Theory of Solid-Propellant Nonsteady Combustion. Vasily B. Novozhilov

system behaviour under constant external conditions is directly related to the problem of turbulence. Variation of one of the control parameters leads to successive bifurcations of combustion regimes, ending up with chaotic behaviour.

In contrast to the majority of studied examples of dynamical systems, the model described above expresses real physical and chemical processes. Within the framework of the presented formulation, such practically important phenomena as propellant burning interaction with the acoustics of the combustion chamber, combustion extinction under depressurization, burning stability under constant pressure, etc. may be investigated.

      Since experimental data on steady‐state combustion are an essential input into the theory, this book begins with an establishing chapter describing the steady‐state burning regime and presenting various fuel property dependencies on external conditions. The theoretical estimations and experimental results provided in the first chapter are necessary for a justification of nonsteady combustion theory. Analytical dependencies of the burning rate and surface temperature on external parameters (i.e. steady‐state burning laws) are useful for quantitative analysis. Their physically sensible form may only be obtained from the current understanding of steady‐state combustion regimes of the simplest systems. These issues are also discussed in the first chapter.

      The second chapter, which is fundamental for the monograph, formulates major assumptions of the theory of nonsteady combustion of solid rocket fuels. A simplified case of constant propellant surface temperature (Zeldovich theory) is considered in detail. The two formulations of the ZN theory, differential and integral, are then presented.

      In the first of these formulations, temperature distribution in the condensed phase is a necessary element of consideration. This profile, however, is rarely used in practice. It turns out that an alternative, integral formulation, involving only the most relevant quantities – pressure and burning rate, as an example – may be developed. Readers interested in a formal mathematical justification of the theory should pay attention to the last section of this chapter.

      The third chapter considers propellant combustion at a constant pressure. In one‐dimensional problem formulation, the stability criterion for the steady‐state combustion regime, relating burning rate and surface temperature derivatives with respect to initial temperature is obtained. The possibility of two‐dimensional instability is discussed using the example of the simplest combustion system. Numerical modelling is used to investigate nonsteady modes of propellant combustion under constant pressure beyond the stability boundary of the steady‐state regime. The simplest propellant combustion model containing just two control parameters is considered. With a fixed value of one of these, the other plays the role of a bifurcation parameter. It is demonstrated that on variation of the bifurcation parameter, the system may transit from the steady‐state to the chaotic combustion regime following the Feigenbaum scenario. A sequence of period doubling bifurcations of the burning rate oscillations, leading eventually to a chaotic combustion regime, is studied. A comparison with experimental data is discussed in the last section of the chapter.

      The following three chapters are devoted to the influence of external parameters, that is, pressure, erosive gas stream, and thermal radiation, on propellant combustion under periodically varying pressure. Practical interest in these processes is due to the need to understand the causes of the development of various nonsteady effects, for example soft or hard excitation of burning rate and pressure oscillations, superimposed on the designed steady‐state solid rocket motor regime.

First, the problem of combustion and acoustics interaction (Chapter 4) is considered in the linear approximation with the burning rate amplitude being proportional to pressure amplitude. An analytical expression for the response function of the burning rate to oscillating pressure is obtained. Furthermore, its properties and relation with the most important property of burning propellant surface (acoustic admittance) are discussed. Then, the notion of response functions of higher orders, with respect to pressure amplitude, is discussed. These would find an application in investigations of sustained and transitional regimes with finite burning rate and pressure amplitudes. A conducted nonlinear analysis reveals a fundamentally new phenomenon: period doubling bifurcations of burning rate oscillations that occur on an increase of amplitude or change of frequency of pressure oscillations. A sequence of bifurcations leading eventually to the chaotic combustion regime is investigated for a propellant combustion model containing a minimal number of parameters in nonsteady burning laws.

      The fifth chapter considers nonsteady propellant burning in the tangential stream of combustion products. Within the framework of the phenomenological theory, the propellant burning rate response to periodically varying pressure and tangential mass flux of combustion products is investigated. An elementary acoustic perturbation in the form of a monochromatic travelling sound wave is considered. Analytical and numerical results are obtained for the simplest propellant model described by a minimal number of parameters. The role of steady‐state and nonsteady erosion contributions at small and large values of the erosion ratio is also revealed.

      The following chapter deals with nonsteady combustion under external radiation. In this case, the heat transfer equation includes an additional source term, while both steady‐state and nonsteady burning laws also change. The stability of the steady‐state combustion regime is investigated in the linear approximation. The response functions of the burning rate to harmonically oscillating pressure in the presence of constant radiative flux, as well as to harmonically oscillating radiative flux, are obtained. In the linear approximation of the ZN theory, an analytical relationship between the response function to oscillating pressure, obtained at a certain initial temperature, and response function to oscillating radiative flux, obtained at the same pressure but different (lower) initial temperature, is established. The difference of initial temperatures satisfies the requirement that steady‐state burning rates with and without radiative flux are equal, and is directly proportional to radiative flux. It is likely that this relationship will be useful for obtaining experimental data on the response function to oscillating pressure.

      Chapter 7 presents theoretical considerations and experimental data related to combustion regimes where pressure varies according to the law, which is different from harmonic oscillations. Such processes include, for example, propellant combustion during transition from one operational regime to another (at higher or lower pressure), extinction on rapid and deep depressurization, and others.

      The eighth chapter describes nonsteady propellant combustion regimes in the combustion chamber of a rocket engine. There are three time scales that are relevant for this problem: the thermal relaxation time of the heated layer of the condensed phase t_c, the acoustic time t_a, and the time of combustion products efflux from the chamber t_ch.

      If the relaxation time of the condensed phase is close to the efflux time, t_c∼t_ch (which occurs in small engines at low pressures), then such regimes may be called nonacoustic. Time scales in such problems are much larger than the acoustic time. This area of research may also be referred to as propellant combustion in semi‐enclosed volume.

On the other hand, over the last few decades a specific and dedicated area of research which may be termed ‘acoustics and combustion’ has taken shape. It deals with the case where acoustic time is close to the condensed phase thermal relaxation time t_a∼t_c, which leads to a possibility of sonic (in the general case nonlinear) oscillations development in the engine. The latter relation between the time scales applies to engines of large size with high pressure values in combustion chambers. The theory of such processes is still at a rudimentary stage. As an example, possible combustion regimes in a solid rocket engine with end burner grain geometry are investigated. A set of equations which allows the interaction between combustion and acoustic processes in a combustion chamber to be modelled is presented. The specific feature of the problem is the existence of the two distinctive time scales, namely the acoustic time and the time of pressure oscillation amplitude variation. These time scales differ by approximately three orders of magnitude, which demands high computational accuracy. A simpler solution method is developed