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:

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

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

Optionally, when a target level for the upstream water level is configured for the weir, the weir height ${z_{w},t}$ at time $t$ can be adjusted automatically during the simulation as followed:

$w_{u,t}=max(w_{l,t},w_{r,t})$

$z_{w,t}^{*}={\begin{cases}z_{th},&{\text{if}}&\|{w_{u,t}-\tau }\|>d_{w}\\z_{w,t-1}^{*},&{\text{if}}&T_{wm}>T\\z_{w,t},&{\text{otherwise}}\end{cases}}$

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_{min} = max ( z_{w,t} - d_w, min (z_{b,l}, z_{b,r} ) ) \\ z_{max} = \begin{cases} z_{w,t} & \text{if} & w_{u,t} < z^{*}_{w,t-1} \\ z_{w,t} + d_w & \text{otherwise} \end{cases} z_{th} = }

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:

$r_{h}={\frac {h_{d}}{h_{s}}}$

$Q={\begin{cases}min(Q_{s},Q_{f}),&{\text{if }}r_{h}>0.5\\Q_{f},&{\text{otherwise}}\end{cases}}$

For free flow, it is calculated directly:

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

For submerged flow, the following calculation is used:

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

Finally the actual amount of water flow is calculated:

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

Where:

• $w_{l,t}$ = The water level on the left side of the weir, relative to datum, at time $t$.
• $w_{r,t}$ = The water level on the right side of the weir, relative to datum, at time $t$.
• $z_{w,t}$ = The WEIR_HEIGHT of the weir, at time $t$.
• $h_{s}$ = The height of the water column relative to the top of the weir, on the side with the highest water level, at time $t$.
• $h_{d}$ = The height of the water column relative to the top of the weir, on the side with the lowest water level, at time $t$.
• $w_{dt}$ = The weir drop threshold of the weir.
• $z_{w,t}$ = The WEIR_HEIGHT of the weir at time $t$.
• $\tau$ = The WEIR_TARGET_LEVEL of the the weir.
• $T$ = Current simulated time in seconds.
• $T_{wm}$ = Moment in time in seconds after which the weir can adjust its height again.
• $t_{wm}$ = The Weir move interval of the water overlay.
• Failed to parse (syntax error): {\displaystyle z_^*{w,t}} = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
• $z_{th}$ = The height of the weir at time t, according to the height adjustment mechanism.
• $z_{b,l}$ = The water bottom elevation at left side of the weir the weir.
• $z_{b,r}$ = The water bottom elevation at the right side of the weir.
• $t_{wd}$ = The total amount of time a weir is dropped until the drop threshold condition is re-evaluated.
• $t_{wd,t}$ = The remaining time the weir is dropped.

• $f_{w}$ = Dutch weir factor, set to 1.7.
• $C_{w}$ = The WEIR_COEFFICIENT of the weir.
• $C_{d}$ = Corresponding Coefficient of discharge. Part of the Dutch weir factor.
• $b$ = The breadth of weir crest, adjustable using the WEIR_WIDTH.
• $Q$ = The potential rate of water flow across the weir.
• $r_{h}$ = 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 due to gravity.
• $\Delta w$ = The water flow which takes place.
• $\Delta t$ = Computational timestep.
• $\Delta x$ = Cell size.

Free flow coefficients

To clarify the free flow formula in comparison with the ISO standard definitions: When $h\approx 0.01$ meters, the following holds.

$f_{w}\cdot c_{w}\approx C_{d}\cdot {\sqrt {g}}$

$C_{d}=0,633\cdot (1-{\frac {0.0003}{h}})^{\frac {3}{2}}$

where:

• $f_{w}$ = Dutch weir factor, set to 1.7.
• $C_{w}$ = The WEIR_COEFFICIENT of the weir, set to a default of 1.1
• $C_{d}$ = Calculated Coefficient of discharge.
• $g$ = acceleration due to gravity.
• $h$ = Weir head; the height of the water column relative to the top of the weir.

For $h$ generally larger than that, the flow can be calculated up to 5.5% smaller than expected when using the second pair of coefficients. However, this can be corrected with the weir coefficient if necessary.

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