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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 18:13:52 +0000Servo-Stabilization of Low-Frequency Oscillations in a Liquid Bipropellant Rocket Motor
https://resolver.caltech.edu/CaltechAUTHORS:20091215-101526652
Authors: Marble, Frank E.; Cox, Dale W., Jr.
Year: 1953
The recent work of H. S. Tsien concerning the servostabilization of rocket motors is extended to the liquid bipropellant rocket motor. It is shown that by use of a feedback system containing a device to sense the combustion chamber pressure, a suitably designed amplifier, and a servomechanism which governs the propellant How, the low-frequency oscillations which occur in the rocket configuration may be stabilized for any value of combustion time lag. A method is given for determining a transfer function of the feedback loop which will assure stable operation. The technique of the Satche diagram is employed in stability analysis.https://authors.library.caltech.edu/records/3a3hb-v4090Ignition and combustion in a laminar mixing zone
https://resolver.caltech.edu/CaltechAUTHORS:20100120-092445763
Authors: Marble, Frank E.; Adamson, Thomas C., Jr.
Year: 1954
The analytic investigation of laminar combustion processes which are essentially two- or three-dimensional present some mathematical difficulties. There are, however, several examples of two-dimensional flame propagation which involve transverse velocities that are small in comparison with that in the principal direction of flow. Such examples occur in thc problem of flame quenching by a cool surface, flame stabilization on a heated flat plate, combustion in laminar mixing zones, etc. In these cases the problem may be simplified by employing what is known in fluid mechanics as the boundary-layer approximation, since it was applied first by Prandtl in his treatment of the viscous flow over a flat plate. Physically it consists in recognizing that if the transverse velocity is small, the variations of flow properties along the direction of main flow are small in comparison with those in a direction normal to the main flow. The analytic description of the problem simplifies accordingly. The present analysis considers the ignition and combustion in the laminar mixing zone between two parallel moving gas streams. One stream consists of a cool combustible mixture, the second is hot combustion products. The two streams come into contact at a given point and a laminar mixing process follows in which the velocity distribution is modified by viscosity, and the temperature and composition distributions by conduction, diffusion, and chemical reaction. The decomposition of the combustible stream is assumed to follow first-order reaction kinetics with temperature dependence according to the Arrhenius law. For a given initial velocity, composition, and temperature distribution, the questions to be answered are: (1) Does the combustible material ignite; and (2) how far downstream of the initial contact point does the flame appear and what is the detailed process of development. Since the hot stream is of infinite extent, it is found that ignition always takes place at some point of the stream. However, when the temperature of the hot stream drops below a certain value, the distance required for ignition increases so enormously that it essentially does not occur in a physical apparatus of finite dimension. The complete development of the laminar flame front is computed using an approximation similar to the integral technique introduced by von Kármán into boundary layer theory.https://authors.library.caltech.edu/records/qycd1-h1z74Servo-stabilization of low-frequency oscillations in liquid propellant rocket motors.
https://resolver.caltech.edu/CaltechAUTHORS:20110118-140949997
Authors: Marble, Frank E.
Year: 1955
DOI: 10.1007/BF01600733
Two points of view may be taken with regard to the undesirable pressure
oscillations in rocket motors which arise from instability of combustion, acoustic resonance, coupled oscillations of chamber pressure and propellant flow
rate, as well as from more obscure sources. One is to eliminate the underlying
cause of instability through change in mechanical design or modification of
propellant properties; this is possible when the mechanism of instability is
understood and its removal is not detrimental to rocket performance. The
alternative is, as was demonstrated by H. S. Tsien [1], to modify the system
dynamics by utilizing a feedback servo control which, for example, senses pressure
fluctuations in the combustion chamber and modifies the propellant feeding
rate at the proper frequency and phase to damp the fluctuation. Servo-stabilization
provides the distinct advantage that stability need not be a major
concern during rocket design, relying upon the feedback system to insure stable
operation. Furthermore these concepts suggest the possibility of eliminating
empirically an undesirable oscillation even when its basic cause is not known.https://authors.library.caltech.edu/records/0kjhw-vy813Propagation of stall in a compressor blade row
https://resolver.caltech.edu/CaltechAUTHORS:20110119-110427631
Authors: Marble, Frank E.
Year: 1955
Recent experimental observations on compressors, in particular those of Rannie and Iura, have clarified some features of the phenomenon of stall propagation. Using these observations as a guide, the process of stall in an airfoil cascade has been characterized by a static pressure loss across the cascade which increases discontinuously at the stall angle, the turning angle being affected in only a minor way. Deductions from this simple model yield the essential features of stall propagation such as dependence of the extent of stalled region upon operating conditions,
the pressure loss associated with stall, and the angular
velocity of stall propagation. Using two-dimensional approximation for a stationary or rotating blade row, free from interference of adjacent blade rows, extent of the stalled region, the total pressure loss and stall propagation speed are discussed in detail for a general cascade characteristic. Employing these results, the effect of stall propagation upon the performance of a single-stage
axial compressor is illustrated and the mechanism of entering the regime of stall propagation is discussed. The essential points of the results seem to agree with experimental evidence.https://authors.library.caltech.edu/records/yptrk-jqv81Flame theory and combustion technology
https://resolver.caltech.edu/CaltechAUTHORS:20110118-074645504
Authors: Marble, Frank E.
