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.

$z_{w,t}^{*}={\begin{cases}z_{th},&{\text{if}}&\|{w_{u,t}-\tau _{w}}\|>\mu {\text{ and }}\tau _{w}>-10000\\z_{w,t-1}^{*},&{\text{if}}&T_{wm}>T\\z_{w,t},&{\text{otherwise}}\end{cases}}$

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 uses either a submerged flow $Q_{s}$ or a free flow $Q_{f}$ formula, 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 level change is calculated:

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

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_S 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 $t$.
• $r_{h}$ = The ratio of water heights on either side of the culvert.
• $Q$ = 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.
• $b$ = The breadth of weir crest, adjustable using the WEIR_WIDTH.
• $f_{loss}$ = 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.
• $g$ = the acceleration due to gravity, set to 9.80665.
• $\Delta w$ = The water level change in meters, which takes place.
• $\Delta t$ = Computational timestep in seconds.
• $\Delta x$ = Cell size in meters.

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.

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

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

where:

• $f_{dw}$ = 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

The following topics are related to this formula.

Structures

Weir

Breach

Models

Surface model

Formulas

Breach flow formula

Test cases

Weir height test case

References

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