Abstract: A matter of increasing interest is finding the best way to integrate the use of powerful computational facilities with the traditional practices of analysis and related disciplines. This is largely a problem in research and development, rather than fabrication of machines, or even development of codes. There is little question that modern manufacturing, for example, cannot be accomplished competitively without computing machinery - in fact, the more the merrier. However, there is a fairly widespread sense that in areas depending on the development and applications of new ideas, and perhaps especially in education, the general emphasis has been skewed too much to rely excessively on computers. This paper has been prepared partly to make the point with examples taken from the author's experiences with unsteady combustion. My claim here is that in many cases, analytical methods (necessarily approximate) offer a path often initially preferable to that presented by numerical methods, which in the cases at hand, mean computational fluid mechanics. The first simple example treated here is the Rijke tube, well known primarily for two reasons: The physical behavior is easy to produce, and for which data may be relatively easily collected; and the necessary analysis seems quite simple, at first glance. Some recently published experimental results will be cited, with an approximate theory based on the known differential equations for one-dimensional motions. The next section is a brief historical summary of Galerkin's method, followed by several sections summarizing the manner in which it may be combined with a perturbation/iteration method to give an effective approximate method. The general approach has been widely used to analyze practical problems of combustion instabilities arising in development of operational systems. Thus a large part of the paper is a review of previously published material, but with considerable clarifications of points that have caused some confusion. The paper ends with a brief discussion answering a serious criticism, of the method, nearly fifteen years old. The basis for the criticism, arising from solution to a relatively simple problem, is shown to be a result of an omission of a term that arises when the average density in a flow changes abruptly. Presently, there is no known problem of combustion instability for which the kind of analysis discussed here is not applicable. The formalism is general; much effort is generally required to apply the analysis to a particular problem. A particularly significant point, not elaborated here, is the inextricable dependence on expansion of the equations and their boundary conditions, in two small parameters, measures of the steady and unsteady flows. Whether or not those Mach numbers are actually ‘small’ in fact, is really beside the point. Work out applications of the method as if they were! Then maybe to get more accurate results, resort to some form of CFD. It is a huge practical point that the approach taken and advocated here cannot be expected to give precise results, but however accurate they may be, they will be obtained with relative ease and will always be instructive. In any case, the expansions must be carried out carefully with faithful attention to the rules of systematic procedures. Otherwise, inadvertent errors may arise from inclusion or exclusion of contributions. I state without proof or further examples that the general method discussed here has been quite well and widely tested for practical systems much more complex than those normally studied in the laboratory. Every case has shown encouraging results. Thus the lifetimes of approximate analyses developed before computing resources became commonplace seem to be very long indeed.

Publication: International Journal of Spray and Combustion Dynamics Vol.: 4 No.: 3 ISSN: 1756-8277

ID: CaltechAUTHORS:20121018-103119474

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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

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Abstract: This paper describes the interim results of a study sponsored by the Keck Institute for Space Studies 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 feasibility hinges on finding an overlap between the smallest NEAs that can be reasonably discovered and characterized and the largest NEAs that can be captured and transported in a reasonable flight time. This overlap appears to be centered on NEAs with a nominal diameter of roughly 7 m corresponding to masses in the range of 250,000 kg to 1,000,000 kg. Trajectory analysis based on asteroid 2008HU4 suggests that such an asteroid could be returned to a high-Earth orbit using a single Atlas V-class launch vehicle and a 40-kW solar electric propulsion system by 2026. The return of such an object could serve as a testbed for human operations in the vicinity of an asteroid. It would provide a wealth of scientific and engineering information and would enable detailed evaluation of its resource potential, determination of its internal structure and other aspects important for planetary defense activities.

Publication: Aerospace Conference, 2012 IEEE
ID: CaltechAUTHORS:20160223-163750505

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Abstract: This work describes new type of combustion instability for which the 3-way coupling between mixing, flame heat release, and acoustics is modified by local buoyancy effects. Measurements of fuel mixture fraction are made for a non-premixed jet flame in a combustion chamber to assess the dynamics of mixing under imposed acoustic oscillations (22–55 Hz). Infrared laser absorption and phase resolved acetone-planar laser induced fluorescence are used to measure the fuel mixture fraction and then the degree of fuel/air mixing is calculated by determining the unmixedness. Results show acoustic excitation causes oscillations in the degree of fuel/air mixing at the driving frequency, which results in oscillatory flame behavior. This oscillatory flame behavior couples to the buoyancy and this in turn affects the mixing. Results also show that the mixing becomes less effective when the excitation frequency is increased or when the flame is present, compared to the non-reacting case. This work describes a key coupling mechanism that occurs when buoyancy is a significant factor in the flow field.

Publication: Combustion and Flame Vol.: 156 No.: 1 ISSN: 0010-2180

ID: CaltechAUTHORS:KANcf09

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Abstract: This note compares the QSHOD (quasi-steady homogeneous one-dimensional) approach and the Zeldovich-Novozhilov (ZN) method for calculating the combustion response of solid propellants. The goal is to draw attention to a certain limitation on the applicability of the ZN method which appears to have been overlooked. It is shown that under certain conditions the ZN and QSHOD approaches produce different expressions for the combustion response function of homogeneous propellants, and the reason for this discrepancy is explained.

Publication: Combustion Theory and Modelling Vol.: 12 No.: 6 ISSN: 1364-7830

ID: CaltechAUTHORS:SHUctm08

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Abstract: This paper extends the classical analytical solution for small perturbation analysis of the pressure-coupled response of a homogeneous propellant to any two-component composite propellant. The solution obtained is general and can be used with any particular model for propellant combustion. As an example, the Cohen and Strand ammonium perchlorate propellant model for a single ammonium perchlorate particle size was used in this work. The results and their mechanistic significance are presented and discussed. It is shown that, for a two-component composite propellant, two forms of pressure exponents arise from the analysis. The significance of the second exponent is that it enables the composite propellant to be viewed as a homogeneous propellant with a frequency-dependent exponent via the coupling coefficients. It is found that the ammonium perchlorate is the main source of instability because of its condensed phase exothermicity and monopropellant flame kinetics. This will be a problem with energetic materials in general. The inert binder provides a stabilizing influence because of its endothermicity and the diffusion flame formed with the ammonium perchlorate. Effects of ammonium perchlorate particle size and pressure stem from the changing flame structure and its effect on burning rate.

Publication: Journal of Propulsion and Power Vol.: 24 No.: 5 ISSN: 0748-4658

ID: CaltechAUTHORS:SHUjpp08

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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

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Abstract: Fuel-air mixing behavior under the influence of imposed acoustic oscillations has been studied by investigating the response of the fuel mixture fraction field. The distribution of local fuel mixture fraction inside the mixing zone, which is expected to evolve into the local equivalence ratio in the flame zone, is closely coupled to unstable and oscillatory flame behavior. The Experiment was performed with an aerodynamically-stabilized non-premixed burner. In this study, acoustic oscillations were imposed at 22, 27, 32, 37, and 55Hz. Phase-resolved acetone PLIF was used to image the flow field of both isothermal and reacting flow cases and this data along with the derived quantities of temporal and spatial unmixedness were employed for analysis. The behavior of the unmixedness factor is compared with the previous measurements of oscillations in the flame zone. This comparison shows that local oscillations (of order millimeters or smaller) in fuel/air mixing are closely related to the oscillatory behavior of the flame. For each driving frequency, the mixture fraction oscillates at that frequency but with a slight phase difference between it and the pressure field/flame intensity, indicating that the fuel mixture fraction oscillation are likely the major reason for oscillatory behaviors of this category of flames and combustor geometry.

ID: CaltechAUTHORS:20110125-074002108

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Abstract: For understanding the fundamental properties of unsteady motions in combustion chambers, and for applications of active feedback control, reduced-order models occupy a uniquely important position. A framework exists for transforming the representation of general behavior by a set of infinite-dimensional partial differential equations to a finite set of nonlinear second-order ordinary differential equations in time. The procedure rests on an expansion of the pressure and velocity fields in modal or basis functions, followed by spatial averaging to give the set of second-order equations in time. Nonlinear gasdynamics is accounted for explicitly, but all other contributing processes require modeling. Reduced-order models of the global behavior of the chamber dynamics, most importantly of the pressure, are obtained simply by truncating the modal expansion to the desired number of terms. Central to the procedures is a criterion for deciding how many modes must be retained to give accurate results. Addressing that problem is the principal purpose of this paper. Our analysis shows that, in case of longitudinal modes, a first mode instability problem requires a minimum of four modes in the modal truncation whereas, for a second mode instability, one needs to retain at least the first eight modes. A second important problem concerns the conditions under which a linearly stable system becomes unstable to sufficiently large disturbances. Previous work has given a partial answer, suggesting that nonlinear gasdynamics alone cannot produce pulsed or 'triggered' true nonlinear instabilities; that suggestion is now theoretically established. Also, a certain form of the nonlinear energy addition by combustion processes is known to lead to stable limit cycles in a linearly stable system. A second form of nonlinear combustion dynamics with a new velocity coupling function that naturally displays a threshold character is shown here also to produce triggered limit cycle behavior.

