The CFD Test Case
Bayesian inference of catalytic efficiencies and plasma jet conditions in reusable thermal protection systems under model error.
During atmospheric re-entry, spacecraft travel at hypersonic velocities, producing strong bow shocks that dissociate air molecules into reactive atomic species (\(\text{N}\) and \(\text{O}\)) and generating surface temperatures exceeding \(2000\,\text{K}\). To withstand these aerothermal environments without ablative surface recession, reusable Thermal Protection Materials (TPMs) are employed. These materials rely on high surface emissivity and low catalytic efficiency to suppress the exothermic recombination of atomic species on the spacecraft surface:
\[\text{N} + \text{N} \xrightarrow{\text{wall}} \text{N}_2 + \Delta H_R, \qquad \text{O} + \text{O} \xrightarrow{\text{wall}} \text{O}_2 + \Delta H_R\]Because surface catalytic efficiencies (\(\gamma_{\text{N}}, \gamma_{\text{O}}\)) cannot be measured directly in ground test facilities, they must be reconstructed by solving an inverse Bayesian problem that matches computational fluid dynamics (CFD) simulations against experimental observables, such as stagnation-point heat flux \(q_w\).
Experimental Testing at the VKI Plasmatron
In collaboration with the von Karman Institute for Fluid Dynamics (VKI), this work investigated supersonic testing in the VKI Plasmatron, the world’s most powerful inductively-coupled-plasma (ICP) wind tunnel. High-purity plasma is generated electrodelessly in a torch, expanded through a convergent-divergent nozzle into a sub-atmospheric chamber, and discharged as a supersonic underexpanded jet over a water-cooled calorimeter probe (\(R = 25\,\text{mm}\)).
Operating and observable quantities include:
- Control Variables: Reservoir pressure \(\mathrm{P}_0\), chamber static pressure \(\mathrm{P}_s\), mass flow rate \(\dot{m}\).
- Measurements: Stagnation heat flux \(q_w\) (\(\pm 5\%\)), probe stagnation pressure \(\mathrm{P}_w\) (\(\pm 1\%\)).
- Unmeasured Nuisance Parameter: The reservoir total temperature \(\mathrm{T}_0\) cannot be measured directly in the high-enthalpy chamber and must be inferred jointly with the catalytic properties.
High-Fidelity & Low-Fidelity Numerical Modeling
- High-Fidelity Solver (
US3D): An unstructured, parallel implicit Navier-Stokes solver simulating 5-species reacting air (\(\text{N}_2, \text{O}_2, \text{NO}, \text{N}, \text{O}\)) in two-temperature thermochemical non-equilibrium (\(T - T_v\)), coupled with finite-rate surface Gas-Surface Interaction (GSI) catalytic boundary conditions. - Low-Fidelity Surrogate (
STAGLINE): A fast coupled model combining a 0D chemical-equilibrium nozzle expansion with a quasi-1D boundary layer solver along the stagnation streamline.
The Danger of Overlooking Model Error
When calibrating catalytic efficiencies without accounting for model discrepancy:
\[\mathbf{q}_w = \tilde{q}_w(\mathbf{X}; \boldsymbol{\gamma}) + \boldsymbol{\epsilon}\]the posterior distributions for \(\gamma_{\text{N}}\) and \(\gamma_{\text{O}}\) collapse onto \(\gamma \approx 1.0\) (predicting a fully catalytic material with severe overconfidence), despite synthetic ground truth being \(\gamma_{\text{N}} \approx 0.07\).
As seen in the residual analysis, all posterior-averaged residuals are systematically positive, proving that missing physical mechanisms (flow non-uniformities, finite-rate gas kinetics, and 1D flow approximations) bias the inference. Introducing an explicit model discrepancy term \(z(\mathbf{x}) \sim \mathcal{GP}(0, k_{\boldsymbol{\psi}})\) eliminates this bias and successfully recovers the true catalytic values.
Physical Sensitivity: Why \(\gamma_{\text{N}}\) Dominates \(\gamma_{\text{O}}\)
The calibration resolves the nitrogen catalytic efficiency \(\gamma_{\text{N}}\) far more accurately than oxygen \(\gamma_{\text{O}}\). Although \(\text{O}_2\) dissociates earlier than \(\text{N}_2\), the formation enthalpy of atomic nitrogen (\(472\,\text{kJ/mol}\)) is nearly twice that of oxygen (\(249\,\text{kJ/mol}\)). Consequently, wall heat flux is overwhelmingly sensitive to nitrogen recombination, ensuring high identifiability for \(\gamma_{\text{N}}\).
Multi-QoI Accelerated Calibration via AS-HGP
To calibrate against high-fidelity CFD without crippling computational overhead, we formulated a 3-observable calibration problem over \(\mathbf{y} = (q_w, \mathrm{P}_w, \dot{m})\) to simultaneously reconstruct reservoir temperature \(\mathrm{T}_0\) and \(\gamma_{\text{N}}\) using the AS-HGP accelerated CMP method.
Performance Comparison (1,000,000 MCMC Samples)
| Method | Offline Time | Time / Sample | Total Time | Speed-Up |
|---|---|---|---|---|
| Exact CMP | 0 s | \(15.15\,\text{ms}\) | \(\approx 4.21\) h | Reference |
| AS-HGP | \(140\,\text{s}\) | \(0.12\,\text{ms}\) | \(\approx 5\) min | \(\approx 50\times\) |
Posterior Predictive Validation
Evaluating the calibrated model at an unseen operating condition (\(\mathrm{P}_0 = 14.14\,\text{kPa}, \mathrm{P}_s = 260\,\text{Pa}\)):
- Uncorrected model (blue): Severely biased and overconfident; fails to overlap the synthetic experimental observation.
- Corrected model (green): By combining parameter posterior uncertainty with the inferred model error term, the predictive distribution centers precisely on the true measurement across all three quantities of interest simultaneously.