Weir formula (Water Overlay): Difference between revisions

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:<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 \\ and \tau_w > -10_000
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}  

Revision as of 15:10, 25 January 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{*}_{w,t} = \begin{cases} z_{th}, & \text{if} & \|{w_{u,t} - \tau_w}\| > \mu \\ and \tau_w > -10_000 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{min,w} = max ( z_{w,t} - \rho, min (z_{b,l}, z_{b,r} ) )}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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} }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{wm} = T + t_{wm}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_s = max(0, max( w_{l,t}, w_{r,t} ) - z_{w,t})}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_h = \frac{h_d}{h_s}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_s = f_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }}

with:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = b \cdot (h_s-h_d)}

Finally the actual amount of water level change is calculated:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w = \frac{\Delta t \cdot Q}{\Delta x \cdot \Delta x}}

Where:

  • = The water level on the left side of the weir, relative to datum, at time .
  • = The water level on the right side of the weir, relative to datum, at time .
  • = The calculated upstream water level, relative to datum, at time .
  • = The height of the weir at time t, according to the height adjustment mechanism.
  • = The WEIR_TARGET_LEVEL of the the weir.
  • = Moment in time in seconds after which the weir can adjust its height again.
  • = The WEIR_HEIGHT of the weir, at time .
  • = The optionally adjusted height of the weir at time t, depending on the weir height adjustment mechanism.
  • = The water bottom elevation at left side of the weir the weir.
  • = The water bottom elevation at the right side of the weir.
  • = The minimum allowed height for weir w.
  • = The maximum allowed height for weir w.
  • = The WEIR_MOVE_STEP_M of the water overlay, applicable to all weirs.
  • = The WEIR_MOVE_RANGE_M of the water overlay, applicable to all weirs.
  • = Current simulated time in seconds.
  • = The WEIR_MOVE_INTERVAL_S of the water overlay, applicable to all weirs.
  • = The height of the water column relative to the top of the weir, on the side with the highest water level, at time .
  • = The height of the water column relative to the top of the weir, on the side with the lowest water level, at time .
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_h} = The ratio of water heights on either side of the culvert.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} = The potential rate of water flow across the weir.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_f} = The potential rate of water flow across the weir, based on a free flow calculation.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_s} = The potential rate of water flow across the weir, based on a submerged calculation.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{dw}} = Dutch weir factor, set to 1.7.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_w} = The WEIR_COEFFICIENT of the weir.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} = The breadth of weir crest, adjustable using the WEIR_WIDTH.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{loss}} = Loss coefficient for submerged weirs, set to 0.9.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} = Flow area, based on the highest water column height relative to the top of the weir, and WEIR_WIDTH of the weir.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} = the acceleration due to gravity, set to 9.80665.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w} = The water level change in meters, which takes place.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} = Computational timestep in seconds.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \approx 0.01} meters, the following holds.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{dw} \cdot C_w \approx C_d \cdot \sqrt{g}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_d=0,633 \cdot (1-\frac{0.0003}{h})^\frac{3}{2}}

where:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{dw}} = Dutch weir factor, set to 1.7.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_w} = The WEIR_COEFFICIENT of the weir, set to a default of 1.1
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_d} = Calculated Coefficient of discharge.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} = acceleration due to gravity.
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} = Weir head; the height of the water column relative to the top of the weir.

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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)
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