Nonlinear Combustion Instability Research Articles

Combustion Simulation of 82-Injector Rocket Engine

A computational study of the nonlinear rocket-engine combustion instability is presented for an engine with 82 coaxial methane–oxygen injector ports, a choked nozzle, and a combustion chamber that models a Rocketdyne experiment. Computations use a three-dimensional unsteady shear-stress transport delayed detached-eddy simulation method. Four different one-step chemical kinetic rates are simulated on the resolved scale to determine the effect of the burning rate. Application of kinetic rates on the resolved scale give faster rates than expected with turbulent combustion where the final mixing and chemical reaction in the cascade occurs on the smallest scales; thus, reduction of the kinetic rate is examined here. When the normalized kinetic rate is 1, implying the nominal Westbrook–Dryer rate, only the longitudinal-mode instability is observed. As the kinetic rate decreases, less of the combustion occurs upstream near the longitudinal-mode pressure antinode, the longitudinal-mode instability becomes weak, and the first-tangential-mode instability emerges, which fits better with experiment. The proximity of the domain of major heat release to the longitudinal mode pressure antinode is found to be critical. Evidence is presented for the need for an improved combustion model that will provide a more accurate and lower burning rate.

A nonlinearly kinematic model of the asymmetrically turbulent premixed slit flame subjected to two-way harmonic disturbances

Transversal and longitudinal combustion instabilities typically occur in gas turbines and rocket engines. Quantifying the nonlinear flame dynamic response of turbulent premixed flames subjected to the two-way disturbances and mean bulk flows is significant to understand the complex nonlinear combustion instability phenomena. A Two-way Nonlinear Unforced Flame Speed Modeling (2W-NUFSM) approach was proposed in this study. The turbulent effects were accounted for in the flame propagation speed term and the steady flame term in the kinematic equation of the flame front; an expansion method was then used to obtain the nonlinear solutions corresponding to the transverse and longitudinal acoustic disturbances. The turbulent effect on the flame consumption speed was also considered in the heat release rate calculation of the perturbed flame. Validations of this approach were conducted by comparing its results to the former credible results obtained from other methods and good agreements were obtained. The influence of turbulent effects on the flame dynamic response in an asymmetrical premixed flame was investigated in detail. The coupling effect of the turbulence and harmonic flow disturbances on the flame nonlinear dynamic response was also quantified.

Nonlinear combustion instability analysis of a bluff body burner based on the flame describing function

Lean premixed combustion is a common form of combustion organization in power equipment and propulsion systems. In order to understand the dynamic characteristics of lean premixed flame and predict and control its combustion instability, it is necessary to obtain its flame describing function (FDF). Based on the open source CFD toolbox, OpenFOAM, the dynamic K-equation model, and the finite rate Partially Stirred Reactor (PaSR) model were used to perform large eddy simulations (LES) of lean premixed combustion, and the response of the unsteady heat release rate to single-frequency harmonic disturbances was studied. The response of the unsteady heat release rate was characterized by the FDF, and the response of the unsteady heat release rate to the two-frequency harmonic disturbance was studied. The results show that the quantitative heat release rate response and flame dynamics have very proper accuracy. In the single-frequency harmonic disturbance, as the forcing frequency increases, the curling behavior of the flame surface and the instantaneous vortex structure change; the nonlinear kinematics effect is manifested by the entrainment of the vortex. At lower forcing frequencies, the heat release response changes linearly with the increase of forcing amplitude; at intermediate frequencies, the heat release response exhibits obvious nonlinear behavior; at high frequencies, the heat release response to amplitude changes decreases. The introduction of the second harmonic disturbance will significantly reduce the response range of the total heat release rate and make the combustion more stable.

Proper orthogonal and dynamic mode decomposition analyses of nonlinear combustion instabilities in a solid-fuel ramjet combustor

