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Acoustic instabilities in gas turbines. There are many ways, in reactive systems, in which a phase change can arise between pressure fluctuations and the induced fluctuations in heat release, to trigger acoustic instabilities via the well known Raleigh criterion. We have described a mechanism that was not previously recognized and reported this in the engineering literature, with the suggestion that it is responsible for the instability known as groaning observed in gas turbines at low speeds. The abstract of this paper is: Slip affects the response time of fuel sprays to acoustic fluctuations in a gaseous flowfield. This article describes how gaseous fuel is released by evaporation as an oscillating response to the acoustic velocity fluctuations, and so contributes to acoustic instability. This article discusses the differences due to the evaporation characteristics of various fuels (JP-4, JP-5, and D-2) as well as the effect of droplet size, inlet air temperature, air speed, and Reynolds number on mass response. For example, it is shown that instability will be driven harder at lower frequencies, higher gas velocities, and increased liquid volatility.

With but minor exceptions, the panel discussion is here reproduced verbatim. Without an indication of the pauses, tone, and body language which, in spoken English, signify erasure, correction, or a new direction, some of the text lacks coherence. Nevertheless, the essential meaning of the speakers is clear.

This paper discusses analytical and numerical work that has been carried out in order to understand flame-balls and related phenomena that have been observed in microgravity experiments. The importance of heat losses is identified, whether from conduction, convection, or radiation. Accurate numerical simulations for hydrogen-air mixtures with radiation losses reveal a flammability limit of 3.5 percent of hydrogen by volume, a value close to the experimental one. The important role of stability analyses is emphasized, with particular attention to the role of three-dimensional instabilities in explaining unsteady spheroidal flames and flame-strings, objects that are observed in the experiments. We speculate that the dynamics of flame-strings is affected by in-depth radiation absorption for mixtures containing SF6, and report on some preliminary calculations in which this phenomenon is accounted for.

A random packing algorithm is implemented to construct numerically models of heterogeneous propellants. These models consist of random distributions of spheres in a periodic cube. The spheres can have various sizes, randomly assigned if desired. Packing fractions are calculated for a bimodal model (spheres of two different sizes) and compare well with old experimental data for steel shot. Slices through the cube dee ne surfaces that represent propellant surfaces, and the stoichiometry of these surfaces is a stochastic variable. Several realizations of the propellant cube are constructed and variations in surface stoichiometry are examined using histograms and Fourierseries.Asamplee amecalculation ispresented andsolved using datadee ned by a singlesurfacerealization.

This paper examines steady linear deflagration of condensates that gasify either by surface pyrolysis or by simple vaporization and undergo a general one-step Arrhenius type gas phase reaction. Heat loss by radiation from the surface of the condensate is taken into account. What distinguishes the study from previous investigations is that rational analytical results are deduced by assuming the activation energy for the gas phase reaction to be very large. For pyrolyzing solids the results agree with experimental observations which show that the burning rate increases with pressure up to a plateau extending beyond a certain critical pressure; that there is a minimum pressure below which deflagration will not occur; that this extinction pressure is lowered by preheating the solid. Extinction is found to occur in two different ways: in true extinction the flame sheet, to which the combustion is confined for large activation energy, suddenly disappears; effective extinction occurs when the flame sheet moves far away from the surface of the condensate, so that surface gasification products effectively pass away unburnt. Effective extinction occurs even for adiabatic deflagration. The results for vaporizing condensates show that the pressure may not be lowered below a minimum without causing the entire condensate to evaporate. At high pressures the burning rate may increase, stay essentially constant or decrease depending upon the relative magnitudes of the pressure exponent and the order of the gas phase reaction. Radiation loss tends to lower the burning rate at a given pressure; in particular Spalding's closed curves are not obtained.

The combustion of a liquid fuel drop immersed in a hot atmosphere is considered when the activation energy is very large. Closed form solutions are obtained which describe a monotone variation of burning rate with Damkohler number, d 1 . For some of the solutions the burning rate varies from the frozen limit at d 1 = 0 , to the Burke-Schumann limit at D 1 = α . For all the others the burning rate varies from the frozen limit to a large D 1 limit distinct from the Burke-Schumann limit.

