Boltzmann rovibrational collisional coarse-grained model for internal energy excitation and dissociation in hypersonic flows

State-to-state study of non-equilibrium recombination of oxygen and nitrogen molecules

Rapidly cooled mixtures are of interest for several applications, including hypersonic flows due to the presence of strong cooling temperature gradients in regions such as hypersonic boundary layers and expanding nozzles. There have been very few studies of rapidly cooled mixtures using the high-fidelity rovibrational databases afforded by ab initio potential energy surfaces. This work makes use of existing rovibrational state-specific databases to study rapidly cooled mixtures. In particular, we seek to understand the importance of thermal non-equilibrium in recombining mixtures using both rovibrational and vibrational state-to-state methods for oxygen and nitrogen molecules. We find that although there is significant non-equilibrium during recombination, it is well captured by the vibrational state-specific approach. Finally, we compare the global recombination rate computed based on the state-specific recombination rate coefficients and the global recombination rate computed based on the time local dissociation rate coefficient, which is reversed using the principle of detailed balance. The local dissociation rate coefficient is computed by weighting the state-specific dissociation rate coefficients with the state-specific distribution of energy states. We find a large difference between these rates, highlighting a potential source of errors in hypersonic flow predictions.

Dissociation cross sections and rates in O2 + N collisions: molecular dynamics simulations combined with machine learning

The collision-induced dissociation reaction of O2 (v, j) + N, a fundamental process in nonequilibrium air flows around reentry vehicles, has been studied systematically by applying molecular dynamics simulations on the 2A', 4A' and 6A' potential energy surfaces of NO2 in a wide temperature range. In particular, we have directly investigated the role of the 6A' surface in this process and discussed the applicability of the simplified approximate rate models proposed by Esposito et al. and Andrienko et al. based on the lowest two surfaces. The present work indicates that the state-selected dissociation of O2 + N is dominated by the 6A' surface for all except for the low-lying O2 states. Furthermore, a complete database of rovibrationally detailed cross sections and rate coefficients is a prerequisite for modeling the relevant nonequilibrium air flows in spacecraft reentry. Here, the combination of the quasi-classical trajectory (QCT) and the neural network (NN) has been proposed to predict all state-selected dissociation cross sections and further construct dissociation parameter sets. All NN-based models established in this work accurately reproduce the results calculated from QCT simulations over a wide range of rovibrational quantum numbers with R2 > 0.99. Compared with the explicit QCT simulations, the computational requirement for predicting cross sections and rates based on the NN models significantly reduces. Finally, thermal equilibrium rate coefficients computed from NN models match remarkably well the available theoretical and experimental results in the whole temperature range explored.

Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows

This work proposes a new machine learning (ML)-based paradigm aiming to enhance the computational efficiency of non-equilibrium reacting flow simulations while ensuring compliance with the underlying physics. The framework combines dimensionality reduction and neural operators through a hierarchical and adaptive deep learning strategy to learn the solution of multi-scale coarse-grained governing equations for chemical kinetics. The proposed surrogate's architecture is structured as a tree, with leaf nodes representing separate neural operator blocks where physics is embedded in the form of multiple soft and hard constraints. The hierarchical attribute has two advantages: (i) It allows the simplification of the training phase via transfer learning, starting from the slowest temporal scales; (ii) It accelerates the prediction step by enabling adaptivity as the surrogate's evaluation is limited to the necessary leaf nodes based on the local degree of non-equilibrium of the gas. The model is applied to the study of chemical kinetics relevant for application to hypersonic flight, and it is tested here on pure oxygen gas mixtures. In 0-[Formula: see text] scenarios, the proposed ML framework can adaptively predict the dynamics of almost thirty species with a maximum relative error of 4.5% for a wide range of initial conditions. Furthermore, when employed in 1-[Formula: see text] shock simulations, the approach shows accuracy ranging from 1% to 4.5% and a speedup of one order of magnitude compared to conventional implicit schemes employed in an operator-splitting integration framework. Given the results presented in the paper, this work lays the foundation for constructing an efficient ML-based surrogate coupled with reactive Navier-Stokes solvers for accurately characterizing non-equilibrium phenomena in multi-dimensional computational fluid dynamics simulations.

