Speaker
Description
Background
Recent years have seen a concerted effort to improve models of plasma formation in hypersonic flows. Validating these models has long been hampered by a lack of experimental data capable of characterizing ionization and its precursor processes -- electronic excitation in particular. Meeting this need requires shock tube experiments that probe the relevant species in suitable test gas mixtures with low detection limits, low measurement uncertainty, and high time resolution.
In previous works, the authors carried out experimental and modeling studies of electronic excitation and ionization in 1–2% N$_2$–Ar and 1% O$_2$–Ar mixtures. Measurements of atomic excited state number densities, together with electron number density in the O$_2$–Ar case, validated collisional-radiative models of these mixtures at a precision not previously achievable [1]. A key limitation, however, was the narrow range of mixtures these studies could access: typically 98–99\% argon, a dilution needed to sustain the high temperatures required for detectable populations of the target electronic levels.
Two recent advances lift this constraint. The development of a ring amplified shock tube in the Hanson Research Group at Stanford University has increased the sensitivity of infrared laser absorption diagnostics by up to a hundredfold, opening a new generation of experiments at lower temperatures, where the heavy-particle excitation processes of greatest interest dominate, and in less dilute, more diverse mixtures. Alongside this gain, a suite of new diagnostics has come online, covering electronically excited states of O, N, Ar, N$_2$, and NO, as well as free electrons and the ground electronic level of N$_2^+$.
In this work, we present a detailed modeling study of data from a recent Stanford shock tube campaign spanning pure argon, O$_2$–Ar, N$_2$–Ar, and N$_2$–O$_2$–Ar mixtures at varying concentrations, with mixtures chosen to isolate key rate coefficients of interest to hypersonic flow modeling. In some of these experiments, time histories of as many as twelve parameters are measured simultaneously, offering an unprecedented opportunity to constrain and improve models of electronic excitation and ionization in shock-heated air.
Methodology
Across the experimental campaign, number densities were measured for free electrons, N$_2^+$(X$^2\Sigma_g^+$), N$_2$(A$^3\Sigma_u^+$), NO(A$^2\Sigma^+$), the fourth through seventh electronic levels of atomic oxygen, the fourth, fifth, and eighth levels of atomic nitrogen, and the second and twelfth electronic levels of argon. Doppler broadening of several atomic lines yields the translational temperature, while Stark effects allow the free electron temperature and number density to be probed. The mixtures span pure argon, 1–30\% N$_2$ in Ar, 1–15\% O$_2$ in Ar, and several N$_2$–O$_2$ mixtures diluted in argon.
The experiments are simulated using the Toolkit for Excitation, Reactions, and Radiation with Applications (TERRA) at the University of Colorado Boulder, employing a recently developed three-temperature collisional-radiative (CR) model [2]. Each simulation includes an enthalpy source term accounting for the non-ideal pressure rise measured in the experiment, following prior findings on the importance of such a term [3].
For each experiment, a sensitivity analysis is performed with more than 1,000 uncertain parameters, each with literature-informed uncertainty intervals. Sobol' indices and mutual information are computed at each data point; the latter incorporates the measurement uncertainty to quantify how much each measurement can reduce the uncertainty in a given rate parameter. Together, these analyses identify which parameters are active in shaping the measured time histories across all experiments and diagnostics, and reveal whether each measurement falls within the model's range of feasible predictions. A measurement that does not fall within the uncertainty intervals of the model signals a clear need for model improvement, potentially through the inclusion of collisional processes not previously accounted for.
TERRA is also equipped with the ability to analytically compute the derivative of output variables with respect to input rate coefficients. This permits gradient-based optimization of rate parameters based on the experimental data. By performing multiple such optimizations of rate coefficients in the kinetic model, starting from different initial parameter values, it is possible to evaluate whether optimized mechanisms are unique.
