Speaker
Description
Key words
State-to-state modeling, hypersonic flow, aerothermodynamics, CFD.
Introduction
During the atmospheric entry of a spacecraft at hypervelocity, the gas in the shock layer goes through high temperatures and through various thermochemical non-equilibrium states. For decades, the scientific community has worked to better understand the behavior of excited species in non-equilibrium plasmas. In the widely used global chemical mechanism approaches -- among them the well-known models developed by Park -- the modeling consists in considering the chemical species as a whole without distinguishing their excited states. The population number density of their excited states is then estimated either by assuming a Boltzmann distribution at an excitation temperature or by assuming the quasi-steady state.
For the past recent years, another methodology has been developed based on a complete description of the flow field with considering every (or several) excited states of each particle as independent species. This approach is very attractive because of its expected accuracy in giving a detailed description of a plasma, but it is very challenging to implement in CFD codes.
In this framework, the present study is a contribution to the progressive work for computing a reactive and non-equilibrium gas flow through the shock layer surrounding an axi-symmetric blunt body entering a Nitrogen atmosphere, with detailed collisional-radiative models. The selected state-to-state model recently implemented in the in-house code PINENS (Parallel Implicit Non-Equilibrium Navier-Stokes) is a state-specific vibrational model, that is a simplified version of CoRaM-N$_2$, a collisional-radiative (CR) model developed for nitrogen [AnnaloroBultelOmaly-2014-JTHT]. This model takes into account the species N$_2$ and N, and is vibrationnally specific for the N$_2$ molecules on their ground electronic state. It allows to simulate the phase of ladder-climbing vibrational excitation and the resulting dissociation of N$_2$.
Two trajectory point for the entry of a simple axi-symmetric body -- a 1-m radius sphere -- are considered. The first point correspond to an altitude of 67 km, where the density of the atmosphere is $1.47 \cdot 10^{-4}$ kg /m$^3$, and the velocity of the sphere is 11.25 km/s. The second point correspond to a lower altitude of 37 km, where the density is $6.0 \cdot 10^{-3}$ kg /m$^3$, and the velocity of the sphere is 6.19 km/s.
Vibrational-state specific model and mathematical formulation
The vibrationally specific model used for the analysis of the non-equilibrium gas in the shock layer takes into account heavy, neutral particles. It considers each of the 68 vibrational states of N$_2$ ($0 \le v \le v_{max} = 67$) on the ground electronic state as well as the ground electronic state of N. According to the CoRaM-N$_2$ model that is used as the basis to derive the present vibrational-state specific model, the vibrational excitation of molecules N$_2$ takes place through vibration--translation processes under molecular impact (VT-m processes), through vibration--vibration processes under molecular impact (VV-m processes), or through vibration--translation processes under atomic impact (VT-a processes). It results 6817 forward collisional elementary processes. The data used for the rate coefficients for these processes were calculated by Armenise et al. and Esposito et al. The backward collisional elementary processes are calculated from the forward rate coefficient and the corresponding equilibrium constant using the detailed balance principle.
The equations to be solved are the usual Navier-Stokes equations, except that a mass conservation equation is written for each vibrational level of N$_2$(v) instead of one conservation equation of mass for N$_2$. As for the equation of vibrational energy of N$_2$ usually used in the global models (Landau-Teller, Millikan-White), it is no longer needed in the detailed model.
Each species $s$ on its excited state $v$ -- named "pseudo-species'' and noted $sv$ -- has a mass density $\rho_{sv}$ and the total mass density of the gas mixture is $\rho = \sum_{sv=1}^{n_{sv}} \rho_{sv}$. For an axi-symmetric flow, one then has to solve 72 equations: 69 conservation equations for the ``species-state'' N and N$_2$(v), two equations for the conservation of the momentum, and one equation for the conservation of the total energy. The set of conservative equations is closed with an adequate equation of state.
Results
For an atmosphere initially composed of 100% of N$_2$ on v=0 vibrational level and, according to the considered vibrational state-to-state model, the molecules N$_2$ on their zero-vibrational level collide, get excited, and -- depending on the type of collisions -- get dissociated across the shock wave. These collisional processes lead to a gas mixture composed of the two species N$_2$ and N, N$_2$ being on various vibrational levels depending on the excitation processes considered. The vibrational-state specific model -- split into several groups of reactions -- is implemented in PINENS code step-by-step in order to show the specific effect of each process group on the population densities of the 68 vibrational levels of N$_2$ and of the atomic species N, throughout the shock layer.
The first group of processes to be considered in the numerical simulations is the 67 (VT-m) processes; then the 4489 (VV-m) exchanges are added, then the single (VT-m-D) process, then the 67 (VV-m-D) processes, then the 2125 (VT-a) processes, and finally the 68 (VT-a-D) processes are added. For each group of processes added, the effect on the N$_2$(v) population densities is noticeable : the population of N$_2$(v=0) decreases to the profit of populating the other vibrational levels of N$_2$, and to the profit of producing N atoms when dissociation processes are considered. However, it has to be noted that the (VV-m) and (VV-m-D) processes generate no visible effect on the population densities except in the boundary layer area. Regarding the global behavior of the shock layer, the introduction of each group leads to reduce the temperature within the shock layer and to reduce the shock layer width. These results are observed for both entry velocities (V$_\infty$ = 11.25 and 6.19 km/s), but noticeable differences is observed on the shock standoff distance and on the flow temperature between the 2 cases.
The Boltzmann diagrams draw the densities of N$_2$(v) versus the energy of the vibrational levels expressed in eV. The Boltzmann distribution is reached when the curves representing the level populations (or mass fractions) -- expressed in logarithmic scale -- versus the energies of the levels -- expressed in linear scale -- are linear. The Boltzmann diagrams then obtained at various locations along the stagnation line across the shock layer, show that the curves are linear only outside the shock wave and outside the boundary layer, meaning that the Boltzmann distribution is reached only outside these zones. This is well known, but it is clearly shown with using the vibrational-state specific model. In some cases, however, when not all the processes are taken into account, the Boltzmann distribution is reached nowhere in the shock layer.
Preliminary conclusion
A vibrational-state specific model issued from the collisional radiative model CoRaM-N$_2$ implemented in the PINENS code is used for simulating the non-equilibrium flow past a sphere entering a Nitrogen atmosphere at 11.25 and 6.19 km/s. The main conclusion is that the vibration-vibration exchanges through molecular impact, leading or not to dissociation, have very little effect on the population densities. On the other hand, clear evidence is shown of the progressive effect of the vibration-translation exchanges through molecular and atomic impacts. The study also showed that the Boltzmann distribution is obtained only in specific positions in the shock layer. Obtaining such results in reasonable computational time showed that it is possible to predict the vibrational level population distributions throughout an axi-symmetric shock layer with a detailed kinetics model.
Acknowledgements
Centre de Calcul Intensif d’Aix-Marseille is acknowledged for granting access to its high performance computing resources.
Reference
[AnnaloroBultelOmaly-2014-JTHT]
Annaloro J., Bultel A., Omaly P. (2014)
``Collisional-Radiative Modeling Behind Shock Waves in Nitrogen'',
J. of Thermophysics and Heat Transfer, Vol.~28, No.~4, pp.~608-622.
doi: 10.2514/1.T4263.
Summary
This abstract presents the analysis of a vibrationnally-excited nitrogen gas flow through the shock layer past a body flying at hypervelocity, modeled with a vibrational-state specific model implemented into a Navier-Stokes code.