Also according to Equation 2.47
y1 cos θ
z1 ~ z1 +

(2.58)
2
and
y 2 cos θ
z2 ~ z2 +

(2.59)
2
With the above expressions for p1, p 2 , z1, and z 2 the energy equation for this control volume
reduces to
y 2 cos θ
α1 V12 y1 cos θ
2
y1 cos θ α 2 V2 y 2 cos θ
+
+ z1 +
=
+
+ z2 +
+ HL
(2.60)
2g
2
2
2g
2
2
or
α1 V12
α 2 V22
+ y1 cos θ + z1 =
+ y 2 cos θ + z 2 + HL
(2.61)
2g
2g
For most natural channels θ is small and y cosθ _ y. The velocity distribution in the vertical is
normally a log function for which α1 _ α2 _ 1. Then the energy equation becomes:
V12
V22
+ y1 + z1 =
+ y 2 + z 2 + HL
(2.62)
2g
2g
and the slopes of the bed, water surface and energy grade line are respectively
(z1  z 2 )
S o = sin θ =
(2.63)
∆L
(z1 + y1 )  (z 2 + y 2 )
Sw =
(2.64)
∆L
and
V12
V22
+ y1 + z1 
+ y2 + z2
HL 2g
2g
Sf =
=
(2.65)
∆L
∆L
Steady uniform flow is an idealized concept for open channel flow and is difficult to obtain even in
laboratory flumes. For many applications, the flow is steady and the changes in width, depth or
direction (resulting in nonuniform flow) are so small or occur over such a long distance that the
flow can be considered uniform.
2.15