Publication: Combustion Science and Technology Vol.: 177 No.: 2 ISSN: 0010-2202

ID: CaltechAUTHORS:20110207-072124993

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Abstract: Beamed-energy launch concepts employing a microwave thermal thruster are feasible in principle, and microwave sources of sufficient power to launch tons into LEO already exist. Microwave thermal thrusters operate on an analogous principle to nuclear thermal thrusters, which have experimentally demonstrated specific impulses exceeding 850 seconds. Assuming such performance, simple application of the rocket equation suggests that payload fractions of 10% are possible for a single stage to orbit (SSTO) microwave thermal rocket. We present an SSTO concept employing a scaled X-33 aeroshell. The flat aeroshell underside is covered by a thin-layer microwave absorbent heat-exchanger that forms part of the thruster. During ascent, the heat-exchanger faces the microwave beam. A simple ascent trajectory analysis incorporating X-33 aerodynamic data predicts a 10% payload fraction for a 1 ton craft of this type. In contrast, the Saturn V had 3 non-reusable stages and achieved a payload fraction of 4%.

No.: 702
ID: CaltechAUTHORS:PARaipcp04a

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Abstract: The microwave thermal thruster heats propellant via a heat-exchanger then expands it through a rocket nozzle to produce thrust. The heat-exchanger is simply a microwave-absorbent structure through which propellant flows in small channels. Nuclear thermal thrusters are based on an analogous principle, using neutrons rather than microwaves, and have experimentally demonstrated specific impulses exceeding 850 seconds. A microwave equivalent will likely have a similar specific impulse, since both nuclear and microwave thermal thrusters are ultimately constrained by material thermal limits, rather than the energy-density limits of chemical propellants. We present the microwave thermal thruster concept by characterizing a novel variation for beamed-energy launch. In reducing the thruster concept to practice, the enabling physical process is microwave absorption by refractory materials, and we examine semiconductor and susceptor-based approaches to achieving this absorption within the heat-exchanger structure.

No.: 702
ID: CaltechAUTHORS:PARaipcp04b

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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

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Abstract: Thermoacoustic instability appears when unsteady heat release is favourably coupled with acoustic pressure perturbations. The important technical applications involving thermoacoustics are combustion instability in rocket motors and low-pollutant lean flames; noisy industrial burners; pulsed combustors; and thermoacoustic engines. The simplest device for studying thermoacoustic instability is a Rijke tube. In this work, a series of experiments is carried out to determine the nonlinear behavior of the transition to instability and the excited regimes for an electrically driven Rijke tube. A hysteresis effect in the stability boundary is observed. A mathematical theory involving heat transfer, acoustics, and thermoacoustic interactions is developed to predict the transition to instability and limit-cycle properties.

Publication: Electronic Journal Technical Acoustics Vol.: 12 No.: 12 ISSN: 1819-2408

ID: CaltechAUTHORS:20101130-144132487

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Abstract: A horizontal Rijke tube with an electric heat source is a system convenient for studying the fundamental principles of thermoacoustic instabilities both experimentally and theoretically. Given the long history of the device, there is a surprising lack of accurate data defining its behavior. In this work, the main system parameters are varied in a quasi-steady fashion in order to find stability boundaries accurately. The chief purposes of this study are to obtain precise values of the system parameters at the transition to instability with specified uncertainties and to determine how well the experimental results can be explained with existing theory. Measurement errors are reported, and the influence of experimental procedures on the results is discussed. A form of hysteresis effect at stability boundaries has been observed. Mathematical modelling is based on a thermal analysis determining the temperature of the heater and the temperature field in the air inside the tube, which, consequently, affects acoustical mode shapes. Solutions of the linearized wave equation for a non-uniform medium, including losses and a heat source term, determine the stability properties of the eigen modes. Calculated results are compared with experimental data and with results of the modelling based on the common assumption of a constant temperature in the tube. The mathematical model developed here can be applied to designing thermal devices with low Mach number flows, where thermoacoustic issue is a concern.

Publication: Journal of Sound and Vibration Vol.: 264 No.: 3 ISSN: 0022-460X

ID: CaltechAUTHORS:20101118-102239874

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Abstract: Vigorous burning of vortices, formed behind flame stabilizers, can drive significant pressure oscillations inside premixed-type combustors. The goal of this work is to derive a reduced-order model for interaction among vortex shedding, chamber acoustics, and combustion process. A dump combustor is considered a general system configuration. Formation of vortices at the sudden expansion in a chamber is affected by the oscillatory flow. A new quasi-steady model is proposed for determining the moment of vortex separation. Vortex burning is assumed to be localized in space and time. A "kicked" oscillator model is utilized for deriving the appropriate dynamical system. The moment of burning and the corresponding vortex location are dependent on the chamber geometry, velocity field, and characteristic chemical and hydrodynamic times. If Rayleigh's criterion is satisfied, acoustic waves can develop in the chamber. Model and experimental results are compared for a chosen configuration. A study of model performance for a realistic system is carried out by variation of parameters where the mean flow velocity and the number of modes are treated as variables.

Publication: Combustion Science and Technology Vol.: 175 No.: 6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101117-101453870

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Abstract: Sir George Cayley invented the conventional configuration of the airplane at the turn of the 19th century. Otto Lilienthal realized that building a successful aircraft meant learning how to fly; he became the first hang glider pilot and also the first flight fatality in 1896. Beginning in the late 1890s, the Wright Brothers absorbed all that was known in aeronautics before them, then added their own discoveries and developed the first successful airplane. Technically, their greatest fundamental achievement was their invention of three-axis aerodynamic control. Less obviously, their success was a consequence of style, their manner of working out their ideas and of progressing systematically to their stunning achievements. They were indeed the first aeronautical engineers, understanding as best they could all aspects of their aircraft and flying. They were thinkers, designers, constructors, analysts, and especially flight-testpilots. Their powers of observation and interpretation of the behavior of their aircraft in flight were remarkable and essential to their development of the airplane. Their work in the period 1899–1905 constitutes the first true research and development program carried out in the style of the 20th century. As the centenary of their first powered flights approaches, the Wright Brothers’ magnificent achievements excite growing admiration and respect for their achievements. The broad features of their accomplishments have long been well known. Only in the past two decades has serious attention been directed to the scientific and technical content of their work, to explain the nature of the problems they faced and how they solved them. After a century’s progress in aeronautics, the principles, understanding,and methods not available to the Wrights provide the basis for interpreting in modern terms the experiences that the Wrights themselves documented so meticulously in their diaries, papers, and correspondence. It is a unique opportunity in the history of technology.

Publication: AIAA Journal Vol.: 41 No.: 6 ISSN: 0001-1452

ID: CaltechAUTHORS:CULaiaaj03

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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

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Abstract: A modified Price–Boggs–Derr model is applied to compute the linear and nonlinear combustion response properties of monopropellant ammonium perchlorate (AP). 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 with the classical theory. Computations using the Levine and Culick boundary condition in the limit of small perturbations were compared with the exact mathematical solution for linear response, and the effect of perturbation amplitude was explored. Then, using the AP model for the boundary condition, various linear and nonlinear 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.

Publication: AIAA Journal Vol.: 40 No.: 4 ISSN: 0001-1452

ID: CaltechAUTHORS:SHUaiaaj02

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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

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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

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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

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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

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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

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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 bulk acoustic modes that simulate unstable combustor conditions in the absence of velocity fluctuations. 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 laser-induced 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 variations with the phase of the chamber pressure lead the OH field variations. This is probably a result of the greater NO sensitivity to temperature.

Publication: Proceedings of the Combustion Institute Vol.: 29 No.: 1 ISSN: 1540-7489

ID: CaltechAUTHORS:20101118-082923771

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Abstract: Although flows in combustors contain considerable noise, arising from several kinds of sources, there is a sound basis for treating organized oscillations as distinct motions. That has been an essential assumption incorporated in virtually all treatments of combustion instabilities. However, certain characteristics of the organized or deterministic motions seem to have the nature of stochastic processes. For example, the amplitudes in limit cycles always exhibit a random character, and even the occurrence of instabilities seems occasionally to possess some statistical features. Analysis of nonlinear coherent motions in the presence of stochastic sources is, therefore, an important part of the theory. We report a few results for organized oscillations in the presence of noise. The most significant deficiency is that, because of the low level of current understanding, the stochastic sources of noise are modeled in ad hoc fashion and are not founded on a solid physical basis appropriate to combustion chambers.

