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 m^1/2/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 f_D. The latter varies typically in the range 0.016 ≤ f_D ≤ 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) Low f = 0.002 Average f = 0.0035 High f = 0.005 Shear stress τ (N/m2) for value of 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) Low f = 0.002 Average f = 0.0035 High f = 0.005 Shear stress τ (lb/ft2) for value of 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 f_D. In practice, the derived formula may be used to relate critical shear stress τ_c  to critical velocity V_c .

References

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