FLAG Lagrangian motion in a turbulent flow living on a Fractal Fourier space

Turbulence arises whenever a fluid is stirred by some external mechanism and there is a large separation of spatial scales between the typical scale of the forcing and the scale at which kinetic energy is transformed into heat. If the scale separation is large enough, such as in laboratory flows, or in geophysical and astrophysical flows, we speak of fully developed turbulence, the regime where chaotic fluctuations dominate both spatial and temporal evolution. In turbulence, a fundamental theory for homogeneous and isotropic turbulent flows is still lacking, notwithstanding almost 70 years of attempts since the pioneering work of A. N. Kolmogorov in 1941. In particular, there is no theory -exception made for phenomenological models- for the observed strong deviation from a Gaussian statistics both for Eulerian and Lagrangian velocity moments, a phenomenon called intermittency. In this Project, we aim to consider to consider a fundamental question of homogeneous and isotropic Lagrangian turbulence : How is Lagrangian strong intermittency modified when the the number of degrees of freedom in the flow is reduced, but the non-linear inviscid invariants of the original Navier-Stokes equations are preserved? By answering this question, we hope to achieve two main objectives: 1) to shed new light on the origin of Lagrangian intermittency; 2) to precisely quantify the role of the number degrees of freedom in developing intermittent temporal fluctuations along tracer trajectories. Recently (Frisch et al. "Turbulence in non-integer dimensions by fractal Fourier decimationâ, Phys. Rev. Lett. 108, 074501 (2012)), it has been proposed to study Navier-Stokes (NS) equations for fluid flows on a Fractal Fourier subspace, i.e. by reducing in Fourier space the number of degrees of freedom in the flow, but preserving the non-linear inviscid invariants of the original NS equations. This is a brand new approach to the study of turbulent interactions in incompressible fluids. Frisch et al. introduced it to study 2D turbulent inverse cascade of energy. Here we propose to use it, for the first time, to study 3D Lagrangian turbulence -i.e. following trajectories of particles seeding the flow. In 2012, thanks to a PRACE grant, we realised DNS of 3D turbulence on a Fractal Foruier sub-space, focusing on the Eulerian statistics of the flow. Now it is crucial to complete this challenging study by performing Direct Numerical Simulations of 3D turbulence, by putting also tracer and inertial particles in the flow. We plan to perform DNS at very high resolution and to reach a satisfactory statistical convergence, to highlight the role of the mean and extreme temporal fluctuations in the flow. Indeed, sometimes DNS of turbulence lack of converged statistics, which might weaken the obtained results. The novelty of the approach is vry high since no other group has attempted to study 3D Lagrangian turbulence on a Fractal Fourier skeleton, our group being the group which already approached the problem from an Eulerian point of view. It is important to recall that the questions tackled in this proposal can be addressed only numerically, and by massive numerical simulations with a very long time integration. Indeed no analytical tool or experimental technique can possibly answer these questions. Morevoer, by studying partice ineractions with coherent structures of the flow, a new understanding of turbulent non linear dynamics and its statistical behavior is expected from this challenging study.