Year: 1956
The study of combustion processes is in a sufficiently early
stage so that there is no strong connection between combustion theory and the technology of combustion chamber development. To clarify such a connection is the principal task of workers engaged in establishing combustion as an engineering science. The equations of aerothermochemistry are reviewed for the case in which temperature and composition gradients are small. Solutions have been obtained in very few cases and under very restrictive circumstances; most detailed considerations are restricted
to the plane laminar flame front. The current situation
in the theory of plane laminar flames is discussed. The few extensions that have been made to two-dimensional problems are then described. Several directions of work which would assist in establishing theoretical results approaching technological requirements appear possible.https://authors.library.caltech.edu/records/900ap-2vf91A mechanism for high-frequency oscillation in ramjet combustors and afterburners
https://resolver.caltech.edu/CaltechAUTHORS:20110114-111632665
Authors: Rogers, Don E.; Marble, Frank E.
Year: 1956
An experimental investigation was made of the behavior
of a small two-dimensional combustion chamber, burning
a uniform mixture of air and fuel vapor under conditions
of high-frequency oscillation or screech. Measurements
were made of the limits of stable screech, the amplitude
and frequency of pressure oscillations over a wide range of
mixture ratio, inlet air temperature, and combustor flow
rate. Spark schlieren photographs and high-speed motion pictures taken of the combustion process showed, in
agreement with other investigations, that the high-frequency
oscillation is accompanied by vortices shed periodically from the flameholder lip with the same frequency as the oscillation. The following mechanism of exciting the oscillations is suggested. A mode of transverse
oscillation is excited as the result of periodic transport
of combustible material, associated with the vortices,
into the hot wake of the flameholder. The vortices, in
turn, are generated at the flameholder lips by the fluctuating transverse velocity. When the ignition time delay lies in the proper range, the phase relationship between oscillations in transverse velocity and combustion intensity is such that the oscillation is amplified.https://authors.library.caltech.edu/records/q1gmz-brp53Constant-temperature magneto-gasdynamic channel flow
https://resolver.caltech.edu/CaltechAUTHORS:20110113-074847570
Authors: Kerrebrock, Jack L.; Marble, Frank E.
Year: 1960
In the course of investigating boundary-layer flow in continuous
plasma accelerators with crossed electric and magnetic
fields, it was found advantageous to have at hand simple
closed-form solutions for the magneto-gas dynamic flow in the
duct which could serve as free-stream conditions for the boundary
layers. Nontrivial solutions of this sort are not available at
present. and in fact, as in the work of Resler and Sears, the
variation of conditions along the flow axis must be obtained
through numerical integration.
Consequently, some simple solutions of magneto-gasdynamic
channel flow were sought, possessing sufficient algebraic simplicity
to serve as free-stream boundary conditions for analytic investigations
of the boundary layer in a physically reasonable accelerator.
In particular, since the cooling of the accelerator tube is likely to
be an important physical problem because of the high gas temperatures
required to provide sufficient gaseous conductivity,
channel flow with constant temperature appears interesting.
Some simple algebraic solutions for the case of a constant temperature
plasma are developed in the following paragraphs.https://authors.library.caltech.edu/records/n62n4-4q295Nozzle contours for minimum particle-lag loss
https://resolver.caltech.edu/CaltechAUTHORS:20110104-120110913
Authors: Marble, Frank E.
Year: 1963
The flow of a gas-particle mixture through a rocket nozzle is analyzed under the approximation that the particle slip velocity is small compared with the average mixture velocity, using one-dimensional gasdynamics, the Stokes drag law, and corresponding approximations for the heat transfer between solid and gas phase. The variational problem defining the pressure distribution giving the minimum impulse loss due to particle lag is formulated and
solved for nozzles of prescribed mass flow, length, and of given exit pressure or area. The throat section of the optimum nozzle is considerably elongated and more gradual than that of the conventional nozzle. The velocity and temperature lags were much lower (about 1/3) in the throat region than those for the conventional nozzle. The impulse loss of the optimum nozzle was, however, reduced only about 30% below that of the conventional nozzle. It is concluded that contouring of the nozzle to improve gas-particle flow performance will result in only very modest gains. As a direct consequence, the impulse losses calculated herein for optimum nozzles can be used as a rough but convenient approximation for the impulse losses in conventional nozzles having the same area ratio or pressure ratio.https://authors.library.caltech.edu/records/q1qy6-9y416Mechanism of particle collision in the one-dimensional dynamics of gas-particle mixtures
https://resolver.caltech.edu/CaltechAUTHORS:MARpof64
Authors: Marble, Frank E.
