Abstract: This report describes the results of a study sponsored by the Keck Institute for Space Studies (KISS) to investigate the feasibility of identifying, robotically capturing, and returning an entire Near-Earth Asteroid (NEA) to the vicinity of the Earth by the middle of the next decade. The KISS study was performed by people from Ames Research Center, Glenn Research Center, Goddard Space Flight Center, Jet Propulsion Laboratory, Johnson Space Center, Langley Research Center, the California Institute of Technology, Carnegie Mellon, Harvard University, the Naval Postgraduate School, University of California at Los Angeles, University of California at Santa Cruz, University of Southern California, Arkyd Astronautics, Inc., The Planetary Society, the B612 Foundation, and the Florida Institute for Human and Machine Cognition. The feasibility of an asteroid retrieval mission hinges on finding an overlap between the smallest NEAs that could be reasonably discovered and characterized and the largest NEAs that could be captured and transported in a reasonable flight time. This overlap appears to be centered on NEAs roughly 7 m in diameter corresponding to masses in the range of 250,000 kg to 1,000,000 kg. To put this in perspective, the Apollo program returned 382 kg of Moon rocks in six missions and the OSIRIS-REx mission proposes to return at least 60 grams of surface material from a NEA by 2023. The present study indicates that it would be possible to return a ~500,000-kg NEA to high lunar orbit by around 2025. The idea of exploiting the natural resources of asteroids dates back over a hundred years, but only now has the technology become available to make this idea a reality. The feasibility is enabled by three key developments: the ability to discover and characterize an adequate number of sufficiently small near-Earth asteroids for capture and return; the ability to implement sufficiently powerful solar electric propulsion systems to enable transportation of the captured NEA; and the proposed human presence in cislunar space in the 2020s enabling exploration and exploitation of the returned NEA. Placing a 500-t asteroid in high lunar orbit would provide a unique, meaningful, and affordable destination for astronaut crews in the next decade. This disruptive capability would have a positive impact on a wide range of the nation’s human space exploration interests. It would provide a high-value target in cislunar space that would require a human presence to take full advantage of this new resource. It would offer an affordable path to providing operational experience with astronauts working around and with a NEA that could feed forward to much longer duration human missions to larger NEAs in deep space. It would provide an affordable path to meeting the nation’s goal of sending astronauts to a near-Earth object by 2025. It represents a new synergy between robotic and human missions in which robotic spacecraft retrieve significant quantities of valuable resources for exploitation by astronaut crews to enable human exploration farther out into the solar system. A key example of this is that water or other material extracted from a returned, volatile-rich NEA could be used to provide affordable shielding against galactic cosmic rays. The extracted water could also be used for propellant to transport the shielded habitat. These activities could jump-start an entire in situ resource utilization (ISRU) industry. The availability of a multi-hundred-ton asteroid in lunar orbit could also stimulate the expansion of international cooperation in space as agencies work together to determine how to sample and process this raw material. The capture, transportation, examination, and dissection of an entire NEA would provide valuable information for planetary defense activities that may someday have to deflect a much larger near-Earth object. Finally, placing a NEA in lunar orbit would provide a new capability for human exploration not seen since Apollo. Such an achievement has the potential to inspire a nation. It would be mankind’s first attempt at modifying the heavens to enable the permanent settlement of humans in space. The report that follows outlines the observation campaign necessary to discover and characterize NEAs with the right combination of physical and orbital characteristics that make them attractive targets for return. It suggests that with the right ground-based observation campaign approximately five attractive targets per year could be discovered and adequately characterized. The report also provides a conceptual design of a flight system with the capability to rendezvous with a NEA in deep space, perform in situ characterization of the object and subsequently capture it, de-spin it, and transport it to lunar orbit in a total flight time of 6 to 10 years. The transportation capability would be enabled by a ~40-kW solar electric propulsion system with a specific impulse of 3,000 s. Significantly, the entire flight system could be launched to low-Earth orbit on a single Atlas V-class launch vehicle. With an initial mass to low-Earth orbit (IMLEO) of 18,000 kg, the subsequent delivery of a 500-t asteroid to lunar orbit represents a mass amplification factor of about 28-to-1. That is, 28 times the mass launched to LEO would be delivered to high lunar orbit, where it would be energetically in a favorable location to support human exploration beyond cislunar space. Longer flight times, higher power SEP systems, or a target asteroid in a particularly favorable orbit could increase the mass amplification factor from 28-to-1 to 70-to-1 or greater. The NASA GRC COMPASS team estimated the full life-cycle cost of an asteroid capture and return mission at ~$2.6B.

ID: CaltechAUTHORS:20190213-143915193

]]>

Abstract: Measurements of the spatial dependence of fuel mixture fraction are made for a non-premixed jet flame in a combustion chamber with imposed acoustic oscillations at frequencies of 22-55 Hz. As part of a set of studies on combustion instabilities and the dynamical behavior of combustion systems, this work is intended to provide a basic understanding of the characteristics of mixing under imposed acoustic oscillations. Infrared laser absorption and phase-resolved acetone PLIF are used to measure the fuel mixture fraction throughout the flow field. The degree of fuel/air mixing is then calculated from the measurements in terms of unmixedness factor, in both temporal and spatial respects. Results show that the acoustic excitation causes oscillations in fuel/air mixing at the driving frequency, which results in oscillatory flame behavior in the flame region. The unmixedness factors for the reacting flow cases exhibit greater overall magnitudes than the cold flow cases, which means that mixing becomes less effective in the presence of flame. Also the degree of mixing decreases with increasing frequency for reacting cases, while, for the cold flows, the mixing tends to be enhanced with frequency.

