Question 10.3: Establish the stage (headwater level)–discharge relationship......

1. H/D ≤1.2. For H<0.6 m, free flow open-channel conditions prevail. Referring to Fig. 10.6 and assuming that a steep slope entry gives entrance control, i.e. the depth at the inlet is critical, for H = 0.2 m, ignoring entry loss y_{ c }=(2 / 3) \times 0.2=0.133\, m \text { and } V_{ c }=1.142 \,m s ^{-1} . This gives the critical slope (V n)^2 / R^{4 / 3}=0.00424 . Therefore the slope of the culvert is mild and hence subcritical flow analysis gives the following results:

\begin{aligned} Q & =1.2 y_0\left[1.2 y_0 /\left(1.2+2 y_0\right)\right]^{2 / 3}(0.001)^{1 / 2} / 0.013 \\ & =2.92 y_0\left[1.2 y_0 /\left(1.2+2 y_0\right)\right] ^{2 / 3} ; \end{aligned}                (i)

Table 1

At the inlet over a short reach,

H=y_0+V^2 / 2 g+K_{ e } V^2 / 2 g  .                        (ii)

The entrance loss coefficient, K_{ e } , is as follows:
for a square-edged entry,            0.5;
for a flared entry,                          0.25;
for a rounded entry,                     0.05;

Table 2

2. H/D≥1.2.
(a) For orifice flow
Q=C_{ d }(1.2 \times 0.6)[2 g(H-D / 2)]^{1 / 2} .               (iii)
With C_{ d } = 0.62 the following results are obtained:

Table 3

(b) For pipe flow the energy equation gives

where

Thus

Q=2.08(H-0.57)^{1 / 2}                   (iv)

Table 4

During rising stages the barrel flows full from H = 0.72 m and during falling stages the flow becomes free-surface flow when H = 0.691 m.
The following table summarizes the results:

Table 5