# Prediction of Hydraulic Jumps on a Triangular Bed Roughness Using Numerical Modeling and Soft Computing Methods

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## Abstract

**:**

^{®}model and experimental data showed that the overall mean value of relative error is 4.1%, which confirms the numerical model’s ability to predict the characteristics of the free and submerged jumps. The SVM model with a minimum of Root Mean Square Error (RMSE) and a maximum of correlation coefficient (R

^{2}), compared with GEP and RF models in the training and testing phases for predicting the sequent depth ratio (y

_{2}/y

_{1}), submerged depth ratio (y

_{3}/y

_{1}), tailwater depth ratio (y

_{4}/y

_{1}), length ratio of jumps (${L}_{j}/{y}_{2}^{*}$) and energy dissipation (ΔE/E

_{1}), was recognized as the best model. Moreover, the best result for predicting the length ratio of free jumps $({L}_{jf/}{y}_{2}^{*}$) in the optimal gamma is γ = 10 and the length ratio of submerged jumps $({L}_{js/}{y}_{2}^{*}$) is γ = 0.60. Based on sensitivity analysis, the Froude number has the greatest effect on predicting the (y

_{3}/y

_{1}) compared with submergence factors (SF) and T/I. By omitting this parameter, the prediction accuracy is significantly reduced. Finally, the relationships with good correlation coefficients for the mentioned parameters in free and submerged jumps were presented based on numerical results.

## 1. Introduction

^{®}software in the numerical investigation of the hydraulic jump. Nikmehr and Aminpour [25] investigated the characteristics of a hydraulic jump over bed roughness with trapezoidal blocks using the CFD model. The results state that increasing the distance and the height of the roughness will decrease the velocity near the bed and increase the shear stress. Ghaderi et al. [26] numerically investigated the characteristics of the hydraulic jumps over various roughness shapes using the FLOW-3D

^{®}model. The results were compared with previous studies. Relationships with good correlation coefficients for the mentioned parameters in free and submerged jumps were presented based on numerical results. Ghaderi et al. [27] studied the effects of triangular microroughness on the characteristics of the submerged jump with the help of the FLOW-3D

^{®}model. To validate the present model, comparisons between numerical simulations and experimental results were performed for the smooth bed and triangular microroughness [27].

## 2. Materials and Methods

#### 2.1. Dimensional Analysis

_{2}) and submerged of the submerged jump depth (y

_{3}) will be a function of the following parameters:

_{1}and y

_{4}are referred to as supercritical of the free jump depth and tailwater of the submerged jump depth; u

_{1}is inlet velocity; and g, ρ, μ, SF, and υ are the gravity acceleration, mass density of water, water dynamic viscosity, submergence factors, and water kinematic viscosity, respectively. T and I are height and distance of roughness, and Fr

_{1}and Re

_{1}are Froude and Reynolds numbers, respectively. The values of the Reynolds number (Re

_{1}) were in the range of 39,884–59,825. For large values of the Reynolds number, viscous effects can be neglected [35,36,37]. Based on the Ead and Rajaratnam [10] and Abbaspour et al. [22] studies, T/y

_{1}does not significantly affect the hydraulic jumps’ depth ratio y

_{2}/y

_{1}and y

_{3}/y

_{1}. Then, relationships (3) and (4) become:

_{jf}/y

_{2}and L

_{js}/y

_{2}), the following relationships are obtained:

#### 2.2. The FLOW-3D^{®} Model

_{F}is the volume fraction of fluid in each cell; A

_{x}, A

_{y}, and A

_{z}are the fractional areas open to flow in the subscript’s direction; ρ is the fluid density; P is the hydrostatic pressure; G

_{i}is the gravitational acceleration in subscript direction; and f

_{i}is the Reynolds stress. In FLOW-3D, free surfaces are modeled with the Volume of Fluid (VOF) technique and developed by Hirt and Nichols [37]. The VOF transport equation is expressed by the following equation:

#### 2.2.1. Turbulence Model

_{k}is the generation of turbulent kinetic energy caused by the average velocity gradient; G

_{b}is the generation of turbulent kinetic energy caused by buoyancy. S

_{k}and S

_{ε}are source terms. ${\alpha}_{k}$, ${\alpha}_{\epsilon}$ and μ

_{eff}, ${C}_{2\epsilon}$, ${C}_{1\epsilon}^{*}$ are model constants is effective viscosity.

