 Froude number for initiation of motioncalculated online Victor M. Ponce and Janaina Da Silva19 April 2018

 Abstract. The classical Shields criterion of sedimentation engineering is expressed in terms of the Froude number and related mean velocity required for initiation of motion in a sand-bed channel. To solve the problem exactly, an iterative algorithm is implemented in the calculator ONLINE SHIELDS VELOCITY, accessible online at http://onlinecalc.sdsu.edu/onlineshieldsvelocity.php

1.  INTRODUCTION

The principle of initiation of motion for flow in a sand-bed channel was pioneered by Shields (American Society of Civil Engineers, 1975; p. 96). The Shields curve is the threshold at which bed-material sediment particles start to move (Fig. 1). This threshold is important in channel design in order to ensure sediment movement and avoid clogging. In this article we express the Shields criterion in terms of the Froude number and related mean velocity required for initiation of motion in a sand-bed channel. To round up the experience, we develop an iterative algorithm to calculate these values online.

2.  THE SHIELDS CRITERION

The Shields criterion for initiation of motion relates the dimensionless shear stress τ* with the boundary Reynolds number R* as shown in 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 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. Fig. 1  The Shields curve for initiation of motion (American Society of Civil Engineers, 1975).

3.  THE FROUDE CRITERION

The Froude number is:

 V F = __________         (g D )1/2 (2)

in which g = gravitational acceleration and D = hydraulic depth.

The quadratic friction formula in open-channel flow is (Ponce, 2014):

 τo = ρ f V 2 (3)

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

Replacing Eqs. 2 and 3 into 1:

 f D F 2 ___________________  ≥  τ*c       [ (γs /γ ) - 1 ] ds (4)

In most cases of practical interest, the ratio γs /γ = 2.65. In this case, the Froude number for initiation of motion is:

 1.65 τ*c (ds / D ) F  ≥  { __________________ } 1/2                           f (5)

As a first approximation, the Shields curve suggests an approximately constant value of critical dimensionless shear stress τ*c = 0.04, applicable for a wide range of boundary Reynolds numbers (Fig. 1). Therefore, Eq. 5 reduces to:

 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 friction factors 0.016 ≤ fD ≤ 0.040. To illustrate, we assume a midrange value f = 0.0035. Therefore, the applicable Froude criterion is:

 F  ≥  4.342 (ds / D )1/2 (7)

For a given particle diameter, relative to hydraulic depth, Eq. 7 states the Froude number that must be exceeded to assure initiation of motion. For instance, for ds = 0.4 mm and hydraulic depth D = 1 m, i.e., (ds / D ) = 0.0004, Eq. 7 reduces to:

 F  ≥  0.087 (8)

Combining Eqs. 2 and 6, the related mean velocity is:

 0.066 g ds V  ≥  [ _____________ ] 1/2                       f (9)

Thus: V = 0.27 m/s.

4.  APPLICATION WITH MANNINGS n

The relationship between friction factor f and Manning's n is (Ponce, 2014):

 g n 2 f = ___________         k 2 R 1/3 (10)

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: RD. Replacing Eq. 10 into Eq. 6, the expression for Froude number in SI units is:

 0.082 D 1/6 (ds / D )1/2 F  ≥  _______________________                           n (11)

For example, given: D = 1 m, ds = 0.4 mm = 0.0004 m, and n = 0.019:

 F  ≥  0.086 (12)

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

 0.067 D 1/6 (ds / D )1/2 F  ≥  ________________________                           n (13)

in which D and ds are given in ft.

For example, given: D = 3.28 ft, ds = 0.4 mm = 0.001312 ft, and n = 0.019:

 F  ≥  0.086 (14)

5.  ITERATIVE ALGORITHM

A more precise calculation of Froude number and related mean velocity may be obtained by using (in Eq. 5) a dimensionless critical shear stress obtained from the actual Shields curve, in lieu of the assumed constant value of 0.04. This procedure, however, requires an iteration. The following algorithm is suggested.

 Algorithm used in ONLINE SHIELDS VELOCITY Assume R* Use Fig. 1 to find τ*c Using Eq. 1, calculate τo Use Eq. 3 to calculate the shear velocity: U* = (τo /ρ)1/2 Calculate the new value of R* = (U* ds / ν), in which ν = kinematic viscosity, a function of temperature. Stop if the new value of R* is close to that assumed in Step 1 (within a certain small tolerance), and use the last value of τ*c (calculated in Step 2) in Eq. 5; Otherwise, return to Step 1 and use the new value of R* (calculated in Step 5) as the assumed value and continue the iteration.

Example.

Given the following channel data:  particle diameter ds = 0.4 mm, hydraulic depth D = 1 m, dimensionless Darcy-Weisbach friction factor f = 0.0035, and water temperature T = 20°C. Calculate the Froude number and related mean velocity at the threshold of initiation of motion using ONLINE SHIELDS VELOCITY. ONLINE CALCULATION. Using the iterative algorithm embedded in ONLINE SHIELDS VELOCITY, the Froude number is F = 0.081 and the mean velocity V = 0.25 m/s. Note that these results are slightly more accurate than those obtained in the approximate direct calculation using Eqs. 6 and 9.

6.  SUMMARY

The classical Shields criterion of sedimentation engineering is expressed in terms of the Froude number and related mean velocity required for initiation of motion in a sand-bed channel. To solve the problem exactly, an iterative algorithm is implemented in the calculator ONLINE SHIELDS VELOCITY, accessible online at http://onlinecalc.sdsu.edu/onlineshieldsvelocity.php

REFERENCES

American Society of Civil Engineers. 1975. Sedimentation Engineering. ASCE Manuals and Reports on Engineering Practice No. 54.

Ponce, V. M. 2014. Fundamentals of open-channel hydraulics. Online textbook.

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