.. include:: /images.rst
FLD07: Steady, 2D fully-developed, turbulent channel flow
---------------------------------------------------------
.. _description-15:
Description
~~~~~~~~~~~
This case uses 2D, fully-developed turbulent channel flow between two horizontal, parallel plates separated by a width, :math:`W`, to assess the single phase *k-ϵ* model in MFIX. Periodic boundaries with a specified pressure drop are imposed in the y-direction as shown in :numref:`fld07fig1`.
.. _fld07fig1:
.. figure:: ../media/fld07-setup.png
:align: center
Turbulent flow in a 2D channel
The pressure drop along the channel is equated to the shear stress at the walls, :math:`\tau_{w}`.
.. math::
W\frac{dP_{g}}{\text{dy}} = {2\tau}_{w}
:label: fld07eq1
The shear stress is related to the gas density, :math:`\rho_{g}`, and friction velocity, :math:`v_{*}`,
.. math::
\tau_{w} = \rho_{g}v_{*}^{2},
:label: fld07eq2
where, the friction velocity, is given by the Reynolds number.
.. math::
\text{Re}_{\tau} = \frac{\rho_{g}v_{*}(W/2)}{\mu_{g}}
:label: fld07eq3
.. _setup-15:
Setup
~~~~~
.. literalinclude:: /subprojects/mfix/tests/fluid/FLD07/mfix.dat
:language: bash
..
Highlighting with bash (not fortran) because FLD07/mfix.dat uses invalid Fortran syntax: DELP_Y = @(0.0543496*0.0543496) ! (Pa)
.. .. _fld07table1:
.. .. csv-table:: FLD-07 Setup, Initial and Boundary Conditions.
.. :widths: auto
.. :header: "Computational/Physical model", " ", " "
..
.. "2D, Unsteady, incompressible", " ", " "
.. "Single-phase (no solids)", " ", " "
.. "No gravity", " ", " "
.. "Thermal energy equation is not solved", " ", " "
.. "Turbulent, standard *k-ϵ* model", " ", " "
.. "Uniform mesh", " ", " "
.. "Superbee discretization scheme", " ", " "
.. "Time step: 0.02 s (fixed)", " ", " "
.. " ", " ", " "
.. "**Geometry**", " ", " "
.. "Coordinate system", "Cartesian", " ", "Grid partitions"
.. "x-length", "2.00", "\(m\)", "6, 12, 18"
.. "y-length", "1.00", "\(m\)", "4"
.. " ", " ", " "
.. "**Material**", " ", " "
.. "Fluid density, :math:`\rho_{g}`", "1.0", "(kg·m\ :sup:`-3`)"
.. "Fluid viscosity, :math:`\mu_{g}`", "1.0E-04", "(Pa·s)"
.. " ", " ", " "
.. "**Initial Conditions**", " ", " "
.. "x-velocity, :math:`u_{g}`", "0.00", "\(m·s\ :sup:`-1`)"
.. "y-velocity, :math:`v_{g}`", "0.00", "\(m·s\ :sup:`-1`)"
.. "Turbulenc kinetic energy, :math:`k_{g}`", "0.01", "\(m\ :sup:`2` \·s\ :sup:`-2`\)"
.. "Turbulenc dissipation rate, :math:`ϵ_{g}`", "0.01", "\(m\ :sup:`2` \·s\ :sup:`-3`\)"
.. " ", " ", " "
.. "**Boundary Conditions**", " ", " ", " "
.. "West wall", "0.0", \(m·s\ :sup:`-1`\), "No-Slip wall"
.. "East wall", "0.0", \(m·s\ :sup:`-1`\), "No-Slip wall"
.. "Cyclic North-South boundary with pressure drop", "2.95E-3", "\(Pa\)", "Specified :math:`\Delta P_{g}`"
.. _results-15:
Results
~~~~~~~
The pressure drop in the y-axial direction, domain length and width, and gas density were chosen to reflect the conditions of Lee and Moser :cite:`Moser2015` for :math:`\text{Re}_{\tau} = 543`. The DNS dataset was accessed on November 10, 2016 from http://turbulence.ices.utexas.edu/channel2015/data/LM_Channel_0550\_mean_prof.dat.
Transient simulations were performed for better numerical stability. The solution was considered converged when the L\ :sub:`2` norms for the gas velocity components, :math:`u_{g}` and :math:`v_{g}`, turbulent kinetic energy, :math:`k_{g}`, and rate of turbulent kinetic energy dissipation, :math:`\epsilon_{g}`, were all less than 10\ :sup:`-10`.
Simulations were conducted for three mesh levels [6, 12, 18] in the x-axial direction. Mesh levels were selected to ensure that the stream-ways velocity components in computational cells adjacent to the wall were located outside the buffer layer. Specifically, the first stream-ways velocity component should be located at least 30 *wall units* from the wall to be consistent with the :math:`k - \epsilon` model wall function implementation.
.. math::
\frac{\Delta x}{2}\frac{\ v_{*}\rho_{g}}{\mu_{g}} > 30
:label: fld07eq4
The MFIX results are shown in :numref:`fld07fig2` along with the direct numerical simulation (DNS) data of Lee and Moser :cite:`Moser2015` for :math:`\text{Re}_{\tau} = 543`. The velocity profiles for the three mesh levels are shown on the left whereas the normalized velocity profiles with respect to wall units are shown on the right.
.. _fld07fig2:
.. figure:: ../media/image45.jpeg
:align: center
2D, fully developed, turbulent channel flow with the DNS data of Lee and Moser :cite:`Moser2015` ; (Left) Velocity profile; (Right) Non-dimensionalized channel width and velocity profile.