SPILLWAYS 215
In this case the maximum depth at the wall yw(important for the free-
board)(occurs at a distance about 1.75y 1 Fr 1 from the point of wall deflec-
tion and is given by
yw/y 1 1 2
Fr 1 (1Fr 1 /4). (4.36)The reduction of shock waves can be achieved by one (or a combina-
tion) of three methods: reduction in the shock number Fr 1 (certainly the
most effective method), wave interference(e.g. in channel contractions –
applicable to one approach flow only) or by bottom and/or sides reduction
elements(shock diffractors). Modification of chute geometry including
‘banking’ and curved inverts is often successful, but not necessarily eco-
nomical. Bottom elements include vanes or oblique steps (Ellis, 1989). For
further details see also ICOLD (1992b) and for a comprehensive treat-
ment of shock waves and flow through chute expansions, contractions and
bends Vischer and Hager (1998).
Gate piers on the crest of spillways also induce shock waves; formerly,
(not very successful) attempts were made to reduce the resulting ‘rooster
tails’ by specially shaped downstream pier ends, but today rectangular pier
ends are often used, accepting the resulting shock waves (Vischer, 1988).
Translatory waves(waves of translation, roll waves) originate under
certain conditions from the structure of supercritical flow and, as their
name implies, move with the flow right into the stilling basin. They again
require a higher freeboard, and by imparting unsteady flow impulses to the
stilling basin may even cause its failure (Arsenishvili, 1965). They occur at
slopes with 0.02S0.35 (which is a very wide range covering most chute
spillways) and with long chutes, but even then can be avoided if the ratio
of the depth to the wetted perimeter is greater than 0.1. Another criterion
for the appearance of translatory waves is the Vedernikov number – Ved;
they form, if Vedk!Fr1, where kis a constant (k2/3) and !the
channel shape factor, ! 1 RdP/dA, and if the chute length Lis greater
thanL 9.2 V 02 /(gS 0 ) ((12/3!/Ved)/(1 Ved)) (Jain 2000). Roll waves
can also be avoided by introducing artificial roughness to the chute surface
which is, of course, contrary to cavitation control. The best method is to
design the spillway with a depth: perimeter ratio larger than 0.1 for the
maximum discharge, and to accept roll waves at low flows when they are
not dangerous to the stilling basin and do not require additional freeboard.
(Self-)aerationis by far the most important feature of supercritical
flow. Although beneficial for energy dissipation and cavitation protection,
it causes an increase of the depth of flow (bulking) and thus requires an
increase of the chute side walls.
The transition from critical flow at the crest, through supercritical
non-uniform non-aerated flow to non-uniform partially and fully aerated
flow to, finally, uniform aerated flow is shown schematically in Fig. 4.11;
the flow depth, ya, can be compared with that of uniform non-aerated flow,