Turbulence in bidimensional fluids

Turbulence in fluid dynamics

Fluid dynamics (the study of liquid and gases behaviour) has had some open problems for centuries. The reason for that is its non-linear equations of motion, which need state-of-the-art mathematical and computational tools to solve and understand the particular behaviours of self-organizing structures in bidimensional turbulent flows.

When I meet God, I am going to ask him two questions: why relativity? And why turbulence? I really believe he will have an answer for the first.
Werner Heisenberg

Through spectral methods, we studied in this project the instability and turbulence in incompressible 2D fluids solving numerically Navier-Stokes equation in the vanishing viscosity limit and, then, comparing the results with data from point-vortex methods, which simulates an ideal fluid (zero viscosity). You can see below both simulations: the solution for NS-Eq and the point-vortex, both experiencing the Kelvin-Helmholtz instability.

Kelvin-Helmholtz instability

Kelvin-Helmholtz instability occurs when the interface of two fluid layers presenting relative motion is perturbed, generating turbulence. In the description of the vorticity field, it happens when a vorticity line is perturbed.

In the case of Kelvin-Helmholtz instability, we could find many similarities in the behaviour of both fluids, although we found differences in the energy spectra of an ideal and viscous fluid. We can see there exists energy and enstrophy transfers between multiple scales of structures.

The theoretical description of bidimensional turbulence is an open problem in classical physics, with great contributions of Kolmogorov and Kraichnan, Leith and Batchelor (KLB) to local forcing and isotropic scenarios. There is yet no theoretical description in the literature of bidimensional turbulence with a spectrally non-localized input of energy and space anisotropy, which is the case of KHI generated turbulence. My final undergraduate thesis described phenomenologically the KHI scenario in an ideal and vanishing-viscosity bidimensional fluids.

Estudo da turbulência bidimensional na instabilidade de Kelvin-Helmholtz: descrição fenomenológica
V. Sudbrack
Federal University of Rio Grande do Sul repository 2018

This work was done under Alexei Mailybaev supervision and in collaboration with Simon Thalabard at the Fluid Lab - IMPA. It was funded by the federal government (CNPq).