ID: CaltechAUTHORS:20110208-092444615

]]>

Abstract: Measurements of fuel mixture fraction are made for a jet flame in an acoustic chamber. Acoustic forcing creates a spatially-uniform, temporally-varying pressure field which results in oscillatory behavior in the flame . Forcing is at 22,27, 32, 37, and 55 Hz. To asses the oscillatory behavior, previous work included chemiluminescence, OH PUF, nitric oxide PUF imaging, and fuel mixture fraction measurements by infrared laser absorption. While these results illuminated what was happening to the flame chemistry, they did not provide a complete explanation as to why these things were happening. In this work, the fuel mixture fraction is measured through PUF of acetone, which is introduced into the fuel stream as a marker. This technique enables a high degree of spatial resolution of fuel/air mixture value. Both non-reacting and reacting cases were measured and comparisons are drawn with the results from the previous work. It is found that structure in the mixture fraction oscillations is a major contributor to the magnitude of the flame oscillations.

ID: CaltechAUTHORS:20110208-083707180

]]>

Abstract: Perhaps the most curious aspect of the Wright Brothers' program to invent and commercialize the airplane is their decision in 1900 to use their novel canard configuration, and to persist with that geometry until 1910 despite the known deficiency that the aircraft were unstable in pitch. The reasons for their initial choice are well-known. Several studies in the part twenty years have proven beyond doubt that the Wrights did not intentionally make their canards unstable. The pitch instability of their machine was an unwitting byproduct of their design chosen partly out of fear of the conventional design and partly (they reasoned) for more positive control. With their great emphasis on control, the Wrights were able to develop a successful aircraft, albeit difficult to fly additionally because the 1903 aircraft also possessed a fast spiral instability. A canard design is not necessarily unstable, but owing chiefly to their airfoil, and an unfortunate fore-and-aft mass distribution, the Wright canards were all unstable. Though easier to fly, their 1909 aircraft was more unstable than the famous 1903 Flyer and the Brothers did not have a stable design until they finally adopted a conventional aft horizontal tail in 1910. Successful control of the canard aircraft depended heavily on large damping-in-pitch. The purpose of this paper is to apply modern analysis of flight mechanics to trace the detailed flying characteristics of their powered aircraft from 1903 to 1910 when they finally gave up the canard. Its a story in which technology, stubborness and commercialization are intimately mingled; we are concerned here only with the technology.

No.: AIAA-2003-0097
ID: CaltechAUTHORS:20101118-085016796

]]>

Abstract: Various techniques have been employed by investigators to measure the response of flames to unsteady changes, but there has been no systematic study of the relative benefits and drawbacks of these competing techniques. The goal of this work is to characterize the performance of two different measurement techniques applied in three ways and to examine the differing insights they offer for the response of a flame in a periodic acoustic field. The burner configuration consists of a jet flame in a partial enclosure that stabilizes the flame approximately 8 cm above the jet exit. This burner is installed in an acoustic chamber that has actively-controlled, frequency-selectable, acoustic forcing. Flame response data for different frequencies obtained with chemiluminescence, OR PLIF, and NO PLIF measurement techniques is the basis for this work. Analysis of the data shows the complexity of the measurement required to achieve a given level of understanding of the flame's behavior. The usefulness of these techniques in flame response measurements individually and taken in combination is discussed, with examples.

ID: CaltechAUTHORS:20101118-080941990

]]>

Abstract: The California Institute of Technology's Combustion Acoustics Facility is used to measure the changes in the creation of NO in a partially premixed jet flame due to acoustic forcing at frequencies ranging from 22 to 55 Hz. The facility generates a quarter-wave mode so that the test flame is in a region where the acoustic velocity is nearly zero. This facility and a similar burner have been previously used to measure the phase-resolved response of the OH field. In this experiment, phase-resolved NO planar laserinduced fluorescence (PLIF) measurements are recorded. The location and phase coupling of the NO field are analyzed, and the production and transport of NO are compared with previously reported OH field measurements. The NO levels increase for frequencies that exhibit stronger acoustic coupling to the flame. The NO concentration field variations lead (in phase space) the OH field variations. This is probably a result of the greater NO sensitivity to temperature (which itself is closely coupled to the chamber pressure).

No.: AIAA-2002-0195
ID: CaltechAUTHORS:20101117-112555177

]]>

Abstract: Future combustors designed for better efficiency and lower pollutant emission are expected to operate closer to their stability boundary, thereby increasing the risk of encountering combustion instability. Onset of combustion instability leads to limit cycle oscillations in the acoustical fluctuations that can often reach amplitudes large enough to cause severe damage. Active control strategies are, therefore, being considered to prevent combustion instabilities, but their development requires nonlinear models that can faithfully capture the combustor system dynamics. A framework for the approximate analysis of the nonlinear acoustics in a combustion chamber exists, which includes all relevant linear contributions and also second order gasdynamic nonlinearities. Nonlinear combustion effects in the form of pressure and velocity coupling models have also been incorporated into the analysis with the aim of capturing the phenomenon of triggered instability, where the acoustical fluctuations are linearly stable to small perturbations, but show a limit cycle behavior for larger perturbations. However, several questions such as those relating to 1) modal truncation of the equations for the acoustic dynamics, 2) absence of triggered limit cycles in the formulation with only second order gasdynamic nonlinearities, and 3) the form of the velocity coupling function, including the need for a threshold character, have not been satisfactorily resolved. In this paper, we address some of these questions on modeling and dynamics of acoustic waves in combustion chambers, using the approximate analysis, that have remained unanswered over the years.

No.: 2002-3592
ID: CaltechAUTHORS:20110204-114636697

]]>

Abstract: Transient behavior of combustion systems has long been a subject of both fundamental and practical concerns. Extreme cases of very rapid changes include the ignition of reacting mixtures and detonation. At the other extreme is a wide range of quasi-steady changes of behavior, for example adjustments of the operating point of a combustion chamber. Between the limiting cases of 'infinitely fast' and 'infinitesimally slow' lie important fundamental problems of time-dependent behavior and a wide array of practical applications. Among the latter are combustion instabilities and their active control, a primary motivation for the work reported in this paper. Owing to the complicated chemistry, chemical kinetics and flow dynamics of actual combustion systems, numerical simulations of their behavior remains in a relatively primitive state. Even as that situation continually improves, it is an essential part of the field that methods of measuring true dynamical behavior be developed to provide results having both fine spatial resolution and accuracy in time. This paper is a progress report of recent research carried out in the Jet Propulsion Center of the California Institute of Technology.

