on the maximum velocity of the PFVSF; however, the ALB and RLB have a slight effect on the maximum velocity with other values of the volume flow rate and OBB. For the sample with volume flow rate, OBB, RLB, and ALB of 1000 L/s, 30 cm, 2.9, and 18°, respectively, its maximum velocity is 1.940 m/s and is greater than that of the other samples. For the sample with volume flow rate, OBB, RLB, and ALB of 400 L/s, 60 cm, 2.9, and 0°, respectively, its maximum velocity is 1.050 m/s and is less than that of the other samples.

The rest area is considered as the region with a velocity less than 0.3 m/s [⁶⁸], and was measured in the total volume of pool 5 in this study. Figure 14 shows the percentage of the rest area in pool 5 of the PFVSF, which changed from 39.8% to 52.2% based on the results. The sample with volume flow rate, OBB, RLB, and ALB of 1000 L/s, 60 cm, 3.875, and 18°, respectively, had the maximum percentage of the rest area. The sample with volume flow rate, OBB, RLB, and ALB of 1000 L/s, 30 cm, 5.5, and 18°, respectively, had the minimum percentage of the rest area. Based on the results of Fig. 14, there is no specific relationship between the variables of the PFVSF and the rest area.

Figure 15 shows the water depths in pool 5 of the PFVSF. Based on Fig. 11(b), the water depth was measured at the center of pool 5. The OBB and volume flow rate had the greatest effects on the value of water depth. The water depth decreased when the OBB and volume flow rate increased. The maximum water depth was determined as 1.88 m for the sample with volume flow rate, OBB, RLB, and ALB of 1000 L/s, 30 cm, 5.5, and 18°, respectively. The minimum water depth was measured as 0.695 m for the sample with volume flow rate, OBB, RLB, and ALB of 400 L/s, 60 cm, 2.9, and 0°, respectively.

The worst and best PFVSFs were determined according to the results of the rest area, as shown by Figs. 16(a) and 16(b), respectively. If the FP is more regular, then the rest area decreased more because the velocity increased. If the FP is irregular, then the flow is more laminar. Figures 16(c) and 16(d) show the velocity at a depth of Z = 0.5H in pool 5 of the worst and best PFVSFs with respect to the percentage of the rest area, respectively. The maximum velocity of the worst PFVSF occurred near the OBBs in pool 5, and was higher than that of the best PFVSF.

Numerical prediction models

The previous section showed that there are five input variables: volume flow rate, OBB, ALB, length of the large baffle, and length of the small baffle; and four outputs: maximum turbulence, maximum velocity, rest area, and water depth expressed as

(8)\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}=\{\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}1,\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}2,\mathrm{...},\mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{s}5\}=\{Q,OBB,ALB,L_1,L_2\},

(9)\mathrm{O}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}=\{T_{\max },T_{\max },RA,H\},

where Q, L₁, and L₂ are the volume flow rate, length of the large baffle, and length of the small baffle, respectively. T_max, V_max, RA, and H are the maximum turbulence, maximum velocity, rest area, and water depth, respectively. The input variables were applied for training and estimating the outputs in each model, which are the maximum turbulence, maximum velocity, rest area, and water depth of the PFVSF. The results and inputs of 70% of the FV models were randomly applied for training the prediction models, as the calibration process. Based on the obtained formulas from the calibration process, the outputs of the remaining PFVSFs were calculated. On the other hand, the estimated values and the obtained values from FV models were compared in 30% of the data set to evaluate the accuracy of prediction models.

A computer with the following specifications: CoreTM i3-3210 CPU @ 3.20 GHz 2.20, RAM 4.00 GB, was employed to simulate the 81 PFVSFs by FV analysis. Analysis of the FV model for each sample took 360–480 min, although the numerical models predicted the maximum turbulence, maximum velocity, rest area, and water depth in just a few seconds. The inputs and outputs of 57 and 24 samples were applied in the numerical models for training and testing, respectively.

Data

Based on the previous section, 81 FV models were developed to evaluate the FP of 81 PFVSF samples. There are five variables in each sample, i.e., volume flow rate, OBB, length of the large baffle, length of the small baffle, and ALB. The variations of these input variables for each sample are shown in Fig. 17.