Efficient quasi-classical trajectory calculations by means of neural operator architectures

An accurate description of non-equilibrium chemistry relies on rovibrational state-to-state (StS) kinetics data, which can be obtained through the quasi-classical trajectory (QCT) method for high-energy collisions. However, these calculations still represent one of the major computational bottlenecks in predictive simulations of non-equilibrium reacting gases. This work addresses this limitation by proposing SurQCT, a novel machine learning-based surrogate for efficiently and accurately predicting StS chemical reaction rate coefficients. The QCT emulator is constructed using three independent components: two deep operator networks (DeepONets) for inelastic and exchange processes and a feed-forward neural network (FNN) for the dissociation reactions. SurQCT is tested on the O₂ + O system, showing a computational speed-up of 85%. Furthermore, we carry out a StS master equation analysis of an isochoric, isothermal heat bath simulation at various temperatures to study how the predicted rate coefficients impact the accuracy of multiple quantities of interest (QoIs) at the kinetics level (e.g., global quasi-steady state (QSS) dissociation rate coefficients and energy relaxation times). For all these QoIs, the master equation analysis relying on SurQCT data shows an accuracy within 15% across the entire temperature regime.

The role of the sextet potential energy surface in O2 + N inelastic collision processes

We have performed molecular dynamics simulations of inelastic collisions between molecular oxygen and atomic nitrogen, employing the quasi-classical trajectory method on the new doublet, quartet, and sextet analytical potential energy surfaces of NO₂. A complete database of vibrationally detailed rate coefficients is constructed in a wide temperature range for high vibrational states up to ν = 25. In particular, the present work shows that the sextet potential energy surface plays a crucial role in the rovibrational relaxation process of O₂ + N collisions. The state-to-state rate coefficients increase by a factor of 2 to 6 when we consider the contribution of this sextet potential energy surface according to the corresponding weight factor, especially for vibrational energy transfer processes in single quantum jumps and/or high-temperature regimes. Furthermore, we also provide Arrhenius-type accurate fits for the vibrational state-specific rate coefficients of this collision system to achieve the flexible application of rate coefficients in numerical codes concerning air kinetics. Our results have implications for understanding the relaxation mechanism of the collision system with degenerate electronic states.

Assessment of Machine Learning Methods for State-to-State Approach in Nonequilibrium Flow Simulations

State-to-state numerical simulations of high-speed reacting flows are the most detailed but also often prohibitively computationally expensive. In this work, we explore the usage of machine learning algorithms to alleviate such a burden. Several tasks have been identified. Firstly, data-driven machine learning regression models were compared for the prediction of the relaxation source terms appearing in the right-hand side of the state-to-state Euler system of equations for a one-dimensional reacting flow of a N2/N binary mixture behind a plane shock wave. Results show that, by appropriately choosing the regressor and opportunely tuning its hyperparameters, it is possible to achieve accurate predictions compared to the full-scale state-to-state simulation in significantly shorter times. Secondly, several strategies to speed-up our in-house state-to-state solver were investigated by coupling it with the best-performing pre-trained machine learning algorithm. The embedding of machine learning algorithms into ordinary differential equations solvers may offer a speed-up of several orders of magnitude. Nevertheless, performances are found to be strongly dependent on the interfaced codes and the set of variables onto which the coupling is realized. Finally, the solution of the state-to-state Euler system of equations was inferred by means of a deep neural network by-passing the use of the solver while relying only on data. Promising results suggest that deep neural networks appear to be a viable technology also for this task.

