Hydrodynamic instabilities and coherent structures. Final technical report, 1990--1997

This project has had two related parts: (1) Nonlinear behavior of wavy film flows, and (2) Linear and nonlinear stability of space-time-periodic flows. In this project, the author has developed novel methods of perturbative derivations of evolution equations governing film flows. In particular, these methods allow one to obtain a priori parametric regions of validity for such approximate descriptions of the nonlinear film dynamics. The methods yield the most general evolution equations, thus eliminating the need for a new derivation every time a new parametric domain is considered (corresponding to one choice of the infinite number of possible different order-of-magnitude relations between the system parameters). New concepts were introduced to advance understanding of these matters. New evolution equations were derived, and their numerical solutions were worked out. The latter uncovered new and unexpected types of nonlinear dynamical phenomena, including particle-like interactions of localized coherent structures having emerged spontaneously and, under certain conditions, proceeding to self-organized process of formation of patterns with unprecedented (at least in fluid dynamics) complexity of order (in an intuitive sense). Exciting agreements were obtained with recent important real-life experiments performed by different groups of researchers. Large-scale instability of space-periodic flow has been studied in connection with such intriguing phenomena in hydrodynamic turbulence as inverse cascade of energy to--and negative eddy viscosity at--large scales of motion and self-organization of large-scale coherent structures. Certain results can be obtained by considering a statistical ensemble of flows such as in Lipscombe, Frenkel and ter Haar 91. However, the main thrust in this project has been the study of instability of deterministic viscous flows sustained by a periodic external forcing. Whereas time-independent flows were considered before, the author included into consideration the periodic dependence on time; it stands to reason that flows periodic in both space and time are more appropriate for turbulence modeling. The conceptual foundation for such studies was worked out in Frenkel 91a. That approach, reducing a differential eigenvalue problem to an algebraic one, proved in the course of studies very useful in clarifying the question of generality of previous conclusions even for steady flows, where only very special, having high symmetry flows had been considered before. The author also emphasized the importance of intermediate-scale instabilities (which cannot be captured by the alternative, differential-equation approach popular in the field), and developed the first nonlinear theory of such instability