The correlation coefficients between the inputs and outputs are listed in Table 3. The OBB and baffle length can be considered as the effective variables for predicting the rest area. The OBB had the greatest effect on the maximum turbulence and velocity of the PFVSF, although the volume flow rate also affected the maximum velocity and turbulence. The ALB is the third most effective parameter for the maximum velocity and turbulence. The results in Table 3 show that the volume flow rate has the greatest effect on the water depth of the PFVSFs. Other parameters have a slight effect on the maximum velocity, maximum turbulence, and water depth.

The correlation coefficient between the input variables and the maximum turbulence, water depth, and the maximum velocity is significant; thus, regression models such as MLR can provide acceptable results. Based on the correlation coefficients between the input variables and the rest area, a global sensitivity analysis is proposed for future studies. Hamdia and Rabczuk proposed a global sensitivity analysis when the coefficient of determination between input variables and outputs is less than 0.7 [⁷²,⁷³].

Method

Several studies have focused on numerical and computational models for predicting the FP of a fluid [⁷⁴–⁸⁰]. Four numerical models: GEP, MLR, MLER, and their combination, were employed to predict the water depth at the center of pool 5, maximum velocity, maximum turbulence, and percentage of rest area of the PFVSFs, which were simulated by the FV models. These models are discussed below.

Multiple Linear Regression

The MLR is created from several linear equations that should be integrated. There are five variables, including volume flow rate, OBB, ALB, length of the large baffle, and length of the small baffle. The maximum turbulence, maximum velocity, rest area, and water depth are considered as dependent variables that should be predicted based on Eq. (10):

(10)Y=b_{\mathrm{1}}+b_{\mathrm{2}}\times X_{\mathrm{2}}+\mathrm{\cdots }+b_k\times X_k+e+c.

The regression technique was employed to determine b₁,b₂,…,b_k, which was implemented using the minimum square error technique; thus, b₁,b₂,…,b_k are determined with the formula in Eq. (11):

(11)b_{\mathrm{i}}={\left[X^{\mathrm{T}}X\right]}^{-\mathrm{1}}\left[X^{\mathrm{T}}Y\right]\mathrm{.}

where X₂,X₃,…,X_k and b₁,b₂,…,b_k are the independent variables and linear regression parameters, respectively; e is the estimated error term obtained from the constant variance and normal distribution of independent random sampling with a zero mean; and Y is the dependent variable. Four equations should be employed for predicting each output separately.

Multiple Log Equation Regression (MLER)

The MLER approach was employed as a developed model from the MLR in this study, which was obtained from several integrated log equations. MLER includes two or more log equations that should be integrated to calculate the outputs. Based on the MLER equation, this model contains a dependent parameter, two or more independent parameters, the specific coefficient for each independent parameter, a constant coefficient, and an error term. In other words, MLER was developed using the MLR technique, and there are several log equations instead of linear equations; therefore, the error term and constant coefficient are determined based on the MLR approach. The generic equation of MLER is as follows:

(12)\mathrm{L}\mathrm{o}\mathrm{g}\left(Y_{\mathrm{M}\mathrm{L}\mathrm{n}\mathrm{E}\mathrm{R}}\right)=B_{\mathrm{1}}+B_{\mathrm{2}}\times \mathrm{L}\mathrm{o}\mathrm{g}\left(X_{\mathrm{2}}\right)+B_{\mathrm{3}}\times \mathrm{L}\mathrm{o}\mathrm{g}\left(X_{\mathrm{3}}\right)+\mathrm{\cdots }+B_K\times \mathrm{L}\mathrm{o}\mathrm{g}\left(X_K\right)+e,

where X₂,X₃,…,X_K are the independent parameters. B₁,B₂,…,B_K are the linear regression parameters, which are defined as the specific coefficient of each log equation of the MLER model. e is the estimated error term obtained from the constant variance and normal distribution of independent random sampling with a zero mean. Y is the dependent parameter. The regression technique was employed to determine the B₁,B₂,…,B_K, constant coefficient, and error term, which was done by employing the minimum square error technique; thus, B₁,B₂,…,B_K are determined by b_i, as presented in Eq. (11). If B₁+ e is equal to logC, then the simple form of Eq. (12) can be rewritten as Eq. (13). If both sides of Eq. (13) are imported as the power of 10, then Eq. (14) is obtained as a power equation.

