Coherent Structure, Chaos, and Turbulence in Fluid Mechanics

Fluids that we interact with on a daily basis (such as water and air) display a remarkable variety of dynamics. At one extreme is chaos and turbulence, wherein the fluid forms complicated fractal-like patterns, the details of which cannot be exactly repeated in experiments due to being very sensitive to small changes in the initial conditions. Turbulence is observed often in fluids, such as in the wakes of aircraft, vehicles, and even obstacles such as buildings and bridges and in the large-scale dynamics of the ocean and atmosphere.

At the opposite extreme is the motion of vortex filaments, the most notable examples being tornadoes and the wing-tip vortices commonly shed from wings and helicopter blades. Vortex filaments tend to move in a predictable manner and maintain their structural integrity for extended periods of time. The goal of this project is to develop a better mathematical understanding of these opposite phenomena in fluid mechanics.

The accurate prediction of vortex filaments and turbulence is crucial in a variety of scientific and industrial applications, including in the design of air, land, and sea vehicles and in the understanding of complex fluid systems such as the climate and weather. Having a firm mathematical foundation could help other applied researchers obtain deeper insights and lead to better modeling.

Further, an understanding of these extremes helps pave the way for a better understanding of the interactions and intermediate regimes, where flows have a mix of structure and chaos. Finally, overcoming the mathematical challenges to these questions will require innovations that will be of interest to the wider mathematical community. The research projects are also integrated with the training of graduate students and younger scientists in mathematics and STEM.

The PI will develop a more mathematically rigorous understanding of two behaviors observed in incompressible fluids at high Reynolds numbers: (1) the coherent motion of vortex filaments; (2) turbulence under "generic", statistically steady forcing. The fundamental questions motivating the PI are: (A) how accurate are the commonly used geometric evolution models such as the Local Induction Approximation (LIA) for the motion of a vortex filament in a fluid with vorticity concentrated on a smooth curve? (B) can we provide a proof for the experimentally observed positive Lyapunov exponents and anomalous dissipation (e.g., as the celebrated Kolmogorov 4/5 law) from the stochastically forced 3D Navier-Stokes in the high Reynolds number limit?

These require a number of unexplored mathematical ideas and currently, there exists no clear way to attack them yet. Instead, the PI has identified several independently interesting problems to build necessary mathematical foundations. For (A), the PI and his collaborators will study vortex filaments in quantum fluids governed by the Gross-Pitaevskii equation.

The quantum case is expected to be easier than the classical case due to the quantization of vorticity and slightly more amenable linearized operators. First, the PI and collaborators will study the stability of vortex solutions in 2d Gross-Pitaevskii, providing necessary ground for understanding the filament core. Next, the PI and collaborators will show that the LIA accurately describes the motion of nearly-straight, and potentially more general, quantum vortex filaments long enough to make useful predictions.

For (B), the PI and his collaborators will: (1) develop novel qualitative theory for Lyapunov exponents in stochastic PDEs inspired by ideas from random dynamical systems; (2) develop better tools for quantitative hypoelliptic regularity and Lyapunov exponent estimation in high dimensional systems; and (3) study quantitative and nonlinear aspects of Lagrangian chaos, that is, how the chaotic dynamics of particles in a fluid can be translated into nonlinear dynamics of the fluid itself.

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.