Free surface flows 91
the flow direction (slightly deviating from the horizontal, since the channel
slope is usually of the order of less than 1%) (Scarlatos 1996a).
The mass (or volume) conservation (continuity) equation is written as
∂
∂
+
∂
∂
=
A
t
Q
x
q L
(5.22)
where q L is the lateral inflow. For a rectangular channel Equation 5.22 is
written as
∂
∂
+
∂
∂
=
y
t B
Q
x
q
B
L
1
(5.23)
Equation 5.23 expresses the evolution of the free surface elevation (or the
depth variation) in terms of the difference between the incoming and outgoing discharges through the infinitesimal control space under consideration (the channel reach of length = δx).
For the formulation of the momentum balance equation, the Newton’s
second law is applied by taking into consideration (Figure 5.4):
• The hydrostatic pressure forces acting on the two sides (upstream and
downstream) of the control space
• The component of the weight of water (W x ) within the control volume
along the flow direction that is parallel to the bed
• The sum of the distributed frictional forces (τ o ) over the wetted perimeter of the channel
y
Water free surface
S o
y
τ o
W
W x
Figure 5.4 Forces acting on the control volume of an open channel.