ID: CaltechAUTHORS:20110208-081441006

]]>

Abstract: The California Institute of Technology's Combustion Acoustics Facility is used to measure the response of a partially premixed jet flame to acoustic forcing at frequencies ranging from 22 Hz to 55 Hz. The facility generates bulk acoustic modes that simulate unstable combustor conditions. This same facility and burner has been previously used to measure the phase-resolved response of the OH PLIF field. In this experiment, phase-resolved chemiluminescence measurements are recorded and analyzed. Flame base oscillations are quantified and compared for two different burner configurations. The chemiluminescence also shows that frequencies that exhibit stronger acoustic coupling to the flame tend to have decreased luminosity in the flame stabilization zone, while frequencies with weaker coupling tend to produce greater luminosity at the base of the flame.

No.: AIAA-2002-0194
ID: CaltechAUTHORS:20101115-105224925

]]>

Abstract: Active control of longitudinal pressure oscillations in a combustion chamber is studied theoretically by means of a low order model obtained by systematic reduction from a complete representation. The formulation is based on the derivation of a generalized wave equation that accommodates the effects of mean flow, combustion, noise and control action. By using spatial averaging, the equations describing the dynamics of the chamber are reduced to a set of coupled ordinary differential equations, representing the motions of a system of coupled oscillators. The form of the resulting equations is particularly convenient for model reduction and for introducing feedback control terms, while retaining all physical processes. The oscillator equations are then rewritten in state-space form. Simulations are carried out to investigate in a unified fashion various aspects of the problem. These include the influences of noise, parameter uncertainties, unmodeled modes and a single timedelay. A criterion is derived that guarantees stability of the controlled closed-loop system in the presence of those quantities. The particular controller used here is based on a standard LQR design, but any design technique can be used as long as the stability criterion is fulfilled.

No.: 2000-3124
ID: CaltechAUTHORS:20101115-085253275

]]>

Abstract: Combustion instabilities exist as consequences of interactions among three classes of phenomena: chemistry and chemical dynamics; combustion dynamics; and combustor dynamics. These dynamical processes take place simultaneously in widely different spatial scales characterized by lengths roughly in the ratios (10^(-3) - 10^(-6)):1:(10^3-10^6). However, due to the wide differences in the associated characteristic velocities, the corresponding time scales are all close. The instabilities in question are observed as oscillations having a time scale in the range of natural acoustic oscillations. The apparent dominance of that single macroscopic time scale must not be permitted to obscure the fact that the relevant physical processes occur on three disparate length scales. Hence, understanding combustion instabilities at the practical level of design and successful operation is ultimately based on understanding three distinct sorts of dynamics.

No.: AIAA-2000-3178
ID: CaltechAUTHORS:20101115-083329034

]]>

Abstract: A modified Price-Boggs-Derr model is applied to compute the linear and non-linear combustion response properties of monopropellant ammonium perchlorate. The kinetics constants were changed to achieve good agreement with response function data as well as with steady-state data. The numerical method was first validated by comparing computations using the Levine & Culick boundary condition in the limit of small perturbations with the exact mathematical solution for linear response. Then, using the AP model for the boundary condition, various linear and non-linear computations were performed. Supplemental mathematical analyses relate the AP model to the basic two parameters of the classical theory and show the key factors determining the nature of the combustion response.

No.: AIAA-2000-3694
ID: CaltechAUTHORS:20101115-101901737

]]>

Abstract: A technique has been devised which can provide insight into the local dynamic response of a flame to an acoustic field. In the experiments, a test chamber is acoustically excited by a pair of low frequency drivers. The response of the flame is visualized by planar laser-induced fluorescence (PLIF) of the hydroxyl (OH) radical, which is a good indicator for heat release in the flame. The resulting images are phase-locked and averaged to yield a qualitative picture of the fluctuation of the heat release. This is correlated with a pressure transducer near the flame, which allows stability to be evaluated using Rayleigh's criterion. Results indicate that the drive frequency and burner configuration have a pronounced effect on the response of the flame. Drive frequencies ranging from 22 Hz to 55 Hz are applied to the jet mixed burner, supplied with a premixed 50/50 mixture of methane and carbon dioxide at a Reynolds number of 20,000. The burner is operated in two configurations; with an aerodynamically stabilized flame, and with a flame stabilized by two protruding bluff-bodies. Results indicate that in general, the bluff-body stabilized flame is less sensitive to chamber acoustic excitation

No.: AIAA-2000-3123
ID: CaltechAUTHORS:20101115-074125126

]]>

Abstract: Considerable data exists suggesting that the response functions for many solid propellants tend to have higher values, in some ranges of frequencies, than predicted by the conventional QSHOD theory. It is a familiar idea that such behavior is associated with dynamical processes possessing characteristic times shorter than that of the thermal wave in the condensed phase. The QSHOD theory, and most of its variants, contains only the dynamics of that process, which normally has a characteristic frequency in the range of a few hundred hertz. Two previous works seeking to correct this deficiency (T'ien, 1972; Lazima and Clavin, 1992) have focused their attention on including the dynamics of the thermal wave in the gas phase. Both include effects of diffusion that complicate the analysis although the second achieves some simplification by applying the ideas of 'activation energy asymptotics'. While their results differ in detail, both works show influences at frequencies higher than those near the broad peak of the response due to the thennal wave. Recent theoretical work and simulations show that a combustion response function based on simple pressure coupling is not enough to explain the characteristics of the instability observed experimentally. Namely, differences in the shape of the response function fail to reproduce the differences observed experimentally in the characteristics of the limit cycle reached by combustion chambers with propellants of different chemical (or physical) composition. On the other hand, velocity coupling in the combustion response seems a promising mechanism able to predict the changes in the unstable modes observed experimentally and to produce considerable effect on the shape of the resulting limit cycle. The Baum and Levine model is used as a starting point in the investigation of velocity coupling. Other models, in which the mass burning rate is modified by some function of the velocity, are also investigated through direct time-simulation and by the use of a continuation method. Modeling of particle damping at high frequency constitutes a serious consideration in the modeling of the interaction of combustion dynamics and chamber acoustics. The effect of particle size distribution is analyzed by considering an experimental particle size distribution. The ultimate goal of this work is to find a link between the global dynamics of the combustion chamber and small changes in the combustion dynamics, caused by differences in propellant chemical composition or physical characteristics (for example, particle size and distribution). Response functions are shown for realistic ranges of the chief parameters characterizing the dynamics of the propellant. The results are also incorporated in the dynamical analysis of a small rocket motor to illustrate the consequences of the combustion dynamics for the stability and nonlinear behavior of unsteady motions in a motor.