#### 2.2.2. Boundary Conditions

#### 2.2.3. Checking Stability and Convergence Criterion

^{3}/s and Q = 0.045 m

^{3}/s. The computational time for the simulations was between 14–18 h using a personal computer with eight cores of a CPU (Intel Core i7-7700K @ 4.20 GHz and 16 GB RAM).

#### 2.2.4. Numerical Domain

#### 2.2.5. Mesh Size Sensitivity Analysis

_{3}/y

_{1}and y

_{2}/y

_{1}ratios at Fr

_{1}= 4.5 for a submerged and free hydraulic jump, numerical solutions for five different mesh sizes at distances close to the computational grid were used. Table 2 provides a summary list of the results for three different mesh sizes. Figure 5 shows that the simulated y

_{3}/y

_{1}and y

_{2}/y

_{1}ratios exhibit better agreement with the measured y

_{3}/y

_{1}and y

_{2}/y

_{1}for the finer cell size of 0.60 cm. In addition, the variation of mean relative errors can be neglected by decreasing the cell size from 0.65 cm to 0.60 cm. As a result, the selected mesh consists of a containing block with 1.3 cm cells and a nested block with 0.65 cm cells. In the present research, the same mesh was utilized for all models to reduce the effect of computational mesh on simulation results. A distance of the first cell from the walls was selected to prevent computations in the viscous sub-layer.

#### 2.3. Artificial Intelligence Methods

#### 2.3.1. Support Vector Machine (SVM)

_{i}and X

_{j}are two vectors in directions i and j, and a, c, and d are Kernel parameters. According to Figure 6, first, the input data is entered into the statistical software. Based on dimensional analysis, the dependent and independent parameters are defined in the software environment by selecting the function (RBF) and entering the main feature of the SVM model of this function (i.e., γ by trial-and-error method). Selecting the appropriate values of γ makes the results accurate and close to reality.

#### 2.3.2. Gene Expression Programming (GEP)

#### 2.3.3. Random Forest (RF)

#### 2.4. Evaluation Criteria

^{2}), Root Mean Square Error (RMSE), Normalized Root Mean Square of Error (NRMSE), and Mean Absolute Percentage Error (MAPE) were used to compare the results of prediction models of hydraulic parameters of hydraulic jumps (Equations (14)–(17)).

_{Pre}and the X

_{Num}are the predicted and the numerical values. It should be noted that the best model is the model in which RMSE is zero and R

^{2}is one, and also NRMSE and MAPE values are less than 10%.

## 3. Results

^{®}model were investigated using SVM, GEP, and RF methods. For this purpose, a total of 620 output data of numerical model were used to predict the parameters (y

_{2}/y

_{1}), (y

_{3}/y

_{1}), (y

_{4}/y

_{1}), (${L}_{j}$/${y}_{2}^{*}$), and (ΔE/E

_{1}) with artificial intelligence methods. To achieve accurate prediction and better results, the training process was repeated several times. Finally, a pattern of 25% data for testing and 75% data for training was used for all methods.

#### 3.1. Validity of the FLOW-3D^{®} Model Results

_{3}/y

_{1}), tailwater ratio (y

_{4}/y

_{1}), and relative jump length (L

_{js}/y

_{1}) of a submerged hydraulic jump and the sequent depth ratio (y

_{2}/y

_{1}) of a free hydraulic jump on a smooth bed have been used to validate the numerical model and are plotted in Figure 9.

