fluid_mechanics

10.4 Uniform Depth Channel Flow 547
Lines of
constant
velocity
y
Centerline
velocity
profiles
y
u
u
(a)
Actual
Uniform
τ w
(b)
τ w = shear stress
distribution
F I G U R E 10.9 Typical velocity and
shear stress distributions in an open channel:
(a) velocity distribution throughout the cross section,
(b) shear stress distribution on the wetted perimeter.
and the fluid velocity is zero on the wetted perimeter, where a wall shear stress, t w , is developed.
This shear stress is seldom uniform along the wetted perimeter, with typical variations as are
indicated in Fig. 10.9b.
Fortunately, reasonable analytical results can be obtained by assuming a uniform velocity
profile, V, and a constant wall shear stress, t w . Similar assumptions were made for pipe flow
situations 1Chapter 82, with the friction factor being used to obtain the head loss.
F l u i d s i n t h e N e w s
Plumbing the Everglades Because of all of the economic development
that has occurred in southern Florida, the natural
drainage pattern of that area has been greatly altered during the
past century. Previously there was a vast network of surface
flow southward from the Orlando area, to Lake Okeechobee,
through the Everglades, and out to the Gulf of Mexico. Currently
a vast amount of freshwater from Lake Okeechobee and
surrounding waterways (1.7 billion gallons per day) is sluiced
into the ocean for flood control, bypassing the Everglades. A
new long-term Comprehensive Everglades Restoration Plan is
being implemented to restore, preserve, and protect the south
Florida ecosystem. Included in the plan are the use of numerous
aquifer-storage-and-recovery systems that will recharge the
ecosystem. In addition, surface water reservoirs using artificial
wetlands will clean agricultural runoff. In an attempt to improve
the historical flow from north to south, old levees will be
removed, parts of the Tamiami Trail causeway will be altered,
and stored water will be redirected through miles of new pipes
and rebuilt canals. Strictly speaking, the Everglades will not be
“restored.” However, by 2030, 1.6 million acres of national
parkland will have cleaner water and more of it. (See Problem
10.77.)
10.4.2 The Chezy and Manning Equations
The basic equations used to determine the uniform flowrate in open channels were derived many
years ago. Continual refinements have taken place to obtain better values of the empirical coefficients
involved. The result is a semiempirical equation that provides reasonable engineering results. A
more refined analysis is perhaps not warranted because of the complexity and uncertainty of the
flow geometry 1i.e., channel shape and the irregular makeup of the wetted perimeter, particularly
for natural channels2.
Under the assumptions of steady uniform flow, the x component of the momentum equation
1Eq. 5.222 applied to the control volume indicated in Fig. 10.10 simply reduces to
F x rQ1V 2 V 1 2 0
For steady, uniform
depth flow in an
open channel
there is no fluid
acceleration.
since V 1 V 2 . There is no acceleration of the fluid, and the momentum flux across section 112 is
the same as that across section 122. The flow is governed by a simple balance between the forces
in the direction of the flow. Thus, F x 0, or
F 1 F 2 t w P/ w sin u 0
(10.15)