Tutorial: Steady Flow Around a Cylinder

.. _TutorialSteadyFlowAroundCylinder:

Steady Flow Around a Cylinder

=============================

This tutorial sets up fluid flow around a fixed circular cylinder based

on the work of Schäfer et al. (1996).  This is a steady, low-Reynolds-number

cylinder-flow example that uses MFIX-Exa's embedded-boundary geometry.

Features

--------

* Incompressible fluid with no particles.

* Steady-state advance

* Quasi-2D domain using a thin, periodic spanwise direction.

* Cylinder represented by a predefined embedded boundary.

* Velocity inlet, pressure outlet, no-slip walls, and a periodic spanwise direction.

* Plotfile output for velocity, pressure, and EB volume fraction.

* Report output for the EB drag force.

Case description

----------------

The flow enters from the low-:math:`x=0.` boundary and exits at the

high-:math:`x=2.46` boundary. The low-:math:`y=-0.20` and high-:math:`y=0.21`

boundaries are no-slip walls. The cylinder has a radius of :math:`0.05` m and

is located at :math:`x=0.2`, :math:`y=0.` so that it is slightly off-centered

with respect to channel height. To mimic the original setup, the channel is

thin and periodic in :math:`z`.

The tutorial uses

.. math::

   \rho = 1,\qquad \mu = 0.001,\qquad D = 0.1,

with the inlet velocity defined by

.. math::

   u(x=0,y) = \frac{1.2(y+0.2)(0.21-y)}{0.41^2} .

Defining the mean velocity as :math:`\overline{u} = 2u(x=0,y=0.41/2-0.2)/3 = 0.2`,

.. math::

   Re = \frac{\rho \overline{u} D}{\mu} = 20.

At this Reynolds number the wake should approach a steady, slightly-asymmetric state.

Important input sections

------------------------

The mesh is a long box with a periodic :math:`z` direction:

.. code-block:: text

   geometry.is_periodic =  0      0      1

   geometry.prob_lo     =  0.00  -0.20  -0.205

   geometry.prob_hi     =  2.46   0.21   0.205

   amr.n_cell = 192  32  32

The cylinder is a predefined embedded boundary.  The

``cylinder.internal_flow = false`` setting means the flow is outside the

cylinder, not inside a pipe and the ``cylinder.height = -1.0`` means that

the cylinder is infinitely long:

.. code-block:: text

   mfix.geometry = "cylinder"

   cylinder.internal_flow = false

   cylinder.radius = 0.05

   cylinder.height = -1.0

   cylinder.direction = 2

   cylinder.center = 0.2  0.0  0.0

The low- and high-:math:`y` faces are no-slip walls.  The low-:math:`x` face

is a mass inflow and the high-:math:`x` face is a pressure outlet:

.. code-block:: text

   bc.regions =  bottom_wall  top_wall  inflow  outflow

   bc.bottom_wall = no-slip

   bc.top_wall = no-slip

   bc.inflow = mi

   bc.inflow.fluid.density = 1.0

   bc.inflow.fluid.inflow_type = velocity

   bc.inflow.fluid.velocity = "1.2*(y+0.2)*(0.21-y)/0.1681"

   bc.outflow = po

   bc.outflow.pressure = 101325.0

The case is run with the pseudo steady-state time advance with a tolerance

of :math:`1.0e-6` and 50,000 maximum iterations. The maximum time step size

is set to ensure :math:`\Delta t \sim (\Delta x)^2`.

.. code-block:: text

   mfix.steady_state = 1

   mfix.steady_state_tol = 1.0e-6

   mfix.steady_state_maxiter = 50000

   mfix.dt_max = 1.64e-4

.. note::

   The complete input file is located in the MFIX-Exa source directory as

   ``tutorials/incompressible-fluid/inputs.steady-around-cylinder``

Post-Processing

---------------

The run writes an AMReX plotfile named ``plt*`` every 5000 iterations into a directory,

``io``, in the run directory. Each plot file saves the fluid:

* velocity field, ``vel_g``,

* pressure, ``p_g``, and

* embedded-boundary volume fraction, ``volfrac``.

The plotfile data can be viewed in ParaView, VisIt, or any other AMReX-compatible

post-processing tool. For visualization, load the final ``plt*`` directory and

display the velocity magnitude on a slice normal to the spanwise direction. Overlaying

velocity vectors helps show the acceleration around the cylinder and the steady

wake behind it.

.. _fig_steady_flow_around_cylinder_vel_mag:

.. figure:: ./images/steady-flow-around-cylinder.png

   :width: 85%

   :align: center

   :alt: Image showing the steady-state velocity magnitude as a color field and 2D vector field at slices in y axis direction.

   Velocity magnitude and selected vector field at steady state.

The setup also enables the **Embedded Boundary (EB) drag report** for the

EB surface captured by the ``eb_cyl`` region. This report region encloses the

physical cylinder so that the EB surface is included in the integration; it does

not redefine the cylinder geometry used by the flow solver.

.. code-block:: text

   regions = ... eb_cyl

   regions.eb_cyl.shape = cylinder

   regions.eb_cyl.cylinder.radius = 0.075

   regions.eb_cyl.cylinder.start = 0.2  0.0  -0.205

   regions.eb_cyl.cylinder.end   = 0.2  0.0   0.205

   mfix.reports.eb_drag.regions = eb_cyl

   mfix.reports.eb_drag.eb_cyl.int = 500

This report calculates the total aerodynamic force by integrating the fluid's

pressure and shear stress over the captured surface:

.. math::

   \vec{F} = \oint \left[-p\vec{n} + \vec{\tau} \cdot \vec{n} \right] dA

The report output is written every 500 iterations and contains the force

components on the embedded boundary. For this two-dimensional benchmark, the

streamwise force :math:`F_x` is used to compute the drag coefficient and the

cross-stream force :math:`F_y` is used to compute the lift coefficient:

.. math::

   C_D = \frac{2 F_x}{\rho \overline{u}^2 D}, \qquad

   C_L = \frac{2 F_y}{\rho \overline{u}^2 D}.

Here :math:`D=0.1` is the cylinder diameter and :math:`\overline{u}=0.2` is the

mean inlet velocity. At convergence, the coefficients should level off to

nearly constant values, indicating that the steady-state solve has reached the

expected asymmetric wake.

.. _fig_steady_flow_around_cylinder_drag_coeffs:

.. list-table::

   :widths: 50 50

   :class: borderless

   * - .. figure:: ./images/steady-flow-around-cylinder-drag-coeff.png

          :width: 100%

          :alt: Figure showing the evolution of the drag coefficient versus iteration

          Drag coefficient vs iteration

     - .. figure:: ./images/steady-flow-around-cylinder-lift-coeff.png

          :width: 100%

          :alt: Figure showing the evolution of the lift coefficient versus iteration

          Lift coefficient vs iteration

Extensions

----------

* Refine ``amr.n_cell`` to run a uniformly finer mesh.

* Enable mesh refinement by increasing ``amr.max_level`` to refine the mesh near the embedded boundary

* Run at higher Reynolds number for longer times to observe unsteady vortex shedding.

Incompressible Fluid

====================

These tutorials describe complete MFIX-Exa cases and the main input choices

used to build them.

.. toctree::

   :maxdepth: 1

   incompressible-fluid/steady-flow-around-cylinder