Saturation of absorption of the powerful radiation of a system of anharmonic oscillators

Recently, the theory of nonequilibrium systems simulated by a set of anharmonic oscillators has received significant development. The investigation of such kinds of systems is especially important in the study of problems associated with the stimulation of chemical reactions and the development of effective molecular lasers. The systematic analysis of the kinetics of anharmonic oscillators assumes the simultaneous solution of a large number of nonlinear equations describing the population balance of the vibrational levels. Realization of this approach is associated with cumbersome numerical calculations and does not permit obtaining a qualitative picture of the behavior of the system as a function of the different parameters (pressure, temperature, etc.). An approximate analytical theory has been formulated in [1, 2] which permits finding the distribution function over the vibrational states with the effects of anharmonicity taken into account. We will employ the approach developed in these papers to describe a system of anharmonic oscillators under conditions of powerful optical pumping. This problem was discussed in [3], where it was found that such a system changes into a saturation mode in the case of high pumping levels. The existence of this mode is explained by the fact that the maximum rate of energy input into a vibrational degree of freedom is determined by the rate of distribution of this energy over all the vibrational levels, i.e., by the constant of V—V-exchange. For sufficiently large pumpings the approximation of the Boltzmann distribution function adopted in [3] in connection with the calculation of the saturation parameters is too crude. The goal of this paper is to derive in explicit form expressions for the vibrational energy supply, the absorbed power, and so on, under saturation conditions without the use of the approximation indicated above [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 10–15, September–October, 1978.     

Thermal effects accompanying nonresonant vibrational exchange

Calculation has been made of the thermal effects attending nonresonant vibrational exchange in a molecular gas in a nonequilibrium state. Study has been made of the binary mixture, modeled as a set of harmonic oscillators of various frequencies. Vibrational exchange, the most rapid relaxation process occurring in the system, was assumed to completely determine the distribution of vibrational energy. It is shown that nonresonant vibrational exchange can lead to either heating or cooling in nonequilibrium states of the gaseous mixture. Calculation has been made of the amount of thermal energy liberated or absorbed in these processes.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 20–23, March–April, 1975.The author would like to thank N. N. Sobolev for his interest in this work and for his valuable advice concerning it.     

Kinetic equations for vibrational relaxation in a mixture of polyatomic gases

Kinetic equations are derived for the relaxation of the vibrational energy in a mixture of polyatomic gases, which are ones with molecules simulated by harmonic oscillators. The most general case is envisaged, where the energy relaxation occurs not only via vibrational-translational transitions but also via multiquantum vibrational exchange involving an arbitrary number of vibrational modes. The analysis also incorporates the possible degeneracy of each mode when the molecules colliding are the same. An expression is derived that extends previous results [1–6] and that relates the vibrational temperatures in the case of quasiequilibrium. Equations are derived for the vibrational relaxation for the CO₂-N₂ case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 29–37, November–December, 1972.We are indebted to L. A. Shelepin for valuable discussions on the results.     

Analysis of Nonequilibrium Dissociation Kinetics in a Boundary Layer Using Various Reaction Models

Values of the nonequilibrium macroscopic reaction rate for a nonisothermal boundary layer of a monatomic diluent gas are calculated using a number of models for thermal dissociation of diatomic molecules — anharmonic Morse oscillators. Analysis is performed for conditions where the diffusive transfer of excited molecules has a significant effect on the population of their upper vibrational levels, which does not only amount to change in vibrational temperature. Under the joint influence of diffusive transfer of molecules, vibrational exchanges, and reactions involving vibrationally excited particles, the local vibrational distribution functions are substantially nonequilibrium. The kinetic models considered take into account the possible contribution of the energy of molecular translational and rotational degrees of freedom to the energy required to overcome the reaction threshold. The effect of multiquantum vibrational—translational exchanges on the distribution of dissociating molecules in their upper vibrational levels is taken into account approximately.     

Physicochemical Model of Hypersonic Rarefied Gas Flow past Bodies

The effect of the internal molecular degrees of freedom on the flow field and heat transfer in hypersonic rarefied gas flow past a cylinder or sphere is investigated using the direct statistical simulation (Monte-Carlo) method. The variable-diameter rough spherical molecule model (VRS-model) is generalized to include the case of energy exchange between the translational and vibrational degrees of freedom. The interaction between diatomic molecules with allowance for vibrational degrees of freedom is simulated as the interaction of classical or quantum-mechanical harmonic and anharmonic oscillators in the external force approximation. A model of the dissociation of a diatomic gas is proposed.     

