SPILLWAYS 213
dM/dxA
the scheme for numerical integration could, for two consecutive sections,
be expressed as follows (Chow, 1983):
∆y ∆V (S 0 Sf)∆x. (4.30)
Equation (4.30) can be solved by trial and error for ∆y, with Q 1 ,V 1 ,S 0 ,∆x,
Q 2 and channel shape known, and an assumed value of y 2 (and therefore
V 2 ) which must agree with ∆yy 2 y 1 given by equation (4.30). The solu-
tion starts from the control section and proceeds upstream; the control
section is either the outflow from the channel or the critical depth section
inside the channel, the position of which is determined from equation
(4.27). Should the computation show that there would be a substantial
length of non-modular outflow in the upstream part of the side-channel
spillway (or, alternatively, that the channel is unnecessarily large) the
design parameters have to be altered.
4.7.3 Chute spillwaysA chute spillway is a steep channel conveying the discharge from a low-
overfall, side-channel, or special shape (e.g. labyrinth) spillway over the
valley side into the river downstream. The design of chute spillways
requires the handling of three problems associated with supercritical flow:
waves of interference, translatory waves, and (self-)aeration.
Interference waves(cross-waves, standing waves) are shock waves
which occur whenever the supercritical flow is interfered with, at inlets
(Fig. 4.10), changes of section, direction, or slope, bridge piers, etc. They
are stationary waves, the position of which depends on discharge; their
main significance is that they require an increased freeboard and higher
chute side walls, as water tends to ‘pile up’ at points where the waves meet
the side walls (e.g. points B and D in Fig. 4.10). The waves can also create
additional difficulties in energy dissipators if they persist so far downstream
(which is rarely the case because once the flow becomes aerated they disap-
pear). They can be minimized by carefully shaping any changes necessary
in cross-section, and making transitions in direction and slope as gradual as
possible. If the chute is relatively long a very gradual reduction in width
commensurate with the flow acceleration can produce some savings in cost;
near its outlet the chute may also be widened gradually to reduce the flow
per unit width and the depth (and cavitation risk), and to improve energy
dissipation. All other forms of interference with the flow are best avoided.
V 2 ∆Q
Q 1Q 1 (V 1 V 2 )
g(Q 1 Q 2 )VQ
gd
dxdy
dx