Weir formula (Water Overlay)

Flow across weirs is calculated differently for free flow and submerged flow. Optionally, the height of the weir can variate based on a provided height values or an automatic adjustment. Therefore, we determined the height of the weir first.

First the upstream water level is calculated as followed:

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

Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) $z_{w,t}$ or it is adjusted according to the weir threshold level. When it is adjusted, a moment in time is set for that weir, during which it cannot be adjusted.

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

The height adjustment range values, defined by the min and max, are determined next:

$z_{min,w}=max(z_{w,t}-\rho ,min(z_{b,l},z_{b,r}))$

$z_{max,w}={\begin{cases}z_{w,t},&{\text{if}}&w_{u,t}<z_{w,t-1}^{*}\\z_{w,t}+\rho ,&{\text{otherwise}}\end{cases}}$

Finally, the adjusted weir height is calculated and stored, as well as the moment in time at which the weir can be adjusted again.

$z_{th}={\begin{cases}min(z_{max,w},max(z_{min,w},z_{w,t-1}^{*}+\mu )),&{\text{if}}&w_{u,t}<\tau _{w}\\min(z_{max,w},max(z_{min,w},z_{w,t-1}^{*}-\mu )),&{\text{otherwise}}\end{cases}}$

$T_{wm}=T+t_{wm}$

$z_{w,t}=z_{w,t}^{*}$

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

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

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

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:

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

For submerged flow, the following calculation is used:

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

with:

$A=b\cdot (h_{s}-h_{d})$

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 $t$.
• $w_{u,t}$ = The calculated upstream water level, relative to datum, at time $t$.

• $z_{th}$ = The height of the weir at time t, according to the height adjustment mechanism.
• $\tau _{w}$ = The WEIR_TARGET_LEVEL of the the weir.
• $T_{wm}$ = Moment in time in seconds after which the weir can adjust its height again.
• $z_{w,t}$ = The WEIR_HEIGHT of the weir, at time $t$.
• $z_{w,t}^{*}$ = The optionally adjusted height of the weir at time t, depending on the weir 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.
• $z_{min,w}$ = The minimum allowed height for weir w.
• $z_{max,w}$ = The maximum allowed height for weir w.
• $\mu$ = The WEIR_MOVE_STEP_M of the water overlay, applicable to all weirs.
• $\rho$ = The WEIR_MOVE_RANGE_M of the water overlay, applicable to all weirs.
• $T$ = Current simulated time in seconds.
• $t_{wm}$ = The Weir move interval of the water overlay, applicable to all weirs.

• $h_{s}$ = 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 $t$.
• $Q$ = The potential rate of water flow across the weir.
• $Q_{f}$ = The potential rate of water flow across the weir, based on a free flow calculation.
• $Q_{s}$ = The potential rate of water flow across the weir, based on a submerged calculation.
• $f_{dw}$ = 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.
• $r_{h}$ = The ratio of water heights on either side of the culvert.
• $f_{loss}$ = 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.

Related

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