(13)\mathrm{L}\mathrm{o}\mathrm{g}\left(Y_{\mathrm{M}\mathrm{L}\mathrm{n}\mathrm{E}\mathrm{R}}\right)=\mathrm{L}\mathrm{o}\mathrm{g}(X_{\mathrm{2}}^{B_{\mathrm{2}}}\times X_{\mathrm{3}}^{B_{\mathrm{3}}}\times \mathrm{\cdots }\times X_K^{B_K}\times C),

(14)Y_{\mathrm{M}\mathrm{L}\mathrm{n}\mathrm{E}\mathrm{R}}=X_{\mathrm{2}}^{B_{\mathrm{2}}}\times X_{\mathrm{3}}^{B_{\mathrm{3}}}\times \mathrm{\cdots }\times X_K^{B_K}\times C.

Gene Expression Program

The GEP was first proposed by Ferreira, and was developed by combining a genetic algorithm (GA) and genetic programming in 1999 [⁸⁰,⁸¹]. In this method, various phenomena are simulated by employing a set of different functions and terminals. A set of different functions includes mathematical functions, trigonometric functions, combination functions, and user-defined functions, which are employed to predict the maximum turbulence, maximum velocity, rest area, and water depth. The set of terminals involves a combination of independent parameters of the problem and constant values. Based on this approach, the GA uses the papulation of data and chooses them to find a function for predicting the maximum turbulence, maximum velocity, rest area, and water depth of the PFVSF. Several operators (genes) are employed to perform genetic variations. Based on the GEP approach, a Roulette wheel is employed to choose the data. Data are reproduced simultaneously utilizing several genetic operators. The inappropriate data are eliminated, and appropriate data are transferred and stored from the present generation to the next generation in the duplication operation. In other words, the main aim of the mutation operator is the internal random optimization of the given chromosomes. All steps of the GEP are illustrated by the flowchart in Fig. 18.

Ensemble model: The combination of GEP, MLER, and MLR

Bates and Granger were the first researchers who introduced ensemble prediction methods in 1969 [⁸²]. The ensemble model was developed by combining GEP, MLER, and MLR. In this model, the outputs of GEP and MLER, and all independent variables were imported as input variables; thus, this model is a combination of linear and two nonlinear models. Figure 19 shows a flowchart that explains the processes in the ensemble method, which employs the MLR technique to predict the outputs.

Results of the application of regression models

In this study, the statistical parameters (i.e., coefficient of determination, RMSE, and NMSE) in both the calibration and validation data sets were first calculated and compared for each model to determine the accuracy of each model on each separate data set. Then, the error terms (i.e., maximum positive and negative errors and MAPE) were employed to compare the accuracy of all prediction models. According to the results of the statistical parameters and error terms, the best model is specified separately for predicting each output. Finally, the error distribution of the best model for each output was evaluated to specify the safety factor for applying to the prediction model. This safety factor was applied in the design of PFVSFs when engineers prefer to use the prediction model instead of FVM.

Predicting maximum turbulence

Table 4 lists the statistical parameters for predicting the maximum turbulence in both the calibration and validation data sets separately. The results of all models are acceptable, although the GEP and ensemble models are more accurate than MLR and MLER. The ensemble model is more accurate than GEP on the calibration data set, although the GEP performs better than the ensemble model on the validation data set with respect to the values of the coefficient of determination and NMSE. Thus, the error terms of these models should be compared to determine the best model for predicting the maximum turbulence of the PFVSF.

Table 5 lists the maximum positive and negative errors and MAPE for each model. GEP shows the best maximum negative error and MAPE, while MLR has the best maximum positive error. On the other hand, the maximum negative error and MAPE of the GEP are better than those of the ensemble model; therefore, the GEP performs better than other models in predicting the maximum turbulence of the PFVSF.

According to the results, GEP was selected as the best model for predicting the maximum turbulence. The following equation was obtained from the calibration process for predicting the maximum turbulence. This equation was then used to calculate the maximum turbulence on the validation data set.

T_{\max }=0.146+0.001\times \theta +1.517e^{-6}\times Q\times L_2\times \theta

+0.422\times L_2\times w_0^2+3.757e^{-8}\times L_2\times Q^2

(15)-0.300\times L_2\times w_0\times L_1^2,

where T_max is the maximum turbulence in pool 5 of the PFVSF. Q, L₁, L₂, W₀, and \thetaare the upstream volume flow rate, the length of the large baffle, the length of the small baffle, OBB, and ALB, respectively.

