A sealed 0.6 × 0.6 × 1.2 m cabinet starts from rest at 25 °C. A 0.4 × 0.3 m patch on one wall is held at −10 °C, with other walls insulated. Natural convection near the patch drives the flow, and its thin thermal boundary layer controls the cooling rate. These are Boreas v0.10.0 results. The energy-balance check used CPU (boreas2); the wall-refinement and time-step ladder used both GPU (boreas2_gpu) and CPU.

Finding: Upwind energy balance closes within 0.5%. Wall heat-transfer rate has not been validated, because cooling does not converge with either wall-cell size or time step in the tested range. This page records that limitation for a thin natural-convection boundary layer on a coarse background mesh. Agreement with the approximate lumped-capacitance model is not claimed.

Model and boundary conditions

A wall patch in the sealed cabinet is held at −10 °C; other walls are insulated. Initial air temperature is 25 °C. Schematic; not to scale.

Reference criteria

Exact energy balance. Heat crosses the cabinet boundary only at the cooling patch, so stored energy must equal integrated wall heat:

Boreas records Q_wall at each step. A nonzero residual indicates numerically created or lost heat.

Lumped-capacitance plausibility model. If cabinet air is assumed well mixed and the patch is treated as a vertical plate of height L = 0.3 m, the laminar natural-convection correlation Nu = 0.59 Ra^{1/4} gives h for θ = T_mean − T_c:

Here c = C_h A/(ρc_pV). Initially θ₀ = 35 K, h = 4.98 W/(m²·K), and the characteristic time is 872 s. This approximate model ignores stratification beneath the patch and is only an order-of-magnitude check. Mesh and time-step convergence, not agreement with this model, determine whether Boreas's cooling rate is resolved.

Simulation setup

Energy balance and cooling rate

Four upwind combinations from the eight uniform-mesh decks. Top: mean temperature and approximate lumped model (black). Bottom: effective h_eff and correlation h(θ). Cooling rate changes substantially between 50 and 25 mm and across turbulence models; h_eff remains about one fifth to two fifths of the correlation value.

• Conservation: All four upwind cases close within 0.5%. Wall heat is calculated consistently with the diffusion solve. Why Chen's 0.35–0.49% residual exceeds laminar's 0.01–0.03% by an order of magnitude remains unresolved. Semi-Lagrangian misses by 3–24% and is not suitable for heat-balance-sensitive runs here.
• Cooling rate: Mean temperature at 1000 s ranges from 4.8 to 18.1 °C across mesh and turbulence settings, far warmer than the lumped model. The estimated thermal-boundary-layer thickness L/Nu ≈ 5 mm is unresolved by 25–50 mm cells; sensitivity to turbulence model is another indication of insufficient resolution.

Wall-refinement and time-step ladder

Wall cells were refined to 12.5, 6.25, and 3.125 mm while dt was reduced from 0.5 to 0.0625 s.

Each panel is a wall-cell size; the horizontal log axis halves dt at each tick and all panels share vertical span. At 6.25 mm, mean temperature at 1000 s shifts by about 0.85 K in the same direction with every dt halving, without shrinking increments. The largest GPU–CPU difference is 0.17 K, smaller than the roughly 0.85 K time-step shifts, so the dependence is unlikely to be GPU specific.

• Wall cells: At dt = 0.125 s, h_eff(250 s) rises 3.34 → 5.29 → 5.69 W/(m²·K) as wall cells shrink. This approaches the correlation's roughly 4.6 at θ = 26 K, but h_eff itself moves with dt (5.55 → 5.08 at 6.25 mm).
• Time step: At 6.25 mm, changes in T_mean(1000 s) after each dt halving are 0.96, 0.95, 0.91 K on GPU and 0.87, 0.81 K on CPU. First-order temporal convergence would halve each increment; dt = 0.0625 s is not in that asymptotic range.
• Instability: With dt = 0.5 s, refined 6.25 and 3.125 mm cases diverge after about 3800 and 1500 s, respectively, as unsuccessful implicit upwind steps accumulate. Refinement requires a smaller dt.
• GPU balance drift: GPU balance error grows with step count from roughly −0.1% to −1.9%. This GPU energy drift was fixed in v0.10.1. CPU has the same dt-dependent cooling rate, so drift does not explain the lack of convergence.

Quantitative results

Limits

• Wall heat transfer and cooling rate are not quantitatively validated for v0.10.0. Neither the three wall resolutions nor four time steps establish a converged value. Agreement of the mean-temperature curve with a lumped model or other benchmark cannot be claimed. An August 2026 statement of qualitative agreement compared only curve shape with an exponential of unspecified h and is withdrawn.
• A working hypothesis is numerical diffusion from temporal discretization. Implicit Euler plus first-order upwind adds streamwise diffusion on the order of u²dt/2. For near-wall descent speeds of 0.1–0.2 m/s, even dt = 0.0625 s gives 15–60 times the air thermal diffusivity (2.1e-5 m²/s), and CFL on 6.25 mm cells is 1–2. Reaching u dt < h/4 may require dt ≲ 0.01 s, around 100,000 cells × 100,000 steps. A next check would compare CPU at 6.25 mm with dt = 0.03 and 0.015 s to see whether increments begin shrinking.
• This limit concerns a thin natural-convection boundary layer on a coarse mesh. For sealed spaces cooled only by an isothermal wall, reduce wall-cell size and dt together and verify convergence before making quantitative predictions.
• The energy-balance check used CPU. GPU Chen wall heat was recorded with a pre-update effective diffusivity and was not used to judge that balance. The time-step ladder gives similar cooling rates on GPU and CPU.