Aerodynamic Heating

Aerodynamic Heating (aerodynamics/high-speed/aerodynamic-heating)

Use when you must estimate the aerodynamic heating at the stagnation point of a hypersonic body: correlation-level convective heating from the Sutton-Graves model, the radiation-equilibrium wall temperature it implies, and the nose-radius bluntness trade that drives thermal protection material selection. This leaf implements the Sutton-Graves stagnation-point correlation and the Stefan-Boltzmann radiation balance in pure Python, stdlib only, for flight at a fixed point (constant freestream density and velocity). It pairs with aerodynamics/high-speed/hypersonic-flow for the hypersonic flight environment context and with structures/thermal-structures/thermal-stress-analysis when the wall temperature feeds a structural thermal analysis.

Domain quick reference

• Sutton-Graves stagnation heat flux: q_s = C_SG * sqrt(rho / R_n) * V**3, with C_SG = 1.83e-4 (air correlation constant, SI units arranged so q_s is in W/m2), rho the freestream density in kg/m3, R_n the nose radius in m and V the flight velocity in m/s. The correlation captures the convective heating of the thin shock layer ahead of a blunt body at hypersonic speed.
• Radiation-equilibrium wall temperature: T_w = (q / (eps * sigma))0.25, from the steady balance q = eps * sigma * T_w4 with sigma = 5.670374419e-8 W/m2/K4 (Stefan-Boltzmann) and eps the surface emissivity, default 0.85 for typical thermal protection surfaces.
• Nose-radius scaling: q scales as sqrt(1 / R_n) at fixed rho and V, so doubling the nose radius divides the flux by sqrt(2) (factor 0.7071) and halving it multiplies the flux by sqrt(2) (factor 1.4142). Blunter bodies run cooler at the stagnation point.
• Scaling laws at fixed radius: q grows with V**3 and with sqrt(rho), so the peak heating on a reentry trajectory sits at the point where the density-velocity product is largest, not at peak velocity.
• Units are SI throughout: W/m2, K, kg/m3, m/s, m.
• NACA TR-824 frames the compressible-flow context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the flight point: freestream density rho (kg/m3), velocity V (m/s) and nose radius R_n (m).
2. Compute the stagnation heat flux with stagnation_heat_flux(rho, V, R_n); the module constant C_SG applies by default.
3. Convert the flux to a wall temperature with radiation_equilibrium_temp(heat_flux, emissivity); supply the surface emissivity or accept EPSILON_DEFAULT = 0.85.
4. Trade nose bluntness with radius_scaling(flux_ref, R_ref, R_new): the flux at the new nose radius at the same rho and V.
5. Run the full assessment with heating_assessment(rho, V, R_n, emissivity): it returns the flux, the temperature and the fluxes at doubled and halved nose radius in one dict.
6. Judge the thermal protection implication: a wall temperature near or above the material limit motivates a blunter nose or a higher emissivity coating.
7. Confirm the deterministic checks with the contract test scripts/test_aerodynamic_heating.py.

Worked example

Heating peak on a reentry body: rho = 0.001 kg/m3, V = 7200 m/s, R_n = 0.5 m, emissivity 0.85.

• Stagnation heat flux: q_s = 1.83e-4 * sqrt(0.001 / 0.5) * 7200**3 = 3.0547e6 W/m2 (spec bound 2.5e6 to 4.0e6 W/m2; hand estimate ~3.1e6).
• Radiation-equilibrium temperature: T_w = (3.0547e6 / (0.85 * 5.670374419e-8))**0.25 = 2821.5 K (spec bound 2600 to 3100 K; hand estimate ~2800 K). This exceeds the usable range of most metallic thermal protection, pointing to an ablator or ceramic tile class.
• Bluntness trade at the same rho and V: R_n = 1.0 m gives 2.1600e6 W/m2, exactly the reference flux divided by sqrt(2); R_n = 0.25 m gives 4.3199e6 W/m2, exactly the reference flux times sqrt(2). Halving the nose radius raises the flux by about 41% and the wall temperature toward 3000 K.

Verification

• stagnation_heat_flux raises ValueError when rho, velocity or nose_radius is not positive; radiation_equilibrium_temp raises ValueError for negative heat flux and for emissivity outside (0, 1]; radius_scaling raises ValueError for any non-positive argument; heating_assessment propagates those errors.
• Exact scaling identities hold to machine precision: doubling R_n gives flux_ref / sqrt(2), halving gives flux_ref * sqrt(2); radius_scaling matches a direct Sutton-Graves evaluation at the new radius.
• Worked example outputs fall inside the spec magnitude bounds: flux in 2.5e6 to 4.0e6 W/m2 and temperature in 2600 to 3100 K.
• Monotonicity: the flux increases with velocity (V**3 law) and with density (sqrt law), and decreases as the nose radius grows.
• Radiation round trip: eps * sigma * T_w**4 recovers the input heat flux; all results are deterministic (no RNG).
• Run the contract test offline: python3 scripts/test_aerodynamic_heating.py (33 tests, deterministic).

Related leaves

• aerodynamics/high-speed/hypersonic-flow: the hypersonic flight environment, Rayleigh pitot pressure and Newtonian impact theory.
• aerodynamics/high-speed/normal-shock and aerodynamics/high-speed/oblique-shock: the flowfield jumps that set the post-shock conditions ahead of the heating estimate.
• aerodynamics/high-speed/prandtl-meyer and aerodynamics/high-speed/shock-expansion-airfoil: expansion fan and surface pressure models on the cold side of the energy balance.
• structures/thermal-structures/thermal-stress-analysis: the wall temperature from this leaf as a load for mechanical thermal stress.

Pitfalls

• Searching for peak heating at peak velocity: q grows as V**3 but only as sqrt(rho), so the worst point on a reentry trajectory is where the density-velocity product is largest, not the fastest point.
• Expecting the wall temperature to fall as fast as the flux: doubling the nose radius divides the flux by sqrt(2), but T_w scales as the fourth root, so the temperature drop is far smaller than the flux drop.
• Trading nose radius across different flight states: radius_scaling is only valid at fixed rho and V; re-run the full assessment when the flight point changes.
• Judging thermal protection from flux alone: in the worked example the 2821 K radiation-equilibrium temperature, not the 3.05e6 W/m2 flux, is what exceeds metallic limits and forces an ablator or ceramic tile class.
• Passing emissivity at the boundary: emissivity must lie in (0, 1]; zero or negative emissivity raises ValueError, as do non-positive density, velocity, radius and heat flux.
• Forgetting the correlation's scope: Sutton-Graves is a stagnation-point, blunt-body, thin-shock-layer estimate; it is not a general surface-heating method for sharp or three-dimensional geometries away from the stagnation point.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_aerodynamic_heating.py

The test covers the worked-example magnitude bounds (flux 2.5e6 to 4.0e6 W/m2, temperature 2600 to 3100 K), the Sutton-Graves flux value and its V**3 and sqrt(rho) scaling laws, the exact sqrt(1/R_n) radius-scaling identities and the direct correlation cross-check, the radiation-equilibrium round trip and emissivity dependence, assessment dict consistency, determinism, and ValueError rejection of every non-physical input (non-positive density, velocity, radius, heat flux and emissivity outside (0, 1]).

Compliance

• Standards referenced, not reproduced: NACA TR-824 is named as the compressible-flow context standard; the Sutton-Graves correlation and Stefan-Boltzmann balance above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.