Quasi-one-dimensional nonequilibrium nozzle flows of reacting air mixture N2/O2/NO/N/O are studied based on the state-to-state description for vibrational and chemical kinetics. Zeldovich exchange reactions of NO formation, dissociation, recombination, and various vibrational energy transitions are considered. The equations for the vibrational-level populations of N2 and O2 molecules are combined with the conservation equations of momentum and total energy and solved numerically for different conditions in the nozzle throat. The vibrational spectra of molecules are simulated based on the anharmonic Morse oscillator. Three nozzle profiles are considered. The variation in the vibrational distributions of nitrogen and oxygen molecules, number densities of species, and the gas temperature along the nozzle axis are studied. The formation of nonequilibrium non-Boltzmann distributions of N2 and O2 molecules with a plateau region at intermediate vibrational levels is shown for different conditions in the throat. The level populations of N2 and O2 molecules obtained in the most accurate state-to-state approximation are compared with those found by using the one-temperature thermal equilibrium approach. The underestimated vibrational-level populations are obtained in terms of the one-temperature simplified kinetic model for the considered conditions. Three kinetic models are used for the Zeldovich exchange reactions of NO formation. The influence of the exchange-reaction models, throat conditions, and the nozzle profile on the gas temperature and vibrational distributions are also studied in the paper.

The review is devoted to the foundation and development of the scientific school of S. V. Vallander at the Leningrad (now St Petersburg) State University. The achievements of the scientific school in the development of the kinetic theory methods for modeling nonequilibrium flows, construction of rigorous self-consistent mathematical models of varying complexity for strong and weak deviations from equilibrium, and the use of the developed models in challenging problems of modern aerodynamics are discussed. Particular attention is paid to the study of non-equilibrium kinetics and transport processes in carbon dioxide, identification of key relaxation mechanisms of polyatomic molecules, derivation of physically based reduced hybrid models, and optimization of non-equilibrium flow numerical simulations using modern machine learning methods. Correct description of electronic excitation in the kinetics and transport processes, models of local equilibrium gas flows with multiple ionization, and features of modeling bulk viscosity in polyatomic gases are discussed.

The review describes the creation and development of the scientific school of Sergei Vasilyevich Vallander at the Leningrad (now St. Petersburg) State University. We discuss the achievements of the scientific school in the development of methods of the kinetic theory of gases for the simulation of nonequilibrium flows, the construction of rigorous self-consistent mathematical models of varying complexity for strong and weak deviations from equilibrium, and the application of the developed models in solving modern problems of aerodynamics. Particular attention is paid to the study of nonequilibrium kinetics and transport processes in carbon dioxide, identifying the key relaxation mechanisms of polyatomic molecules, the development of physically reasonable reduced hybrid models, and the optimization of numerical simulation of flows using modern machine-learning methods. We discuss the problems of correctly accounting for electronic excitation in modeling the kinetics and transport processes, models of equilibrium gas flows with multiple ionization, and the peculiarities of simulating bulk viscosity in polyatomic gases.

In the present study, a new hybrid approach is proposed to modeling coupled vibrational and chemical kinetics in carbon dioxide (CO2) and products of its decomposition. The study develops and completes our previous work carried out for a single-component CO2 gas. The model is based on self-consistent implementation of state-to-state chemical and energy production rates into the equations of multi-temperature CO2 kinetics. It distinguishes vibrational temperatures of all CO2 modes and diatomic species and thus takes into account multiple relaxation mechanisms including intra-mode, inter-mode, and inter-molecular energy transitions as well as state-specific dissociation and exchange reactions. Other advantages of the proposed full multi-temperature approach are the possibility of capturing strong non-equilibrium effects in a flow, straightforward implementation of the chemical-vibrational coupling terms, easy update for new models of state-specific reaction rates. Comparisons with the results obtained in the frame of a detailed but numerically demanding state-to-state approach for the problem of spatially homogeneous relaxation showed good accuracy of the new model under the wide range of initial conditions; at the same time, traditional multi-temperature approaches failed to provide accurate predictions of non-equilibrium flow parameters under arbitrary deviations from equilibrium. Effects of chemical reaction models and selective mode excitation are assessed. The numerical efficiency of the developed model is found acceptable compared to that of the state-to-state approach.

A four-temperature kinetic-theory approach for modeling vibrationally non-equilibrium carbon dioxide flows is developed. The model takes into account all kinds of vibrational–translational energy transitions and inter-mode vibrational energy exchange between symmetric, bending, and asymmetric CO2 modes. The key feature of the model is using the averaged state-resolved relaxation rates instead of conventional Landau–Teller expressions. Spatially homogeneous CO2 vibrational relaxation is studied using the state-to-state, new four-temperature and commonly used three-temperature models. Excellent agreement between four-temperature and state-to-state solutions is found, whereas using the three-temperature model with the Landau–Teller production rates leads to significant loss of accuracy. Numerical efficiency of various approaches is discussed as well as the ways for its improvement.