Vibrational relaxation of diatomic molecules in a non-Boltzmann thermostat

The action of resonance IR laser radiation on a molecular gas leads, at high-power absorption intensity, to a breakdown in the equilibrium (Boltzmann) energy distribution in the internal degrees of freedom [1]. Under realistic conditions, molecular gases usually are (due to small amounts of impurities or isotopic components) multicomponent systems. In this case resonance IR laser radiation (or other methods of selective action), disturbing the distribution function of the primary gas, does not interact directly with impurities. The problem thus arises of determining the distribution function of the impurity gas interacting with the nonequilibrium (non-Boltzmann) thermostat. The present paper, devoted to the solution of this problem, treats the distribution function of harmonic oscillators A, consisting of a small amount of impurities in a system of harmonic oscillators B with given nonequilibrium distribution functions of vibrational energy. The behavior of a system in a nonequilibrium thermostat was first considered in [2, 3] where, as well as in [4, 5], it was shown that in a non-Maxwellian thermostat with a small amount of harmonic oscillator impurities, a Boltzmann distribution in harmonic oscillator vibrational energies is established under stationary conditions, with a temperature differing from the gas-kinetic temperature of the thermostat, defined in terms of the mean-square velocity. The behavior of a small amount of impurities (heavy monoatomic particles and harmonic oscillators) in a non-Maxwellian thermostat of a light gas was further investigated in [6–8]. Unlike the papers mentioned, the present one considers the behavior of a small amount of harmonic oscillator impurities in a thermostat with a Maxwellian velocity distribution and with a nonequilibrium (non-Boltzmann) distribution in vibrational energies.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1978.     

An extension of CVDV model to Zeldovich exchange reactions for hypersonic non-equilibrium air flows

A new preferential vibration-dissociation-exchange reactions coupling model – labelled CVDEV – resulting from an extension of the well-known Treanor and Marrone CVDV model, has been derived to take into account the coupling between the vibrational excitation of the and molecules and the two Zeldovich exchange reactions. Analytical expressions for the exchange reactions coupling factor and for the average vibrational energy lost – or gained – by a molecule through an exchange reaction have been developed. The influence of such a coupling has been shown by means of numerical simulations of hypersonic air flows through normal and bow shock waves. Code-to-code comparisons between our model and other recent approaches have been conducted. The infrared radiation of nitric oxide behind a normal shock wave resulting from computations with the CVDEV model has been compared with other coupling model results and to recent shock tube experimental data. These comparisons have shown a good agreement of our model results with the experimental data. In this context, the results show the prominent influence of vibration coupling on the first Zeldovich reaction, and the absence of vibration coupling effects on the second Zeldovich reaction. Received 30 June 1997 / Accepted 3 December 1997     

Vibrational-translational relaxation of anharmonic oscillators at low temperatures

An approximate analytic solution is found to the problem of the vibrational-translational relaxation of anharmonic oscillators at translational temperatures which are small compared with the energy difference between adjacent levels of the oscillator. The deviation of the obtained distribution from the Boltzmann distribution in the relaxation process is analyzed. A study is made of the behavior of the vibrational energy near equilibrium at temperatures such that dissociation has only a small effect on the rate of vibrational relaxation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 3–8, March–April, 1976.The author wishes to thank M. B. Zheleznyak and A. Kh. Mnatsakanyan for a useful discussion of the work.     

Diffusion equation in the theory of vibrational relaxation of diatomic molecules

A kinetic equation is obtained for the distribution function of anharmonic oscillators with respect to the vibrational energy; it enables one in the diffusion approximation to describe the vibrational relaxation of diatomic molecules in a medium of inert gas when there is a weak interaction between the oscillators and a thermal bath. The main difference from the equation for harmonic oscillators is in the appearance in the diffusion coefficient of an adiabaticity function that characterizes the variation of the adiabaticity factor because of the anharmoni-city of the vibrations. It follows from the form of this function that the greatest difference between the relaxation of anharmonic and harmonic oscillators is to be expected in the case of adiabatic interaction of oscillators with particles of the inert gas.     

