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^[1] 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.

References

↑ 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)