In ramjet propulsion systems, syelf-excited combustion instability is highly undesirable. It may lead to violent structural vibration and even mission failure. For this, self-excited combustion instabilities in such SFRJ are investigated numerically by 2D unsteady Reynolds-Average Navier-Stokes (URANS) model and varying the inlet mass flow rate ṁair. When combustion instability occurs, large-amplitude temperature, velocity, and reacting species wavy motions are observed along the axial flow direction. To shed light on the flow features and the energy distribution among the unstable eigenmodes, proper orthogonal (POD) and dynamic mode decomposition (DMD) analyses of nonlinear combustion instabilities in a solid-fuel Ramjet (SFRJ) combustor are conducted. The POD studies reveal that the fluctuation energy of the first six modes contributes to 99% of the total fluctuation energy of all modes. It means that the main features of the SFRJ combustors could be captured by using the first six modes. Unlike POD studies, DMD analysis captures the frequency information and spatial structures in the entire flow field. Examining the DMD growth rate reveals that all dominant DMD eigenmodes are marginally stable (growth rate = 0) or unstable (positive growth rate). And the frequency of each dominant mode could be enhanced by 3–10 Hz as theṁair is increased by 0.2 kg/s. The mode energy decreases with the increase of the mode order, with the decrease up to around 90%. The DMD analysis on velocity field indicated that the flow separation could be observed near the backward facing step. And the DMD mode structure of the temperature field can significantly affect the mode structures of Mach number and YCO2 field. In general, the present work provides detailed analyses of the SFRJ flow field in the presence of self-excited combustion instability by using POD and DMD approaches.

Combustion Dynamics Simulation of a 30-Injector Rocket Engine

ABSTRACT A computational study of the nonlinear rocket-engine combustion instability is presented for an engine with 30 coaxial methane-oxygen injector ports, a choked nozzle, and a combustion chamber. Three combustion chamber lengths of 18 cm, 24 cm, and 30 cm are simulated with differing instability modes depending on length. Computations using a three-dimensional unsteady k − ω shear-stress transport delayed detached-eddy simulation method provide detailed time-resolved information about the combustion instability. Our flame-index analysis explains why the premixed-flame structures dominate over diffusion flame-structures in the driving mechanism for the instability. This causes the engine to become more unstable as the mixture becomes more fuel rich. Spontaneous longitudinal-mode and tangential-mode instabilities are observed. The oscillation amplitude of the two instability modes vary in alternating fashion during the simulation time. The alternating behavior of the two instability modes can be modified by pulsing the injector mass-flux on the mixture ratio at a particular frequency.

Characterizing nonlinear dynamic features of self-sustained thermoacoustic oscillations in a premixed swirling combustor

To meet stringent NOx emission requirements, lean premixed swirling combustion technology is widely applied in power generation and propulsion systems. However, such combustion systems are more susceptible to nonlinear combustion instability due to the fluid flow-acoustics-combustion interaction. It is typically self-exited and characterized by large-amplitude pulsating oscillations. By applying experimental measurements and theoretical modelling, we explore the rich physics of how a methane-burnt swirling flame sustains periodic pulsating combustion oscillations, and its nonlinear dynamics features via recurrence plots (RP) and 0–1 chaotic test. The effects of (1) the equivalence ratio Φ, (2) the volume flow rate Va of inlet air and (3) the swirling number SN are examined. Hopf Supercritical and period-doubling bifurcation behaviours are experimentally observed with increased Φ from lean to rich combustion. Same nonlinear features are theoretically modelled by using ‘kick-oscillator’ model representing combustion pulses and to capture periodic limit cycle behaviours followed by period-doubling bifurcation and transition to chaos. The deterministic or chaotic nature of the swirling combustor could be experimentally identified using classical approaches such as probability density functions (PDFs) and acoustics power spectrum in presence of combustion-sustained periodic fluctuations and 0–1 chaotic test method. When the combustor is experimentally operated under either lean (Φ ≤ 0.6) or rich (Φ ≥ 1.1) conditions, no self-sustained periodic acoustic fluctuations are generated. Furthermore the combustor is found to be more chaotic. The flame shapes (M- or V-shaped), colour, brightness and volumes are found to depend on SN strongly. The present findings are physically insightful on understanding nonlinear features of swirling flow-acoustics-flame interaction.

Nonlinear Combustion Instability in a Multi-Injector Rocket Engine

A computational study is presented of the nonlinear combustion instability of a multi-injector rocket engine. The study addresses a choked nozzle and a combustion chamber with 10 and 19 coaxial injector ports. Computations using a three-dimensional unsteady shear-stress transport delayed detached-eddy simulation method provide detailed time-resolved information about the combustion instability. The spontaneous longitudinal-mode instability is observed for the 10- and 19-injector geometries with a combustion-chamber diameter of 28 cm. The triggered tangential and longitudinal instability modes are obtained for the 19-injector geometry with a combustion-chamber diameter of 43 cm by pulsing the injector mass flux. It is shown that an oscillating combustion-chamber flow can be triggered to a new mode with a larger disturbance amplitude. The streamwise vorticity created by the instability impacts the heat release in a manner that supports the instability. A preliminary study of injector-size scaling effects is performed by comparison of the 10- and 19-injector combustors.