No.: AIAA-2000-3187
ID: CaltechAUTHORS:20101115-094319661

]]>

Abstract: A non-steady model for the combustion of ammonium perchlorate composite propellants has been developed in order to be incorporated into a comprehensive gasdynamics model of solid rocket motor flow fields. The model including the heterogeneous combustion and turbulence mechanisms is applied to nonlinear combustion instability analyses. This paper describes the essential mechanisms and features of the model and discusses the methodology of non-steady calculations of the combustion instabilities of solid rocket motors.

No.: AIAA-99-2804
ID: CaltechAUTHORS:20101118-132511549

]]>

Abstract: This paper is a report of work in progress, part of the Caltech MURI Program: Novel Energetic Materials to Stabilize Rocket Motors. The primary technical objective of the MURI Program is to understand the connections between propellant composition and chemistry, and the dynamical behavior observed in solid propellant rocket motors. Here we are concerned with the theoretical framework in which chamber dynamics are investigated; and certain aspects of combustion dynamics represented by the response function which is ultimately the macroscopic realization of the propellant chemistry and combustion. Some results are given to illustrate possible influences of the frequency spectrum of the response function on linear and nonlinear motions in a solid rocket. A simple model is described which is extended eventually to provide a way to model phenomenologically some of the observed characteristics of the combustion dynamics of a burning solid propellant.

No.: 98-3704
ID: CaltechAUTHORS:20101118-144631059

]]>

Abstract: A numerical analysis of unsteady motions in solid rocket motors with a nozzle has been conducted. The formulation treats the complete conservation equations for the gas phase and the one-dimensional equations in the radial direction for the condensed phase. A fully coupled implicit scheme based on a dual time-stepping integration algorithm has been adopted to solve the governing equations and associated boundary conditions. After obtaining a steady state solution, periodic pressure oscillations are imposed at the head end to simulate acoustic oscillations of a traveling-wave motion in the combustion chamber. The amplitude of the pressure oscillation is 1.0 % of the mean pressure and the frequency is 1790 Hz, corresponding to the twice of the fundamental frequency of the chamber. Magnitude and phase of pressure and axial velocity fluctuations are influenced by the upstream reflecting wave from the nozzle wall. Axial velocity near surface region oscillates in phase advance manner with reference to the acoustic pressure. Large vorticity fluctuations are observed in near surface region. The mass-flow-rate at the nozzle exit periodically oscillates with a time delay compared to the imposed pressure oscillations at the head end.

No.: 98-0253
ID: CaltechAUTHORS:20101118-141630231

]]>

Abstract: A numerical analysis of unsteady motions in solid rocket motors has been conducted. The formulation considers a 2-D axisymmetric combustion chamber and a choke nozzle, and treats the complete conservation equations accounting for turbulence closure and finiterate chemical kinetics in the gas phase and subsurface reactions. A fully coupled implicit scheme based on a dual time-stepping integration algorithm has been adopted to solve the governing equations and associated boundary conditions. Results of the steady-state calculations indicate that the distributions of pressure in the motor and Mach number in the nozzle are one-dimensional along the axial direction. Vorticity contours show similar pattern to those of Mach number in the combustion chamber. The nozzle has an influence on the flow and temperature fields in the combustion chamber. A narrow pressure pulse is imposed at the head end to simulate unsteady acoustic oscillations in the combustion chamber. When the front of the pulse reaches near the nozzle throat, pressure near the nozzle throat increases and blocks the hot gas flow from passing through the nozzle throat. Self-generated oscillations have similar frequencies to those of standing waves of the combustion chamber. Large vorticity fluctuations are observed in near surface region. The luminous flame zone responds to low-frequency pressure wave rather than highfrequency one. Temperature fluctuations in the primary flame zone of the head end oscillates independently of the imposed pressure oscillations while temperature fluctuations in downstream region show pressure-dependent oscillations.

ID: CaltechAUTHORS:20101119-074948591

]]>

Abstract: Chiefly for improved efficiency, the trend to increasing use of gas turbine engines in stationary powerplants has been firmly established. The requirement for minimum NOx production has motivated operation as close as practically possible near the lean flammability limit, to reduce flame temperatures and consequently reduce formation of nitrogen oxides via the Zeldovich thermal mechanism. However, experience has shown that under these conditions, stability of the chamber is compromised, often leading to the presence of sustained oscillations in the combustor. That possibility raises the problem of the influence of oscillatory motions on the production of nitrogen oxides. Numerically calculating these influences for a complex geometry gas turbine combustor is too computationally expensive at this ?me. Nonlinear analytical methods making use of these influences are a promising direction for simplei ways to design and develop operational gas turbine combustors. However, this analysis needs results on which to base unsteady models of the interaction between nonlinear oscillations and species production within a gas turbine combustor. In this paper, two methods are explored briefly as an initial step. The first is based on a configuration of perfectly stirred and plug flow reactors to approximate the flow in a combustion chamber. A complete representation of the chemical processes is accommodated, but the geometry is simplified. The second is a full numerical simulation for a realistic geometry, but at this stage the chemistry is simplified.

