Mechanics of Fluids

With the same assumptions used in deriving the expression for the
discharge through a rectangular notch, we have
u2 ={2g(H − z2 + u 2 1 /2g)}1/2
ForaVnotch, however, the cross-sectional area of the approach channel is
usually so much greater than that of the notch that u2 1 /2g may be neglected.
The idealized discharge through an element of the notch (as in Fig. 3.27) is
therefore bδz2{2g(H − z2)} 1/2 and, if each side makes an angle θ/2 with the
vertical, b = 2z2 tan(θ/2). The total idealized discharge is therefore
where h = H − z2. Hence
Qideal = 2 tan θ �
(2g)
2
= 2 tan θ �
(2g)
2
Qideal = 2 tan θ �
(2g)
2
= 8
15
� H
0
� H
0
θ �
5/2
tan (2g)H
2
z2(H − z2) 1/2 dz2
(H − h)h 1/2 dh
�
2
3 Hh3/2 − 2
5 h5/2
�H 0
The actual discharge Q is therefore
Q = 8
15 Cd tan θ �
5/2
(2g)H
2
Simple applications of Bernoulli’s equation 129
(3.26)
The angle θ is seldom outside the range 30 ◦ –90 ◦ . If the head H is sufficient
for the nappe to spring clear of the notch plate and the width of the approach
channel is at least four times the maximum width of the nappe, C d is about
0.59 for water flow. As a result of viscosity and surface tension effects the
value increases somewhat as the head falls.
Fig. 3.26
Fig. 3.27