# Experimental Analysis of Incipient Motion for Uniform and Graded Sediments

^{1}

^{2}

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^{5}

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## Abstract

**:**

_{c}*) was higher for the graded sediment than for the three finer uniform sediment sizes. The finer fractions of the mixture have a higher particle Froude number than their corresponding uniform sediment value, while the coarser fractions of the mixture showed a lower stability than their corresponding uniform sediment value. Results demonstrated that the reduction in the particle Froude number was more evident in lower relative roughness conditions. The current study provides a clearer insight into the interaction between initial sediment transport and flow characteristic, especially particle Froude number for incipient motion in natural rivers where stream beds have different gravel size distribution.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Effective Variables

_{c}), water depth (Y), median diameter of particles (d

_{50}), bed slope (S), fluid density (${\rho}_{w}$), dynamic viscosity ($\mu $), particle submerged density (${\rho}_{s}-{\rho}_{w}$) and gravitational acceleration (g) as follow:

_{1}is a function to be determined experimentally, F

^{*}is the particle densimetric Froude number or particle stability parameter, d

_{50}/Y is relative roughness (or particle’s size parameter), and Re is the Reynolds number. For high Re number fully developed turbulent flow (typical of river flows), the Re parameter can be neglected. Comparing Equation (2) with Equation (1), it can be seen that there is an additional dependence on S in our analysis, which we have considered in our experiments.

#### 2.2. Flume Set-Up

_{50}prior to each experiment; the bed was carefully leveled using screed-board riding on the flume rails. To prevent local scour and backwater, fixed roughened bed sections with the same size in a movable bed each experiment, were placed for 4 m near the inlet and 2.8 m near the outlet of the flume. The length of the movable bed was 5 m and a trap 0.5 m long and 0.3 m wide was placed at the end of the movable bed. More details on the experimental runs are presented in Khosravi et al. [29].

_{50}and mean size of 12.5 mm and 13.57 mm, respectively. An angular uniform sediment with mean size of 10.35 mm was also used (Table 1). The rounded sediment had a specific gravity of 2.567 and the angular sediment had a specific gravity of 2.4.

#### 2.3. Experimental Procedure

_{*}is the shear velocity, d is the sediment diameter and $\upsilon $ is the kinematic viscosity of water ($\upsilon =\mu /\rho $ where μ is water dynamic viscosity). The shear velocity was calculated as ${V}_{*}=\sqrt{\tau /\rho}$ with shear stress obtained as $\tau =\rho g{R}_{b}S$ where R

_{b}is the hydraulic radius which is calculated based on Shvidchenko and Pender [9] where the side wall effect is considered. The critical Shield stress for incipient motion ($\tau *$) was calculated as follows:

## 3. Results

#### 3.1. Determination of the Most Effective Factors on Sediment Transport

#### 3.2. Critical Shear Velocity

_{c}; for the 10.35 mm particles, an increase in S from 0.01 to 0.03 induced a V*c increase of 20.75%; for the 14 mm sediment, an increase in S from 0.015 to 0.03 led to a 6.2% increase in V*c; and for the 20.7 mm a slope increasing from 0.03 to 0.035 resulted in V*c decreasing by 18.35%. For the angular 10.35 mm size, an increase of slope from 0.015 to 0.03 led to the critical shear velocity increase by 9.5%; and, finally, Vc increased by 3.4% for the graded bed sediments and with slopes ranging from 0.015 to 0.03.

#### 3.3. Ratio of Critical Flow Velocity to Critical Shear Velocity (V_{c}/V_{c}*)

_{c}/V

_{c}* with increasing grain Reynolds number for fixed slopes of 0.015 and 0.03 (Figure 3). The increase in Re* corresponds to a sediment size increase from 5.17 to 14 mm. However, with a further increase in Re* (and an increase in sediment size from 14 to 20.7 mm), the ratio of V

_{c}/V

_{c}* increased. For a given grain size sediment and incipient motion condition, an increase in bed slope led to a reduction in critical flow velocity (Figure 3). In addition, with increasing bed slope (from 0.015 to 0.03), the value of V

_{c}/V

_{c}* was reduced by 12.4% and 17% for bed sediment sizes of 10.35 and 14 mm, respectively.

#### 3.4. Critical Shields Stress for Graded and Uniform Sediment

#### 3.5. Particle Densimetric Froude Number

## 4. Discussion

#### 4.1. Effect of Flow Velocity, Shear Velocity and Shield Stress on Incipient Motion Condition

_{c}/V

_{c}* would imply a non-constant roughness value. The relationship between the particle Froude number and bed slope showed that for a given bed sediment, the steeper the bed slope, the lower the particle Froude number. Moreover, for a given bed sediment diameter, the higher the relative roughness, the lower the particle Froude number.

#### 4.2. Comparison between Uniform Sediment and Counterpart Sediment Fractions of Graded Bed

_{50}of 12.5 mm) has a lower mean size but higher critical flow velocity than the 14 mm material. This is due to the fact that the coarse fraction of 20.7 mm caused graded bed sediment to be more stable, thus showing a higher critical flow velocity for incipient motion in comparison to rounded bed sediment, even with a higher sediment diameter (e.g., sediment of 14 mm).

