Weir formula (Water Overlay): Difference between revisions

Revision as of 10:04, 15 April 2019

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)

h_d = max(0, min( w_l, w_r ) - z_w)

For free flow, capacity is calculated directly:

C_free = f_w,d * c_w * w_w * ( h_s - h_d )^3/2

For submerged flow, a culvert-like calculation is used:

C_submerged = U_loss * A * sqrt( 2 * g * ( h_s - h_d ) )

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 = h_d / h_s

C = min( C_submerged , C_free) if h_ratio > 0,5

C = C_free otherwise

Finally the actual amount of water flow is calculated:

Δw = Δt * C / cell

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.
z_w = The WEIR_HEIGHT attribute of the weir.
f_w,d = Dutch weir factor, set to 1.7.
c_w = The WEIR_COEFFICIENT attribute of the weir.
w_w = The WEIR_WIDTH attribute of the weir.
C_free = The potential rate of water flow across the weir, based on a free flow calculation.
C_submerged = The potential rate of water flow across the weir, based on a submerged calculation.

C = The potential rate of water flow across the weir.
h_ratio = The ratio of water heights on either side of the culvert.

U_loss = Loss coefficient for submerged weirs (0,9).
A = Flow area, based on the highest water column height relative to the top of the weir, and the weir width, defined by the WEIR_WIDTH attribute of the weir
g = Acceleration factor of gravity.

Δw = The water flow which takes place.
Δt = Computational timestep.
cell = Cell size.

@@ Line 4: / Line 4: @@
 : ''h<sub>s</sub> = max(0, max( w<sub>l</sub>, w<sub>r</sub> ) - z<sub>w</sub>)''
 : ''h<sub>d</sub> = max(0, min( w<sub>l</sub>, w<sub>r</sub> ) - z<sub>w</sub>)''
-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<sub>ratio</sub> = h<sub>d</sub> / h<sub>s</sub>''
-:: C = min( C<sub>submerged</sub> , C<sub>free</sub>) if h<sub>ratio</sub> &gt; 0,5
-:: C = C<sub>free</sub> otherwise
 For free flow, capacity is calculated directly:
@@ Line 15: / Line 10: @@
 For submerged flow, a culvert-like calculation is used:
 : ''C<sub>submerged</sub> = U<sub>loss</sub> * A * sqrt( 2 * g * ( h<sub>s</sub> - h<sub>d</sub> ) )''
+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<sub>ratio</sub> = h<sub>d</sub> / h<sub>s</sub>''
+:: C = min( C<sub>submerged</sub> , C<sub>free</sub>) if h<sub>ratio</sub> &gt; 0,5
+:: C = C<sub>free</sub> otherwise
 Finally the actual amount of water flow is calculated:
@@ Line 20: / Line 20: @@
 Where:
-* Δw = The water flow which takes place.
+* h<sub>s</sub> = The height of the water column relative to the top of the weir, on the side with the highest water level.
-* Δt = Computational timestep.
+* h<sub>d</sub> = The height of the water column relative to the top of the weir, on the side with the lowest water level.
-* cell = Cell size.
+* w<sub>l</sub> = The water level on the left side of the weir, relative to [[Datum|datum]].
+* w<sub>r</sub> = The water level on the right side of the weir, relative to datum.
+* z<sub>w</sub> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] attribute of the weir.
+* f<sub>w,d</sub> = Dutch weir factor, set to 1.7.<!--<ref name="duflowfactor" />-->
+* c<sub>w</sub> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] attribute of the weir.
+* w<sub>w</sub> = The [[Weir width (Water Overlay)|WEIR_WIDTH]] attribute of the weir.
+* C<sub>free</sub> = The potential rate of water flow across the weir, based on a free flow calculation.
+* C<sub>submerged</sub> = The potential rate of water flow across the weir, based on a submerged calculation.
 * C = The potential rate of water flow across the weir.
 * h<sub>ratio</sub> = The ratio of water heights on either side of the culvert.
-* C<sub>free</sub> = The potential rate of water flow across the weir, based on a free flow calculation.
-* C<sub>submerged</sub> = The potential rate of water flow across the weir, based on a submerged calculation.
-* h<sub>s</sub> = The height of the water column relative to the top of the weir, on the side with the highest water level.
-* h<sub>d</sub> = The height of the water column relative to the top of the weir, on the side with the lowest water level.
 * U<sub>loss</sub> = Loss coefficient for submerged weirs (0,9).
 * A = Flow area, based on the highest water column height relative to the top of the weir, and the weir width, defined by the [[Weir width (Water Overlay)|WEIR_WIDTH]] attribute of the weir
-* g = Acceleration factor of gravity
+* g = Acceleration factor of gravity.
-* f<sub>w,d</sub> = Dutch weir factor, set to 1.7<!--<ref name="duflowfactor" />.-->
-* c<sub>w</sub> = The [[Weir coefficient (Water Overlay)|WEIR_COEFFICIENT]] attribute of the weir.
-* z<sub>w</sub> = The [[Weir height (Water Overlay)|WEIR_HEIGHT]] attribute of the weir.
+* Δw = The water flow which takes place.
-* w<sub>w</sub> = The [[Weir width (Water Overlay)|WEIR_WIDTH]] attribute of the weir.
+* Δt = Computational timestep.
-* w<sub>l</sub> = The water level on the left side of the weir, relative to [[Datum|datum]].
+* cell = Cell size.
-* w<sub>r</sub> = The water level on the right side of the weir, relative to datum.
 ==See also==
 * [[Weir_(Water_Overlay)|Weir]]
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Weir formula (Water Overlay): Difference between revisions

Revision as of 10:04, 15 April 2019

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