# An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy

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## Abstract

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## 1. Introduction

## 2. Tsallis Entropy for Velocity Lag

#### 2.1. Definition of the Tsallis Entropy

#### 2.2. Constraint Equations

#### 2.3. Maximization of Entropy

#### 2.4. Determination of Lagrange Multipliers

#### 2.5. Hypothesis for the Cumulative Distribution Function of Velocity Lag

#### 2.6. Derivation of Velocity Lag

#### 2.7. Reparameterization

## 3. Comparison with Experimental Data and Other Models and Discussion

#### 3.1. Selected Experimental Data

#### 3.2. Some Deterministic Models

#### 3.3. Comparison with Experimental Data and Other Models

#### 3.4. Physical Explanation

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the velocity lag between the sediment particle and the surrounding fluid in sediment-laden flow. In this figure, ${u}_{r}\left(y\right)$ is velocity lag at a vertical distance $y$ from the channel bottom, and $H$ is the water depth of sediment-laden flow, respectively. Here the velocity of the sediment particle is the bulk velocity of the sediment particles (i.e., it is defined statistically as the average velocity of many sediment particles), which is different from the velocity of individual sediment particles.

**Figure 2.**Variation of the non-dimensional velocity lag with vertical distance from the channel bottom with different $G$ values.

**Figure 3.**Comparison of the derived expression for the Tsallis entropy (Equation (17)), the deterministic model of Greimann et al. [12] (Equation (19)), the deterministic model of Cheng [1] (Equation (20)), the deterministic model of Pal et al. [14] (Equation (21)), as well as the Shannon entropy-based model derived by Kumbhakar et al. [15], with twenty-two collected experimental datasets (T1–T22). In each figure, ED: experimental data; TM: the Tsallis entropy-based model (Equation (17)); PM: the deterministic model of Pal et al. [14] (Equation (21)); SM: the Shannon entropy-based model derived by Kumbhakar et al. [15]; GM: the deterministic model of Greimann et al. [12] (Equation (19)); CM: the deterministic model of Cheng [1] (Equation (20)).

**Figure 4.**Vertical velocity lag distribution for three different values of friction velocity of the flow. Other parameter values are presented in the test case T1 of Table 1.

**Figure 5.**Vertical velocity lag distribution for three different values of particle diameter. Other parameter values are presented in the test case T1 of Table 1.

**Figure 6.**Vertical velocity lag distribution for three different values of particle specific gravity. Other parameter values are presented in the test case T1 of Table 1.

Test Number | Reference | Particle Material | $\mathbf{Particle}\mathbf{Diameter}\mathit{d}$ (mm) | $\mathbf{Particle}\mathbf{Specific}\mathbf{Gravity}\mathit{s}$ | $\mathbf{Shear}\mathbf{Velocity}{\mathit{u}}_{\ast}(\mathbf{cm}/\mathbf{s})$ | $\mathbf{Flow}\mathbf{Depth}\mathit{H}$ (cm) | $\mathbf{Kinematic}\mathbf{Viscosity}\mathbf{of}\mathbf{the}\mathbf{Fluid}{\mathit{\nu}}_{\mathit{f}}({\mathbf{cm}}^{2}/\mathbf{s})$ | $\mathbf{Reynolds}\mathbf{Number}{\mathbf{Re}}^{\ast}$ |
---|---|---|---|---|---|---|---|---|

T1 | Rashidi et al. [7] | Polystyrene | 0.12 | 1.03 | 0.9 | 2.75 | 0.0084 | 1.286 |

T2 | 0.22 | 1.03 | 0.9 | 2.75 | 0.0084 | 2.357 | ||

T3 | 0.65 | 1.03 | 0.9 | 2.75 | 0.0084 | 6.964 | ||

T4 | 1.10 | 1.03 | 0.9 | 2.75 | 0.0084 | 11.786 | ||

T5 | Kaftori et al. [36] | Polystyrene | 0.1 | 1.05 | 1.28 | 3.25 | 0.008 | 1.600 |

T6 | 0.275 | 1.05 | 1.29 | 3.27 | 0.0079 | 4.491 | ||

T7 | 0.9 | 1.05 | 1.34 | 3.27 | 0.0081 | 14.889 | ||

T8 | 0.1 | 1.05 | 1.6 | 3.52 | 0.0081 | 1.975 | ||

T9 | 0.275 | 1.05 | 1.6 | 3.51 | 0.008 | 5.500 | ||

T10 | 0.9 | 1.05 | 1.55 | 3.77 | 0.0078 | 17.885 | ||

T11 | Best et al. [6] | Glass | 0.125 | 2.6 | 3.4 | 5.75 | 0.0083 | 5.120 |

T12 | 0.175 | 2.6 | 3.4 | 5.75 | 0.0083 | 7.169 | ||

T13 | 0.225 | 2.6 | 3.4 | 5.75 | 0.0083 | 9.217 | ||

T14 | 0.275 | 2.6 | 3.4 | 5.75 | 0.0083 | 11.265 | ||

T15 | Muste and Patel [5] | Natural sand | 0.23 | 2.65 | 3.02 | 12.9 | 0.0103 | 6.744 |

