.. include:: /images.rst
FLD05: Steady, 2D Couette flow
------------------------------
.. _description-13:
Description
~~~~~~~~~~~
Couette flow is a laminar flow of a viscous fluid between two parallel plates separated by a distance, :math:`H`, with the upper wall moving at velocity, :math:`U`. Different velocity distributions are obtained depending on the pressure gradient applied to the flow field. The schematic of the problem is shown in :numref:`fld05fig1`.
.. _fld05fig1:
.. figure:: ../media/fld05-setup.png
:align: center
Couette flow between two flat plates of length L, separated by a distance H with the upper wall moving at velocity U.
In this problem, the Navier-Stokes equations reduce to a second order, linear, ordinary differential equation (ODE),
.. math::
\frac{d^{2}u_{g}}{dy^{2}} = \frac{1}{\mu_{g}}\frac{\text{dp}}{\text{dx}}
:label: fld05eq1
where :math:`\mu_{g}` is the fluid viscosity, and :math:`\text{dp}/\text{dx}` is the prescribed pressure drop across the length of the pipe. The no-slip and partial-slip boundary conditions are specified by
.. math::
:label: fld05eq2
&u_{g}\left( 0 \right) = 0,\\
&u_{g}\left( H \right) = \text{U.}
The analytical solution is given by,
.. math::
u_{g}\left( y \right) = \frac{1}{2\mu}\frac{\text{dp}}{\text{dx}}\left( y^{2} - yH \right) + U\frac{y}{H}
:label: fld05eq3
.. _setup-13:
Setup
~~~~~
.. literalinclude:: /subprojects/mfix/tests/fluid/FLD05/mfix.dat
:language: fortran
.. .. _fld05table1:
.. .. csv-table:: FLD-05 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", " ", " "
.. "Superbee discretization scheme", " ", " "
.. " ", " ", " "
.. "**Geometry**", " ", " "
.. "Coordinate system", "Cartesian", " ", "Grid partitions"
.. "x-length", "1.00", "\(m\)", "4"
.. "y-length", "0.01", "\(m\)", "8,16,32,64"
.. " ", " ", " "
.. "**Material**", " ", " "
.. "Fluid density, :math:`\rho_{g}`", "1.0", "(kg·m\ :sup:`-3`)"
.. "Fluid viscosity, :math:`\mu_{g}`", "5.0E-06", "(Pa·s)"
.. " ", " ", " "
.. "**Initial Conditions**", " ", " "
.. "x-velocity, :math:`u_{g}`", "0.0", "\(m·s\ :sup:`-1`)"
.. "y-velocity, :math:`v_{g}`", "0.0", "\(m·s\ :sup:`-1`)"
.. " ", " ", " "
.. "**Boundary Conditions**", " ", " ", " "
.. "North wall", "10.0", \(m·s\ :sup:`-1`\), "Partial-Slip wall"
.. "South wall", "0.0", \(m·s\ :sup:`-1`\), "No-Slip wall"
.. "East and west boundaries", "Cyclic", ":math:`\Delta P_{g}` varied"
.. _results-13:
Results
~~~~~~~
Simulations were conducted for seven pressure drops, :math:`\lbrack - 3.0,\ - 2.0,\ - 1.0,\ \ 0.0,\ 1.0,\ 2.0,\ 3.0\rbrack` Pa, specified across the x-axial cyclic boundaries. Four mesh levels [8, 16, 32, 64] in the y-axial direction were used to assess discretization error.
The analytical and numerical solutions for the zero pressure drop case are shown in :numref:`fld05fig2`. For clarity, only a subset of the numerical solutions is presented, resulting in a slight offset/shift in displayed data points. Note that the analytical solution reduces to a linear variation in velocity between the lower and upper walls when the specified pressure drop is zero. For this case, the absolute error in velocity is bounded above by :math:`10^{- 6}\ \mathrm{m \cdot sec}^{- 1}` and is observed for the finest grid resolution (64 mesh). Further investigation (not presented) indicated that the increase in numerical error is attributed to the solution mechanism of the linear equation system. This error can be reduced by modifying the default linear equation solver settings (e.g., tighten convergence criteria, increase number of iterations, etc.).
.. _fld05fig2:
.. figure:: ../media/image42.png
:align: center
Couette flow with a zero pressure gradient with four grid resolutions.
The analytical and numerical solutions for the adverse and favorable pressure drops are shown in :numref:`fld05fig3` and :numref:`fld05fig4`. Again, only a subset of the numerical solutions is presented resulting in a slight offset/shift in displayed data points. These cases demonstrate a second order rate of convergence with respect to grid size which is attributed to the second-order discretization of the viscous stress term.
.. _fld05fig3:
.. figure:: ../media/image43.jpeg
:align: center
Adverse pressure gradient (-1, -2, -3 Pa) Couette flow with four grid resolutions. Absolute error and observed order of accuracy only shown for -3 Pa pressure gradient.
.. _fld05fig4:
.. figure:: ../media/image44.jpeg
:align: center
Favorable pressure gradient (1, 2, 3 Pa) Couette flow with four grid resolutions. Absolute error and observed order of accuracy only shown for 3 Pa pressure gradient.