.. _vc_turb_channel_flow:

======================
Turbulent Channel Flow
======================

A turbulent channel flow consists of a transient pressure-driven flow between parallel plates that occurs at high Reynolds number, as illustrated below

.. figure:: /_static/img/solver/validation/cases/turbulent_channel.svg
    :width: 80%
    :align: center

where :math:`u_{\mathrm{w}}` is the friction velocity, which 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: vc_friction_velocity

This kind of flow presents eddies that dissipate the energy. The vorticity generation is measured through the flow enstrophy. 
In spite of this kind of flow being inherently transient, it does present a statistically developed state, to which average and standard deviation velocities are solely dependent on the flow Reynolds number.

The characterization of the flow is usually performed from its friction Reynolds number, given by:

.. math::
    \mathrm{Re}_{\tau}=\frac{u_{\mathrm{w}}\delta}{\nu}
    :label: vc_friction_reynolds

For a turbulent channel flow, the length scale for calculation of the eddy turnover time is :math:`l=\delta`.

A well established numerical validation of this kind of flow was performed by :footcite:t:`KimMoinMoiser1987-177` for a :math:`\mathrm{Re}_{\tau} = 180`. 
This work compares its DNS results with the experiment from :footcite:t:`Eckelmann1974-65` (:math:`\mathrm{Re}_{\tau} = 142`) and supplies useful data for the validation of a turbulent numerical model. 
Further results for a :math:`\mathrm{Re}_{\tau} = 520` are found in :footcite:t:`Germaine2014-749`, which performs a numerical model validation using experimental data from :footcite:t:`Lepore2011`.

For an accurate DNS, a grid resolution that accounts for effects of small scale eddies is required. :footcite:t:`Eggels1994`; :footcite:t:`Lammers2006-1137` suggest that three grid nodes inside the viscous sublayer (:math:`y^{+} \leq 8`) give a sufficient spatial resolution, hence an accurate DNS design requires :math:`\Delta y^{+} \leq 2.3`, where:

.. math::
    \Delta y^{+}=\Delta y \frac{u^{*}}{\nu}
    :label: vc_delta_y

The plots from :footcite:t:`Eckelmann1974-65` and :footcite:t:`Germaine2014-749` respectively are shown below to illustrated the expected statistics profiles:

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

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

   Turbulen Channel (Re 142) <04.1_turbulent_channel_142.ipynb>
   Turbulen Channel (Re 180) <04.2_turbulent_channel_180.ipynb>
   Turbulen Channel (Re 2003) <04.3_turbulent_channel_2003.ipynb>


.. footbibliography::