Turbulence and weak solutions of the Euler equations

Undergraduate thesis in physics, Universidad de los Andes (2022)

In this article, a study and discussion is carried out from a different approach of classical mechanics, specifically working with the model for a non-viscous and incompressible fluid described by the Euler equations. To do this, it begins with a motivation by giving examples applied to the harmonic oscillator and an elastic membrane, where the concept of weak formulation is introduced. Subsequently, a preliminary introduction is made to familiarize oneself with the mathematical tools required to deeply address the weak formulation that arises from the variational formulation of Lagrange, defining concepts such as weak derivative, Sobolev spaces, and compact support functions. Then, a demonstration is made of the relationship between weak and strong formulation, showing their equivalence in some cases. In addition, the variational development of the Euler equations is carried out to obtain the weak formulation of the fluid’s dynamic equations, and an idea is given of how to obtain a solution by linearizing the equation and including certain restrictions on the variables. Afterwards, a discussion is made of the importance of vorticity to determine the transition from a laminar regime to a turbulent one, and how the principle of energy conservation presents difficulties when approaching the singular regions of the fluid. Finally, a theoretical framework developed by the mathematician Andrei Kolmogorov is approached from a somewhat more qualitative and less rigorous perspective, to understand what happens to the energy at different scales of a fluid, and then to conclude with some conclusions and development of the ideas studied during this work.

Grupo de Física Estadística

Departamento de Física

Edificio Ip

Carrera 1E # 18A-10

Bogotá, Colombia

Universidad de los Andes | Vigilada Mineducación
Reconocimiento como Universidad: Decreto 1297 del 30 de mayo de 1964.
Reconocimiento personería jurídica: Resolución 28 del 23 de febrero de 1949 Minjusticia.

Web design and programming © Gabriel Téllez