AETHER · Multi-objective shape study of a hypersonic re-entry capsule heat shield · Planned Apr 2026, May to Aug 2026
A capsule needs drag to slow down and must survive the heating that comes with it. AETHER asks how three shape parameters of an Apollo-like capsule, the nose radius, the shoulder radius and the rear-body cone angle, change its drag coefficient and the peak convective heat flux on its heat shield, at one fixed hypersonic condition: the clean Apollo command-module test of NASA TM X-1243, Mach 10.18, ReD = 1.1 × 10⁶, cold wall.
Twenty shapes were computed with laminar perfect-gas Navier–Stokes (SU2 v8.5.0) on body-fitted structured meshes generated from one parameter table, validated against the Apollo pressure and heating distributions and the Wieting cylinder heating benchmark, and assessed with a three-level mesh study. The selection rule, the lowest peak heating among shapes keeping 98% of the baseline drag, was written down before any result.
An AI assistant drafted and helped debug parts of the Python pipeline, explained solver options and reviewed drafts, under a policy written at the start. The geometry family, the condition, the objectives, the selection rule, the benchmarks, the mesh families, the convergence rules and the physical checks that exposed the carbuncle branch were mine; every number is produced by the code in the repository and checked against reference data, a mesh study or a hand calculation. The log of assistance used, and how each output was checked, is in the repository.
CD over the nose-radius sweep, Rn/D from 0.6 to 1.6. Nose radius is the drag lever: the face pressure approaches the flat-disc limit as the nose flattens. The peak heating has a minimum near Rn/D = 1.0. Below it the stagnation region takes over from the shoulder; above it the shoulder peak grows as the flow accelerates harder around the toroid, even as the face plateau falls.
The face carries nearly the pitot pressure whatever its curvature, so drag is set by how much of the maximum-diameter disc the high-pressure face covers: a large nose radius and a small shoulder radius. CD falls almost linearly from 1.50 at Rs/D = 0.03 to 1.24 at 0.18, because the toroid replaces part of the face.
The heating is flat over the face at a level set by the nose radius and peaks at the cap/toroid tangency, sharper and higher the smaller the shoulder: S03 113, B00 111, K03 103 kW/m². Lowering the peak therefore costs drag, and the sweep measures the exchange rate.
The cone half-angle, 25° to 41°, moves drag and peak heating by less than 0.4%, inside the ranking margins. At Mach 10 the afterbody pressure is about 1% of the stagnation pressure, so the cone contributes 1–2% of the drag and nothing to the shield heating. The angle sets length and volume (L/D from 0.80 to 0.51), which is where that design choice actually lives.
What broke. On eight of eighteen shapes the first-order scheme converged, from any start, to a carbuncle branch: a numerical shock mode with the stagnation pressure 12% above anything physics allows. The residuals said done; the pitot pressure said no. Low-CFL retries then produced slow transients that passed every monitor.
A physical check, stagnation pressure against the Rayleigh pitot value and the face plateau against the nose-radius scaling, now decides for every solution. The fix was in the mesh: cells elongated along the bow shock. The fourteen shapes converged on both mesh families agree to 0.05% in drag and 0.8% in peak heating.
The nondominated set is the sharp shoulder S03 (+2.8% drag, +2.5% heating), the baseline family, N10 (Rn/D = 1.0: −1.8% drag, −5.3% heating) and K03 (Rn/D = 1.0 with Rs/D = 0.12: −9.3%, −6.9%). The stated rule selects N10: 5.3% less peak heating for 1.8% less drag, four times the ranking margin established by refinement (0.24% in drag, 1.4% in heating). At baseline drag nothing beats the Apollo shape beyond the margins.
rms difference in p/pt2 from the 14 measured Apollo stations; the stagnation pressure is within 0.5% of the pitot value, and the face heating is flat as measured.
grid-convergence index of the peak heating at the sweep level: 112.8, 110.6, 109.3 kW/m² over three nested meshes, observed order 1.5. Drag moves within 0.15%.
what the absolute heating level is trusted to, the perfect-gas gap to the shock-tunnel data. Differences between shapes are trusted to the ranking margins, and only those are claimed.
Where it came from, and how it was built
Two years ago my project was a model Formula One car: shape it, simulate it, race it, and find out whether the numbers were right. It left one habit, change one thing at a time and measure. Then, in March 2026, two days at ISRO's Satish Dhawan Space Centre. In front of a re-entry capsule the first thing you notice is how blunt it is; to someone who had spent a year chasing streamlines it looked wrong. Blunt is the point: the flat face pushes the shock away and keeps the heat off the shield. I went home with the question this chapter answers, and with the F1 habit as the method.