FDF-based combustion instability analysis for evolution of mode shapes and eigenfrequency in the multiple flame burner

In this work, the FDF (flame describing function)-based combustion instability analysis together with the Helmholtz solver has been made for the wide range of velocity perturbation and plenum length in the multiple premixed burner. The Helmholtz solver is based on the commercial software COMSOL and the present numerical analysis is made for the two-dimensional axisymmetric geometry. This study has been mainly motivated to systematically analyze the effects of the velocity perturbation on the detailed evolution of eigenfrequency and mode transformation characteristics versus the variation of the plenum length. Due to the realistic treatment for the acoustic boundary conditions as well as the pressure jump condition across the perforated plate, the present study is able to predict the continuous transformation of the mode shapes according to the variation of the plenum length. To precisely analyze the nonlinear combustion instability phenomena in the multiple flame combustor, computations are made for the wide range of velocity perturbation and plenum length. In terms of stable and unstable ranges of frequency, numerical results yield the comparable results with measurements. Moreover, the present numerical results for the higher velocity perturbation ratios clearly reveal the frequency locking phenomenon which does not change the eigenfrequency even for the condition transforming the acoustic modes several times. Based on numerical results, the detailed discussions are made for effects of the velocity perturbation on the evolution of eigenfrequency and mode transformation characteristics versus the variation of the plenum length.

Experimental and theoretical study on characteristics of pulse excitation in T-burners

Pulse excitation is the key to measure the pressure-coupling response function of composite propellant. It is also a key trigger factor for nonlinear combustion instability. This paper aims at understanding characteristics of pulse excitation in T-burners. Pulse excitation is provided by black powder (BP). D2 law is used to calculate BP burning properties. Firstly, the experimental pressure history of a pulse excitation is analyzed. Pressure pulse and mean pressure increment are introduced to describe pulse excitation. Secondly, the modified zero-dimension model and one-dimension model of pressure pulse are established based on energy conservation and modification. The results of models indicate that the modified zero-dimensional model can accurately predict the pressure pulse. The modified zero-dimension model demonstrates that the pressure pulse is determined by pulse build-up time threshold, volume coefficient, effective weight fraction of BP, weight of BP et. al. When burning time of BP is larger than the threshold, volume coefficient is equal to 2, and effective weight fraction of BP is less than 1. The pressure pulse is approximately linear correlation with weight and effective weight fraction of BP. Otherwise, volume coefficient is larger than 2, and effective weight fraction of BP is equal to 1. The pressure pulse is approximately linear correlation with volume coefficient and BP weight. Thirdly, a zero-dimensional prediction model of mean pressure is established based on conservations of energy and mass. The prediction models of pressure pulse and mean pressure are validated by T-burner experiments. Finally, effects of BP burning properties on pressure pulse and mean pressure increment are studied. The results show that both pressure pulse and mean pressure increment increase with increasing BP weight, linearly. The pressure pulse is more sensitivity to the variations of burning time of BP. As burning time of BP decreases, the mean pressure increment gradually increases to the maximum, and the pressure pulse can become a very large value.

Nonlinear Combustion Instabilities Analysis of Azimuthal Mode in Annular Chamber

Thermo-acoustic combustion instabilities are due to the coupling mechanism between acoustic pressure oscillations and heat release rate fluctuations. This phenomenon is of great interest because combustion instabilities cause recurrent problems in modern gas turbines. The case of annular chambers largely diffused in gas turbines for power generation is of particular interest. In this work, the azimuthal modes of a three-dimensional annular combustion chamber are studied by means of a finite element (FEM) code. For each burner, the coupling between heat release fluctuations ▪ and the velocity fluctuations ▪ is modeled by means of a non-linear analytical Flame Describing Function (FDF) consisting of a third order and fifth order polynomial expression. A delay time between ▪and ▪is considered. The dissipation of acoustic energy of the system due to viscous and thermal boundary layer is directly modeled in the code. The weakly nonlinear analysis is performed to determining the bifurcation diagrams for these flame models, considering the acoustic-combustion interaction index n as the control parameter. The influence on the bifurcations diagram of the time delays (τ) and of the damping coefficient (ζ) is analyzed.

