# Evaluation of Incipient Motion of Sand Particles by Different Indirect Methods in Erosion Function Apparatus

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Threshold Shear Stress in Erosion Function Apparatus (EFA)

#### 1.2. Threshold Shear Stress in Other Experimental Facilities

#### 1.3. Importance of Resistance Coefficients in EFA and Other Experimental Facilities

## 2. Materials and Methods

#### 2.1. Experiment and Procedure

^{3}, was used as the test material. Cohesionless sand was packed in the sand box as the sediment bed of the conduit.

#### 2.2. Indirect Methods to Estimate Shear Velocity

#### 2.2.1. Log-Law Method

#### 2.2.2. Reynolds Shear Stress Method

#### 2.2.3. Turbulence Intensity Method

## 3. Results and Discussion

#### 3.1. Streamwise Flow Velocity in Closed Conduit

#### 3.2. Universal Characteristics of Turbulence Flow in Closed Conduit

#### 3.2.1. Logarithmic Distributions of Flow Profiles

#### 3.2.2. Reynolds Shear Stress Distribution of Flow Profiles

#### 3.2.3. Turbulence Intensity Distribution of Flow Profiles

#### 3.3. Shear Velocity and Bed Shear Stress of Flow

#### 3.4. Determination of Critical Shear Velocity

#### 3.5. Determination of Dimensionless Critical Bed Shear Stress

^{−4}–1.01 × 10

^{−3}m and 6.67–16.48, respectively. It means that the turbulent flow is in the range of transition which is close to the hydraulically smooth range [47]. Since the upstream edge of the sand zone is physically smooth, the downstream component of the fluid force exerted on the boundary can result only from the action of the viscous shear stresses, because the pressure forces do not have any component in the direction of flow [48]. Thus, the boundary shear stress acting on the upstream edge of the sand zone is the only component of the viscous shear stress which may cause higher values of dimensionless critical bed shear stress.

#### 3.6. Determination of Resistance Coefficients

_{m}, and the water depth, d, can be calculated as follows:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Vertical distributions of primary mean velocity with log-law: (

**a**) sand zone; and (

**b**) smooth zone.

**Figure 5.**Vertical distributions of turbulence intensities in x-direction: (

**a**) sand zone; and (

**b**) smooth zone.

**Figure 6.**Vertical distributions of turbulence intensities in y-direction: (

**a**) sand zone; and (

**b**) smooth zone.

**Figure 7.**Shear velocity estimated by different indirect methods: (

**a**) sand zone; and (

**b**) smooth zone.

**Figure 8.**Bed shear stress estimated by different indirect methods in the sand zone and smooth zone. The error bars represent the 95% confidence intervals (±two standard deviations).

**Figure 9.**Estimation of critical shear velocity from shear velocity calculated by different indirect methods at sand zone: (

**a**) regression analysis of universal logarithmic law; (

**b**) Reynolds shear stress; (

**c**) turbulence intensity considering x-component of flow; and (

**d**) turbulence intensity considering y-component of flow.

**Figure 10.**Estimation of critical shear velocity from shear velocity calculated by different indirect methods at upstream edge of sand zone: (

**a**) curve fitting to universal logarithmic law; (

**b**) Reynolds shear stress; (

**c**) turbulence intensity considering x-component of flow; and (

**d**) turbulence intensity considering y-component of flow.

**Figure 12.**The Manning roughness coefficient estimated by different indirect methods in the sand zone and smooth zone.

**Figure 13.**The Darcy–Weisbach friction coefficient estimated by different indirect methods in the sand zone and smooth zone.

Flow Rate, Q lit/min | Reynolds Number, Re_{Q} | Froude Number, F_{r} | Sediment Transport Rate, $\dot{\mathit{\epsilon}}$ m ^{3}/s/m |
---|---|---|---|

79 | 15,419 | 0.37600 | 1.188 × 10^{−8} |

80 | 16,028 | 0.38076 | 3.053 × 10^{−8} |

81 | 15,395 | 0.38552 | 3.296 × 10^{−8} |

83 | 16,199 | 0.39504 | 2.768 × 10^{−8} |

84 | 15,965 | 0.39980 | 4.551 × 10^{−8} |

86 | 16,346 | 0.40932 | 6.506 × 10^{−8} |

88 | 16,280 | 0.41883 | 2.229 × 10^{−7} |

89 | 16,916 | 0.42359 | 2.341 × 10^{−7} |

Streamwise Flow Velocity (m/s) at y = 0.1h | Streamwise Flow Velocity (m/s) at y = 0.3h | Streamwise Flow Velocity (m/s) at y = 0.5h | |||||||
---|---|---|---|---|---|---|---|---|---|

Q | Sand Zone | Smooth Zone | % Increase | Sand Zone | Smooth Zone | % Increase | Sand Zone | Smooth Zone | % Increase |

