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Description
Background of the study
The dense and stable atmosphere characterizing Titan [1], Saturn's largest moon, make it a celestial body of great interest for the scientific community, as it may give insights into the formation of life [2], also due to the presence of organic compounds, such as methane.
With the scheduled launch of the NASA mission Dragonfly [3], aiming to deliver a nuclear-powered octocopter on Titan's surface, it is now essential to properly analyze the critical phase of the atmospheric entry, estimated at approximately 6 km/s.
The criticality of crossing the atmosphere at such high velocities lies in the collision with its molecules that generates a strong shock wave on the capsule's forebody, influencing the body's aerodynamic behavior and heating. For this reason, specifically designed computational fluid dynamics (CFD) solvers are adopted to predict flow behavior and involved phenomena such as chemical reactions, ionization, radiation and wall catalysis and ablation [4].
Polytechnic of Bari, in collaboration with the Institute for Plasma Science and Technology (ISTP) of CNR in Bari, developed a finite-volume multi-GPU code, Damiso [5], to run hypersonic high enthalpy gas dynamic simulation. The code exploits the GPU inherent parallelizing capabilities to solve the Navier-Stokes equations and contemporarily take into account non-equilibrium phenomena by means of thermochemical models such as the Multi-Temperature or the State-to-State [6].
The difference between the two models lies in the description of the internal energy. In the Multi-Temperature model species internal energies are defined by means of statistical distributions driven by characteristic temperatures and are accounted for introducing additional energy transport equations [7]. The State-to-State, on the other hand, solves the internal states of the most relevant particles by treating them as pseudo-species and formulating a continuity equation for each of them [8]. The more accurate description obtained entails a higher computational cost, but becomes essential under marked non-equilibrium conditions.
Methodology
The code adopts a finite-volume approach. The solution is obtained by splitting time integration into two different steps: a first one solving the fluid dynamics and a second step, to account for the thermochemical non-equilibrium.
In the first step, the system of Navier Stokes equations is solved under the perfect gas assumption by means of a third-order accurate Runge-Kutta scheme [9]. Inviscid fluxes are discretized by means of a Steger and Warming flux vector splitting approach [10] and reconstructed with a second order MUSCL scheme [11], whereas viscous fluxes are solved with a second order cell-centered scheme.
The chemical step adopts the implicit Gauss-Seidel method [12] to account for non-equilibrium by evaluating mass loss or production of each species by means of the Arrhenius relation for the Multi-Temperature model or more complex analytical forms for the State-to-State model.
As regards wall catalysis, it is evaluated by means of the $\gamma$ model [13] by defining a recombination efficiency $\gamma_c$ which represents the ratio between recombining particles and particles impinging on the wall, in terms of mass fractions.
To demonstrate the reliability of the tool, the verification is performed reproducing a literature test case of a capsule atmospheric entry simulation by NASA [14].
Two different mixtures are implemented: a neutral 12-species mixture ($CH_4$, $CH_3$, $CH_2$, $N_2$, $C_2$, $H_2$, $CH$, $NH$, $CN$, $N$, $C$ and $H$) and an 18-species one (which also includes cherged particles as $N_2^+$, $CN^+$, $C^+$, $N^+$, $H^+$ and $e^-$) to evaluate ionization effects.
Two-dimensional simulations are performed adopting the flow conditions corresponding to an altitude of 269 km, which is related to the peak convective heating condition along the reference entry trajectory [14].
The freestream composition is made of molecular nitrogen $N_2$ for the 95% and methane $CH_4$ for the remaining 5% by volume.
Two different Multi-Temperature models are implemented, namely Park [15] and Nelson [16], and are compared with a developed simplified State-to-State model.
Given the high computational cost of a full StS model and the major presence of $N_2$ and $CN$ downstream of the bow shock, the adopted StS formulation only adopts a detailed description of internal vibrational
energy levels for those two molecules, whereas all the other molecular species are described using a single vibrational level (and atomic species have no vibrational mode at all).
Results
Code verification involves primarily the analysis of temperature along the stagnation line and wall heat flux.
The first step involves a comparison of reference data with results obtained using Park's model with the 18-species ionized mixture.
Given the good agreement of the results, the model is considered verified and used as benchmark to assess ionization effects comparing 18-species simulation with a 12-species neutral mixture simulation.