Figure 20(a) shows the error distribution of the obtained maximum turbulence from the GEP. The absolute errors of all samples, except samples 10, 15, and 16, are less than 20%. The absolute error of the 55 samples is less than 10%.

Figure 20(b) shows the comparison of the obtained maximum turbulence from FLOW-3D and the predicted maximum turbulence by GEP. The predicted values are close to the obtained values from FLOW-3D. The maximum turbulence is one of the important parameters in fishway design; therefore, a safety factor should be applied to Eq. (15) for designing the PFVSF. The value of this safety factor should be considered as 1.31. Equation (16) is proposed for applying the safety factor.

(16)T_{\max }^{\text{'}}=T_{\max }\times SF\mathrm{,}

where T_max, T'_max, and SF are the predicted maximum turbulence, modified maximum turbulence, and safety factor, respectively.

Predicting maximum velocity

Table 6 lists the statistical parameters for each model on the calibration and validation data sets. All models are acceptable with respect to the coefficient of determination. Moreover, the accuracy of all models is acceptable according to the results of the RMSE and NMSE. The ensemble model has the lowest RMSE and NMSE and also, the best coefficient of determination; therefore, the ensemble model was selected as the best model for predicting the maximum velocity of the PFVSF.

The error terms were calculated to specify the maximum positive and negative errors and MAPE, as shown in Table 7. The ensemble model performs better than other models with respect to the maximum positive and negative errors, although GEP performs better with respect to MAPE.

According to the results, the ensemble model was selected as the best model for predicting the maximum velocity of the PFVSF. According to Fig. 19 and the features of the ensemble model, the volume flow rate, OBB, ALB, length of large baffle, length of small baffle, and the outputs of MLER and GEP should be imported as input variables. Therefore, the maximum velocities from MLER and GEP should first be obtained; then, they should be imported as the input variables. Equations (17), (18), and (19) are used to calculate the maximum velocity by MLER, GEP, and the ensemble model, respectively.

V_{\mathrm{M}\mathrm{L}\mathrm{E}\mathrm{R}}=0.399\times Q^{0.192}\times L_1^{-1.500}\times L_2^{-0.375}

(17)\times {\theta }^{0.029}\times w_0^{-0.327},

V_{\mathrm{G}\mathrm{E}\mathrm{P}}=1.108+0.003\times \theta +0.001\times Q

+3.468e^{-7}\times Q\times {\theta }^2-2.0823\times w_0^4

(18)-4.570e^{-10}\times w_0\times Q^3,

V_{\max }=-\mathrm{0.664}\times Q-\mathrm{1.8}{e}^{-\mathrm{4}}\times L_{\mathrm{1}}-\mathrm{0.018}\times L_{\mathrm{2}}

+\mathrm{0.018}\times \theta -\mathrm{3.010}{e}^{-\mathrm{3}}\times w_{\mathrm{0}}+\mathrm{0.461}\times V_{MLER}

(19)+\mathrm{0.408}\times V_{\mathrm{G}\mathrm{E}\mathrm{P}}+\mathrm{1.037},

where V_MLER, V_GEP, and V_max are the predicted maximum velocity by MLER, GEP, and the ensemble model in pool 5 of the PFVSF, respectively. Q, L₁, L₂, W₀, and \thetaare the upstream volume flow rate, length of the large baffle, length of the small baffle, OBB, and ALB, respectively. Only in Eq. (17), ALB of 1° should be applied instead of 0°.

Figure 21(a) shows the error distribution of the predicted maximum velocity in pool 5 of the PFVSF by the ensemble model. The absolute errors of all samples except sample 15 were less than 10%. The error of sample 15 was −18.9%. The maximum positive error was calculated as 7.1%. The error distribution results show that the ensemble model is an accurate model for predicting the maximum velocity. Figure 21(b) shows the obtained maximum velocity from FLOW-3D and the predicted maximum velocity by the ensemble model. The predicted values are close to the obtained maximum velocity from FLOW-3D.

The maximum velocity is one of the important parameters for designing the fishway. Thus, a safety factor is proposed to apply to Eq. (19) for designing the PFVSF. The value of this safety factor should be considered as 1.07, and Eq. (20) is suggested for applying the safety factor.