Publication: AIAA Journal Vol.: 38 No.: 8 ISSN: 0001-1452

ID: CaltechAUTHORS:BURaiaaj00

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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

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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

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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

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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

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Abstract: Triggering of oscillations in rocket motors is the subject of an ongoing investigation at the California Institute of Technology. Some issues that have been previously addressed in several recent publications by the authors [1-3] have more recently been examined by Wicker et al.[4] Whereas the latter paper provides a thorough investigation of the equations as derived, it apparently overlooks the possible effects of two key approximations that were used in the derivation of these equations. Some of the possible effects of these assumptions have been addressed previously, [3] and the authors suggest that the work by Wicker et al. might have benefited from our earlier and accessible conclusions.

Publication: Journal of Propulsion and Power Vol.: 16 No.: 1 ISSN: 0748-4658

ID: CaltechAUTHORS:BURjpp00

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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

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Abstract: The chief purpose of this paper is to investigate the influences of noise, more generally stochastic sources of any sort, on linear and nonlinear unsteady motions in combustion chambers. To be definite, our primary applications here relate to combustion instabilities in solid propellant rockets, particularly linear stability. Two aspects are especially relevant to practical applications: the direct effects of noise on stability; and extraction of information about stability margin from noisy pressure records taken for stable motors. However, the formulation and results are relevant to combustors generally. A fundamental issue is the distinction between and relative importance of self-excited (linearly unstable) oscillations on the one hand and forced oscillation on the other. This has been a controversial and occasionally misunderstood topic for many years. The essential ideas can be clarified unambiguously within the context of global dynamics treated here. We are not so concerned with the details of data processing as with the physical interpretation of the results.

ID: CaltechAUTHORS:20101118-105039121

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Abstract: Most of the previous work on periodic limit cycles in linearly unstable combustors has been carried out for the case of purely longitudinal classical modes whose natural frequencies are integral multiples of the fundamental. The contrary situation of transverse modes in a cylindrical chamber has led to unexpectedly high amplitudes in the periodic limit cycles, the reasons for which have not been thoroughly understood. Some results reported here serve to clarify the two chief consequences of nonlinear coupling between modes: the first is obvious, for the nonlinear coupling causes excitation of higher modes, and hence energy transfer, when, say, the first mode is unstable; the second has long been known, namely that the coupling must also cause shifts of the frequencies from their linear non-integral values to the integral values required for a periodic limit cycle, but part of the significance of this property has been unclear. In particular, the necessity for the frequency shifts reduces the efficiency at which energy is transfered between modes and leads to higher amplitude oscillations. Our analysis shows that the larger are the differences of the linear frequencies from perfect integral values, the larger are the amplitudes in the limit cycle. These conclusions suggest a way to reduce the amplitudes of transverse oscillations, and examples are presented showing this possibility.

Publication: Combustion Science and Technology Vol.: 144 No.: 1-6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101118-135546896

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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

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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

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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

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Abstract: A numerical analysis of unsteady motions in solid rocket motors has been conducted. 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. A narrow pressure pulse is imposed at the head end to simulate unsteady acoustic oscillations in the combustion chamber. Pressure increases when the front of the pulse reaches near the nozzle area. Self-generated oscillations with frequency of standing wave propagates upstream in the combustion chamber. Investigation of transient response of gas-phase dynamics to traveling pressure wave and its effects on propellant combustion reveals several aspects: Combustion responses have a strong relationship with vorticity fluctuations in case of high turbulent intensity on the propellant surface. Temperature fluctuations of the propellant surface in the head end region seem to be very unstable and independent of the pressure wave. Surface temperature without turbulence effect looks more sensitive to temperature fluctuations in the primary flame zone. Stability of surface temperature is strongly related to turbulent intensity on the propellant surface.

Vol.: 2 No.: 662
ID: CaltechAUTHORS:20110128-075156436

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Abstract: Hysteretic behaviour, as found in the dump combustor facility at GALCIT, allows nonlinear active control of the instability, demonstrated recently by Knoop et al. (1996). As in that work, pulses of secondary fuel, based on a simple on/off control law, have been successfully used to drive the transition between the two modes present in the hysteretic region, thereby reducing the amplitude of the pressure oscillations with minimal use of fuel. In order to clarify the origin of the phenomenon, high speed shadowgraph images of the flowfield during the transition between 'unstable' and stable burning have been taken, showing distinctive features that may help in modeling the observed behaviour. A preliminary parametric study (type of injector, duration of pulses, type of secondary flow) has also been conducted, showing that the transition can be obtained over a broad range of conditions.

Publication: Combustion Science and Technology Vol.: 126 No.: 1-6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101118-151443293

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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

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Abstract: In the stunning rush of inventions at the end of the 19th century, the year 1896 held events remarkably significant for the development of the airplane and aeronautics in the early part of the 20th century. The three major figures responsible for those pioneering aeronautical events were the German mechanical engineer Otto Lilienthal (1848-1896) and two Americans: Samuel P. Langley (1834-1906), a self-educated physicist and Secretary of the Smithsonian Institution, and Octave Chanute (1832-1910) an eminent civil engineer best known for his participation in development of the railway system in the midwest United States but in 1896 devoting his energies almost totally to invention of the flying machine.

Publication: AIAA Journal Vol.: 35 No.: 2 ISSN: 0001-1452

ID: CaltechAUTHORS:20140527-081613688

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Abstract: Formal analysis of combustion instabilities has been based on expansion of the equations of motion in two small parameters, March numbers characterizing the average and fluctuating flows. The procedure is reviewed and extended in this note explain how new terms involving nonlinear mean flow/acoustic interactions may be incorporated. As a further example of the ordering procedure, the reasoning is given to show why coupling of modes, linear in the average March number, cannot legitimately be retained when the equations have been expanded only to first order in Mach number.

Publication: Combustion Science and Technology Vol.: 126 No.: 1 ISSN: 0010-2202

ID: CaltechAUTHORS:20101118-150130234

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Abstract: Experimental results for instabilities in dump combustors have long shown the presence of some sort of hysteresis, in the vicinity of the stability boundary defined by flow speed and mixture ratio. This note reports measurements of the precise character of hysteresis loops found when the mixture ratio is varied with flow speed held constant. The upper branch represents relatively high amplitude limit cycles and the lower branch contains low level oscillations possibly driven by stochastic sources within the chamber. Existence of the hysteresis loops suggests the possibility of causing a transition from the upper unstable branch to the lower stable branch. This has been accomplished by injecting short pulses of secondary fuel either in the boundary layer upstream of the dump plane or in the recirculation zone. Only a single pulse is required if its flow rate and width are suitably chosen. These results demonstrate the feasibility of a novel method of active control of combustion instabilities if hysteresis is present, apparently the first example of true nonlinear control of combustor dynamics. The main advantage is the small amount of secondary fuel required. More generally, this work illustrates the value, for active control, of understanding the dynamics of the uncontrolled system.

Publication: Combustion Science and Technology Vol.: 123 No.: 1-6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101119-150022314

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Abstract: The results given in this paper constitute a continuation of progress with nonlinear analysis of coherent oscillations in combustion chambers. We are currently focusing attention on two general problems of nonlinear behavior important to practical applications: the conditions under which a linearly unstable system will execute stable periodic limit cycles; and the conditions under which a linearly stable system is unstable to a sufficiently large disturbance. The first of these is often called 'soft' excitation, or supercritical bifurcation; the second is called 'hard' excitation, 'triggering,' or subcritical bifurcation and is the focus of this paper. Previous works extending over more than a decade have established beyond serious doubt (although no formal proof exists) that nonlinear gasdynamics alone does not contain subcritical bifurcations. The present work has shown that nonlinear combustion alone also does not contain subcritical bifurcations, but the combination of nonlinear gasdynamics and combustion does. Some examples are given for simple models of nonlinear combustion of a solid propellant but the broad conclusion just mentioned is valid for any combustion system. Although flows in combustors contain considerable noise, arising from several kinds of sources, there is sound basis for treating organized oscillations as distinct motions. That has been an essential assumption incorporated in virtually all treatments of combustion instabilities. However, certain characteristics of the organized or deterministic motions seem to have the nature of stochastic processes. For example, the amplitudes in limit cycles always exhibit a random character and even the occurrence of instabilities seems occasionally to possess some statistical features. Analysis of nonlinear coherent motions in the presence of stochastic sources is therefore an important part of the theory. We report here a few results of power spectral densities of acoustic amplitudes in the presence of a subcritical bifurcation associated with nonlinear combustion and gasdynamics.