ID: CaltechAUTHORS:20101119-141141495

]]>

Abstract: A pulsed or "triggered" instability occurs when pressure oscillations develop in a linearly stable combustion system after being subjected to a sufficiently large disturbance. Such true nonlinear instabilities usually occur as subcritical bifurcations in dynamical systems theory. Understanding which nonlinear processes can lead to subcritical bi~urcations is the focus of this work. Earlier work with the approximate analysis used here has shown convincingly that nonlinear acoustics alone does not contain the phenomenon of pulsed instabilities; evidently some other nonlinear contribution must also be included. An extensive experimental and numerical investigation conducted by Baum and Levine strongly suggests that nonlinear combustion is required. Using models of pressure and velocity coupling, the current work studies the effect of nonlinear combustion on the behavior of the system.

No.: AIAA-95-2430
ID: CaltechAUTHORS:20101119-133446852

]]>

Abstract: The second order nonlinear longitudinal acoustics in a cylindrical combustion chamber are studied for the case of an unstable second mode. A modal analysis is undertaken and a continuation method is used to determine the limit cycle behavior of the time dependent amplitudes of the acoustic modes as functions of the linear stability of the unstable acoustic mode. It is shown that if an insufficient number of modes are included in the truncated system, bifurcations of the primary limit cycle occur. The energy in the limit cycles is analyzed and the bifurcations are shown to occur as a means of increasing the amount of energy transfer out of the unstable acoustic mode and into the stable acoustic modes through the nonlinear terms.

No.: 94-3190
ID: CaltechAUTHORS:20101123-135307146

]]>

Abstract: 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.

No.: 93-0114
ID: CaltechAUTHORS:20101122-100414304

]]>

Abstract: N/A

ID: CaltechAUTHORS:20110207-085016646

]]>

Abstract: 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.

No.: 90-3927
ID: CaltechAUTHORS:20101129-112004053

]]>

Abstract: Dynamical systems theory has been used to study the nonlinear dynamics of the F-14. An eight degree of freedom model that does not include the control system present in operational F-14's has been analyzed. The aerodynamic model, supplied by NASA, includes nonlinearities as functions of the angles of attack and sideslip, the rotation rate, and the elevator deflection. A continuation method has been used to calculate the steady states of the F-14 as continuous functions of the control surface deflections. Bifurcations of these steady states have been used to predict the onset of wing rock, spiral divergence, and jump phenomena which cause the aircraft to enter a spin. A simple feedback control system was designed to eliminate the wing rock and spiral divergence instabilities. The predictions were verified with numerical simulations.

No.: AIAA-90-0221
ID: CaltechAUTHORS:20101130-140634171

]]>

Abstract: This paper consists of two parts summarizing two portions of the ONR/NAVAIR Research Option. The option began in 1983 and continued for five years, involving 11 organizations. Simultaneously, similar or related programs supported by other agencies or institutions were being carried out in several other places. Results of those programs have been briefly summarized in five papers collected in a document to be published by C.P.L.A. This paper contains two of the five papers in that document. Here we cover the subjects of approximate analyses and stability; and large-scale structures and passive control. The first is concerned chiefly with an analytical framework constructed on the basis of observations; it is intended to provide a means of correlating and interpreting data, and predicting the stability of motions in a combustion chamber. The second is a summary of recent experimental work directed to understanding the flows in dump combustors of the sort used in modern ramjet engines. Much relevant material is not included here, but may be found in the remaining papers of the document cited above. For completeness, we note briefly the substance of those reports. In their summary "Spray Combustion Processes in Ramjet Combustion Instability," Bowman (Stanford), Law (University of California, Davis) and Sirignano (University of California, Irvine) review several aspects of spray combustion relevant to combustion instabilities. The objectives of the works were: (1) to determine the effect of spray characteristics on the energy release pattern in a dump combustor and the subsequent effects on combustion instability; (2) to gain a fundamental understanding of the coupling of the spray vaporization process with an unsteady flow field; and (3) to investigate methods for controlling and enhancing spray vaporization rates in liquid-fueled ramjets. During the past five years considerable progress has been made in applying methods of computational fluid dynamics to the flow in a dump combustor including consequences of energy release due to combustion processes. Jou has summarized work done at Flow Research, Inc. and at the Naval Research Laboratory in his paper "A Summary Report on Large-Eddy Simulations of Pressure Oscillations in a Ramjet Combustor." The serious effects of combustion instabilities on the inlets of ramjet engines were discovered in the late 1970's in experimental work at the Aeropropulsion Laboratory, Wright Field, the Naval Weapons Center and the Marquardt Company. The most thorough laboratory work on the unsteady behavior of inlets has been accomplished at the McDonnell-Douglas Research Laboratory by Sajben who has reviewed the subject in his paper "The Role of Inlet in Ramjet Pressure Oscillations."

ID: CaltechAUTHORS:20101203-091515515

]]>

Abstract: This paper is a summary of work carried out during the past five years as part of an ONR/NAVAIR Research Initiative. The effort has been devoted to developments and applications of an approximate analysis intended to provide understanding of observed behavior and guidelines for design and development. Although most of the work has been carried out for longitudinal modes, quite extensive results have been obtained for transverse modes in a circular chamber, for both second and third order nonlinear gasdynamics. Especially important are the useful conclusions based on the simplified two-mode approximation. Preliminary results obtained for the influences of stochastic sources suggest that some of the observed behavior of the amplitudes of oscillations may be attributed to random inputs, such as flow separation and turbulence. Some important aspects of the problem of existence and stability of limit cycles in linearly stable systems ("triggering") remain unclear.

