chapter 3 hydraulics of open channel flow

3.16 Chapter Three
occurs in a critical state can be found: that is, a slope such that normal flow occurs with
Fr � 1. The slope obtained is the critical slope, but it also is a normal slope. The smallest
critical slope, for a specified channel shape, roughness, and discharge is termed the limiting
critical slope. The critical slope for a given normal depth is
gn2D
Sc � �
�2RN
4/
N
� (3.49)
3
where the subscript N indicates the normal depth value of a variable and, for a wide channel,
gn2
Sc � �� �2y
1/3
c
(3.50)
3.4.6.2 Sheetflow. A special but noteworthy uniform flow condition is that of sheetflow.
From the viewpoint of hydraulic engineering, a necessary condition for sheetflow is that
the flow width must be sufficiently wide so that the hydraulic radius approaches the depth
of flow. With this stipulation, the Manning’s equation, Eq. (3.48), for a rectangular channel
becomes
Q � � �
n � TyN 5/3 �S� (3.51)
where T � sheetflow width and yN � normal depth of flow. Then, for a specified flow rate
and sheetflow width, Eq. (3.51) can be solved for the depth of flow, or
nQ
yN � ⎛ ⎜ �
⎝ �T �S�
�⎞ 3/5
⎟
⎠
(3.52)
The condition that the value of the hydraulic radius approaches the depth of flow is not
a sufficient condition. That is, this condition specifies no limit on the depth of flow, and
there is general agreement that sheetflow has a shallow depth of flow. Appendix 3.A summarizes
Manning’s n values for overland and sheetflow.
3.4.6.3 Superelevation. When a body of water moves along a curved path at constant
velocity, it is acted for a force directed toward the center of the curvature of the path.
When the radius of the curve is much larger than the top width of the water surface, it can
be shown that the rise in the water surface at the outer channel boundary above the mean
depth of flow in a straight channel (or superelevation) is
∆y � � V2T
� (3.53)
2gr
where r � the radius of the curve (Linsley and Franzini, 1979). It is pertinent to note that
if the effects of the velocity distribution and variations in curvature across the channel are
considered, the superelevation may be as much as 20 percent more than that estimated by
Eq. (3.53) (Linsley and Franzini, 1979). Additional information regarding superelevation
is available in Nagami et al., (1982) and U.S. Army Corps of Engineers (USACE, 1970).
3.5 GRADUALLY AND SPATIALLY VARIED FLOW
3.5.1 Introduction
HYDRAULICS OF OPEN CHANNEL FLOW
The gradual variation in the depth of flow with longitudinal distance in an open channel
is given by Eq. (3.19), or
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