Year: 1964
DOI: 10.1063/1.1711372
A theory is developed for the one-dimensional flow of a gas, containing solid particles of two different sizes, in which the effect of particle collisions is accounted for as well as the interaction between the particles and the gas. It is assumed that the particles behave as smooth elastic spheres, that they follow the Stokes drag law and exchange heat with the gas at a Nusselt number of unity. It is shown that there exists a range of parameters which provides that (i) the viscous flow fields about each particle do not interfere during collision, and (ii) the random velocities imparted by one collision are damped before either particle suffers another collision. Using the assumption of small particle slip, the one-dimensional flow problem is solved explicitly up to first order terms in the small slip. It is found, of course, that the tendency of collisions is to cause the two particle-slip speeds to have more nearly the same value than they would in the absence of interparticle collision. It appears that, although the physical assumptions restrict the magnitude of the interparticle forces, the model does provide the proper limit for very strong particle interaction and can conceivably be applied in this range also without gross error.https://authors.library.caltech.edu/records/g47jd-r8g20Reply to comments by S. L. Soo
https://resolver.caltech.edu/CaltechAUTHORS:MARpof65
Authors: Marble, Frank E.
Year: 1965
DOI: 10.1063/1.1761500
If one takes due regard for the condition under which my collision model is valid, explicitly stated by Eq. (15), Ref. 1, the difficulties experienced by Soo(2) will not arise.https://authors.library.caltech.edu/records/7gkvs-8x881Droplet Agglomeration in Rocket Nozzles Caused by Particle Slip and Collision
https://resolver.caltech.edu/CaltechAUTHORS:20101222-073250075
Authors: Marble, Frank E.
Year: 1967
Droplet Agglomeration in Rocket Nozzles Caused by Particle Slip and Collision. The development
of the particle mass spectrum in a rocket nozzle is investigated under the assumption that droplet growth
by collision and agglomeration is the dominant mechanism subsequent to initial appearance of particles
in the rocket chamber. Collisions are calculated on the basis oflinearized particle slip theory and a spectral
integral equation is derived describing the development of particle mass spectrum during the flow process
along the nozzle. This agglomeration process continues until the droplet temperature falls below the freezing
point of the material.
A solution is obtained for the approximate growth in the average particle size during the expansion
process. The results show that, according to this model, the particle size is strongly dependent on the
initial pressure in the rocket chamber and is independent of nozzle geometry.
These results suggest that the collision-agglomeration process is at least one of the critical factors that
accounts for the size of solid particles in rocket exhausts.https://authors.library.caltech.edu/records/30246-2c925Some Gasdynamic Problems in the Flow of Condensing Vapors
https://resolver.caltech.edu/CaltechAUTHORS:20101221-140000406
Authors: Marble, Frank E.
Year: 1969
Some Gasdynamic Problems in the Flow of Condensing Vapors. The general problem of the flow
of a wet vapor, with or without an inert diluent is formulated under the assumption that the liquid phase
is finely divided and dispersed throughout the gaseous component in droplets whose radii are nearly
constant in any local region. The processes of momentum transfer, heat transfer between phases are
assumed to take place according to Stokes law and Nusselt number of unity, respectively. The mass transfer
process is treated as diffusion governed in the presence of an inert diluent and kinetic governed for two
phases of a pure substance.
The physical understanding of such problems, in contrast with those of conventional gas dynamics,
rests largely in the role played by the relaxation times or equilibration lengths associated with these three
processes. Consequently, both simple and coupled relaxation processes are examined rather carefully by
specific examples. Subsequently, the problem of near-equilibrium flow in a nozzle with phase change is solved
under the small-slip approximation. The structure of the normal shock in a pure substance is investigated
and reveals three rather distinct zones: the gasdynamic shock, the vapor relaxation zone, and the thermal
and velocity equilibration zone. The three-dimensional steady flow of the two-phase condensing continuum
is formulated according to first order perturbation theory, and the structure of waves in such supersonic flow
is examined. Finally, the attenuation of sound in fogs is formulated and solved accounting for the important
effects of phase change as well as the viscous damping and heat transfer which have been included in previous
analyses.https://authors.library.caltech.edu/records/1y97c-ae981Dynamics of Dusty Gases
https://resolver.caltech.edu/CaltechAUTHORS:20101220-141547545
Authors: Marble, Frank E.
