This article contains a review of combustion instabilities coupled by pressure waves and of the related pulse combustor and active control principles and applications. The basic mechanisms leading to combustion oscillations are first identified from experimental evidence gathered in recent years. It is shown that in the presence of pressure waves, the flow features large scale motions which drive the instability. The dynamics of the flame is then dominated by processes of hydrodynamic instability, vortex roll-up, vortex interactions, front and reacting stream pulsations, periodic extinctions and reignitions, self acceleration. Knowledge accumulated on these various aspects has provided guidelines in combustor design. Progress accomplished in the special domain of pulse combustors is surveyed. These devices which operate under sustained pressure oscillations have been the subject of intensive research with as objective the understanding of the dynamics of the flow in various geometrical situations in order to improve the heat transfer efficiency and diminish the level of pollutant formation. While pulse combustors use pressure oscillations in normal operation, in most circumstances the presence of pressure waves causes problems of vibration, enhanced heat fluxes to the walls, cyclic fatigue, and in some cases important damage to the combustor. One generally wishes to eliminate or attenuate these waves. This is usually achieved by changes in the geometrical design and in the operation of the system. New possibilities based on active control have been investigated in recent years. The basic principles and the initial demonstrations of active instability control are first reviewed. It is then shown that such methods constitute a promising technology and that they are also a unique tool in the experimental investigation of combustion instability and pressure related phenomena.

Combustion instabilities coupled by pressure waves and their active control

This volume brings together international scientists in the field of solid rocket propulsion. Thirty-nine papers present in-depth coverage on a wide range of topics including: advanced materials and non-traditional formulations; the chemical aspects of organic and inorganic components in relation to decomposition mechanisms, kinetics, combustion and modelling; safety issues, hazards and explosive characteristics; and experimental and computational interior ballistics research, including chemical information and the physics of the complex flow field.

Solid propellant chemistry, combustion, and motor interior ballistics

Conditions of high energy densities and low losses in combustion chambers encourage the excitation and
sustenance of organized unsteady motions generically called combustion instabilities The fluctuations, common in propulsion systems, often reach sufficient amplitudes to cause excessive rates of heat transfer to exposed
surfaces and unacceptable structural vibrations, causing failure in extreme cases In many cases, to avoid the
occurrence of instabilities, combustion chambers are operated below their maximum performance Considerable effort has been spent, for more than four decades, on experimental and analytical programs devoted to solving problems of combustion instabilities Much of the work has been required to measure quantities which, because of the complex processes involved, cannot be predicted accurately from first principles Analytical work has been concerned largely with linear behavior, the chief purpose being to predict stability of small disturbances in combustion chambers Many useful results have been obtained, serving in practice to help design experiments, correlate data, and predict the stability of new systems However, linear behavior is only a small part of the general problem A combustion chamber is an isolated system so far as its stability is concerned, and unstable disturbances evolve as 'self-excited' motions Hence their amplitudes will grow indefinitely unless nonlinear processes are effective Complete understanding of observed behavior will therefore be reached only by treating nonlinear behavior In the recent past, increased attention has been paid to nonlinear combustion instabilities It is particularly important for practical purposes to explain the existence of limit cycles and the occurrence of unstable motions in linearly stable systems exposed to large initial disturbances These matters are far from closed, and although substantial progress has been accomplished, little impact has been made on the development of new systems The chief purpose of this paper is to provide a brief review of work on nonlinear combustion instabilities, largely in the framework of an approximate analysis Some connections will be made with the modern theory of nonlinear dynamical systems, including very recent and incomplete attempts by others to assess the possible chaotic behavior observed in laboratory tests

/pdf/some-recent-results-for-nonlinear-acoustics-in-combustion-nl4m93mic5.pdf

Some recent results for nonlinear acoustics in combustion chambers

a = speed of sound, Eq. (23) Ab = defined in Eq. (92b) An = defined in Eq. (58) Anij = defined in Eq. (75) Bn = defined in Eq. (58) Bnij = defined in Eq. (75) C = specific heat of particle material Cm = mass fraction of particles, Cm = p^/pg Cp — mass-averaged specific heat, Eq. (20) Cv = mass-averaged specific heat, Eq. (20) Dni = defined in Eq. (75) e0 = stagnation energy, gas phase El = defined in Eq. (47b) Eni = defined in Eq. (75) ("&} = time-averaged energy / defined in Eq. (42) /! defined in Eq. (52b)

Prediction of the stability of unsteady motions in solid-propellant rocket motors