Publication: International Journal of Energetic Materials and Chemical Propulsion Vol.: 4 No.: 1+6 ISSN: 2150-766X

ID: CaltechAUTHORS:20101119-151322582

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Abstract: Instabilities of motions in a combustion chamber are consequences of the coupled dynamics of combustion processes and of the flow in the chamber. The extreme complexities of the problem always require approximations of various sorts to make progress in understanding the mechanisms and behavior of combustion instabilities. This paper covers recent progress in the subject, mainly summarizing efforts in two areas: approximate analysis based on a form of Galerkin's method, particularly useful for understanding the global linear and nonlinear dynamics of combustion instabilities and numerical simulations intended to accommodate as fully as possible fundamental chemical processes in both the condensed and gaseous phases. One purpose of current work is to bring closer together these approaches to produce more comprehensive and detailed realistic results applicable to the interpretation of observations and for design of new rockets for both space and military applications. Particularly important are the goals of determining the connections between chemical composition and instabilities; and the influences of geometry on nonlinear behavior.

Vol.: 2
ID: CaltechAUTHORS:20101119-113430324

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Abstract: The purpose of this paper is to give a broad overview of the field of combustion instabilities in propulsion systems. Virtually all of the material included here has appeared elsewhere, either in primary research reports or in reviews. None of the propulsion systems are covered in great detail, but sufficiently to establish the fundamental point that while there are obvious practical differences among the systems, for understanding and treating combustion instabilities, much is to be gained by treating the various phenomena within a common framework. In that context, the systems are distinguished chiefly by geometry and the kinds of propellants used. On that basis, a general framework can be constructed to serve both practical and theoretical purposes.

Vol.: 306
ID: CaltechAUTHORS:20110128-112158907

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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

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Abstract: A decoupled adaptive scheme is proposed for the trajectory control of robots. Under moderate constraints, the scheme is stable and robust in the presence of feedback delay and signal hold due to digital computation in the control loop. The design procedure involves a search for compensators with the aim of minimizing a quadratic performance index in order to minimize the maximum tracking error. Stability analysis of the proposed control law (the selected compensators) then provides stability bounds of disturbances, control and adaptation gains, and desired trajectories and their time-derivatives. The region of attraction is local due to unmodelled dynamics.

Publication: International Journal of Control Vol.: 61 No.: 2 ISSN: 0020-7179

ID: CaltechAUTHORS:20101122-080359296

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Abstract: N/A

No.: 169
ID: CaltechAUTHORS:20101122-074028101

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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

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Abstract: Two important approximations have been incorporated in much or the work with approximate analysis or unsteady motions in combustion chambers: 1) truncation of the series expansion to a finite number or modes, and 2) time-averaging. A major purpose or the present analysis is to investigate the limitations or those approximations. A continuation method Is used to determine the limit cycle behavior or the time-dependent amplitudes or the longitudinal acoustic modes in a combustion chamber. The results show that time-averaging works well only when the system Is slightly unstable. In addition, the stability boundaries predicted by the twomode 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 birurcations are found to exist: 1) a pitchfork bifurcation leading to a new branch of limit cycles, and 2) a torus bifurcation leading to quasiperiodic motions.

Publication: Journal of Propulsion and Power Vol.: 10 No.: 4 ISSN: 0748-4658

ID: CaltechAUTHORS:20101124-101555564

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Abstract: Bifurcation theory has been used to study Ihe nonlinear dynamics of the F-14. An 8 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 nonlinearlties as functions of the angles of attack and sideslip, the rotation rate about the velocity vector, and the elevator deflection. A continuation method has been used to calculate the steady states of the F -14 as continuous functions of the elevator deflection. Bifurcations of these steady states have been used to predict the onset of wing rock, spiral divergence, and jump phenomena that 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.

Publication: Journal of Aircraft Vol.: 31 No.: 1 ISSN: 0021-8669

ID: CaltechAUTHORS:20101123-102858346

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Abstract: N/A

Publication: AIAA Journal Vol.: 32 No.: 1 ISSN: 0001-1452

ID: CaltechAUTHORS:20101123-094949189

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Abstract: In this article, a robust adaptive control scheme for robotic manipulators is designed based on the concept of performance index and Lyapunov's second method. Compensators are selected for a given feedback system by using a quadratic performance index. Then the stability of the system is proven based on Lyapunov's method, where a Lyapunov function and its time-derivative are derived from the selected compensators. In the process of stabilization, stability bounds are obtained for disturbances, control gains, adaptation gains, and desired trajectories, in the presence of feedback delay due to digital computation and first-order hold in the control loop.

Publication: Journal of Robotic Systems Vol.: 11 No.: 4 ISSN: 0741-2223

ID: CaltechAUTHORS:20101123-141347474

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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

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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

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Abstract: N/A

ID: CaltechAUTHORS:20110207-085016646

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Abstract: Numerical solutions of viscous, swirling flows through circular pipes of constant radius and circular pipes with throats have been obtained. Solutions were computed for several values of vortex circulation, Reynolds number and throat/inlet area ratio, under the assumptions of steady flow, rotational symmetry and frictionless flow at the pipe wall. When the Reynolds number is sufficiently large, vortex breakdown occurs abruptly with increased circulation as a result of the existence of non-unique solutions. Solution paths for Reynolds numbers exceeding approximately 1000 are characterized by an ensemble of three inviscid flow types: columnar (for pipes of constant radius), soliton and wavetrain. Flows that are quasi-cylindrical and which do not exhibit vortex breakdown exist below a critical circulation, dependent on the Reynolds number and the throat/inlet area ratio. Wavetrain solutions are observed over a small range of circulation below the critical circulation, while above the critical value, wave solutions with large regions of reversed flow are found that are primarily solitary in nature. The quasi-cylindrical (QC) equations first fail near the critical value, in support of Hall's theory of vortex breakdown (1967). However, the QC equations are not found to be effective in predicting the spatial position of the breakdown structure.

Publication: Journal of Fluid Mechanics Vol.: 242ISSN: 0022-1120

ID: CaltechAUTHORS:20101129-102843529

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Abstract: This paper is concerned with some aspects of non-linear behavior of unsteady motions in combustion chambers. The emphasis is on conditions under which organized oscillations having discrete frequencies may exist in the presence of random motions. In order to treat the two types of motions together, and particularly to investigate coupling between noise and combustion instabilities, the unsteady field is represented as a synthesis of acoustic modes having time-varying amplitudes. Each of the amplitudes are written as the sum of two parts, one associated with the random field and the remainder representing the organized oscillations. After spatial averaging, the general problem is reduced to solution of a set of second-order ordinary differential equations whose structure depends on the sorts of nonlinear processes accounted for. This formulation accommodates any physical process; in particular, terms are included to represent noise sources, although only limited modeling is discussed. Our results suggest that random sources of noise have only small effects on combustion instabilities and seem not to be a cause of unstable motions. However, the coupling between the two sorts of unsteady motions may be important as an essential process in a proposed scheme for noise control. It is now a familiar observation that many nonlinear deterministic systems are capable of exhibiting apparently random motions called 'chaos.' This is a particularly interesting possibility for systems which also executed non-deterministic random motions. In combustion chambers, a nonlinear deterministic system (acoustical motions) exists in the presence of noise produced by flow separation, turbulent motions, and energy released by combustion processes. The last part of the paper is directed to the matter of discovering whether or not chaotic motions exist in combustion systems. Analysis has not progressed sufficiently far to answer the question. We report here recent results of processing data taken in one combustor to determine the dimensions of any attractors in the motions. No evidence has been found for chaos in the strict sense, but the method seems to be an important means of investigating the nonlinear behavior of combustion systems.

No.: 512
ID: CaltechAUTHORS:20110121-134650696

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Abstract: N/A

No.: 143
ID: CaltechAUTHORS:20110125-152359532

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Abstract: Organized oscillations excited and sustained by high densities of energy release in combustion chambers have long caused serious problems in development of propulsion systems. The amplitudes often become sufficiently large to cause unacceptable structural vibrations. Because the oscillations are self-excited, they reach limiting amplitudes (limit cycles) only because of the action of nonlinear processes. Traditionally, satisfactory behavior has been achieved through a combination of trial-and-error design and testing, with control always involving passive means: geometrical modifications, changes of propellant composition, or devices to enhance dissipation of acoustic energy. Active control has been applied only to small-scale laboratory devices, but the limited success suggests the possibility of serious applications to full-scale propulsion systems. Realization of that potential rests on further experimental work, combined with deeper understanding of the mechanisms causing the oscillations and of the physical behavior of the systems. Effective design of active control systems will require faithful modeling of the relevant processes over broad frequency ranges covering the spectra of natural modes. This paper will cover the general character of the linear and nonlinear behavior of combustion systems, with special attention to acoustics and the mechanisms of excitation. The discussion is intended to supplement the paper by Doyle et al. concerned primarily with controls issues and the observed behavior of simple laboratory devices.