Year: 1970
This review deals with a certain restricted portion of the mechanics of heterogeneous media. The volume fraction of the solid-particle or droplet cloud is considered to be so small that the interaction between individual particles may be neglected or highly simplified. This limitation applies to the individual flow fields about the particles as well as to collisions, and to heat and mass transfer as well as to momentum exchange between phases. Under this circumstance, the problem of detailed transport processes between particles and gas may be treated independently of the complete dynamical problem, and this aspect, being a study of its own, will be suppressed to a considerable extent here. There are problems, such as the impact
of particles on walls, the concentration separation in boundary layers or pipe flow, in which the distortion of the particle flow field due to a solid wall or another particle is the central physical issue. These problems therefore lie outside the scope of the review. On the other hand, the structure of shock waves, sound attenuation, and many flow-field problems can be treated within our present restrictions. The basic equations and exchange processes will be introduced first, together with the physical parameters that indicate the relative importance of the particle cloud and the limitations of the dusty-gas concept. Then several different problems will be discussed that lead to some of the significant results in the field and illustrate analytical techniques that have proven useful.https://authors.library.caltech.edu/records/fp41b-fs587Sound attenuation in a condensing vapor
https://resolver.caltech.edu/CaltechAUTHORS:MARpof70
Authors: Marble, Frank E.; Wooten, David C.
Year: 1970
DOI: 10.1063/1.1692847
The process of acoustic attenuation in a condensing medium is investigated using a continuumlike formulation that allows for the phase-exchange process. The liquid phase is assumed sufficiently disperse so that the field may be treated as a continuum. The elementary relaxation processes associated with droplet velocity, temperature, and vapor pressure equilibration are equally important in determining the attenuation when vapor, liquid, and inert gas mass fractions are of the same order. When the liquid mass fraction is small, however, a strong attenuation band appears at low frequencies. This attenuation process involves a coupled relaxation process in which heat transfer and vaporization processes combine to change the temperature of the relatively large gas mass. This attenuation band (i) centers on a frequency that is proportional to the concentration of liquid, and (ii) has a maximum value that varies directly as the concentration of condensible vapor and roughly as the square of the latent heat of vaporization. When the concentrations of liquid and condensible vapor are both small, the low-frequency attenuation band is nearly isolated and may be described in a convenient analytical manner.https://authors.library.caltech.edu/records/r094r-b2s22Kinetic Theory of Transient Condensation and Evaporation at a Plane Surface
https://resolver.caltech.edu/CaltechAUTHORS:20101220-151620237
Authors: Shankar, P. N.; Marble, Frank E.
Year: 1971
DOI: 10.1063/1.1693464
The phenomenon of transient condensation onto, or evaporation from, a liquid sheet in contact
with its pure vapor is treated from a kinetic theory viewpoint. The Maxwell moment method is used
to formulate the detailed transient problem. A steady surface mass flux rate exists for times large in
comparison with the collision time, that is, in the continuum regime, and explicit formulas are given
for this limit. The complete gasdynamic field, however, is nonsteady for all times. The calculations are
carried out utilizing four moments, and the effects of incorporating additional moments are negligible.
Finally, the analysis is extended to incorporate imperfect mass and temperature accommodation.
Examination of the transient solution and a matched asymptotic "quasisteady" solution shows that
the gasdynamic field consists of a diffusion process near the liquid surface coupled through an expansion
or compression wave to the constant far field state.https://authors.library.caltech.edu/records/7w1b7-kvd85Nitric oxide formation in turbulent diffusion flames
https://resolver.caltech.edu/CaltechAUTHORS:20101214-141112621
Authors: Quan, Victor; Marble, Frank E.; Kliegel, James R.
Year: 1973
DOI: 10.1016/S0082-0784(73)80078-5
Combustion and NO formation are investigated in the turbulent mixing region between
parallel fuel and oxidant streams. Chemical reactions are divided into two classes: (i) the fast,
diffusion-limited combustion reaction, and (ii) the relatively slow, rate-limited NO formation.
For the fast reaction, the turbulent mixing zone contains fuel, oxidant, and reaction products.
The formation of NO is calculated separately as a trace species, since it has negligible effect
on the flowfield.
Transport of momentum, enthalpy, and chemical species is calculated, using a mixing-length
theory. Because NO generation is highly temperature sensitive, the history of combustion
product gases, subsequent to their formation, is decisive in determining the total NO production.
Upper and lower bounds on NO production are obtained by considering that: (i) the combustion
products remain undiluted and intact in the form of eddies as the turbulent field
transports them throughout the mixing layer, and (ii) the combustion products are locally
mixed with cool oxidizer or fuel. These yield upper and lower limits, respectively.
The time-averaged velocity, temperature, and concentrations of fuel, oxidant, products,
and NO distributions, are illustrated. Molecular mixing of turbulent eddies is shown to have
a great influence on the amount of NO formed, although its effect on the time-averaged fluid
properties is negligible. For a sample problem, the NO concentration obtained by assuming
complete local molecular mixing is nearly an order-of-magnitude lower than the value predicted
for no mixing.https://authors.library.caltech.edu/records/prxwz-qvr95Acoustic Attenuation in Fans and Ducts by Vaporization of Liquid Droplets
https://resolver.caltech.edu/CaltechAUTHORS:20101216-132206855
Authors: Marble, Frank E.; Candel, Sebastien M.
