CEM88(V) : Weir / Undershot gate (small sill elevation)

Coupe longitudinale vanne
Figure 18. Device schematic view

Weir - Free-flow

Q=\mu_f L \sqrt{2g} h_1^{3/2}

Weir - Submerged

Q=k_F \mu_F L \sqrt{2g} h_1^{3/2} [17]

with k_F coefficient of reduction for submerged flow.

The flow reduction coefficient is a function of \frac{h_2}{h_1} and of the value {\alpha} of this ratio at the instant of the free-flow/submerged transition. The submerged conditions are obtained when \frac{h_2}{h_1}>\alpha. The law of variation of the k_F coefficient has been derived from experimental results (\alpha= 0.75).

Let x = \sqrt{1-\frac{h_2}{h_1}}:

If x > 0.2 : k_F = 1 - \left(1 - \frac{x}{\sqrt{1-\alpha}}\right)^\beta

If x \leq 0.2 : k_F = 5x \left(1 - \left(1 - \frac{0.2}{\sqrt{1-\alpha}} \right)^\beta \right)

With \beta = -2\alpha + 2.6

One calculates an equivalent coefficient for free-flow conditions as before.

Undershot gate - Free-flow

Q = L  \sqrt{2g} \left(\mu h_1^{3/2} - \mu_1 (h_1 - W)^{3/2} \right)                                [18]

It has been established experimentally that the undershot gate discharge coefficient increases with \frac{h_1}{W}. A law of variation of \mu of the following form is adopted:

\mu = \mu_0 - \frac{0.08}{\frac{h_1}{W}} avec : \mu_0 \simeq 0.4

Hence, \mu_1 = \mu_0 - \frac{0.08}{\frac{h_1}{W}-1}

In order to ensure the continuity with the open channel free-flow conditions for \frac{h1}{W} = 1, we must have: \mu_F = \mu_0 - 0.08

Hence, \mu_F = 0.32 for \mu_0 = 0.4

Undershot gate - Submerged

Partially submerged flow

Q = L  \sqrt{2g} \left[k_F \mu h_1^{3/2} - \mu_1 \left(h_1 - W \right)^{3/2} \right] [19]

k_F being the same as for open channel flow.

The following free-flow/submerged transition law has been derived on the basis of experimental results:

\alpha = 1 - 0.14 \frac{h_2}{W}

0.4 \leq \alpha  \leq0.75

In order to ensure continuity with the open channel flow conditions, the free-flow/submerged transition under open channel conditions has to be realized for \alpha = 0.75 instead of 2/3 in the weir/orifice formulation.

Totally submerged flow

Q = L \sqrt{2g} \left(k_F \mu h_1^{3/2} - k_{F1} \mu_1 \left(h_1 - W \right)^{3/2} \right) [20]

The k_{F1} equation is the same as the one for k_{F} where h_2 is replaced by h_2-W (and h_1 by h_1-W) for the calculation of the x coefficient and {\alpha} (and therefore for the calculation of k_{F1}).

The transition to totally submerged flow occurs for:

h_2  > \alpha_1 h_1 +  (1 - \alpha_1) W

with: \alpha_1 = 1 - 0.14 \frac{h_2 - W}{W}
(\alpha_1 = \alpha (h_2 - W))

The functioning of the weir / undershot gate device is represented by the above equations and displayed in figure 20. Whatever the conditions of the pipe flow, one calculates an equivalent free-flow discharge coefficient, corresponding to the classical equation for the free-flow undershot gate.

C_F = \frac{Q}{L\sqrt{2g} W \sqrt{h_1}}

The reference coefficient introduced for the device is the classic C_G coefficient of the free-flow undershot gate, usually close to 0.6. It is then transformed to \mu_0 = \frac{2}{3} C_G which allows to compute \mu and \mu_1 from equation [18] for the free-flow undershot gate.

Remark: it is possible to get C_F \neq  C_G, even under free flow conditions, since the discharge coefficient increases with the \frac{h_1}{W} ratio.

Graphique h2/w = f(h1/w) déversoir / vanne de fond

(12): Weir - Free flow
(19): Undershot gate - Partially submerged
(17): Weir - Submerged
(20): Undershot gate - Totally submerged
(18): Undershot gate - Free flow
Figure 20. Weir - Undershot gate

Equations are also available in a Matlab script file (function Qouvrage) here.