高温非平衡流动中的氧分子振动态精细分析

高超声速流动在头激波压缩后常处于高 温条件下的热化学非平衡状态. 本文采用态-态方法和双温度模型计算分析了一维正激波后和高超声速钝体绕流驻点线上的氧气热化学非平衡流动. 态-态方法将氧气的每个振动能级当成独立的组分,通过耦合 Euler 方程或驻点线上的降维 Navier-Stokes 方程,数值求解得 到了高温流动中的精细热化学非平衡状态. 而双温度模型假设氧气的振动能级服从 Boltzmann 分布,通过求解振动能方程得到振动温度. 一维正激波后热化学松弛过程的计算结果表明,态-态计算预测的温度分布和氧原子浓度分布较好地吻合了文 献中的实验结果,而经典的双温度模型的预测结果误差较大,且不同双温度模型的计算结果比较发散. 态-态方法详细地给出了所有振动能级的变化过程. 无论是正激波还是脱体激波后的流场,都是高振动能级首先得到激发;但是数密度大的低振动能级先达到热平衡,而高能级 分子要经过很长距离后才能达到热平衡. 在驻点附近,复合反应生成的氧气分子处于高振动能级,导致高振动能级分子数密度显著高于平衡分布. 计算还发现,经典双温度模型的离解反应速率明显偏离态-态计算结果,无法准确体现振动离解耦合效应对离解反应 速率的影响,但是 Park 双温度模型将离解失去的振动能取为 0.3$\sim 高超声速流动在头激波压缩后常处于高 温条件下的热化学非平衡状态. 本文采用态-态方法和双温度模型计算分析了一维正激波后和高超声速钝体绕流驻点线上的氧气热化学非平衡流动. 态-态方法将氧气的每个振动能级当成独立的组分,通过耦合 Euler 方程或驻点线上的降维 Navier-Stokes 方程,数值求解得 到了高温流动中的精细热化学非平衡状态. 而双温度模型假设氧气的振动能级服从 Boltzmann 分布,通过求解振动能方程得到振动温度. 一维正激波后热化学松弛过程的计算结果表明,态-态计算预测的温度分布和氧原子浓度分布较好地吻合了文 献中的实验结果,而经典的双温度模型的预测结果误差较大,且不同双温度模型的计算结果比较发散. 态-态方法详细地给出了所有振动能级的变化过程. 无论是正激波还是脱体激波后的流场,都是高振动能级首先得到激发;但是数密度大的低振动能级先达到热平衡,而高能级 分子要经过很长距离后才能达到热平衡. 在驻点附近,复合反应生成的氧气分子处于高振动能级,导致高振动能级分子数密度显著高于平衡分布. 计算还发现,经典双温度模型的离解反应速率明显偏离态-态计算结果,无法准确体现振动离解耦合效应对离解反应 速率的影响,但是 Park 双温度模型将离解失去的振动能取为 0.3$\sim $0.5 倍分子离解能是比较合理的.     

Nonequilibrium dissociation of diatomic molecules in flows with convective and diffusive particle transport

The nonequilibrium effects in the kinetics of the thermal dissociation which occurs in a streaming gas of diatomic molecules are investigated. Expressions are obtained for the macroscopic reaction rate and the vibrational energy distribution of the molecules, taking into account the influence of the gas motion. Cases of flows with convective and diffusive particle transport are considered. The dissociating molecules are simulated by cutoff harmonic oscillators. The vibrational kinetics is described in the framework of the so-called diffusion approximation.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 147–153, November–December, 1985.     

