# Assessing Machine Learning versus a Mathematical Model to Estimate the Transverse Shear Stress Distribution in a Rectangular Channel

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## Abstract

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## 1. Introduction

_{1}= 0.20 [5]. The symbols g, H and S denote gravity, water level and channel slope, respectively, whereas u is the velocity at height z, u

_{*}is the shear velocity, k is von Karman constant and z

_{0}is the roughness length.

## 2. Materials and Methods

#### 2.1. Data Collection

_{w}is the water surface slope, ${k}_{s}$ is the roughness height, (Re) is the Reynolds number and (Fr) is the Froude number.

#### 2.2. Tsallis Entropy

_{0}and λ

_{1}are the Lagrange multipliers. By ∂L/∂(τ) = 0 to maximize entropy, the f(τ) yields as:

_{1}and λ′ are Lagrange multipliers that can be derived by trial and error from two implicit equations that follow. Indeed, by inserting and integrating Equation (10) into two constraints (Equations (8) and (9)), two Equations (15) and (16) are returned as:

_{1}and λ′). To estimate the SSD, a pair of mean and maximum shear stresses is required. The results of the Lashkar-Ara and Fatahi [9] studies have been used for this reason in order to estimate the values of ${\tau}_{\mathrm{max}}$ and $\overline{\tau}$. They adjusted the slope of the bed flume at $9.58\times {10}^{-4}$. The shear stress carried by the walls and bed was measured for a different aspect ratio (B/H = 2.86, 4.51, 5.31, 6.19, 7.14, 7.89, 8.96, 10.71, 12.24 and 13.95). For each aspect ratio, the distribution of shear stress in the bed and wall was measured by a Preston tube. The best fit equation was obtained for ${\tau}_{\mathrm{max}}$ and $\overline{\tau}$ separately for wall and bed in aspect ratio 2.89 < B/H < 13.95 by assuming a fully turbulent and subcritical regime among all the experimental results. Relationships are shown in Equations (17)–(20).

#### 2.3. Genetic Programming (GP)

#### 2.4. Adaptive Neuro Fuzzy Inference System (ANFIS)

#### 2.5. Criteria for Statistical Assessment

_{i}is the observed parameter value, P

_{i}predicted parameter value, $\overline{O}$ is the mean value observed parameter value and n number of samples.

## 3. Results

#### 3.1. Modeling of GP

#### 3.2. ANFIS Modeling

_{b}) and Figure 6 shows the performance of the ANFIS model to estimate the wall SSD (τ

_{w}), 30% of the data, which were not used in the training stage would be used to evaluate the performance of the model. The results of statistical indexes for modeling shear stress with ANFIS are summarized in Table 4. As well, the estimating bands of the four above parameters using to determine the shear stress are shown in Figure 5. Skewness results obtained from statistical prediction dimensionless parameters.

#### 3.3. Comparison of the GP Model, Tsallis Entropy and ANFIS

## 4. Conclusions

- The effect of different input variable on the result was investigated to find the best input combination.
- In the present study B/H had the highest effect on the prediction power.
- For bed shear stress predictions, the GP model, with an average RMSE of 0.0893 performed better than the Tsallis entropy-based equation and ANFIS model with RMSE of 0.0714 and 0.138 respectively.
- To estimate the wall shear stress distribution the proposed ANFIS model, with an average RMSE of 0.0846 outperformed the Tsallis entropy-based equation with an RMSE of 0.0880 followed by the GP model with an RMSE of 0.0904.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematics of local shear stress distribution coordinates in the rectangular channel wall and bed.

