Culvert formula (Water Overlay): Difference between revisions

Latest revision as of 13:20, 5 March 2024

Flow through culverts is based on an open channel flow calculation.

The actual datum height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:

B_{c}=\max(T_{c},B_{l},B_{r})

For circular culverts, the flow height $h_{f}$ is:

h_{f}=\max(0,\min(D,\max(w_{l},w_{r})-B_{c})))

For rectangular culverts, the flow depth h is:

h_{f}=\max(0,\min(H,\max(w_{l},w_{r})-B_{c})))

The loss coefficient for the culvert is calculated:

U={\sqrt {\frac {1}{1+{\frac {2g\cdot n^{2}\cdot length}{R_{h}^{\frac {4}{3}}}}}}}

The hydraulic radius $R_{h}$ is calculated as:

R_{h}={\frac {A}{P_{w}}}

For circular culverts, the flow area $A$ and the wetted perimeter $P_{w}$ is calculated using the formula's in the image below.

Where the radius of the culvert:

r={\frac {D}{2}}

For rectangular culverts, the flow area and wetted perimeter is calculated as followed:

A=h_{f}\cdot D

$P_{w}={\begin{cases}D+2h_{f},&{\text{if }}h_{f}<H\\2D+2H,&{\text{if }}h_{f}\geq H\end{cases}}$

The potential flow through the culvert is then calculated:

Q=U\cdot K\cdot {\sqrt {2g\cdot \Vert w_{l}-w_{r}\Vert }}

Finally the actual amount of water flow is calculated:

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

Where:

$D$ = The CULVERT_DIAMETER attribute of the culvert, representing either the inside diameter of a circular culvert or the inside width of a rectangular culvert.
$H$ = The CULVERT_RECTANGULAR_HEIGHT attribute of the culvert, representing the inside height of a rectangular culvert.
$h_{f}$ = The (flow) height of the water inside the culvert.
$T_{c}$ = The CULVERT_THRESHOLD attribute of the culvert.
$B_{c}$ = The datum height of the base of the culvert.
$w_{l}$ = The water level on the left end of the culvert, relative to datum.
$w_{r}$ = The water level on the right end of the culvert, relative to datum.
$R_{h}$ = The hydraulic radius in the culvert^[1].
$R_{w}$ = The wetted perimeter.
$A$ = The flow area.
$K$ = Circular flow area, based on the height of the water in the (circular) culvert.
$g$ = the acceleration due to gravity, set to 9.80665.
$L$ = The length of the culvert, calculated as the distance between the culvert's endpoints.
$U$ = Loss coefficient for culverts.
$n$ = The CULVERT_N attribute of the culvert.
$Q$ = The potential rate of water flow through the culvert in $m^{3}s^{-1}$ .
$\Delta f$ = The water flow which takes place.
$\Delta t$ = Computational timestep in seconds.
$\Delta x$ = Cell size in meters.

References

↑ Hydraulic Radius Equations Formulas Calculator ∙ found at: https://www.ajdesigner.com/phphydraulicradius/hydraulic_radius_equation_pipe.php ∙ (last visited 2019-02-11)

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[hydradius-1] Hydraulic Radius Equations Formulas Calculator ∙ found at: https://www.ajdesigner.com/phphydraulicradius/hydraulic_radius_equation_pipe.php ∙ (last visited 2019-02-11)

[1]