(20)V_{\max }^{\text{'}}=V_{\max }\times SF,

where V_max, V_{\max }^{\text{'}}, and SF are the predicted maximum velocity, modified maximum velocity, and safety factor, respectively.

Predicting the rest area

Table 8 shows the results of the RMSE, NMSE, maximum positive error, maximum negative error, and MAPE for each model. Based on the results of the RMSE and NMSE in both the calibration and validation data sets, the ensemble model performs better than other models, although GEP performs better than other models with respect to the results of the maximum negative error. The maximum positive error of the ensemble model is less than that of other models.

According to the results, the ensemble model was selected as the best model for predicting the rest area in pool 5 of the PFVSF. To present a formula for using the ensemble model to predict the rest area, the outputs of the predicted rest area by GEP and MLER should be calculated. In other words, the outputs of both models, the volume flow rate, OBB, ALB, the length of the large baffle, and the length of the small baffle should be imported as input variables in the ensemble model. Equations (21), (22), and (23) are used for calculating the rest area by MLER, GEP, and the ensemble model, respectively.

(21){AR}_{\mathrm{M}\mathrm{L}\mathrm{E}\mathrm{R}}=70.628\times Q^{-0.009}\times L_1^{-1.306}\times L_2^{-0.257}\times {\theta }^{0.003}\times w_0^{0.078},

(22){AR}_{\mathrm{G}\mathrm{E}\mathrm{P}}=50.163+1.459e^{-11}\times w_0\times Q^4+\sin (5042.351\times L_1+1.459e^{-11}\times Q^5\times w_0^2)-\sin {(5042.351\times L_1+1.459e^{-11}\times Q^5\times w_0^2)}^{\theta }\times \cos \left(\frac{L_1}{w_0}\right),

(23)AR=-\mathrm{57.73}\times Q+\mathrm{2.7}{e}^{-\mathrm{3}}\times L_{\mathrm{1}}+\mathrm{5.070}\times L_{\mathrm{2}}-\mathrm{5.07}\times \theta +\mathrm{0.035}\times w_{\mathrm{0}}-\mathrm{9.988}\times V_{\mathrm{M}\mathrm{L}\mathrm{E}\mathrm{R}}+\mathrm{0.873}\times V_{\mathrm{G}\mathrm{E}\mathrm{P}}+\mathrm{1.339},

where AR_MLER, AR_GEP, and AR are the predicted rest area by MLER, GEP, and the ensemble model in pool 5 of the PFVSF, respectively. Q, L₁, L₂, W₀, and \thetaare the upstream volume flow rate, the length of the large baffle, the length of the small baffle, OBB, and ALB, respectively. Only in Eq. (21), ALB of 1° should be applied instead of 0°.

Figure 22(a) shows the error distribution for predicting the rest area using the ensemble model. The absolute errors of all samples except samples 12 and 66 were less than 10%. The rest area is one of the important parameters for designing fishways. Figure 22(b) shows the obtained rest area from FLOW-3D and the predicted rest area by the ensemble model. The results show the predicted values are close to the obtained results from the FV model.

The error of samples 12 and 66 was calculated as −11.5% and 11.4%, respectively. Therefore, a safety factor for calculating the rest area should be applied to Eq. (23) to increase the accuracy of the ensemble model. On the other hand, the safety factor decreases the percentage of the rest area and increases the accuracy of Eq. (23). The value of the safety factor is considered as 1.15.

(24)A{R}^{\text{'}}={\displaystyle \frac{AR}{SF}},

where AR, A{R}^{\text{'}}, and SF are the predicted rest area, the modified rest area, and the safety factor, respectively.

Predicting water depth

Table 9 lists the statistical parameters for predicting the water depth by MLR, MLER, GEP, and the ensemble model. All models can effectively predict the water depth in pool 5 of the PFVSF. The ensemble model performs better than other models with respect to the coefficient of determination and RMSE in both the calibration and validation data sets. Although the lowest NMSE in the calibration data set is related to the ensemble model, the lowest NMSE in the validation data set is related to the GEP. Thus, to select the best model, the error terms should be evaluated for all models.

The maximum positive and negative errors and MAPE were calculated for all models, as shown in Table 10. Based on the results, the GEP has the lowest MAPE, although the ensemble model performs better than the GEP with respect to the maximum positive and negative errors. Thus, the ensemble model was selected as the best model for predicting the water depth in pool 5 of the PFVSF.

