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:

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

Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) ${\displaystyle z_{w,t}}$ 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.

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

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

${\displaystyle z_{min,w}=max(z_{w,t}-\rho ,min(z_{b,l},z_{b,r}))}$
${\displaystyle z_{max,w}={\begin{cases}z_{w,t},&{\text{if}}&w_{u,t}

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

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

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

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

Based on the relative water heights, the weir uses either a submerged flow ${\displaystyle Q_{s}}$ or a free flow ${\displaystyle Q_{f}}$ formula, based on the following ratio:

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

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

For submerged flow, the following calculation is used:

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

with:

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

Finally the actual amount of water level change is calculated:

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

Where:

• ${\displaystyle w_{l,t}}$ = The water level on the left side of the weir, relative to datum, at time ${\displaystyle t}$.
• ${\displaystyle w_{r,t}}$ = The water level on the right side of the weir, relative to datum, at time ${\displaystyle t}$.
• ${\displaystyle w_{u,t}}$ = The calculated upstream water level, relative to datum, at time ${\displaystyle t}$.
• ${\displaystyle z_{th}}$ = The height of the weir at time t, according to the height adjustment mechanism.
• ${\displaystyle \tau _{w}}$ = The WEIR_TARGET_LEVEL of the the weir.
• ${\displaystyle T_{wm}}$ = Moment in time in seconds after which the weir can adjust its height again.
• ${\displaystyle z_{w,t}}$ = The WEIR_HEIGHT of the weir, at time ${\displaystyle t}$.
• ${\displaystyle z_{w,t}^{*}}$ = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
• ${\displaystyle z_{b,l}}$ = The water bottom elevation at left side of the weir the weir.
• ${\displaystyle z_{b,r}}$ = The water bottom elevation at the right side of the weir.
• ${\displaystyle z_{min,w}}$ = The minimum allowed height for weir w.
• ${\displaystyle z_{max,w}}$ = The maximum allowed height for weir w.
• ${\displaystyle \mu }$ = The WEIR_MOVE_STEP_M of the water overlay, applicable to all weirs.
• ${\displaystyle \rho }$ = The WEIR_MOVE_RANGE_M of the water overlay, applicable to all weirs.
• ${\displaystyle T}$ = Current simulated time in seconds.
• ${\displaystyle t_{wm}}$ = The WEIR_MOVE_INTERVAL_S of the water overlay, applicable to all weirs.
• ${\displaystyle h_{s}}$ = The height of the water column relative to the top of the weir, on the side with the highest water level, at time ${\displaystyle t}$.
• ${\displaystyle h_{d}}$ = The height of the water column relative to the top of the weir, on the side with the lowest water level, at time ${\displaystyle t}$.
• ${\displaystyle r_{h}}$ = The ratio of water heights on either side of the culvert.
• ${\displaystyle Q}$ = The potential rate of water flow across the weir.
• ${\displaystyle Q_{f}}$ = The potential rate of water flow across the weir, based on a free flow calculation.
• ${\displaystyle Q_{s}}$ = The potential rate of water flow across the weir, based on a submerged calculation.
• ${\displaystyle f_{dw}}$ = Dutch weir factor, set to 1.7.
• ${\displaystyle C_{w}}$ = The WEIR_COEFFICIENT of the weir.
• ${\displaystyle b}$ = The breadth of weir crest, adjustable using the WEIR_WIDTH.
• ${\displaystyle f_{loss}}$ = Loss coefficient for submerged weirs, set to 0.9.
• ${\displaystyle A}$ = Flow area, based on the highest water column height relative to the top of the weir, and WEIR_WIDTH of the weir.
• ${\displaystyle g}$ = the acceleration due to gravity, set to 9.80665.
• ${\displaystyle \Delta w}$ = The water level change in meters, which takes place.
• ${\displaystyle \Delta t}$ = Computational timestep in seconds.
• ${\displaystyle \Delta x}$ = Cell size in meters.

Free flow coefficients

To clarify the free flow formula in comparison with the ISO[1] standard definitions: When ${\displaystyle h\approx 0.01}$ meters, the following holds.

${\displaystyle f_{dw}\cdot C_{w}\approx C_{d}\cdot {\sqrt {g}}}$
${\displaystyle C_{d}=0,633\cdot (1-{\frac {0.0003}{h}})^{\frac {3}{2}}}$

where:

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

For ${\displaystyle h}$ 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

1. ISO FDIS 4360, 2020 Edition, March 3, 2020 - HYDROMETRY - OPEN CHANNEL FLOW MEASUREMENT USING TRIANGULAR PROFILE WEIRS ∙ Technical Committee: ISO/TC 113/SC 2 Flow measurement structures ∙ Found at: https://www.iso.org/standard/70915.html ∙ (last visited: 28-11-2022)