Year: 1975
DOI: 10.2514/3.49777
A cloud of small water droplets in saturated air attenuates acoustic disturbances by viscous drag, heat transfer,
and vapor exchange with the ambient gas. The viscous and heat transfer phenomena attenuate at frequencies
above 104 Hz for I-J.l droplets. The processes associated with phase exchange attenuate at a much lower frequency
that may he controlled by choice of the liquid mass fraction. The strength of this attenuation is proportional to the mass of water vapor in the air, a factor controlled by air temperature. For plane waves, the attenuation
magnitude e~ceeds 5 db!m ~t a temperature of 25°C with a cloud of 0.7 J.l radius droplets constituting 1 % of the
gas mass. ThiS attenuation mcreases to more than 7 dbjm at frequencies above 1000 Hz where viscous and heat
transfer mechanisms contribute significantly. The attenuation of higher order duct modes is strongly increased over the above values, similarly to the attenuation by duct lining. When the droplet cloud occupies only a fraction of the duct height close to the walls, the droplet clond may be up to twice as elfective as the uniform cloud, and a significant saving is possible in the water required to saturate the air and furnish the water droplets.https://authors.library.caltech.edu/records/vva2e-1yd89The Effect of Strain Rate on Diffusion Flames
https://resolver.caltech.edu/CaltechAUTHORS:20101216-133642630
Authors: Carrier, G. F.; Fendell, F. E.; Marble, F. E.
Year: 1975
DOI: 10.1137/0128038
Several steady state and time-dependent solutions to the compressible conservation laws describing direct one-step near-equilibrium irreversible exothermic burning of initially unmixed gaseous reactants, with Lewis-Semenov number unity, are presented. The quantitative investigation first establishes the Burke-Schumann thin-flame solution using the Shvab-Zeldovich formulation. Real flames do not have the indefinitely thin reaction zone associated with the Burke-Schumann solution. Singular perturbation analysis is used to provide a modification of the thin-flame solution which includes a more realistic reaction zone of small but finite thickness. The particular geometry emphasized is the un bounded counterflow such that there exists a spatially constant rate of strain along the flame. While the solutions
for diffusion flames under a finite tangential strain rate may be of interest in and of themselves for laminar flow, the problems are motivated by the authors' belief that they are pertinent to the study of diffusion-flame burning in transitional and turbulent shear flows.https://authors.library.caltech.edu/records/36jdn-22095Core noise from gas turbine exhausts
https://resolver.caltech.edu/CaltechAUTHORS:20101214-073031680
Authors: Cumpsty, N. A.; Marble, F. E.
Year: 1977
DOI: 10.1016/0022-460X(77)90031-1
There is abundant evidence to show that the exhaust noise from gas turbines contains
components which exceed the jet mixing noise at low jet velocities. This paper describes
results of a theory developed to calculate the acoustic power produced by tempetature
fluctuations from the combustor entering the turbine. With the turbine Mach numbers and
flow directions at blade mid-height, and a typical value for the fluctuation in temperature,
as parameters it has been possible to predict the acoustic power due to this mechanism for
three different engines. In all three cases the agreement with measurements of acoustic power
at low jet velocities is very good. Similarly, based on a measured spectrum of the temperature
fluctuation, the prediction of the acoustic power spectrum agrees quite well with that
measured.https://authors.library.caltech.edu/records/tfn80-39y50Core noise from gas turbine exhausts
https://resolver.caltech.edu/CaltechAUTHORS:20101214-073031680
Authors: Cumpsty, N. A.; Marble, F. E.
Year: 1977
DOI: 10.1016/0022-460X(77)90031-1
There is abundant evidence to show that the exhaust noise from gas turbines contains
components which exceed the jet mixing noise at low jet velocities. This paper describes
results of a theory developed to calculate the acoustic power produced by tempetature
fluctuations from the combustor entering the turbine. With the turbine Mach numbers and
flow directions at blade mid-height, and a typical value for the fluctuation in temperature,
as parameters it has been possible to predict the acoustic power due to this mechanism for
three different engines. In all three cases the agreement with measurements of acoustic power
at low jet velocities is very good. Similarly, based on a measured spectrum of the temperature
fluctuation, the prediction of the acoustic power spectrum agrees quite well with that
measured.https://authors.library.caltech.edu/records/3asec-psr22Acoustic disturbance from gas non-uniformities convected through a nozzle
https://resolver.caltech.edu/CaltechAUTHORS:20101214-073952475
Authors: Marble, F. E.; Candel, S. M.
