Weir formula (Water Overlay): Difference between revisions

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Flow across [[Weir (Water Overlay)|weir]]s is calculated differently for free flow and submerged flow.
Flow across [[Weir (Water Overlay)|weir]]s 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.  


The height of the water at each end of the weir, relative to the weir, is calculated:
First the upstream water level is calculated as followed:
: <math>h_s = max(0, max( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
: <math>h_d = max(0, min( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
 
Optionally, when a target level for the upstream water level is configured for the weir, the weir height <math>z_{w,t}</math> at time <math>t</math> can be adjusted automatically during the simulation as followed.


First the upstream water level is calculated:
:<math>w_{u,t} =  max( w_{l,t}, w_{r,t})</math>
:<math>w_{u,t} =  max( w_{l,t}, w_{r,t})</math>


Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) <math>z_{w,t}</math> 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.   
Next, the adjusted weir height is determined. It either originates from the supplied weir height(s) <math>z_{w,t}</math> 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.   
:<math>z^{*}_{w,t} =  
:<math>z^{*}_{w,t} =  
\begin{cases}
\begin{cases}
z_{th}, & \text{if} & \|{w_{u,t} - \tau_w}\| > \mu \\
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-1}, & \text{if} & T_{wm} > T \\
z_{w,t}, & \text{otherwise}  
z_{w,t}, & \text{otherwise}  
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</math>
</math>


The height adjustment range is determined next:  
The height adjustment range values, defined by the min and max, are determined next:  
:<math>z_{min} = max ( z_{w,t} - \rho, min (z_{b,l}, z_{b,r} ) )</math>


:<math>z_{max} =  
:<math>z_{min,w} = max ( z_{w,t} - \rho, min (z_{b,l}, z_{b,r} ) )</math>
 
:<math>z_{max,w} =  
\begin{cases}
\begin{cases}
z_{w,t}, & \text{if} &  w_{u,t} < z^{*}_{w,t-1} \\
z_{w,t}, & \text{if} &  w_{u,t} < z^{*}_{w,t-1} \\
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</math>
</math>


Finally, the adjusted weir height is calculated and store, as well as the moment in time at which the weir can be adjusted again.
Finally, the adjusted weir height is calculated and stored, as well as the moment in time at which the weir can be adjusted again.
:<math>z_{th} =
:<math>z_{th} =
\begin{cases}  
\begin{cases}  
min(z_max, max( z_{min}, 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{if} & w_{u,t} < \tau_w \\
min(z_max, max( z_{min}, z^{*}_{w,t-1}- \mu )), & \text{otherwise}
min(z_{max,w}, max( z_{min,w}, z^{*}_{w,t-1} - \mu )), & \text{otherwise}
\end{cases}
\end{cases}
</math>
</math>
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:<math>z_{w,t} = z^{*}_{w,t}</math>
:<math>z_{w,t} = z^{*}_{w,t}</math>


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:
Now knowing the height of the weir, the height of the water at each end of the weir, relative to the weir, is calculated:
: <math>h_s = max(0, max( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
: <math>h_d = max(0, min( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
 
 
Based on the relative water heights, the weir uses either a submerged flow <math>Q_s</math> or a free flow <math>Q_f</math> formula, based on the following ratio:
: <math>r_h = \frac{h_d}{h_s}</math>
: <math>r_h = \frac{h_d}{h_s}</math>
:<math>
:<math>
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For free flow, it is calculated directly:
For free flow, it is calculated directly:
: <math>Q_f = f_{w} \cdot c_w \cdot b \cdot ( h_s - h_d )^{3/2}</math>
: <math>Q_f = f_{dw} \cdot C_w \cdot b \cdot ( h_s - h_d )^{3/2}</math>


