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This IMA Volume in Mathematics and its Applications DYNAMICAL ISSUES IN COMBUSTION THEORY is based on the proceedings of a workshop which was an integral part of the 1989-90 IMA program on "Dynamical Systems and their Applications." The aim of this workshop was to cross-fertilize research groups working in topics of current interest in combustion dynamics and mathematical methods applicable thereto. We thank Shui-Nee Chow, Martin Golubitsky, Richard McGehee, George R. Sell, Paul Fife, Amable Liiian and Foreman Williams for organizing the meeting. We especially thank Paul Fife, Amable Liiilin and Foreman Williams for editing the proceedings. We also take this opportunity to thank those agencies whose financial support made the workshop possible: the Army Research Office, the National Science Foundation and the Office of Naval Research. Avner Friedman Willard Miller, Jr. ix PREFACE The world ofcombustion phenomena is rich in problems intriguing to the math­ ematical scientist. They offer challenges on several fronts: (1) modeling, which involves the elucidation of the essential features of a given phenomenon through physical insight and knowledge of experimental results, (2) devising appropriate asymptotic and computational methods, and (3) developing sound mathematical theories. Papers in the present volume, which are based on talks given at the Workshop on Dynamical Issues in Combustion Theory in November, 1989, describe how all of these challenges have been met for particular examples within a number of common combustion scenarios: reactiveshocks, low Mach number premixed reactive flow, nonpremixed phenomena, and solid propellants.



Bifurcation, Pattern Formation and Chaos in Combustion

Problems in gaseous combustion and in gasless condensed phase combustion are studied both analytically and numerically. In gaseous combustion we consider the problem of a flame stabilized on a line source of fuel. We find both stationary and pulsating axisymmetric solutions as well as stationary and pulsating cellular solutions. The pulsating cellular solutions take the form of either traveling waves or standing waves. Transitions between these patterns occur as parameters related to the curvature of the flame front and the Lewis number are varied. In gasless condensed phase combustion both planar and nonplanar problems are studied. For planar condensed phase combustion we consider two models: (i) accounts for melting and (ii) does not. Both models are shown to exhibit a transition from uniformly to pulsating propagating combustion when a parameter related to the activation energy is increased. Upon further increasing this parameter both models undergo a transition to chaos: (i) by intermittency and (ii) by a period doubling sequence. In nonplanar condensed phase combustion the nonlinear development of a branch of standing wave solutions is studied and is shown to lead to relaxation oscillations and subsequently to a transition to quasi-periodicity.
Alvin Bayliss, Bernard J. Matkowsky

Mathematical Investigation of the Cold Boundary Difficulty in Flame Propagation Theory

Our aim in this paper is to describe a rigorous mathematical answer to the well-known paradox called the “cold boundary difficulty” in flame propagation theory. We essentially report here on some work done by J. M. Roque-joffre (see [23], [24], [25] where it appears in detail) in collaboration with the first two authors.
H. Berestycki, B. Larrouturou, J. M. Roquejoffre

Nonlinear Development of Low Frequency One-Dimensional Instabilities for Reacting Shock Waves

The classical theory of von Neumann, Zeldovich, and Doering (the ZND theory) from the early 1940’s postulates that detonation waves are steady traveling waves with a quasi one-dimensional structure of an ordinary fluid dynamic shock followed by a reaction zone. In contrast, the detonations observed in many experimental circumstances demonstrate extremely complicated unstable wave patterns (see Oppenheim & Soloukhin [1], Fickett & Davis [2], Lee & Moen [3]). The nature of these instabilities ranges from complex transverse Mach stems in gases to regular and chaotically irregular pulsating fronts in both gaseous and condensed phases. Understanding the mechanisms responsible for the correspondingly higher pressures in unstable detonations has obvious practical implications.
A. Bourlioux, A. Majda, V. Roytburd

Dynamics of Laminar Triple-Flamelet Structures in Non-Premixed Turbulent Combustion

In the spirit of laminar-flamelet modelling of non-premixed turbulent combustion, a diffusion flamelet is studied. However, the flamelet is also taken to end at a finite position. Such an end of a diffusion flame exhibits fuel-rich and fuel-lean premixed elements as well as the diffusion flame-sheet itself—a structure that is known as a triple-flame and which has the property of being able to propagate. A counterflow geometry with shear becomes the most relevant situation in which to picture ends of diffusion flames in a turbulent flow. In an equidiffusive system, the speed of propagation of the end-point is demonstrated to be positive only for relatively limited values of the strain or scalar dissipation rate and becomes large and negative towards the higher finite value at which a diffusion flame would extinguish uniformly. The implications of these findings for the behaviour of turbulent diffusion flames are discussed.
J. W. Dold, L. J. Hartley, D. Green