_{1}. The overall mean value of relative error is 4.1%, which confirms the ability of the numerical model to predict the specifications of free and submerged jumps. In general, the CFD model is in excellent agreement with the experimental data [56].

#### 3.2. Sequent Depth Ratio in the Free Jump (y_{2}/y_{1})

_{2}/y

_{1}, which somehow represents the height of the jump, is directly related to the changes in the Fr

_{1}and the distance of the roughness element. By increasing these parameters, the value y

_{2}/y

_{1}is increased. According to the results of the FLOW-3D

^{®}model, the most significant decrease y

_{2}/y

_{1}with increasing Froude number compared to the smooth bed is at T/I = 0.50 with 17.83% as mean. The results showed that the y

_{2}/y

_{1}for the jump on the bed roughness was smaller than that of the corresponding jumps on a smooth bed [26,27]. Table 5 summarizes the results of estimating the y

_{2}/y

_{1}. Comparing the results of three models, the SVM model with the lowest RMSE = 0.2075 and the highest R

^{2}= 0.9966 for the training phase and RMSE = 0.2990 and R

^{2}= 0.9960 for the testing phase in predicting the y

_{2}/y

_{1}as a model the best was selected.

^{®}model and the SVM model to estimate the y

_{2}/y

_{1}in the training and testing phase. It can be seen that the SVM model has a good performance in predicting this parameter, and the output results of the SVM model are in good agreement with the FLOW-3D

^{®}values and were recognized as the best model. It is also observed that during predicting y

_{2}/y

_{1}in the testing phase, the SVM model estimates higher values at maximum points than the FLOW-3D

^{®}model.

_{2}/y

_{1}in the free jump with a correlation coefficient equal to 0.997 is expressed as:

#### 3.3. Submerged Depth Ratio in Submerged Jump (y_{3}/y_{1})

_{3}/y

_{1}) and the tailwater ratio (y

_{4}/y

_{1}) depend on the Fr

_{1}, T/I, and SF. According to the results of the FLOW-3D

^{®}, the most significant decrease y

_{3}/y

_{1}and y

_{4}/y

_{1}with increasing Froude number compared to the smooth bed are at T/I = 0.50 with 20.88% and 23.34% as mean, respectively [26,27]. Comparing the results of the three models presented in Table 6 shows that among the three models, for the y

_{3}/y

_{1}, the SVM model with values of RMSE = 0.3391 and R

^{2}= 0.9964 for the testing phase is close to the FLOW-3D

^{®}numerical model. The SVM model also performed better in predicting y

_{4}/y

_{1}and had very little error. After the SVM model, the GEP model also provided acceptable results in estimating (y

_{3}/y

_{1}) and (y

_{4}/y

_{1}).

^{®}model and predicting the SVM, GEP, and RF models in the testing phase (y

_{3}/y

_{1}) and (y

_{4}/y

_{1}). According to the graphs, it is clear that the SVM model has a better prediction than the other two models. At the maximum and minimum points, the (y

_{3}/y

_{1}) and (y

_{4}/y

_{1}), always accompanied by turbulence in the water surface, it can be seen that the SVM model has the highest efficiency and the lowest error over other models. The predicted values of these parameters by the SVM model have good adaptation. They overlap with the output values of the numerical model.

_{3}/y

_{1}and y

_{4}/y

_{1}in the submerged jump with a correlation coefficient equal to 0.993 and 0.989, respectively, on the triangular bed roughness was obtained:

#### 3.4. The Length Ratio of Jumps (${L}_{j}$/${y}_{2}^{*}$)

^{®}model, the (${L}_{j}$/${y}_{2}^{*}$) for the bed roughness is less than the smooth bed, and for the submerged jump it is larger than the free jump. For T/I = 0.5, the ratio length of free and submerged jumps decreases by about 25.52% and 21.65% as a mean, respectively [27]. Estimating the jump length reduces the volume of construction operations and ultimately reduces the project’s overall cost. Therefore, an accurate estimation of the hydraulic jump length is essential to design the length of the stilling basin based on this parameter. The results of predicting (${L}_{j}$/${y}_{2}^{*}$) along with the evaluation criteria are presented in Table 7. According to the results, the SVM model has good statistical criteria among other models and has high accuracy in predicting the relative length of free and submerged hydraulic jumps.

