fluid_mechanics

10.3 Energy Considerations 545
COMMENT If the flow conditions upstream of the ramp
were supercritical, the free-surface elevation and fluid depth
would increase as the fluid flows up the ramp. This is indicated in
Fig. E10.2c along with the corresponding specific energy diagram,
as is shown in Fig. E10.2d. For this case the flow starts at
112 on the lower 1supercritical2 branch of the specific energy curve
and ends at 122 on the same branch with y 2 7 y 1 . Since both y and
z increase from 112 to 122, the surface elevation, y z, also
increases. Thus, flow up a ramp is different for subcritical than it
is for supercritical conditions.
y
V 2 > c 2
y 2 > y 1
y
V 1 > c 1
1 0.5 ft
y 2
y 1
(2)
0.5 ft
(1)
E
(c)
F I G U R E E10.2
(Continued)
(d)
10.3.2 Channel Depth Variations
By using the concepts of the specific energy and critical flow conditions 1Fr 12, it is possible to
determine how the depth of a flow in an open channel changes with distance along the channel.
In some situations the depth change is very rapid so that the value of dydx is of the order of 1.
Complex effects involving two- or three-dimensional flow phenomena are often involved in such
flows.
In this section we consider only gradually varying flows. For such flows, dydx 1 and it
is reasonable to impose the one-dimensional velocity assumption. At any section the total head is
H V 2 2g y z and the energy equation 1Eq. 10.52 becomes
H 1 H 2 h L
where h L is the head loss between sections 112 and 122.
As is discussed in the previous section, the slope of the energy line is dHdx dh L dx S f and
the slope of the channel bottom is dzdx S 0 . Thus, since
we obtain
or
dH
dx d dx aV2 2g y zb V dV
g dx dy
dx dz
dx
dh L
dx V dV
g dx dy
dx S 0
V dV
g dx dy
dx S f S 0
For a given flowrate per unit width, q, in a rectangular channel of constant width b, we have
or by differentiation
dV
dx q dy
y 2 dx V dy
y dx
(10.12)
V qy