**Figure 3.**Comparison to the estimate of ${\tau}_{b}/{\overline{\tau}}_{b}$ between the observed and predicted GP for (

**a**) B/H = 2.86, (

**b**) B/H = 4.51, (

**c**) B/H = 7.14 and (

**d**) B/H = 13.95.

**Figure 4.**Comparison to the estimate of ${\tau}_{w}/{\overline{\tau}}_{w}$ between the observed and predicted GP for (

**a**) B/H = 2.86, (

**b**) B/H = 4.51, (

**c**) B/H = 7.14 and (

**d**) B/H = 13.95.

**Figure 5.**Comparison to the estimate of ${\tau}_{b}/{\overline{\tau}}_{b}$ $\frac{{\tau}_{b}}{\overline{\tau}}$ between the observed and predicted adaptive neuro-fuzzy inference system (ANFIS) for (

**a**) B/H = 2.86, (

**b**) B/H = 4.51, (

**c**) B/H = 7.14, and (

**d**) B/H = 13.95.

**Figure 6.**Comparison to the estimate of ${\tau}_{w}/{\overline{\tau}}_{w}$ between the observed and predicted ANFIS for (

**a**) B/H = 2.86, (

**b**) B/H = 4.51, (

**c**) B/H = 7.14, and (

**d**) B/H = 13.95.

**Figure 7.**The dimensionless bed shear stress distribution for (

**a**) B/H = 2.86, (

**b**) B/H = 4.51, (

**c**) B/H = 7.14 and (

**d**) B/H = 13.95.

**Figure 8.**The dimensionless wall shear stress distribution for (

**a**) B/H = 2.86, (

**b**) B/H = 4.51, (

**c**) B/H = 7.14 and (

**d**) B/H = 13.95.

Parameters | Variable Definition | Minimum | Maximum | Mean |
---|---|---|---|---|

H (m) | Flow depth | 0.043 | 0.21 | 0.0928 |

B/H | aspect ration | 2.86 | 13.95 | 7.98 |

Q (L/s) | Discharge | 11.06 | 102.38 | 34.795 |

V (m/s) | Velocity | 0.429 | 0.813 | 0.568 |

Fr | Froude number | 0.66 | 0.566 | 0.618 |

Re $\times {10}^{4}$ | Reynolds number | 6.4 | 39.87 | 16.418 |

${\mathrm{Re}}_{*}$ | Shear Reynolds | 0.322 | 0.609 | 0.426 |

γHS | Total shear stress | 0.442 | 2.162 | 0.955 |

Parameter | Definition | Value (Model 1) | Value (Model 2) | Value (Model 3) |
---|---|---|---|---|

P_{1} | Function set | +, −, *, √, ^2, cos, sin, exp | +, −, *, √, ^2, cos, sin, exp | +, −, *, √, ^2, cos, sin, exp |

P_{2-1} | The terminal set for ${\tau}_{b}/{\overline{\tau}}_{b}$ | b/B, B/H, Fr, Re | b/B, B/H, Fr | b/B, B/H |

P_{2-2} | The terminal set for ${\tau}_{w}/{\overline{\tau}}_{w}$ | z/H, B/H, Fr, Re | z/H, B/H, Fr | z/H, B/H |

B/H | Input Variable | Bed | Input Variable | Wall | ||||||
---|---|---|---|---|---|---|---|---|---|---|

ME | MAE | RMSE | NSE | ME | MAE | RMSE | NSE | |||

2.86 | b/B, B/H, Fr, Re | 0.2338 | 0.0837 | 0.1062 | 0.947 | z/H, B/H, Fr, Re | 0.0363 | 0.0617 | 0.0516 | 0.8021 |

2.86 | b/B, B/H, Fr | 0.2445 | 0.1038 | 0.1206 | 0.9456 | z/H, B/H, Fr | 0.0693 | 0.0257 | 0.0821 | 0.7759 |

2.86 | b/B, B/H | 0.2259 | 0.0713 | 0.1051 | 0.9382 | z/H, B/H | 0.0728 | 0.0217 | 0.0870 | 0.7277 |

4.51 | b/B, B/H, Fr, Re | 0.0927 | 0.0473 | 0.0526 | 0.9911 | z/H, B/H, Fr, Re | 0.0546 | 0.0202 | 0.0701 | 0.8548 |

4.51 | b/B, B/H, Fr | 0.1019 | 0.0642 | 0.0638 | 0.9903 | z/H, B/H, Fr | 0.0818 | 0.0302 | 0.0890 | 0.8972 |

