1. Relation between shear stress and mean velocity
The Chezy formula is the following (Chow, 1959):
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:
Multiplying and dividing by the gravitational acceleration g:
Defining the dimensionless Chezy friction factor f:
The shear stress is defined as follows (Chow, 1959):
Combining Eqs. 5 and 6, the quadratic equation for shear stress is obtained:
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:
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 2 shows a similar table in U.S. Customary units.
A general relation between shear stress and mean velocity in open-channel flow
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
Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, New York.