Unformatted text preview: approach, used here, is to make a one-dimensional-flow approximation, as in Fig. 10.2.
Since the liquid density is nearly constant, the steady-flow continuity equation reduces
to constant-volume flow Q along the channel
Q V(x)A(x) const (10.1) where V is average velocity and A the local cross-sectional area, as sketched in Fig. 10.2.
A second one-dimensional relation between velocity and channel geometry is the
energy equation, including friction losses. If points 1 (upstream) and 2 (downstream)
are on the free surface, p1 p2 pa, and we have, for steady flow,
2g z1 z2 hf (10.2) where z denotes the total elevation of the free surface, which includes the water depth
y (see Fig. 10.2a) plus the height of the (sloping) bottom. The friction head loss hf is
analogous to head loss in duct flow from Eq. (6.30):
hf f x2 x1 V2v
2g Dh hydraulic diameter 4A
P (10.3) where f is the average friction factor (Fig. 6.13) between sections 1 and 2. Since channels are irregular in shape, their “...
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- Spring '08
- Fluid Dynamics, Fig, e-Text Main Menu, subcritical flow