Year: 1977
DOI: 10.1016/0022-460X(77)90596-X
The non-steady flow generated by convection of gas containing non-uniform temperature regions or "entropy spots" through a nozzle is examined analytically as a source of acoustic disturbance. The first portion of the investigation treats the "compact nozzle", the case
where all wave lengths are much longer than the nozzle. Strengths of transmitted and reflected one-dimensional waves are given for supersonic and subsonic nozzles and for one configuration of supersonic nozzle with normal shock at the outlet. In addition to a wave reflected from the nozzle inlet, the supersonic nozzle discharges two waves, one facing upstream and the other facing downstream. For reasonable values of the nozzle inlet Mach number, the pressure amplitude of each wave increases directly as the discharge Mach number. The acoustic perturbations from a supercritical nozzle of finite length, in which the
undisturbed gas velocity increases linearly through the nozzle, are analyzed for several inlet and discharge Mach number values and over a wide frequency range. The results,
which agree with the compact analysis for low frequency, deviate considerably as the frequency rises, achieving pressure fluctuation levels of several times the compact values. It is shown that this result originates in a phase shift between the two waves emitted downstream and that the pressure fluctuations for moderate frequencies may be approximated from the compact analysis with an appropriate phase shift. In all cases, the pressure fluctuations caused by a 2% fluctuation in absolute inlet temperature are large enough to require consideration in acoustic analysis of nozzles or turbine blade channels.https://authors.library.caltech.edu/records/g6z34-27q91Acoustic disturbance from gas non-uniformities convected through a nozzle
https://resolver.caltech.edu/CaltechAUTHORS:20101214-073952475
Authors: Marble, F. E.; Candel, S. M.
Year: 1977
DOI: 10.1016/0022-460X(77)90596-X
The non-steady flow generated by convection of gas containing non-uniform temperature regions or "entropy spots" through a nozzle is examined analytically as a source of acoustic disturbance. The first portion of the investigation treats the "compact nozzle", the case
where all wave lengths are much longer than the nozzle. Strengths of transmitted and reflected one-dimensional waves are given for supersonic and subsonic nozzles and for one configuration of supersonic nozzle with normal shock at the outlet. In addition to a wave reflected from the nozzle inlet, the supersonic nozzle discharges two waves, one facing upstream and the other facing downstream. For reasonable values of the nozzle inlet Mach number, the pressure amplitude of each wave increases directly as the discharge Mach number. The acoustic perturbations from a supercritical nozzle of finite length, in which the
undisturbed gas velocity increases linearly through the nozzle, are analyzed for several inlet and discharge Mach number values and over a wide frequency range. The results,
which agree with the compact analysis for low frequency, deviate considerably as the frequency rises, achieving pressure fluctuation levels of several times the compact values. It is shown that this result originates in a phase shift between the two waves emitted downstream and that the pressure fluctuations for moderate frequencies may be approximated from the compact analysis with an appropriate phase shift. In all cases, the pressure fluctuations caused by a 2% fluctuation in absolute inlet temperature are large enough to require consideration in acoustic analysis of nozzles or turbine blade channels.https://authors.library.caltech.edu/records/arngp-3s557The Interaction of Entropy Fluctuations with Turbine Blade Rows; A Mechanism of Turbojet Engine Noise
https://resolver.caltech.edu/CaltechAUTHORS:20101213-114306465
Authors: Cumpsty, N. A.; Marble, F. E.
Year: 1977
DOI: 10.1098/rspa.1977.0171
The theory relating to the interaction of entropy fluctuations ('hot spots'), as well as vorticity and pressure, with blade rows is described. A basic feature of the model is that the blade rows have blades of sufficiently short chord that this is negligible in comparison with the wavelength of the disturbances. For the interaction of entropy with a blade row to be important, it is essential that the steady pressure change across the blade row should be large, although all unsteady perturbations are assumed small. A number of idealized examples have been calculated, beginning with isolated blade rows, progressing to single and then to several turbine stages. Finally, the model has been used to predict the low-frequency rearward-radiated acoustic power from a commercial turbojet engine. Following several assumptions, together with considerable empirical data, the correct trend and level are predicted, suggesting the mechanism to be important at low jet velocities.https://authors.library.caltech.edu/records/t2737-xyc09An analytical study of the non-steady behavior of large combustors
https://resolver.caltech.edu/CaltechAUTHORS:20110121-130330771
Authors: Marble, Frank E.; Candel, Sebastien M.
Year: 1979
DOI: 10.1016/S0082-0784(79)80074-0
The transient response of large burners depends primarily upon fluid mechanical adjustment rather than in time delays associated with transient chemical response. Examples of this behavior are the non-steady behavior of burners in utility boilers, and the low-frequency response of after burners in aircraft gas turbines. The non-steady behavior of a flame stabilized by a single-flame holder at the center of a long two-dimensional duct is investigated analytically when it is excited by periodic acoustic
disturbances that approach the flame zone from either the upstream or downstream direction. The flame zone itself is considered acoustically compact. The problem is treated by an integral technique in which relevant equations are integrated across high-density and low-density portions of the gas separately; the two fields are then coupled across the thin flame front, the determination of its shape being part of the solution. Transmission and reflection coefficients were calculated for a range of flame velocities, burner inlet flow velocities, combustion temperature ratio and imposed acoustic frequency.
