chapter 3 hydraulics of open channel flow

3.24 Chapter Three
HYDRAULICS OF OPEN CHANNEL FLOW
5. Every gradually varied flow profile exemplifies the principle that subcritical flows
are controlled from the downstream while supercritical flows are controlled from
upstream. Gradually varied flow profiles would not exist if it were not for the
upstream and downstream controls.
6. In channels with horizontal and adverse slopes, the term “normal depth of flow”
has no meaning because the normal depth of flow is either negative or imaginary.
However, in these cases, the numerator of Eq. (3.60) is negative and the shape of
the profile can be deduced.
Any method of solving a gradually varied flow situation requires that cross sections be
defined. Hoggan (1989) provided the following guidelines regarding the location of cross
sections:
1. They are needed where there is a significant change in flow area, roughness, or longitudinal
slope.
2. They should be located normal to the flow.
3. They should be located in detail—upstream, within the structure, and downstreamat
structures such as bridges and culverts. They are needed at all control structures.
4. They are needed at the beginning and end of reaches with levees.
5. They should be located immediately below a confluence on a main stem and immediately
above the confluence on a tributary.
6. More cross sections are needed to define energy losses in urban areas, channels
with steep slopes, and small streams than needed in other situations.
7. In the case of HEC-2, reach lengths should be limited to a maximum distance of
0.5 mi for wide floodplains and for slopes less than 38,550 m (1800 ft) for slopes
equal to or less than 0.00057, and 370 m (1200 ft) for slopes greater than 0.00057
(Beaseley, 1973).
3.6 GRADUALLY AND RAPIDLY VARIED UNSTEADY FLOW
3.6.1 Gradually Varied Unsteady Flow
Many important open-channel flow phenomena involve flows that are unsteady.
Although a limited number of gradually varied unsteady flow problems can be solved
analytically, most problems in this category require a numerical solution of the governing
equations. Examples of gradually varied unsteady flows include flood waves, tidal
flows, and waves generated by the slow operation of control structures, such as sluice
gates and navigational locks.
The mathematical models available to treat gradually varied unsteady flow problems
are generally divided into two categories: models that solve the complete Saint Venant
equations and models that solve various approximations of the Saint Venant equations.
Among the simplified models of unsteady flow are the kinematic wave, and the diffusion
analogy. The complete solution of the Saint Venant equations requires that the equations
be solved by either finite difference or finite element approximations.
The one dimensional Saint Venant equations consist of the equation of continuity
� ∂y
� � y �
∂t
∂v
� � u �
∂x
∂y
� � 0 (3.70a)
∂x
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