The quasi-steady decomposition of a monopropellant is considered when the monopropellant reacts with the ambient atmosphere in a one-step irreversible process. The solution is characterized by the variation of the vaporization rate M with two Damkohler numbers, D 1 and D 2, the first a measure of the monopropellant reaction rate and the second a measure of the bipropellant reaction rate. Three possible asymptotic limits are described in the double-equilibrium limit D 1→∞, D 2→∞, and necessary and sufficient conditions on the various parameters of the problem are given for these limits to exist. Large activation-energy asymptotics is used to elucidate one of these limits which would otherwise be intractable. The relation of the three limits to known results for pure monopropellant and pure bipropellant burning is clarified by an analysis when just one of the Damkohler numbers is infinite. In particular a complete discussion of the variation of M with D 2 in the limit D 1→ ∞ is given when the bipro...

We consider the deformation of a thin thread of viscous liquid (viscida) as its ends are slowly moved together. Equations are deduced which are capable of describing the motion of the thread when the displacement of the axis from a straight line is either on the scale of the thread thickness (problem 1) or on the much larger scale of the thread length (problem 2). In the former case it is shown analytically that an arbitrary initial displacement evolves in such a way that, as the appropriately scaled time τ becomes large, the first mode of the disturbance emerges in a dominant role with an amplitude that is proportional to τ½ and independent of the initial amplitude. This provides the initial condition for problem 2, for which a numerical description is obtained.In addition, we analyse the situation when the ends of the viscida are slowly pulled apart. In this case the high frequency end of the spectrum dominates as an arbitrary disturbance decays.

A model is constructed for intumescent paints in which it is assumed that the transition to the swollen state occurs at a very thin zone or front. Across this front there is a discontinuity in the density, the velocity, and the temperature gradient. The temperature is continuous and fixed at the front at a value that is a property of the paint. All of the mass loss through outgassing occurs at the front, and this gives rise to a large increase in the volume of the paint particles as they are traversed by the front. On each side of the front, the physical processes consist exclusively of heat conduction. In the context of this model, a finite coating subject to a heat flux contains two moving boundaries: the front, and the outer surface of the coating at which the flux is applied. Through an elementary mathematical transformation, this problem is reduced to one in which the outer surface is fixed and only the inner front moves. This simplified problem is of Stefan type, easily amenable to numerical analysis. The time-dependent substrate temperature, when calculated in this way, tends to level off at a constant value until the front has completely traversed the coating; then it sharply increases again. This is a characteristic of much of the available experimental data and, combined with visual evidence, lends credence to our model.

An analysis is undertaken, both mathematical and numerical, of the quenching of two-dimensional premixed flames under a variety of circumstances. The discussion is divided into two parts, the first of which deals with quenching due to proximity to a surface through which there are heat losses, the second, quenching due to a shear flow of the kind experienced by a flame attached to a wire. The discussion includes a critical appraisal of flame stretch and the role sometimes claimed for it in intuitive explanations of flame quenching.

Tribrachial (three-armed) flames are generated when a premixed flame propagates into a nonuniform mixture, and consist of a fuel-rich branch joined to a fuel-lean branch, with a diffusion flame trailing from the junction (the stoichiometric point). They are important in the context of ‘laminar flamelet’ models of turbulent non-premixed flames. We provide a theoretical description of these flames, valid when the upstream concentration gradient is small. In addition, we assume that the Lewis number of the oxidizer is 1, but that of the fuel (Le f ) is greater than 1. Solutions are sought corresponding to steady, unbounded propagation into unburnt mixture, where we would expect—nominally—that all three arms of the flame trail behind the junction, so that to the left of some plane perpendicular to the oncoming flow there is everywhere no reaction. Instead, we find that the fuel-rich branch necessarily extends ahead of any plane, penetrating to upstream infinity. This corresponds to an anchored flame rather than one that is freely propagating, and appears to be a consequence of the stretch effects that are possible when Le f >1; a decrease in the mixture strength does not necessarily lead to a decrease in the local flame speed if flame curvature generates negative stretch. If we assume that extinction occurs at sufficiently low flame temperatures, the flame is bounded and our solution can be reinterpreted as a freely-propagating flame, albeit with unexpected shape.

