Non-Equilibrium Reacting Gas Flows

For the recent decades, the kinetic theory of gases has encountered a necessity to solve some new physical problems. One of them is modeling of gas flows under the conditions of strong deviations from thermodynamic equilibrium. Such conditions occur, for example, near surfaces of nonexpendable space vehicles in their reentry into the Earth atmosphere, in experiments carried out with high-enthalpy facilities, in supersonic gas flows in nozzles and jets, for various types of energy pumping into vibrational and electronic degrees of freedom.

In high-temperature and hypersonic flows of gas mixtures, the energy exchange between translational and internal degrees of freedom, chemical reactions, ionization and radiation may result in noticeable violation of thermodynamic equilibrium. In particular, it was found that a large section of descending trajectory of modern space vehicles in the course of their reentry into the Earth atmosphere lies at large heights (60 to 70 km), where the gas is substantially rarefied (its pressure there is of the order of 10^− 2 atm) and the characteristic times of kinetic and gas-dynamic processes are comparable. Therefore the non-equilibrium effects become of importance for a hypersonic flow and should be taken into account for the prediction of the flow parameters.

In the present monograph, on the basis of the methods of kinetic theory, we study non-equilibrium flows of multi-component mixtures consisted of molecular gases with translational, rotational, and vibrational degrees of freedom. Collisions between particles can result in the translational energy exchange, internal energy transitions, bimolecular chemical reactions, dissociation, and recombination. Our consideration is restricted by the conditions when electronic excitation, ionization, and radiation can be neglected. A quasi-classical approach is applied. In the framework of this approach, the translational degrees of freedom of gas particles are treated classically whereas the rotational and vibrational energy spectra are assumed to be quantized; the pure quantum effects of the diffraction and particles collective are neglected. The gas description is based on one-particle distribution functions of molecules over the velocity, internal energy and chemical species. The quasi-classical method is well suited for the solution of molecular supersonic aerodynamic problems in a wide temperature and pressure range. The quantum description becomes necessary for low temperatures and rapid processes in strong fields, as well as for light gases. Such conditions are not considered in the monograph.

In this chapter, multi-component reacting gas mixture flows are studied under the conditions of strong vibrational and chemical non-equilibrium. For such flows, the zero-order and first-order distribution functions are derived in the frame of the modified Chapman-Enskog method. A set of macroscopic equations is obtained including the conservation equations coupled to the equations for the vibrational level populations and number densities of atomic species. The proposed approach makes it possible to develop the most detailed model of physical gas dynamics taking into account state-to-state vibrational and chemical kinetics. On the basis of the first-order distribution function, the kinetic theory of transport processes accounting for detailed non-equilibrium kinetics is established, and the features of dissipative processes under the conditions of strong deviations from the thermal and chemical equilibrium are discussed.

The approach proposed in the previous Chapter makes it possible to develop the most rigorous model of reacting gas mixture dynamics under the conditions of strong vibrational and chemical non-equilibrium, since it takes into account the detailed state-to-state vibrational and chemical kinetics for the definition of the gas flow parameters. However, practical implementation of this method leads to serious difficulties. The first important problem encountered in the realization of the stateto-state model is its computational cost. Indeed, the solution of the fluid dynamics equations coupled to the equations of the state-to-state vibrational and chemical kinetics requires numerical simulation of a great number of equations for the vibrational level populations of all molecular species. Moreover, in the viscous gas approximation, numerical simulations require the calculation of a large number of transport coefficients, particularly, diffusion coefficients in each space cell and at each time step, which significantly complicates the study of specific flows.

In the present Chapter, we consider a chemically reacting gas mixture flow under the condition (1.45), which suggests that the characteristic time for the relaxation of all internal energy modes is considerably smaller than that for chemical reactions. Such conditions often provide a subject for chemical kinetics studies [126], when non-equilibrium chemical reactions are simulated on the basis of the maintaining thermal equilibrium distributions over the internal energy. A kinetic description for the tempered reaction regime can be found in Refs. [249, 76, 60]. Thus, in the monograph [76], mathematical aspects of transport theory in reacting mixtures with slow chemical reactions are considered in detail, and efficient algorithms for the solution for linear transport systems are developed. In this Chapter, we briefly discuss the zero- and first-order distribution functions, governing equations, and the structure of transport and production terms in the one-temperature approach.

