Wind calculation model (Heat Overlay): Difference between revisions

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==Implementation==
==Implementation==
Our implementation differs slightly from the implementation described by {{Template:Heat_DPRA_Report}}. It does not do cubic interpolation from coarse grids to more detailed grids. We do all operations on a detailed grid.
The calculation implemented in the {{software}} differs from the implementation described by the {{Template:Heat_DPRA_Report}}. Specifically, rather than performing the described cubic interpolation from coarse grids to more detailed grids, all operation are done on a detailed grid.


Our implementation is as followed:
===Averaging window===
===Average window===
{{main|Average calculation model (Heat Overlay)#Small window|Average calculation model (Heat Overlay)}}
First of all, the wind window has to be selected. If the wind speed is lower than 1.5 m/s, a window of 175m x 175m is used for averaging. Otherwise a window of 140m x 280m will be used. See the image below:
The average wind speed is based on an average of wind speeds across a larger area. The window for averaging is established based on wind speed and direction. For slow wind speeds (below 1.5m/s), a wind speed of 175x175m is used. For high wind speeds, a window of 280x140m if used.
 
[[File:Wind_window_sq(Heat_Overlay).png|250px]][[File:Wind_window_(Heat_Overlay).png|250px]]


===Per grid cell===
===Per grid cell===
Per wind window grid cell:
Per grid cell in the averaging window:
# Determine the average obstacle height per square meter. Buildings and trees lower than 2 meters are not taken into account. It is divided by the area of the search window.
# Determine the average obstacle height per square meter. Buildings and trees lower than 2 meters are not taken into account. It is divided by the area of the search window.
# Determine the average vertical building surface perpendicular to the wind direction. Again this is divided by the area of the search window.
# Determine the average vertical building surface perpendicular to the wind direction. Again this is divided by the area of the search window.
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# Obtain the wind speed at <math>u_{1,2}</math> by multiplying the wind translation factor with the <math>u_{10}</math> measured at the weather station. Since the wind speed can still be too low, apply a correction formula to <math>u_{1,2}</math> to obtain the end result.
# Obtain the wind speed at <math>u_{1,2}</math> by multiplying the wind translation factor with the <math>u_{10}</math> measured at the weather station. Since the wind speed can still be too low, apply a correction formula to <math>u_{1,2}</math> to obtain the end result.


====Step 1====
====Determine average object height====
For each cell in the wind direction window the average obstacle height of the buildings and foliage are calculated. Only foliage and buildings with a minimum height of 4.0 area taken into account.
For each cell in the wind direction window the average obstacle height of the buildings and foliage are calculated. Only foliage and buildings with a minimum height of 4.0 area taken into account.
If no obstacles were present, the average height will still be at the minimum of 4 meters.
If no obstacles were present, the average height will still be at the minimum of 4 meters.
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<math>H_{obstacles} = \frac{\sum_{c=0}^n\sum_{r=0}^m (H_{r,c}>2.0) \cdot \max(H_{r,c}, 4.0)}{\sum_{c=0}^n\sum_{r=0}^m (H_{r,c}>2.0)} </math>
<math>H_{obstacles} = \frac{\sum_{c=0}^n\sum_{r=0}^m (H_{r,c}>2.0) \cdot \max(H_{r,c}, 4.0)}{\sum_{c=0}^n\sum_{r=0}^m (H_{r,c}>2.0)} </math>


====Step 2====
====Determine average vertical surface perpendicular to wind direction====
[[File:Wind_grid_slices.png|350px|thumb|right|Evaluated horizontal slizes from top to bottom, for a wind window of a wind coming from the west]]
[[File:Wind_grid_slices.png|350px|thumb|right|Evaluated horizontal slizes from top to bottom, for a wind window of a wind coming from the west]]
For each strip in the direction of the wind:
For each strip in the direction of the wind:
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For <math>\lambda_{foliage}</math> the same process is repeated, but then with [[Foliage_height_calculation_model_(Heat_Overlay)|foliage heights]] instead of Building heights.
For <math>\lambda_{foliage}</math> the same process is repeated, but then with [[Foliage_height_calculation_model_(Heat_Overlay)|foliage heights]] instead of Building heights.


====Step 3====
====Determine the combined frontal surface compactness factor====
First, calculate <math>\lambda_{buildings}</math> and <math>\lambda_{foliage}</math> in a similar way, described in step 2.
First, calculate <math>\lambda_{buildings}</math> and <math>\lambda_{foliage}</math> in a similar way, described in step 2.


