.. include:: /images.rst

FLD08: Steady, 2D turbulent pipe flow
-------------------------------------

.. _description-16:

Description
~~~~~~~~~~~

This case uses turbulent flow in a pipe of length :math:`L` and radius :math:`R` to assess the single phase *k-ϵ* model in MFIX. A 2D axisymmetric domain is used to define the pipe geometry, and pressure boundaries are used to induce flow in the positive y-axial direction as shown in :numref:`fld08fig1`. The results are compared with the experimental data of Zagarola and Smits :cite:`Zagarola1998`.

.. _fld08fig1:
.. figure:: ../media/fld08-setup.png
   :align: center

   Turbulent flow in a pipe

.. _setup-16:

Setup
~~~~~

.. literalinclude:: /subprojects/mfix/tests/fluid/FLD08/mfix.dat
      :language:  fortran

.. .. _fld08table1:
.. .. csv-table:: FLD-08 Setup, Initial and Boundary Conditions.
..    :widths: auto
..    :header: "Computational/Physical model", " ", " "
..
..    "2D axisymmetric, Unsteady, incompressible", " ", " "
..    "Single-phase (no solids)", " ", " "
..    "No gravity", " ", " "
..    "Thermal energy equation is not solved", " ", " "
..    "Turbulent, standard *k-ϵ* model", " ", " "
..    "Uniform mesh", " ", " "
..    "First-order upwind discretization scheme", " ", " "
..    "Time step: 0.01 s (fixed)", " ", " "
..    " ", " ", " "
..    "**Geometry**", " ", " "
..    "Coordinate system", "Cartesian", " ", "Grid partitions"
..    "x-length", "8.00", "\(m\)", "100"
..    "y-length", "0.06468", "\(m\)", "16"
..    " ", " ", " "
..    "**Material**", " ", " "
..    "Fluid density, :math:`\rho_{g}`", "1.1620", "(kg·m\ :sup:`-3`)"
..    "Fluid viscosity, :math:`\mu_{g}`", "1.8487E-05", "(Pa·s)"
..     " ", " ", " "
..    "**Initial Conditions**", " ", " "
..    "x-velocity, :math:`u_{g}`", "0.00", "\(m·s\ :sup:`-1`)"
..    "y-velocity, :math:`v_{g}`", "5.00", "\(m·s\ :sup:`-1`)"
..    "Turbulenc kinetic energy, :math:`k_{g}`", "0.047", "\(m\ :sup:`2` \·s\ :sup:`-2`\)"
..    "Turbulenc dissipation rate, :math:`ϵ_{g}`", "0.213", "\(m\ :sup:`2` \·s\ :sup:`-3`\)"
..    " ", " ", " "
..    "**Boundary Conditions**", " ", " ", " "
..    "West wall", "0.0", \(m·s\ :sup:`-1`\), "No-Slip wall"
..    "East wall", "0.0", \(m·s\ :sup:`-1`\), "No-Slip wall"
..    "South wall, Pressure inflow", "20.684", "\(Pa\)", "Specified :math:`P_{g}`"
..    "  Turbulent kinetic energy, :math:`k_{g}`", "0.047", "\(m\ :sup:`2` \·s\ :sup:`-2`\)"
..    "  Turbulent dissipation rate, ϵ_{g}", "0.213", "\(m\ :sup:`2` \·s\ :sup:`-3`\)"
..    "North wall, Pressure outflow", "0.0", "\(Pa\)"

.. _results-16:

Results
~~~~~~~

Pressure drop in the y-axial direction, domain width, gas density and viscosity were chosen to reflect the conditions of :cite:`Zagarola1998` for :math:`Re = 41727`. A transient simulation was 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`.

The simulation was conducted with 16 cells in the x-axial direction. The mesh level ensures 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 was 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: fld08eq1

Here, the friction velocity, :math:`v_{*}`, is given by the Karman number, :math:`R^{+}` :cite:`Zagarola1998`,

.. math::
   R^{+} = \frac{Dv_{*}}{\nu}
   :label: fld08eq2

where :math:`D` is pipe diameter, and :math:`\nu` is the kinematic viscosity.

The MFIX results are shown in :numref:`fld08fig2` along with the experimental data of Zagarola and Smits :cite:`Zagarola1998` for :math:`Re = 41727`. The experimental dataset was accessed on November 10, 2016 from http://www.princeton.edu/~gasdyn/Superpipe_data/4.1727E+04.txt

The velocity profile is shown on the left, and the normalized velocity profile with respect to wall units is shown on the right. The velocity profile is given for two locations near the pipe exit, 7.2 m and 8.0 m respectively, with the maximum difference less than 10\ :sup:`-2` m·sec\ :sup:`-1`, indicating that the flow is fully developed. The largest discrepancy between the experimental measurements and the simulation results occurs at the centerline of the domain where the simulation under-predicts the observed velocity by 0.3 m·sec\ :sup:`-1`.

.. _fld08fig2:
.. figure:: ../media/image46.jpeg
   :align: center

   2D, turbulent pipe flow with the experimental data of DNS data of Zagarola and Smits :cite:`Zagarola1998`; (Left) Velocity profiles; (Right) Nondimensionalized channel width and velocity profile.