^{2}and RMSE versus different gammas are presented for the best model of the ${L}_{jf}$/${y}_{2}^{*}$ and the ${L}_{js}$/${y}_{2}^{*}$ in the testing phase (Figure 14). In the support vector machine, selecting the appropriate gamma is one of the main parameters in determining the best model, which has been done by trial and error. Finally, the best result for predicting the ${L}_{jf}$/${y}_{2}^{*}$ in the optimal gamma is 10 (γ = 10), and for ${L}_{js}$/${y}_{2}^{*}$ in the optimal gamma it is 0.60 (γ = 0.60).

^{®}and the predicted models of ${L}_{jf}$/${y}_{2}^{*}$ and the ${L}_{js}$/${y}_{2}^{*}$ data for the best SVM model in the training and testing phases. According to Figure 15, it can be seen that when the values of the ${L}_{jf}$/${y}_{2}^{*}$ reach the maximum and minimum points, the prediction accuracy of the SVM model decreases. In other words, when the ${L}_{jf}$/${y}_{2}^{*}$ reaches the maximum and minimum jump values, the prediction error of the SVM model increases. Moreover, as shown in Figure 16 for the ${L}_{js}$/${y}_{2}^{*}$, it can be seen that the SVM model always has values close to the FLOW-3D

^{®}model and has a better performance compared to the ${L}_{jf}$/${y}_{2}^{*}$. On the other hand, most SVM model errors in both parameters occurred in the initial range of testing data. In the middle to the end of the data, the prediction error decreased.

#### 3.5. The Energy Dissipation (ΔE/E_{1})

_{1}, E

_{2}, E

_{3}, and E

_{4}are specific energies upstream and downstream of the free and submerged jumps, respectively (see Figure 1). According to the results of the FLOW-3D

^{®}, the ΔE/E

_{1}increases with increasing the Fr

_{1}. The highest ΔE/E

_{1}occurs with T/I = 0.50 in the free and submerged jumps compared to other distances between the roughnesses of the corresponding T/I ratios [26,27]. Determining the amount of ΔE/E

_{1}that occurs due to hydraulic jumps will lead to the stilling basin’s more efficient and economical design. The results of predicting energy dissipation due to free jump (ΔE/E

_{1})

_{f}and submerged jump (ΔE/E

_{1})

_{S}are presented in Table 8. The results showed that for energy dissipation for (ΔE/E

_{1})

_{f}, the SVM model with R

^{2}= 0.9848 and RMSE = 0.0313, and for the testing phase (ΔE/E

_{1})

_{S}, R

^{2}= 0.9843 and RMSE = 0.0238, these were recognized as the best models. Therefore, the best prediction with the least possible error among the three models is obtained by the SVM model.

^{2}and RMSE of energy dissipation due to free and submerged jumps are presented for the testing phase (Figure 17). Radar graphs can show the accuracy of predictions of different models compared to each other. It can be seen that the SVM model has provided acceptable performance and has a much better prediction than the GEP and RF models. Furthermore, because RMSE values are small and their changes are not visible in the graph, by multiplying the RMSE by 10, the range of changes became broader and more precise.

_{1}and Fr

_{1}with a correlation coefficient equal to 0.963 and 0.946, respectively, for the free and submerged jumps:

#### 3.6. Sensitivity Analysis

_{3}/y

_{1}). The parameter that had the most impact was identified, and its results are presented in Table 9.

