--------------
Turbulent flow
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A turbulent flow is characterized by chaotic fluctuations of pressure and velocity such that it can only be assessed statistically. 
The presence of small perturbations in a high Reynolds number flow can result in a laminar to turbulent transition.

For a instant :math:`t_{1}` to which the flow is statistically developed, the average velocity field is given by:

.. math::
    \bar{u}\left(\mathbf{x}\right) = \frac{1}{N}\sum_{n=0}^{N-1}u\left(\mathbf{x},t_{1}+n\Delta t\right)
    :label: vc_average_velocity

While the averaged root mean square-fluctuation velocity is given by:

.. math::
    \bar{u}_{\mathrm{rms}}\left(\mathbf{x}\right) = \sqrt{\frac{1}{N}\sum_{n=0}^{N-1}\left[u\left(\mathbf{x},t_{1}+n\Delta t\right) - \bar{u}\left(\mathbf{x}\right)\right]^{2}}
    :label: vc_rms_velocity

where the :math:`\Delta t` has be sufficiently small, and the amount of time steps :math:`N` must be enough to correctly represent the flow statistics. 
The interval between each velocity measurement is usually estimated from the flow's eddy turnover time (:math:`\mathrm{ETT}=l/u^{*}`). 

The :math:`ETT` is the typical time scale for an eddy of length scale :math:`l` to undergo significant distortion.
The friction velocity is representative of the mean wall shear stress :math:`\sigma_{\mathrm{w}}`, and is calculated with:

.. math::
    u^{*}=\sqrt{\frac{\sigma_{\mathrm{w}}}{\rho}}
    :label: friction_velocity

Where :math:`\sigma_{\mathrm{w}}` is the wall shear-stress.

^^^^^^^^^^^^^^^^
Wall coordinates
^^^^^^^^^^^^^^^^

It is common practice to work with wall normalized scales in turbulent flows, those are identified by the superscript :math:`^{+}`.
The normalization is performed according to flow's friction (:math:`u^{*}`) velocity and fluid's viscosity (:math:`\nu`).
Hence, the length :math:`y^{+} = y u^{*}/\nu` is the normalized distance from wall (:math:`y^{+} = 0` at the wall).
The velocity is normalized with the wall velocity :math:`u^{+} = u/u^{*}`.

The use of such units is mostly used to compare a numerical solution with the logarithmic law of the wall, a self similar solution for the mean velocity parallel to the wall, valid for flows at high Reynolds numbers.
For a turbulent channel flow, the average flow profile can be generalized as below

.. figure:: /_static/img/theory/LBM/log_law_plot.svg
    :width: 70%
    :align: center

The viscous sublayer is the first near wall region in a turbulent flow, in which flow retains aspects of a laminar flow, with viscous shear-stresses predominant over turbulent shear-stresses.
It has the highest gradients of time averaged velocity.
The buffer layer is also often called turbulent-generation layer, since it stands between both viscous and inertial dominated regions of the flow, it also presents a strong gradient of the time averaged velocity.
Beyond :math:`\Delta y^{+} \approx 30` there is the log-law of the wall region, in which the flow is characterized by small average velocity gradient and in which the average velocity of a turbulent flow can be approximated by a logarithmic law written in terms of :math:`y^{+}`. 

When dealing of numerical models, the normalization of length measurement with :math:`\nu/u^{*}` can be applied to determine the resolution of the numerical simulation 

.. math::
    \Delta y^{+} = \Delta y u^{*}/\nu
    :label: delta_y

The smallest length scale in a turbulent flow is given by the Kolmogorov microscale :math:`k^{+}`, which for a Newtonian turbulent channel flow is around 1.5.
Which means that for a :math:`\Delta y^{+}>1.5`, the numerical simulation will not be able to reproduce all near a wall eddies. Since this would require a overwhelming amount of elements for high Reynolds simulations, workarounds such as the use of Large-eddy simulations and wall models are adopted to overcome this numerical limitations.

.. note::
    A uniform mesh channel with 144 nodes at wall direction performing a simulation with :math:`Re_{\mathrm{b}} \approx 54000` would present a :math:`\Delta y^{+} \approx 30`.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
How to generate turbulent flow
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Computationally, a stable turbulent flow can be generated by setting a high Reynolds number and using a temporary body to generate the perturbations. 
In this case, the body stays in the computational domain for a limited amount of time, and is removed after sufficient turbulence is produced.

After the removal of the solid body, additional time-steps must be conducted before a statistically developed turbulent flow is obtained. 
The average and standard deviation velocities and average density are calculated from the subsequent time-steps.

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

   Turbulence Spectra <turbulence_spectra.rst>