The results showed that a considerably stronger pressure wave passed upstream of the flame than downstream, in the sense that could be expected from the different acoustic impedences of the hot and cold gas. Of most significance, however, was the very large (active) response of the burner at certain characteristic frequencies which corresponded to well-defined values of ωL/u_o where L is the length of the flame zone and u_o is the flow velocity upstream of the burner. It is indicated that these energetic response modes result from vorticity shed from the distorted flame which induces a propagating wave along the flame front.https://authors.library.caltech.edu/records/kafrt-byz19Study of a Diffusion Flame in a Stretched Vortex
https://resolver.caltech.edu/CaltechAUTHORS:20101201-080248569
Authors: Karagozian, Ann R.; Marble, Frank E.
Year: 1986
DOI: 10.1080/00102208608923842
The time dependent interaction of a laminar diffusion flame with a single plane vortex and with a stretched line vortex is examined with the aim of determining the flame configuration and the augmentation to the reactant consumption rate resulting from the interaction. Elements
of the resulting curved flame sheets behave essentially as isolated flames until the neighboring flame sheets become so closely spaced that they interact and consume the intervening reactant. This process creates a core of combustion products with external isolated flame surfaces. The augmentation of the reactant consumption rate results both from the local straining of the flame in its own plane and from the overall increase in flame surface area. Three examples are treated in detail. The first is the plane problem in which an initially straight flame is distorted by a vortex. In the second, the situation is similar except that the problem is expanded to three dimensions
and the vortex line is being stretched along its own axis. Finally, the effects of the density change resulting from the heat release are examined.https://authors.library.caltech.edu/records/knxj3-vw617The Effect of Strain Rate on a Premixed Laminar Flame
https://resolver.caltech.edu/CaltechAUTHORS:20101203-093502119
Authors: Darabiha, N.; Candel, S. M.; Marble, F. E.
Year: 1986
DOI: 10.1016/0010-2180(86)90057-X
The structure of a strained premixed laminar flame is examined. The flame is formed in the vicinity of a
stagnation point established by the counterflow of fresh mixture and hot combustion products. This ideal
configuration analyzed by Libby and Williams [18] with activation energy asymptotics is here examined
numerically. This allows an exact description of flame and flow structure and a calculation of the mass rate of
reaction per unit flame area for the whole range of strain rates. Previous results obtained for intermediate and
high strain rates are confirmed. However, for low strain rates the mass rate of reaction per unit flame area
differs from that determined with large activation energy asymptotics. The present calculations also provide
the exact value of the strain rate (or Damkiihler number) for which the partial extinction regime appears. If the
strain rate is increased beyond that value the flame front develops on the hot side of the stagnation point. The
reactive front first moves away from the stagnation point and then moves back toward that point for the very
large values of the strain rate.https://authors.library.caltech.edu/records/gwfaj-34k40Structure and behavior of diffusion flames in a pressure gradient
https://resolver.caltech.edu/CaltechAUTHORS:20101202-142309487
Authors: Marble, Frank E.; Hendricks, Gavin J.
Year: 1988
DOI: 10.1016/S0082-0784(88)80363-1
The structure of a diffusion flame embedded in a flow field parallel to the flame is studied under conditions where this external flow imposes an adverse pressure gradient. It is convenient to think of the physical problem as a flame lying along the flow direction of a divergent channel.
The mathematical problem is reduced to a set of ordinary differential equations by (i) employing the Howarth transformation to eliminate the variable density and (ii) introducing a similarity solution somewhat in the manner of the Falkner-Skan treatment of boundary layer flows.
Because the low-density gas near the flame responds more readily to the pressure gradient than does the higher density gas, a reverse flow develops in the low density region which severely affects both the structure of the flame and the fuel consumption rate. For a flame with unit stoichiometry, the reverse flow eventually leads to extinction of the flame by separating the two shear layers that bound the fuel and oxidizer streams. For stoichiometry
corresponding to methane-air, the flame situates itself near the oxidizer side of the reverse flow and has no tendency toward extinguishment.https://authors.library.caltech.edu/records/twwae-s8609Swirling flows in an annular-to-rectangular transition section
https://resolver.caltech.edu/CaltechAUTHORS:20101203-094947244
Authors: Sobota, Thomas H.; Marble, Frank E.
Year: 1989
DOI: 10.2514/3.23157
Mechanisms for generation of axial vorticity by swirling flows in rectangular nozzles have been investigated
experimentally and computationally. A detailed experimental investigation is described that demonstrates the
formation of axial vortices in the nozzle is dependant on the vorticity distribution at the turbine exhaust. Further,
mechanisms providing for the formation of axial vortices are identified. A parallel computational investigation was
carried out that not only confirmed the relationship between the turbine exhaust vorticity and the vortex patterns
formed in the nozzle, but also provided details of the flowfield between the turbine discharge and the nozzle exit.
