(wall_model_IBM)=

# Wall Model - IBM

This is the second of the two coupling routes introduced on the {ref}`LBM boundary-condition page <wall_model_LBM_BC>`.
Use it when the solid surface is an immersed STL body that does not align with the Cartesian lattice, which is the common case in CWE: the curved roofs, inclined facades and sharp edges of buildings and vehicles.
The wall function is the same as for the boundary-condition route; the difference is that here the modeled stress is applied as a velocity correction, spread back to the fluid through the immersed-boundary interpolation rather than written directly into a wall-node tensor.

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 Lagrangian mesh, some auxiliary 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
---
align: center
width: 50%
---
```

where the point P is the reference point $\mathrm{P_{N}}$ at normal direction in which the velocity will be linearly interpolated from nearest points.
It is usual to fix a distance $H$ of at least $3\Delta x$ from the wall, which will be located in a point external to IBM diffusive layer.

```{admonition} A subtle trap: keep the reference point outside the diffusive layer
---
class: warning
---
P must sit beyond the IBM diffusive layer, where the velocity field is already free of the spreading kernel's smoothing. Sampling P inside that layer reads a velocity the immersed-boundary forcing has artificially damped, so the wall function is fed a corrupted matching-point state. The $H \geq 3\Delta x$ rule of thumb places P clear of the 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 $h=0.1 - 0.25\Delta x$.

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

```{admonition} Activate the wall model only after the flow has settled
---
class: important
---
Let the simulation reach a stable flow with the standard IBM before switching on the wall model. Once activated, the wall model replaces the standard IBM, which ceases execution. Enabling it on a transient start-up field feeds the wall function an unconverged matching-point state.
```

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

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

$$
\mathbf{u}_{n} = \left(\mathbf{u}\cdot \mathbf{n}\right)\mathbf{n}
$$ (wm_normal_velocity)

$$
\mathbf{u}_{t} = \mathbf{u}-\mathbf{u}_{n}
$$ (wm_tangential_velocity)

- The magnitude of $\mathbf{u}_{t}$ is then assumed as $u_{\mathrm{t,P_{N}}}$ and replaced in the {ref}`wall function <wall_model_wall_functions>` for the calculation of tangential velocity at Q. Concretely, for the log-law closure the friction velocity is recovered from the matching-point tangential velocity via {math:numref}`wm_friction_velocity_log`,

$$
u^{*} = \kappa\, u_{\mathrm{t,P_{N}}} \left[\mathrm{ln}\left(\frac{z_{\mathrm{P_{N}}}}{z_{0}}\right)\right]^{-1},
$$

and the tangential velocity at the near-wall point Q follows from evaluating the log law {math:numref}`wm_velocity_terrain` at $z_{\mathrm{Q}}$:

$$
u_{\mathrm{t,Q}} = \frac{u^{*}}{\kappa}\,\mathrm{ln}\left(\frac{z_{\mathrm{Q}}}{z_{0}}\right).
$$

- The magnitude from $u_{\mathrm{t,Q}}$ is multiplied by the unit vector $\mathbf{u}_{t}/ |\mathbf{u}_{t}|$ to obtain $\mathbf{u}_{\mathrm{t,Q}}$
- The normal velocity at Q is given by a quadratic relation ({footcite:t}`du2022wall`):

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

- Then, the target velocity at Q is given by $\mathbf{u}_{\mathrm{Q}}=\mathbf{u}_{\mathrm{t,Q}}+\mathbf{u}_{\mathrm{n,Q}}$.
- 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}|`.

```{eval-rst}
.. footbibliography::
```