High-amplitude pressure oscillations in solid propellant rocket motor combustion chambers display nonlinear effects including: 1) limit cycle behavior in which the fluctuations may dwell for a considerable period of time near their peak amplitude, 2) elevated mean chamber pressure (DC shift), and 3) a triggering amplitude above which pulsing will cause an apparently stable system to transition to violent oscillations. Along with the obvious undesirable vibrations, these features constitute the most damaging impact of combustion instability on system reliability and structural integrity. The physical mechanisms behind these phenomena and their relationship to motor geometry and physical parameters must, therefore, be fully understood if instability is to be avoided in the design process, or if effective corrective measures must be devised during system development. Predictive algorithms now in use have limited ability to characterize the actual time evolution of the oscillations, and they do not supply the motor designer with information regarding peak amplitudes or the associated critical triggering amplitudes. A pivotal missing element is the ability to predict the mean pressure shift; clearly, the designer requires information regarding the maximum chamber pressure that might be experienced during motor operation. In this paper, a comprehensive nonlinear combustion instability model is described that supplies vital information. The central role played by steep-fronted waves is emphasized. The resulting algorithm provides both detailed physical models of nonlinear instability phenomena and the critically needed predictive capability. In particular, the true origin of the DC shift is revealed.

http://majdalani.eng.auburn.edu/publications/pdf/2007%20-%20Flandro-Fischbach-Majdalani%20-%20PoF%20-%20%20Nonlinear%20Rocket%20Motor%20Stability%20Prediction%20%20Limit%20Amplitude,%20Triggering,%20and%20Mean%20Pressure%20Shift.pdf

Nonlinear Rocket Motor Stability Prediction: Limit Amplitude, Triggering, and Mean Pressure Shift

An existing nonlinear instability model has been extended and improved to allow it to more realistically treat the longitudinal shock-wave type of combustion instability frequently encountered in tactical solid rocket motors. Results obtained utilizing this model to investigate limiting amplitude and velocity-coupling phenomena in solid rocket motors are presented. An advanced finite difference integration technique capable of accurately describing shocks and contact discontinuities has been incorporated into the computer program, as well as an improved heat conduction solution, an heuristic velocity-coupling model, and a spectral analysis capability. Solutions demonstrating that limiting amplitude is independent of the characteristics of the initial disturbance are presented, as are results of a preliminary study of nonlinear velocity coupling demonstrating such phenomena as triggering, mean pressure shift, modulated limit cycles, and threshold velocity effects.

A numerical study of nonlinear instability phenomena in solid rocketmotors

The results of an investigation to select a satisfactory numerical method for calculating the propagation of steep-fronted, shock-like waveforms in a solid rocket motor combustion chamber are presented. A number of different numerical schemes were evaluated by comparing the results obtained for three related but simpler problems: the shock-tube problem, the linear wave equation, and nonlinear wave propagation in a closed tube. The most promising method, a combination of the Lax-Wendroff, Hybrid, and Artificial Compression techniques, was incorporated into an existing nonlinear combustion instability program. The capability of the modified program to treat steep-fronted wave instabilities in low-smoke tactical motors was verified by solving a number of motor test cases with disturbance amplitudes as high as 80% of the mean pressure.

Numerical techniques for solving nonlinear instability problems in solid rocket motors

A comprehensive model of nonlinear longitudinal combustion instability in solid rocket motors has been developed. The two primary elements of this stability analysis are a finite difference solution of the two-phase flow in the combustion chamber and a coupled solution of the nonlinear transient propellant burning rate. Although quasi-one dimensional, the model has been generalized to treat realistic variable cross-section and partial length grains. An excellent finite difference shock capturing technique—a combination of the Lax-Wendroff, Hybrid and Artificial Compression schemes—gives the analysis the ability to treat the multiple shock-wave type of instabilities that are frequently observed in reduced smoke solid rocket motors. Ad hoc velocity coupling models were also incorporated into the analysis. Solutions are presented demonstrating that pressure oscillations in unstable solid rocket motors (with metallized as well as unmetallized propellants) reach the same limit cycle (amplitude and waveform) independent of the characteristics of the initiating disturbance. Results obtained with the velocity coupling models demonstrate the ability to analytically predict triggering, DC pressure shifts, modulated amplitude limit cycle, and strongly nonlinear waveforms; phenomena that have all been observed in actual solid rocket motor firings.

Joseph D. Baum

Papers

A numerical study of nonlinear instability phenomena in solid rocketmotors

Numerical techniques for solving nonlinear instability problems in solid rocket motors

Modeling of nonlinear combustion instability in solid propellant rocket motors

Numerical techniques for solving nonlinear instability problems in solid rocket motors