(vc_lowmach_bubble)=

# Variable-density low-Mach warm bubble (sanity check)

```{note}
This is a **sanity check**, not a benchmark validation. There is no closed-form
or cited reference solution to compare against; the case confirms that the
variable-density low-Mach closure runs **stably** over a sustained run and
reproduces the **qualitatively correct** buoyant physics. The quantitative
benchmark validation for this closure is the Sandia helium plume.
```

## Why this case matters

The variable-density low-Mach thermal closure {footcite:t}`taha2024fire` slaves the
density to temperature through the equation of state $\rho = P/(rT)$ and carries
the reduced hydrodynamic pressure $\theta^h$ in the fluid zeroth moment. That
pressure relaxes acoustically, with no elliptic projection, so the dilatation
produced by the energy field can excite an under-damped, highest-wavenumber
(checkerboard / acoustic) mode, so the closure is numerically delicate. This case
exercises the closure end-to-end with the {ref}`in-collision bulk-viscosity stabilization <thermo_stability_lowmach>` (`models.LBM.bulk_viscosity`)
enabled, and checks that the run stays stable and physical.

The stabilization is necessary: with the bulk viscosity off the identical setup
diverges within a few hundred steps; with it on the run completes. The case
therefore also guards against regressions in the stabilization.

## Setup

A smooth warm Gaussian temperature bubble is seeded low in a periodic box:

```{math}
---
label: vc_lowmach_bubble_ic
---
T(x,y,z) = T_\mathrm{ref}\left[1 + A\,\exp\!\left(-\frac{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}{2s^2}\right)\right]
```

with amplitude $A = 0.05$ and width $s \approx 6$ lattice units, centred low at
$(x_0,y_0,z_0) = (24,24,24)$ in a $48\times48\times96$ domain. The EOS makes the
warm core lighter ($\rho = P/(rT)$), and gravity along $-z$ lifts it. There is no
continuous heat source: the bubble simply diffuses and rises.

### Simulation parameters

| Parameter                  | Value                                                        |
| -------------------------- | ------------------------------------------------------------ |
| Domain                     | $48 \times 48 \times 96$                                     |
| Velocity set / operator    | D3Q27 / HRR-BGK                                              |
| LES / SGS constant         | Smagorinsky / $C_S = 0.1$                                    |
| $\tau$                     | $0.6$                                                        |
| Gravity                    | $[0, 0, -2\times10^{-4}]$                                    |
| EOS                        | $r = 1$, $P = 1$, $T_\mathrm{ref} = 1$, $\mathrm{Pr} = 0.71$ |
| `bulk_viscosity` ($\nu_b$) | $0.16667 = 1/6$ (i.e. $\omega_\mathrm{bulk} = 1.0$)          |
| Steps                      | $6000$                                                       |

### Validation metrics

The notebook reads the volume snapshots and asserts the qualitative success
criteria of a healthy run:

- **No divergence** - every snapshot is finite (no NaN in $\rho$, $\mathbf{u}$).
- **Bounded, subsonic velocity** - $\max|\mathbf{u}|$ stays far below
  $\mathrm{Ma} = 0.1$ and shows no exponential growth (the checkerboard mode is
  suppressed). Observed peak $\approx 1.1\times10^{-3}$ ($\mathrm{Ma} \approx
  2\times10^{-3}$).
- **Physical buoyant rise** - the mean vertical velocity is positive and grows
  smoothly.
- **Diffusive relaxation** - the peak temperature decays toward ambient and the
  warm core stays EOS-consistently lighter than ambient.

## References

```{footbibliography}
```

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