Geometric Structure of the Turbulent Cascade

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The majority of fluid flows in nature and in engineering applications are turbulent. But despite being so commonplace, understanding and modeling turbulence remains a significant challenge. Turbulent flows are distinct from more general unsteady flows because they display a characteristic cascade of energy from the large length scales at which energy is generated to the small length scales on which it is dissipated by viscosity. Typically, this process is described in an abstract way that makes it difficult to appreciate the underlying physics. The goal of this proposed research is to recast the turbulent energy cascade as a mechanical process, where large scales do work on the small scales. This model immediately emphasizes the important role played by geometry, since if the forces provided by the large scales are misaligned with the flow direction, they can do no work. In this proposed research, the geometry of the turbulent cascade will be studied in detail and its links to other turbulent processes will be clarified. Ultimately, the results of this research may lead to new strategies for turbulence modeling. The proposed research will support the education and training of graduate students, and the results will be folded into existing graduate courses. Additionally, the supported scientists will participate in educational outreach activities coordinate through Stanford's Bob and Norma Street Environmental Fluid Mechanics Laboratory. The objective of this proposed research is to gain a deeper understanding of how the geometric properties of scale-dependent turbulent stresses and strain rates and, in particular, their relative alignment control the turbulent cascade. Specifically, the alignment and possible spatial ordering of the turbulent stress and strain rate will be characterized, an understanding of how this alignment is modulated by advection will be developed, and potential links between alignment and intermittency will be investigated. These questions will be studied via theoretical work and analysis of data sets for three-dimensional turbulence, two-dimensional turbulence, and unsteady, random, and multiscale but non-turbulent velocity fields. By basing this research on sound and transparent mechanical principles of work and energy transfer, the results will bring new clarity to the origins of the detailed structure of turbulence that is more interpretable than abstract approaches such as multifractality.