In order to define transport terms in the governing equations, it is necessary to calculate the coefficients of viscosity, thermal conductivity, diffusion, and relaxation pressure. In the present Chapter, algorithms for the calculation of transport coefficients are considered for different levels of accuracy of the description of non-equilibrium flows: in the state-to-state and quasi-stationary approaches. In contrast to weak nonequilibrium gases, the transport coefficients under the strong non-equilibrium conditions depend not only on the gas temperature but also on other parameters of the flow, which characterize the deviation from equilibrium, namely, on the vibrational level populations in the state-to-state approach and on the temperature of non-equilibrium modes in the quasi-stationary approximations. This results in some peculiarities in the transport coefficients discussed in this Chapter.

An important feature of modeling of real gas flows under the conditions of strong deviations from the equilibrium is that the set of governing equations for macroscopic parameters includes not only the conservation equations for the momentum and total energy, but also the relaxation equations. The latter equations contain the rates of slow physical-chemical processes proceeding on the gas-dynamic time scale. It is necessary to know the rate coefficients for these processes in order to solve the equations of non-equilibrium aerodynamics and calculate the flow parameters taking into account vibrational and chemical kinetics.

In Chapters 7-9, we consider the applications of the above-developed kinetic theory of transport and relaxation processes to some particular problems of non-equilibrium gas dynamics. In this Chapter, the peculiarities of the relaxation zone behind strong shock waves occurring in a hypersonic gas flow are considered. The rapid gas compression within a thin shock front with the characteristic length of about several mean free paths, results in a temperature jump which, due to the significant difference in relaxation times, occurs essentially without a variation in the mixture composition and molecular distributions over the vibrational energy. After that, in the relaxation zone behind the shock front, the excitation of vibrational degrees of freedom and chemical reactions take place, and as a result of relaxation processes, the total thermal and chemical equilibrium is established. The length of the relaxation zone reaches many tens and even hundreds mean free paths. The gas state in the unperturbed flow before the shock front is usually supposed to be equilibrium, and therefore, as a consequence of the gas propagation through the shock wave and relaxation zone, one equilibrium state is transformed to another. In the vibrational relaxation and chemical reactions behind the shock wave, equilibrium or weakly non-equilibrium distributions over the translational and rotational degrees of freedom established in the shock front are maintained. The assumed invariance of the Maxwell distribution corresponds to the Euler approximation for an inviscid nonconducting gas flow. Taking into account deviations from the Maxwell distribution makes it possible to study dissipative processes in a non-equilibrium viscous gas.

In this Chapter, we consider non-equilibrium vibrational distributions, macroscopic flow parameters, and dissipative properties of gas mixtures behind strong shock waves, in an A2/A binary mixture and in air, within the framework of the state-to-state, multi-temperature, and one-temperature kinetic models.

Close to the surface of a hypersonic space vehicle reentering the Earth atmosphere, strongly non-equilibrium conditions are established.While crossing the bow shock, the gas is substantially compressed and heated. Due to the high temperature in the flow, the dissociation rate increases and strong vibrational excitation occurs. In the vicinity of the cold surface, the temperature of translational and rotational degrees of freedom drops sharply, and the vibrational temperature appears to be higher than that of the gas. Under such conditions, the distribution of molecules over the vibrational levels significantly differs from equilibrium, dissociation near the high-temperature external edge of the boundary layer as well as recombination near the cold surface are substantially non-equilibrium, which, in turn, results in the formation of the non-Boltzmann vibrational distributions. These strongly non-equilibrium distributions may noticeably affect the transport processes.

Establishing of strongly non-equilibrium conditions in molecular gas flows in nozzles and jets is connected with rapid gas cooling and essential differences between the relaxation times for translational and internal degrees of freedom. Under the conditions of sharp expansion of a preliminarily heated gas, the flow is accelerated, whereas its pressure and temperature drop dramatically. Due to the significant discrepancy between the relaxation rates of the translational and internal modes, the vibrational energy in each elementary volume of a moving gas varies considerably slower than the translational energy, and cannot achieve the value corresponding to the equilibrium state at a local gas temperature. As a result, the vibrational energy appears to exceed translational substantially, and the relaxation processes display the regime of strong vibrational excitation. In expanding flows, the non-equilibrium distributions of molecules over the vibrational levels occur also as a result of chemical reactions, particularly recombination, which becomes more important with a decrease in the temperature.