chapter 3 hydraulics of open channel flow

3.6 Chapter Three
or
and
Q2
1 � �� gA3
� dA
Q2
T V2
� � 1 � �� dy
gA2
�� � 1 � �g � � 0
A D
V2
�� � �
2g
D
� (3.9)
2
�
V
� � Fr � 1 (3.10)
�g�D�
which is the definition of critical flow. Therefore, minimum specific energy occurs at the
critical hydraulic depth and is the minimum energy required to pass the flow Q. With this
information, the portion of the curve AC in Fig. 3.1 is interpreted as representing supercritical
flows, where as AB represents subcritical flows.
With regard to Fig. 3.1 and Eq. (3.6), the following observations are pertinent. First,
for channels with a steep slope and α ≠ 1, it can be shown that
V
��
Fr � (3.11)
�� g � D � c
�α o � s(
� θ � ) ��
Second, E – y curves for flow rates greater than Q lie to the right of the plotted curve,
and curves for flow rates less than Q lie to the left of the plotted curve. Third, in a rectangular
channel of width b, y � D and the flow per unit width is given by
q � � Q
� (3.12)
b
and
and
HYDRAULICS OF OPEN CHANNEL FLOW
V � � q
�
y
Then, where the subscript c indicates variable values at the critical point,
(3.13)
yc � �� q2
�� g
1/3
(3.14)
� V
2
c
� � �
2g
yc
�
2
(3.15)
yc � � 2
3 � Ec (3.16)
In nonrectangular channels when the dimensions of the channel and flow rate are specified,
critical depth is calculated either by the trial and error solution of Eqs. (3.8), (3.9),
and (3.10) or by use of the semiempirical equations in Table 3.3.
3.2.3 Variation of Depth with Distance
At any cross section, the total energy is
V2
H � �� � y � z (3.17)
2g
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