Wind calculation model (Heat Overlay)

The wind speed on streetlevel (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 streetlevel. 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

Our implementation differs slightly from the implementation described by DPRA Heat stress report. It does not do cubic interpolation from coarse grids to more detailed grids. We do all operations on a detailed grid.

Our implementation is as followed:

Average window

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:

Per grid cell

Per wind window grid cell:

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 $\lambda$, separately for buildings and trees. Combine these to obtain $\lambda _{cor}$.
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 $d$, $z_{w}$, $z_{0}$, $A$ 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.

Step 1

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)} }

Step 2

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.

Step 3

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} .

Step 4

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.

Step 5

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

Step 6

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))}

Step 7

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 main directions: North, east, south, west.

See also

• Wind speed result type

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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