Stability and dynamics of shock, detonation, and boundary layers

Indiana University, Bloomington IN

Investigators

Abstract

The principal investigator proposes to study a variety of problems in stability and behavior of shock and boundary layer type solutions of the equations of compressible gas dynamics, with an emphasis on global (i.e., amplitude-independent) analysis; new functional-analytic tools suitable for treatment of time-periodic, discrete, and or kinetic waves; and practical numerical testing of stability and bifurcation, especially for the large systems resulting from inclusion of viscous heat- and magnetic- or electromagnetic-inductive, phase-transitional, and reactive effects, or by Fourier transform/truncation in transverse modes of genuinely multidimensional solutions such as nonplanar (i.e., varying in transverse, or nonaxial, coordinates) flow in a cylindrical duct. The latter is expected to provide quantitative information of interest to physical practitioners in shock and detonation theory. A larger goal is to move beyond simple stability analysis to the study of nontrivial dynamics including bifurcation, interaction, and behavior of complex flows. The project involves interesting and nonstandard issues in singular perturbation theory, dynamical systems and bifurcation, spectral theory of linear operators, and nonlinear partial differential equations, and should result in the development of new mathematical tools of general application. The plan of attack centers around Evans function and related spectral techniques developed recently by the investigator and various collaborators. The stability of regular flow patterns is an old and central topic in fluid, gas, and plasma dynamics, deciding which (stable) patterns will typically be observed, and which unstable) are only mathematical and not physically observable solutions. The transition from stability to instability is of particular importance, since it usually signals the appearance of alternative, more complicated flow patterns close to the original (now unstable) one- this is a way to understand complicated flows by the study of simpler and better-understood ones. Our goal is to move existing theory from the qualitative to the quantitative regime, obtaining new information of use to practitioners at the same time that we advance the mathematical theory. The planned activities expected to strengthen and extend existing networks of cooperation across field, and to aid in training of graduate and postdoctoral students. The ultimate aim of these investigations, of quantitative predictions of transition to instability, would, if achieved, be of direct and practical use at the level of engineering, in chemical, manufacturing, and other processes.