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Description
1. Background of the study
The renewed interest in space exploration has increased the need for accurate and affordable models of atmospheric-entry flows. During planetary entry, hypersonic vehicles travel at velocities of the order of 10 km/s, generating a detached bow shock and severe thermal loads on the surface. Reliable prediction of this aerothermodynamic environment is therefore essential for thermal protection system design [1].
Across the shock layer, kinetic energy is converted into internal energy, and temperatures may reach values of the order of 10 000 K. In this regime, vibrational excitation, dissociation, ionization, and communication blackout may occur. Since fluid-dynamic, relaxation, and chemical time scales become comparable, local thermodynamic and chemical equilibrium assumptions are generally not valid [2].
A common approach for thermochemical non-equilibrium is the two-temperature model, where translational and rotational modes are described by the gas temperature, while vibrational and electronic modes share a separate temperature [3]. Although efficient, this model relies on prescribed Boltzmann distributions and empirical relaxation laws, which may be inaccurate when vibrational populations strongly depart from equilibrium.
A more detailed alternative is the vibrationally resolved state-to-state approach, where molecular vibrational levels are treated as independent pseudo-species [4]. This avoids imposing a predefined internal distribution, but the resulting number of species and kinetic processes makes the method expensive for multidimensional simulations. Reduced-order models based on StS data, such as the multi-internal-temperature (MiT) approach considered here [5, 6], aim to retain the main kinetic features while reducing the computational cost.
2. Methodology
The flow is modeled in the continuum regime by solving the Euler equations for a reactive neutral air mixture composed of N$_2$, O$_2$, NO, N, and O. The formulation includes conservation of species mass, momentum, and total energy, with additional variables depending on the thermochemical model.
Two descriptions are considered. The StS model explicitly resolves the vibrational levels of N$_2$ and O$_2$, treating each level as a pseudo-species evolving through vibration–translation, vibration–vibration, dissociation–recombination, and exchange processes. This provides a detailed reference solution but requires a large kinetic mechanism.
The reduced MiT model groups the vibrational levels of N$_2$ and O$_2$ into a limited number of subsets. Each group is described by a local Boltzmann distribution at its own internal temperature. Mass and vibrational-energy source terms are obtained by averaging the StS rate coefficients over the levels in each group, so that the reduced model remains consistent with the detailed kinetics.
Different numbers of groups and grouping strategies are tested to assess the trade-off between accuracy and cost. In particular, strategies with increased resolution in the high-energy part of the vibrational distribution are considered, since these levels strongly affect dissociation.
The governing equations are solved using a cell-centered finite-volume method on structured grids. Inviscid fluxes are computed with the upwind flux-vector-splitting scheme of Steger and Warming, while MUSCL reconstruction with a limiter provides second-order spatial accuracy near shocks. Time integration is performed through operator splitting: a frozen-flow step advances the fluid dynamics explicitly, whereas a chemical step updates relaxation and reaction source terms implicitly. Reduced kinetic rates are precomputed and stored in look-up tables.
3. Results
The model is first assessed on 0D reactors of pure oxygen, pure nitrogen, and five-species air under high-temperature non-equilibrium conditions. For O$_2$ and N$_2$, the agreement between MiT and StS improves as the number of groups increases. The single-group model cannot properly describe the underpopulation of high vibrational levels, whereas multi-group formulations progressively recover this effect. The five-group model provides temperature and mass-fraction evolutions close to the StS reference.
The air reactor confirms the same trend. Translational temperature is well reproduced by the MiT models, while some discrepancies remain in the species evolution, mainly related to oxygen dissociation. Higher-energy groups relax more slowly than low-energy groups, showing that a single vibrational temperature is not sufficient to capture the internal non-equilibrium distribution.
The model is then applied to a two-dimensional inviscid axisymmetric hypersonic flow past a sphere, representative of the experiment by Nonaka et al. [7]. The StS model accurately predicts the shock stand-off distance, while MiT solutions approach the StS result as the number of groups increases. The one-group case behaves similarly to a classical Park-type formulation.
The grouping strategy also affects the solution. Assigning more levels to the high-energy portion of the distribution improves the agreement with StS, especially for the shock stand-off distance, because it provides a more appropriate description of vibrational energy redistribution. From the computational point of view, the MiT approach yields CPU speed-ups of the order of 10$^3$ with respect to the full StS model, while GPU simulations also show significant acceleration (3.9 · 10$^1$ − 2.21 · 10$^2$).
4. Conclusions
A consistent multi-internal-temperature model has been implemented for hypersonic flows in thermochemical non-equilibrium. The method reduces the complexity of state-to-state kinetics by grouping vibrational levels into Boltzmann-like subsets, while deriving source terms from the detailed kinetic database.
The results show that the MiT formulation offers a good compromise between accuracy and efficiency. Increasing the number of groups improves agreement with the StS reference solution in both homogeneous reactors and multidimensional hypersonic-flow simulations. Multi-group models capture the delayed equilibration of high-energy vibrational levels and its impact on dissociation and shock-layer structure.
Overall, the MiT approach substantially reduces computational cost while preserving the main physical features of detailed non-equilibrium kinetics. It therefore represents a promising tool for high-fidelity simulations of atmospheric-entry flows, with future extensions toward viscous flows, complex geometries, and gas–surface interactions.
References
[1] Peter A Gnoffo. Planetary-entry gas dynamics. Annual Review of Fluid Mechanics, 31(1):459–494, 1999.
[2] John D Anderson Jr. Hypersonic and high-temperature gas dynamics. American Institute of Aeronautics and Astronautics, 2006.
[3] Chul Park. Nonequilibrium hypersonic aerothermodynamics. John Wiley & Sons, 1989.
[4] G Colonna, F Bonelli, and G Pascazio. Impact of fundamental molecular kinetics on macroscopic properties of high-enthalpy flows: The case of hypersonic atmospheric entry. Physical Review Fluids, 4(3):033404, 2019.
[5] Aurélien Guy, Anne Bourdon, and Marie-Yvonne Perrin. Consistent multi-internal-temperatures models for nonequilibrium nozzle flows. Chemical Physics, 420:15 – 24, 2013.
[6] Francesco Bonelli, Davide Ninni, Antonio Narracci, Gianpiero Colonna, and Giuseppe Pascazio. Assessment of a consistent multi-internal-temperature kinetic model for hypersonic neutral air flows using a finite volume solver. Computers Fluids, 301:106796, 2025.
[7] Satoshi Nonaka, Hiroyasu Mizuno, Kazuyoshi Takayama, and Chul Park. Measurement of shock standoff distance for sphere in ballistic range. Journal of thermophysics and heat transfer, 14(2):225–229, 2000.
Summary
This work presents a multi-internal-temperature kinetic model for the simulation of hypersonic neutral-air flows in thermochemical non-equilibrium. The model is derived from detailed state-to-state kinetics by grouping molecular vibrational levels into a limited number of Boltzmann-like subsets, each characterized by its own internal temperature. In this way, the main features of vibrational non-equilibrium are retained while significantly reducing the computational cost of the full state-to-state approach. The model is assessed through homogeneous reactor cases and a two-dimensional hypersonic flow past a sphere. Results show that increasing the number of internal groups improves the agreement with the state-to-state reference solution, especially in the prediction of vibrational relaxation, dissociation, and shock-layer structure. The proposed formulation provides a substantial computational speed-up and represents a promising tool for efficient simulations of atmospheric-entry flows.