Victor Roytburd

Theoretical and numerical structure for reacting shock waves

Several remarkable theoretical and computational properties of reacting shock waves are both documented and analyzed. In particular, for sufficiently small heat release or large reaction rate, we demonstrate that the reacting compressible Navier–Stokes equations have dynamically stable weak detonations which occur in bifurcating wave patterns from strong detonation initial data. In the reported calculations, an increase in reaction rate by a factor of 5 is sufficient to create the bifurcation from a spiked nearly Z-N-D detonation to the wave pattern with a precursor weak detonation. The numerical schemes used in the calculations are fractional step methods based on the use of a second order Godunov method in the inviscid hydrodynamic sweep; on sufficiently coarse meshes in inviscid calculations, these fractional step schemes exhibit qualitatively similar but purely numerical bifurcating wave patterns with numerical weak detonations. We explain this computational phenomenon theoretically through a new class of nonphysical discrete travelling waves for the difference scheme which are numerical weak detonations. The use of simplified model equations both to predict and analyze the theoretical and numerical phenomena is emphasized.

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.

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.

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.

Numerical Modeling of the Initiation of Reacting Shock Waves

The transition to detonation in gases is a very complicated multifaceted process. Turbulent mixing, interaction of acoustic waves with underlying chemical reactions, formation of regularly spaced Mach stem structures—this is just a partial list of phenomena taking part in the transition from deflagration to a self-sustained detonation. (See the review article [7] for an experimentalist’s summary). In this paper through carefully documented numerical experiments we investigate one aspect of the transition process which is also related to the direct initiation of reacting shock waves.

A Hopf bifurcation for a reaction-diffusion equation with concentrated chemical kinetics

Abstract A reaction-diffusion equation related to some mathematical models of gasless combustion of solid fuel is studied. A formal bifurcation analysis by B. J. Matkowsky and G. I. Sivashinsky (SIAM J. Appl. Math. 35 (1978), 465–478) shows that solutions demonstrate behavior typical for the Hopf bifurcation. A rigorous treatment of this phenomenon is developed. The problem is considered as an evolution equation in a Banach space. To circumvent difficulties involving a possible resonance with the continuous spectrum, appropriate weighted norms are introduced. A suitable version of the Hopf bifurcation theorem is developed and the existence of time periodic solutions is proved for values of the parameter near some critical value.

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.

A free boundary problem modeling thermal instabilities: well-posedness

In this paper, the authors analyze a simple free boundary model associated with solid combustion and some phase transition processes. There is strong evidence that this “one-phase” model captures many salient features of dynamical behavior of more realistic (and complicated) combustion and phase transition models. The main result is a global existence and uniqueness theorem whose proof is based on a uniform a priori estimate on the growth of solutions. The techniques employed are quite elementary and involve some maximum principle type estimates as well as parabolic potential estimates for the equivalent integral equation.