ID: CaltechAUTHORS:20101130-075321569

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Abstract: In 1878, Lord Rayleigh formulated his criterion to explain several examples of acoustic waves excited and maintained by heat addition. It is a qualitative explanation successfully capturing the essence of the phenomena but not providing a basis for quantitative predictions. The widespread appeal of Rayleigh's criterion merits placing this important result on a more rigorous basis. To do so requires careful formulation grounded in the theory of small amplitude motions in a compressible fluid. In this chapter, we review the construction of an approximate analysis and establish the equivalence of Rayleigh's criterion and the condition for linear stability of small amplitude motions. Thus Rayleigh's criterion is formulated explicitly in the context of an analysis applicable to any combustion chamber. Some results are discussed for both linear and nonlinear motions. Recent experimental results discussed by others suggest that the criterion may offer a practical means for investigating the causes of instabilities in propulsion systems.

ID: CaltechAUTHORS:20110127-073132186

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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

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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

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Abstract: Nonlinear pressure oscillations in combustion chambers have been studied using an approximate analysis which treats the unsteady motions as a collection of coupled nonlinear oscillators, each corresponding to an acoustic mode of the chamber. Based on spatial and temporal averaging of the conservation equations, the formulation provides a convenient means for analyzing the nonlinear behavior of acoustic wave motions. The work addresses the linear and nonlinear characteristics of transverse acoustic waves in a cylindrical chamber, with attention focused on the conditions for the existence and stability of limit cycles. As a result of the special structure of the equations, both general and precise conclusions can be reached. Explicit results are given for the special cases of two and three standing modes. Spinning wave motions will be treated in a subsequent paper.

Publication: Combustion Science and Technology Vol.: 72 No.: 1-3 ISSN: 0010-2202

ID: CaltechAUTHORS:20101129-110317027

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Abstract: Based on spatial and temporal averaging of the conservation equations, an approximate analysis provides a convenient vehicle for analyzing the nonlinear behavior of unsteady motions in combustion chambers. Recent work has been concerned with the conditions for existence and stability of limit cycles, and in particular their dependence on the order of nonlinear terms contained in the equations of motion. It seems to be generally true that if only nonlinear gasdynamic terms to second order are accounted for, a stable nontrivial limit cycle is unique. This implies that no initial disturbance will cause a linearly stable system to execute a limit cycle, a result contrary to prior experience with, for example, solid propellant rocket motors. In this paper, we study some conditions under which true nonlinear instabilities may be found, with special attention focused on the effect of nonlinear gasdynamics. Our results indicate that third-order terms in the acoustic motions do not lead to triggering to either a stable or an unstable limit cycle; they only modify the stability domain of the system. However, the interactions between mean flow and nonlinear acoustic waves may trigger a linearly stable system to an unstable limit cycle. The influence of nonlinear combustion response may have quite different consequences and will be discussed in Part II of this work.

Publication: Combustion Science and Technology Vol.: 72 No.: 4-6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101130-073713187

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Abstract: The behavior of unsteady pressure fluctuations in combustion chambers is examined. An approximate method used in the derivation of the amplitude equations is based on the spatial and time domain averaging of the conservation equations, and follows the analytical framework introduced by Culick (1976a,b). The first order perturbation terms retained in the analysis correspond to linear contributions from the combustion processes, gas/particle interactions, mean flow and boundary conditions, as well as second order nonlinear gas dynamics terms. Further simplification of these equations is obtained by an appropriate change of variables. Following this step, the analysis based on two longitudinal modes is reduced to the solution of a three-dimensional system of nonlinear equations. This enables derivation of exact results for the existence, stability and the amplitude of the limit cycle, in the general case of frequency shifted periodic oscillations. Consideration is also given to the transfer of energy within the spectrum of the acoustic modes. In agreement with experimental observations, it is shown analytically that the preferred direction of energy transfer is from the lower to the higher acoustic modes. Validation of the results is accomplished by comparison with numerical results obtained when higher numbers of modes are treated. Finally, it is shown that combustion instabilities can be treated analytically using the center manifold theory.

Publication: Combustion Science and Technology Vol.: 65 No.: 1-3 ISSN: 0010-2202

ID: CaltechAUTHORS:20101203-081753345

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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

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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

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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

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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

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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

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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

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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

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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

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Abstract: N/A

Publication: Combustion Science and Technology Vol.: 56 No.: 4-6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101202-151716445

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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

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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

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Abstract: Longitudinal combustion instabilities in liquid-fueled ramjet engines have been investigated with attention focused on determination of the acoustic mode structures. Detailed pressure measurements, including both amplitude and phase, were made at ten positions spanning the length of the engine. The experimental data have been analyzed using two linear acoustic models. Four distinct modes were observed for various inlet/combustor combinations. These results help identify the mechanisms exciting low-frequency pressure oscillations in ramjet engines.

Publication: Journal of Propulsion and Power Vol.: 2 No.: 2 ISSN: 0748-4658

ID: CaltechAUTHORS:20101201-085332901

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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

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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

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Abstract: Unsteady motions in combustion chambers have previously been treated with an approximate analysis in which an acoustic field is represented as a collection of coupled nonlinear oscillators constructed in one-to-one correspondence to the acoustic modes. Two parameters characterize the linear behavior of each oscillator; a single parameter arises from the nonlinear acoustics carried out to second order in small fluctuations. The formal results are used here as the basis for studying the existence and stability of limit cycles for longitudinal modes. Owing to the special structure of the equations, explicit and precise conclusions can be reached. Existence and stability depend only on the parameters defining the linear motions. The nonlinear gasdynamics influence the amplitudes of motion in the limit cycle. At least one of the acoustic modes must be linearly unstable to produce a nontrivial limit cycle. Generally, energy flows both up and down among the modes, but there are exceptional cases when limit cycles exist only if the fundamental mode is unstable. Explicit results are given for the special cases of two and three modes; the analysis is extendible to any number of modes.

Publication: Combustion Science and Technology Vol.: 46 No.: 3-6 ISSN: 0010-2202

ID: CaltechAUTHORS:20101201-083223351

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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

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Abstract: The Los Angeles Chapter of the American Institute of Aero and Astronautics is building two replicas of the 1903 Wright Flyer airplane; one to wind-tunnel test and display, and a modified one to fly. As part of this project the aerodynamic characteristics of the Flyer are being analyzed by modern wind-tunnel and analytical techniques. Tnis paper describes the Wright Flyer Project, and compares key results from small-scale wind-tunnel tests and from vortex-lattice computations for this multi-biplane canard configuration. Analyses of the stability and control properties are summarized and their implications for closed-loop control by a pilot are derived using quasilinear pilot-vehicle analysis and illustrated by simulation time histories. It is concluded that, although the Wrights were very knowledgeable and ingenious with respect to aircraft controls and their interactions (e.g., the good effects of their wing-warp-to-rudder linkage are validated), they were largely ignorant of dynamic stability considerations. The paper shows that the 1903 Flyer was readily controllable about all axes but was intrinsically unstable in pitch and roll, and it could barely be stabilized by a skilled pilot.

No.: WF 84/09-1
ID: CaltechAUTHORS:20101207-140100221

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Abstract: A finite difference scheme with a shock-fitting algorithm has been used to investigate unsteady inviscid now with a shock in an inlet diffuser. The flowfield consists of three different regions: the supersonic and the subsonic regions, and a region containing both air and liquid fuel droplets, separated by a normal shock wave and a fuel injection system. The analysis is based on a two-phase, quasi-one-dimensional model. The response of a shock wave to various disturbances has been studied, including large-amplitude periodic oscillations and pulse perturbations.

Publication: Journal of Propulsion and Power Vol.: 1 No.: 3 ISSN: 0748-4658

ID: CaltechAUTHORS:20110125-143851324

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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

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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

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Abstract: An analytical and experimental study is being made of the role of combustion in large vortical structures in the mechanism of unsteady and unstable burning in air-breathing engines. A large body of experimental evidence supports the contention that these periodic fluctuations are themselves generated by the nonsteady flow over the flame holders and other surfaces. The mechanism itself is relatively independent of the acoustic configuration of the powerplant and its installation and hence constitutes the fundamental element of the combustion instability process. Whether or not the mechanism is excited does, however, depend upon the detailed acoustic properties of the combustion chamber and its environment and in many circumstances it is apparent that non-linear acoustics plays an essential role. As a consequence, the program includes detailed analytical studies of linear and non-linear acoustics in combustion configurations as a means of coupling the instability mechanism to a particular environment. The effective separation of the instability process into i) its mechanism and ii) its environment is aimed at eventually providing means of rational scaling of laboratory size experiments.

ID: CaltechAUTHORS:20141114-143821228

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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

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Abstract: The frequency response of a normal shock in a diverging channel is calculated for application to problems of pressure oscillations in ramjet engines. Two limits of a linearized analysis arc discussed: one represents isentropic flow on both sides of a shock wave; the other may be a crude appr'l'I;imation to the influence of flow separation induced hy the wave. Numerical results arc given, and the influences of the shock wave on oscillations in the engine are discus,ed.