On the basis of this more detailed understanding, it is now possible to "tailor" the vortex distribution at the nozzle
exit by design of the turbine discharge and the intervening passage.https://authors.library.caltech.edu/records/yzft0-jps07Response of a Thin Airfoil Encountering a Strong Density
Discontinuity
https://resolver.caltech.edu/CaltechAUTHORS:20101123-132225533
Authors: Marble, F. E.
Year: 1993
DOI: 10.1115/1.2910183
Airfoil theory for unsteady motion has been developed extensively assuming the undisturbed medium to be of uniform density, a restriction accurate for motion in the atmosphere, Glauert (1929), Burgers (1935), Theodorsen (1935), Kussner (1936), Karman and Sears (1938), Kinney and Sears (1975). In some instances, notably for airfoils comprising fan, compressor and turbine blade rows, the undisturbed medium may carry density variations or "spots," resulting from non-uniformaties in temperature or composition, of a size comparable to the blade chord. This condition existsfor turbine blades, Marble (1975), Giles and Krouthen (1988), immediately downstream of the main burner of a gas turbine engine where the density fluctuations of the order of 50 percent may occur. Disturbances of a somewhat smaller magnitude arise from the ingestion of hot boundary layers into fans, Wortman (1975), and exhaust into hovercraft. Because these regions of non-uniform density convect with the moving medium, the airfoil experiences a time varying load and moment which we propose to calculate.https://authors.library.caltech.edu/records/m125p-ja609Aerospace Propulsion Technology—A Fertile Source of Issues in Basic Fluid Mechanics
https://resolver.caltech.edu/CaltechAUTHORS:20101123-134419710
Authors: Marble, Frank E.
Year: 1993
DOI: 10.1115/1.2910176
[No abstract]https://authors.library.caltech.edu/records/j4w57-8hy62Gasdynamic enhancement of nonpremixed combustion
https://resolver.caltech.edu/CaltechAUTHORS:20101123-150925263
Authors: Marble, Frank E.
Year: 1994
DOI: 10.1016/S0082-0784(06)80621-1
To promote efficient performance of very high speed air-breathing propulsion systems, the combustor Mach number must be of the order of six for a flight Mach number of 18. Because of this high gas speed through the combustor, mixing rates of hydrogen fuel with air must be very rapid in order to allow a combustor of reasonable length. It is proposed to enhance the rate of mixing and combustion of hydrogen and air, and thereby reduce combustor length, through the introduction of streamwise vorticity generated by the interaction of a weak oblique shock wave with the density gradient between air and a cylindrical jet of hydrogen.
Because of the high Mach number flow in the combustor, the oblique shock traverses the jet at a small angle with respect to the free stream direction, and the principle of slender body theory allows one conceptually to replace the three-dimensional steady flow with a two-dimensional unsteady flow. As a consequence, two-dimensional time-dependent computational studies and an extensive experimental shock tube investigation were employed to assess mixing rates for the steady flow in the combustor. The results indicated that under realistic conditions, adequate mixing could be accomplished within 1 ms, a rate that was technologically interesting.
Encouraged by these experiments, a "practical" injector, utilizing shock-enhanced mixing, was designed for a combustor having a free stream Mach number of 6.0. A detailed aerodynamic and mixing investigation was carried out in the Mach 6 High Reynolds Number Tunnel at the NASA-Langley Research Center. The results confirmed both the details and the overall effectiveness of the shock-enhanced mixing concept.https://authors.library.caltech.edu/records/en7k6-b5996Numerical Simulations of High-Speed Chemically Reacting Flow
https://resolver.caltech.edu/CaltechAUTHORS:20101123-142807198
Authors: Ton, V. T.; Karagozian, A. R.; Marble, F. E.; Osher, S. J.; Engquist, B. E.
Year: 1994
DOI: 10.1007/BF00312347
The essentially nonoscillatory (ENO) shock-capturing scheme for the solution of hyperbolic
equations is extended to solve a system of coupled conservation equations governing
two-dimensional, time-dependent, compressible chemically reacing flow with full chemistry. The
thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are
separated from the fluid motion by a fractional step algorithm. The methodology is used to study
the concept of shock-induced mixing and combustion, a process by which the interaction of a
shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity
generation. Test cases with and without chemical reaction are explored here. Our results indicate
that, in the temperature range examined, vorticity generation as well as the distribution of atomic
species do not change significantly with the introduction of a chemical reaction and subsequent
heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process.
This suggests that the fluid mechanics of this problem may be successfully decoupled from the
combustion processes, and that computation of the mixing problem (without combustion chemistry)
can elucidate much of the important physical features of the flow.https://authors.library.caltech.edu/records/fq7kp-sgw33