The evolution of turbulence theories and the need for continuous wavelets

Abstract

In the first part of this article, I summarise two centuries of research on turbulence. I also critically discuss some of the interpretations that are still in use, as turbulence remains an inherently non-linear problem that is still unsolved to this day. In the second part, I tell the story of how Alex Grossmann introduced me to the continuous wavelet representation in 1983, and how he instantly convinced me that this is the tool I was looking for to study turbulence. In the third part, I present a selection of results I obtained in collaboration with several students and colleagues to represent, analyse and filter different turbulent flows using the continuous wavelet transform. I have chosen to present both these theories and results without the use of equations, in the hope that the reading of this article will be more enjoyable.

Figures (24)

• Figure 1: Vorticity field of a two-dimensional turbulent flow, computed by direct numerical simulation of Navier--Stokes equation with periodic boundary conditions, using a pseudo-spectral method with resolution $N=512^2$. It is visualised with a cavalier representation Farge 1987. We observe the cusp-like vortices which drive the flow evolution.
• Figure 2: Vorticity field of a two-dimensional turbulent flow, computed by direct numerical simulation of Navier--Stokes equation with periodic boundary contitions, using a pseudo-spectral method with resolution $N=512^2$. It is visualised with a cartographic representation Farge 1987 where strong positive vorticity is red, strong negative vorticity is blue, weak vorticity is grey and zero vorticity is yellow. We observe the merging of same-sign vortices and the binding of opposite sign vortices to form vortex dipoles.
• Figure 3: Evolution of the vorticity field from an initial random distribution, computed by direct numerical simulation of the two-dimensional Navier--Stokes equation with periodic boundary contitions, using a pseudo-spectral method with resolution $N=512^2$. It is visualised with a cartographic representation Farge 1987, with the same colour-scale as [Figure \ref{['Figure2']}]. We observe the formation of coherent vortices emerging out of the initial random flow, followed by the merging of same sign vortices which eject vorticity filaments into the background flow, together with the binding of opposite sign vortices which form vortex dipoles.
• Figure 4: Evolution of the vorticity field from an initial random distribution, computed by direct numerical simulation of the two-dimensional Navier--Stokes equation in a circular container with no-slip boundary conditions, using a pseudo-spectral method and volume penalisation to take into account the solid wall, with resolution $N=1024^2$. It is visualised with a cartographic representation Farge 1987 with the same colour-scale as [Figure \ref{['Figure2']}]. We observe the formation of coherent vortices at the solid wall, due to Kelvin-Helmholtz instability, which spread into the bulk flow and exhibit same sign vortex merging and opposite sign vortex binding, as we have previously noticed.
• Figure 5: Satellite photograph of the Pacific Ocean from NASA. The clouds reveal the presence of two-dimensional vortices, because at large scale the Earth's rotation constraints the atmospheric turbulent flow to be two-dimensional. We observe a large vortex dipole and many smaller ones generated by the von Karman vortex street in the wake of the Guadalupe Islands (Baja California, Mexico).