bijective/capsule-aeroheating

Capsule forebody aeroheating, Newtonian and Navier–Stokes CFD

The capsule family with Newtonian heating, meshed like the dataset's walls; an illustration at Mach 10 and 60.6 km.

Laminar, perfect-gas heat flux and pressure on the heat shield of a parametric entry capsule, at two fidelities. There are 1,024 samples, each a unique shape and flight condition, from a Varda-like 0.8 m to beyond Orion's 5 m and from Mach 6 to 15. Every capsule has a solution from the Newtonian panel method, a fast engineering model, and 32 of them also have a second-order laminar Navier–Stokes CFD solution from Eilmer. The meshes were created using TopoLink, a topology-linked, hybrid-cell, structured mesh generator. Every wall has the same nodes, so node i is the same point of the heat shield on every capsule and at both fidelities.

One CFD case (capsule A below) at Mach 13.2 and 53 km: SU2 (Eilmer's starting field), Eilmer on its shock-fitted grid, and the wall heat flux of all three methods.

The whole dataset is stored in one Gamut file, vehicle_capsule_multifidelity.gamut, readable from Python, MATLAB, Julia, C++, C, Rust and the command line.

At a glance

Members	1,024 sampled capsules (label member), each one shape at one flight condition
Runs	1,056: 1,024 Newtonian (fidelity = newtonian_panel) and 32 Eilmer (fidelity = eilmer_second_order)
Inputs	diameter, nose radius, forebody angle and shoulder radius (shape); Mach number and altitude (flight)
Wall	each capsule's own TopoLink CFD wall, the full heat shield to mid-shoulder: 2,361 nodes and 4,640 triangles, the same on every run; coordinates in metres at real size
Fields	heat flux and pressure at the nodes; heat flux on the wall faces; arc length from the stagnation point
Scalars per run	heat load (the heat flux integrated over the wall at the run's flight condition, in W; not over a trajectory); stagnation, peak and mean heat flux; stagnation pressure; freestream state
Ranges, Newtonian	stagnation heat flux 2.49–1,884 kW/m², heat load 2.89–8,705 kW
Ranges, Eilmer	stagnation heat flux 10.4–461 kW/m², heat load 15.1–3,186 kW
Held fixed	calorically perfect air, laminar flow, a 300 K wall, zero angle of attack
File	vehicle_capsule_multifidelity.gamut, 50.6 MB; gamut info lists its contents, and its source attribute describes the models and checks

Two fidelities on one wall

The Newtonian panel method (fidelity = newtonian_panel) is an engineering model: modified Newtonian pressure, Fay–Riddell stagnation-point heating (its perfect-gas form) at the velocity gradient of the nose radius, and the Lees laminar distribution elsewhere. A run takes seconds, so every capsule has one. It cannot see how the shoulder changes the flow over the face, and it under-predicts the heating peak on the shoulder.

Eilmer (fidelity = eilmer_second_order) is CFD: the laminar Navier–Stokes equations, second order, on a 232,000-cell grid fitted to the bow shock around the whole heat shield. A case takes

1.3–3.7 hours on six cores, including the SU2

start, so only a sample of the capsules has one.

The cheap model maps the heating over the whole design; the expensive one shows how far off it is, and where. A field such as heat_flux is one array of runs × nodes, so the difference between the fidelities is a subtraction at each node, with no interpolation between meshes.

How good is each fidelity

Along the heat shield of three of the CFD cases:

Both fidelities on the same nodes; the shoulder is grey, and Eilmer's curve is a running median over 25 nodes.

On all 32 CFD cases, the Newtonian value minus Eilmer's, as a percentage of Eilmer's:

One point per CFD case; A, B and C are the capsules above.

• Heat load. Below Eilmer's on every case, by 1.5–15.4 % (median 5.9 %), most on the largest noses: the three largest (1.25 D and above) are12.8–15.4 % low. The closest 5 are less than 2 % low; the uncertainty in Eilmer's own heat load was not measured.
• Stagnation heat flux. From +32.2 % (capsule_0745, a 0.27 D nose on a 70° cone) to−12.5 % (capsule_0588, a 1.45 D nose). It is low on 7 of the 8 noses above 0.9 D and within 0.3 % on the other; on smaller noses it is high or low depending on the face. This is consistent with the panel method taking the stagnation velocity gradient from the nose radius alone, where in the flow the whole forebody sets it.
• Shoulder peak. The panel method's highest heat flux on the shoulder (from the ring of nodes where the cone meets it) is 14–33 % below Eilmer's (median 23 %). The shoulder is the hottest point of the heat shield on10 cases in Eilmer and on 4 in the panel method.
• Eilmer itself. At zero incidence the heating should be the same all the way round the axis. Over 3–80 % of the arc, Eilmer's varies round the axis by a standard deviation of at most0.98 % at each distance from the stagnation point.

Inputs

Name	Range	Units	Meaning
diameter	0.8–5.5	m	maximum diameter D, sampled log-uniform
nose_radius	0.25–1.6	× D	radius of the spherical nose cap
forebody_angle	55–75	degree	half-angle of the cone between the nose cap and the shoulder
shoulder_radius	0.025–0.075	× D	radius of the toroidal shoulder
mach	6–15	1	freestream Mach number
altitude	40–80	km	freestream from the 1976 US Standard Atmosphere

Held fixed for every run: zero angle of attack, a 300 K isothermal wall, and calorically perfect air (γ = 1.4, R = 287.05 J/(kg K), Sutherland viscosity, Pr = 0.72), laminar.