For submerged flow, the following calculation is used:
For submerged flow, the following calculation is used:
: <math>Q_s = U_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }</math>
: <math>Q_s = f_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }</math>
 
with:
: <math>A = b \cdot (h_s-h_d)</math>


Finally the actual amount of water flow is calculated:
Finally the actual amount of water level change is calculated:
: <math>\Delta w = \frac{\Delta t \cdot Q}{\Delta x}</math>
: <math>\Delta w = \frac{\Delta t \cdot Q}{\Delta x \cdot \Delta x}</math>


Where:
Where:
* <math>w_{l,t}</math> = The water level on the left side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>w_{l,t}</math> = The water level on the left side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>w_{r,t}</math> = The water level on the right side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>w_{r,t}</math> = The water level on the right side of the weir, relative to {{datum}}, at time <math>t</math>.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
* <math>w_{u,t}</math> = The calculated upstream water level, relative to {{datum}}, at time <math>t</math>.
* <math>h_s</math> = The height of the water column relative to the top of the weir, on the side with the highest [[Surface water level formula (Water Overlay)|water level]], at time <math>t</math>.
 
* <math>h_d</math> = The height of the water column relative to the top of the weir, on the side with the lowest [[Surface water level formula (Water Overlay)|water level]], at time <math>t</math>.
* <math>z_{th}</math> = The height of the weir at time t, according to the height adjustment mechanism.
* <math>w_{dt}</math> = The [[Weir drop threshold (Water Overlay)|weir drop threshold]] of the weir.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir at time <math>t</math>.
* <math>\tau_w</math> = The [[Weir target level (Water Overlay)|WEIR_TARGET_LEVEL]] of the the weir.
* <math>\tau_w</math> = The [[Weir target level (Water Overlay)|WEIR_TARGET_LEVEL]] of the the weir.
* <math>\mu</math> = The [[Weir move step m model attribute (Water Overlay)|WEIR_MOVE_STEP_M]] of the water overlay, applicable to all weirs.
* <math>\rho</math> = The [[Weir move range m model attribute (Water Overlay)|WEIR_MOVE_RANGE_M]] of the water overlay, applicable to all weirs.
* <math>T</math> = Current simulated time in seconds.
* <math>T_{wm}</math> = Moment in time in seconds after which the weir can adjust its height again.
* <math>T_{wm}</math> = Moment in time in seconds after which the weir can adjust its height again.
* <math>t_{wm}</math> = The [[Weir move interval model attribute (Water Overlay)|Weir move interval]] of the water overlay, applicable to all weirs.
* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
* <math>z^{*}_{w,t}</math> = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
* <math>z^{*}_{w,t}</math> = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
* <math>z_{th}</math> = The height of the weir at time t, according to the height adjustment mechanism.
* <math>z_{b,l}</math> = The water bottom elevation at left side of the weir the weir.
* <math>z_{b,l}</math> = The water bottom elevation at left side of the weir the weir.
* <math>z_{b,r}</math> = The water bottom elevation at the right side of the weir.
* <math>z_{b,r}</math> = The water bottom elevation at the right side of the weir.
* <math>t_{wd}</math> = The [[Weir drop time s (Water Overlay)|total amount of time]] a weir is dropped until the drop threshold condition is re-evaluated.
* <math>z_{min,w}</math> = The minimum allowed height for weir w.
* <math>t_{wd,t}</math> = The remaining time the weir is dropped.
* <math>z_{max,w}</math> = The maximum allowed height for weir w.
* <math>\mu</math> = The [[Weir move step m (Water Overlay)|WEIR_MOVE_STEP_M]] of the water overlay, applicable to all weirs.
* <math>\rho</math> = The [[Weir move range m (Water Overlay)|WEIR_MOVE_RANGE_M]] of the water overlay, applicable to all weirs.
* <math>T</math> = Current simulated time in seconds.
* <math>t_{wm}</math> = The [[Weir move interval s (Water Overlay)|WEIR_MOVE_INTERVAL_S]] of the water overlay, applicable to all weirs.