ID: CaltechAUTHORS:20110207-095215895

]]>

Abstract: This paper summarizes work accomplished chiefly during the past five years on analysis of stability related to recent experimental results on combustion instabilities in dump combustors. The primary purpose is to provide the information in a form useful to those concerned with design and development of operational systems. Thus most substantial details are omitted and the material is presented in a qualitative fashion.

ID: CaltechAUTHORS:20110207-112022288

]]>

Abstract: An analytical analysis has been developed to investigate the behavior of unsteady transverse pressure oscillations in combustion chambers. The model extends the previous analysis for the second-order standing wave motions and accommodates spinning oscillations and third-order nonlinearities. The influences of various parameters and initial conditions on the limit cycles and triggering of pressure oscillations are discussed. Results indicate that the existence of spinning oscillations depends strongly on the number of modes included in the analysis and on the initial conditions. The third-order acoustics has little influence on the triggering of instability. It only modifies the characteristics of limiting amplitudes.

No.: AIAA-88-0152
ID: CaltechAUTHORS:20101203-144226944

]]>

Abstract: Dynamical systems theory has been used to study nonlinear aircraft dynamics. A six degree of freedom model that neglects gravity has been analyzed. The aerodynamical model, supplied by NASA, is for a generic swept wing fighter and includes nonlinearities as functions of the angle of attack. A continuation method was use to calculate the steady states of the aircraft, and bifurcations of these steady states, as functions of the control deflections. Bifurcations were used to predict jump phenomena and the onset of periodic motion for roll coupling instabilities and high angle of attack maneuvers. The predictions were verified with numerical simulations.

No.: AIAA-88-4372
ID: CaltechAUTHORS:20101203-102240185

]]>

Abstract: This is a summary of a lecture intended primarily as a progress report of the Los Angeles AIAA Section Wright Flyer Project. Begun in late 1978, the project is devoted chiefly to construction of two replicas of the 1903 Wright 'Flyer.' The first, now being covered, is an exact replica intended for full-scale wind tunnel tests. The second will be a flying replica, incorporating minimal modifications to produce a less unstable aircraft. Partly in preparation for the second aircraft, considerable attention has been given to the aerodynamics, stability, and control of the 1903 'Flyer.' Wind tunnel tests have been conducted with a 1/6 flexible model, and a 1/8 scale steel model tested at full-scale Reynolds numbers. The data have provided basis for analyzing both the closed-loop and open-loop performance of the aircraft. Another aspect of the Project has been concerned with the history of early aeronautics, especially as related to the Wright Brothers' work. Thus a significant portion of the lecture is given to aeronautical history both before and after 1903, to provide a better appreciation for the Wrights' achievements and a clearer perspective of their work in the context of aeronautical progress.

No.: AIAA 88-0094
ID: CaltechAUTHORS:20101220-104426212

]]>

Abstract: Combustion of reactants in a confined volume favors excitation of unsteady motions over a broad range of frequencies. A relatively small conversion of the energy released will produce both random fluctuations or noise, and, under many circumstances, organized oscillations generically called combustion instabilities. Owing to the high energy densities and low losses in combustion chambers designed for propulsion systems, the likelihood of combustion instabilities is high. The accompanying heat transfer to exposed surfaces, and structural vibrations are often unacceptable, causing failure in extreme cases. This paper is a brief review of combustion instabilities in liquid-fueled propulsion systems-rockets, ramjets, and thrust augmentors-with emphasis on work accomplished during the past decade. To provide a common framework for discussing the wide range of works, a theory of two-phase flow is reviewed as the basis for an approximate analysis of combustion instabilities. The analysis is directed primarily to treatment of linear stability; it is sufficiently general to accommodate all processes occurring in actual systems. A new result has been obtained for an extended form of Rayleigh's criterion and its relation to the growth constant for unstable waves. The chief mechanisms for combustion instabilities in liquid-fueled systems are reviewed, followed by a summary of the common methods of analysis and applications to the three classes of propulsion systems. Control of instabilities by passive and active means is examined briefly.

ID: CaltechAUTHORS:20110204-151853189

]]>

Abstract: A one-dimensional analytical model is presented for calculating the longitudinal acoustic modes of idealized "dump-type" ramjet engines. The model contains the matching required to place an oscillating flame sheet in the interior of a combustion chamber with mean flow. The linear coupling of the acoustic and entropy waves at the inlet shock, flame sheet, and exit nozzle along with acoustic admittances at the inlet and exit are combined to determine the stability of the system as well as the acoustic modes. Since the acoustic and entropy waves travel at different velocities, the geometry is a critical factor in determining stability. Typical values of the admittances will produce damped solutions when the entropy is neglected, but, as the ratio of the entropy to acoustic fluctuations is increased, the coupling can either feed acoustic energy into or out of different modes independently. This transfer of energy has a destabilizing or stabilizing effect on the acoustic modes of the system depending upon the phase of the energy transfer.

No.: AIAA-87-1872
ID: CaltechAUTHORS:20101202-105114390

]]>

Abstract: In August 1908, Wilbur Wright gave the first public demonstration of the Wright airplane, at Le Mans, France. Two weeks later Orville was the first to fly publicly a powered man-carrying aircraft in the United States, at Fort Myer, Virginia. Those astonishing flights were the beginnings of the Wrights' programs to fulfill the requirements of contracts the Brothers had made with the French and U.S. governments. At the time nobody else had a practical aircraft fully controllable and capable of being maneuvered at the pilot's command.

ID: CaltechAUTHORS:20110204-135939767

]]>

Abstract: An analytical analysis has been developed to investigate the behavior of unsteady motions in combustion chambers. The model accommodates the third-order nonlinear acoustics and a second-order combustion response. Ths influences of various linear and nonlinear parameters on the limit cycles and triggering of pressure osoillations are disoussed in detail. Results indicate that the third-order acoustics has little influence on the triggering of instability. It only affects the limiting amplitudes and the stability domains of limit cycles. The nonlinear combustion response plays an essential role in determining the characteristics of triggering.

