INITIATION OF MOTION BASED ON FROUDE NUMBER

1. Initiation of motion

1.1
Initiation of motion is the threshold at which bed-material sediment particles start to move in open-channel 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

               τo
τ* =  _______________  ≥  τ*c
          (γs - γ) ds

2.5
in which τo = bottom shear stress, γs = specific weight of sediment particles, γ = specific weight of water, ds = 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

τo =  f ρ v 2

3.3
in which ρ = mass density of water, f = a friction factor for open-channel flow, equal to 1/8 of the Darcy-Weisbach friction factor fD, and v = mean velocity.

4. The Froude criterion

4.1
The Froude number is:

4.2

              v
F =  _____________
           (gd)1/2

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

                 f F2
 ___________________________  ≥  τ*c
   [(γs / γ) - 1]  (ds / d)

5.3
In most cases, the ratio γs/γ = 2.65.

5.4
Therefore, the Froude criterion reduces to:

5.5

             1.65 τ*c (ds / d)
F  ≥  [  _____________________ ] 1/2
                        f

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

             0.066 (ds / d)
F  ≥  [  ___________________ ] 1/2
                        f

5.8
The friction factor varies normally in the range 0.002 ≤ f ≤ 0.005, which corresponds with Darcy-Weisbach's friction factor 0.016 ≤ fD ≤ 0.040.

5.9
We assume a midrange value of f = 0.0035, corresponding to fD = 0.028.

5.10
The Froude criterion reduces to:

5.11

F  ≥  [ 18.86 (ds / d) ] 1/2

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 ds / d = 0.0004, i.e., a particle size of 0.4 mm in 1 m of depth, this equation reduces to:

5.14

F  ≥  0.087

5.15
For ds / d = 0.0003, this equation reduces to:

5.16

F  ≥  0.075

5.17
As an example, assume: (a) a friction factor f = 0.005 (a high value), and (b) a relative particle diameter ds / d = 0.0005 (a high value).

5.18
The application of this equation [Eq. 6] results in:

5.19

F  ≥  0.081

5.19
In terms of Manning friction, for a hydraulically wide channel in SI units, the Froude criterion for initiation of motion is:

F  ≥  (0.082 / n) d1/6 (ds / d) 1/2

5.20
For example, for flow depth d = 1 m, mean diameter ds = 0.4 mm, and Manning's n = 0.020, the Froude criterion is:

F  ≥  0.082

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 bed-material sediment motion in open-channel 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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