Publication: AIAA Journal Vol.: 21 No.: 10 ISSN: 0001-1452

ID: CaltechAUTHORS:20101208-111845159

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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

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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

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Abstract: An experimental investigation has been carried out for acoustic oscillations sustained by flow through a duct containing two baffles. Pure acoustic tones corresponding to longitudinal resonant modes of the duct are produced when certain flow and geometrical conditions are satisfied. The conditions are such as to ensure close coincidence between the frequency of vortex shedding from the forward baffle, and a natural frequency of the duct. Flow visualization has shown that under these conditions a stable vortex structure exists between the baffles, containing at all times an integral number of vortices.

Publication: Journal of Sound and Vibration Vol.: 84 No.: 2 ISSN: 0022-460X

ID: CaltechAUTHORS:20101209-093924977

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Abstract: Chemiluminescence from vibrationally excited carbon monoxide formed by the reaction CS+O-->CO(v)+S was observed in CS2/O2 and CS2/O2/N2O flames to which an electric discharge was applied. Although the total amount of chemiluminescence increased with increasing discharge current probably due to enhanced reaction rates as a result of radical formation, the vibrational distribution was quenched, becoming thermal in character. The thermalization is attributed to superelastic electron collisions [e+CO(v)-->e+CO(v−1)]. The technique demonstrates a sensitive method for detecting collisional transfers between excited states by separating the perturbation (electron collisions) from the initial excitation mechanism (chemical reactions).

Publication: Journal of Applied Physics Vol.: 52 No.: 6 ISSN: 0021-8979

ID: CaltechAUTHORS:KUSjap81

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Abstract: This paper describes an experimental investigation of interactions between acoustic waves and a non-uniform steady flow field. Data have been obtained for a resonance tube having a vent at the center in the lateral boundary, an average flow being introduced at the ends. Experiments have been done for both circular and slot vents, over ranges of both frequency and Mach number. According to the one-dimensional linear stability analysis, the interactions between the longitudinal acoustic field and the mean flow in the vicinity of the vent cause a net transfer of acoustic energy from the average flow to the acoustic field. This result has been verified by the experiments reported here. The gain of energy measured is less than that deduced from a one-dimensional analysis.

Publication: Journal of Sound and Vibration Vol.: 75 No.: 4 ISSN: 0022-460X

ID: CaltechAUTHORS:20101209-102019799

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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

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Abstract: A model which completely describes the Cu/CuCI double pulse laser is presented. The dissociation discharge pulse and afterglow are simulated and the results are used as initial conditions for an analysis of the pumping discharge pulse and laser pulse. Experimental behavior including the minimum, optimum, and maximum delays between pulses, and the dependence of laser pulse energy on dissociation energy are satisfactorily reproduced. An optimum tube temperature is calculated, and the dependence of laser pulse energy on tube temperature (i.e., CuCI vapor pressure) is explained for the first time.

Publication: Journal of Applied Physics Vol.: 51 No.: 6 ISSN: 0021-8979

ID: CaltechAUTHORS:20101208-101900961

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Abstract: A model which completely describes the Cu/CuCl double pulse laser is presented. The dissociation discharge pulse and afterglow are simulated and the results are used as initial conditions for an analysis of the pumping discharge pulse and laser pulse. Experimental behavior including the minimum, optimum, and maximum delays between pulses, and the dependence of laser pulse energy on dissociation energy are satisfactorily reproduced. An optimum tube temperature is calculated, and the dependence of laser pulse energy on tube temperature (i.e., CuCl vapor pressure) is explained for the first time.

Publication: Journal of Applied Physics Vol.: 51 No.: 6 ISSN: 0021-8979

ID: CaltechAUTHORS:KUSjap80

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Abstract: Characteristics of the pumping discharge pulse and laser pulse in a Cu/CuCl double pulse laser have been measured as a function of time delay, buffer gas pressure, and tube temperature. We have found that for otherwise fixed discharge conditions, pumping rates decrease as these quantities are increased. The shape of the laser pulse as a function of time delay is shown to be dependent on the rate of current rise of the pumping pulse. The length of time required by the pumping pulse to achieve threshold is found to be a function of time delay, buffer gas pressure, and tube temperature. The implications of this behavior for the role of metastable copper and its mode of relaxation are discussed.

Publication: IEEE Journal of Quantum Electronics Vol.: 16 No.: 6 ISSN: 0018-9197

ID: CaltechAUTHORS:KUSieeejqe80

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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

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Abstract: A continuous glow discharge was applied to a Cu/CuCl double pulse laser. Maximum laser pulse energy was observed to increase as much as 35 percent at low buffer gas pressure and 3.5 percent at optimum buffer gas pressure. Minimum and optimum time delays decreased with increasing glow discharge current. The greater pulse energy may be due to increased rate of current rise of the pumping discharge pulse, and decreased contribution to the population of metastable copper from ion recombination.

Publication: IEEE Journal of Quantum Electronics Vol.: 15 No.: 9 ISSN: 0018-9197

ID: CaltechAUTHORS:KUSieeejqe79

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Abstract: Large solid propellant rocket motors are conveniently assembled with segmented grains. The interfaces of the grains are coated with inert, slow-burning material. As the propellant burns radially during a firing, the inert material is exposed in the form of annular rings oriented normal to the axis of the chamber. The flow through a ring will produce periodic shedding of vortices over a broad range of conditions.

Publication: Journal of Sound and Vibration Vol.: 64 No.: 3 ISSN: 0022-460X

ID: CaltechAUTHORS:20101213-084217131

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Abstract: The Wright brothers' "Flyer" of 1903 was not just a lucky effort by two bicycle mechanics from Dayton but the outcome of an intensive program of research, engineering and testing

Publication: Scientific American Vol.: 241 No.: 1 ISSN: 0036-8733

ID: CaltechAUTHORS:20101213-074304929

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Abstract: Electron-density measurements have been made in a CuCl/Ne discharge using a CO2 laser interferometer. A local maximum followed by a local minimum in electron density have been found to move a higher tube temperature with higher buffer pressure. Extrema in Cu/CuCl double-pulsed laser energy may be correlated with the extrema in electron density. An analysis based on rate equations yields qualitative agreement with measured laser-pulse energies. The results support the idea that ionization of copper is primarily responsible for the decrease of laser output as the tube temperature is increased.

Publication: Applied Physics Letters Vol.: 33 No.: 8 ISSN: 0003-6951

ID: CaltechAUTHORS:KUSapl78

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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

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Abstract: A one-dimensional thermal theory is applied to a low-pressure, laminar, premixed, CS_2/O_2 flame. The spatial distribution of the species concentrations and rate of production of carbon monoxide are calculated. The flame speed is calculated as a function of the initial O_2/CS_2 ratio at a pressure of 1 kPa.

Publication: Combustion and Flame Vol.: 33 No.: 2 ISSN: 0010-2180

ID: CaltechAUTHORS:20101210-120127838

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Abstract: The rate of vibrational population of carbon monoxide in a steady CS2/O2 flame has been determined from CO overtone emission. A steady-state analysis indicates that, for a flame, the fifteenth vibrational level has the largest rate of population.

Publication: Journal of Chemical Physics Vol.: 67 No.: 5 ISSN: 0021-9606

ID: CaltechAUTHORS:VETjcp77

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Abstract: The rate of vibrational population of carbon monoxide in a steady CS_2/O_2 flame has been determined from CO overtone emission. A steady-state analysis indicates that, for a flame, the fifteenth vibrational level has the largest rate of population.

Publication: Journal of Chemical Physics Vol.: 67 No.: 5 ISSN: 0021-9606

ID: CaltechAUTHORS:20101213-112042239

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Abstract: The flame speed for a low speed premixed CS_2/O_2 flame has been experimentally determined as a function of O_2/CS_2 ratio. The data, which are given in Fig. 1, were taken for lean flames at 1 kPa, under conditions which are similar to those found in flame lasers.

Publication: Combustion and Flame Vol.: 30 No.: 1 ISSN: 0010-2180

ID: CaltechAUTHORS:20101210-102405671

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Abstract: N/A

Publication: Combustion Science and Technology Vol.: 15 No.: 3 ISSN: 0010-2202

ID: CaltechAUTHORS:20101214-075406344

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Abstract: The response functions of composite propellants are theoretically derived, including explicitly the pressure-dependent degradation term in the condensed phase, hypothesizing that the overall rate-limiting reactions are in the relatively low temperature condensed phase. The method of "inner" and "outer" expansions with the reduced activation energy as the singular perturbation parameter has been employed. The results cover subsurface reactions with and without surface reactions, adiabatic and uniform-combustion models for the quasisteady gas phase processes. The response functions display several experimentally observed features like dependence on mean pressure and the instability behavior of some "zero-n" propellants.