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<math>\lambda_{cor} = 0.6 \cdot \lambda_{buildings} + 0.3 \cdot \lambda_{foliage} + 0.015</math>.
<math>\lambda_{cor} = 0.6 \cdot \lambda_{buildings} + 0.3 \cdot \lambda_{foliage} + 0.015</math>.


====Step 4====
====Determine average tree height====
Calculate the average obstacle height <math>H</math>, where a minimum height threshold of 4 is used. This means it is always at least 4 meters high.
Calculate the average obstacle height <math>H</math>, where a minimum height threshold of 4 is used. This means it is always at least 4 meters high.


====Step 5====
====Select applicable formula parameters from McDonald 2000 table====
Select the function parameters <math>d</math>, <math>z_w</math>, <math>z_0</math>, <math>A</math> and <math>B</math> from the table using <math>H</math> and <math>\lambda_{cor}</math>:  
Select the function parameters <math>d</math>, <math>z_w</math>, <math>z_0</math>, <math>A</math> and <math>B</math> from the table using <math>H</math> and <math>\lambda_{cor}</math>:  


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|}
|}


====Step 6====
====Caclulate wind translation factor====
Calculate the translated intermediate wind speeds:
Calculate the translated intermediate wind speeds:


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<math>u_{1.2} = u_{H} \exp(9.8 \cdot \lambda \cdot (\frac{1.2}{H}-1))</math>
<math>u_{1.2} = u_{H} \exp(9.8 \cdot \lambda \cdot (\frac{1.2}{H}-1))</math>


====Step 7====
====Obtain final corrected wind speed====
The obtained wind speed <math>u_{1.2}</math> can be too low. Therefore it is corrected with the following formula:
The obtained wind speed <math>u_{1.2}</math> can be too low. Therefore it is corrected with the following formula:


<math>u_{1.2 corr} = u_{10} \cdot ((u_{1.2} - 0.0796) \cdot 0.9175 + 0.1254)</math>
<math>u_{1.2 corr} = u_{10} \cdot ((u_{1.2} - 0.0796) \cdot 0.9175 + 0.1254)</math>


This corrected value is used in the formula for [[PET_formulas_(Heat_Overlay)|PET]].
This corrected value is used in the formula for [[PET formulas (Heat Overlay)|PET]].


==Notes==
==Notes==
* The model currently only calculates the adjusted wind speeds in the 4 main directions: North, east, south, west.
* The model currently only calculates the adjusted wind speeds in the 4 cardinal directions: north, east, south, west.


==See also==
==See also==
* [[Wind_speed_result_type_(Heat_Overlay)|Wind speed result type]]
* [[Wind speed result type (Heat Overlay)|Wind speed result type]]


{{Template:HeatOverlay_formula_nav}}
{{Template:HeatOverlay formula nav}}


{{Template:Heat_Module_buttons}}
{{Template:Heat Module buttons}}

Revision as of 14:12, 18 December 2019

Frontal surface and average obstacle heights are both parameters for the wind translation to 1.2 meters above ground.

The wind speed on street level (1.2 meter) is calculated using a translation of the 10 meter wind speed measured at a weather station. A single reduction field per wind direction can be calculated, which in turn can be multiplied with the 10 meter wind speed to obtain the wind speed at street level. The calculation model is based on the McDonald (2000) method for wind in cities. It uses the vertical surfaces of buildings perpendicular to the wind direction to determine in what way the wind speed is slowed down. The DPRA Heat stress report has extended this method with perpendicular vertical tree surfaces as well.

Implementation

The calculation implemented in the Tygron Platform differs from the implementation described by the DPRA Heat stress report. Specifically, rather than performing the described cubic interpolation from coarse grids to more detailed grids, all operation are done on a detailed grid.

Averaging window

Main article: Average calculation model (Heat Overlay)

The average wind speed is based on an average of wind speeds across a larger area. The window for averaging is established based on wind speed and direction. For slow wind speeds (below 1.5m/s), a wind speed of 175x175m is used. For high wind speeds, a window of 280x140m if used.

Per grid cell

Per grid cell in the averaging window:

  1. Determine the average obstacle height per square meter. Buildings and trees lower than 2 meters are not taken into account. It is divided by the area of the search window.
  2. Determine the average vertical building surface perpendicular to the wind direction. Again this is divided by the area of the search window.
  3. Determine the frontal surface compactness factor 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 \lambda} , separately for buildings and trees. Combine these to obtain 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 \lambda_{cor}} .
  4. Determine the average height of trees, with 4 meters as minimum value.
  5. Using these values, select the correct formula parameters from the McDonald 2000 table 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 d} , 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} , 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_0} , 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} and 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}
  6. Using the obtained parameters, calculate the wind translation factor.
  7. Obtain the wind speed at 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 u_{1,2}} by multiplying the wind translation factor with the 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 u_{10}} measured at the weather station. Since the wind speed can still be too low, apply a correction formula to 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 u_{1,2}} to obtain the end result.

