Weir Formula Structure

The Weir Formula structure models the discharge through a structure with standard weir formulas.

Weir Formula 1

The Weir Formula 1 is based on a standard weir expression simplified according to the Villemonte formula:

Figure: Definition of terms used in the three Weir Formula structures.

\[ \begin{array}{r l}{}&{{}W\!\!=\!C\!\cdot\!(H_{u s}\!-\!H_{w})^{k}\!\cdot\!\bigg[1\!-\!\bigg(\frac{H_{d s}\!-\!H_{w}}{H_{u s}\!-\!H_{w}}\bigg)^{k}\bigg]^{0.38!}}\end{array} \]

Equation: Discharge through a weir structure with contraction and submergence effects.

Where Q is discharge through the structure, W is width, C is weir coefficient, k is the weir exponential coefficient, H_us is upstream water level, H_ds is downstream water level and H_w is weir level. (See the definition of terms used in the three Weir Formula structures).

Weir Formula 2 (Honma)

The Weir Formula 2 is the Honma formula:

\[ Q = \begin{cases} C_{1}W\left(H_{us} - H_{w}\right)\sqrt{H_{us} - H_{w}} & \text{for } \left(H_{ds} - H_{w}\right)/H_{us} < 2/3 \\[1.5ex] C_{2}W\left(H_{ds} - H_{w}\right)\sqrt{H_{us} - H_{ds}} & \text{for } \left(H_{ds} - H_{w}\right)/H_{us} \geq 2/3 \end{cases} \]

Equation: Discharge formula for a broad-crested weir, free-flowing or submerged depending on the downstream/upstream depth ratio.

Where Q is the discharge through structure, W is the width, C₁ is the first weir coefficient, \(C_2 = \frac{3}{2} \sqrt{3} C_1\) is the second weir coefficient, H_us is the upstream water level, H_ds is the downstream water level and H_w is weir level. (See the definition of terms used in the three Weir Formula structures).

Weir Formula 3 (extended Honma)

The Weir Formula 3 uses the extended Honma formula. The discharge is calculated in three flow regimes: 1) perfect flow, 2) imperfect flow, and 3) submerged overflow. The active regime is defined by the ratio between downstream and upstream water depth above the crest.

\[ Q = \begin{cases} C_{0}Wh_{u}^{3/2} & \text{for } h_{d}/h_{u} \leq \left(h_{d}/h_{u}\right)_{i} \\[1.5ex] C_{0}\left(\alpha \cdot \left(h_{d}/h_{u}\right)^{q} - \beta\right)Wh_{u}^{3/2} & \text{for other cases} \\[1.5ex] C_{0}\left(\gamma \cdot \left(h_{d}/h_{u}\right)^{r} - \delta\right)Wh_{d}\left(h_{u} - h_{d}\right)^{1/2} & \text{for } h_{d}/h_{u} \geq \left(h_{d}/h_{u}\right)_{s} \end{cases} \]

Equation: Discharge formula for the weir, with free-flowing, transitional and submerged cases.

where

\(C_{0}~=~{ a}\cdot(h_{u}/H)^{p}+b\)

W is weir width

H is weir height above cross section invert

h_d is the upstream water level above the crest

h_u is the down stream water level above the crest

(h_d/h_u)_i is a user specified depth ratio limit between perfect and imperfect flow regime.

(h_d/h_u)_s is a user specified depth ratio limit between imperfect and submerged flow regime.

a, b, p, q, r, \(\alpha\) , \(\frac{h}{H}\) , \(\gamma\) , and \(\delta\) are user specified parameters.

NOTE: The MIKE 1D engine does not check if there is a continuous transition from one flow regime to the next. This has to be ensured by the user through proper selection of parameters.