Michael L. Frankel

On self-acceleration of outward propagating wrinkled flames

Abstract Numerical simulations of hydrodynamically unstable outward propagating premixed flames are presented. In a qualitative agreement with many experimental observations it is shown that the expanding wrinkled flames enjoy an appreciable acceleration. The morphology of expanding wrinkled flames appears to be essentially different from that occuring in non-expanding flames. In the latter case short-wavelength corrugations merge forming a single cusp, whose scale is controlled by the overall size of the system. In expanding flames the tendency to merge is balanced by the overall stretch. As a result the flame interface appears to be more or less uniformly wrinkled. In contrast to hydrodynamically unstable flames, expanding flames, subject exclusively to the effect of diffusive instability, do not indicate any disposition towards acceleration.

A Sequence of Period Doublings and Chaotic Pulsations in a Free Boundary Problem Modeling Thermal Instabilities

A simplified one-sided model associated with combustion and some phase transitions has been solved numerically. The results show a transition from the basic uniform motion of the free boundary to chaotic pulsations via periodic oscillations and a clearly manifested sequence of period doublings. For both numerical and rigorous treatment, the free boundary problem presents clear advantages over the two commonly used classes of models: the free interface (two-sided) models and the models with distributed kinetics. It is argued that, in view of its generic nature and relative simplicity, the problem may serve as a canonical example of the thermal instability leading to a variety of self-oscillatory regimes.

Galloping and spinning modes of subsonic detonation

A reduced model for a pressure-driven subsonic combustion wave spreading through an inert porous medium (subsonic detonation) is derived. It is shown that the associated planar travelling wave solution may lose its stability assuming a galloping or spinning structure as occurs in supersonic free-space detonation. The problem of subsonic detonation is found to be dynamically akin to the problem of gasless combustion known for its rich pattern-forming dynamics.

Complex dynamics generated by a sharp interface model of self-propagating high-temperature synthesis

This paper presents results of a numerical study of a free-interface problem modelling self-propagating high-temperature synthesis (solid combustion) in a one-dimensional infinite medium. Evolution of the free interface exhibits a remarkable range of dynamical scenarios such as finite and infinite sequences of period doubling; the latter leading to chaotic oscillations, reverse sequences and infinite period bifurcation that may replace the supercritical Hopf bifurcation for some interface kinetics. Solutions were verified by using different numerical methods, including reduction to an integral equation for which convergence to the solutions has been demonstrated rigorously. Therefore, the ability of the free-interface model to generate the dynamical scenarios observed previously in models with a distributed reaction rate should be regarded as firmly established.

On the equation of a curved flame front

Abstract A strongly nonlinear, spatially invariant equation for the dynamics of premixed flame is derived, on the assumption that the curvature of the flame front is small. The equation generalizes the corresponding weakly nonlinear equation, obtained previously near the stability threshold.

On the nonlinear evolution of a solid-liquid interface

Abstract A strongly nonlinear, spatially invariant equation for the dynamics of interface is derived, using a free-boundary problem with the linear attachment kinetics and zero anisotropy. The equation turns out to be qualitatively identical to that describing the thermo-diffusive propagation of flame fronts.

Finite-dimensional dynamical system modeling thermal instabilities

We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary problem modeling solid combustion and rapid solidification, and is capable of generating its major dynamical patterns. These patterns include a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. A computer-assisted bifurcation analysis uncovers some novel mechanisms of stability exchange. The most striking of them is an infinite period bifurcation which resembles the classical Shilnikov bifurcation, but instead of a funnel-shaped spiral along which the period is continually increasing, the continuation produced a series of isolas. Each isola is a closed branch of solutions of roughly the same period, and with the same number of oscillations. The isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other, and appear to accumulate on a saddle-focus homoclinic connection of Shilnikov type. ©2000 Elsevier Science B.V. All rights reserved.

DYNAMICAL PORTRAIT OF A MODEL OF THERMAL INSTABILITY: CASCADES, CHAOS, REVERSED CASCADES AND INFINITE PERIOD BIFURCATIONS

The paper presents results of numerical simulations on a model free boundary problem which is qualitatively equivalent to the free interface problems describing solid combustion and exothermic phase transitions. The model problem has been recently shown to exhibit transition to chaotic oscillations via a sequence of period doubling, assuming an Arrhenius type boundary kinetics. In the present paper we demonstrate that for a slightly different class of kinetics the behavior pattern, while retaining the above scenario, may undergo a drastic change. This behavior is characterized by slowly expanding oscillations followed by a powerful burst, after which the system returns to near equilibrium and the scenario is repeated periodically. As the bifurcation parameter approaches the stability threshold, the total period tends to infinity due to an increasingly prolonged “accumulation phase.” Additional scenarios corresponding to increasing supercriticality of the bifurcation parameter include finite period doublin...

Thermo-kinetically controlled pattern selection

Through a combination of asymptotic and numerical approaches we investigate bifurcation and pattern formation for a free boundary model related to a rapid crystallization of amorphous films and to the self-propagating high-temperature synthesis (solid combustion). The unifying feature of these diverse physical phenomena is the existence of a uniformly propagating wave of phase transition whose stability is controlled by the balance between the energy production at the interface and the energy dissipation into the medium. For the propagation on a two-dimensional strip with thermally insulated edges, we develop a multi-scale weakly-nonlinear analysis that results in a system of ordinary differential equations for the slowly varying amplitudes. We identify a nonlinear parameter which is responsible for the pattern selection, and utilize the amplitude system for predicting the evolving patterns. The pattern selection is confirmed by direct numerical simulations on the free boundary problem. Some numerical results on strongly nonlinear regimes are also presented.

Finite-dimensional model of thermal instability

Abstract Using the collocation method, we derive a finite-dimensional (3-D) dynamical system whose dynamics mimic behavior of a free boundary problem for the heat equation that describes solid combustion and some phase transitions and has been shown to develop complex dynamical patterns. Numerical simulations on the 3-D system demonstrate a variety of dynamical scenarios depending on boundary kinetic functions inherited from the original problem. The multitude of nontrivial dynamical features as well as a clear physical origin of the system make it interesting in its own right.