Necessary and Sufficient Conditions for the Transfer of Kinetic Energy at Asymptotically Large Reynolds Numbers

Abstract

At high Reynolds numbers, an incompressible fluid will become turbulent --- a phenomenon where the fluid is sensitive to external noise, develops chaotic time dynamics, and can develop eddy-type structures which break apart into smaller and smaller versions of themselves to dissipate energy as heat. In the field of mathematical physics, turbulent flows are typically modeled using stochastic partial differential equations to model the apparent randomness of the turbulent flow. Moreover, from a physics perspective, this method accounts for external noise on the system, such as the vibrations of the table holding the cup of coffee. Parameterizing these solutions by the viscosity (or the inverse of the Reynolds number) we can then study the behavior of the flow in the inviscid limit --- or as the viscosity decreases toward 0. One striking feature of three-dimensional turbulence is the presence of anomalous dissipation, or that the mean rate of energy dissipation is bounded below by a positive number in the inviscid limit. This is thought to be due to the convective acceleration acting in part like a dissipation mechanism instead of a pure transport mechanism at asymptotically large Reynold's numbers (i.e. in the inviscid limit). Moreover the amount of anomalous dissipation dictates how fast the various eddy structures in the flow can break apart into smaller and smaller versions of themselves known as an energy cascade. In 1941, Kolmogorov predicted the rate of the energy cascade to be $\frac{4}{5}\varepsilon \ell$ where $\ell$ is the size of the eddy structure and $\varepsilon$ is the amount of anomalous dissipation. Kolmogorov's work has been experimentally verified and simulated in numerous studies, but has faced serious mathematical obstacles in its analysis. In the first part of this dissertation we focus on finding necessary and sufficient conditions for Kolmogorov's flux laws on the movement of kinetic energy. We complete this over a torus in both two and three dimensions and discusses both the physical and mathematical differences encountered due to the dimension.

In the second part of this dissertation we examine the anomalous dissipation assumption itself. Here we consider the case of a bounded domain, subject to the Navier slip condition and show that the existence of (global) anomalous dissipation --- anomalous dissipation over the entire domain --- can be caused in a linear problem through lack of control over the tangential component of the velocity at the boundary.