4.51 | b/B, B/H | 0.1450 | 0.0962 | 0.0995 | 0.9889 | z/H, B/H | 0.0874 | 0.0530 | 0.0972 | 0.8987 |

7.14 | b/B, B/H, Fr, Re | 0.0889 | 0.0466 | 0.0533 | 0.9958 | z/H, B/H, Fr, Re | 0.0422 | 0.0424 | 0.0507 | 0.8982 |

7.14 | b/B, B/H, Fr | 0.0851 | 0.0408 | 0.0493 | 0.9962 | z/H, B/H, Fr | 0.0330 | 0.0321 | 0.0617 | 0.8566 |

7.14 | b/B, B/H | 0.0826 | 0.0348 | 0.0468 | 0.9955 | z/H, B/H | 0.0589 | 0.1153 | 0.0648 | 0.9049 |

13.95 | b/B, B/H, Fr, Re | 0.2005 | 0.1269 | 0.1376 | 0.8667 | z/H, B/H, Fr, Re | 0.0612 | 0.0720 | 0.1045 | 0.7916 |

13.95 | b/B, B/H, Fr | 0.2678 | 0.1398 | 0.1566 | 0.8511 | z/H, B/H, Fr | 0.0559 | 0.0264 | 0.1117 | 0.8097 |

13.95 | b/B, B/H | 0.1619 | 0.0908 | 0.1059 | 0.8534 | z/H, B/H | 0.0716 | 0.0926 | 0.1126 | 0.7758 |

B/H | Bed | Wall | ||||||
---|---|---|---|---|---|---|---|---|

ME | MAE | RMSE | NSE | ME | MAE | RMSE | NSE | |

2.86 | 0.2559 | 0.0991 | 0.1268 | 0.9279 | 0.0383 | 0.0314 | 0.0492 | 0.8026 |

4.51 | 0.1728 | 0.1240 | 0.1266 | 0.9744 | 0.0870 | 0.0959 | 0.1004 | 0.9033 |

7.14 | 0.2157 | 0.1699 | 0.1724 | 0.9871 | 0.0868 | 0.0634 | 0.0745 | 0.907 |

13.95 | 0.2278 | 0.1048 | 0.1271 | 0.8482 | 0.1792 | 0.0909 | 0.1145 | 0.7752 |

B/H | Bed | Wall | ||||||
---|---|---|---|---|---|---|---|---|

ME | MAE | RMSE | NSE | ME | MAE | RMSE | NSE | |

2.86 | 1.252 | 0.0531 | 0.0706 | 0.9276 | 1.3145 | 0.0622 | 0.0797 | 0.7721 |

4.51 | 1.476 | 0.0522 | 0.0625 | 0.9425 | 1.3741 | 0.0749 | 0.0894 | 0.7632 |

7.14 | 1.538 | 0.0672 | 0.0685 | 0.9310 | 1.6254 | 0.0631 | 0.0738 | 0.8275 |

13.95 | 1.511 | 0.0643 | 0.0840 | 0.8426 | 1.2562 | 0.0893 | 0.1094 | 0.8398 |

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**MDPI and ACS Style**

Lashkar-Ara, B.; Kalantari, N.; Sheikh Khozani, Z.; Mosavi, A.
Assessing Machine Learning versus a Mathematical Model to Estimate the Transverse Shear Stress Distribution in a Rectangular Channel. *Mathematics* **2021**, *9*, 596.
https://doi.org/10.3390/math9060596

**AMA Style**

Lashkar-Ara B, Kalantari N, Sheikh Khozani Z, Mosavi A.
Assessing Machine Learning versus a Mathematical Model to Estimate the Transverse Shear Stress Distribution in a Rectangular Channel. *Mathematics*. 2021; 9(6):596.
https://doi.org/10.3390/math9060596

**Chicago/Turabian Style**

Lashkar-Ara, Babak, Niloofar Kalantari, Zohreh Sheikh Khozani, and Amir Mosavi.
2021. "Assessing Machine Learning versus a Mathematical Model to Estimate the Transverse Shear Stress Distribution in a Rectangular Channel" *Mathematics* 9, no. 6: 596.
https://doi.org/10.3390/math9060596