Weir formula (Water Overlay): Difference between revisions

Revision as of 15:32, 15 December 2022

Flow across weirs is calculated differently for free flow and submerged flow.

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

h_{s}=max(0,max(w_{l},w_{r})-z_{w,t})

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

Optionally, when a Weir drop threshold is configured, the weir height $z_{w},t$ at time $t$ can update depending on the water level and the remaining weir drop time:

z_{w,t}={\begin{cases}z_{b},&{\mbox{if }}t_{wd,t}>0\\z_{w},&{\mbox{otherwise }}\end{cases}}

t_{wd,t}={\begin{cases}t_{wd},&{\mbox{if }}max(w_{l},w_{r})>w_{dt}{\mbox{ and }}t_{wd,t}<=0\\t_{wd,t-1}-\Delta t,&{\mbox{otherwise}}\end{cases}}

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:

h_{ratio}={\frac {h_{d}}{h_{s}}}

Q={\begin{cases}min(Q_{submerged},Q_{free})&{\mbox{if }}h_{ratio}>0.5\\Q_{free},&{\mbox{otherwise}}\end{cases}}

For free flow, capacity is calculated directly:

Q_{free}=f_{w,d}\cdot c_{w}\cdot w_{w}\cdot (h_{s}-h_{d})^{3/2}

For submerged flow, the following calculation is used:

Q_{submerged}=U_{loss}\cdot A\cdot {\sqrt {2\cdot g\cdot (h_{s}-h_{d})}}

Finally the actual amount of water flow is calculated:

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

Where:

$h_{s}$ = The height of the water column relative to the top of the weir, on the side with the highest water level.
$h_{d}$ = The height of the water column relative to the top of the weir, on the side with the lowest water level.
$w_{l}$ = The water level on the left side of the weir, relative to datum.
$w_{r}$ = The water level on the right side of the weir, relative to datum.
$w_{dt}$ = The weir drop threshold of the weir.
$z_{w,t}$ = The height of the weir at time t, depending on drop mechanism.
$z_{w}$ = The WEIR_HEIGHT of the weir.
$z_{b}$ = The water bottom elevation at the weir.
$t_{wd}$ = The total amount of time a weir is dropped until the drop threshold condition is re-evaluated.
$t_{wd,t}$ = The remaining time the weir is dropped.
$f_{w,d}$ = Dutch weir factor, set to 1.7.
$c_{w}$ = The WEIR_COEFFICIENT of the weir.
$w_{w}$ = The WEIR_WIDTH of the weir.
$Q$ = The potential rate of water flow across the weir.
$h_{ratio}$ = The ratio of water heights on either side of the culvert.
$Q_{free}$ = The potential rate of water flow across the weir, based on a free flow calculation.
$Q_{submerged}$ = The potential rate of water flow across the weir, based on a submerged calculation.
$U_{loss}$ = 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.
$g$ = Acceleration factor of gravity, set to 9.80665.
$\Delta w$ = The water flow which takes place.
$\Delta t$ = Computational timestep.
$\Delta x$ = Cell size.

@@ Line 2: / Line 2: @@
 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_s = max(0, max( w_l, w_r ) - z_{w,t})</math>
-: <math>h_d = max(0, min( w_{l,t}, w_{r,t} ) - z_{w,t})</math>
+: <math>h_d = max(0, min( w_l, w_r ) - z_{w,t})</math>
+Optionally, when a Weir drop threshold is configured, the weir height <math>z_w,t</math> at time <math>t</math> can update depending on the water level and the remaining weir drop time:
+:<math>z_{w,t} =
+\begin{cases}
+z_b, & \mbox{if }t_{wd,t} > 0 \\
+z_w, & \mbox{otherwise }
+\end{cases}
+</math>
+:<math>t_{wd,t} =
+\begin{cases}
+t_{wd}, & \mbox{if } max( w_l, w_r) > w_{dt} \mbox{ and } t_{wd,t} <= 0 \\
+t_{wd, t-1} - \Delta t, & \mbox{otherwise}
+\end{cases}
+</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:
-: <math>r_h = \frac{h_d}{h_s}</math>
+: <math>h_{ratio} = \frac{h_d}{h_s}</math>
-:<math>
+: <math>Q =
-Q =
+\begin{cases}
-\begin{cases}
+min ( Q_{submerged}, Q_{free}) & \mbox{if } h_{ratio} > 0.5 \\
-min ( Q_s , Q_f), & \text{if } r_h > 0.5 \\
+Q_{free}, & \mbox{otherwise}
- Q_f, & \text{otherwise}
 \end{cases}
 </math>
-For free flow, it is calculated directly:
+For free flow, capacity is calculated directly:
-: <math>Q_f = f_{w} \cdot c_w \cdot b \cdot ( h_s - h_d )^{3/2}</math>
+: <math>Q_{free} = f_{w,d} \cdot c_w \cdot w_w \cdot ( h_s - h_d )^{3/2}</math>
 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_{submerged} = U_{loss} \cdot A \cdot \sqrt{ 2 \cdot g \cdot ( h_s - h_d ) }</math>
 Finally the actual amount of water flow is calculated:
-: <math>\Delta w = \frac{\Delta t \cdot Q}{\Delta x}</math>
+: <math>\Delta w = \Delta t \cdot \frac{Q}{\Delta x}</math>
 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>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]].
-* <math>w_{r,t}</math> = The water level on the right side of the weir, relative to {{datum}}, 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]].
-* <math>z_{w,t}</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir, at time <math>t</math>.
+* <math>w_l</math> = The water level on the left side of the weir, relative to {{datum}}.
-* <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>w_r</math> = The water level on the right side of the weir, relative to {{datum}}.
-* <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>w_{dt}</math> = The [[Weir drop threshold (Water Overlay)|weir drop threshold]] of the weir.
-* <math>f_{w}</math> = Dutch weir factor, set to 1.7.
+* <math>z_{w,t}</math> = The height of the weir at time t, depending on drop mechanism.
-* <math>C_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
+* <math>z_w</math> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] of the weir.
-* <math>C_d</math> = Corresponding Coefficient of discharge. Part of the Dutch weir factor.
+* <math>z_b</math> = The water bottom elevation at the weir.
-* <math>b</math> = The breadth of weir crest, adjustable using the [[Weir width (Water Overlay)|WEIR_WIDTH]].
+* <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>t_{wd,t}</math> = The remaining time the weir is dropped.
+* <math>f_{w,d}</math> = Dutch weir factor, set to 1.7.
+* <math>c_w</math> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] of the weir.
+* <math>w_w</math> = The [[Weir width (Water Overlay)|WEIR_WIDTH]] of the weir.
 * <math>Q</math> = The potential rate of water flow across the weir.
-* <math>r_h</math> = The ratio of water heights on either side of the culvert.
+* <math>h_{ratio}</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>g</math> = acceleration due to gravity.
+* <math>g</math> = Acceleration factor of gravity, set to 9.80665.
 * <math>\Delta w</math> = The water flow which takes place.
 * <math>\Delta t</math> = Computational timestep.
 * <math>\Delta x</math> = Cell size.
-===Free flow coefficients===
-To clarify the free flow formula in comparison with other definitions:
-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>C_d=0,633 \cdot (1-\frac{0.0003}{h})^\frac{3}{2}</math>
-For <math>h</math> 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.
-where:
-* <math>f_{w}</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_d</math> = Calculated Coefficient of discharge.
-* <math>g</math> = acceleration due to gravity.
-* <math>h</math> = Weir head; the height of the water column relative to the top of the weir.
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Weir formula (Water Overlay): Difference between revisions

Revision as of 15:32, 15 December 2022

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