CAREER: Nonlinear stability mechanisms and boundary layer singularities in fluid flows

New York University, New York NY

Investigators

Abstract

The project concerns the mathematical analysis of fluid flows in the vicinity of physical boundaries, a fundamental problem in fluid dynamics. The realization that drag forces exerted on a body moving through air (or water) take place on a thin layer around the body lead to the development of boundary layer theory. The practical importance of boundary layer theory in aerodynamics and hydrodynamics is immense: computing the drag on an object and the energy dissipation rate nearby, with implications for fuel efficiency; understanding the problem of wing stall, the dependence of the lift on the angle of attack; and even the enhancement of heat transfer near solid walls. Mathematically, these questions concern the behavior of solutions to the Navier-Stokes equations in the vanishing viscosity limit, and the stability of the ensuing boundary layers. This project will develop new analytical tools to study the validity of the vanishing viscosity limit, the formation of singularities in boundary layers, and to explore the nonlinear stability mechanisms by which these lead to dynamic boundary layer separation. This theoretical information will lead to more accurate reduced models and finer predictions about real fluid flows. The project will train undergraduate, graduate, and post-graduate researchers, in modern research problems in applied mathematics and the tools to study them. This project aims to develop new mathematical tools that will further our understanding of the vanishing viscosity limit: the question whether solutions of the incompressible Navier-Stokes equations converge to solutions of the incompressible Euler equations as the viscosity approaches zero, in the presence of a characteristic physical boundary. For fixed external parameters, the vanishing viscosity limit is equivalent to the infinite Reynolds number limit, and thus this problem is of vital importance to the study of the onset of turbulence in fluid flows. The PI and collaborators will address the emergence of singularities in the Prandtl boundary layer equations by establishing more robust finite time blow-up scenarios. The stability of nearly laminar boundary layers will be studied via a hypocoercive analysis of the Prandtl system and of higher order models. Finally, the PI will develop new energy methods that intertwine Eulerian and Lagrangian approaches to prove the vanishing viscosity limit in high regularity regimes. The goal is to develop nonlinear, solution-adapted methods. Graduate students and post-doctoral fellows will be mentored and included in the research activities.