하이브리드로켓 연소실의 와류발생과 연소압력 진동

The similarity in internal flow of solid and hybrid rocket suggests that hybrid rocket combustion can be susceptible to instability due to vortex sheddings and their interaction. This study focuses on the evolution of interaction of vortex generated in pre-chamber with other types of vortex in the combustor and the change of combustion characteristics. Baseline and other results tested with disks show that there are five different frequency bands appeared in spectral domain. These include a frequency with thermal lag of solid fuel, vortex shedding due to obstacles such as forward, backward facing step and wall vortices near surface. The comparison of frequency behavior in the cases with disk 1 and 3 reveals that vortex shedding generated in pre-chamber can interact with other types of vortex shedding at a certain condition. The frequency of Helmholtz mode is one of candidates resulting to a resonance when it was excited by other types of oscillation even if this mode was not discernable in baseline test. This selective mechanism of resonance may explain the reason why non-linear combustion instability occurs in hybrid rocket combustion.

Velocity Coupling from Pulse-Test Firings

There has been much dispute about triggering of non-linear combustion instability(CI). The norm has been to use ad hoc models to include second order combustion terms most notably velocity coupling. This term arose due to classical linear analyses predicting stable motors necessitating another driving function. However there is no practical universally accepted experimental method that can obtain the velocity coupling response function. The solid rocket motor (SRM) designer is left to his own devices as to what the contribution of velocity coupling may be. This has led to much confusion and rocket motors that have been predicted stable but in practice were not. Flandro and co workers provide an alternative interpretation into the cause of non-linear combustion instability by including rotational terms into the linear analysis. This has been shown to improve the linear stability prediction by predicting unstable systems i.e. CI is always present but at low amplitudes growing too slowly to form a steep fronted wave. A pulse in this context needs to be of high enough amplitude to induce rapid growth. No criterion exists to predict such an amplitude. This study uses a newly developed analysis methodology using tubular grain SRMs to investigate both interpretations i.e. a linearly stable system with velocity coupling or linearly unstable system. This method when combined with response data can be used to obtain a velocity coupling response function from pulse tests. From this analysis it was possible to obtain reasonable velocity coupling response function. When using these constants to predict it was possible to predict a trigger point. Interestingly this point changes significantly with the initial higher order mode amplitudes. It then becomes vital to understand the likely wave structure of a pulse and what influence it has on the triggering amplitude. The system dynamics did not follow that of the experimental data but this may be due to truncation errors and assumptions made in the mathematical model. However the limiting amplitude is predicted correctly. The Flandro interpretation can be evaluated using this new method and applying a systematic study to evaluate criteria for the triggering amplitude. The linearly unstable values seem to capture the system dynamics better. However, in theory any pulse should cause CI. Regardless whether it is true or perceived triggering there seems to be an amplitude value at which non-linear CI is induced. The boundary layer pumping term here seems to be the key to interpreting triggering. The boundary layer acts as a piston that drives the acoustic oscillations when in phase. Thus for non-linear CI to occur the pulse must be of such an amplitude to cause the boundary layer to oscillate and maintain a high enough amplitude until the boundary layer starts to oscillate in phase. Regardless of interpretation this method shows great promise in allowing SRM designers to obtain quantitative response data and investigate triggering quickly and cheaply.

Nonlinear combustion instability analysis based on the flame describing function applied to turbulent premixed swirling flames

Instability analysis of swirling flames is of importance in the design of advanced combustor concepts for aircraft propulsion and powerplant for electricity production. Thermoacoustic instabilities are analyzed here by making use of a nonlinear representation of flame dynamics based on a describing function. In this framework, the flame response is determined as a function of frequency and amplitude of perturbations impinging on the combustion region. This model is adapted to the case of confined swirling flames comprising an upstream manifold, an injection unit equipped with a swirler and a cylindrical flame tube. The flame describing function is experimentally determined and is combined with an acoustic transfer matrix representation of the system to provide growth rates and oscillation frequencies as a function of perturbation amplitude. These data can be used to determine regions of instability, frequency shifts with respect to the acoustic eigenfrequencies and they also yield amplitude levels when self-sustained oscillations of the system have reached a limit cycle. This equilibrium is obtained when the amplitude dependent growth rate equals the damping rate in the system. This requires an independent determination of this last quantity which is here based on measurements of the combustor resonance response curve, together with numerical estimates of the flame contribution to the system response. The geometrical parameters of the upstream manifold and flame tube are varied and the corresponding operating regimes are compared with those predicted with the FDF framework. The present demonstration of the FDF framework in a generic configuration indicates that this can be used in more general situations of technological interest.

