This project is a study of qualitative and quantitative properties of solutions of the partial differential equations of fluid mechanics including the Navier-Stokes equations. The latter 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 many areas of applied physical sciences and engineering and present a number of outstanding challenges for mathematical physics. The investigations will be carried out utilizing modern applied analysis, computation, and numerical simulation in collaboration with graduate students performing doctoral research and postdoctoral researchers working under the direction of the PI. The project has three major components.
Advection: Mathematical methods developed by the PI and collaborators will be applied to the advection-diffusion equation and turbulent mixing. This analysis will place absolute limits on diffusive enhancements for passive scalar fields in terms of bulk and statistical features of the stirring flows, and indicate particularly efficient or inefficient stirring strategies. New searches for optimal stirring strategies will be undertaken, and the mixing effectiveness of turbulence will be investigated. Convection: Theoretical issues in thermal convection will be studied using rigorous analysis and numerical simulation. Differences between convective turbulence sustained by fixed heat flux and fixed temperature conditions will be investigated. The analytical techniques of the PI will be developed and applied to surface tension driven convection. Energy dissipation and enstrophy production: Work will continue to determine how maximum enstrophy generating flow-field configurations are related to structures observed in fully developed turbulence. Variational approaches for the derivation of a priori bounds on energy dissipation rates for complex and turbulent flows will be extended to flow configurations relevant to geophysical and astrophysical applications.
Knowledge gained from this project will contribute to our fundamental understanding of mathematical models in fluid dynamics, of direct relevance in the applied physical sciences and engineering. With regard to this activitys broader impacts in education, it provides frontier dissertation research opportunities for doctoral students and support for postdoctoral researchers at the University of Michigan. This research also involves extensive collaborations and interactions with investigators from institutions worldwide. In the long term this research will aid the development of practical techniques for applications ranging from aeronautics to astrophysics, and meteorology to materials manufacturing.
This research project in mathematical physics and applied analysis was 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. The project had three major components: • Advection: Mathematical methods developed by principal investigator and collaborators wereapplied to the advection-diffusion equation and turbulent mixing. This analysis placed absolute limits on diffusive enhancements for passive scalar fields in terms of bulk and statistical features of the stirring flows, and indicated key features of particularly efficient or inefficient stirring strategies. New searches for optimal stirring strategies were undertaken, and the mixing effectiveness of turbulence was investigated. • Convection: Theoretical and mathematical issues in thermal convection were studied via rigorous analysis and direct numerical simulation. Fundamental differences between convective turbulence sustained by fixed heat flux and fixed temperature conditions were investigated, and enhancements of the analytical techniques pioneered by the principal investigator were extended and applied to surface tension driven convection. • Energy dissipation and enstrophy production: Variational approaches for the derivation of a priori bounds on energy dissipation rates for complex and turbulent flows were extended to flow configurations relevant to geophysical and astrophysical applications, including surface-stress driven flows. The key scientific outcomes were documentd and published in the refereed scientific literature. The Principal Investigator, postdocs, and graduate students involved in this project also presented various components of the work at professional conferences and workshops. This work was carried in collaboration with graduate students performing doctoral research and postdoctoral researchers working under the direction of Principal Investigator Charles R. Doering at the University of Michigan. Several doctoral students participated in this project. Indeed, three of those who received support from this award recently received their PhD degrees and moved on to postdoctoral positions. The postdoc supported on this project is now a tenure-line Asstistant Professor of Mathematics at the U.S. Naval Academy.
