.. _bc_solid_wall:

==========
Solid Wall
==========

Solid wall boundary conditions are those which impose no-slip at a wall. The rate-of-strain can be either calculated from a finite difference scheme or a wall model.

.. _bc_halfway_bounce_back:

-------------------
Halfway Bounce-Back
-------------------

For the halfway bounce-back (HWBB) the unknown populations are obtained by reflecting back the opposite post-collision populations. 

.. figure:: /_static/img/theory/LBM/wall.svg
    :width: 70 %
    :align: center

Therefore, with the halfway bounce-back boundary condition, the unknown populations at streaming are given by:

.. math::
    f_{\bar{i}}\left(t+\Delta t\right)=f_{i}^{*}\left(t\right)
    :label: hbbounceback

where :math:`\bar{i}` is the opposite :math:`i`-direction.

.. admonition:: Use Case

    Halfway bounce-back is used to represent a wall 

.. note::
    This is a fully local boundary condition.

-----------------------
Regularized Bounce-Back
-----------------------

There are more appropriate methods to implement boundary conditions (BC) within the RRBGK collision operator, as mentioned in :footcite:t:`Malaspinas2015-1048`. The most adequate way is to implement a BC through direct reconstruction of populations using pre-determined macroscopics. 
For a wall boundary condition, this translates to :math:`\mathbf{u}_{b} = 0`.

However, both the fluid density and rate-of-strain can vary in the wall during the flow evolution, hence their values are estimated using the neighboor nodes. The density at the wall is assumed to be the same as first node from boundary towards normal direction. 
The rate-of-strain is estimated through a second-order forwards finite-difference scheme:

.. math::
    S_{\alpha\beta}=\frac{1}{2}\left(\frac{\Delta u_{\alpha}}{\Delta x_{\beta}}+\frac{\Delta u_{\beta}}{\Delta x_{\alpha}}\right),
    :label: bc_finite_difference

where:

.. math::
    \frac{\Delta u}{\Delta x_{n}}\left(x_{0}\right)=\frac{-3u\left(x_{0}\right)+4\left(x_{0}+\Delta x\right)-u\left(x_{0}+2\Delta x\right)}{2 \Delta x}
    :label: bc_2nd_finite_difference

With the density, velocity (:math:`\mathbf{u}_{\mathrm{b}}=\mathbf{0}`), and rate-of-strain known, the equilibrium and non-equilibrium populations can be built through a third-order Hermite polynomial approximation.


.. only:: internal

    .. include:: bc.wall_model.rst.inc

.. footbibliography::