.. _wall_model_IBM:

****************
Wall Model - IBM
****************

To couple the wall model to fluid flow simulation, the immersed boundary method (IBM) operates with a velocity correction calculated from the selected :ref:`wall function <wall_model_wall_functions>`.

From the a Lagrangian mesh some auxiliar points must be defined as illustrated below ( :footcite:t:`shi2019wall`, :footcite:t:`du2022wall`, :footcite:t:`wang2024wall`):

.. figure:: /_static/img/theory/wall_model/ibm_wall_model.svg
    :width: 50%
    :align: center

where the point P is the reference point :math:`\mathrm{P_{N}}` at normal direction in which the velocity will be linearly interpolated from nearest points.
It is usual to fix a distance :math:`H` of at least :math:`3\Delta x` from the wall, which will be located in a point external to IBM diffusive layer.
This velocity profile between P and the wall is modeled and the velocity at Q is calculated.
The distance from W to Q is usually fixed :math:`h=0.1 - 0.25\Delta x`.

The point Q is also distanced from the wall towards normal direction and is an auxiliar point where a calculated velocity will be imposed.
The correction will be applied using a interpolation-spread procedure from IBM at Q point.

.. important::
    It is important that the simulation first reaches an stable flow with the standard IBM before activating the wall model.
    After activating the wall model, the standard IBM ceases its execution and is replaced by the wall model.

Hence the algorithm for the WM-IB-LBM executes the following sequence of operations:

* Reference velocity :math:`\mathbf{u}` is interpolated at P using a linear interpolation between nearest nodes and decomposed in its tangential :math:`t` and normal :math:`n` components with:

.. math::
    \mathbf{u}_{n} = \left(\mathbf{u}\cdot \mathbf{n}\right)\mathbf{n}
    :label: wm_normal_velocity

.. math::
    \mathbf{u}_{t} = \mathbf{u}-\mathbf{u}_{n}
    :label: wm_tangential_velocity

* The magnitude of :math:`\mathbf{u}_{t}` is then assumed as :math:`u_{\mathrm{t,P_{N}}}` and replaced in the wall function for the calculation of tangential velocity at Q.
* The magnitude from :math:`u_{\mathrm{t,Q}}` is multiplied by the tensor :math:`\mathbf{u}_{t}/ |\mathbf{u}_{t}|` to obtain :math:`\mathbf{u}_{\mathrm{t,Q}}`
* The normal velocity at Q is given by a quadratic relation (:footcite:t:`du2022wall`):

.. math::
    \mathbf{u}_{\mathrm{n,Q}}=\left(\frac{h}{H}\right)^{2}\mathbf{u}_{n}
    :label: wm_normal_velocity_q

* Then, the target velocity at Q is given by :math:`\mathbf{u}_{\mathrm{Q}}=\mathbf{u}_{\mathrm{t,Q}}+\mathbf{u}_{\mathrm{n,Q}}`.
* The velocity correction is implemented using a traditional interpolation-spread procedure from Q.

.. The implementation of a wall model using the immersed boundary method (IBM) follows the same principles as that from the regularized bounce-back BC.
.. The friction velocity is calculated using the log-law of the wall:

.. .. math::
..     u^{*}=\frac{\kappa u\left(\mathbf{x}_{1,\mathrm{out}}\right)}{\mathrm{ln}\left(y/y_{0}\right)}
..     :label: wm_ibm_friction_velocity

.. where :math:`\kappa=0.41` is the Von Karman constant, :math:`u\left(\mathbf{x}_{1,\mathrm{out}}\right)` is the tangential to surface velocity after the diffusive layer from IBM (which is where the IB forces are spread into), consisted of the blue nodes in the figure below:

.. .. figure:: /_static/img/theory/wall_model/ibm_wall_model.svg
..     :width: 50%
..     :align: center

.. The velocity at the reference point is found using IBM interpolation.

.. The subgrid viscosity of the nodes contained within the IBM diffusive layer is then adjusted using the Van Driest damping function :footcite:t:`ma2019dynamic`:

.. .. math::
..     \nu_{\mathrm{SGS}}=\nu \kappa y_{n}^{+}\left(1- e^{-y_{n}^{+}/A}\right)^{2}
..     :label: wm_ibm_sgs_viscosity

.. where :math:`A=19`, and :math:`y_{n}^{+}` denotes the normal distance to the wall on local wall viscosity units.
.. For this adjustment with IBM, a spreading operation of the subgrid viscosity is performed as:

.. .. math::
..     \nu_{\mathrm{SGS}}=\sum_{\mathrm{fluid\;nodes}}\nu_{\mathrm{SGS}}\left(r_{n}\right)\phi_{1}\left(r_{t}\right)
..     :label: wm_viscosity_spread

.. where :math:`\nu_{\mathrm{SGS}}\left(r_{n}\right)` is the subgrid viscosity calculated with the normal distance from the Lagrangian node, and :math:`\phi_{1}\left(r_{t}\right)` is a 1-point kernel function with :math:`r_{t}` being the tangential distance to the referred Lagrangian node:

.. .. math::
..     \phi\left(r\right)=
..     \begin{cases}
..         1 - |r| & \text{, $|r| \leq 1.0$}\\
..         0 & \text{, $1.0\leq |r|$}\\
..     \end{cases}
..     :label: wm_deltaf

.. The distance between LBM grid node :math:`\mathbf{x}` and IB mesh point :math:`\mathbf{X}` is given by :math:`\mathbf{r}=\mathbf{x}-\mathbf{X}`.
.. The normal distance is :math:`r_{n}=\mathbf{r}\cdot\mathbf{n}`, and tangential distance :math:`r_{t}=|\mathbf{r}-r_{n}\mathbf{n}|`.

.. footbibliography::