Publication: Combustion Science and Technology Vol.: 15 No.: 5 ISSN: 0010-2202

ID: CaltechAUTHORS:20101213-105529799

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Abstract: The electron temperature as a function of laser tube temperature of a copper vapor laser utilizing copper chloride as a lasant has been obtained by measuring the microwave incoherent radiation from the plasma in the laser tube. An unexpected increase of electron temperature at higher laser tube temperature may be due mainly to Penning ionization of the Cu atom by the metastable state of He or Ne buffer gases. The results obtained in this work provide part of the basis for understanding the behavior of the laser output as a function of tube temperature.

Publication: Journal of Applied Physics Vol.: 47 No.: 10 ISSN: 0021-8979

ID: CaltechAUTHORS:SOVjap76

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Abstract: This paper summarizes analytical studies and the interpretation of experimental results for the compression and rarefaction waves generated in the cavity of a pulsed CO electric discharge laser. A one-dimensional analysis of acoustic waves is applied to a transversely excited laser. The influences of heating in the cathode fall, heat transfer to the cathode, flow through both the anode and cathode, and bulk heating of the plasma are included. The analysis is used to relate the bulk heating rate to observable features of the pressure and density waves. Data obtained from interferograms and reported elsewhere are used to infer the bulk heating rates in a pulsed CO laser. Results are presented for CO/Ar, CO/N2, and N2 plasmas. Comparison of the data with recent theoretical results for the heating due to electron/ neutral collisions and the anharmonic defect associated with V-V energy transfer shows substantial differences at lower values of total energy deposition. The change of heating with E/N is in fairly good agreement with predicted values.

Publication: IEEE Journal of Quantum Electronics Vol.: 12 No.: 10 ISSN: 0018-9197

ID: CaltechAUTHORS:CULieeejqe76

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Abstract: The approximate analysis developed in Part I of Ihis work is apllied to severa1 specific problems. One purpose is to illustrate the use of the formalism, and second is to demostrate the validity of the method by comparing results with numerical solutions, obtained elsewhere, for the "exact" equations. A simple problem is treated first, the decay of a standing wave in a box containing a mixture of gas and suspended particles; one example of the steepening of an initially sinusoidal wave in pure gas is included. Viscous losses on an inert surface are treated essentially according to classical linear theory; recent experimental results are used as the basis for incorporating approximately the influence of nonlinear heat transfer in unsteady flow. All of the preceding results are combined in calculations of two examples of unstable motions in a solid propellant rocket motor and in a T-burner.

Publication: Acta Astronautica Vol.: 3 No.: 9-10 ISSN: 0094-5765

ID: CaltechAUTHORS:20101217-103254887

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Abstract: This paper 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.

Publication: Acta Astronautica Vol.: 3 No.: 9 ISSN: 0094-5765

ID: CaltechAUTHORS:20101217-100432116

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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

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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

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Abstract: The problem oflinearized three-dimensional motions in a non-uniform flowfield is re-examined. Several modifications of the general analysis are effected: The influence of particulate matter is acounted for, to zeroth order, and certain boundary processes treated in earher one-dimensional computations are incorporated in an analysis applicable to any geometry. All processes occurring in combustion chambers are accommodated. As a specific example, the results are applied to a problem of linear stability in solid propellant rocket motors.

Publication: Combustion Science and Technology Vol.: 10 No.: 3 ISSN: 0010-2202

ID: CaltechAUTHORS:20101216-154702705

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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

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Abstract: Two independent methods were used to measure the rate of heat transfer from a hot gas to the walls of a cylindrical combustion chamber in which large-amplitude combustion-driven oscillations accompanied a mean flow. The results obtained from the two methods were in good agreement and indicated a definite increase in the heat transfer in the presence of oscillations. The increase was found to vary approximately as the square root of the oscillation amplitude and as the fourth root of the frequency. A correlation in terms of dimensionless variables was obtained using the thickness of the acoustic boundary layer as the appropriate length scale.

Publication: Combustion Science and Technology Vol.: 9 No.: 1/2 ISSN: 0010-2202

ID: CaltechAUTHORS:20101216-110120321

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Abstract: This work is aimed at experimentally exploring the effects of minor compositional variables upon the combustion behavior of composite solid propellants. More specifically, it is intended to determine the influence, if any, of ingredients that improve the mechanical properties upon the oscillatory combustion characteristics. Tests are carried out in the familiar Crawford bomb, a low-pressure L^* burner, and a high-pressure T-burner. Two families of propellants are investigated; each family consists of two propellants with a minor compositional variation between them. In the family that shows a decreasing (steady state) pressure index (n) with increasing pressures, all of the combustion characteristics are found to be very similar although the mechanical properties are widely different. In the other family, which shows an increasing n with increasing pressures, unmistakable differences are found between the two propellants in the low-pressure L ^* instability behavior (along with the differences in the mechanical properties), while the other combustion characteristics are almost identical. The results are interpreted to be consistent with a theory that highlights the importance of condensed phase heat-transfer effects.

Publication: Combustion Science and Technology Vol.: 8 No.: 4 ISSN: 0010-2202

ID: CaltechAUTHORS:20101216-101742743

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Abstract: The problem of linearized one-dimensional motions in a non-uniform flow field is re-examined. Earlier work is clarified, and some assumptions previously used are relaxed. The formalism accommodates all processes occurring in combustion chambers, including sources of mass, momentum, and energy at the lateral boundary. The work is intended partly to provide some results required for subsequent analyses of linear and nonlinear three-dimensional unsteady motions.

Publication: Combustion Science and Technology Vol.: 7 No.: 4 ISSN: 0010-2202

ID: CaltechAUTHORS:20101214-143437599

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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

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Abstract: Due to non-linear loss or gain of energy, unstable oscillations in combustion chambers cannot grow indefinitely. Very often the limiting amplitudes are sufficiently low that the wave motions appear to be sinusoidal without discontinuities. The analysis presented here is based on the idea that the gasdynamics throughout most of the volume can be handled in a linear fashion. Non-linear behavior is associated with localized energy losses, such as wall losses and particle attenuation, or with the interaction between the oscillations and the combustion processes which sustain the motions. The formal procedure describes the non-linear growth and decay of an acoustic mode whose spatial structure does not change with time. Integration of the conservation equations over the volume of the chamber produces a single non-linear ordinary differential equation for the time-dependent amplitude of the mode. The equation can be solved easily by standard techniques, producing very simple results for the non-linear growth rate, decay rate, and limiting amplitude. Most of the treatment is developed for unstable motions in solid propellant rocket chambers. Other combustion chambers can be represented as special cases of the general description.

Publication: Combustion Science and Technology Vol.: 3 No.: 1 ISSN: 0010-2202

ID: CaltechAUTHORS:20101214-084430395

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Abstract: Of the various unstable motions observed in solid propellant rocket chambers, the most troublesome currently are those involving oscillatory motions parallel to the axis. Such instabilities are found to arise particularly in larger rockets using propellants which contain aluminum. The problem is formulated here in one-dimensional form and solved for the case of small amplitude standing waves. Both pressure and velocity coupling may be accommodated, although the proper description of the response function for velocity coupling is not yet known. In addition to several special cases, the stability boundary is discussed for a straight chamber having variable cross section. The influences of the mean flow field, the nozzle, particulate matter, and motions of the solid propellant grain are taken into account.

Publication: Combustion Science and Technology Vol.: 2 No.: 4 ISSN: 0010-2202

ID: CaltechAUTHORS:20101220-152647813

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Abstract: A calculation of the burning rate for a composite solid propellant is based on a model in which the distribution of combustion in the gas phase is specified. The formulation of the problem can accommodate an arbitrary distribution, but only the simplest case of uniform combustion is treated here. An inverse numerical computation is carried out, as the magnitude of the reaction rate, and its dependence on pressure, are calculated for several specified burning rates. Simultaneously, flame stand-off distances and thicknesses are determined as part of the solution. The numerical results for those characteristics of the flame appear to be comparable to observed values. However, the principal value of this treatment of the problem is due to the ease with which one can study and understand the influence of the various physical properties and combustion characteristics.

Publication: Combustion Science and Technology Vol.: 1 No.: 3 ISSN: 0010-2202

ID: CaltechAUTHORS:20101221-134757282

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Publication: AIAA Journal Vol.: 7 No.: 7 ISSN: 0001-1452

ID: CaltechAUTHORS:20101221-094400998

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Abstract: It has long been a cherished hope of some people that measurements of unsteady burning in the laboratory should be applicable to the very important practical problem of instabilities in a rocket chamber. In this Note, we wish to describe what appears to be the first experimental verification of this idea.