Reactive, Inelastic, and Dissociation Processes in Collisions of Atomic Nitrogen with Molecular Oxygen

Collisions of atomic nitrogen with molecular oxygen have been treated with the quasiclassical trajectory method (QCT) in order to obtain a complete database of vibrationally detailed cross sections and rate coefficients for reactive, inelastic, and dissociation processes. For reaction rate coefficients, the agreement with experimental and theoretical data in the literature is excellent on the whole available interval 300-5000 K, with reliable extension to 20,000 K. For the inelastic case and for dissociation, no comparisons are available; therefore, a study of QCT reliability is proposed. In the inelastic case, it is found that "purely inelastic" and "quasireactive" collisions show not only different mechanisms but also different QCT levels of reliability at low energy. For dissociation, similar considerations bring to the conclusion that for the present collisional system, the QCT method is appropriate on the whole energy range studied. Rate coefficients for all the processes studied are provided in the electronic form.

Data-Inspired and Physics-Driven Model Reduction for Dissociation: Application to the O2 + O System

This work presents an in-depth discussion on the nonequilibrium dissociation of O₂ molecules colliding with O atoms, combining quasi-classical trajectory calculations, master equation, and dimensionality reduction. A rovibrationally resolved database for all of the elementary collisional processes is constructed by including all nine adiabatic electronic states of O₃ in the QCT calculations. A detailed analysis of the ab initio data set reveals that for a rovibrational level, the probability of dissociating is mostly dictated by its deficit in internal energy compared to the centrifugal barrier. Because of the assumption of rotational equilibrium, the conventional vibrational-specific calculations fail to characterize such a dependence. Based on this observation, a new physics-based grouping strategy for application to coarse-grained models is proposed. By relying on a hybrid technique made of rovibrationally resolved excitation coupled to coarse-grained dissociation, the new approach is compared to the vibrational-specific model and the direct solution of the rovibrational state-to-state master equation. Simulations are performed in a zero-dimensional isothermal and isochoric chemical reactor for a wide range of temperatures (1500-20,000 K). The study shows that the main contribution to the model inadequacy of vibrational-specific approaches originates from the incapability of characterizing dissociation, rather than the energy transfers. Even when constructed with only twenty groups, the new reduced-order model outperforms the vibrational-specific one in predicting all of the QoIs related to dissociation kinetics. At the highest temperature, the accuracy in the mole fraction is improved by 2000%.

State-to-State Master Equation and Direct Molecular Simulation Study of Energy Transfer and Dissociation for the N2-N System

We present a detailed comparison of two high-fidelity approaches for simulating non-equilibrium chemical processes in gases: the state-to-state master equation (StS-ME) and the direct molecular simulation (DMS) methods. The former is a deterministic method, which relies on the pre-computed kinetic database for the N₂-N system based on the NASA Ames ab initio potential energy surface (PES) to describe the evolution of the molecules' internal energy states through a system of master equations. The latter is a stochastic interpretation of molecular dynamics relying exclusively on the same ab initio PES. It directly tracks the microscopic gas state through a particle ensemble undergoing a sequence of collisions. We study a mixture of nitrogen molecules and atoms forced into strong thermochemical non-equilibrium by sudden exposure of rovibrationally cold gas to a high-temperature heat bath. We observe excellent agreement between the DMS and StS-ME predictions for the transfer rates of translational into rotational and vibrational energy, as well as of dissociation rates across a wide range of temperatures. Both methods agree down to the microscopic scale, where they predict the same non-Boltzmann population distributions during quasi-steady-state dissociation. Beyond establishing the equivalence of both methods, this cross-validation helped in reinterpreting the NASA Ames kinetic database and resolve discrepancies observed in prior studies. The close agreement found between the StS-ME and DMS methods, whose sole model inputs are the PESs, lends confidence to their use as benchmark tools for studying high-temperature air chemistry.

Coarse-grained modeling of thermochemical nonequilibrium using the multigroup maximum entropy quadratic formulation

This work addresses the construction of a reduced-order model based on a multigroup maximum entropy formulation for application to high-enthalpy nonequilibrium flows. The method seeks a piecewise quadratic representation of the internal energy-state populations by lumping internal energy levels into groups and by applying the maximum entropy principle in conjunction with the method of moments. The use of higher-order polynomials allows for an accurate representation of the logarithm of the distribution of the low-lying energy states, while preserving an accurate description of the linear portions of the logarithm of the distribution function that characterize the intermediate- and high-energy states. A comparison of the quadratic and the linear reconstructions clearly demonstrates how the higher-order reconstruction provides a more accurate representation of the internal population distribution function at a modest increase in the computational cost. Numerical simulations carried out under conditions relevant to hypersonic flight reveal that the proposed model is able to capture the dynamics of the nonequilibrium distribution function using as few as three groups, thereby reducing the computational costs for simulations of nonequilibrium flows.