In the present paper non-equilibrium flows of CO 2 /CO/O 2 /O/C mixture behind shock waves is studied taking into account vibrational excitation of CO 2 , CO, O 2 molecules. The kinetic scheme includes vibrational energy transitions, dissociation, recombination and exchange chemical reactions. The problem is solved in the five-temperature kinetic theory approximation. The equations for vibrational temperatures of symmetric–bending and asymmetric CO 2 modes and for diatomic species are coupled to the gas dynamic equations and applied to study flow parameters in the relaxation zone behind shock waves. Different models for one-temperature and multi-temperature rates of chemical reactions are used in calculations and the influence of exchange reactions and reaction models on flow parameters in the relaxation zone behind the shock wave is studied in the paper.

In this paper new multi-temperature models for rate coefficients of non-equilibrium chemical reactions in mixtures containing CO2 molecules are derived on the basis of the kinetic theory. The models are obtained by averaging of state-dependent reaction rate coefficients, found previously, over multi-temperature vibrational distributions. Five-temperature, three-temperature and two-temperature non-equilibrium distributions are considered and comparison of reaction rate coefficients derived using these distributions as well as the thermal equilibrium one-temperature model is presented. The proposed rate coefficients are used in the governing equations for vibrational and chemical relaxation in the five-component mixture CO2/CO/O2/C/O. The solution of these equations is obtained in the five-temperature, three-temperature, two-temperature and one-temperature approaches. Finally, the influence of chemical reaction models on macroscopic mixture parameters is discussed in the paper.

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a computationally complex problem requiring the solution of a huge system of stiff differential equations. In the present study, we use parallel numerical scheme, based on the extended backward differential formula with adaptive timestep strategy. Using this scheme, we study the contribution of various types of energy exchanges to the solution of kinetics equations. The comparison of the results obtained with the use of the ”cut” and full state-to-state approaches is also shown in the paper.

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a challenging time-consuming computational task that involves solving a huge system of stiff differential equations and requires optimized methods to solve it. In the present study, we propose and analyse optimizations for the Extended Backward Differential Formula (EBDF) scheme. Using adaptive timesteps instead of fixed ones reduces the number of steps in the algorithm many thousands of times, although with an increase in step complexity. The use of parallel computations to calculate relaxation terms allows one to further reduce the computation time. Numerical experiments on the modeling of spatially homogeneous carbon dioxide vibrational relaxation were performed for optimized computational schemes of different orders. Based on them, the most optimal algorithm of calculations was recommended: a parallel EBDF scheme of fourth-order with an adaptive timestep. This method takes less computational time and memory costs and has the high stability.

Numerical modeling of nonequilibrium state-to-state carbon dioxide kinetics is a challenging time-consuming computational task that involves solving a huge system of stiff differential equations and requires optimized methods to solve it. In the present study, we propose and analyse optimizations for the Extended Backward Differential Formula (EBDF) scheme. Using adaptive timesteps instead of fixed ones reduces the number of steps in the algorithm many thousands of times, although with an increase in step complexity. The use of parallel computations to calculate relaxation terms allows one to further reduce the computation time. Numerical experiments on the modeling of spatially homogeneous carbon dioxide vibrational relaxation were performed for optimized computational schemes of different orders. Based on them, the most optimal algorithm of calculations was recommended: a parallel EBDF scheme of fourth-order with an adaptive timestep. This method takes less computational time and memory costs and has the high stability.

Carbon dioxide is a key species for many fundamental and applied problems. Using state-resolved models gives a deep insight into the physics of vibrationally excited states, but at the same time, it is computationally very expensive. A study examines the routes to simplify the simulation of CO${}_{2}$ vibrational relaxation by revelation of main vibrational relaxation channels and by using the reduced-order (multitemperature) models

Shock wave structure in carbon dioxide is studied on the basis of several continuum models and compared to the solution obtained using the kinetic approach. The problem is solved in the frame of one- and two-temperature Euler equations as well as Navier-Stokes equations accounting for the bulk viscosity. The Rankine-Hugoniot relations with constant specific heat ratio fail to predict accurately the final equilibrium state in polyatomic gases. A good qualitative agreement of the solutions obtained using the continuum and kinetic approaches is shown. Taking into account the bulk viscosity leads to a considerable increase in the shock wave width; its variation in a flow modifies the profiles of gas-dynamic parameters. In the multi-temperature approach, solving the Euler equations coupled to the relaxation equation for the vibrational energy provides the results similar to those obtained within the kinetic approach taking into account the effect of bulk viscosity.

