Resolve "Documentation de la loi Cunge80"

Closes #403 (closed)

docs/en/calculators/structures/cunge_80.md

-# Cunge 80
+# Cunge 1980 formula
 
-This flow law corresponds to the equations described by Cunge in his book [1], or in more detail in an article by Mahmood and Yevjevich [2]. This law takes into account the different flow conditions: drowned, dewatered, free surface and loaded as well as the equations [Cemagref 1988](cem_88_d.md), but there is no continuity between free surface and loaded flow conditions. This can lead to computational problems in the vicinity of this transition.
+This flow law corresponds to the equations described by Cunge in his book [^1], or in more detail in an article by Mahmood and Yevjevich [^2]. This law is a compilation of the classical laws taking into account the different flow conditions: submerged, free flow, free surface and in charge as well as the equations [CEM88(D) : Weir / Orifice (important sill)](cem_88_d.md) and [CEM88(D) : Weir / Orifice (low sill)](cem_88_v.md). However, contrary to these equations, it does not provide any continuity between free surface and in charge flow conditions. This can lead to design problems in the vicinity of this transition.
 
-*[1] Cunge, Holly, Verwey, 1980, "Practical aspects of computational river hydraulics", Pitman, p. 169 for weirs and p. 266 for gates*.
+This law is suitable for a broad-crested rectangular weir, possibly in combination with a valve. The default discharge coefficient \(C_d = 1\) corresponds to the following discharge coefficients for the classical equations:
 
-*[2] Mahmood K., Yevjevich V., 1975, "Unsteady flow in open channels, Volume 1 and 2", Water resources publications, Fort Collins, USA, 923 p*.
+- \(C_d = 0.385\) for [the free flow weir](seuil_denoye.md).
+- \(C_d = 1\) for [the submerged weir](seuil_noye.md).
+- \(C_d = 1\) for [the submerged gate](vanne_noyee.md).
+- \(C_d = 0.7\) for [the free flow gate](vanne_denoyee.md) but the equation used here is different (see below) especially for an opening \(W\) greater than the critical height.
+
+## Free flow / submerged regime detection
+
+The flow regime is in free flow as long as the downstream water level is below critical height:
+
+$$ (Z_2 - Z_{dv}) < \dfrac{2}{3} (Z_1 - Z_{dv})$$
+
+with \(Z_1\) upstream water elevation, \(Z_2\) downstream water elevation, et \(Z_{dv}\) apron or sill elevation of the hydraulic structure.
+
+## Free flow / in charge flow detection
+
+The water level at the gate when the gate is open is considered here to be equal to:
+
+- the critical height in the case of a free flow regime;
+- the downstream water height in the case of a submerged regime.
+
+The flow becomes in charge when the gate touches the surface of the water at this point.
+
+In free flow, the flow is in charge when:
+
+$$ W \leq \dfrac{2}{3} (Z_1 - Z_{dv})$$
+
+In submerged flow, the condition becomes:
+
+$$ W \leq Z_2$$
+
+## Discharge equations
+
+The free flow gate equation is slightly different than the classical formulation:
+
+$$ Q = C_d L W \sqrt{2g} h_1 \sqrt{(h_1 - W)}$$
+
+For all other flow regimes, used equations here are the following:
+
+|        | Free surface | In charge |
+|--------|---------------|-----------|
+| Free flow | [Free flow weir](seuil_denoye.md) | Voir ci-dessus |
+| Submerged | [Submerged weir](seuil_noye.md) | [Submerged gate](vanne_noyee.md) |
+
+[^1]: Cunge, Holly, Verwey, 1980, "Practical aspects of computational river hydraulics", Pitman, p. 169 for weirs and p. 266 for gates.
+
+[^2]: Mahmood K., Yevjevich V., 1975, "Unsteady flow in open channels, Volume 1 and 2", Water resources publications, Fort Collins, USA, 923 p.