_{3}/y

_{1}) is when all three parameters of Fr

_{1}, T/I, and SF are involved in the prediction. The Fr

_{1}has the greatest effect on predicting the (y

_{3}/y

_{1}) based on sensitivity analysis. By omitting this parameter, the prediction accuracy is significantly reduced. The SF and T/I are also involved in the study of (y

_{3}/y

_{1}), but the impact of each is less than the Fr

_{1}.

## 4. Conclusions

^{®}software was used. Key findings of the comparative analysis are given below:

- By comparing the results of the two experiments (physical and numerical), the FLOW-3D
^{®}software can accurately predict the characteristics of free and submerged hydraulic jumps. The overall mean value of relative error between numerical results and experimental data is 4.1%, which confirms the numerical model’s ability to predict the characteristics of the free and submerged jumps. - The SVM model with the RMSE = 0.2075 and R
^{2}= 0.9966 for the training phase and RMSE = 0.2990 and R^{2}= 0.9960 for the testing phase in predicting the y_{2}/y_{1}is the best model and close to the FLOW-3D^{®}result. - For the y
_{3}/y_{1}, the SVM model with values of RMSE = 0.3391 and R^{2}= 0.9964 for the testing phase is close to the FLOW-3D^{®}model. The SVM model also performed better in predicting y_{4}/y_{1}and had very little error. After the SVM model, the GEP model also provided acceptable results in estimating (y_{3}/y_{1}) and (y_{4}/y_{1}). - The SVM model demonstrated better statistical criteria among other models (i.e., GEP and RF) and has high accuracy in predicting the relative length of free and submerged hydraulic jumps. Furthermore, the best result for predicting the ${L}_{jf}/{y}_{2}^{*}$ in the optimal gamma is 10 (γ = 10) and the ${L}_{js}/{y}_{2}^{*}$ in the optimal gamma is 0.60 (γ = 0.60).
- For energy dissipation due to (ΔE/E
_{1})_{f}and (ΔE/E_{1})_{S}, for the testing phase, SVM model with R^{2}= 0.9848 and RMSE = 0.031 as well as R^{2}= 0.9843 and RMSE = 0.0238 were recognized as the best models, respectively. - The Fr
_{1}has the greatest effect on predicting the (y_{3}/y_{1}) based on sensitivity analysis. By omitting this parameter, the prediction accuracy is significantly reduced. The SF and T/I are also involved in the (y_{3}/y_{1}), but the impact of each is less than the Fr_{1}. - Relationships with good correlation coefficients for the mentioned parameters in free and submerged hydraulic jumps were presented based on numerical results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

Q | Discharge (L^{3}T^{−1}) |

d | Gate opening (L) |

E_{1}, E_{2} | Specific energy at the beginning and after the free jump (L) |

E_{3}, E_{4} | Specific energy at the beginning and after the submerged jump (L) |

ΔE | Energy dissipation (L) |

y_{1} | Inlet depth of the hydraulic jump (L) |

y_{2} | Sequent depth of the free jump (L) |

y_{3} | Submerged depth (L) |

y_{4} | Tailwater depth (L) |

L_{jf} | Length of the free jump (L) |

L_{js} | Length of the submerged jump (L) |

u_{1} | Inlet horizontal velocity (LT^{−1}) |

g | Gravitational acceleration (LT^{−2}) |

I | Distance of triangular roughness (L) |

T | Roughness height (L) |

Fr_{1} | Inlet Froude number (-) |

Re_{1} | Inlet Reynolds number (-) |

SF | Submergence factor (-) |

t | Time (T) |

p | Pressure (ML^{−1}T^{−2}) |

F | Fraction function |

ρ | Mass density of water (ML^{−3}) |

$\nu $ | Kinematic viscosity of water (LT^{−1}) |

μ | Dynamic viscosity of fluid (ML^{−1}T^{−1}) |

k | Turbulence kinetic energy (L^{2}T^{−3}) |

ε | Turbulence dissipation rate (L^{2}T^{−3}) |

µ_{eff} | Effective viscosity (ML^{−1}T^{−1}) |

G_{k} | The generation of turbulent kinetic energy caused by the average velocity gradient |