The combustion of a liquid fuel droplet is considered when the Damkohler number is very large, and a solution is constructed that is quite distinct from the Burke-Schumann (B-S) limit. For a certain range of parameters this solution coexists with the B-S limit and is probably unstable. However, there are other values of the parameters for which the new solution exists but the B-S solution is excluded, and in this case the new solution is probably stable.

This paper completes an extensive study of quasi-steady decomposition of a monopropellant droplet in an inert, stagnant atmosphere for constant gas properties and one-step, first-order Arrhenius-type gas phase kinetics. Response curves of vaporization rate versus Damkohler number are again examined in the limit of large activation energy, so as to determine their weak- and strong-decomposition branches. A graph already existing in the fuel-oxidiser literature is shown to locate the ignition point at the end of the former, when there is one; and an explicit formula is derived for the extinction or ignition/extinction point at the end of the latter, when there is one.

A simple model of a low Mach number deflagration wave is studied, with emphasis on the effects of a phenomenological heat loss term. It is shown by explicit construction that two quite different solutions, a fast wave and a slow wave, are possible in the adiabatic limit. In both cases the chemical reaction goes to completion and all the fuel is consumed, but for the fast wave the temperature increases from the ambient value to the usual adiabatic flame temperature, whereas the slow wave is isothermal. An analysis for finite heat loss is carried out in the realistic limit of infinite activation energy. In general this leads to a much simplified numerical problem, but for values of the flame speed of greatest physical interest an analytical description is possible. This predicts quenching when the heat loss is too great or the reaction rate too small, and yields simple explicit quenching criteria.

We examine a premixed flame located in a counterflow of fresh cold mixture and hot burnt gas, the latter at a temperature close to that of adiabatic deflagration. A constant density, high activation energy model is adopted, and the steady state response depends essentially on two parameters, the Lewis number, and the temperature of the remote hot gas. If the second is sufficiently small, the response of flame position to changes in straining rate is a variation on the familiar S-shaped response for systems that display ignition and extinction. The possible implications for a flammability-limit model are discussed. The stability of the steady state under the influence of one-dimensional disturbances is examined in detail, using the numerical method of weighted residuals.

Pursuing ideas of Sivashinsky, we consider distinguished limit perturbations of the classical one-dimensional deflagration wave in the limit of large activation energy in order to investigate unsteady effects, three-dimensional effects, and the effects of heat losses and area changes.

The paper considers the linear stability of the reaction zone structure that has its origins in the large activation energy description of the counterflow diffusion flame. The steady reaction zone structure holds in the premixed flame regime and is known in the literature as Linan’s structure problem. The stability of the reaction zone was first studied numerically by Peters [Combustion and Flame, 33 (1978), pp. 315–318]. The present paper gives an analytical derivation of the stability results obtained by Peters that is valid when the flame is nearly adiabatic.

This paper treats thrust bearings in which the lubricant is a ferroliquid confined by an applied magnetic field to the gap between two plates, one fixed and one mounted on a rotating shaft carrying an axial load. For plates with plane, parallel surfaces, the lubricant extends beyond the edges of the plates and axially along their sides. Since this is undesirable, other plate geometries are considered, for which the ferroliquid is confined to the gap between the plates. Plate surfaces which are considered are cones, paraboloids and frustrums, i.e. plane, parallel surfaces out to some radius and conical, diverging surfaces beyond. Results indicate that paraboloids and frustrums are suitable, but cones are not. FHD thrust bearings with magnetic plates are stiff, i.e. large changes in axial load produce relatively small changes in gap width. With currently available ferroliquids, FHD thrust bearings are limited to relatively small axial loads.