79 | 0.26551 | 0.27041 | 1.81262 | 0.31508 | 0.31384 | −0.39417 | 0.33174 | 0.33291 | 0.35159 |

80 | 0.26749 | 0.26908 | 0.59237 | 0.32124 | 0.32613 | 1.49949 | 0.34062 | 0.34212 | 0.43965 |

81 | 0.27569 | 0.28101 | 1.89063 | 0.32900 | 0.33291 | 1.17555 | 0.34084 | 0.34540 | 1.32219 |

83 | 0.28509 | 0.28829 | 1.11192 | 0.32967 | 0.33512 | 1.62635 | 0.34966 | 0.35272 | 0.86715 |

84 | 0.28174 | 0.28187 | 0.04605 | 0.33515 | 0.34073 | 1.63793 | 0.34877 | 0.35135 | 0.73566 |

86 | 0.28345 | 0.28691 | 1.20682 | 0.33945 | 0.34327 | 1.11349 | 0.35914 | 0.36232 | 0.87720 |

88 | 0.28540 | 0.29273 | 2.50730 | 0.34715 | 0.35344 | 1.77765 | 0.36852 | 0.37292 | 1.17770 |

89 | 0.30106 | 0.30128 | 0.07154 | 0.35669 | 0.35916 | 0.68581 | 0.36971 | 0.37168 | 0.53083 |

Average | 0.28068 | 0.28395 | 0.33418 | 0.33807 | 0.35112 | 0.35393 | |||

St. dev. | 0.01058 | 0.01014 | 0.01267 | 0.01360 | 0.01281 | 0.01327 |

Zone | Q (lit/min) | Shear Velocity (m/s) | Bed Shear Stress (pa) | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{u}}_{*\mathit{l}}$ | ${\mathit{u}}_{*\mathit{r}}$ | ${\mathit{u}}_{*\mathit{t}\mathit{x}}$ | ${\mathit{u}}_{*\mathit{t}\mathit{y}}$ | ${\mathit{\tau}}_{*\mathit{l}}$ | ${\mathit{\tau}}_{*\mathit{r}}$ | ${\mathit{\tau}}_{*\mathit{t}\mathit{x}}$ | ${\mathit{\tau}}_{*\mathit{t}\mathit{y}}$ | ||

Sand zone | 79 | 0.01583 | 0.01703 | 0.01701 | 0.01586 | 0.25059 | 0.29002 | 0.28934 | 0.25154 |

80 | 0.01845 | 0.01761 | 0.01755 | 0.01414 | 0.34040 | 0.31011 | 0.30800 | 0.19994 | |

81 | 0.01566 | 0.01638 | 0.01745 | 0.01477 | 0.24524 | 0.26830 | 0.30450 | 0.21815 | |

83 | 0.01652 | 0.01797 | 0.01812 | 0.01521 | 0.27291 | 0.32292 | 0.32833 | 0.23134 | |

84 | 0.01919 | 0.01809 | 0.01842 | 0.01393 | 0.36826 | 0.32725 | 0.33930 | 0.19404 | |

86 | 0.02034 | 0.01662 | 0.01871 | 0.01523 | 0.41372 | 0.27622 | 0.35006 | 0.23195 | |

88 | 0.02206 | 0.01612 | 0.01960 | 0.01501 | 0.48664 | 0.25985 | 0.38416 | 0.22530 | |

89 | 0.02214 | 0.01719 | 0.01910 | 0.01536 | 0.49018 | 0.29550 | 0.36481 | 0.23593 | |

Average | 0.01877 | 0.01713 | 0.01825 | 0.01494 | 0.35849 | 0.29377 | 0.33356 | 0.22353 | |

St. dev. | 0.00246 | 0.00068 | 0.00083 | 0.00060 | 0.09289 | 0.02335 | 0.03025 | 0.01778 | |

${\mathit{u}}_{*\mathit{f}}$ | ${\mathit{u}}_{*\mathit{r}}$ | ${\mathit{u}}_{*\mathit{t}\mathit{x}}$ | ${\mathit{u}}_{*\mathit{t}\mathit{y}}$ | ${\mathit{\tau}}_{*\mathit{f}}$ | ${\mathit{\tau}}_{*\mathit{r}}$ | ${\mathit{\tau}}_{*\mathit{t}\mathit{x}}$ | ${\mathit{\tau}}_{*\mathit{t}\mathit{y}}$ | ||

Smooth zone | 79 | 0.01708 | 0.01703 | 0.01755 | 0.01446 | 0.29173 | 0.29002 | 0.30800 | 0.20909 |

80 | 0.01781 | 0.01680 | 0.01687 | 0.01545 | 0.31720 | 0.28224 | 0.28460 | 0.23870 | |

81 | 0.01802 | 0.01553 | 0.01667 | 0.01542 | 0.32472 | 0.24118 | 0.27789 | 0.23778 | |