G_{b} | The generation of turbulent kinetic energy caused by buoyancy |

S_{k}, S_{ε} | Source terms |

SVM | Support Vector Machine |

GEP | Gene Expression Programming |

RF | Random Forest |

R^{2} | Correlation coefficient |

RMSE | Root Mean Square Error |

NRMSE | Normalized Root Mean Square of Error |

MAPE | Mean Absolute Percentage Error |

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**Figure 1.**Definition sketch of the free and submerged hydraulic jumps on a triangular bed roughness after Ghaderi et al. [26].

**Figure 2.**The boundary conditions governing the simulation, (

**a**) smooth bed, (

**b**) the triangular bed roughness.

**Figure 3.**CFD flow discharge time variation in the inlet and outlet boundaries, (

**a**) Q = 0.03 m

^{3}/s, (

**b**) Q = 0.045 m

^{3}/s.

**Figure 4.**Structured rectangular hexahedral mesh with two different mesh blocks, (

**a**) smooth bed, (

**b**) the triangular bed roughness.

**Figure 5.**Variations of the relative error of y

_{3}/y

_{1}and y

_{2}/y

_{1}at Fr

_{1}versus cell size.

**Figure 9.**Numerical versus basic experimental parameters of submerged and free hydraulic jumps. (

**a**) y

_{3}/y

_{1}, (

**b**) y

_{4}/y

_{1}, (

**c**) L

_{js}/y

_{1}, and (

**d**) y

_{2}/y

_{1}.

**Figure 12.**Comparison of the numerical results and the predicted models of (y

_{3}/y

_{1}) for the testing phase.

**Figure 13.**Comparison of the numerical results and the predicted models of (y

_{4}/y

_{1}) for the testing phase.

**Figure 17.**Radar graphs of R

^{2}and RMSE for energy dissipation due to free and submerged jumps in the testing phase.

Bed Type | Q (m ^{3}/s) | I (cm) | T (cm) | d (cm) | y_{1}(cm) | y_{4}(cm) | Fr_{1} | SF |
---|---|---|---|---|---|---|---|---|

Smooth | 0.03, 0.045 | - | - | 5 | 1.62–3.83 | 9.64–32.10 | 1.7–9.3 | 0.26–0.50 |

Triangular roughness | 0.03, 0.045 | 4–8–12–16–20 | 4 | 5 | 1.62–3.84 | 6.82–30.08 | 1.7–9.3 | 0.21–0.44 |

Test No. | Coarser Cells Size (cm) | Finer Cells Size (cm) | Total Cells | (y_{3}/y_{1})_{Num} | (y_{3}/y_{1})_{Exp} | (y_{2}/y_{1})_{Num} | (y_{2}/y_{1})_{Exp} | MAPE ^{1}-y_{3}/y_{1} (%) | MAPE-y_{2}/y_{1} (%) |
---|---|---|---|---|---|---|---|---|---|

T1 | 2.00 | 0.95 | 910,358 | 8.55 | 6.88 | 7.43 | 5.88 | 26.36 | 24.27 |

T2 | 1.70 | 0.85 | 1,285,482 | 7.85 | 6.88 | 6.91 | 5.88 | 17.51 | 14.09 |

T3 | 1.50 | 0.75 | 1,871,649 | 7.38 | 6.88 | 6.44 | 5.88 | 9.52 | 7.26 |

T4 | 1.30 | 0.65 | 2,908,596 | 7.17 | 6.88 | 6.20 | 5.88 | 5.44 | 4.21 |

T5 | 1.15 | 0.60 | 3,812,035 | 7.10 | 6.88 | 6.08 | 5.88 | 3.41 | 3.34 |

^{1}Mean Absolute Percentage Error = $100\times \frac{1}{\mathrm{n}}{\displaystyle \sum _{1}^{\mathrm{n}}}\left|\frac{{\mathrm{X}}_{\mathrm{Exp}}-{\mathrm{X}}_{\mathrm{Num}}}{{\mathrm{X}}_{\mathrm{Exp}}}\right|$. X

_{Exp}: the experimental value of X; X

_{Num}: the numerical value of X; and n: the total amount of data.

