.. include:: /images.rst

DEM02: Bouncing particle
------------------------

This case provides a comparison between the MFIX-DEM linear spring-dashpot collision model and the *hard sphere* model where collisions are instantaneous. The hard sphere model can be seen as the limiting case where the normal spring coefficient is large, :math:`k_{n} \rightarrow \infty`. This case was originally reported in :cite:`Garg2012`.

.. _description-dem02:

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

A smooth (frictionless), spherical particle falls freely under gravity from an initial height, :math:`h_{0}`, and bounces upon collision with a fixed wall :numref:`dem01fig1`. Assuming that the collision is instantaneous, the maximum height the particle reaches after the first collision (bounce), :math:`h_{1}^{\mathrm{\max}}`, is given by

.. math::
   h_{1}^{\mathrm{\max}} = \left( h_{0} - r_{p} \right)e_{n}^{2}
   :label: dem02eq1

where :math:`r_{p}` is the particle radius, and :math:`e_{n}` is the restitution coefficient. A general expression for the maximum height following the :math:`k^{\text{th}}` bounce is

.. math::
   h_{k}^{\mathrm{\max}} = \left( h_{0} - r_{p} \right)e_{n}^{2k} + r_{p}.
   :label: dem02eq2

.. _setup-dem02:

Setup
~~~~~

.. _dem01table2:
.. csv-table:: DEM-02 Setup, Initial and Boundary Conditions.
   :widths: auto
   :header: "Computational/Physical model", " ", " "

   "1D, Transient", " ", " "
   "Granular flow (no gas)", " ", " "
   "Gravity", " ", " "
   "Thermal energy equation is not solved", " ", " "
   " ", " ", " "
   "**Geometry**", " ", " "
   "Coordinate system", "Cartesian", " ", " "
   "x-length", "1.0", "\(m\)", " "
   "y-length", "1.0", "\(m\)", " "
   "z-length", "1.0", "\(m\)", " "
   " ", " ", " "
   "**Solids Properties**", " ", " "
   "Normal spring coefficient, :math:`k_{n}`", "*varied*", "(N·m\ :sup:`-1`)"
   "Restitution coefficient, :math:`e_{n}`", "*varied*", "\( \)"
   "Friction coefficient, :math:`\mu`", "0.0", "\( \)"
   " ", " ", " "
   "**Solids 1 Type**", "DEM", " "
   "Diameter, :math:`d_{p}`", "0.2", "\(m\)"
   "Density, :math:`\rho_{s}`", "2600", "(kg·m\ :sup:`-3`)"
   " ", " ", " "
   "**Boundary Conditions**", " ", " ", " "
   "All boundaries", "Solid walls", " "

.. _results-dem02:

Results
~~~~~~~

Simulations of a freely-falling particle dropped from an initial height of 0.5m were conducted for three normal spring coefficients, :math:`\lbrack 0.5,\ 5.0,\ 50.0\rbrack \times 10^{5}` N·m\ :sup:`-1`, and six restitution coefficients, [0.5, 0.6, 0.7, 0.8, 0.9, 1.0]. All simulations employed the Adams-Bashforth time-stepping method. The maximum height attained after the k\ :sup:`th` collision for all cases is shown in :numref:`dem02fig1`.

.. _dem02fig1:
.. figure:: ../media/image52.png
   :align: center

   Comparison between the analytic solution from a hard-sphere model (solid lines) and MFIX-DEM (symbols) of the maximum height reached after the k\ :sup:`th` wall collision for a freely falling particle. Three values for the normal spring coefficient are used (left to right) with six restitution coefficients.

:numref:`dem02fig2` illustrates the percent relative difference between the analytical solution for a hard-sphere model and the MFIX-DEM simulation. In the limit of the hard-sphere model (shown left to right by an increasing spring coefficient), the difference between the two collision models decreases.

.. _dem02fig2:
.. figure:: ../media/image53.png
   :align: center

   Percent relative difference between the analytic solution for a hard-sphere model and MFIX-DEM of the maximum height reached after the k\ :sup:`th` wall collision for a freely falling particle. Three values for the normal spring coefficient are used (left to right) with six restitution coefficients.