.. include:: /images.rst

FLD04: Gresho vortex problem
----------------------------

.. _description-12:

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

The Gresho vortex problem :cite:`Gresho1990` involves a stationary rotating vortex for which the centrifugal forces are exactly balanced by pressure gradients. The angular velocity and pressure distribution varies with radius as given by :eq:`fld04eq1`,:eq:`fld04eq2` :cite:`Liska2003` while the radial velocity is zero everywhere and the density is one everywhere.

.. math::
   :label: fld04eq1

   u_{\phi}(r) = \begin{matrix} 5r, &0 \leq r < 0.2 \\
   2 - 5r, &0.2 \leq r < 0.4 \\
   0, &0.4 \leq r \\
   \end{matrix};

.. math::
   :label: fld04eq2

   \quad p(r) = \begin{matrix} 5+12.5r^{2}, &0 \leq r < 0.2 \\
   9 - 4ln0.2 + 12.5r^{2} - 20r + 4lnr, &0.2 \leq r < 0.4 \\
   3 + 4ln2, &0.4 \leq r \\
   \end{matrix}

.. _fld04fig1:
.. figure:: ../media/image40.png
   :align: center

   Exact solution for the Gresho vortex problem (shown for :math:`\left( \mathbf{x,y} \right)\mathbf{\in}\left( \mathbf{0.5,1} \right)\mathbf{\times}\left( \mathbf{0.5,1} \right)`)

This problem is setup as a time-independent solution to the incompressible, homogeneous Euler equations. The exact solution is symmetric about the horizontal and the vertical axes and is shown for the quadrant of :math:`\left( x,y \right) \in \left( 0.5,1 \right) \times \left( 0.5,1 \right)` in Figure 3‑11, where :math:`\left( 0.5,0.5 \right)` is the center of the vortex. The simulation is initialized with the exact solution and periodic conditions on all boundaries of a 2D domain of unit dimensions (i.e., :math:`\left( x,y \right) \in \left( 0,1 \right) \times \left( 0,1 \right)`)). Different numerical schemes in MFIX are used to find the numerical solution after three seconds which are then compared with an exact solution to assess the quality of the results.

.. _setup-12:

Setup
~~~~~

.. literalinclude:: /subprojects/mfix/tests/fluid/FLD04/mfix.dat
      :language:  fortran

.. .. _fld04table1:
.. .. csv-table:: FLD-04 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", " ", " "
..    "Turbulence equations are not solved (Laminar)", " ", " "
..    "Uniform mesh", " ", " "
..    "Different discretization schemes", " ", " "
..    " ", " ", " "
..    "**Geometry**", " ", " "
..    "Coordinate system", "Cartesian", " ", "Grid partitions"
..    "x-length", "1.0", "\(m\)", "40"
..    "y-length", "1.0", "\(m\)", "40"
..    " ", " ", " "
..    "**Time-stepping**", " ", " "
..    "Initial time", "0.0", "\(s\)", " "
..    "Final time", "3.0", "\(s\)", " "
..    "Time step", "0.01", "\(s\)", " "
..    "Variable time-stepping disabled", " ", " ", " "
..     " ", " ", " ", " "
..    "**Material**", " ", " "
..    "Fluid density, :math:`\rho_{g}`", "1.0", "(kg·m\ :sup:`-3`)"
..    "Fluid viscosity, :math:`\mu_{g}`", "0.0", "(Pa·s)"
..     " ", " ", " "
..    "**Initial Conditions**", " ", " "
..    "x-velocity, :math:`u_{g}`", ":eq:`fld04eq1`", "\(m·s\ :sup:`-1`)"
..    "y-velocity, :math:`u_{g}`", ":eq:`fld04eq1`", "\(m·s\ :sup:`-1`)"
..    "Pressure, :math:`P_{g}`", ":eq:`fld04eq1`", "(Pa)"
..     " ", " ", " "
..    "**Boundary Conditions**", " ", " ", " "
..    "All boundaries", "Cyclic"

.. _results-12:

Results
~~~~~~~

MFIX simulations of the Gresho vortex problem were carried out with nine spatial discretization schemes. The final flow vorticity is illustrated in :numref:`fld04fig2` with the exact vorticity provided for reference at left. FOUP and FOUP using downwind factors are identical as expected, therefore only results for FOUP are shown. FOUP clearly fails to capture the vorticity distribution over the entire domain; Minmod and QUICKEST fail to accurately capture this distribution in the region of :math:`0.1\ m \leq r \leq 0.3\ m` (e.g., :math:`0.6m \leq x \leq 0.8m` along :math:`y = 0.5`).

.. _fld04fig2:
.. figure:: ../media/image41.png
   :align: center

   Comparison of vorticity by different numerical schemes with the exact solution (at :math:`\mathbf{T = 3\ s}`).

The total kinetic energy of the flow is included in :numref:`fld04table2`. FOUP has the greatest loss of kinetic energy, followed by QUICKEST, and Minmod. Central scheme maintains the best agreement followed by SMART and MUSCL.

.. _fld04table2:
.. csv-table:: Total kinetic energy of the flow field compared to the exact (initial) value for various spatial discretization schemes.
   :widths: auto
   :header: "Scheme", "Calculated TKE", "Abs. Error", "% Rel. Error"

   "FOUP", "42.64", "91.25", "68.15"
   "FOUP w/DWF", "42.64", "91.25", "68.15"
   "Superbee", "144.57", "10.66", "7.69"
   "SMART", "130.46", "3.44", "2.57"
   "QUICKEST", "93.47", "40.43", "30.19"
   "MUSCL", "128.45", "5.45", "4.07"
   "van Leer", "125.79", "8.11", "6.05"
   "Minmod", "112.95", "20.94", "15.64"
   "Central", "133.70", "0.20", "0.15"

As a final measure of solution accuracy, the average L\ :sub:`2` Norm is shown in Table 3‑3. Again, FOUP, QUICKEST, and Minmod demonstrate greatest amount of solution error whereas Central, SMART, and Superbee have the least amount of error.

.. _fld04table3:
.. csv-table:: Average L2 Norms for the gas pressure (Pg), x-axial velocity (Ug) and y-axial velocity (Vg) for various spatial discretization schemes.
   :widths: auto
   :header: "Scheme", ":math:`P_{g}` L\ :sub:`2`-norm", ":math:`U_{g}` L\ :sub:`2`-norm", ":math:`V_{g}` L\ :sub:`2`-norm"

   "FOUP", "0.1430", "0.1468", "0.1468"
   "FOUP w/DWF", "0.1430", "0.0822", "0.0822"
   "Superbee", "0.0184", "0.0182", "0.0182"
   "SMART", "0.0078", "0.0163", "0.0163"
   "QUICKEST", "0.0647", "0.0612", "0.0612"
   "MUSCL", "0.0109", "0.0200", "0.0200"
   "van Leer", "0.0149", "0.0107", "0.0107"
   "Minmod", "0.0343", "0.0408", "0.0408"
   "Central", "0.0076", "0.0161", "0.0161"