**Table 3.**Types of kernel functions [50].

Function | Expression |
---|---|

Linear Kernel | $K\left({x}_{i},{x}_{j}\right)=\left({x}_{i},{x}_{j}\right)$ |

Polynomial Kernel | $K\left({x}_{i},{x}_{j}\right)={\left(({x}_{i},{x}_{j})\hspace{0.17em}+\hspace{0.17em}1\right)}^{d}$ |

Radial Basis Kernel | $K\left({x}_{i},{x}_{j}\right)=exp\left(-\frac{{\Vert {x}_{i}\hspace{0.17em}-\hspace{0.17em}{x}_{j}\Vert}^{2}}{2{\sigma}^{2}}\right)$ |

Sigmoid Kernel | $K\left({x}_{i},{x}_{j}\right)\hspace{0.17em}=\hspace{0.17em}tanh\left(-a({x}_{i},{x}_{j})\hspace{0.17em}+\hspace{0.17em}c\right)$ |

**Table 4.**Basic flow variables for the numerical and physical models after Ahmed et al. [15].

Models | Bed | Q (m^{3}/s) | d (cm) | y_{1} (cm) | u_{1} (m/s) | Fr_{1} |
---|---|---|---|---|---|---|

Numerical and physical | Smooth | 0.045 | 5 | 1.62–3.83 | 1.04–3.70 | 1.7–9.3 |

Training | Testing | |||||||
---|---|---|---|---|---|---|---|---|

Model | R^{2} | RMSE | NRMSE (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.9953 | 0.2356 | 4.03 | 5.35 | 0.9933 | 0.3335 | 5.56 | 8.83 |

RF | 0.9682 | 0.5924 | 10.97 | 11.91 | 0.9275 | 1.0811 | 14.73 | 11.26 |

SVM | 0.9966 | 0.2075 | 3.5481 | 5.07 | 0.9960 | 0.2990 | 4.98 | 8.46 |

**Table 6.**Prediction results for the submerged depth ratio (y

_{3}/y

_{1}) and the tailwater depth ratio (y

_{4}/y

_{1}).

Training | Testing | |||||||
---|---|---|---|---|---|---|---|---|

y_{3}/y_{1} | R^{2} | RMSE | NRMSE (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.9903 | 0.4016 | 5.96 | 8.63 | 0.9895 | 0.5379 | 7.61 | 9.92 |

RF | 0.9815 | 0.5679 | 8.43 | 11.97 | 0.9750 | 0.7804 | 11.04 | 23.03 |

SVM | 0.9978 | 0.2024 | 3.01 | 2.67 | 0.9964 | 0.3391 | 4.80 | 4.93 |

y_{4}/y_{1} | R^{2} | RMSE | NRMSE (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.9972 | 0.2811 | 3.41 | 3.34 | 0.9963 | 0.3923 | 4.54 | 6.84 |

RF | 0.9901 | 0.5157 | 6.26 | 9.68 | 0.9899 | 0.6462 | 7.48 | 10.04 |

SVM | 0.9991 | 0.1639 | 1.99 | 1.63 | 0.9988 | 0.2806 | 3.25 | 5.08 |

Training | Testing | |||||||
---|---|---|---|---|---|---|---|---|

${L}_{jf}$/${y}_{2}^{*}$ | R^{2} | RMSE | NRMSE (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.829 | 0.234 | 4.39 | 3.66 | 0.766 | 0.249 | 4.54 | 4.54 |