Monte Carlo analysis of dissociation and recombination behind strong shock waves in nitrogen

Computations are presented for the relaxation zone behind strong, one-dimensional shock waves in nitrogen. The analysis is performed with the direct simulation Monte Carlo method (DSMC). The DSMC code is vectorized for efficient use on a supercomputer. The code simulates translational, rotational and vibrational energy exchange and dissociative and recombinative chemical reactions. A new model is proposed for the treatment of three body recombination collisions in the DSMC technique which usually simulates binary collision events. The new model represents improvement over previous models in that it can be employed with a large range of chemical rate data, does not introduce into the flow field troublesome pairs of atoms which may recombine upon further collision (pseudo-particles) and is compatible with the vectorized code. The computational results are compared with existing experimental data. It is shown that the derivation of chemical rate coefficients must account for the degree of vibrational nonequilibrium in the flow. A nonequilibrium chemistry model is employed together with equilibrium rate data to compute successfully the flow in several different nitrogen shock waves.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Model of the physico-chemical kinetics behind the front of a very intense shock wave in air

A model of the physico-chemical kinetics of the reactions taking place behind the front of an intense shock wave propagating in air with a speed of 9–14 km/s is proposed. The problem of describing the chemical reactions, namely, molecular dissociation and exchange reactions involving vibrationally excited molecules in the absence of vibrational equilibrium, is solved. The vital role of the vibrational excitation delay in the dissociation of oxygen and nitrogen is established. The rate of the exchange reaction between nitrogen molecules and oxygen atoms in the shock wave depends only slightly on the vibrational excitation level. It is demonstrated that the rate constants for thermally nonequilibrium dissociation reactions can be represented within the framework of the one-temperature approximation at constant vibrational temperatures of the dissociating species satisfying quasi-stationary conditions.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 169–182, March–April, 1995.     

Dissociation of anharmonic molecules by means of high-power infrared emission

The kinetics of nonequilibrium molecular dissociation when vibrations are excited by high-power infrared emission is investigated for a model of anharmonic oscillators. The case when exchange of vibrational quanta during collision with molecules in the lower states plays a fundamental role in the formation of the vibratory distributionfunction at the upper level is analyzed. Dependences of the “vibration temperature” and the rate constant for nonequilibrium dissociation, as a function of the optical pumping probability are obtained for different pumping conditions. The results are compared with similar calculations for a harmonic model.     

Thermal dissociation of diatomic molecules in a nonisothermal boundary layer in the presence of exchange reactions

The effect of an inhomogeneous temperature field in a boundary layer on the kinetics of dissociation of diatomic molecules simulated by truncated harmonic oscillators is considered in a multicomponent mixture in the presence of exchange reactions which take place at lower vibrational levels as compared with dissociation.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–172, January–February, 1995.     

The effect of gas motion on the reaction kinetics of vibrationally excited molecules

The effect is considered of gas motion on the kinetics of reactions whose energy threshold is overcome as the result of vibrational excitation of the reactant molecules. The conditions are determined for which such an effect may be realized. An expression is obtained for the rate of thermal dissociation of diatomic molecules considered as harmonic oscillators representing a small impurity in a monoatomic inert gas; the expression depends explicitly and nonlinearly on the divergence of the flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 100–105, November–December, 1984.     

About a class of nonlinear oscillators with amplitude-independent frequency

This work is concerned with nonlinear oscillators that have a fixed, amplitude-independent frequency. This characteristic, known as isochronicity/isochrony, is achieved by establishing the equivalence between the Lagrangian of the simple harmonic oscillator and the Lagrangian of conservative oscillators with a position-dependent coefficient of the kinetic energy, which can stem from their mass that changes with the displacement or the geometry of motion. Conditions under which such systems have an isochronous center in the origin are discussed. General expressions for the potential energy, equation of motion as well as solutions for a phase trajectory and time response are provided. A few illustrative examples accompanied with numerical verifications are also presented.     

耦合阻尼对非保守耦合振子能量分布与功率流的影响

作为非保守耦合系统统计能量分析的基础,本文研究了耦合阻尼对非保守耦合系统能量分布与功率流的影响.在给出各项有关的损耗因子和耦合损耗因子的定义后,本文从理论上推导了非保守耦合振子间各项功率流与振子平均振动能量之间关系的理论表达式,以及功率平衡方程式和振子能量比的表达式.理论分析和数值计算的结果表明,非保守耦合振子之间的原始功率流和附加功率流以及总功率流不仅取决于两振子的平均振动能量之差,而且取决于振子的平均振动能量之和,总功率流的方向即与两振子能量相对大小有关,也与耦合性质有关;小耦合阻尼是非保守耦合的特例,由此特例不足以得到非保守耦合情况的一般特点。