The flame as a hydrodynamic discontinuity

The plane premixed flame discussed in Chapters 2 and 3 is an idealization seldom approximated, since in practice the flame is usually curved. A Bunsen burner flame is inherently so; but even under circumstances carefully chosen to nurture a plane state, instabilities can precipitate a multidimensional structure. Such flames have been extensively studied using a hydrodynamical approach (Markstein 1964, p. 7); a brief description of it will provide an introduction to our subject.

On a scale that is large compared to its nominal thickness λ/cpMr, the flame is simply a surface across which there are jumps in temperature and density subject to Charles's law (as appropriate for an isobaric process). Deformation of the surface from a plane is associated with pressure variations in the hydrodynamic fields of the order of the square of the Mach number (see section 1.5). These small pressures jump across the surface to conserve normal momentum flux. Because Euler's equations for small Mach number hold outside the flame (cf. the end of section 3), the temperature and density do not change along particle paths; so that for a flame traveling into a uniform gas the temperature and density ahead are constant and the flow is irrotational. The flow behind is stratified, however, since flame curvature generates both vorticity and nonuniform temperature jumps. Variations in temperature from the adiabatic flame temperature are usually neglected everywhere (a matter we shall treat later); but vorticity generation cannot be ignored so that, even though Euler's equations apply behind the flame, the flow is not potential there.

Models of mixtures

In general, two-reactant flames can be classified as diffusion or premixed. In a premixed flame the reactants are constituents of a homogeneous mixture that burns when raised to a sufficiently high temperature. In a diffusion flame the reactants are of separate origin; burning occurs only at a diffusion-blurred interface.

Both kinds of flames can be produced by a Bunsen burner. If the air hole is only partly open, so that a fuel-rich mixture of gas and air passes up the burner tube, a thin conical sheet of flame (typically blue-green in color) stands at the mouth; this is a premixed flame. Any excess gas escaping downstream mixes by diffusion with the surrounding atmosphere and burns as a diffusion flame (typically blue-violet). If the air hole is closed, only the diffusion flame is seen; if all the oxygen is removed from the atmosphere (but not from the air entering the hole), only the premixed flame is observed. (Yellow coloring due to carbon-particle luminosity may also be seen.)

A premixed flame can also be obtained by igniting a combustible mixture in a long uniform tube; under the right conditions, a flame propagates down the tube as a (more or less) steady process, called a deflagration wave. The Bunsen burner brings such a wave to rest by applying a counterflow and stabilizes it with appropriate velocity and thermal gradients.

Synopsis

Ignition has been discussed in three contexts: burning of a linear condensate (Chapter 4), spherical diffusion flames (Chapter 6), and spherical premixed flames (Chapter 7). Based upon the S-shaped response curves determined by steady-state analyses it is argued that the burning rate will jump from a weak, almost extinguished, level to a vigorous one as the pressure (and, hence, the Damköhler number) is increased through some critical value (corresponding to the lower bend of the S). The occurrence of an intrinsically unsteady phenomenon, appropriately called ignition, is thereby inferred from results inherent in the steady state.

Such analyses have a fairly long history in combustion investigations, the simplest example being the thermal theory of spontaneous combustion identified with the name of its originator, Frank-Kamenetskii (1969). Section 2 presents a mathematical version of that theory, which shows an early appreciation of activation-energy asymptotics (though not in the formal sense of this monograph).

Even though steady-state analyses are undoubtedly useful, descriptions of the unsteady state are also needed; providing them is the main goal of this chapter. The three contexts already considered are too complicated for our purposes; we concentrate instead on a simpler problem containing the essential feature of ignition: the evolution of a deflagration wave from an unburnt state.

Consider a thermally insulated enclosure containing a mixture at sufficiently low temperature for the reaction to be very weak. Since the heat released cannot escape, the temperature must rise, albeit very slowly.

Cylindrical flames

Although it has not been studied to the same extent as the plane premixed flame, the cylindrical premixed flame is (in principle) almost as easy to produce. The reacting mixture is supplied through the surface of a circular cylinder and is induced to flow radially by means of end plates sufficiently close together. The flame then forms a coaxial cylinder that can be observed through the end plates, which should be transparent and good thermal insulators.

Analytically the cylindrical flame lies between the plane and spherical flames. The structure of its reaction zone is the same as that of the plane flame, with temperature constant beyond, so that there is none of the curvature effect of the spherical flame. Like the spherical flame it does not exhibit the cold-boundary difficulty: the mixture must be introduced at a finite radius, which can be so small that a line source is effectively formed. In their attempt to treat curved flames Spalding & Jain (1959) use plane-flame analysis on the spherical flame, where it is never valid, and neglect the cylindrical flame, where it is always valid. The results, however, are qualitatively correct (Ludford 1976).

Sections 2 and 3 show how cylindrical geometry modifies a premixed flame (Ludford 1980). For simplicity we shall consider a single reactant under going a first-order decomposition and take ℒ = 1. Spherical premixed flames, which have many similarities to spherical diffusion flames, will be treated in sections 4–6.

Scope

Flame stability is a very broad subject; both the mechanisms and manifestations of instability assume many forms. A whole monograph could be devoted to it if a less analytical approach than ours were adopted. It has been studied since the 1777 observations of Higgins (1802) on singing flames, a phenomenon involving interaction between an acoustic field and an oscillating flame of the kind described in Chapter 5. Such interactions were considered to be merely laboratory curiosities for many years, but lately they have assumed technological importance in the development of rocket motors and large furnaces. This type of instability is well understood qualitatively (Chu 1956). Other combustion phenomena thought to be due to instability are, in general, poorly understood; in some cases instability is merely suspected of playing a role but no certain evidence is yet available. The following examples are representative of the abundance of instability phenomena in combustion.

