The dynamics of partly 2D/partly 3D turbulence: an experimental and theoretical study in the Low-Rm MHD framework

Abstract

This thesis aims at clarifying the role of the solenoidal component of the Lorentz force in fixing the topological dimensionality, and the ensuing dynamics of low-Rm MHD turbulent flows confined between electrically insulating and no-slip Hartmann walls. The work pre- sented here breaks down into two main parts: an analytical investigation carried out in the weakly inertial limit on the one hand, and an experimental study of fully developed turbu- lence on the other hand. The analytical investigation was performed on a single, steady and axisymmetric electrically driven vortex confined between no-slip and electrically insulating walls perpendicular to a uniform magnetic field. Thanks to an asymptotic expansion of the Navier-Stokes equations valid for any Hartmann number, we show that the topological di- mensionality of the leading order is fully imposed by a single parameter. More precisely, this parameter quantifies the competition between the solenoidal component of the Lorentz force and viscous friction, over the distance separating the two no-slip planes. This study high- lights two inertial mechanisms capable of introducing a velocity component in the direction of the field at first order, by means of recirculations in the meridional plane: direct and/or inverse Ekman pumping. An experimental platform has also been designed and built from the ground up during this project, to investigate the dynamics of liquid metal turbulence subject to high magnetic fields (up to 10 T). The statistically steady turbulence sustained in our experiment was forced electrically by imposing a DC current through a square peri- odic array of electrodes. We show that the statistics of the turbulent fluctuations generated by this setup are homogeneous and axisymmetric to a satisfactory level, despite the forc- ing mechanism being inhomogeneous and anisotropic. By comparing the energy densities measured along the walls perpendicular to the magnetic field, we confirm that the physical processes at stake in the 3D inertial range of wall-bounded MHD turbulence at low-Rm are the solenoidal component of the Lorentz force on the one hand, and inertia on the other hand. Using a statistical analysis in scale space, we show that the kinematics of the flow driven in our experimental setup follows a universal law, which turns out to be fully de- scribed by only two lengthscales: first, the forcing scale in the direction perpendicular to the magnetic field, and second, the range of action of the Lorentz force before it is balanced out by inertial transfers, in the direction parallel to the field. We prove that the ratio of this latter scale to the height of the channel in fact segregates kinematically quasi-2D from kine- matically 3D turbulent structures. By computing the actual flux of perpendicular turbulent kinetic energy along horizontal scales, we show that it always flows towards larger turbulent scales regardless of their topology. In other words, the existence of an inverse cascade of perpendicular kinetic energy does not necessarily require horizontal turbulent structures to be topologically quasi-2D in the inertial range.

Date of Award	Dec 2016
Original language	English
Awarding Institution	Coventry University
Sponsors	Université Grenoble Alpes
Supervisor	Alban Potherat (Supervisor)