Weir formula (Water Overlay)

Flow across weirs is calculated differently for free flow and submerged flow.

The height of the water at each end of the weir, relative to the weir, is calculated:

$h_{s}=max(0,max(w_{l},w_{r})-z_{w,t})$

$h_{d}=max(0,min(w_{l},w_{r})-z_{w,t})$

Optionally, when a Weir drop threshold is configured, the weir height $z_{w}$ can update depending on the water level and weir drop time:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{w,t} = z_b \if max( w_l, w_r) > w_{dt} \or t_{wd} > 0 }

$z_{w,t}=z_{w}$ otherwise

$z_{w,t}={\begin{cases}z_{b},&{\mbox{if }}max(w_{l},w_{r})>w_{dt}{\mbox{ or }}t_{wd}>0\\z_{w},&{\mbox{otherwise }}\end{cases}}$

Based on the relative water heights, the weir is judged to have either a submerged flow or a free flow, based on the following ratio:

$h_{ratio}={\frac {h_{d}}{h_{s}}}$

$Q=min(Q_{submerged},Q_{free})$ if $h_{ratio}>0.5$

$Q=Q_{free}$otherwise

For free flow, capacity is calculated directly:

$Q_{free}=f_{w,d}\cdot c_{w}\cdot w_{w}\cdot (h_{s}-h_{d})^{3/2}$

For submerged flow, the following calculation is used:

$Q_{submerged}=U_{loss}\cdot A\cdot {\sqrt {2\cdot g\cdot (h_{s}-h_{d})}}$

Finally the actual amount of water flow is calculated:

$\Delta w=\Delta t\cdot {\frac {Q}{\Delta x}}$

Where:

• $h_{s}$ = The height of the water column relative to the top of the weir, on the side with the highest water level.
• $h_{d}$ = The height of the water column relative to the top of the weir, on the side with the lowest water level.
• $w_{l}$ = The water level on the left side of the weir, relative to datum.
• $w_{r}$ = The water level on the right side of the weir, relative to datum.
• $z_{w,t}$ = The height of the weir at time t, depending on drop mechanism.
• $z_{w}$ = The WEIR_HEIGHT of the weir.
• $z_{b}$ = The water bottom elevation at the weir.
• $t_{wd}$ = The weir drop timeout.
• $f_{w,d}$ = Dutch weir factor, set to 1.7.
• $c_{w}$ = The WEIR_COEFFICIENT of the weir.
• $w_{w}$ = The WEIR_WIDTH of the weir.
• $Q$ = The potential rate of water flow across the weir.
• $h_{ratio}$ = The ratio of water heights on either side of the culvert.
• $Q_{free}$ = The potential rate of water flow across the weir, based on a free flow calculation.
• $Q_{submerged}$ = The potential rate of water flow across the weir, based on a submerged calculation.
• $U_{loss}$ = Loss coefficient for submerged weirs, set to 0.9.
• $A$ = Flow area, based on the highest water column height relative to the top of the weir, and WEIR_WIDTH of the weir.
• $g$ = Acceleration factor of gravity, set to 9.80665.
• $\Delta w$ = The water flow which takes place.
• $\Delta t$ = Computational timestep.
• $\Delta x$ = Cell size.

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