* <math>f_{w}</math> = Dutch weir factor, set to 1.7.  
* <math>h_s</math> = The height of the water column relative to the top of the weir, on the side with the highest [[Surface water level formula (Water Overlay)|water level]], at time <math>t</math>.
* <math>h_d</math> = The height of the water column relative to the top of the weir, on the side with the lowest [[Surface water level formula (Water Overlay)|water level]], at time <math>t</math>.
* <math>r_h</math> = The ratio of water heights on either side of the culvert.
* <math>Q</math> = The potential rate of water flow across the weir.
* <math>Q_f</math> = The potential rate of water flow across the weir, based on a free flow calculation.
* <math>Q_s</math> = The potential rate of water flow across the weir, based on a submerged calculation.
* <math>f_{dw}</math> = Dutch weir factor, set to 1.7.  
* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
* <math>C_d</math> = Corresponding Coefficient of discharge. Part of the Dutch weir factor.
* <math>b</math> = The breadth of weir crest, adjustable using the [[Weir width (Water Overlay)|WEIR_WIDTH]].
* <math>b</math> = The breadth of weir crest, adjustable using the [[Weir width (Water Overlay)|WEIR_WIDTH]].
* <math>Q</math> = The potential rate of water flow across the weir.
* <math>f_{loss}</math> = Loss coefficient for submerged weirs, set to 0.9.
* <math>r_h</math> = The ratio of water heights on either side of the culvert.
* <math>Q_{free}</math> = The potential rate of water flow across the weir, based on a free flow calculation.
* <math>Q_{submerged}</math> = The potential rate of water flow across the weir, based on a submerged calculation.
* <math>U_{loss}</math> = Loss coefficient for submerged weirs, set to 0.9.
* <math>A</math> = Flow area, based on the highest water column height relative to the top of the weir, and [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
* <math>A</math> = Flow area, based on the highest water column height relative to the top of the weir, and [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
* <math>g</math> = acceleration due to gravity.
* <math>g</math> = {{gravity}}.
* <math>\Delta w</math> = The water flow which takes place.
* <math>\Delta w</math> = The water level change in meters, which takes place.
* <math>\Delta t</math> = Computational timestep.
* <math>\Delta t</math> = Computational [[Timestep formula (Water Overlay)|timestep]] in seconds.
* <math>\Delta x</math> = Cell size.
* <math>\Delta x</math> = Cell size in meters.


===Free flow coefficients===
===Free flow coefficients===
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When <math>h \approx 0.01</math> meters, the following holds.
When <math>h \approx 0.01</math> meters, the following holds.


:<math>f_{w} \cdot c_w \approx C_d \cdot \sqrt{g}</math>
:<math>f_{dw} \cdot C_w \approx C_d \cdot \sqrt{g}</math>


:<math>C_d=0,633 \cdot (1-\frac{0.0003}{h})^\frac{3}{2}</math>
:<math>C_d=0,633 \cdot (1-\frac{0.0003}{h})^\frac{3}{2}</math>


where:
where:
* <math>f_{w}</math> = Dutch weir factor, set to 1.7.  
* <math>f_{dw}</math> = Dutch weir factor, set to 1.7.  
* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir, set to a default of 1.1
* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir, set to a default of 1.1
* <math>C_d</math> = Calculated Coefficient of discharge.
* <math>C_d</math> = Calculated Coefficient of discharge.
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The following topics are related to this formula.
The following topics are related to this formula.
; Structures
; Structures
: [[Weir_(Water_Overlay)|Weir]]
: [[Weir (Water Overlay)|Weir]]
: [[Breach (Water Overlay)|Breach]]
; Models
; Models
: [[Surface model (Water Overlay)|Surface model]]
: [[Surface model (Water Overlay)|Surface model]]
; Formulas
: [[Breach flow formula (Water Overlay)|Breach flow formula]]
; Test cases
: [[Weir height test case (Water Module)|Weir height test case]]


{{Article end
{{Article end

Latest revision as of 13:21, 5 March 2024

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.

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)
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