Results
Comparisons with the measured time histories are given for the argon, O$_2$–Ar, N$_2$–Ar, and N$_2$–O$_2$–Ar data. In the N$_2$–O$_2$–Ar and O$_2$–Ar mixtures, the model overpredicts the measurements of oxygen, argon, and nitrogen excited state number densities by up to a factor of three across much of the dataset. Predictions of N$_2$(A) and NO(A), by contrast, agree well with the measurements, except at very early times, when the models overpredict the transient peak in these states.
Electron number density predictions in N$_2$–O$_2$–Ar likewise exceed the measurements, with charge exchange involving N$_2^+$ playing a major role in the overall ionization rate. The new N$_2^+$ measurements offer direct insight on this process, allowing models of charge exchange to be substantially improved and the electron number density data to be leveraged further for validating ionization reactions in hypersonic air flows.
Gradient-descent optimization of rate coefficients in the full N$_2$–O$_2$–Ar kinetic mechanism is attempted using 17 experiments spanning all four test gas combinations. Mixed agreement is achieved, with optimized model predictions agreeing with some diagnostics and disagreeing severely with others. Optimizations are then performed that focus on a reduced set of five experiments, covering just NO(A), N$_2$(A), and N$_2^+$ in N$_2$–Ar and N$_2$–O$_2$–Ar mixtures, and yield much better agreement with the data for all optimized parameter sets. However, across the ten optimizations performed, only a small number of sensitive parameters appear to be uniquely constrained, with the remaining parameters taking widely different optimized values across runs.
Making meaningful parameter inferences from the experiments requires a rate parametrization that carefully balances physical fidelity with identifiability from the available experimental data. To study this tradeoff in greater detail on a simpler system, experiments in pure argon are analyzed next. In this case, ten independent gradient-descent optimizations yield nearly identical agreement with the data, but the optimized values of the most sensitive parameters scatter across multiple orders of magnitude. Ongoing work is focusing on developing reduced parameterizations of the kinetic mechanism that maintain its essential physical fidelity while permitting unique inferences to be made from the available experimental data where possible.
Conclusion
Recent experimental data gathered at Stanford offer an unprecedented opportunity to advance the state of the art in kinetic models of excitation and ionization behind strong shock waves. Through detailed comparison, sensitivity analysis, and principled parameter inference, we aim to exploit this extensive dataset to reduce the uncertainty in existing kinetic models for ionized hypersonic flows.
An approach for analyzing large experimental datasets and determining parameter combinations they may constrain has been presented. Work will continue by focusing on the argon experiments, aiming to determine a parameter set whose values are uniquely determined by the experiments. Once the active parameter set is established, the final step is a Bayesian inference of the most sensitive rate parameters using the same surrogates built for the sensitivity analysis. Marginalization over nuisance variables will also be included to quantify the effect of experimental condition uncertainties on the rate coefficient posteriors. Analyses in pure argon will be followed by analyses in O$_2$–Ar, N$_2$–Ar, and eventually N$_2$–O$_2$–Ar, likely leveraging data from the literature or from previous test campaigns to constrain additional rates in the model, such as those for dissociation.
References
[1] Aiken, T.T. and Boyd, I.D., “State-resolved modeling of electronic excitation in weakly ionized oxygen mixtures,” Physical Review E, Vol. 109, No. 4, pp. 045203, 2024.
[2] Aiken, T.T., Boyd, I.D., and Adamovich, I.V., “Three-Temperature Collisional-Radiative Model of Ionization and Recombination in Hypersonic Air Plasmas,” Physics of Plasmas, Vol. 32, No. 10, pp. 103512, 2025.
[3] Aiken, T.T. and Boyd, I.D., “Collisional-radiative modeling of shock-heated nitrogen mixtures,” Journal of Applied Physics, Vol. 135, No. 9, pp. 093301, 2024.
Summary
A comprehensive study of electronic excitation and ionization in N2-O2-Ar using recent reflected shock experimental data from the Hanson Group at Stanford University.