The capsules are a Latin hypercube over the six inputs, with the nose radius drawn as a fraction of the largest buildable at each forebody angle and shoulder radius (nose_radius_fraction); that limit is the empty corner of the middle panel. The CFD cases are 32 capsules matched, by optimal assignment, to the points of a 32-point Latin hypercube over the same inputs; they spread over the whole design but are not themselves a Latin hypercube.

Quick start

pip install bijective-gamut
hf download bijective-systems/capsule-aeroheating --repo-type dataset --local-dir capsule
gamut info capsule/vehicle_capsule_multifidelity.gamut

In Python, choosing runs by fidelity and pairing each Eilmer run with the Newtonian run of the same member:

import numpy as np
import gamut

with gamut.open("capsule/vehicle_capsule_multifidelity.gamut") as study:
    runs = study.table()                                    # inputs, results and labels, one entry per run
    cfd = study.select("fidelity == eilmer_second_order")   # positions of the Eilmer runs
    newt = study.select("fidelity == newtonian_panel")      # positions of the Newtonian runs, one per member
    row_of = dict(zip(runs["member"][newt], newt))
    pair = np.array([row_of[m] for m in runs["member"][cfd]])   # each Eilmer run's Newtonian run
    q_cfd = study.get_field("heat_flux", rows=cfd)          # (Eilmer runs, nodes), W m-2
    q_newt = study.get_field("heat_flux", rows=pair)        # the same members, node for node
    xyz = study.get_points(int(cfd[0]))                     # wall nodes of the first CFD member, m
    cells = study.get_cells()                               # the shared triangles, as VTK gives them

difference = q_newt / q_cfd - 1        # Newtonian against Eilmer at every node of every CFD case

From the command line, without writing code:

# the Eilmer runs and the Newtonian runs as two CSV tables; join them on member
gamut get capsule/vehicle_capsule_multifidelity.gamut member diameter nose_radius mach altitude heat_load stagnation_heat_flux --where "fidelity == eilmer_second_order" -o eilmer.csv
gamut get capsule/vehicle_capsule_multifidelity.gamut member diameter nose_radius mach altitude heat_load stagnation_heat_flux --where "fidelity == newtonian_panel" -o newtonian.csv

# a smaller study file of the Eilmer runs alone, then one .vtu per run, with every field, for ParaView
gamut get capsule/vehicle_capsule_multifidelity.gamut --where "fidelity == eilmer_second_order" -o eilmer.gamut
pip install "bijective-gamut[vtk]"
gamut-vtk export eilmer.gamut eilmer_vtk/

gamut-vtk export capsule/vehicle_capsule_multifidelity.gamut capsule_vtk/ writes every run the same way. Gamut has libraries for MATLAB, Julia, C++, C and Rust as well; see the documentation.

How it was made

Each capsule's CAD comes from the vehicle_capsule parametric family and its wall mesh from TopoLink, with the same nodes and triangles on every capsule. The Newtonian panel method ran on the wall of every capsule. For each CFD case, a first-order laminar SU2 solution on a grid extruded from the wall served only as Eilmer's starting field: it reads the stagnation heat flux 5.5–17.9 % above Eilmer's and is not in the file. Eilmer then ran on two grids fitted to the shock: on the first until its CFL number reached 1,000 or for 150 steps, on the second until the wall heat flux and the heat load were steady. The flow-field figure near the top shows one case.

Limitations

• The Newtonian method's errors. Its heat load is low on every CFD case, its stagnation heat flux is high or low depending on the shape, and it under-predicts the shoulder peak.
• Few CFD cases. 32 cases over six inputs show the size and pattern of the Newtonian errors, not the CFD heating everywhere in the design.
• No error bars on the CFD. Eilmer's results rest on its grids and convergence checks; the grid was not refined case by case.
• Simplified physics. Calorically perfect gas (no chemistry or vibrational excitation, which matter over most of the Mach range, increasingly toward its top), laminar flow, zero incidence, a 300 K wall, and the heat shield alone, with no afterbody or wake.

Versions

• Version 2 (11 November 2026) adds vehicle_capsule_multifidelity.gamut.
• Version 1 (4 November 2026), vehicle_capsule_newtonian.gamut, is unchanged: the same 1,024 capsules with the Newtonian panel method alone, on TopoLink's family mesh of 2,929 nodes and 2,880 quads. The two files' Newtonian heat loads agree within 0.3 % and their stagnation heat fluxes are the same. Version 2 adds the label fidelity, the field heat_flux_cell and the result stagnation_pressure, and drops version 1's label kind and result stagnation_velocity_gradient. Version 1's card is in this repository's history.

Licence and citation

The data and figures are licensed under CC BY 4.0.

@misc{bijective_capsule_aeroheating_2026_v2,
  title        = {Capsule forebody aeroheating, Newtonian and Navier--Stokes {CFD}},
  author       = {{Bijective Systems}},
  year         = {2026},
  version      = {2},
  howpublished = {\url{https://huggingface.co/datasets/bijective-systems/capsule-aeroheating}}
}

Version 1 alone (vehicle_capsule_newtonian.gamut) keeps its original entry, bijective_capsule_aeroheating_2026, titled "Capsule forebody aeroheating, engineering model (demo)".

About

Made by Bijective Systems with TopoLink (meshes for parametric geometry families) and Gamut (an open file format for parametric simulation studies). The CFD used Eilmer, with starting fields from SU2. Proximant, a Bijective Systems product, chose the design and fits surrogate models to studies like this one.