Under conditions of saturated magnetization ferroliquids obey Boussinesq equations in which the body force term is a combination of the ordinary gravitational force and a magnetic force that is proportional to the gradient of the magnetic field and the local temperature fluctuation. Hence non-isothermal flows of ferroliquids can be stabilized or destabilized by the application of a magnetic field. In this paper we examine the effects of magnetic stabilization which can be used to delay the transition from laminar to turbulent flow, or to suppress the level of turbulent transport in fully turbulent flows. The analogy between ferroliquid flows and stratified flows is developed to permit the application of results from studies of stratified flows. Using this analogy the magnitude of turbulence suppression is examined for a limiting case of fully developed turbulent channel flow with constant transverse heat flux. It is found that the channel friction factor can be reduced perceptably with attainable magnetic field strengths and moderate heat fluxes, and that reductions exceeding one order of magnitude can be achieved in extreme conditions. The friction factor reduction is largest for small Reynolds number and decreases with increasing flow rate when the heat flux is held constant.

Radially propagating spherical premixed flames are studied for mixtures characterized by a Lewis number (Le) that is less than unity. It is known that small flames (Zeldovich flames) have a flame temperature that exceeds the remote gas temperature by ∼ ( Q C p )·( Y ∞ Le ) , where Y∞ is the remote reactant concentration; and large flames are slowly varying (SVFs) with a flame temperature close to the adiabatic flame temperature, so that the increment ∼ ( Q C p ) Y∞. It is shown, using asymptotic methods, that there is an intermediate class of flames, characterized by intermediate flame temperatures. The radius of each of these flames grows, essentially, like t , behavior that has been observed in microgravity ignition experiments. When volumetric heat losses are included these flames can exhibit abrupt extinction at some critical time and this, superficially, also agrees with experimental observations.

Overdriven detonations for which the chemistry is modeled by one-step Arrhenius kinetics are examined in the asymptotic limit θ → ∞ where θ is the activation energy. A stability analysis is carried out in which the disturbances have a wavelength ∼ θδ and evolve on the time scale θδ u f , where δ is the detonation wave thickness and uf its speed. Instability is identified in which the growth rate is a monotonically decreasing function of wavenumber. A similar analysis carried out previously for wavelengths ∼θδ predicts a growth rate that is a monotonically increasing function of wavenumber. It follows that there is an intermediate wavelength for which the growth rate is a maximum, and it is shown that this has magnitude ∼θ 2 3 δ . It is argued that this wavelength will be selected from the continuum of values present in an arbitrary initial disturbance and will define the initial spacing of the triple points characteristic of unstable detonations.

The unsteady creeping motion of a thin sheet of viscous liquid as it advances over a gently sloping dry bed is examined. Attention is focused on the motion of the leading edge under various influences and four problems are discussed. In the first problem the fluid is travelling down an open channel formed by two straight parallel retaining walls placed perpendicular to an inclined plane. When the channel axis is parallel to the fall line there is a progressive-wave solution with a straight leading edge, but inclination of the axis generates distortions and these are calculated. In the second problem a sheet with a straight leading edge travelling over an inclined plane penetrates a region where the bed is uneven, and the subsequent deformation of the leading edge is followed. The third problem considers the flow down an open channel of circular cross-section (a partially filled pipe) and the time-dependent shape of the leading edge is calculated. The fourth problem is that of flow down an inclined plane with a single curved retaining wall. These problems are all analysed by assuming that a length characteristic of the geometry is large compared with the fluid depth divided by the bed slope, and all the solutions display extreme sensitivity to the data.

We propose a model for flammability limits based on flame stretch that is applicable to upward propagation through lean CH4/air mixtures in a round tube. Predictions of the model are in good agreement with experiment.

