SIC^2: Simulation and Integration of Control for Canals

CEM02(V) et CEM02(D) : Trapezoidal sills and gates

David Dorchies — 2015-04-13T13:38:54Z

These 2 equations are the sum of one of "Cem88(V)" and "CEM88(D)" equations with a triangular sill.

Kindsvater-Shen formula [1] is used for the triangular part :

$Q=C_e\frac{8}{15}\mathrm{tan}\frac{\alpha}{2}\sqrt{2g} h^{5/2}$

The discharge coefficient $C$ used in SIC is $C=C_e\frac{8}{15}\sqrt{2g}$. $C=\approx1,37$ for $C_e=0,58$ as suggested by the norm [1].

[1] Norme NF X 10-311 : Mesure de débit de l'eau dans les canaux découverts au moyen de déversoirs en mince paroi

Cross structure equations

David Dorchies — 2009-10-22T15:33:50Z

When cross structures exist on the canal (singular section) the water surface profile equation cannot be used locally to calculate the water surface elevation upstream of the structure.

The hydraulic laws of the different devices present in the section must be applied.

The modeling of these devices is a delicate problem to solve when developing open channel mathematical models. The equations used to represent the hydraulic devices are numerous and do not cover all the possible operating conditions.

In particular, it is rather difficult to maintain the continuity between the different formulations, for example, at the instant of transition between free-flow conditions and submerged conditions, or between open-channel conditions and pipe-flow conditions.

A distinction has been made between devices with a high sill elevation (called hereafter Weir / Orifice) and devices with a low sill elevation (called hereafter Weir / Undershot gates).

CEM88(D) : Weir / Orifice (high sill elevation)

David Dorchies — 2009-10-22T15:33:50Z

Figure 18. Device schematic view

Weir - Free flow ($h_1 < W$ and $h_2 \leq \frac{2}{3} h_1$)

$Q = \mu _F L \sqrt{2g} h_1^{3/2} $ [12]

Classical equation for the free flow weir ($\mu_F \simeq 0.4$).

Weir - Submerged ($h_1 < W$ and $h_2 \geq \frac{2}{3} h_1$)

$Q = \mu _S L \sqrt{2g} (h_1-h_2)^{1/2} h_2$ [13]

Classical formulation for the submerged weir.

The free-flow/submerged transition takes place for $h_2 = \frac{2}{3} h_1$

Thus,
$\mu_S = \frac{3 \sqrt{3}}{2} \mu_F$ for $\mu_F = 0.4 \Rightarrow \mu_S = 1.04$

The equivalent free-flow coefficient can be calculated:

$ \mu_{F} = \frac{Q}{L \sqrt{2g} h_{1}^{3/2}} $

It indicates the degree of submergence of the weir by comparing it to the introduced free-flow coefficient $\mu_F$. In effect, the reference coefficient of the device considered is that corresponding to the free-flow weir ($\mu_F$ close to $0.4$).

Orifice - Free flow ($h_1 \geq W$ and $h_2 \leq \frac{2}{3} h_1$)

An equation of the following type is applied:

$ Q = \mu L \sqrt{2g} \left( h_1^{3/2} - (h_1 - W)^{3/2} \right)$ [14]

This formulation is applicable to large width rectangular orifices.

The continuity towards the open-channel flow is assured when:

$\frac{h1}{W} = 1$, one then has: $\mu = \mu_F$.

Orifice - Submerged

Two formulations exist, according to whether the flow is partially submerged or completely submerged.

Partially submerged flow ($h_1 \geq W$ and $\frac{2}{3} h_1 < h_2 < \frac{2}{3} h_1 + \frac{W}{3}$)

$Q = \mu_F L \sqrt{2g} \left[ \frac{3\sqrt{3}}{2} \left( \left( h_1 - h_2 \right)^{1/2} h_2 \right) - \left(h_1 - W \right)^{3/2} \right]$ [15]

Totally submerged flow ($h_1 \geq W$ and $\frac{2}{3} h_1 + \frac{W}{3} < h_2$)

$Q = \mu` L \sqrt{2g} (h_1-h_2)^{1/2} \left[ h_2 - (h_2 - W) \right]$
$\Rightarrow$ $Q = \mu` L \sqrt{2g} (h_1-h_2)^{1/2} W$ [16]

This is the classic equation of the submerged orifice with $\mu` = \mu_S$.

The operation of the weir/orifice device is represented by the above equations and in figure 19. Whatever the conditions of pipe flow, one calculates an equivalent free-flow coefficient, corresponding to the free-flow orifice:

$C_F = \frac{Q}{L \sqrt{2g} W (h_1 - 0.5 W)^{1/2}}$

(12): Weir - Free flow
(15): Orifice - Partially submerged
(13): Weir - Submerged
(16): Orifice - Totally submerged
(14): Orifice - Free flow
Figure 19. Weir - Orifice

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

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

David Dorchies — 2009-10-22T15:33:50Z

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.