Once the effects of ionization on the flow have been characterized, the subsequent analyses use the 12-species mixture due to the significantly higher computational cost of the ionized mixture.
In this framework, Park's model is first compared with Nelson's model to assess the differences between the two formulations and then adopted to determine the effects of catalysis on the flow behavior by comparing simulations with a fully catalytic wall ($\gamma$ = 1) and with a non-catalytic wall ($\gamma$ = 0)
Finally, Park's results are compared with the outcome of a StS simulation, resulting in a moderate difference between the two models.
Conclusion
Hypersonic atmospheric entry remains a major challenge in aerospace aerodynamics, requiring continuous research and development efforts.
The developed code is compared to a literature case by NASA to verify the implemented Multi-Temperature models in the case of an atmospheric entry at Titan. The solver is also employed to evaluate ionization effects in the flow and the impact of wall catalysis on the shock wave and on the heat flux on the capsule wall. Finally, a State-to-State formulation is adopted and its results compared to those of the Multi-Temperature models
Results demonstrate a good reliability of the model in capturing the main features of the flow, whereas future work should focus on a further improvement of the thermochemical model, enhancing the implemented simplified State-to-State model.
References
[1] Nixon, C. A., “The Composition and Chemistry of Titan’s Atmosphere,” ACS Earth and Space Chemistry, Vol. 8, No. 3, 2024, p. 406–456. https://doi.org/10.1021/acsearthspacechem.2c00041.
[2] Niemann, H. B., Atreya, S. K., Demick, J. E., Gautier, D., Haberman, J. A., Harpold, D. N., Kasprzak, W. T., Lunine, J. I., Owen, T. C., and Raulin, F., “Composition of Titan’s lower atmosphere and simple surface volatiles as measured by the Cassini-Huygens probe gas chromatograph mass spectrometer experiment,” Journal of Geophysical Research: Planets, Vol. 115, No. E12, 2010. https://doi.org/10.1029/2010JE003659.
[3] Jet Propulsion Laboratory, “Solar System Exploration: This Is the 2006 Solar System Exploration Roadmap for NASA’s Science Mission Directorate,” Technical Report JPL D-35618, National Aeronautics and Space Administration, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, Sep. 2006.
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[6] Bonelli, F., Ninni, D., Narracci, A., Colonna, G., and Pascazio, G., “Assessment of a consistent multi-internal-temperature kinetic model for hypersonic neutral air flows using a finite volume solver,” Computers & Fluids, Vol. 301, 2025, p. 106796. https://doi.org/10.1016/j.compfluid.2025.106796.
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[10] Steger, J. L., and Warming, R., “Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods,” Journal of Computational Physics, Vol. 40, No. 2, 1981, pp. 263–293. https://doi.org/10.1016/0021-9991(81)90210-2.
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[12] Verwer, J. G., “Gauss-Seidel Iteration for Stiff ODEs from Chemical Kinetics,” SIAM Journal on Scientific Computing, Vol. 15, No. 5, 1994, pp. 1243–1250. https://doi.org/10.1137/0915076.
[13] Goulard, R., “On catalytic recombination rates in hypersonic stagnation heat transfer,” Journal of Jet Propulsion, Vol. 28, No. 11, 1958, pp. 737–745. https://doi.org/10.2514/8.7444.
[14] Takashima, N., Hollis, B., Olejniczak, J., Wright, M., and Sutton, K., “Preliminary Aerothermodynamics of Titan Aerocapture Aeroshell,” 36th AIAA Thermophysics Conference, 2003, p. 4952. https://doi.org/10.2514/6.2003-4952.
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Summary
Atmospheric entry remains a critical phase of planet exploration: the high temperatures involved during the descent may represent a risk for the safety of the capsule and the bow shock developed in front of the body involves non-equilibrium phenomena which can be non-trivial to predict.
To account for such effects, computational fluid dynamics (CFD) simulations adopt thermochemical models, as the implemented Multi-Temperature (mT) and State-to-State (StS) models.
The present work presents the verification of those models in the case of an atmospheric entry on Titan, Saturn's largest moon. Two different mT models (Park and Nelson) are implemented to reproduce a NASA test case and to estimate the presence of ionization effects and the influence of a catalytic wall on the flow. Finally, Park model is compared with the in-house developed StS model to assess accuracy.