fluid_mechanics

548 Chapter 10 ■ Open-Channel Flow
τ = 0
y 1 = y 2
Control surface
F 1
V 1
(1)
V 2 = V 1
θ
τ P
w
y 2
F 2
θ
= γ A
sin θ
(2)
x
F I G U R E 10.10
Control volume for uniform flow
in an open channel.
p(y)
y 1
y
y 2 = y 1
Equal pressure
distributions
where F 1 and F 2 are the hydrostatic pressure forces across either end of the control volume, as shown
by the figure in the margin. Because the flow is at a uniform depth 1y 1 y 2 2, it follows that F 1 F 2
so that these two forces do not contribute to the force balance. The term wsin u is the component of
the fluid weight that acts down the slope, and t w P/ is the shear force on the fluid, acting up the slope
as a result of the interaction of the water and the channel’s wetted perimeter. Thus, Eq. 10.15 becomes
t w w sin u
P/
w S 0
P/
where we have used the approximation that sin u tan u S 0 , since the bottom slope is typically
very small 1i.e., S 0 12. Since w gA/ and the hydraulic radius is defined as R h AP, the
force balance equation becomes
f
For uniform depth,
channel flow is governed
by a balance
between friction
and weight.
e
f = f D
Re
Wholly
turbulent
(10.16)
Most open-channel flows are turbulent rather than laminar. In fact, typical Reynolds numbers
are quite large, well above the transitional value and into the wholly turbulent regime. As was
discussed in Chapter 8, and shown by the figure in the margin, for very large Reynolds number pipe
flows 1wholly turbulent flows2, the friction factor, f, is found to be independent of Reynolds number,
dependent only on the relative roughness, eD, of the pipe surface. For such cases, the wall shear
stress is proportional to the dynamic pressure, rV 2 2, and independent of the viscosity. That is,
where K is a constant dependent upon the roughness of the pipe.
It is not unreasonable that similar shear stress dependencies occur for the large Reynolds
number open-channel flows. In such situations, Eq. 10.16 becomes
or
t w gA/S 0
P/
t w Kr V 2
gR h S 0
Kr V 2
2 gR hS 0
V C 2R h S 0
(10.17)
where the constant C is termed the Chezy coefficient and Eq. 10.17 is termed the Chezy equation.
This equation, one of the oldest in the area of fluid mechanics, was developed in 1768 by A. Chezy
11718–17982, a French engineer who designed a canal for the Paris water supply. The value of the
Chezy coefficient, which must be determined by experiments, is not dimensionless but has the
dimensions of 1length2 1 2
per time 1i.e., the square root of the units of acceleration2.
From a series of experiments it was found that the slope dependence of Eq. 10.17 1V S 12 0 2
23
is reasonable, but that the dependence on the hydraulic radius is more nearly V R h rather than
V R 12 h . In 1889, R. Manning 11816–18972, an Irish engineer, developed the following somewhat
modified equation for open-channel flow to more accurately describe the dependence:
2
R h
V R2 3 12
h S 0
n
(10.18)