Coupling of state-resolved rovibrational coarse-grain model for nitrogen to stochastic particle method for simulating internal energy excitation and dissociation

We propose to couple a state-resolved rovibrational coarse-grain model to a stochastic particle method for simulating internal energy excitation and dissociation of a molecular gas. A coarse-grained model for a rovibrational reaction mechanism of an ab initio database developed at the NASA Ames Research Center for the N₂-N system is modified based on variably spaced energy bins. The thermodynamic properties of the modified coarse-grained model allow us to closely match those obtained with the full set of rovibrational levels over a wide temperature range, while using a number of bins significantly smaller than the complete mechanism. The chemical-kinetic behavior of equally and variably spaced bin formulations is compared by simulating internal energy excitation and dissociation of nitrogen in an adiabatic, isochoric reactor. We find that the variably spaced formulation is better suited for reproducing the dynamics of the full database at conditions of interest in the Earth atmospheric entry. Also in this paper, we discuss the details of our particle method implementation for the uniform rovibrational collisional bin model and describe changes to the Direct Simulation Monte Carlo (DSMC) collision algorithm, which become necessary to accommodate our state-resolved reaction mechanism for excitation and dissociation reactions. The DSMC code is then verified against equivalent master equation calculations. In these simulations, state-resolved cross sections are used in analytical form. These cross sections verify micro-reversibility relations for the rovibrational bins and allow for fast execution of the DSMC code. In our verification calculations, we obtain very close agreement for the concentrations profiles of N and N₂, as well as the translational and rovibrational mode temperatures obtained independently through both methods. In addition to macroscopic moments, we compare discrete internal energy populations predicted at selected time steps via DSMC and the master equations. We observe good agreement between the two sets of results within the limits imposed by statistical scatter, which is inherent to particle-based DSMC solutions. As future work, the rovibrational coarse-grain model coupled to the particle method will allow us to study 3D reentry flow configurations.

Principal component analysis acceleration of rovibrational coarse-grain models for internal energy excitation and dissociation

The present work introduces a novel approach for obtaining reduced chemistry representations of large kinetic mechanisms in strong non-equilibrium conditions. The need for accurate reduced-order models arises from compression of large ab initio quantum chemistry databases for their use in fluid codes. The method presented in this paper builds on existing physics-based strategies and proposes a new approach based on the combination of a simple coarse grain model with Principal Component Analysis (PCA). The internal energy levels of the chemical species are regrouped in distinct energy groups with a uniform lumping technique. Following the philosophy of machine learning, PCA is applied on the training data provided by the coarse grain model to find an optimally reduced representation of the full kinetic mechanism. Compared to recently published complex lumping strategies, no expert judgment is required before the application of PCA. In this work, we will demonstrate the benefits of the combined approach, stressing its simplicity, reliability, and accuracy. The technique is demonstrated by reducing the complex quantum N₂(Σg+1)-N(Su4) database for studying molecular dissociation and excitation in strong non-equilibrium. Starting from detailed kinetics, an accurate reduced model is developed and used to study non-equilibrium properties of the N₂(Σg+1)-N(Su4) system in shock relaxation simulations.