The stability of burner flames depends on an appropriate interaction between the flame and its surroundings, in particular, a flux of heat from the flame to the burner rim. The role of this flux in anchoring the flame and preventing blow-off has already been mentioned in Chapter 9.

Propagation limits are important questions in the study of premixed flames, with steady, sustained combustion being possible only for a certain range of the fuel to oxidant ratio. If a mixture is too rich or too lean it will not burn, and instability may well be involved.

The Earth's rotation has occupied the interest of astronomers, mathematicians and geophysicists for at least the last 200 yr. This continued involvement, in what must have initially been thought of as a straightforward problem, is a consequence of the multitude of factors that perturb the rotational motion from what it would be if the Earth were wholly rigid. Forces and deformations in the atmosphere, oceans, crust, mantle and core all perturb the rotation to varying degrees from the idealized rigid body motion, and a complete discussion requires one to delve into many aspects of the Earth and planetary sciences. It is undoubtedly this interdisciplinary aspect that has drawn astronomers, oceanographers, meteorologists and solid Earth physicists to the subject.

Geophysical studies of the Earth's rotation have their roots in the works of Lord Kelvin, Sir George Howard Darwin and Sir Harold Jeffreys amongst others. Munk & MacDonald, in 1960, thoroughly reviewed the subject in their monograph The Rotation of the Earth; a geophysical discussion. Their work has dominated the subject ever since and it is unusual to find any aspect of the problem that they have not touched upon. Yet since 1960, and probably because of their very considerable effort at clarifying the subject, much new information, both of an observational and of a geophysical nature, has become available. Some of this is collected in symposia proceedings, in particular those edited by Marsden & Cameron (1966) and by Mansinha, Smylie & Beck (1970).

In this chapter I discuss the fluctuations in the Earth's rotation in the frequency range from about 0.2 cycle yr–1 to very low frequencies. These are often referred to loosely as the decade fluctuations, although they cover a much longer-period range from, say, 5 yr to Newcomb's Great Empirical Term of 300 yr. The astronomical evidence for these fluctuations in both wobble and l.o.d. has been discussed in chapter 5, and it is in the l.o.d. that they are most pronounced and well above the observational noise level. Most of the discussion will, in consequence, concern the l.o.d. decade variations. Observed since the early nineteenth century, these changes are illustrated in figure 5.3 in the form of m3 and m3. The characteristic time constant of the changes is 10-20 yr. Since the introduction of atomic time in 1955, the improved observations are indicative of a shorter time constant, about 5 yr. The wobble observations suggest a secular drift of the pole upon which an approximately 20-yr oscillation appears to be superimposed. But the reality of this term is open to question and may be a characteristic of the observing process itself, rather than of some geophysical excitation (see section 5.3).

The geophysical origin of the decade fluctuations has been subject to considerable debate and most discussions have centred around the possible role of the core.

The Chandler wobble, discovered in 1891 after a long and fruitless search for a 10-month period in astronomical latitude observations, is still associated with almost as much controversy today as it was then, and many of the questions that were raised by Chandler, Newcomb, Kelvin, Volterra, Larmor, Darwin and others are still with us. These questions relate to the three essential problems associated with the Chandler wobble, (i) Can the lengthening of the period, from the 305 d predicted for a rigid Earth to the observed 434 d, be explained quantitatively? (ii) Being a free motion, the Chandler wobble will ultimately be damped out but the astronomical record of near 150 yr does not show any indication of a gradually diminishing amplitude. What maintains the motion against damping? (iii) If damping occurs, where is the rotational energy dissipated? These are also the questions that we wish to discuss once again in this chapter.

Astronomical evidence for the Chandler wobble has been discussed in chapter 5. The main characteristic is a period of about 434 sidereal days. The broad spectral peak (figure 5.14) is very suggestive of damping and, if a linearly damped oscillation is assumed, the relaxation time is of the order of 25–40 yr; the wobble amplitude would decay to e–l of its original value in something like 25–40 yr. The associated Chandler wobble Q would be of the order of 60–100.

Polar wander

The concept of polar wander, the large-scale wandering of the Earth's axis of rotation throughout geological time, goes back more than 100 yr and has its origin in observations of fossil plant and animal distributions, and in the scars, tillites and moraines of past glaciations. For example, the discovery that a subtropical climate existed in Spitsbergen at a time when Central and Southern Europe were subject to a tropical climate and when extensive glaciations occurred in Southern Africa, led to the conclusion that, in the western hemisphere, the Carboniferous equator must have lain far to the north of the present equator. A further degree of freedom to the interpretation of the paleontological and paleoclimatic data was introduced by Wegener and by F. B. Taylor with their concept of continental drift, in which the continents are postulated to have moved relative to each other over large distances throughout geologic time. For lack of convincing observational evidence and for an absence of compelling theoretical arguments, the notions of polar wander and of continental drift both remained at the periphery of scientific responsibility until rescued from this limbo by two important paleomagnetic discoveries: that large changes have occurred in the mean direction of the geomagnetic field and that this field has periodically reversed itself. Evidence for both changes is found throughout the Phanerozoic and Proterozoic, permitting some conclusions to be drawn about the reality of drift and wander for much of the geologic record.