Speaker
Description
1. Introduction
The radiative heat flux on a capsule during atmospheric re-entry represents a significant fraction of the total flux on the backshell, and depends on the population of the excited energy levels in the plasma around the capsule. This population does not follow a Boltzmann distribution due to non-equilibrium recombination in the afterbody region.
The Quasi-Steady State approximation has been used extensively to calculate this distribution of energy states, due to its lower computational cost. It consists in assuming that the production/depletion rates of excited species are far greater than their densities’ evolution over time, which can then be neglected. This means assuming that excited states adapt instantly to the changes in ground level densities, which are instead simulated.
In recent work, [Johnston and Panesi, 2018] showed that this hypothesis is not necessarily respected in all the flow field around the capsule. We test the validity of QSS for simulating the experiment ran by [Tibère-Inglesse et al., 2018] at the EM2C laboratory, on a nitrogen-argon plasma forced to recombine under nonequilibrium conditions at atmospheric pressure. We will use a State-to State collisional-radiative model for $N_2 / Ar$ developed as a vibronic-specific model by [Mariotto, 2023] and subsequently reduced to an electronic-specific model by [Dubuet, 2024].
The results show very good agreement between a full time-dependent simulation and the QSS approximation.
2. The Quasi-Steady State hypothesis
To use the QSS approximation, we must first simulate the 2D or 3D flow field, calculating its temperature and pressure, but only calculating the densities of the ground states for all chemical species with the appropriate time-dependent lagrangian master equation:
$\frac{dn_i}{dt} = S_{\text{kin}}(i) + S_{\text{rad}}(i) + {S}_{\text{diff}}(i)$
This accounts for the production / depletion of the specie through chemical reactions, radiative processes and diffusion. The densities of the specie $i$ depends then on the densities of all the other species included in the system.
We can then use the QSS hypothesis to calculate the densities of all excited states, which are necessary to estimate the radiative heat flux emitted towards the capsule. This turns all ordinary differential equations into algebraic equations, making the system significantly easier to solve. It only requires as inputs the temperature, pressure and densities of the ground states, which in our case of $N_2/Ar$ we took as:
$N ({}^4S), N^+({}^3P), N_2(X), N_2^+(X), Ar({}^1S), e^-$
3. Methodology and results
A full time-dependent simulation of the condition of Tibère-Inglesse’s experiment was run as “benchmark”, with both the electronic and vibronic-specific model, and concentrations were compared to experimental data.
20 points along the plasma torch were then chosen to run a QSS simulation, using the ground state densities from the time-dependent simulation. The resulting excited states densities were compared and expressed as a percentage of the “benchmark” result, for a select set of species of interest. Another analysis was made to verify which are the levels that deviate further percentually from the benchmark.
The results for the electronic-specific model show a very good agreement between QSS and time-dependent simulations, with interest species predicted with 98% accuracy and all species within 95%. Another simulation was conducted with a more relaxed set of constraints that did not include ions $N^+, N_2^+$ and this showed larger discrepancies, especially for ionized species, of up to 8%.
The vibronic-specific model was also tested by only imposing the density of the first vibronic state for molecular species: $N_2(X, v=0), N_2^+(X, v=0)$. The agreement for these results was worse, but still within an order of magnitude.
References
Dubuet, U., “Kinetics of a Nonequilibrium Recombining Air/Argon Plasma,” Theses. Université Paris-Saclay, 2024.
Mariotto, P., “Kinetics of Nonequilibrium Recombining Nitrogen-Argon Plasmas,” Paris-Saclay University, 2023. https://doi.org/10.5281/zenodo.18845333
Tibère-Inglesse, A., Mcguire, S., Mariotto, P., and Laux, C., “Validation Cases for Recombining Nitrogen and Air Plasmas,” Plasma Sources Science and Technology, Vol. 27, No. 11, 2018, p. 115010. https://doi.org/10.1088/1361-6595/aada61
Johnston, C. O., and Panesi, M., “Impact of State-Specific Flowfield Modeling on Atomic Nitrogen Radiation,” Physical Review Fluids, Vol. 3, No. 1, 2018, p. 013402. https://doi.org/10.1103/PhysRevFluids.3.013402
Dubuet, U., Mariotto, P., Laux, C. O., and Perrin, M.-Y., “Electronic-Specific Modeling of a Nonequilibrium Recombining N2/Ar Plasma and Comparison with Experiments,” Plasma Sources Science and Technology, Vol. 35, No. 4, 2026, p. 045007. https://doi.org/10.1088/1361-6595/ae50b5
Summary
Numerical assessment of the validity of the Quasi-Steady State hypothesis for a recombining nitrogen-argon plasma at atmospheric pressure. The results of QSS simulations were compared to time-dependent simulations of the same case, showing agreement within 5%.