(vc_poiseuille_channel_flow)=

# Poiseuille Channel Flow

## Why this case matters

Pressure-driven channel flow is the laminar counterpart to Couette flow: instead of a moving wall it is driven by a body force or a pressure gradient between two stationary no-slip plates. It is the canonical test for the LBM force term, since the parabolic profile in {eq}`vc_poiseuille_periodic_vel_profile` only emerges if the Guo forcing scheme injects momentum correctly throughout the bulk. The case also confirms that the two no-slip walls and the periodic (or inlet/outlet) treatment combine to give the right mean flow rate, linking the prescribed forcing to the analytical average velocity in {eq}`vc_poiseuille_channel_avg_velocity`. In computational wind engineering this validates the machinery that later sustains a turbulent boundary layer against wall friction. The analytical solution is a standard result of viscous flow theory {footcite:t}`Pozrikidis2011-do`.

Poiseuille periodic flow can be described as a flow between two stationary plates, caused by a pressure gradient $\mathrm{d}p/\mathrm{d}x$ parallel to the plates. This is illustrated in the figure below.

```{figure} /_static/img/solver/validation/cases/poiseuille_channel.svg
---
width: 60%
align: center
---
```

We're mainly interested in the velocity profile from $y=0$[ to :math:\`y=1]{.title-ref} (top plate) in the flow direction. This profile is analytically defined by equation {eq}`vc_poiseuille_periodic_vel_profile`

```{math}
---
label: vc_poiseuille_periodic_vel_profile
---
\frac{u(y)}{u_{\mathrm{avg}}}=6\left(y - y^2\right)
```

And $u_{\mathrm{avg}}$ is the average velocity, which can be obtained numerically, calculating its value using the simulation data, or using the analytical equation:

```{math}
---
label: vc_poiseuille_channel_avg_velocity
---
u_{\mathrm{avg}}=\left(-\frac{\mathrm{d}p}{\mathrm{d}x}\right)\frac{h^{2}}{12\rho\nu}
```

where the pressure gradient can be set through pressure inlet/outlet boundary conditions, or adding a external force to the bulk $F_{x}=-\mathrm{d}p/\mathrm{d}x$.

```{toctree}
---
hidden:
---
02.1_poiseuille_channel_periodic.ipynb
02.2_poiseuille_channel_regularized.ipynb
02.3_poiseuille_channel_vel_neumann.ipynb
02.4_poiseuille_channel_multilevel.ipynb
```

```{footbibliography}
```