Turbulence is generally associated with the formation of vortices in a fluid. There are numerous experiences in daily life where one can note the presence of turbulence: the movements of a river downstream of an obstacle, the smoke escaping through a chimney, vortical motions of the air, or the turbulence zones that we sometimes cross by plane. Since it is not necessary to use powerful microscopes or telescopes to study turbulence one could conclude that it is probably not difficult to understand it. Unfortunately that is not the case! Although significant progress has been made since the middle of the twentieth century, several important questions remain unanswered and it is clear that at the beginning of the twenty-first century turbulence remains a central research topic in physics.

The first theoretical bricks of turbulence were laid from the moment physicists started to tackle the non-linearities of the hydrodynamic equations. As we will see, it is in this context that the first fundamental law of turbulence was established: this was the statistical law of Kolmogorov (1941). Nowadays the theoretical treatment of turbulence is partly based on numerical simulations which, accompanied by very powerful tools of visualization, allow us to tackle this difficult problem from a different angle and stimulate new questions. The purpose of this chapter is to present concepts and fundamental results on fully developed turbulence. This chapter is devoted to hydrodynamics, from which some foundations of the theory of turbulence have emerged. The two other chapters in this part of the book will be devoted to MHD turbulence.

What is Turbulence?

Unpredictability and Turbulence

It is not easy to define turbulence quantitatively because to do this one requires knowledge of a number of concepts that will be defined partly in this chapter.

Without going into the details, we can notice that the disordered – or chaotic – aspect seems to be the main characteristic of turbulent flows. It is often said that a system is chaotic when two points originally very close to each other in phase space separate exponentially over time. As we will see later, this definition can be extended to the case of fluids.