Hydraulic Structures: Fourth Edition

Equation (4.24) can be solved by numerical methods, and also results in

the differential equation of flow:

S 0 Sf 

g

2

A

Q

 2 /(1 Fr

(^2) ). (4.25)
The flow in the channel will, certainly in the upstream part, be subcritical,
but at the downstream end may change into supercritical. Critical flow
occurs at Fr1, but as dy/dxhas a finite value for critical flow
S 0 Sf0.
For Q2/3Cd 2
g1/2H3/2xqx, dQ/dxq (with q2/3 Cd 2
g1/2H3/2
constant).
ForFr^2 Q^2 B/gA^3 1 (Bis the water surface width in the channel),
the above condition leads to
S 0 
1/3
(4.26)
whenCis the Chézy coefficient.
ForPPcandBBc(critical section) we can calculate from equation
(4.26) the critical slope S0cwith control at the outflow from the channel at
xL. For a critical section to occur inside the channel, either the slope S 0
must be larger than the value given by equation (4.26) for xL,PPc, and
BBc, or the length Lof the spillway (channel) must be larger than
x 8 q^2 gB^2 cS (^0) 
3

1

. (4.27)

Equation (4.25) can be integrated for a rectangular channel section and for

S 0 Sf0, resulting in

x/L  1   (^) 
y
y
L

3

2 F

1

rL

 (4.28)

giving the relationship between xand the depth of flow in the channel y;

indexLrefers to the outflow (end) section (xL). For a critical depth yc

atxL, equation (4.28) yields

x/L

y

y

c

1.5 0.5

y

y

c



2

. (4.29)

Returning to equation (4.24), after substituting for

1



2 Fr^2 L

y



yL

gPc



C^2 Bc

q^2 B



gx

2



B

gP



C^2 B

dQ



dx

2 Q



gA^2

dQ



dx

dy



dx

212 DAM OUTLET WORKS