Mechanisms for Energy Conservation in Onsager Supercritical Fluids

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The complexity of turbulent motion of fluids like water presents many theoretical as well as technological challenges. From the practical standpoint laws of turbulence are crucial in many real-life applications. They stand behind the modern design of a plane airfoil or development of weather and climate forecast models. One of the features of turbulence is called anomalous energy dissipation. This phenomenon arises when the motion of a fluid is so chaotic or rough that the the classical laws of smooth dynamics no longer apply. Anomalous energy dissipation is harnessed in many commonly used energy-dumping mechanisms, such as automobile wheel struts. Common though it is, in some cases energy dissipation does not occur even in what otherwise would be considered a flow turbulent enough to facilitate such dissipation. It has been observed that in various natural phenomena, such as vortex sheets that develop behind the wing of a plane, energy dissipation does not occur until motion reaches a supercritical state. The project goal is to isolate several mechanisms responsible for energy preservation or dissipation in fluid motion that is turbulent or nearly so. A main focus is on investigation of the role of symmetries in energy conservation. Students are included in the project. The investigator studies weak solutions to the Euler equation and its viscous Navier-Stokes counterpart in the vanishing viscosity limit regime, by considering the role of energy conservation or dissipation in the fluid flows described by these equations. The equations have been shown to describe turbulence rather accurately from a numerical point of view, although theoretically they present many challenges. Following Onsager, in terms of regularity a solution reaches its turbulent state when smoothness of the flow is reduced to a third of one full derivative, also known as Onsager regularity. In that regularity regime the investigator examines four main mechanisms as candidates responsible for energy conservation or dissipation: Hamiltonian structure of the underlying governing equation, incompressibility condition, basic scaling symmetries and transport nature of the motion, and the vanishing viscosity limit in the two-dimensional setting. The last point connects energy dissipation to regularity of solutions of the Euler equations. The project involves active participation of students.