(vc_poiseuille_pipe_flow)=

# Poiseuille Pipe Flow

## Why this case matters

Pipe flow extends the laminar channel test to a curved, axisymmetric geometry, and is where the staircased representation of a round wall on a Cartesian lattice is first put under pressure. The parabolic profile in {eq}`vc_poiseuille_pipe_vel_profile` has an exact analytical form, so deviations expose how faithfully the boundary conditions reproduce a circular no-slip wall and how much spurious anisotropy the velocity set introduces. Passing it in the laminar regime is a prerequisite for the turbulent pipe case that follows, where lattice isotropy becomes critical. The analytical solution is a classical result of viscous flow theory {footcite:t}`Pozrikidis2011-do`.

Poiseuille pipe flow can be described as a pressure driven axissimetric flow that occurs in a circular cross-section pipe, as illustrated below.

```{figure} /_static/img/solver/validation/cases/poiseuille_pipe.svg
---
width: 80%
align: center
---
```

This results in an unidirectional flow to which an analytical solution of the velocity profile with $0 \leq r \leq 1$ is given by {eq}`vc_poiseuille_pipe_vel_profile`

```{math}
---
label: vc_poiseuille_pipe_vel_profile
---
\frac{u(r)}{u_{\mathrm{avg}}}=2\left(1 - r^2\right)
```

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

```{math}
---
label: vc_poiseuille_pipe_avg_velocity
---
u_{\mathrm{avg}}=\left(-\frac{\mathrm{d}p}{\mathrm{d}x}\right)\frac{\left(0.5d\right)^{2}}{8\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:
---
03.1_poiseuille_pipe_periodic.ipynb
03.2_poiseuille_pipe_vel_neumann.ipynb
```

```{footbibliography}
```