Hydraulic Structures: Fourth Edition

for 0.16S/q1/51.4. For smooth spillways,

C0.5027(S/q2/3)0.385 (4.45)

for 0.23S/q2/32.3 (qin equations (4.44) and (4.45) is in m^2 s^1 ). When

applying any one of these equations it is important not to exceed the

(experimental) limits given above; even so, the equations tend to under-

estimateCat the lower end of the range.

From equation (4.38) it is evident that y 0 should really be determined

from the water discharge component of the water and air mixture (assum-

ing both parts to have equal velocities). As seen from equations (4.43) and

(4.48) the aerated flow friction coefficient is smaller than the non-aerated

one; this means that the water component depth y 0 ∫ 0 ∞(1 C)dyis smaller

thany 0 for C0. According to Straub and Anderson for 0C0.7,

1 y 0 /y 0 0.75, with significant departures of y 0 /y 0 from 1 only for C0.4

(C0.4,y 0 /y 0 0.95).

Falvey (1980) developed an equation for 0C0.6 and smooth

spillways (slope ) which includes the effect of surface tension in the form

of Weber number We(V/(/ y 0 )1/2);

C0.05Fr (sin)1/2We/(63Fr). (4.46)

For relatively narrow spillways, Hall (1942) suggested the use of

Manning’s equation for aerated flow:

V

R

n

a2/3

S1/2 (4.47)

and

V

R

n

2

a

/3

S1/2 (4.48)

wherenan(equation (4.43)) and



1

(^1)
(^1)
c 1 c 2 (4.49)
where for smooth concrete surfaces c 1 0.006 and c 2 0.
Writing Rby/(b 2 y)q/(VK), where K 2 q/b, equations
(4.47)–(4.49) result in
V 
2/3
^1 c^1 (VK)
2/3

^1  
2/3

. (4.50)

2 c 1 V^2



gb

V^2



gq

q



VK

S1/2



n

V^2



gR

218 DAM OUTLET WORKS