Studies in Mathematical Physics: Advection, Convection and Turbulent Transport

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This research project in mathematical physics and applied analysis is a study of qualitative and quantitative properties of solutions of the partial differential equations of fluid mechanics including the Navier-Stokes equations. The Navier-Stokes equations constitute the basic mathematical model of fluid flow and are believed to contain turbulence among their solutions. Turbulent transport and mixing have important applications in, and implications for, many areas of applied physical sciences and engineering and present a number of outstanding challenges for theoretical physics and applied mathematics. The investigations will be carried out utilizing modern applied analysis, computation and numerical simulation with graduate students and postdoctoral researchers working under the direction of Principal Investigator Charles R. Doering at the University of Michigan. The project has three major components: . Mathematical methods previously developed by principal investigator and collaborators for the study of turbulent transport in the Navier-Stokes and related equations will be extended and applied to the advection-diffusion equation and turbulent mixing. This analysis will place limits on mixing efficiencies for passive scalar fields in terms of bulk and statistical features of the applied flows, and indicate key features of particularly efficient or inefficient stirring strategies. . Theoretical and mathematical issues in thermal convection will be studied via rigorous analysis and direct numerical simulation. Modern enhancements of the analytical techniques pioneered by the principal investigator will be developed and applied to open problems including homogeneous convection, Rayleigh-Benard convection with free-slip boundaries, infinite Prandtl number models and flows driven by internal heating with applications in geophysics. . The turbulent energy cascade and enstrophy generation will be investigated for solutions of the incompressible Navier-Stokes equations. Variational approaches capable of bounding turbulence driven by time-independent body-forces will be extended and applied to time-dependent and broadband (fractal) forcing. Work in progress will continute to determine maximum enstrophy generating flow-field configurations, how they are related to structures observed in fully developed turbulence, and their role in the development of singularities. With regard to the intellectual merit of this activity, knowledge gained from this project will contribute to fundamental understandings of mathematical models in fluid dynamics that are of direct relevance to many branches of applied science and engineering. In the long term this research wll aid the development of practical techniques for simulation, prediction and control of physical processes with applications ranging from meteorology to materials manufacturing. With regard to this activity's even broader impacts, there are several significant advanced training aspects to this project: it provides frontier dissertation research opportunities for graduate students in Michigan's Ph.D. program in Applied & Interdisciplinary Mathematics and support and guidance for postdoctoral researchers at the University of Michigan. This research also involves collaborations and interactions with investigators, including graduate students and postdoctoral researchers, from other institutions.