(vc_sandia_helium_plume)= # Sandia helium plume (Taha 2024 Case 1) ```{important} **Status: planned - not yet validated against reference data.** This is the non-reactive benchmark Taha et al. (2024) used to validate the variable-density low-Mach closure, and it is the closure's own validation target. The setup, dimensionless matching and reference data are documented below; the comparison notebook and digitized reference data are added once the prerequisites in the Prerequisites subsection are met. The primary validation quantity is the plume **puffing frequency**. ``` ## Why this case matters The closed and Boussinesq buoyant cases ({ref}`warm bubble ` and the {ref}`Steckler room fire `) prove the buoyancy coupling in the small-density-variation regime, where the density swing is a linear perturbation about a reference state. Real fire-induced flows are not in that regime: the density varies by a factor of several, so the buoyancy must come from the exact equation of state, not its Boussinesq linearization. Before stacking combustion and radiation on top (the {ref}`methane pool fire `), the variable-density low-Mach closure {footcite:t}`taha2024fire` must be validated on its own. The Sandia FLAME **helium plume** is the clean check for exactly that. A light gas (helium mixture, mean molecular weight `W ~ 5.45 g/mol`) rises into air at a density ratio of order five, with **no combustion and no radiation**. It isolates the Tier 3 closure - the equation-of-state density slaving `rho = P/(rT)`, the reduced-pressure split, and the exact `(rho - rho_inf) g` buoyancy term - from any chemistry. It is Case 1 of Taha et al. (2024), who validated it against the Sandia measurements of O'Hern et al. (2005) and Tieszen et al.; reproducing their result confirms our implementation of the same model. The plume is also a **dynamic** check, not just a mean-profile one. The buoyant shear layer is unstable and the plume self-oscillates at a well-defined **puffing frequency** (~1.37 Hz in the experiment), which is the headline acceptance quantity. ## Physical description A `1 m` diameter circular source at the bottom centre of an `8 x 8 x 4 m` domain injects the helium mixture vertically at `0.325 m/s`. The source is surrounded by a `0.51 m` wide solid floor (the ground plane) and a `2.4 m` diameter chimney above. Air co-flows outside the source annulus at `0.01 m/s`. The injected gas is a mixture of `96.4% He`, `1.9% O2` and `1.7% acetone` by volume, with mean molecular weight `W_He = 5.45 g/mol`; the ambient is air at `T_inf = 285 K`, `p_inf = 80900 Pa`. The light core is buoyant, rises, and the surrounding shear layer rolls up into the periodic puffing structures that dominate the near field. Because the gas is light rather than hot, the density contrast here is driven by **composition** (molecular weight), not temperature: the helium mass fraction is transported as a scalar and the local mixture molecular weight sets the density through the equation of state. This composition-driven density is the central modeling requirement for the case (see Prerequisites). ## Governing equations The density is slaved to the ideal-gas equation of state rather than read from the population sum, ```{math} --- label: vc_helium_eos --- \rho = \frac{P}{r\, T} ``` and the buoyancy enters the fluid as the exact body force ```{math} --- label: vc_helium_buoyancy --- F_\alpha = (\rho - \rho_\infty)\, g_\alpha ``` with gravity along `-z`, of which the Boussinesq force is the linearized small-density-variation limit. For the helium plume the local density contrast comes from the mixture molecular weight: a helium-rich parcel has a lower mean `W` (equivalently a larger effective `r = R_universal / W`) and is therefore lighter than ambient air at the same `P` and `T`, so {eq}`vc_helium_buoyancy` lifts it. The composition is carried by a helium mass-fraction scalar transported by advection-diffusion and coupled back into the density. ## Dimensionless numbers | Dimensionless group | Sandia / Taha 2024 | This case (target) | Matched | | -------------------------------------------------------------- | ------------------ | ------------------ | ------- | | Reynolds number `Re = U D / nu` | `~3220` | `~3220` | target | | Richardson number `Ri = g (rho_inf - rho_0) D / (rho_inf U^2)` | `~75.4` | `~75.4` | target | | Prandtl number `Pr = nu / alpha` | `0.7` | `0.7` | target | | Helium Schmidt number `Sc_He = nu / D_He` | `0.2` | `0.2` | target | | Density ratio `rho_inf / rho_0` | `~5` | `~5` | target | | Mach number `Ma = U / c_s` | `<< 0.1` | `<< 0.1` | target | The driving group is the Richardson number `Ri ~ 75.4` (strongly buoyancy dominated, as expected for a low-momentum light-gas plume), with `Re ~ 3220` from the `1 m` source diameter and `0.325 m/s` inlet. The `~5` density ratio is what takes the case out of the Boussinesq regime and makes it a genuine variable-density test. Reaching the target density ratio requires the composition-driven density described under Prerequisites. ## Simulation setup The reference mesh study (Taha et al. 2024) used a coarse (`dx = 2 cm`, ~2M cells), medium (`dx = 1 cm`, ~4M cells) and fine (`dx = 0.8 cm`, ~7.8M cells) Cartesian mesh with refinement toward the source, converged at the medium level. LES uses Smagorinsky with `Cs = 0.1`. The reference runs ~26 s of physical time with the first ~13 s discarded as the initial transient and the remaining ~13 s used for statistics. | Parameter | Value | | ----------------------- | ------------------------------------------------------------------------- | | Domain | `8 x 8 x 4 m` | | Source | `1 m` diameter inlet, floor centre; `0.51 m` solid floor; `2.4 m` chimney | | Inlet | helium mixture `W ~ 5.45 g/mol` at `0.325 m/s`; air co-flow `0.01 m/s` | | Ambient | `T_inf = 285 K`, `p_inf = 80900 Pa` | | Velocity set / operator | D3Q27 / RRBGK | | Closure | variable-density low-Mach (`models.low_mach`) | | Turbulence model | LES Smagorinsky, `Cs = 0.1` | | Transport | `mu = 1.88768e-5 kg/m/s` (constant); `Pr = 0.7`; `Sc_He = 0.2` | | Reference resolution | coarse `2 cm` / medium `1 cm` / fine `0.8 cm`; converged at medium | | Physical time | `~26 s` (`~13 s` transient + `~13 s` statistics) | The `02_sandia_helium_plume.nassu.yaml` config carries the parts of this setup that the configuration surface supports: the domain, the `models.low_mach` closure and the energy / EOS block. The composition-driven density needed to reach the target helium-air density ratio is listed under Prerequisites. ## Reference and acceptance Reference: Taha et al. (2024) Case 1 (Sec. 3, Figs. 2-12); the underlying Sandia FLAME measurements of O'Hern et al. (2005) and Tieszen et al. See `reference/REFERENCES.md` for provenance and the digitization status. A passing result reproduces: - the **puffing frequency** within the experimental band (`1.37 +/- 0.1 Hz`), from the FFT of the centreline axial velocity at `z = 0.5 m` (primary quantity), - the **-5/3 inertial-range slope** in the temporal energy spectrum of axial velocity at `z = 0.5 m`, - the **mean and rms axial velocity** along the centreline up to `z = 0.8 m`, - the **mean and rms helium mass fraction** along the centreline (rms at a looser tolerance), - the **radial profiles** of mean / rms axial and cross-stream velocity at `z = 0.2, 0.4, 0.6 m`. ## Prerequisites The case depends on one solver capability beyond the variable-density low-Mach closure: - **Composition-driven density (helium-air mixture).** The density must depend on a transported composition scalar, so that the local mixture molecular weight sets the density through the equation of state (a mixture `r(Y_He)` or `W(Y_He)`-driven EOS, coupled to the helium mass-fraction field). This is what carries the helium-air `~5` density ratio that defines the case. ## Results ```{note} The quantitative comparison notebook (puffing-frequency FFT, energy spectrum, and the centreline / radial velocity and helium-mass-fraction profiles against Taha et al. (2024) and the Sandia measurements) and the digitized reference data are added once the Prerequisites are met and a GPU validation run is committed. ``` ```{footbibliography} ```