Publication: AIAA Journal Vol.: 7 No.: 6 ISSN: 0001-1452

ID: CaltechAUTHORS:20101221-083201081

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Abstract: An Elementary Calculation of the Combustion of Solid Propellants. The problem of the burning of a solid propellant is treated in an approximate manner by prescribing the spatial distribution of energy rdease in the gas phase. Adoption of this inverse approach converts the usual non-linear problem to a linear eigenvalue problem which is easily solvable. Results for concentrated combustion (a flame front) and special cases of distributed energy release are given. Qualitative aspects such as flame stand-off distance, flame thickness, and various components of heat transfer within the combustion region are clearly shown. Examination of the influence of energy loss by radiation from the solid-gas interface gives satisfactory agreement with previous computations based on the non-linear formulation. An application to the problem of unsteady burning, and the response to pressure fluctuations is discussed briefly. The ease with which useful results can be obtained seems to justify the imperfect nature of the analysis.

Publication: Astronautica Acta Vol.: 14 No.: 2 ISSN: 0004-6205

ID: CaltechAUTHORS:20101221-102902637

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Abstract: Experimental data has been obtained from two T-burners and an L^* burner over a range of frequencies from 10 cps to 8000 cps and from 25 psia to 1000 psia. From these data the response of the propellants to a pressure perturbation has been calculated and has been found to be consistent between burners. The results have been interpreted as a response function surface with burning rate and frequencies as independent variables. The data indicates that for a given pressure level the response function has a peak value at a frequency that increases as pressure increases. The magnitude of this peak height varies with pressure so that there is a localized peak in the response function surface for a given propellant. The results are in qualitative agreement with theoretical studies in that the models also predict a maximum value in the response function. However, they also predict that this magnitude should be approximately a constant and not dependent on the pressure and frequency to the extent observed in the present study. The results obtained for an aluminized propellant indicate that presence of metal in the combustion zone greatly influences both the magnitude of the response function peak as well as its location.

Publication: Symposium (International) on Combustion Vol.: 12 No.: 1 ISSN: 0082-0784

ID: CaltechAUTHORS:20101221-114030246

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Publication: AIAA Journal Vol.: 6 No.: 12 ISSN: 0001-1452

ID: CaltechAUTHORS:20101223-101546101

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Abstract: Calculation of the Admittance Function for a Burning Surface. A thorough analysis of pressure oscillations in a solid propellant rocket requires specification of the response of the burning solid. Indeed, for the case of small amplitude waves, this is the most crucial aspect of the problem; unfortunately, it is also poorly understood. The admittance function is merely a convenient expression of the response which contains the primary mechanism for driving waves. In the work reported here, the usual one-dimensional approximation is made, and three main regions are distinguished: the solid phase being heated, the solid phase involving decomposition (a thin region near the surface), and the gas phase. The problem reduces simply to the solution of appropriate ordinary differential equations and satisfaction of boundary conditions, which include matching at interfaces. The most significant differences from previous work are incorporation of a decomposition region and the treatment of the gas phase. A greatly simplified analysis of the latter leads essentially to the same results found elsewhere, but with substantially less labor. Only a quasistatic analysis, valid for frequencies less than a few thousand cycles per second, is covered, but it can be extended to higher frequencies. Laboratory measurements have shown that the response consists generally of a single peak in the range of frequency for which the quasi static approximation appears to be accurate. The qualitative aspects of such peaks, and their connection with 'self-excited' oscillations, are discussed. In particular, the influence of decomposition and pressure sensitivity of the various chemical reactions is examined. Limited numerical results are included. Eventually, the aim of calculations is principally to gain some understanding of the unsteady combustion process and to aid in classifying propellants. The regions involved in the burning are separately characterized by a small number of dimensionless groups. It appears that the effects represented by these parameters may be distinguished in the response function; one may therefore be able, by use of experimental results, to determine at least qualitative connections between the response to pressure oscillations and changes of composition. In this regard, observations made in both T -burners and L * burners should prove useful.

Publication: Astronautica Acta Vol.: 13 No.: 3 ISSN: 0004-6205

ID: CaltechAUTHORS:20101222-075221402

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Abstract: Acoustic Oscillations in Solid Propellant Rocket Chambers. Among the various kinds of periodic motions observed in rocket combustion chambers, the most common and simplest to analyze are those related to classical acoustic modes. If the amplitudes are small, the main perturbations of the familiar standing or travelling waves in a closed chamber are proportional to the Mach number of the mean flow. The correct equations describing the problem are here obtained from the general equations of motion by a limit process which will also provide equations for studying waves of finite amplitude. Subsequently, a single non-homogeneous wave equation is deduced, and solved by an iteration· perturbation procedure. The principal result is a simple formula for the complex frequency showing explicitly the effects of burning, suspended particles in the gases, the exhaust nozzle, and viscous wall forces as well as the mean flow itself. The last is particularly interesting since, owing primarily to the flow inward from the burning surface, the mean flow, if it is irrotational, never acts to damp modes which do not involve axial oscillations. As a particular application, the extensive data taken by BROWNLEE and MARBLE are interpreted to the extent that the linear analysis permits. A stability boundary was obtained from 250 firings of small cylindrical rockets, the principal variables being initial port diameter and length. The propellant did not contain metal particles, and it appears that the observations cannot be explained by the supposition that viscous damping associated with particles in the product gases was the main source of energy loss. Apparently dissipation at the head end, such as that associated with tangential wall shear forces, was an important loss. On the other hand, there is little doubt that if the combustion produces particles, the consequent dissipation is adequate to damp small amplitude waves.

Publication: Astronautica Acta Vol.: 12 No.: 2 ISSN: 0004-6205

ID: CaltechAUTHORS:20110103-101232276

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Publication: AIAA Journal Vol.: 4 No.: 6 ISSN: 0001-1452

ID: CaltechAUTHORS:20110103-100121099

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Publication: AIAA Journal Vol.: 4 No.: 8 ISSN: 0001-1452

ID: CaltechAUTHORS:20110103-102744600

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Abstract: A limiting case of the interaction between particles in a viscous fluid is described by a linear form of Boltzmann's equation. The distribution function, f(v), is found for small particles falling under the action of gravity and viscous drag, and colliding with large particles. Several approximate procedures for computing f(v) and the connection with flows of gases containing solid particles are examined.

Publication: Physics of Fluids Vol.: 7 No.: 12 ISSN: 0031-9171

ID: CaltechAUTHORS:CULpof64

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Abstract: N/A

Publication: Zeitschrift für Angewandte Mathematick und Physik Vol.: 15 No.: 2 ISSN: 0044-2275

ID: CaltechAUTHORS:20110104-142506543

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Publication: AIAA Journal Vol.: 1 No.: 11 ISSN: 0001-1452

ID: CaltechAUTHORS:20110104-112109595

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Abstract: The problem of determining the stability of high-frequency pressure oscillations in rocket combustion chambers is treated explicitly as a perturbation of the classical acoustics problem. On the basis of previous experimental results, the energy addition hy combustion is emphasized, and an analysis is developed for tbe stability of stationary pressure waves in chammers using either liquid or gaseous propellants. The formulation is for the three-dimensional case; computations are carried out in detail for a cylindrical chamber in which the mean velocity is parallel to the axis and varies only with axial position. The principal result is a formula for the imaginary part of the complex frequency associated with each of the natural modes of the chamber. This yields a single dimensionless group as a measure of the stability of individual chamber modes. Because of the present lack of sufficiently detailed experimental evidence, quantitative interpretation seems impossible, but qualitative agreement with observations can be demonstrated.

Publication: AIAA Journal Vol.: 1 No.: 5 ISSN: 0001-1452

ID: CaltechAUTHORS:20110104-082306038

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Publication: AIAA Journal Vol.: 1 No.: 6 ISSN: 0001-1452

ID: CaltechAUTHORS:20110104-093958992

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Abstract: Subsequent to puncturing at a point, a horizontal soap film develops a hole whose edge, owing to surface tension, propagates outward from the point of puncture at apparently constant velocity. Measurements by Ranz [1] yielded results roughly 10% lower than those calculated on the basis of a simple energy conservation suggested by Rayleigh [2]. The discrepancy was attributed to an additional retarding viscous stress not included in the analysis. It appears, however, that the energy balance quoted [1] neglects an important contribution, indeed related to th viscous effect noted by Ranz, but which reduces the calculated values to 20% below those measured. A more detailed analysis of the motion of the edge gives this result; the neglected contribution arises from inelastic acceleration of the undisturbed fluid up to the velocity of the edge. The concomitant loss in mechanical energy may be identified with viscous dissipation which is estimated to be confined to a relatively thin region. Lack of agreement between calculated and measured values of the edge velocity seems to be causes by a second-order effect in the method used [1] to determine the thickness of the film.

Publication: Journal of Applied Physics Vol.: 31 No.: 6 ISSN: 0021-8979

ID: CaltechAUTHORS:CULjap60

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