Wind logarithm and Power law profiles Basim Alknani Vertical Wind Profile in boundary layer Wind speed is zero at the surface and increases sharply in the near surface layer such that the wind speed increases exponentially with height . The
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Presentation Transcript
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Wind logarithm and Power law profiles Basim Alknani<br>
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Vertical Wind Profile in boundary layer
Wind speed is zero at the surface and increases sharply in the near surface layer such that the wind speed increases exponentially with height . The shape of the profile is influenced by the nature of the underlying surface, the surrounding topography and nearby obstacles. Wind speed shows an increase with respect to height and as such it is generally beneficial to discharge pollutants at a greater height as dispersion is greater. The rate of change with height (known as wind shear) is greatest near the surface. شكل 2 شكل 1 تغيركبير تغيرصغير<br>
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Aerodynamic Roughness Length:
the aerodynamic roughness length, z0, is defined as the height where the wind speed becomes zero. the aerodynamic roughness length is determined for a particular surface, it does not change with wind speed, stability, or stress. it can change if the roughness elements on the surface change such as caused by changes in the height and coverage of vegetation, manufacture of fences, construction of houses, deforestation or lumbering, etc. Letta (1969) suggested a method for estimating the aerodynamic roughness length based on the average vertical extent of the roughness elements (𝒉∗), the average vertical cross –section area obtainable to the wind by one element (S𝒔), and [ SL = (total ground surface area / number of elements)].
𝑧0 = 0.5 ℎ∗ (S𝑠/ 𝑆𝐿) ………………………………… (1.1) This relationship is acceptable when the roughness elements are evenly spaced, not too close together, and have similar height and shape.<br>
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The shearing stress exerted on a surface by fluid flow is generated within the boundary layer and transmitted downwards to the surface in the form of a momentum flux. ( shearing stress can be expressed , force per unit area , or momentum per unit area per unit time ) .<br>
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There are two mathematical models that are widely used to model the vertical profile of wind speed over regions of homogeneous and flat terrain, which are :
a- Logarithmic Law profile.
b- Power Law profile.<br>
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Where (L) is the Obukhov length and the function Ψ (𝑧/𝐿) is given for stable conditions (z/L > 0) by :
Ψ (𝑧/𝐿) = 4.7 (𝑧/ 𝐿) …………………….…………… 1.8
And for unstable (z/L < 0) by:
Ψ (𝑧/𝐿) = −2ln[(1+𝑥)/ 2] −ln[(1+𝑥2)/ 2] +2𝑡𝑎𝑛−1(𝑥)-𝜋 /2………………. 1.9
Where 𝑥 = [1− (15 𝑧/𝐿)]1/4.<br>
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There is another method used to estimate the shear exponent, which is:<br>
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HOMEWORK 1
Q1 ) write a mathematical expression of the following:
1- Wind speed logarithmic equation in stable conditions.
2- Wind speed logarithmic equation in unstable conditions.
3- Wind speed logarithmic equation in neutral conditions.
4- power-law equation for wind profile<br>
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HOMEWORK 2
Q2 ) write a mathematical expression of the following:
1- The aerodynamic roughness length equation when the roughness elements are evenly spaced.
2- The aerodynamic roughness length equation which uses in an urban city
3- Displacement distance equation .
4- Friction velocity equation .
5- 𝑑𝑟𝑎𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 equation .<br>
