.. include:: /images.rst

FLD03: Steady, lid-driven square cavity
---------------------------------------

.. _description-11:

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

Lid-driven flow in a 2D square cavity in the absence of gravity is illustrated in :numref:`fld03fig1`. The problem definition follows the work of Ghia et al. :cite:`Ghia1982` where the domain is bounded on three sides with stationary walls while one wall, the lid, is prescribed a constant velocity. The cavity is completely filled with a fluid of selected viscosity and the flow is assumed to be incompressible and laminar.

.. _fld03fig1:
.. figure:: ../media/fld03-setup.png
   :align: center

   Schematic of the lid-driven square cavity.

.. _setup-11:

Setup
~~~~~

.. literalinclude:: /subprojects/mfix/tests/fluid/FLD03/mfix.dat
      :language:  fortran

.. .. _fld03table1:
.. .. csv-table:: FLD-03 Setup, Initial and Boundary Conditions.
..    :widths: auto
..    :header: "Computational/Physical model", " ", " "
..
..    "2D, Pseudo Steady-state, incompressible", " ", " "
..    "Single-phase (no solids)", " ", " "
..    "No gravity", " ", " "
..    "Thermal energy equation is not solved", " ", " "
..    "Turbulence equations are not solved (Laminar)", " ", " "
..    "Uniform mesh", " ", " "
..    "Superbee discretization scheme", " ", " "
..    " ", " ", " "
..    "**Geometry**", " ", " "
..    "Coordinate system", "Cartesian", " ", "Grid partitions"
..    "x-length", "1.0", "\(m\)", "128"
..    "y-length", "1.0", "\(m\)", "128"
..    " ", " ", " "
..    "**Material** :sup:`†`", " ", " "
..    "Fluid density, :math:`\rho_{g}`", "1.0", "(kg·m\ :sup:`-3`)"
..    "Fluid viscosity, :math:`\mu_{g}`", "*varied*", "(Pa·s)"
..     " ", " ", " "
..    "**Initial Conditions**", " ", " "
..    "x-velocity, :math:`u_{g}`", "-0.01", "\(m·s\ :sup:`-1`)"
..    "y-velocity, :math:`u_{g}`", "0.00", "\(m·s\ :sup:`-1`)"
..     " ", " ", " "
..    "**Boundary Conditions**", " ", " ", " "
..    "North boundary", "1.0", "\(m·s\ :sup:`-1`)", "Partial-Slip wall"
..    "All other boundaries", "0.0", "\(m·s\ :sup:`-1`)", "No-Slip wall"

.. _results-11:

Results
~~~~~~~

Numerical solutions were obtained on a 128x128 grid mesh for Reynolds numbers of 100 and 400 by specifying fluid viscosities of 1/100 and 1/400 Pa·s, respectively. A time step of 0.01 second was used and the simulations considered converged when the average L\ :sub:`2` Norms for the x-axis and y-axis velocity components, :math:`u_{g}` and :math:`v_{g}`, were less than 10\ :sup:`-8`.

The horizontal velocity at the vertical centerline (:math:`x = 0.5`) and the vertical velocity at the horizontal centerline (:math:`y = 0.5`) are compared with those of Ghia et al. :cite:`Ghia1982` in :numref:`fld03fig2` and :numref:`fld03fig3`.

.. _fld03fig2:
.. figure:: ../media/image36.jpeg
   :align: center

   Comparison of velocities at the vertical (x=0.5) and horizontal centerlines (y=0.5) of the cavity with Ghia et al. :cite:`Ghia1982` for Reynolds number of 100 (128x128 grid).

.. _fld03fig3:
.. figure:: ../media/image37.jpeg
   :align: center

   Comparison of velocities at the vertical (x=0.5) and horizontal centerlines (y=0.5) of the cavity with Ghia et al. :cite:`Ghia1982` for Reynolds number of 400 (128x128 grid).

Similarly, numerical solutions were obtained on a 128x128 grid mesh for Reynolds numbers of 1000 and 3200 by specifying fluid viscosities of 1/1000 and 1/3200 Pa·s, respectively. The horizontal velocity at the vertical centerline (:math:`x = 0.5`) and the vertical velocity at the horizontal centerline (:math:`y = 0.5`) are compared with those of Ghia et al. :cite:`Ghia1982` in :numref:`fld03fig4` and :numref:`fld03fig5`. These cases are not included in the continuous integration server test suite.

.. _fld03fig4:
.. figure:: ../media/image38.jpeg
   :align: center

   Comparison of velocities at the vertical (x=0.5) and horizontal centerlines (y=0.5) of the cavity with Ghia et al. :cite:`Ghia1982` for Reynolds number of 1000 (128x128 grid).

.. _fld03fig5:
.. figure:: ../media/image39.jpeg
   :align: center

   Comparison of velocities at the vertical (x=0.5) and horizontal centerlines (y=0.5) of the cavity with Ghia et al. :cite:`Ghia1982` for Reynolds number of 3200 (128x128 grid).