No.: AIAA-87-1873
ID: CaltechAUTHORS:20101203-142326424

]]>

Abstract: A new approach for linear analysis of the stability of jets with arbitrary mean velocity profiles is presented. This method is based on utilizing Green function technique a to transform the Rayleigh equation into an integral equation form. The integral equation is then solved numerically using a type of finite element approximation to determine the eigenvectors and complex wave numbers of various instability modes. In order to demonstrate this method's capability of handling arbitrary jet mean velocity profiles, a comparison is made to the elliptic jet case where generally good agreement 1s apparent. A brief discussion on how the effects of compressibility and temperature variation in flows car be incorporated within the formalism is presented.

No.: AIAA-86-0542
ID: CaltechAUTHORS:20101122-083110170

]]>

Abstract: Pressure oscillations in ramjet engines have been studied using an approximate method which treats the flow fields in the inlet and the combustor separately. The acoustic fields in the combustor are expressed as syntheses of coupled nonlinear oscillators corresponding to the acoustic modes of the chamber. The influences of the inlet flow appear in the admittance function at the inlet /combustor interface, providing the necessary boundary condition for calculation of the combustor flow. A general framework dealing with nonlinear multi-degree-of-freedom system has also been constructed to study the time evolution of each mode. Both linear and nonlinear stabilities are treated. The results obtained serve as a basis for investigating the existence and stabilities of limit cycles for acoustic modes. As a specific example, the analysis is applied to a problem of nonlinear transverse oscillations in ramjet engines.

Vol.: 1 No.: AIAA-1986-1
ID: CaltechAUTHORS:20110126-104752906

]]>

Abstract: Undesirable sound generation in the combustion chambers of solid propellant rocket motors previously has been attributed to vortex shedding from obstructions that are uncovered as the propellant burns back. An experimental investigation of the phenomenon has re-conf irmed this observation and extended the understanding of the mechanism by which the process is self-sustaining. A pair of aluminum baffles within a lucite duct through which air is drawn models the important aspects which enable the sound generation mechanism to operate. The baffles form an edgetone system which interacts with the longitudinal acoustic modes of the chamber. Pure acoustic tones occur spontaneously, at frequencies equal to the acoustic resonances, when the spacing between the baffles satisfies certain criteria. Flow visualization using smoke and a strobe light triggered by the pressure oscillation indicates that vortex shedding occurs at the upstream baffle in phase with the acoustic velocity oscillation there. Based on the results of the present experiments and others reported in the literature, a mechanism is postulated which explains the observed behavior. It is suggested that pressures induced on the downstream baffle by the vortices convected past by the freestream drive the acoustic resonance. In turn, the acoustic velocity at the upstream baffle serves as the perturbation triggering the formation of vortices in the shear layer growing from the separation point at that location. The amplitude is limited by the nonlinearity in the growth of the vortices in the shear layer. A lIodel based on the proposed mechanism is formulated and written as a computer program. The results predict the behavior of the experilllental apparatus well, confirming that the postulated mechanism is correct.

No.: 86-000183
ID: CaltechAUTHORS:20101207-081629419

]]>

Abstract: This paper summarizes work accomplished during the past five years on analysis of stability related to recent experimental results on combustion instabilities in dump combustors. The primary purpose is to provide the information in a form useful to those concerned with design and development of operational systems. Thus most substantial details are omitted; the material is presented in a qualitative fashion.

ID: CaltechAUTHORS:20110207-100340630

]]>

Abstract: The acoustic/vortical interaction in a twodimensional free shear layer has been studied. The flowfield is represented by division into two parts: the vortical and the compressible flows. Each field is treated separately and linked with the other through the Bernoulli enthalpy. Acoustic waves are identified as unsteady compressible motions free of vorticity. Calculations have been carried out for the flowfields with and without externally imposed disturbances. Preliminary results reported here indicate that the motion of large coherent flow structures contributes significantly to sound generation. In particular, the formation of large structure and subsequent pairing appear globally as a quadrupole source.

No.: AIAA 85-0043
ID: CaltechAUTHORS:20101208-072047856

]]>

Abstract: Pressure oscillations in a side-dump ramjet engine have been studied, using a one-dimensional numerical analysis. The engine is treated in two parts; the inlet section, including a region of two-phase flow downstream of fuel injection, and a dump combustor. Each region is treated separately and matched with the other. Following calculation of the mean flow field, the oscillatory characteristics of the engine are determined by its reponse to a disturbance imposed on the mean fiow. Results have shown favorable comparison with experimental data obtained at the Naval Weapons Center, China Lake.

No.: AIAA-1984-365
ID: CaltechAUTHORS:20101209-115108188

]]>

Abstract: In this paper, we discuss two problems. First, using a second order expansion in the pressure amplitude, some analytical results on the existence, stability and amplitude of limit cycles for pressure oscillations in combusticm chambers are presented. A stable limit cycle seems to be unique. The conditions for existence and stability are found to be dependent only on the linear parameters. The nonlinear parameter affects only the wave amplitude. The imaginary parts of the linear responses, to pressure oscillations, of the different processes in the chamber play an important role in the stability of the limit cycle. They also affect the direction of flow of energy among modes. In the absence of the imaginary parts, in order for an infinitesimal perturbation in the flow to reach a finite amplitude, the lowest mode must be unstable while the highest must be stable; thus energy flows from the lowest mode to the highest one. The same case exists when the imaginary parts are non-zero, but in addition, the contrary situation is possible. There are conditions under which an infinitesimal perturbation may reach a finite amplitude if the lowest mode is stable while the highest is unstable. Thus energy can flow "backward" from the highest mode to the lowest one. It is also shown that the imaginary parts increase the final wave amplitude. Second, the triggering of pressure oscillations in solid propellant rockets is discussed. In order to explain the triggering of the oscillations to a nontrivial stable: limit cycle, the treatment of two modes and the inclusion in the combustion response of either a second order nonlinear velocity coupling or a third order nonlinear pressure coupling seem to be sufficient.