Construction of a coarse-grain quasi-classical trajectory method. I. Theory and application to N2-N2 system

This work aims to construct a reduced order model for energy transfer and dissociation in non-equilibrium nitrogen mixtures. The objective is twofold: to present the Coarse-Grain Quasi-Classical Trajectory (CG-QCT) method, a novel framework for constructing a reduced order model for diatom-diatom systems; and to analyze the physics of non-equilibrium relaxation of the nitrogen molecules undergoing dissociation in an ideal chemical reactor. The CG-QCT method couples the construction of the reduced order model under the coarse-grain model framework with the quasi-classical trajectory calculations to directly construct the reduced model without the need for computing the individual rovibrational specific kinetic data. In the coarse-grain model, the energy states are lumped together into groups containing states with similar properties, and the distribution of states within each of these groups is prescribed by a Boltzmann distribution at the local translational temperature. The required grouped kinetic properties are obtained directly by the QCT calculations. Two grouping strategies are considered: energy-based grouping, in which states of similar internal energy are lumped together, and vibrational grouping, in which states with the same vibrational quantum number are grouped together. A zero-dimensional chemical reactor simulation, in which the molecules are instantaneously heated, forcing the system into strong non-equilibrium, is used to study the differences between the two grouping strategies. The comparison of the numerical results against available experimental data demonstrates that the energy-based grouping is more suitable to capture dissociation, while the energy transfer process is better described with a vibrational grouping scheme. The dissociation process is found to be strongly dependent on the behavior of the high energy states, which contribute up to 50% of the dissociating molecules. Furthermore, up to 40% of the energy required to dissociate the molecules comes from the rotational mode, underscoring the importance of accounting for this mode when constructing non-equilibrium kinetic models. In contrast, the relaxation process is governed primarily by low energy states, which exhibit significantly slower transitions in the vibrational binning model due to the prevalence of mode separation in these states.

Adaptive coarse graining method for energy transfer and dissociation kinetics of polyatomic species

A novel reduced-order method is presented for modeling reacting flows characterized by strong non-equilibrium of the internal energy level distribution of chemical species in the gas. The approach seeks for a reduced-order representation of the distribution function by grouping individual energy states into macroscopic bins, and then reconstructing state population using the maximum entropy principle. This work introduces an adaptive grouping methodology to identify and lump together groups of states that are likely to equilibrate faster with respect to each other. To this aim, two algorithms have been considered: the modified island algorithm and the spectral clustering method. Both methods require a measure of dissimilarity between internal energy states. This is achieved by defining "metrics" based on the strength of the elementary rate coefficients included in the state-specific kinetic mechanism. Penalty terms are used to avoid grouping together states characterized by distinctively different energies. The two methods are used to investigate excitation and dissociation of N₂ (Σg+1) molecules due to interaction with N(Su4) atoms in an ideal chemical reactor. The results are compared with a direct numerical simulation of the state-specific kinetics obtained by solving the master equations for the complete set of energy levels. It is found that adaptive grouping techniques outperform the more conventional uniform energy grouping algorithm by providing a more accurate description of the distribution function, mole fraction and energy profiles during non-equilibrium relaxation.

Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method

A rovibrational collisional model is developed to study the internal energy excitation and dissociation processes behind a strong shock wave in a nitrogen flow. The reaction rate coefficients are obtained from the ab initio database of the NASA Ames Research Center. The master equation is coupled with a one-dimensional flow solver to study the nonequilibrium phenomena encountered in the gas during a hyperbolic reentry into Earth's atmosphere. The analysis of the populations of the rovibrational levels demonstrates how rotational and vibrational relaxation proceed at the same rate. This contrasts with the common misconception that translational and rotational relaxation occur concurrently. A significant part of the relaxation process occurs in non-quasi-steady-state conditions. Exchange processes are found to have a significant impact on the relaxation of the gas, while predissociation has a negligible effect. The results obtained by means of the full rovibrational collisional model are used to assess the validity of reduced order models (vibrational collisional and multitemperature) which are based on the same kinetic database. It is found that thermalization and dissociation are drastically overestimated by the reduced order models. The reasons of the failure differ in the two cases. In the vibrational collisional model the overestimation of the dissociation is a consequence of the assumption of equilibrium between the rotational energy and the translational energy. The multitemperature model fails to predict the correct thermochemical relaxation due to the failure of the quasi-steady-state assumption, used to derive the phenomenological rate coefficient for dissociation.