Small Scale Dynamics in Fluid Mechanics

About This Grant

The fluids we interact with on a daily basis (for example, air and water) display a wide variety of multi-scale phenomena, that is, interesting and important dynamics occurring at very different scales in space and time. For example, the thin boundary layer of fluid along the wing of an aircraft or a moving car can be only a few millimeters thick, and yet the dynamics of this thin layer can have a dramatic effect on the flight characteristics of the aircraft in terms of maneuverability, speed, and fuel efficiency. These so-called viscous boundary layers are one very important small scale in fluids. Another example are the complicated set of whorls and vortices one observes when the fluid is undergoing turbulence, like for example in the wake of a vehicle. In this situation there are many different active scales and the statistics of these patterns has a profound effect on the overall behavior of the fluid, for example, often creating a great deal of additional air drag on the vehicle. A third more subtle example is in the interaction between wave-like motion and a background flow, for example surface water waves traveling across a large gyre. While the small scales may not be immediately visible, in fact they play an important role in determining the interaction between the waves and large scale vortex structures. Accurate understanding of boundary layers, turbulence, and other such small scales, are crucial in a variety of scientific and industrial applications, including in the design of air, land, and sea vehicles and in the understanding of more complicated fluid-like systems such as the weather or confined fusion plasmas. The principal investigator (PI) will work to build firm mathematical foundations for understanding these phenomena more precisely, which could help other applied researchers obtain deeper insights, lead to better modeling, and better computational methods for these problems. Further, a better understanding of these extremes helps pave the way for a better understanding of the interactions and intermediate situations, for example, flows that have a mix of structure and chaos. Finally, overcoming the inherent mathematical challenges to these questions will require multiple innovations that will be of interest to the wider mathematical and scientific community. The research projects are also integrated with the training and mentorship of graduate students, post-doctoral scholars, and younger scientists in mathematics and STEM at large in order to help build and maintain a strong scientific and engineering expertise in the United States. The PI will work to develop a more mathematically rigorous understanding of three behaviors observed in incompressible fluids at high Reynolds numbers and in similar systems: (A) chaos and turbulence in statistically stationary models subjected to random forcing; (B) the structure and stability of sharp transition layers, such as boundary layers and shock layers; (C) the interaction of large scale waves (such as atmospheric Rossby waves) with the small-scale filamentation created by disturbing a large background shear flow or vortex. The primary long term goal motivating Program (A) is to provide a proof for the observed positive Lyapunov exponents and anomalous dissipation (e.g. as the celebrated Kolmogorov 4/5 law) from the stochastically-forced three-dimensional (3D) Navier-Stokes equations or other similar turbulent models. For program (B) it is to determine the dynamics of laminar boundary layers over long times in the presence of large scale flows. For program (C), it is to provide mathematical analysis of filamentation/inviscid damping dynamics interacting with large-scales waves in a nonlinear system, as this frequently arises in many applications, such as in weather dynamics and plasma physics. The PI has chosen several specific problems that will advance the theory towards significant insights while being ambitious yet achievable in the near term. Projects have been selected to train and integrate graduate students and post-doctoral scholars into the research, including the PI's current and future mentees. The PI and his collaborators have planned a number of activities to integrate the professional development of younger scientists, including participating seminars, conferences, and lectures series. 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.