(vc_lid_driven_cavity)=

# Lid-Driven Cavity

## Why this case matters

The lid-driven cavity is the canonical benchmark for validating **moment-based boundary conditions** in the lattice Boltzmann method {footcite:t}`mohammed2017lid`. A square cavity with three stationary no-slip walls and a single moving lid produces a recirculating primary vortex plus secondary corner vortices whose strength and structure depend only on the Reynolds number, so the centreline velocity profiles are a sharp, geometry-free test of the wall treatment. It is the natural validation for the moment-based wall closure introduced in the boundary-condition rework (see {ref}`Moment-Based Boundary Conditions <bc_moment_based>`): the lid is the moment-based moving wall `RegularizedVelocityWall`, and the three static walls are the no-slip `RegularizedHWBB`, which is the zero-velocity case of the same closure. No population bounce-back is used. Because the corner singularities at the lid make this a demanding wall test, reproducing the benchmark profiles confirms that imposing the hydrodynamic moments at the boundary captures both the bulk vortex and the near-wall behaviour.

The accepted reference is the high-accuracy multigrid Navier-Stokes solution of {footcite:t}`ghia1982high`.

### Setup

A square cavity of side $L$ (2-D, D2Q9 lattice) is filled with fluid initially at rest. The top wall (the lid) translates in the $+x$ direction at a constant speed $U$; the bottom and the two side walls are stationary no-slip walls.

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

The Reynolds number is

```{math}
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label: vc_lid_driven_cavity_re
---
\mathrm{Re} = \frac{U L}{\nu}.
```

It is swept over $\mathrm{Re} \in \{100, 400, 1000\}$ at fixed grid ($L = 256$) and fixed lid speed ($U = 0.05$, so $\mathrm{Ma} = U\sqrt{3} \approx 0.087 < 0.1$), varying only the viscosity through $\tau = 3\nu + \tfrac{1}{2}$.

### Reference solution

There is no closed form. The benchmark is the tabulated solution of {footcite:t}`ghia1982high`, given as two centreline profiles:

- the streamwise velocity $u(y)$ along the vertical centreline $x = L/2$, and
- the vertical velocity $v(x)$ along the horizontal centreline $y = L/2$,

both normalised by the lid speed $U$. The digitised tables (Re = 100, 400, 1000) are committed under `reference/` with their provenance in `reference/REFERENCES.md`.

### Simulation parameters

| Parameter              | Value                                                          |
| ---------------------- | -------------------------------------------------------------- |
| Lattice                | D2Q9, RR-BGK                                                   |
| Cavity side $L$        | 256                                                            |
| Lid speed $U$          | 0.05                                                           |
| Reynolds numbers       | 100, 400, 1000 (`!unroll`, `sim_id` 0/1/2)                     |
| Relaxation time $\tau$ | 0.8840 / 0.5960 / 0.5384                                       |
| Lid BC                 | `RegularizedVelocityWall` ($u = (U, 0)$, extrapolated density) |
| Side / bottom BC       | `RegularizedHWBB` (no-slip)                                    |
| Steps to steady state  | 300k / 500k / 800k                                             |

### Validation metrics

At steady state the simulated centreline profiles $u(y)$ at $x = L/2$ and $v(x)$ at $y = L/2$ are compared against the Ghia tables at each Reynolds number. Success criteria: the profiles overlay the benchmark points (including the sign reversals of the primary vortex and the near-wall extrema), and the mean absolute deviation from the tabulated points stays within about one percent of the lid speed across Re = 100-1000. The wall density is taken zero-normal-gradient from the interior (the default for the moment-based walls); pinning it to a uniform constant instead would impose a spurious uniform wall pressure and roughly quintuple the deviation at Re = 1000, since the cavity wall pressure genuinely varies.

```{toctree}
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hidden:
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05_lid_driven_cavity.ipynb
```

```{footbibliography}
```