In this paper considers algorithms for calculating vibrational energy exchange rate coefficients for collisions of carbon dioxide molecules are considered. For numerical modeling of CO 2 vibrational kinetics in the state-to-state approach, it is necessary to solve a system of several thousands of differential equations for level populations of three vibrational CO 2 modes at an each step of calculations. The right hand parts of these kinetic equations contain energy exchange rate coefficients for collisions of molecules from different levels of three vibrational CO 2 modes. There are hundreds of thousands of these coefficients due to the large number of energy exchanges. From the numerical point of view, this modeling qualifies as Big Data and requires the development of rapid numerical methods or pre-calculations. Such an amount of data also requires a quick access data structure. Until now for the state-to-state description of CO 2 vibrational relaxation has used only simplified kinetic schemes with a limited numbers of vibrational levels and energy transitions have been used. In the present paper the problem is solved in the complete formulation. An effective scheme for calculating the coefficients is proposed on the basis of parallel computations and convolute code optimization, together with an optimal data structure for their storage.

The flows of CO2/CO/O-2/O/C mixture in the relaxation zone behind shock waves are studied on the basis of the kinetic theory multi-temperature approaches taking into account vibrational and chemical non-equilibrium. The vibrational temperatures are introduced for the coupled (symmetric-bending) and asymmetric modes of CO2 as well as for diatomic species. Governing equations of the flow are written in the five-temperature approximation and solved numerically for different conditions in the free stream. Vibrational energy transitions, dissociation, recombination and exchange reactions are included to the kinetic scheme. The obtained results are compared with those found with the use of simplified three-temperature, two-temperature and one-temperature kinetic models. The influence of vibrational distributions on gas flow parameters and dissociation rates behind the shock front is shown. Along with the equilibrium conditions before the shock front the case of weakly non-equilibrium CO2 distributions in the free stream is also considered in calculations. The peculiarities of the results obtained for this case are discussed.

Shock wave structure in carbon dioxide is studied using different continuum models within the framework of one-temperature thermal equilibrium flow description. Navier-Stokes and Euler equations as well as commonly used Rankine-Hugoniot equations with different specific heat ratios are used to find the gas-dynamic parameters behind the shock wave. The accuracy of the Rankine-Hugoniot relations in polyatomic gases is assessed, and it is shown that they give a considerable error in the predicted values of fluid-dynamic variables. The effect of bulk viscosity on the shock wave structure in CO2 is evaluated. Taking into account bulk viscosity yields a significant increase in the shock wave width; for the complete model, the shock wave thickness varies non-monotonically with the Mach number.

One-dimensional non-equilibrium air flows in nozzles are studied on the basis of the state-to-state description of vibrational-chemical kinetics. Five-component mixture N-2/O-2/NO/N/O is considered taking into account Zeldovich exchange reactions of NO formation, dissociation, recombination and vibrational energy transitions. The equations for vibrational and chemical kinetics in a flow are coupled to the conservation equations of momentum and total energy and solved numerically for different conditions in a nozzle throat. The vibrational distributions of nitrogen and oxygen molecules, number densities of species as well as the gas temperature and flow velocity along a nozzle axis are analysed using the detailed state-to-state flow description and in the frame of the simplified one-temperature thermal equilibrium kinetic model. The comparison of the results showed the influence of non-equilibrium kinetics on macroscopic nozzle flow parameters. In the state-to-state approach, non-Boltzmann vibrational distributions of N-2 and O-2 molecules with a plateau part at intermediate levels are found. The results are found with the use of the complete and simplified schemes of reactions and the impact of exchange reactions, dissociation and recombination on variation of vibrational level populations, mixture composition, gas velocity and temperature along a nozzle axis is shown.

The paper deals with modeling of vibration-chemical coupling in mixtures containing CO2 molecules. The temporal vibrational and chemical relaxation in the five-component and three-component space-homogeneous mixtures CO2/CO/O2/C/O and CO2/CO/O is studied taking into account vibrational energy transitions, dissociation, recombination and exchange chemical reactions. The vibrational-chemical coupling in considered mixtures is studied on the basis of three-temperature, two-temperature and one-temperature vibrational CO2 distributions. Governing equations for macroscopic mixture parameters in the three-temperature, two-temperature and one-temperature approximations are solved numerically for different initial conditions. Comparison of the results obtained in three approaches showed the in uence of non-equilibrium vibrational distributions on chemical reaction rates and on the temporal variation of the gas temperature, mixture composition and vibrational temperatures in two considered mixtures. The role of exchange chemical reactions in relaxation processes is also discussed in the paper.