#### 4.3. Comparison with Previous Results

_{50}/Y between the current study and the literature results, probably due to differences in bed sediment diameter and bed slope, as coarser sediment was used for the current study. However, the F* values for both sediments were very close to each other (Figure 10).

_{50}but different mixtures of sizes, and also different uniform particles with the same densities but different diameters, to allow comparison of the effects on flow hydraulic and incipient motion of sediment.

## 5. Conclusions

_{c}/V

_{c}* against the grain Reynolds number suggested that a graded bed sediment on a given bed slope has a higher V

_{c}/V

_{c}* than uniform sediments of 5.17, 10.35, and 14 mm. In incipient motion condition, the finer the fraction, the higher the Shields stress and relative submergence. Particle Froude number analysis showed that the greater the bed slope and relative roughness, the lower the particle Froude number will be for a given bed sediment. Comparison between uniform sediment and its counterpart fraction of the same size revealed that finer fractions in graded sediment are more stable than their uniform counterparts; however, coarser fractions in graded sediment are less stable than their uniform counterparts. The reduction in the particle Froude number was more marked in lower relative roughness.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Experimental flume set-up (not to scale) [29].

**Figure 3.**Relationship between V

_{c}/V

_{c}* and Re* for different bed sediments in fixed slopes of 0.015 and 0.03.

**Figure 4.**Critical Shields stress against of sediment diameter in bed slope of 0.015 (

**a**) and 0.03 (

**b**).

**Figure 5.**Critical Shields stress as a function of bed slope (

**a**) and relative submergence (

**b**) for each fraction in graded sediment.

**Figure 6.**Comparison between particle Froude number for uniform rounded and angular sediments of 10.35 mm.

**Figure 7.**Comparison between the particle Froude number of the uniform and graded fractions corresponding to 5.17, 10.35, and 14 mm in a bed slope of 0.015.

**Figure 8.**Comparison between particle Froude number for sediment sizes of 10.35, 14, and 20.7 mm in the fractions of the graded and uniform beds with a slope of 0.03.

**Figure 9.**Particle Froude number as a function of bed slope (

**a**) and relative roughness (

**b**) for uniform and graded bed sediment.

Sediment Type | Fraction, mm | d_{50} | σ_{g} | Sediment Size (d), mm | Density, kg/m^{3} | Porosity | Grain Shape |
---|---|---|---|---|---|---|---|

Fine gravel | 4.75–5.6 | - | - | 5.17 | 2391 | 0.4 | Rounded |

Lower Medium Gravel | 9.5–11.2 | - | - | 10.35 | 2375 | 0.4 | Rounded |

Lower Medium Gravel | 9.5–11.2 | - | - | 10.35 | 2400 | 0.45 | Angular |

Higher Medium Gravel | 13–15 | - | - | 14 | 2900 | 0.45 | Rounded |

Coarse gravel | 19–22.4 | - | - | 20.7 | 2552 | 0.43 | Rounded |

Graded (mixture) | 4.75–22.4 | 12.5 | 1.7 | 13.57 | 2567 | 0.37 | Rounded |

Sediment Size (mm) | Slope (m/m) | Flow Depth (cm) | Mean Velocity (m/s) | Froude Number | q^{*}(*10^{−5}) |
---|---|---|---|---|---|

5.17 | 0.005, 0.0075, 0.01, 0.015 | 9, 6, 4, 3.5 | 0.92, 0.83, 0.75, 0.68 | 0.97–1.6 | 6, 8, 3, 53 |

10.35-R | 0.01, 0.015, 0.03 | 8, 7, 4 | 1.11, 1.10, 1.05 | 1.2–1.65 | 31, 7, 50 |

10.35-A | 0.015– 0.0175, 0.03 | 8, 7, 4 | 1.3, 1.2, 1.07 | 1.28–1.71 | 13, 28, 52 |

14 | 0.015, 0.02, 0.03 | 8.5, 6.5, 4.5 | 1.30, 1.19, 1.10 | 1.4–1.8 | 18, 6, 17 |

20.7 | 0.03, 0.0325, 0.035 | 8, 6, 5 | 1.66, 1.42, 1.35 | 1.8–1.93 | 25, 33, 15 |

Graded | 0.015, 0.02, 0.03 | 10, 7, 5 | 1.51, 1.25, 1.25 | 1.52 | 13, 19, 29 |

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**MDPI and ACS Style**

Khosravi, K.; Chegini, A.H.N.; Mao, L.; Rodriguez, J.F.; Saco, P.M.; Binns, A.D.
Experimental Analysis of Incipient Motion for Uniform and Graded Sediments. *Water* **2021**, *13*, 1874.
https://doi.org/10.3390/w13131874

**AMA Style**

Khosravi K, Chegini AHN, Mao L, Rodriguez JF, Saco PM, Binns AD.
Experimental Analysis of Incipient Motion for Uniform and Graded Sediments. *Water*. 2021; 13(13):1874.
https://doi.org/10.3390/w13131874

**Chicago/Turabian Style**

Khosravi, Khabat, Amir H. N. Chegini, Luca Mao, Jose F. Rodriguez, Patricia M. Saco, and Andrew D. Binns.
2021. "Experimental Analysis of Incipient Motion for Uniform and Graded Sediments" *Water* 13, no. 13: 1874.
https://doi.org/10.3390/w13131874