.. include:: /images.rst
FLD01: Steady, 2D Poiseuille flow
---------------------------------
.. _description-9:
Description
~~~~~~~~~~~
Plane Poiseuille flow is defined as a steady, laminar flow of a viscous fluid between two horizontal parallel plates separated by a distance, :math:`H`. Flow is induced by a pressure gradient across the length of the plates, :math:`L`, and is characterized by a 2D parabolic velocity profile symmetric about the horizontal mid-plane as illustrated in :numref:`fld01fig1`.
.. _fld01fig1:
.. figure:: ../media/fld01-setup.png
:align: center
Plane Poiseuille flow between two flat plates of length L, separated by a distance H.
In this problem, the Navier-Stokes equations reduce to a second order, linear, ordinary differential equation (ODE),
.. math::
\mu_{g}\frac{d^{2}u_{g}}{dy^{2}} = \frac{dP_{g}}{\text{dx}},
:label: fld01eq1
where :math:`\mu_{g}` and :math:`P_{g}` are correspondingly the fluid viscosity and pressure, and :math:`u_{g}` and :math:`v_{g}` are respectively the :math:`x` and :math:`y` velocity components. Furthermore, it is assumed that gravitational forces are negligible, the pressure gradient is constant, i.e., :math:`dP_{g}/dx = C`, and all velocity components are zero at the channel walls. The resulting analytical solution to :eq:`fld01eq1` is given as
.. math::
u_{g}\left( y \right) = - \frac{dP_{g}}{\text{dx}}\frac{1}{2\mu_{g}}y\left( H - y \right).
:label: fld01eq2
.. _setup-9:
Setup
~~~~~
.. literalinclude:: /subprojects/mfix/tests/fluid/FLD01/mfix.dat
:language: fortran
.. .. _fld01table1:
.. .. csv-table:: FLD-01 Setup, Initial and Boundary Conditions.
.. :widths: auto
.. :header: "Computational/Physical model", " ", " "
..
.. "2D, 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", "0.2", "\(m\)", "8, 16, 32, 64"
.. "y-length", "0.01", "\(m\)", "8, 16, 32, 64"
.. " ", " ", " "
.. "**Material** :sup:`†`", " ", " "
.. "Fluid density, :math:`\rho_{g}`", "1.0", "(kg·m\ :sup:`-3`)"
.. "Fluid viscosity, :math:`\mu_{g}`", "0.001", "(Pa·s)"
.. " ", " ", " "
.. "**Initial Conditions**", " ", " "
.. "Pressure *(gauge)*, :math:`P_{g}`", "101325", "(Pa)"
.. "x-velocity, :math:`u_{g}`", "10.0", "(m·s\ :sup:`-1`)"
.. "y-velocity, :math:`v_{g}`", "0.0", "(m·s\ :sup:`-1`)"
.. " ", " ", " "
.. "**Boundary Conditions**", " ", " "
.. "South boundary", "0.0", "(m·s\ :sup:`-1`)", "No-Slip wall"
.. "North boundary", "0.0", "No-Slip wall"
.. "Cyclic West-East boundary with pressure drop", ":math:`\Delta P_{g}`", "240.0", "\(Pa\)"
.. _results-9:
Results
~~~~~~~
The analytical and numerical solutions for x-velocity, :math:`u_{g}`, are shown in :numref:`fld01fig2`. Only a subset of the numerical solution data points are plotted causing the appearance of a slight shift in presented data points. The observed error demonstrates a second-order rate of convergence with respect to grid size in the y-axial direction. This is attributed to the second-order discretization of the viscous stress term as convection/diffusion terms do not contribute to the solution.
.. _fld01fig2:
.. figure:: ../media/image33.png
:align: center
Steady, 2D channel flow x-velocity profile (left), absolute error in x-velocity solution (center), and observed order of accuracy (right) using four grid levels (JMAX = 8, 16, 32, 64).
The fluid pressure, :math:`P_{g}`, varies linearly along the length of the plates as shown in :numref:`fld01fig3`. The largest observed absolute error is bounded above by :math:`10^{- 12}` and occurs for the finest mesh. This error is attributed to the convergence criteria of the linear equation system.
.. _fld01fig3:
.. figure:: ../media/image34.png
:align: center
:figwidth: 75%
Steady, 2D channel flow pressure profile (left) and absolute error in pressure solution (right) using four grid levels (IMAX = 8, 16, 32, 64).