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.

(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.