@@ Line 1: / Line 1: @@
 Flow through [[culvert (Water Overlay)|culvert]]s is based on an open channel flow calculation.
-The actual height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:
+The actual datum height of the culvert is at least the height of the terrain on either end of the culvert and the provided threshold height:
 : <math>B_c = \max( T_c , B_l , B_r )</math>
-The radius of the culvert:
+For circular culverts, the flow height <math>h_f</math> is:
-: <math>r = \frac{D}{2}</math>
+: <math>h_f = \max(0,\min(D,\max (w_{l}, w_{r})-B_c)))</math>
+For rectangular culverts, the flow depth h is:
-The height of the water column at either end of the culvert, relative to the culvert, is calculated:
+: <math>h_f = \max(0,\min(H,\max (w_{l}, w_{r})-B_c)))</math>
-: <math>w_{l} = \max(0,w_{l}-B_c)</math>
-: <math>w_{r} = \max(0,w_{r}-B_c)</math>
-[[File:Hydraulic_radius.png|right]]
-Flow depth d is:
-: <math>d = \max(0,\min(D,\max (w_{l}, w_{r})-B_c)))</math>
 The loss coefficient for the culvert is calculated:
 : <math>U = \sqrt{ \frac{1}{ 1 + \frac{2g \cdot n^2 \cdot  length }{  R_{h}^{\frac{4}{3}}} }}</math>
-The hydraulic radius R_h is calculated using the formula's in the image on the right.
+The hydraulic radius <math>R_h</math> is calculated as:
+: <math>R_h = \frac{A}{P_w}</math>
+For circular culverts, the flow area <math>A</math> and the wetted perimeter <math>P_w</math> is calculated using the formula's in the image below.
+[[File:Hydraulic_radius.png|left]]{{clear}}
+Where the radius of the culvert:
+: <math>r = \frac{D}{2}</math>
+For rectangular culverts, the flow area and wetted perimeter is calculated as followed:
+: <math>A = h_f \cdot D</math>
+<math>
+P_w =
+\begin{cases}
+D + 2h_f, & \text{if } h_f < H \\
+D + 2H, & \text{if } h_f \ge H
+\end{cases}
+</math>
 The potential flow through the culvert is then calculated:
-: <math>C = U \cdot K \cdot \sqrt{ 2g \cdot \Vert w_{l} - w_{r} \Vert }</math>
+: <math>Q = U \cdot K \cdot \sqrt{ 2g \cdot \Vert w_{l} - w_{r} \Vert }</math>
 Finally the actual amount of water flow is calculated:
-: <math>\Delta f = \frac{\Delta t \cdot C} { \Delta x}</math>
+: <math>\Delta f = \frac{\Delta t \cdot Q} { \Delta x}</math>
 Where:
-* <math>D</math> = The [[Culvert diameter (Water Overlay)|CULVERT_DIAMETER]] attribute of the culvert.
+* <math>D</math> = The [[Culvert diameter (Water Overlay)|CULVERT_DIAMETER]] attribute of the culvert, representing either the inside diameter of a circular culvert or the inside width of a rectangular culvert.
+* <math>H</math> = The [[Culvert diameter (Water Overlay)|CULVERT_RECTANGULAR_HEIGHT]] attribute of the culvert, representing the inside height of a rectangular culvert.
+* <math>h_f</math> = The (flow) height of the water inside the culvert.
 * <math>T_c</math> = The [[Culvert threshold (Water Overlay)|CULVERT_THRESHOLD]] attribute of the culvert.
-* <math>B_c</math> = The [[Terrain height (Water Overlay)|surface height]] of the base of the culvert.
+* <math>B_c</math> = The [[Terrain height (Water Overlay)|datum height]] of the base of the culvert.
 * <math>w_{l}</math> = The [[Surface water level formula (Water Overlay)|water level]] on the left end of the culvert, relative to {{datum}}.
 * <math>w_{r}</math> = The [[Surface water level formula (Water Overlay)|water level]] on the right end of the culvert, relative to {{datum}}.
 * <math>R_h</math> = The hydraulic radius in the culvert<ref name="hydradius"/>.
-* <math>K</math> = Flow area, based on the height of the water in the (circular) culvert.
+* <math>R_w</math> = The wetted perimeter.
-* <math>g</math> = Acceleration factor of [[Gravity (Water Overlay)|GRAVITY]], defined for the Water Overlay.
+* <math>A</math> = The flow area.
+* <math>K</math> = Circular flow area, based on the height of the water in the (circular) culvert.
+* <math>g</math> = {{gravity}}.
 * <math>L</math> = The length of the culvert, calculated as the distance between the culvert's endpoints.
 * <math>U</math> = Loss coefficient for  culverts.
 * <math>n</math> = The [[Culvert n (Water Overlay)|CULVERT_N]] attribute of the culvert.
-* <math>C</math> = The potential rate of water flow through the culvert.
+* <math>Q</math> = The potential rate of water flow through the culvert in <math>m^3s^{-1}</math>.
 * <math>\Delta f</math> = The water flow 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.
 ==Related==
@@ Line 49: / Line 64: @@
 : [[Surface model (Water Overlay)|Surface model]]
-==References==
+{{article end
+|references=
 <references>
 <ref name="hydradius">Hydraulic Radius Equations Formulas Calculator ∙ found at: https://www.ajdesigner.com/phphydraulicradius/hydraulic_radius_equation_pipe.php ∙ (last visited 2019-02-11)</ref>
 </references>
+}}
 {{WaterOverlay formula nav}}

Culvert formula (Water Overlay): Difference between revisions

Latest revision as of 13:20, 5 March 2024

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