T16 | 0.23 | 2.65 | 3.05 | 12.9 | 0.0103 | 6.811 | ||

T17 | 0.23 | 2.65 | 3.13 | 12.9 | 0.0105 | 6.856 | ||

T18 | Righetti and Romano [30] | Glass | 0.1 | 2.6 | 3.29 | 2.3 | 0.009 | 3.656 |

T19 | 0.2 | 2.6 | 3.97 | 2 | 0.0094 | 8.447 | ||

T20 | Muste et al. [37] | Natural sand | 0.23 | 2.65 | 4.2 | 2.1 | 0.0093 | 10.387 |

T21 | 0.23 | 2.65 | 4.2 | 2.1 | 0.0096 | 10.063 | ||

T22 | 0.23 | 2.65 | 4.2 | 2.1 | 0.0091 | 10.615 |

**Table 2.**Comparison result of the derived expression for the Tsallis entropy (Equation (17)), the Greimann et al. [12] model (Equation (19)), the Cheng [1] model (Equation (20)), the Pal et al. [14] model (Equation (21)), and the Shannon entropy-based model proposed by Kumbhakar et al. [15] with twenty-two collected laboratory datasets. In each row, the symbol *** corresponds to the minimum error for each case.

Test Number | $\mathbf{Fitting}\mathbf{Result}:\mathit{R}$ Value | ||||
---|---|---|---|---|---|

Tsallis Entropy-Based Model | Greimann et al. [12] Model | Cheng [1] Model | Pal et al. [14] Model | Shannon Entropy-Based Model [15] | |

T1 | 8.38 *** | 98.48 | 95.79 | 21.65 | 8.69 |

T2 | 9.85 *** | 96.16 | 94.04 | 22.88 | 9.98 |

T3 | 15.03 *** | 82.86 | 89.46 | 32.68 | 15.51 |

T4 | 17.60 *** | 69.98 | 87.53 | 42.75 | 17.96 |

T5 | 63.44 | 97.81 | 89.16 | 57.24 *** | 59.55 |

T6 | 41.52 *** | 75.94 | 53.04 | 59.21 | 44.08 |

T7 | 18.28 *** | 49.08 | 59.15 | 18.43 | 18.52 |

T8 | 47.92 | 98.23 | 87.37 | 54.29 | 42.39 *** |

T9 | 56.02 | 87.02 | 63.59 | 52.88 | 50.36 *** |

T10 | 36.91 | 47.88 | 44.83 | 44.42 | 33.70 *** |

T11 | 90.82 *** | 147.92 | 115.36 | 94.38 | 97.19 |

T12 | 68.09 | 38.92 *** | 48.46 | 58.20 | 60.81 |

T13 | 61.13 *** | 65.69 | 61.79 | 61.96 | 62.07 |

T14 | 54.16 *** | 117.68 | 57.86 | 69.26 | 55.78 |

T15 | 46.69 *** | 193.70 | 56.01 | 67.96 | 50.00 |

T16 | 38.45 *** | 90.44 | 47.19 | 50.31 | 40.94 |

T17 | 50.96 *** | 104.97 | 51.73 | 71.01 | 55.60 |

T18 | 25.17 | 65.54 | 64.08 | 30.31 | 21.68 *** |

T19 | 28.52 | 61.72 | 57.65 | 15.71 *** | 49.53 |

T20 | 30.85 | 52.87 | 59.14 | 26.30 *** | 26.64 |

T21 | 22.45 | 65.27 | 72.40 | 26.31 | 18.60 *** |

T22 | 19.80 | 65.23 | 72.08 | 27.08 | 12.23 *** |

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**MDPI and ACS Style**

Zhu, Z.; Yu, J.; Dou, J.; Peng, D.
An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy. *Entropy* **2019**, *21*, 522.
https://doi.org/10.3390/e21050522

**AMA Style**

Zhu Z, Yu J, Dou J, Peng D.
An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy. *Entropy*. 2019; 21(5):522.
https://doi.org/10.3390/e21050522

**Chicago/Turabian Style**

Zhu, Zhongfan, Jingshan Yu, Jie Dou, and Dingzhi Peng.
2019. "An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy" *Entropy* 21, no. 5: 522.
https://doi.org/10.3390/e21050522