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Flow over the surface
Concept 
Wind corresponds to a mass of air moving parallel to the surface, and due to its interaction with the surface, it exhibits a velocity gradient. This gradient is null at the surface and increases as one moves away from it. The relationship between the wind speed with height ($V_z$) and the height above ground ($z$) follows a logarithmic function:
$V_z \propto \ln z$
The shape of this function depends on the roughness of the surface, represented by the parameter the roughness length ($z_o$), which in turn depends on the surface structure:
Additionally, the wind speed at height depends on the speed measured at 10 meters above ground level, characterized by the friction speed ($u$). The velocity profile with height is modeled as follows:
| $ V_z = \displaystyle\frac{2}{5} u \ln\left(\displaystyle\frac{ z }{ z_o }\right)$ |
ID:(762, 0)
Tension on the surface
Concept
As the mass of air moves over the body, it tends to drag the surface, creating a surface friction shear stress ($\sigma_a$):
This depends on the air density ($\rho_a$) and on a velocity associated with friction, which we'll denote as the friction speed ($u$), as follows:
| $ \sigma_a = \rho_a u ^2$ |
ID:(763, 0)
Pressure on the surface
Concept
The pressure on the surface ($p_z$), which acts vertically to the surface:
and is generally smaller than the atmospheric pressure ($p_0$) due to the effects of air displacement from the air density ($\rho_a$) with the wind speed with height ($V_z$).
In this case, we can model it using the Bernoulli equation with its kinetic energy term:
$\displaystyle\frac{1}{2} \rho_a V_Z^2$
This factor is corrected with a factor depending on the shape of the body, the aerodynamic shape factor ($C_a$), and a factor originating from fluctuations due to turbulence vortices, the surface pressure reduction ($q_z$), resulting in:
| $ q_z = \displaystyle\frac{1}{2} \rho_a V_z ^2 C_d C_a$ |
Thus, the pressure on the surface ($p_z$) can be calculated resulting in:
| $ p_z = p_0 - q_z $ |
ID:(768, 0)
Aerodynamic shape factor
Concept
The effect of wind on a structure depends on the direction in which it strikes the surface. If the wind direction is represented by an arrow, we have the following scenarios:
| W | windward (against the wind) |
| S | side |
| L | leeward (with the wind) |
| U | roof slope against the wind |
| R | roof slope with crosswind |
| D | roof slope with the wind |
To better understand these, examples provided in the AS/NZS 1170.2 standard can be consulted:
In summary, the aerodynamic shape factor can be modeled by a factor that depends on the angle between the surface normal and the wind direction. Based on experimental data from various shapes, this factor can be approximated by a shape curve:
It's important to realize that the function implies the existence of areas of positive pressure as well as areas of negative pressure, which correspond to areas where the wind literally sucks.
ID:(760, 0)
Model
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Variable 
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Target : Equation To be used 
Equations
$ C_a = 0.3445 \theta ^2 - 1.4961 \theta + 0.8$
C_a = 0.3445* theta ^2 - 1.4961* theta + 0.8
$ p_z = p_0 - q_z $
p_z = p_0 - q_z
$ q_z = \displaystyle\frac{1}{2} \rho_a V_z ^2 C_d C_a$
q_z = rho_a * V_z ^2* C_d * C_a /2
$ \sigma_a = \rho_a u ^2$
sigma_a = rho_a * u ^2
$ V_z = \displaystyle\frac{2}{5} u \ln\left(\displaystyle\frac{ z }{ z_o }\right)$
V_z = 2* u * log( z / z_o )/5
ID:(764, 0)
Speed with height
Equation
The wind speed with height ($V_z$) depends on the height above ground ($z$). Typically, it is virtually negligible at the surface and reaches the value reported in meteorological reports at a height of 10 meters. Its variation is influenced by the terrain roughness, expressed by the roughness length ($z_o$), and by the friction speed ($u$), as follows:
| $ V_z = \displaystyle\frac{2}{5} u \ln\left(\displaystyle\frac{ z }{ z_o }\right)$ |
ID:(757, 0)
Tension on the surface
Equation
The surface friction shear stress ($\sigma_a$) is generated by the mass of air moving over the surface. This is described by the friction speed ($u$) and the air density ($\rho_a$) according to the following equation:
| $ \sigma_a = \rho_a u ^2$ |
ID:(758, 0)
Reduction of pressure on the surface
Equation
The surface pressure reduction ($q_z$) is the pressure per unit area by which it decreases on the surface of the body. It is modeled as a modification of the Bernoulli's principle, characterized by the air density ($\rho_a$) and the wind speed with height ($V_z$), correcting for dynamics with the dynamic response factor ($C_d$), and geometry with the aerodynamic shape factor ($C_a$):
| $ q_z = \displaystyle\frac{1}{2} \rho_a V_z ^2 C_d C_a$ |
ID:(759, 0)
Pressure on the surface of a body
Equation
The pressure on the surface ($p_z$) is equal to the atmospheric pressure ($p_0$) reduced by the surface pressure reduction ($q_z$):
| $ p_z = p_0 - q_z $ |
ID:(776, 0)
Aerodynamic form factor model
Equation
The aerodynamic shape factor ($C_a$) can be modeled as a function of the wind angle ($\theta$) to estimate the contributions to the pressure from the different surfaces of the object. This model is based on constants derived from measurements taken on various different objects:
| $ C_a = 0.3445 \theta ^2 - 1.4961 \theta + 0.8$ |
ID:(761, 0)