From Self-similar Solutions to Turbulent Cascades

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Turbulent fluid flows are both ubiquitous and immensely important in the modern society. They shape climate and weather, control transmission of many infectious diseases and the spreading of pollution, and determine the fuel-efficiency of our cars, airplanes, and ships. Yet, many of turbulent flows’ properties remain mysterious despite centuries of systematic research. In particular, we do not fully understand the principles that control the motion of fluids such at air and water at very small scales. As a result, despite continuing advances in computing power, numerical simulations resolving the small-scale structure of turbulent flows remains out of reach, limiting our ability to make sustained improvements in numerous civilian and military applications. This project harnesses recent numerical and theoretical advances to describe how fluid motions at one scale can generate motions at another (often much larger or much smaller) scale, yielding a hierarchy of eddies responsible for the unique and beautiful structure of fluid turbulence. This research project uses a combination of novel numerical methods and innovative theoretical approaches to addresses several fundamental problems in fluid turbulence that remain largely unsolved despite almost a century of concerted effort. One of the oldest, most fundamental, and least understood issues in fluid turbulence is the multi-scale structure of the flow at high Reynolds numbers. Such structure emerges due to cascades of various quantities, such as energy, vorticity, and helicity, describing the fluid flow. Our understanding of the physical mechanisms of respective cascades is very limited. For instance, it is not entirely clear what determines the direction of the cascades, i.e., whether the flux of a particular quantity is towards large scales (inverse cascade) or small scales (direct cascade). This project aims to develop a dynamical description of direct and inverse cascades in two spatial dimensions. Advanced numerical methods for finding self-similar solutions to inviscid governing equations are used to uncover the fundamental physical mechanisms describing scale interaction in fluid flows that is responsible for the multi-scale nature of developed turbulence. Established tools from dynamical systems theory are leveraged to develop a comprehensive quantitative theory of turbulent cascades that does not rely on unproven assumptions and approximations that underlie existing statistical theories of turbulence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.