Despite their importance in many areas of engineering, nonequilibrium systems remain difficult to analyze, to control, to design, or to predict because of the nonlinear way that spatiotemporal patterns affect the transport of energy and matter, which in turn modifies the spatiotemporal patterns. Examples include the weather and climate, the efficiency of combustion and chemical reactions, the convection of biological organisms in the oceans, heart dynamics, crystal growth from a melt, and fluid turbulence. A particular challenge is to understand spatiotemporal chaos, a commonly observed behavior of nonequilibrium systems where properties of the system evolve aperiodically in time and space. New fundamental insights into the spatiotemporal chaos of spatially-extended nonequilibrium systems will be obtained through a detailed numerical investigation of Rayleigh-Benard convection (a thin horizontal layer of fluid heated uniformly from below). The PI has developed parallel numerical methods providing accurate simulations for the precise conditions of experiment. This research builds upon these successes to explore spatiotemporal chaos in large-aspect-ratio convective domains to make predictions that can be verified by experiment. These predictions are only possible by the recent convergence of increased computing power and improved numerical algorithms, including the continuing research progress of the PI. The research will probe the origins and basic building blocks of spatiotemporal chaos to quantify the number, size, and dynamics of the individual chaotic degrees of freedom. Numerical simulations will also shed new insight upon transport in a chaotic flow field. As examples, an exploration of the enhancement of combustion efficiency in premixed gases by complex fluid velocity fields will directly affect energy production and consumption; and an understanding of the fluid convection driven by the activity of biological organisms suspended in oceans will improve models of the climate. The education program is tightly coupled with this research to provide extensive opportunities for students at all levels to participate in state-of-the-art engineering research. The PI's pre-college outreach program is focused upon exposing a large group of students, with special emphasis on under-represented groups, to challenges facing engineering today with the goal of attracting, retaining, and eventually graduating a more diverse group of world-class engineers. The PI is working closely with the Virginia Tech Center for the Enhancement of Engineering Diversity to develop and implement programs that will reach over 400 pre-college students each year. The PI will develop, organize, and lead problem solving sessions that are guided by hands-on interactive numerical experiments. The numerical experiments will be directly related to this research and will spark the interests of young students with such subjects as the difficulty of weather prediction and the scientific meaning of the popular phrase "the Butterfly Effect." The interactive programs will be written in Java and publicly available on a computational science and engineering web site established for this purpose. The PI will mentor undergraduate students each academic year and each summer on projects related to this research. Students will be selected from the Multicultural Academic Opportunities Program (MAOP) and a NSF funded Summer Undergraduate Research Program (SURP). A new multidisciplinary graduate course will be developed entitled "Spatiotemporal Chaos." A major theme of the course will be the quantitative link between theory and experiment provided by the computations of this research. The education program will be carefully assessed and improved through a close collaboration with the Virginia Tech Engineering Education Department.
The research of this CAREER grant explored the complex and chaotic dynamics of very large systems that are driven externally to be far-from-equilibrium. An important goal of this research was to use large-scale numerical simulation to gain new physical insights. Example problems of interest include: the complex dynamics of the weather and climate; the collective motion of microorganisms in the oceans and rivers; the spread of forest fires; and the combustion of premixed gases in a complex flow field. This was accomplished using large-scale computations and highly-parallel numerical algorithms to integrate complex systems of governing nonlinear equations. Significant effort and attention was given to perform these numerical simulations for the precise conditions of experiment. This research used numerical simulation to provide a quantitative link between available theoretical descriptions and experimental measurements to generate new fundamental insights. The specific problems explored were the chaotic dynamics of a thin layer of fluid heated uniformly from below in a gravitational field (Rayleigh-Benard convection), the collective motion of a large population of swimming microorganisms (bioconvection), the dynamics of a reacting and propagating front in a complex flow field, the dynamics of a model equation describing fluid convection (Swift-Hohenberg equation), and the dynamics of a simple model with relevance to weather prediction (Lorenz-96 equation). Using these systems we explored a wide range of parameters that yielded complex pattern dynamics and chaos. We computed theoretically important diagnostics that are currently inaccessible to both theory and experiment for these systems. This allowed us to quantify and probe the origins and implications of nonlinear dynamics and chaos for large spatially-extended systems of practical importance. The new physical insights gained from this research will be used to guide future theoretical and experimental efforts to further improve our understanding of the complex dynamics of nonlinear and chaotic systems that are driven far-from-equilibrium. This research supported the training of 4 graduate students and 3 undergraduate students in the growing and important field of computational science and engineering. These students have continued on in positions in academia and industry. The results of this research was widely disseminated through presentations at national and international conferences, student theses, and published journal articles. Hands-on numerical workshops exploring chaos and fractals were developed, and conducted, for over 250 middle school and high school students with the goal of capturing the interests of the next generation of engineers and scientists. A new graduate course called "Chaos and Nonlinear Dynamics" was developed at Virginia Tech and has generated broad interest.
