Extreme Events and Optimal Closures in Fluid Mechanics

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This application seeks funding for Dr. Protas' ongoing research program which will offer a new perspective on a number of classical and emerging fundamental problems in the area of theoretical fluid dynamics. By performing carefully designed and executed numerical computations we will shed light on how well the rigorous mathematical analysis can describe actual flow behaviours. We will also develop optimal ways of simplifying the mathematical description of complex fluid flows.The first class of open questions we will investigate addresses forms of extreme behavior possible in flows of incompressible fluids. Such "extreme behavior" may concern the spontaneous growth of energy-like quantities characterizing the flow structure and might potentially manifest itself in the formation of singularities in finite time. Related questions arise, for instance, in the characterization of the maximum possible mixing of passive scalars or convective heat transfer. In addition to assessing the fundamental performance limitations of various flow processes in science and engineering, these issues are in the first place relevant for our basic understanding of how the mathematical models of fluid flow behave. An example of such a question is one of the Clay Institute's "Millennium Problems" for the mathematical community concerning the well-posedness of the Navier-Stokes system in 3D which remains notoriously unresolved. The ultimate goal of the proposed research program is to determine systematically, via a suitable process of numerical optimization, smooth Navier-Stokes flows in 3D exhibiting a worst-case behavior in order to understand whether such behavior may be compatible with singularity formation in finite time. As intermediate objectives, we will address analogous questions for a range of related simplified flow models in which such singular behavior is known, or suspected, to exist. These results will provide key new insights about the sharpness of the mathematical analysis and the properties of mathematical fluid models with ubiquitous applications, bridging in this way mathematical analysis with large-scale scientific computations.The second class of open problems is motivated by the development of reduced-order models for turbulent flows. Given the multiscale complexity exhibited by such flows, their complete resolution in numerical computations will always be a challenge and some form of approximation is needed to represent unresolved dynamics. Although such problems have been studied for a long time, almost all of the work has been based on purely empirical approaches. Mathematically optimal solutions to this so-called "closure problem" are the second major open question we will address. In addition to offering an entirely new way to design such closure strategies, our work will also answer questions about the fundamental performance limitations of these approaches.A common theme in this proposed research effort is that these different questions will be framed in terms of variational optimization problems amenable to solutions using modern methods of numerical optimization. During this project we will offer uniquely broad interdisciplinary training to a large number of trainees at different levels. It will span applied analysis and large-scale scientific computation, and will be combined with applied fields such as fluid mechanics and model reduction.