Determine average object height

For each cell in the wind direction window the average obstacle height of the buildings and foliage are calculated. Only foliage and buildings with a minimum height of 4.0 area taken into account. If no obstacles were present, the average height will still be at the minimum of 4 meters.

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_{obstacles} = \frac{\sum_{c=0}^n\sum_{r=0}^m (H_{r,c}>2.0) \cdot \max(H_{r,c}, 4.0)}{\sum_{c=0}^n\sum_{r=0}^m (H_{r,c}>2.0)} }

Determine average vertical surface perpendicular to wind direction

Evaluated horizontal slizes from top to bottom, for a wind window of a wind coming from the west

For each strip in the direction of the wind:

Check if the height of buildings is higher than the previous cell: 

  if 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_{curr} > H_{prev}}
 and 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 > 2.0}

     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 S_{frontal} = S_{frontal} + (H_{curr} - H_{prev}) \cdot w_{cell}}

  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_{prev} = H_{curr}}


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 w_{cell}} is the width of a cell. So we only calculate additional heights.

Next, divide 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 S_{frontal}} by n, the amount of cells visited to obtain 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 \lambda} .

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 \lambda_{buildings} = S_{frontal} / n_{cells}}

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 \lambda_{foliage}} the same process is repeated, but then with foliage heights instead of Building heights.

Determine the combined frontal surface compactness factor

First, calculate 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 \lambda_{buildings}} and 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 \lambda_{foliage}} in a similar way, described in step 2.

Next, calculate the 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 \lambda_{cor}} using the 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 \lambda_{buildings}} and 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 \lambda_{foliage}} .

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 \lambda_{cor} = 0.6 \cdot \lambda_{buildings} + 0.3 \cdot \lambda_{foliage} + 0.015} .

Determine average tree height

Calculate the average obstacle height 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} , where a minimum height threshold of 4 is used. This means it is always at least 4 meters high.

Select applicable formula parameters from McDonald 2000 table

Select the function parameters 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 d} , 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} , 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_0} , 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} and 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} from the table using 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} and 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 \lambda_{cor}} :

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 \lambda_{cor}} 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 \frac{d}{H}} 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 \frac{z_w}{H}} 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 \frac{z_0}{H}} 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 \frac{A}{H}} 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}
0.05 (<0.08) 0.066 2 0.048 -0.35 0.56
0.11 (<0.135) 0.26 2.5 0.071 -0.35 0.50
0.16 (<0.18) 0.32 2.7 0.084 -0.34 0.48
0.20 (<0.265) 0.47 1.5 0.08 -0.56 0.66
0.33 (>=0.265) 0.57 1.2 0.077 -0.85 0.92

Caclulate wind translation factor

Calculate the translated intermediate wind speeds:

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 u_{zw} = u_{60} \cdot \frac{ \ln (\frac{z_w-d}{z_0})}{ \ln (\frac{60-d}{z_0})}}

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 u^* = 0.4 \cdot \frac{u_{60}}{\ln (\frac{60 - d}{z_0})}}

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 u_{H} = \frac{-u^*}{B} \ln \frac{A+B_{zw}}{A+B_H} +U_{zw}}

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 u_{1.2} = u_{H} \exp(9.8 \cdot \lambda \cdot (\frac{1.2}{H}-1))}

Obtain final corrected wind speed

The obtained wind speed 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 u_{1.2}} can be too low. Therefore it is corrected with the following formula:

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 u_{1.2 corr} = u_{10} \cdot ((u_{1.2} - 0.0796) \cdot 0.9175 + 0.1254)}

This corrected value is used in the formula for PET.

Notes

  • The model currently only calculates the adjusted wind speeds in the 4 cardinal directions: north, east, south, west.

See also

Heat Module
Benchmarks button.png
Heat introduction button.png
Heat Module
Benchmarks button.png
Getting started button.png
Tutorials
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Overview button.png
Overview
Benchmarks button.png
Theory button.png
Theory
Benchmarks button.png
Benchmarks button.png
Benchmarks
Group
Formulas
UHI • Temperatures • Diffuse radiation • PET
Calculation models
Foliage height • Sky view factor • Shade • Average vegetation and sky view factor
Related
Calculation models and formulas • Feature attributes • Model attributes • Result types



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Heat introduction button.png
Heat Module
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Getting started button.png
Tutorials
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Overview button.png
Overview
Benchmarks button.png
Theory button.png
Theory
Benchmarks button.png
Benchmarks button.png
Benchmarks
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