Simulation numérique de la turbulence axisymétrique
Wouter Bos - Co-directeur de thèse
Claude Cambon - Examinateur
Bérengère Dubrulle - Rapporteur
Aurore Naso - Co-directeur de thèse
Kai Schneider - Examinateur
Eric Serre - Rapporteur
Antoine Venaile - Examinateur
Axisymmetric turbulence is investigated using direct numerical simulations. A fully spectral method is implemented using Chandrasekhar-Kendall eigenfunctions of the curl-operator.The numerical domain is a periodic cylinder with no-penetration and partial slip conditionsat the wall.
Numerical simulations are first carried out for freely decaying axisymmetric turbulence,starting from a variety of initial conditions. The simulations indicate that the global angularmomentum is the most robust invariant of the system. It is further observed that large-scalecoherent structures emerge, as in 2D isotropic turbulence. Energy decays more slowly thanhelicity, and the toroidal kinetic energy decays faster than its poloidal part. In the casewhere the toroidal kinetic energy becomes negligible, a quasi-two dimensional turbulencein the poloidal plane is obtained, with a behavior compatible with predictions of statisticalmechanics theories.
Forced and decaying simulations are then carried out to assess the cascade-behavior of the different invariants. The existence of an inverse cascade is shown to explain the robustness of the angular momentum and the possible ‘spontaneous generation’ of this quantity and of circulation in the flow. In helical flows, the existence of a dual cascade is confirmed, with a scenario compatible with the existence of an inverse energy cascade towards the large scales, and a direct cascade of helicity towards the small scales. The inverse energy cascade seems to be mainly associated with the poloidal velocity field. Using a helical decomposition of the flow, it is shown that the direct cascade of helicity seems to subsist even in the absence of net helicity, when the ‘cascade’ of the helicity contained in oppositely polarized modes is considered individually. The scaling of the energy spectra associated with the energy cascade
is compatible with elementary dimensional arguments, whereas the scaling of the inverse (presumably helicity) cascade yields an anomalously steep slope. It is shown that this slope adjusts to the value predicted by dimensional analysis when the spectra are computed from a filtered velocity field in which strong intermittent regions of velocity are not accounted for.
Finally, a preliminary (but unfortunately unfruitful) attempt is presented to apply a
variational principle to the description of turbulent scalar mixing in three-dimensional