To present a formula for employing the ensemble model to predict the water depth, the water depth should first be predicted by GEP and MLER. Then, the outputs of both models, the volume flow rate, OBB, ALB, length of the large baffle, and length of the small baffle, are imported as input variables in the ensemble model. Equations (25), (26), and (27) are employed for calculating the water depth by MLER, GEP, and the ensemble model, respectively.

(25)H_{\mathrm{M}\mathrm{L}\mathrm{E}\mathrm{R}}=4.09e^{-3}\times Q^{0.838}\times L_1^{0.124}\times L_2^{-0.026}\times {\theta }^{0.003}\times w_0^{-0.182},

(26)H_{\mathrm{G}\mathrm{E}\mathrm{P}}=1.021+0.001\times Q+0.002\times L_2\times \theta +0.903\times w_0^2+2.0516e^{-7}\times Q^2-L_2\times w_0-0.250\times L_1-0.860\times w_0,

(27)H=-\mathrm{0.083}\times Q-\mathrm{4.8}{e}^{-\mathrm{4}}\times L_{\mathrm{1}}-\mathrm{0.036}\times L_{\mathrm{2}}+\mathrm{0.036}\times \theta -\mathrm{3.1}{e}^{-\mathrm{4}}\times w_{\mathrm{0}}+\mathrm{0.118}\times V_{MLER}+\mathrm{0.312}\times V_{\mathrm{G}\mathrm{E}\mathrm{P}}+\mathrm{1.018},

where H_MLER, H_GEP, and H are the predicted water depth by MLER, GEP, and the ensemble model in pool 5 of the PFVSF, respectively. Q, L₁, L₂, W₀, and \thetaare the upstream volume flow rate, length of the large baffle, length of the small baffle, OBB, and ALB, respectively. Only in Eq. (25), ALB of 1° should be applied instead of 0°. According to the maximum negative error, the predicted water depth should be decreased to improve the confidence of the model. The following equation is proposed for applying the safety factor:

(28)H^{\text{'}}={\displaystyle \frac{H}{SF}},

where H, H^{\text{'}}, and SF are the predicted water depth, modified water depth, and safety factor, respectively. The value of the safety factor was considered as 1.06 for predicting the water depth.

Figure 23(a) shows the obtained error distribution from the comparison of the FV models and prediction results; the absolute errors were less than 10%. Figure 23(b) shows the predicted water depth by the ensemble model for each sample and the obtained water depth from the FV models. According to the results, the predicted values are close to the results, which were obtained from FLOW-3D.

Akaike’s information criterion (AIC) was proposed to evaluate the accuracy of multiple linear models [⁸³]. Table 10 lists the AIC values for each model in predicting each output. The results show that the GEP is the best model for predicting the maximum turbulence in pool 5 of the PFVSF. The ensemble model can predict the maximum velocity, rest area percentage, and the water depth better than the other models.

In the present study, 70% of the data set was randomly used to train the models, which is defined as the model calibration. Then, the calibrated model was utilized to predict the remained data set. The results show that the accuracy of the best model for each output in both the calibration and validation data sets, is the same. Thus, the best model can be used for the data, which was not used in this study because the prediction models find and learn the effect of each input variable on the output. On the other hand, they learned how to calculate the outputs based on the input variables. As a result, the best prediction model can be used for a wider range of variations than the 81 samples.

Conclusions

This study is divided into two parts: evaluating the parametric study and presenting prediction models. The OBB, length of the large baffle, length of the small baffle, and volume flow rate were considered as the input variables. The FV model was developed to simulate 81 PFVSFs. According to the FV models of the PFVSFs, the maximum turbulence, maximum velocity, rest area, and water depth of pool 5 in the fishway were determined as the outputs. Three single models, including MLR, MLER, GEP, and an ensemble model, were employed to predict the outputs of all samples. The results are summarized as follows.

1) The volume flow rate and OBB had the greatest effect on the maximum turbulence, maximum velocity, and water depth.

2) Changes in the ALB and the RLB have a slight effect on the maximum turbulence, maximum velocity, and water depth.

3) GEP is proposed for predicting the maximum velocity, although the ensemble model can predict the maximum velocity, rest area, and water depth better than other models.

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