Gaseous fuel is injected at a constant rate M from the bottom of a narrow insulated chamber and oxidant diffuses from the top. The object is to determine the response curve, i.e. the unburnt fuel fraction R as a function of M for fixed Damkohler number (representing either the constant chamber pressure or the chamber's length). The conclusions we draw are based on activation-energy asymptotics. Explicit response curves are obtained over the whole range of D. Two critical values Da, D0 are uncovered and determined explicitly. If D D0 the response is S-shaped: the top of the curve lies on R = 1 and extends to M → ∞; the bottom lies on R = 0 and persists however small M becomes. The isola has two extinction points, corresponding to the fuel being supplied too slowly or too fast to a flame at the bottom or top of the chamber. The S has just the latter, but also has an ignition point; as M is decreased through the corresponding value the flame suddenly appears. The asymptotic analysis completes the inadequate picture obtained previously by numerical methods. It spans the whole range of parameter values and enables explicit formulas to be written for the ignition and extinction points.

—The title problem for a spherical droplet is considered when the activation energy is large. Asymptotic solutions are obtained which, when heat is gained from the atmosphere, describe a monotonic increase of the burning rate M as the Damkohler number D increases from 0 to ∞. If the specific heat of vaporization is larger than the heat of decomposition there is a value of D beyond which M stays constant; otherwise M increases indefinitely, with a convective-diffusive boundary layer forming at the surface of the droplet. When heat is lost to the atmosphere, an S-shaped response curve of M versus D is obtained, exhibiting auto-ignition and extinction, on which M always increases indefinitely with D.

An extensive experimental program has been conducted on intumescent systems where the components have been varied systematically. Small plates were coated with the various formulations, exposed to a heat source typical of aviation fuel fires, and the temperature-time history of the substrate recorded. Concurrently, a simplified mathematical model was developed for an intumes cing system which has given insight to the fundamental mechanisms of in tumescence. The model and the results of the experimental program demonstrate certain desirable features of intumescent systems.

The quasi-steady combustion of a liquid fuel drop is considered in the limit of infinite activation energy when the ambient mass fraction of oxygen is very small. It is shown that under these nearly asphyxiated conditions ignition and extinction phenomena do not necessarily occur even for an exothermic combustion field.

This paper is an investigation of MHD boundary layers in a transverse magnetic field when the magnetic Reynolds number (Rm) is small. The main purpose is to understand something about the suppression of separation by a strong magnetic field, with particular emphasis on the behaviour near a rear stagnation point. Given anO(1) inviscid flow it is shown that there is a critical value ofN, the interaction parameter, to completely suppress separation. This value is one half that proposed by Leibovich (1967), a discrepancy that is due to the non-regularity of the boundary-layer equations at a rear stagnation point, a possibility that Leibovich did not consider in his solution. Model linear equations suggest the true role of Leibovich's solution. The possibility of a viscous wake leaving the rear stagnation point is considered and it is suggested that one doesnotarise from vorticity generated in the boundary layer.

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The equations of inviscid magnetohydrodynamics, appropriate when the induced magnetic field can be neglected, are studied when the interaction parameter N is very large. The limit equations, in which the nonlinear inertia terms do not appear, cannot fully describe general inviscid flow situations suggesting the existence of inviscid boundary layers in which the missing terms must be reinstated. Similarity solutions are obtained for two kinds of layers, one perpendicular to the magnetic field and the other parallel to the magnetic field. The parallel layers are backward jets and constitute our most important result. The existence of the perpendicular layers, which look like reversed shock layers, may have implications for flow in ducts with discontinuous area changes.

The flow close to separation is studied using a coordinate expansion in order to determine the effects of transverse curvature on the structure. The simplest expansion using the Goldstein variables leads to a Stewartson alternative in which either the curvature must vanish or the skin friction must be regular. This difficulty is overcome by the addition of logarithmic terms when the curvature is negative, but is unresolved for positive curvature. The analogy with the flat‐plate compressible boundary layer is pointed out (with heat transfer analogous to curvature), and it is concluded that it is misleading to suggest that the difficulties for a hot wall can be resolved by setting the heat transfer equal to zero.