(bc_moving_wall)=

# Moving Wall

Moving wall boundary conditions are those in which a sliding wall is represented, its implementation is usually similar to solid wall implementations, but with a non-null fixed velocity.

(bc_velocity_bounce_back)=

## Velocity Bounce-Back

The velocity bounce-back boundary condition imposes a velocity profile with no transversal components.

```{figure} /_static/img/theory/LBM/inlet.svg
---
align: center
width: 50 %
---
```

The unknown populations are obtained by reflecting back post-collision populations that entered the wall, just like the HWBB, and then correcting them by

$$
f_\bar{i}(t + \Delta t) = f_i^* + \frac{2 w_i \rho_\mathrm{b}\, c_{i\alpha}\, u_{\mathrm{w},\alpha}}{c_s^2}
$$ (bc_vhwbb)

where $\bar{i}$ is the opposite $i$-direction and $u_{\mathrm{w},\alpha}$ is the prescribed wall velocity, projected onto the lattice direction through $c_{i\alpha} u_{\mathrm{w},\alpha}$ summed over all components $\alpha$ (the generic wall-velocity projection, {footcite:t}`Kruger2016-cv` Eq. 5.26). The correction is added to the opposite (reflected) population $f_{\bar{i}}$.
Notice that the density at the boundary ($\rho_\mathrm{b}$) has to be calculated someway.
Nassu extrapolates it from the first two fluid nodes along the inward wall normal as $\rho_\mathrm{b} = \frac{1}{2}\left(3\rho_{b+1} - \rho_{b+2}\right) = \rho_{b+1} + \frac{1}{2}\left(\rho_{b+1} - \rho_{b+2}\right)$, where $\rho_{b+1}$ and $\rho_{b+2}$ are the densities at the first and second fluid nodes along the inward wall normal.

```{admonition} Use case
Velocity bounce-back forces a tangential velocity at a wall without restricting the pressure, so it serves both a sliding (moving) wall and a velocity inlet.
```

```{important}
Velocity bounce-back steadily adds density to the domain, which can drift the average density and eventually trigger instabilities.
This makes it a poor choice at an inlet, where the bias accumulates over the whole run.
Where a constant-density inlet is acceptable, use the {ref}`Uniform velocity BC <vc_poiseuille_pipe_flow>` instead.
```

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