83 | 0.01815 | 0.01590 | 0.01767 | 0.01531 | 0.32942 | 0.25281 | 0.31223 | 0.23440 | |

84 | 0.01831 | 0.01531 | 0.01787 | 0.01526 | 0.33526 | 0.23440 | 0.31934 | 0.23287 | |

86 | 0.01853 | 0.01810 | 0.01850 | 0.01617 | 0.34336 | 0.32761 | 0.34225 | 0.26147 | |

88 | 0.01872 | 0.01816 | 0.01950 | 0.01488 | 0.35044 | 0.32979 | 0.38025 | 0.22141 | |

89 | 0.01878 | 0.01830 | 0.01970 | 0.01630 | 0.35269 | 0.33489 | 0.38809 | 0.26569 | |

Average | 0.01818 | 0.01689 | 0.01804 | 0.01541 | 0.33060 | 0.28662 | 0.32658 | 0.23768 | |

St. dev. | 0.00052 | 0.00114 | 0.00105 | 0.00057 | 0.01868 | 0.03847 | 0.03816 | 0.01756 |

Zone | Critical Shear Velocity (m/s) | ||||
---|---|---|---|---|---|

${\mathit{u}}_{*\mathit{c}\mathit{l}}$ | ${\mathit{u}}_{*\mathit{c}\mathit{f}}$ | ${\mathit{u}}_{*\mathit{c}\mathit{r}}$ | ${\mathit{u}}_{*\mathit{c}\mathit{t}\mathit{x}}$ | ${\mathit{u}}_{*\mathit{c}\mathit{t}\mathit{y}}$ | |

Sand zone | 0.01599 | - | 0.01630 | 0.01749 | 0.01337 |

Smooth zone | - | 0.01816 | 0.01662 | 0.01753 | 0.01497 |

Zone | Dimensionless Critical Bed Shear Stress | ||||
---|---|---|---|---|---|

${\mathit{\tau}}_{*\mathit{c}\mathit{l}}$ | ${\mathit{\tau}}_{*\mathit{c}\mathit{f}}$ | ${\mathit{\tau}}_{*\mathit{c}\mathit{r}}$ | ${\mathit{\tau}}_{*\mathit{c}\mathit{t}\mathit{x}}$ | ${\mathit{\tau}}_{*\mathit{c}\mathit{t}\mathit{y}}$ | |

Sand zone | 0.02740 | - | 0.02847 | 0.03277 | 0.01949 |

Smooth zone | - | 0.03533 | 0.02960 | 0.03294 | 0.02403 |

**Table 6.**The estimated Manning roughness coefficient and friction coefficient for the Darcy–Weisbach formula considering different indirect methods.

Zone | Manning Roughness Coefficient | ||||
---|---|---|---|---|---|

${\mathit{n}}_{\mathit{l}}$ | ${\mathit{n}}_{\mathit{f}}$ | ${\mathit{n}}_{\mathit{r}}$ | ${\mathit{n}}_{\mathit{c}\mathit{t}\mathit{x}}$ | ${\mathit{n}}_{\mathit{c}\mathit{t}\mathit{y}}$ | |

Sand zone | 0.01110 | - | 0.01132 | 0.01214 | 0.00936 |

Smooth zone | - | 0.01260 | 0.01154 | 0.01217 | 0.01040 |

Friction Coefficient for Darcy–Weisbach Formula | |||||

${\mathit{f}}_{\mathit{l}}$ | ${\mathit{f}}_{\mathit{f}}$ | ${\mathit{f}}_{\mathit{r}}$ | ${\mathit{f}}_{\mathit{c}\mathit{t}\mathit{x}}$ | ${\mathit{f}}_{\mathit{c}\mathit{t}\mathit{y}}$ | |

Sand zone | 0.02625 | - | 0.02727 | 0.03139 | 0.01867 |

Smooth zone | - | 0.03384 | 0.02836 | 0.03155 | 0.02302 |

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**MDPI and ACS Style**

Jewel, A.; Fujisawa, K.; Murakami, A. Evaluation of Incipient Motion of Sand Particles by Different Indirect Methods in Erosion Function Apparatus. *Water* **2021**, *13*, 1118.
https://doi.org/10.3390/w13081118

**AMA Style**

Jewel A, Fujisawa K, Murakami A. Evaluation of Incipient Motion of Sand Particles by Different Indirect Methods in Erosion Function Apparatus. *Water*. 2021; 13(8):1118.
https://doi.org/10.3390/w13081118

**Chicago/Turabian Style**

Jewel, Arif, Kazunori Fujisawa, and Akira Murakami. 2021. "Evaluation of Incipient Motion of Sand Particles by Different Indirect Methods in Erosion Function Apparatus" *Water* 13, no. 8: 1118.
https://doi.org/10.3390/w13081118