Prediction of unsteady non-linear combustion instability in solid rocket motors

The influence of the surrounding motor structure on the appearance of unsteady non-linear axial combustion instability symptoms in a solid rocket motor internal combustion chamber is investigated. The numerical simulation model consists of three coupled physical components, including the combined propellant and motor casing structure, the motor core fluid flow, and the corresponding propellant combustion. A transient frequency-dependent burning-rate model and an updated erosive burning-rate component model are incorporated in the current overall numerical simulation model. The results of simulating a hot-flow unsteady motor firing indicate that structural vibrations directly affect the burning rate and the axial pressure wave development within the chamber that appears as a primary symptom of combustion instability. Comparison of the predicted results with the experimental test results indicates that a good correlation exists between the two, providing support for the present simulation model.

A unified framework for nonlinear combustion instability analysis based on the flame describing function

Analysis of combustion instabilities relies in most cases on linear analysis but most observations of these processes are carried out in the nonlinear regime where the system oscillates at a limit cycle. The objective of this paper is to deal with these two manifestations of combustion instabilities in a unified framework. The flame is recognized as the main nonlinear element in the system and its response to perturbations is characterized in terms of generalized transfer functions which assume that the gain and phase depend on the amplitude level of the input. This ‘describing function’ framework implies that the fundamental frequency is predominant and that the higher harmonics generated in the nonlinear element are weak because the higher frequencies are filtered out by the other components of the system. Based on this idea, a methodology is proposed to investigate the nonlinear stability of burners by associating the flame describing function with a frequency-domain analysis of the burner acoustics. These elements yield a nonlinear dispersion relation which can be solved, yielding growth rates and eigenfrequencies, which depend on the amplitude level of perturbations impinging on the flame. This method is used to investigate the regimes of oscillation of a well-controlled experiment. The system includes a resonant upstream manifold formed by a duct having a continuously adjustable length and a combustion region comprising a large number of flames stabilized on a multipoint injection system. The growth rates and eigenfrequencies are determined for a wide range of duct lengths. For certain values of this parameter we find a positive growth rate for vanishingly small amplitude levels, indicating that the system is linearly unstable. The growth rate then changes as the amplitude is increased and eventually vanishes for a finite amplitude, indicating the existence of a limit cycle. For other values of the length, the growth rate is initially negative, becomes positive for a finite amplitude and drops to zero for a higher value. This indicates that the system is linearly stable but nonlinearly unstable. Using calculated growth rates it is possible to predict amplitudes of oscillation when the system operates on a limit cycle. Mode switching and instability triggering may also be anticipated by comparing the growth rate curves. Theoretical results are found to be in excellent agreement with measurements, indicating that the flame describing function (FDF) methodology constitutes a suitable framework for nonlinear instability analysis.

Nonlinear rocket motor stability prediction: Limit amplitude, triggering, and mean pressure shift

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 origin of the DC shift is revealed.

Reply by the Authors to V. S. Burnley and F. E. C. Culick

2Burnley, V. S., Swenson, G., and Culick, F. E. C., “Pulsed Instabilities in Combustion Chambers,” 31st AIAA/ASME/SAE/ASEE Joint Propulsion Conference, San Diego, CA, AIAA, Washington, DC, 1995; also AIAA Paper 95-2430, 1995. 3Burnley, V. S., “Nonlinear Combustion Instabilities and Stochastic Sources,” Ph.D. Dissertation, California Inst. of Technology, Pasadena, CA, 1996. 4Wicker, J. M., Greene, W. D., Kim, S.-I., and Yang, V., “Triggering of Longitudinal Combustion Instabilities in Rocket Motors: Nonlinear Combustion Response,” Journal of Propulsion and Power, Vol. 12, No. 6, 1996, pp. 1148–1158. 5Jahnke, C. C., and Culick, F. E. C., “An Application of Dynamical Systems Theory to Nonlinear Combustion Instabilities,” Journal of Propulsion and Power, Vol. 10, No. 4, 1994, pp. 508–517. 6Kim, S. I., “Nonlinear Combustion Instabilities in Combustion Chambers,” Ph.D. Dissertation, Dept. of Mechanical Engineering, Pennsylvania State Univ., University Park, PA, May 1989. 7Greene, W. D., “Triggering of Longitudinal Combustion Instabilities in RocketMotors,”M.S.Thesis, Dept. ofAerospace Engineering,Pennsylvania State Univ., University Park, PA, Dec. 1990. 8Ma, Y., Van Moorhem,W. K., and Shorthill,R. W., “Experimental Investigation of Velocity Coupling in Combustion Instability,” Journal of Propulsion and Power, Vol. 7, No. 5, 1991, pp. 692–699.

