♦ Initiation of motion ♦

Initiation of motion is the threshold at which sediment particles start to move in open-channel flow. This threshold is important in channel design in order to assure sediment movement and avoid clogging. The Froude number F = 0.08 has often been taken as a minimum value to prevent clogging. 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.

♦ Shields criterion ♦

The Shields criterion for initiation of motion relates the dimensionless shear stress τ_* with the boundary Reynolds number R_*, as shown in the above figure.¹ The solid curve separates motion above the curve from no motion below the curve. The Shields criterion is:

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

(1)

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.

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_*.

♦ Quadratic friction ♦

The quadratic friction formula is: ²

τo = ρ f v 2

(2)

in which ρ = mass density of water, f = friction factor, equal to 1/8 of the Darcy-Weisbach friction factor f_D, and v = mean velocity.

♦ Froude criterion ♦

The Froude number is:

v F = _________          (gd)1/2

(3)

in which g = gravitational acceleration and d = flow depth.

♦ Froude criterion for initiation of motion ♦

Replacing Eqs. 2 and 3 into 1:

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

(4)

In most cases, the ratio γ_s/γ = 2.65. Therefore, the Froude criterion reduces to:

1.65 τ*c (ds / d) F  ≥  { __________________ }1/2                           f

(5)

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:

0.066 (ds / d) F  ≥  { ________________ }1/2                          f

(6)

The friction factor varies normally in the range 0.002 ≤ f ≤ 0.005, which corresponds with Darcy-Weisbach's friction factor 0.016 ≤ f_D ≤ 0.040. We assume a midrange value of f = 0.0035, corresponding to f_D = 0.028. The Froude criterion reduces to:

F  ≥  [ 18.86 (ds / d) ]1/2

(7)

which reduces to:

F  ≥  4.342 (ds / d)1/2

(8)

For a given particle diameter, relative to the flow depth, Eq. 8 states the Froude number that must be exceeded to assure initiation of motion. For instance, for d_s / d = 0.0004, i.e., a particle size of 0.4 mm in 1 m of depth, Eq. 8 reduces to:

F  ≥  0.087

(9)

For d_s / d = 0.0003, Eq. 8 reduces to:

F  ≥  0.075

(10)

♦ 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). The application of Eq. 6 results in:

F  ≥  0.081

(11)

♦ Application with Manning's n ♦

The relationship between friction factor f and Manning's n is:

g n2 f = ___________          k2 R1/3

(12)

in which g = gravitational acceleration, n = Manning's n, R = hydraulic radius, and k = a constant equal to 1 in SI units and 1.486 in U. S. Customary units.

For a hydraulically wide channel: R ≅ d. Replacing Eq. 12 into Eq. 6, the expression for Froude number in SI units is:

0.082 d1/6 (ds / d)1/2 F  ≥  _______________________                           n

(13)

For example, for d = 1 m, d_s = 0.4 mm = 0.0004 m, and n = 0.020:

F  ≥  0.082

(14)

In U.S. Customary units, the expression for Froude number is:

0.067 d1/6 (ds / d)1/2 F  ≥  _______________________                           n

(15)

in which d and d_s are given in ft.

For example, for d = 1 m = 3.28 ft, d_s = 0.4 mm = 0.0004 m = 0.001312 ft, and n = 0.020:

F  ≥  0.082

(16)

♦ Summary ♦

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. 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.