(vc_couette_flow)=

# Couette Flow

## Why this case matters

Couette flow is the simplest shear flow with an exact, time-dependent analytical solution, which makes it the first quantitative check a solver should pass. It isolates the response of the collision operator to a pure shear stress, with no pressure gradient and no advection, so any error maps directly to the viscous and boundary-condition behaviour of the LBM. The transient term in {eq}`vc_startupcouette` exercises momentum diffusion from a no-slip wall, while the steady state in {eq}`vc_couette` verifies that a moving-wall boundary condition reproduces a linear velocity profile to machine precision. For computational wind engineering this is the building block of every near-wall shear layer, so getting it exactly right is a prerequisite for the more demanding turbulent cases. The reference solution is the classical series expansion reported by {footcite:t}`Pozrikidis2011-do`.

Couette flow can be defined as a flow between two plates, initially at rest, in which the top plate starts moving with a constant velocity $U$ while the bottom plate is kept with a null velocity. This configuration is illustrated below:

```{figure} /_static/img/solver/validation/cases/couette.svg
---
width: 70%
align: center
---
```

This flow consists of a unidirectional flow with a velocity field that varies only at $y$-direction. An analytical solution for it can be found in {footcite:t}`Pozrikidis2011-do`, and through this solution the velocity can be written as:

```{math}
---
label: vc_startupcouette
---
u\left(y,t\right)=U\frac{y}{h}-2\frac{U}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}e^{-n^{2}\pi^{2}\frac{\nu t}{h^{2}}}\mathrm{sin}\left[n\pi\left(1-\frac{y}{h}\right)\right]
```

After a sufficient time, the flow reaches a steady state Couette flow with a linear velocity profile:

```{math}
---
label: vc_couette
---
u\left(y\right)=U\frac{y}{h}
```

```{toctree}
---
hidden:
---
01_startup_couette.ipynb
```

```{footbibliography}
```