INITIATION OF MOTION BASED ON FROUDE NUMBER
1. Initiation of motion 1.1 Initiation of motion is the threshold at which bedmaterial sediment particles start to move in openchannel flow. 1.2 This threshold is important in channel design in order to assure sediment movement and avoid clogging. 1.3 The Froude number F = 0.08 has often been taken as a minimum value to prevent clogging. 1.4 Herein we derive a general Froude criterion for initiation of motion, confirming the validity of F = 0.08 for a wide range of practical applications. 2. The Shields criterion 2.1 The Shields criterion for initiation of motion relates the dimensionless shear stress τ_{*} with the boundary Reynolds number R_{*}, as shown in this figure. 2.2 The solid curve separates motion above the curve from no motion below the curve. 2.3 The Shields criterion is: 2.4
2.5 in which τ_{o} = bottom shear stress, γ_{s} = specific weight of sediment particles, γ = specific weight of water, d_{s} = sediment particle diameter, and τ_{*c} = dimensionless critical shear stress. 2.6 For practical applications, the dimensionless critical shear stress τ_{*c} can be taken as a constant for a wide range of boundary Reynolds numbers R_{*}. 3. The quadratic friction formula 3.1 The quadratic friction formula of fluid mechanics is: 3.2
3.3 in which ρ = mass density of water, f = a friction factor for openchannel flow, equal to 1/8 of the DarcyWeisbach friction factor f_{D}, and v = mean velocity. 4. The Froude criterion 4.1 The Froude number is: 4.2
4.3 in which g = gravitational acceleration and d = flow depth. 5. The Froude criterion for initiation of motion 5.1 Replacing this equation [Eq. 2] and this equation [Eq. 3] into this equation [Eq. 1]: 5.2
5.3 In most cases, the ratio γ_{s}/γ = 2.65. 5.4 Therefore, the Froude criterion reduces to: 5.5
5.6 As a convenient approximation, the Shields curve suggests a value of critical dimensionless shear stress τ_{*c} = 0.04 for a wide range of boundary Reynolds numbers. Therefore: 5.7
5.8 The friction factor varies normally in the range 0.002 ≤ f ≤ 0.005, which corresponds with DarcyWeisbach's friction factor 0.016 ≤ f_{D} ≤ 0.040. 5.9 We assume a midrange value of f = 0.0035, corresponding to f_{D} = 0.028. 5.10 The Froude criterion reduces to: 5.11
5.12 For a given particle diameter, relative to the flow depth, this equation states the Froude number that must be exceeded to assure initiation of motion. 5.13 For instance, for d_{s} / d = 0.0004, i.e., a particle size of 0.4 mm in 1 m of depth, this equation reduces to: 5.14
5.15 For d_{s} / d = 0.0003, this equation reduces to: 5.16
5.17 As an example, assume: (a) a friction factor f = 0.005 (a high value), and (b) a relative particle diameter d_{s} / d = 0.0005 (a high value). 5.18 The application of this equation [Eq. 6] results in: 5.19
5.19 In terms of Manning friction, for a hydraulically wide channel in SI units, the Froude criterion for initiation of motion is:
5.20 For example, for flow depth d = 1 m, mean diameter d_{s} = 0.4 mm, and Manning's n = 0.020, the Froude criterion is:
6. Summary 6.1 The Shields criterion is expressed in terms of the Froude number, enabling the calculation of the minimum Froude number for initiation of bedmaterial sediment motion in openchannel flow. 6.2 Given critical dimensionless shear stress, friction factor, and relative particle diameter, the developed relation confirms the validity of Froude number F = 0.08 as a convenient descriptor of initiation of motion in most cases of practical interest. Narrator: Victor M. Ponce Music: Fernando Oñate Graphics: Flor Perez Editor: Aleksandr Gostomelskiy Credits: initiation.sdsu.edu
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