No.: AIAA-1983-576
ID: CaltechAUTHORS:20101209-094958257

]]>

Abstract: Pressure oscillations in ramjet engines are studied. within quasi one-dimensional linear acoustics. The flow field in the dump combustor is approximated by division into three parts: a flow of reactants, a region containing combustion products, and a recirculation zone, separated by a flame sheet and a dividing streamline. The three zones are matched by considering kinematic and conservation relations. Acoustic fields in the inlet section and in the combustion chamber are coupled to provide an analytical equation for the complex wave number characterizing the linear stability. The calculated results are compared with the experimental data reported by the Naval Weapons Center. Reasonable agreements are obtained.

No.: AIAA 83-0574
ID: CaltechAUTHORS:20101207-145019986

]]>

Abstract: Two problems crucial to the stability of longitudinal acoustic waves in solid rocket motors are examined experimentally. The first is the dissipation of energy associated with an average flow inward at the lateral boundary. Measurements reported here, though subject to considerable experimental error, show that the actual losses are much larger than predicted by the approximate one dimensional analysis. The second problem is the attenuation of waves accompanying reflection by the nonuniform flow in a choked exhaust nozzle. Empahsis in this work has been on technique, to provide data relatively easily and inexpensively. It appears that good results can be obtained in a routine manner using small supersonic wind tunnel operated as an open cycle. At least for Mach numbers up to 0.04 at the nozzle entrance, difficulties with signal/noise are satisfactorily overcome with a tracking filter.

No.: NTRS: 2005-10-10
ID: CaltechAUTHORS:20101208-114247352

]]>

Abstract: Pressure oscillations in ramjet engines are approximated as one-dimensional motions and treated within linear acoustics. The exhaust nozzle is represented by the admittance function for a short choked nozzle. New results have been obtained for the quasi-steady response of a normal shock wave in the diffuser. Acoustic fields in the inlet region and in the combustion chamber are matched to provide an analytical expression of the criterion for linear stability. Combustion processes are accommodated but not treated in detail. As examples, data are discussed for two liquid-fueled engines, one having axial dump and one having side dumps.

No.: AIAA 80-1192
ID: CaltechAUTHORS:20101209-074709263

]]>

Abstract: An experimental program has been conducted to study interactions between acoustic waves and non-uniform flow fields. Data have been taken in an impedance tube, modified to accommodate an average flow, and in a resonance tube. Results are given for the influence of flow through a subsonic exhaust vent on an acoustic field. Measurements for circular and slot vents of several sizes and for two frequencies show that the flow through the vent causes a transfer of energy from the average flow field to the acoustic field: The vent acts as a source or gain of acoustic energy.

ID: CaltechAUTHORS:20141017-143920713

]]>

Abstract: This paper summarizes recent work on the approximation to nonlinear pressure oscillations using two modes. A large part of the effort has been concerned with the consequences of gasdynamic nonlinearities of second order in the fluctuations. It appears that the two-mode approximation is valid over a broad range of the linear parameters that govern the global qualitative behavior, particularly if the lower mode is the unstable mode. Exact solutions have been found for the existence and stability of limit cycles, allowing meaningful comparison with numerical solutions obtained with larger numbers of modes considered. The nonlinear analysis to second order does not contain "triggering" - nonlinear instability of a linearly stable system - because the nonlinear processes order terms involving the mean flow speed, or DC shifts in the amplitudes of oscillation do produce triggering.

ID: CaltechAUTHORS:20110207-102353008

]]>

Abstract: This report is concerned with the general problem of the nonlinear growth and limiting amplitude of acoustic waves in a combustion chamber. The analysis is intended to provide a formal framework within which practical problems can be treated with a minimum of effort and expense. There are broadly three parts. First, the general conservation equations are expanded in two small parameters, one characterizing the mean flow field and one measuring the amplitude of oscillations, and then combined to yield a nonlinear inhomogeneous wave equation. Second, the unsteady pressure and velocity fields are expressed as syntheses of the normal modes of the chamber, but with unknown time-varying amplitudes. This procedure yields a representation of a general unsteady field as a system of coupled nonlinear oscillators. Finally, the system of nonlinear equations is treated by the method of averaging to produce a set of coupled nonlinear first order differential equations for the amplitudes and phases of the modes. These must be solved numerically, but results can be obtained quite inexpensively. Subject to the approximations used, the analysis is applicable to any combustion chamber. The most interesting applications are probably to solid rockets, liquid rockets, or thrust augmentors on jet engines. The discussion of this report is oriented towards solid propellant rockets.

ID: CaltechAUTHORS:20141017-154356385

]]>

Abstract: At the present time, there are three general analytical techniques available to study problems of unsteady motions in rocket motors: linear stability analysis; approximate nonlinear analysis, founded on examining the behavior of coupled normal modes; and numerical calculations based on the conservation equations for one-dimensional flows. The last two yield the linear results as a limit. It is the main purpose of this paper to check the accuracy of the approximate analysis against the numerical analysis for some special cases. The results provide some justification for using the approximate analysis to study three dimensional problems.

No.: AIAA 74-201
ID: CaltechAUTHORS:20101216-103529506

]]>

Abstract: Unstable motions must be anticipated as a possible problem in solid-propellant rocket motors; the characteristics of an instability depend primarily on the geometry of the motor and composition of the propellant. It is the purpose of this paper to review mainly the current state of analyses of combustion instability in solid-propellant rocket motors, but appropriate measurements and observations are cited. The work discussed has become increasingly important, both for the interpretation of laboratory data and for predicting the transient behavior of disturbances in full-scale motors. Two central questions are addressed: linear stability and nonlinear behavior. Several classes of problems are discussed as special cases of a general approach to the analysis of combustion instability. Application to motors, and particularly the limitations presently understood, are stressed.

No.: AIAA 72-1049
ID: CaltechAUTHORS:20101214-091516347

]]>