and

q = (gy 3 )

1/ 2

(2.139)

c

or

1/ 3

q2

Vc2

yc =

=2

(see Equation 2.147)

(2.140)

g

2g

Note that

Vc2 = y c g

(2.141)

or

Vc

=1

(2.142)

gy c

but

V

= Fr

(2.143)

gy

also

Vc2

3

=

+ y c = y c (see Equation 2.147)

Hmin

(2.144)

2g

2

Thus, flow at minimum specific energy has a Froude number equal to one. Flows with velocities

larger than critical (Fr > 1) are called rapid or supercritical and flow with velocities smaller than

critical (Fr < 1) are called tranquil or subcritical. These flow conditions are illustrated in Figure

2.21, where a rise in the bed causes a decrease in depth when the flow is tranquil and an

increase in depth when the flow is rapid. Furthermore, there is a maximum rise in the bed for a

given H1 where the given rate of flow is physically possible. If the rise in the bed is increased

beyond ∆zmax for Hmin then the approaching flow depth y1 would have to increase (increasing H)

or the flow would have to be decreased. Thus, for a given flow in a channel, a rise in the bed

level can occur up to a ∆zmax without causing backwater.

For a constant H, Equation 2.137 can be solved for y as a function of q. By plotting y as a

function of q, Figure 2.22 is obtained and for any discharge smaller than a specific maximum, two

depths of flow are possible (Rouse 1946).

2.40

Integrated Publishing, Inc. |