Inclusion of friction term in weir formulae

The Villemonte equation (Weir Formula 1) is given as:

\[ Q = w \cdot C \cdot h_{1}^{k} \cdot \left\lbrack 1 - \left( \frac{h_{2}}{h_{1}} \right)^{k} \right\rbrack^{0.385} \]

Equation: Villemonte weir equation, Weir Formula 1.

where \(w\) is the width, \(C\) is the weir coefficient, \(k\) the exponential coefficient (accepted value is generally \(k = 1.5\)), \(h_{1}\) is the depth of water above the weir level downstream (\(H_{us} - H_{w}\)) and \(h_{2}\) is the water depth above weir level downstream (\(H_{ds} - H_{w}\)). This equation is a free overflow term (\(w \cdot C \cdot h^{k}\)) combined with a scaling term for submergence (\(\lbrack\ldots\rbrack^{0.385}\)) that approaches 0 as \(h_{1}\) approaches \(h_{2}\).

The Honma equation (Weir Formula 2) is given as:

\[ Q = w \cdot C \cdot h_{1} \cdot \sqrt{h_{1}}, \quad \frac{h_{2}}{h_{1}} \leq \frac{2}{3} \]

Equation: Honma weir equation for free overflow conditions.

\[ Q = w \cdot \frac{3}{2} \cdot \sqrt{3} \cdot C \cdot h_{2} \cdot \sqrt{h_{1} - h_{2}}, \quad \frac{h_{2}}{h_{1}} > \frac{2}{3} \]

Equation: Honma weir equation for submerged conditions.

The first equation is for free overflow (identical to Weir Formula 1 with free overflow), while the second is for submerged conditions. Both equations are derived from the equation for form loss across a structure:

\[ \Delta h = z \cdot \frac{v^{2}}{2g} \]

Equation: Form loss across a structure.

Where \(v\) represents the flow velocity. If a friction loss is added this becomes:

\[ \Delta h = (z + F) \cdot \frac{v^{2}}{2g} \]

Equation: Combined form and friction loss across a structure.

Where:

\[ F = \frac{2 g L n^{2}}{h_{1}^{4/3}} \]

Equation: Friction term added to the form loss coefficient.

and \(n\) is Manning's number. The length \(L\) is calculated as the length between opposite corners in a surface element:

\[ L = \sqrt{\left( \Delta X_{S} \right)^{2} + \left( \Delta Y_{S} \right)^{2}} \]

Equation: Length between opposite corners of a surface element.

Using the fact that area \(= wh_{1}\) as well as \(Q =\) area \(\cdot\) velocity, we obtain:

\[ Q = w \cdot \sqrt{\frac{2g}{z + F}} \cdot h_{1} \cdot \sqrt{\Delta h} \]

Equation: Discharge across a weir including the friction term.

This equation is similar in form to Weir Formulae 1 and 2 for free overflow, except for the weir coefficient \(C\) and the additional friction:

\[ C = \sqrt{\frac{2 g}{z}} \Leftrightarrow z = \frac{2g}{C^{2}} \]

Equation: Relation between the weir coefficient and the form loss coefficient.

\[ C_{f} = \sqrt{\frac{2 g}{z + F}} = \sqrt{\frac{2 g}{\frac{2 g}{C^{2}} + F}} \]

Equation: Friction-corrected weir coefficient expressed through the form loss coefficient.

So, the existing equations for weir flow calculations can be used, but the weir coefficient is modified according to:

\[ C_{f} = \sqrt{\frac{2 g}{\frac{2 g}{C^{2}} + \frac{2 g L n^{2}}{h_{1}^{4/3}}}} \]

Equation: Friction-corrected weir coefficient in full.

This equation is applied to the weir calculations in the lateral link specifications.