Nonlinear Stability Testing of Full-Scale Tactical Motors

The U.S. Naval Air Warfare Center has participated in a program to develop an improved understanding of linear and nonlinear combustion instability in solid propellant rocket motors. One goal of this program was to develop a systematic database of motor stability data. This paper describes the nonlinear aspects of the motor e rings and analysis. The motors used had diameters of 127 mm and were 1.7 m long. The majority were loaded with an 88% solids reduced-smoke ammonium perchlorate propellant with a nominal burning rate of 6.1 mm/s at 6.9 MPa. Motor pressures ranged from 3.45 to 10.34 MPa and various grain geometries were tested. In addition, motors have been e red that contain 1% 8- mm aluminum oxide, 90- mm aluminum oxide, and 3- mm zirconium carbide as stability additives in place of 1% ammonium perchlorate. This paper discusses experimental methods of motor pulsing and various theoretical approaches to predict pulse amplitudes in motors. The paper also examines nonlinear acoustic motor response to various amplitudes of acoustic pulsing and the resultant characteristics of sustained nonlinear oscillations. Finally, some theoretical interpretations are presented from the experimental motor data. The results show that there is a direct relationship between the dc pressure shift and the magnitude of the acoustic oscillations.

Stability Testing of Full-Scale Tactical Motors

The U.S. Naval Air Warfare Center has participated in a program to develop an improved understanding of linear and nonlinear combustion instability in solid propellant rocket motors. One goal of this program was to develop a systematic database of motor and stability data. This paper describes the linear aspects of the motor e rings and analysis. The motors that were used had diameters of 127 mm and were 1.7 m long. The majority were loaded with an 88% solids reduced-smoke ammonium perchlorate propellant with a nominal burning rate of 6.1 mm/s at 6.9 MPa. In addition, motors have been e red that contain 1% 8- mm aluminum oxide, 90- mm aluminum oxide, and 3- mm zirconium carbide as stability additives in place of 1% ammonium perchlorate. Motor pressures ranged from 3.45 to 10.34 MPa and various grain geometries were tested. Pressure-coupled combustion response measurements were made at the nominal motor operating pressures. Motor performance and stability calculations were made using the Air Force Solid Performance Program and the Standard Stability Prediction Program for the motor cone gurations that were e red. The stability predictions were compared to the data obtained from the motor e rings. The results indicate that it is possible to theoretically predict linear motor stability for the class of motors discussed in this paper. It was also learned that higher-pressure motors tend to be less

Application of dynamical systems theory to nonlinear combustion instabilities

Two important approximations have been incorporated in much of the work with approximate analysis of unsteady motions in combustion chambers: truncation of the series expansion to a finite number of modes, and time averaging. A major purpose of the analysis reported in this paper has been to investigate the limitations of those approximations. In particular two fundamental problems of nonlinear behavior are discussed: the conditions under which stable limit cycles of a linearly unstable system may exist; and conditions under which bifurcations of the limit cycle may occur. A continuation method is used to determine the limit cycle behavior of the equations representing the time dependent amplitudes of the longitudinal acoustic modes in a cylindrical combustion chamber. The system includes all linear processes and second-order nonlinear gas dynamics. The results presented show that time averaging works well only when the system is, in some sense, only slightly unstable. In addition, the stability boundaries predicted by the two-mode approximation are shown to be artifacts of the truncation of the system. Systems of two, four, and six modes are analyzed and show that more modes are needed to analyze more unstable systems. For the six-mode approximation with an unstable second mode two bifurcations are found to exist. A pitchfork bifurcation causes a new branch of limit cycles to exist in which the odd acoustic modes are excited. This new branch of limit cycles then undergoes a torus bifurcation that causes the system to exhibit stable quasi-periodic motions.