RF | 0.752 | 0.278 | 5.08 | 3.92 | 0.741 | 0.319 | 6.32 | 5.21 |

SVM | 0.919 | 0.169 | 3.16 | 2.74 | 0.881 | 0.174 | 3.19 | 2.90 |

${L}_{js}$/${y}_{2}^{*}$ | R^{2} | RMSE | NRMSE (%) | MAPE% | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.878 | 0.273 | 3.88 | 3.17 | 0.867 | 0.336 | 4.71 | 4.33 |

RF | 0.787 | 0.316 | 4.37 | 3.51 | 0.764 | 0.425 | 6.50 | 5.46 |

SVM | 0.961 | 0.154 | 2.19 | 1.93 | 0.940 | 0.212 | 2.97 | 2.37 |

Training | Testing | |||||||
---|---|---|---|---|---|---|---|---|

(ΔE/E_{1})_{f} | R^{2} | RMSE | NRMSE (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.980 | 0.029 | 6.76 | 6.59 | 0.977 | 0.040 | 10.19 | 16.04 |

RF | 0.855 | 0.069 | 16.5 | 25.61 | 0.801 | 0.072 | 16.32 | 21.22 |

SVM | 0.985 | 0.027 | 6.37 | 6.58 | 0.984 | 0.031 | 7.80 | 9.04 |

(ΔE/E_{1})_{S} | R^{2} | RMSE | NRMSE (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

GEP | 0.980 | 0.025 | 7.22 | 8.85 | 0.969 | 0.033 | 10.07 | 11.63 |

RF | 0.912 | 0.051 | 12.94 | 13.83 | 0.916 | 0.047 | 13.53 | 13.43 |

SVM | 0.985 | 0.022 | 6.22 | 6.43 | 0.984 | 0.023 | 7.25 | 9.05 |

Training | Testing | ||||||||
---|---|---|---|---|---|---|---|---|---|

Input parameter | Omitted parameter | R^{2} | RMSE | NRME (%) | MAPE (%) | R^{2} | RMSE | NRMSE (%) | MAPE (%) |

Fr_{1}, SF, T/I | - | 0.999 | 0.163 | 1.99 | 1.63 | 0.998 | 0.280 | 3.25 | 5.08 |

SF, T/I | Fr_{1} | 0.787 | 1.639 | 24.33 | 34.63 | 0.731 | 2.417 | 34.21 | 36.69 |

Fr_{1}, T/I | SF | 0.986 | 0.479 | 7.12 | 10.83 | 0.984 | 0.664 | 9.41 | 15.9 |

Fr_{1}, SF | T/I | 0.989 | 0.415 | 6.17 | 9.85 | 0.988 | 0.548 | 7.76 | 14.66 |

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## Share and Cite

**MDPI and ACS Style**

Dasineh, M.; Ghaderi, A.; Bagherzadeh, M.; Ahmadi, M.; Kuriqi, A.
Prediction of Hydraulic Jumps on a Triangular Bed Roughness Using Numerical Modeling and Soft Computing Methods. *Mathematics* **2021**, *9*, 3135.
https://doi.org/10.3390/math9233135

**AMA Style**

Dasineh M, Ghaderi A, Bagherzadeh M, Ahmadi M, Kuriqi A.
Prediction of Hydraulic Jumps on a Triangular Bed Roughness Using Numerical Modeling and Soft Computing Methods. *Mathematics*. 2021; 9(23):3135.
https://doi.org/10.3390/math9233135

**Chicago/Turabian Style**

Dasineh, Mehdi, Amir Ghaderi, Mohammad Bagherzadeh, Mohammad Ahmadi, and Alban Kuriqi.
2021. "Prediction of Hydraulic Jumps on a Triangular Bed Roughness Using Numerical Modeling and Soft Computing Methods" *Mathematics* 9, no. 23: 3135.
https://doi.org/10.3390/math9233135