.. _vc_atmospheric_boundary_layer:

==========================
Atmospheric Boundary Layer
==========================

The atmospheric boundary layer is localized on the lowest part of the atmosphere, where the wind flow is highly influenced by the proximity to the ground, which generates a turbulent flow, as illustrated below:

.. figure:: /_static/img/solver/validation/cases/atm_boundary_layer.svg
    :width: 70%
    :align: center

When assessing the wind load in constructions it's necessary to consider an atmospheric turbulent flow condition. The setup of a numerical model that reproduces this kind of flow mimics an experimental wind-tunnel.

.. note::
    The wind-tunnel may reproduce some aspects of a atmospheric flow, however it does not give a completely accurate representation. Contrary to the atmospheric flow, the wind tunnel has well defined boundary conditions. Also, it is not able to represent the Coriolis forces or achieve the high Reynolds number present on the atmosphere.

The height and geografical density of obstacles in the terrain, such as trees and constructions, affect directly the caracteristics of the local athmospheric boundary layer.
In order to account for the different profiles caractetistically found in certain region, such as lakes, rural areas, urban areas, etc., 
technical standards for the assessment of forces caused by the wind over buildings present a set of categories or exposures:

+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+
|**Terrain Category**|**Description**                                                                                                                                                                     |:math:`z_{0}` **(m)**|:math:`z_{\mathrm{min}}` **(m)**|
+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+
|0                   |Sea or coastal area exposed to the open sea                                                                                                                                         |0.003                |1                               |
+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+
|I                   |Lakes or flat and horizontal are with negligible vegetation and without obstacles                                                                                                   |0.01                 |1                               |
+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+
|II                  |Area with low vegetation such as grass and isolated obstacles (tress, buildings) with separations of at least 20 obstacle heights                                                   |0.05                 |2                               |
+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+
|III                 |Area with regular cover of vegetation or buildings or with isolated obstacles with separations of maximum 20 obstacle heights (such as villages, suburban terrain, permanent forest)|0.3                  |5                               |
+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+
|IV                  |Area in which at least 15% of the surface is covered with buildings and their average height exceedes 15 m                                                                          |1.0                  |10                              |
+--------------------+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------+--------------------------------+

Where :math:`z_{0}` is the roughness length, the main constant defining the shape of the logarithmic profile applied, 
and :math:`z_{\mathrm{min}}` is the minimum height above which the logarithmic law can be applied to estimate the velocity.
Altough :math:`z_{0}` cannot be interpreted as a direct measure of obstacles mean height, some empirical relations can approximatelly 
relate both notions, as it will be seen further.
From :footcite:t:`en2005eurocode`, the time-averaged velocity profile is calculated with:

.. math::
    \bar{u}\left(z\right)=c_{r}\left(z\right)c_{0}\left(z\right)U_{\infty}
    :label: vc_atmospheric_vel

where :math:`c_{0}` is the topography factor, usually assumed as 1.0, and :math:`c_{r}` is the roughness factor, calculated as:

.. math::
    c_{r}\left(z\right)=k_{r}\mathrm{ln}\left(\frac{z}{z_{0}}\right) \quad \mathrm{for} \quad &z_{\mathrm{min}}\leq z \leq z_{\mathrm{max}} \\
    c_{r}\left(z\right)= c_{r}\left(z_{\mathrm{min}}\right) \quad \mathrm{for} \quad &z \leq z_{\mathrm{min}}
    :label: vc_roughness_coefficient

where :math:`z_{\mathrm{max}}=200` m, and :math:`k_{r}=0.19\left(z_{0}/z_{0,II}\right)^{0.07}`.

Another quantity of great importance is the turbulence intensity :math:`I_{v}`, whose reference standard values are calculated as:

.. math::
    I_{v}\left(z\right)=\frac{k_{l}}{c_{0}\left(z\right)\mathrm{ln}\left(z/z_{0}\right)}
    :label: vc_turbulence_intensity

In order to reproduce such profiles of mean velocity and turbulence intensity, experimental wind tunnels must position a pattern of obstacles 
so as to simulate an equivalent terrain's roughness condition. A general wind tunnel setup is illustrated below:

.. figure:: /_static/img/solver/validation/cases/wind_tunnel_example.svg
    :width: 70%
    :align: center

The building in which the wind load will be measured is positioned in a rotative table. 
To build the correct wind profile, the fan is usually followed by a grid to produce a turbulent inflow. 
The illustrated setup follow :footcite:t:`wittwer2013statistical`, which uses a castellated barrier right at beginning of the flow, 
followed by elliptical vortex generators and a sequence of obstacles. 
This is the Counihan method for the simulation of atmospheric boundary layer. 
The obstacles are built following the proposal from :footcite:t:`ricci2018towards` in which the following relation is applied:

.. math::
    z_{0}=0.5h\frac{A_{r}}{A_{t}}
    :label: vc_obstacles

in which, :math:`h` is the obstacles height, :math:`A_{r}` is the roughness element area perpendicular to wind direction, and :math:`A_{t}` is the ground area per roughness element. The velocity profiles normalized to 10 meters and the turbulence intensity for the mentioned terrain categories is shown in the plot below:

.. list-table::

    * - .. figure:: /_static/img/solver/validation/cases/velocity_categories.svg
            :align: center
            :width: 100%

            Velocities for each category

      - .. figure:: /_static/img/solver/validation/cases/intensity_categories.svg
            :align: center
            :width: 100%

            Turbulence intensity for each category

.. toctree::
   :maxdepth: -1
   :hidden:

   Atmospheric Boundary Layer Flow <06.1_atm_BL.ipynb>

.. footbibliography::