Free Boundary Problems and Dynamical Geometry Associated with Flames

We introduce several simplified free boundary problems capable of generating basic dynamical patterns that are peculiar to flame propagation. The evolution of free boundaries can in turn be modeled by appropriate equations of dynamical geometry that relate the normal velocity (or higher “normal” time derivatives) of the surface to its instantaneous geometrical characteristics. The discussion is aimed to initiating numerical simulation and rigorous study of these models.
Michael L. Frankel

On The Dynamics of Weakly Curved Detonations

Recent progress in detonation theory involves systematic perturbation analyses based on the fact that the radii of front curvature are typically large compared to an average reaction zone thickness. The present paper contributes to the theory for near-Chapman-Jouguet waves. It explains weakly nonlinear acoustic effects in the near-sonic burnt gas flow which act on time scales much shorter than the overall evolution time of the front.
The transonic flow behind a diverging wave approaches a quasisteady limit solution and a relation, D = D(κ), between the front curvature and the detonation speed emerges. In this way, the results of earlier quasisteady theories are recovered. We add an analysis of equilibrium effects in the small curvature limit. In contrast to the quasisteady behavior of diverging waves, focusing waves undergo a continous acceleration on the acoustic time scale according to D/D CJ = 1+C(κt)2+…
Up to this point, a quasi-onedimensional treatment of the reaction zone is sufficient but the accelerating focusing waves eventually violate the underlying asymptotic assumptions and extensions of the theory are required. The paper considers a regime of small transverse length scales where multidimensional effects become important in the transonic flow before the detonation speed exceeds the Chapman-Jouguet value by order O(1). The wave evolution is then governed by the unsteady multidimensional transonic flow equations in a halfspace with an interesting coupling to the boundary which arises due to curvature effects. Important for the dynamics of gasphase detonations are temperature sensitive induction reactions that precede the main heat releasing steps. The paper concludes with an outline of a regime where changes in the induction zone length directly interact with the multidimensional transonic layer flow.
Rupert Klein

Simplified Equations for Low Mach Number Combustion with Strong Heat Release

The familiar system of reaction-diffusion equations
$$ {\rho _0}{C_p}\frac{{DT}}{{Dt}} = \Delta T + {q_0}{\rho _0}K{e^{ - A/AT}}Z\frac{{DZ}}{{Dt}} = {(Le)^{ - 1}}\Delta Z - K{e^{ - A/T}}Z $$
describes combustion processes at low Mach numbers provided that the heat release is sufficiently weak ([1]); with weak heat release, the hydrodynamic flow field decouples from the equations in (0.1) to leading order.
Andrew Majda, Kevin G. Lamb

Attractors And Turbulence for Some Combustion Models

A relevant tool to study the long-time behavior of evolution equations is the concept of universal at tractor. This at tractor is the natural mathematical object describing the permanent regime and, according to ideas first introduced by Smale [18], Ruelle-Takens [16], its complicated structure is the cause (or one of the causes) of the chaotic or turbulent behaviors. Therefore, an understanding of these sets provides a better understanding of chaos. In particular, studying their dimensions gives information on the complexity of the flow which can be very relevant from a physical point of view. For example, in the case of the Navier-Stokes equations, one recovers in that way the estimate of the number of degrees of freedom predicted by the Kolmogorov theory of turbulence (see Constantin-Foias-Manley-Temam [6]).
Martine Marion

Linear Stability of One-Dimensional Detonations

We examine the one-dimensional stability of plane detonations characterized by one-step Arrhenius kinetics, using numerical techniques. A pseudo-spectral method, specifically a collocation scheme with Tchebychev polynomials, provides an approximation to the unstable discrete spectrum, in addition to permitting the calculation of stability boundaries in parameter space. We show that behavior predicted by Buckmaster and Neves (Physics of Fluids 31 (12) December 1988, pp. 3571–6) using activation energy asymptotics, can occur for physically realistic values of the activation energy. That is, the growth rate of the modes (real part of the eigenvalue) can be a non-monotone function of the frequency (imaginary part). The present work forms part of the Ph.D. thesis of the first author to be submitted to the faculty at the University of Bordeaux.
G. S. Namah, C. Brauner, J. Buckmaster, C. Schmidt-Laine

Discrete Modeling of Beds of Propellant Exposed to Strong Stimulus

This paper is a description of experimental and theoretical modeling concepts that are being developed to describe the behavior of confined propellant beds made of discrete particles, when subjected to a strong stimulus such as the impact of a penetrating jet. A principal goal is to discover basic mechanisms within the bed that control a reactive wave that either fails and extinguishes, or propagates as a rapid burning or detonating wave. This paper is an extended version of a lecture given by D. S. Stewart at the Institute for Mathematics and its Application in November of 1989.
D. Scott Stewart, Blaine W. Asay
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