 Critical shear stress vscritical velocity Victor M. Ponce10 March 2014

 Abstract. A general relation between shear stress and mean velocity in open-channel flow is derived. The relation is a function solely of the dimensionless Chezy friction factor f, which is equal to 1/8 of the Darcy-Weisbach friction factor fD. The derived formula may be used to relate critical shear stress τc  to critical velocity Vc .

1.  Relation between shear stress and mean velocity

The Chezy formula is the following (Chow, 1959):

 V  =  C R 1/2 S 1/2 (1)

in which: V = mean flow velocity, in m/s; R = hydraulic radius, in m; S = channel slope, in m/m, and C = Chezy coefficient, in m1/2/s. From Eq. 1:

 V 2  =  C 2 R S (2)

Multiplying and dividing by the gravitational acceleration g:

 C 2 V 2  =  ______  g R S                g (3)

Defining the dimensionless Chezy friction factor f:

 g f  =  _______             C 2 (4)

 1 V 2  =  ____  g R S              f (5)

The shear stress is defined as follows (Chow, 1959):

 τ  =  γ R S (6)

Combining Eqs. 5 and 6, the quadratic equation for shear stress is obtained:

 τ  =  ρ f V 2 (7)

in which ρ = γ/g = mass density of water.

2.  Dimensionless Chezy friction factor

It can be shown that the dimensionless Chezy friction factor f of Eq. 4 is equal to 1/8 of the Darcy-Weisbach friction factor fD. The latter varies typically in the range 0.016 ≤ fD ≤ 0.040 (Chow, 1959). Therefore, the typical range of variation of the dimensionless Chezy friction factor f is:  0.002 ≤ f ≤ 0.005.

Since ρ = 1000 N s2/m4, Eq. 7 can be expressed as follows:

• For the low value f = 0.002:

 τ  =  2 V 2 (8)

• For the average value f = 0.0035:

 τ  =  3.5 V 2 (9)

• For the high value f = 0.005:

 τ  =  5 V 2 (10)

in which τ is in N/m2 and V is in m/s.

3.  Shear stress versus mean velocity

Table 1 shows values of shear stress τ as a function of mean velocity V  for three values of friction factor: low, average, and high. The mean velocities vary between 1 and 6 m/s; the associated shear stresses vary from 2 to 180 N/m2.

 Table 1  Shear stress τ as a function of mean velocity V and dimensionless friction factor f. V(m/s) Lowf = 0.002 Averagef = 0.0035 Highf = 0.005 Shear stress τ (N/m2) for valueof f indicated above 1 2 3.5 5 2 8 14.0 20 3 18 31.5 45 4 32 56.0 80 5 50 87.5 125 6 72 126.0 180

Table 2 shows a similar table in U.S. Customary units.

 Table 2  Shear stress τ as a function of mean velocity V and dimensionless friction factor f. V(fps) Lowf = 0.002 Averagef = 0.0035 Highf = 0.005 Shear stress τ (lb/ft2) for valueof f indicated above 3 0.0349 0.0611 0.0873 6 0.1397 0.2444 0.3492 9 0.3143 0.5500 0.7857 12 0.5587 0.9778 1.3968 15 0.8730 1.5278 2.1825 18 1.2571 2.2000 3.1428

4.  Conclusions

A general relation between shear stress and mean velocity in open-channel flow is derived. This relation is referred to as the quadratic equation for shear stress. The relation is a function solely of the dimensionless Chezy friction factor f, which is equal to 1/8 of the Darcy-Weisbach friction factor fD. In practice, the derived formula may be used to relate critical shear stress τc  to critical velocity Vc .

References

Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, New York.

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