This paper presents a model for quasi two-dimensional MHD flows between two planes with small magnetic Reynolds number and constant transverse magnetic field orthogonal to the planes. A method is presented that allows to take 3D effects into account in a 2D equation of motion thanks to a model for the transverse velocity profile. The latter is obtained by using a double perturbation asymptotic development both in the core flow and in the Hartmann layers arising along the planes. A new model is thus built that describes inertial effects in these two regions. Two separate classes of phenomena are thus pointed out : the one related to inertial effects in the Hartmann layer gives a model for recirculating flows and the other introduces the possibility of having a transverse dependence of the velocity profile in the core flow. The "recirculating" velocity profile is then introduced in the transversally averaged equation of motion in order to provide an effective 2D equation of motion. Analytical solutions of this model are obtained for two experimental configurations : isolated vortices aroused by a point electrode and axisymmetric parallel layers occurring in the MATUR (MAgneticTURbulence) experiment. The theory is found to give a satisfactory agreement with the experiment so that it can be concluded that recirculating flows are actually responsible for both vortices core spreading and excessive dissipative behavior of the axisymmetric side wall layers. arXiv:2006.15468v1 [physics.flu-dyn]

Magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number is experimentally investigated by studying a liquid metal flow in a cubic domain. We focus on the mechanisms that determine whether the flow is quasi-two-dimensional, three-dimensional or in any intermediate state. To this end, forcing is applied by injecting a DC current I through one wall of the cube only, to drive vortices spinning along the magnetic field. Depending on the intensity of the externally applied magnetic field, these vortices extend part or all of the way through the cube. Driving the flow in this way allows us to precisely control not only the forcing intensity but also its dimensionality. A comparison with the theoretical analysis of this configuration singles out the influences of the walls and of the forcing on the flow dimensionality. Flow dimensionality is characterised in several ways. First, we show that when inertia drives three-dimensionality, the velocity near the wall where current is injected scales as U b ∼ I 2/3 . Second, we show that when the distance l z over which momentum diffuses under the action of the Lorentz force (Sommeria & Moreau, J. Fluid Mech., vol. 118, 1982, pp. 507-518) reaches the channel width h, the velocity near the opposite wall U t follows a similar law with a correction factor (1 − h/l z ) that measures three-dimensionality. When l z < h, by contrast, the opposite wall has less influence on the flow and U t ∼ I 1/2 . The central role played by the ratio l z /h is confirmed by experimentally verifying the scaling l z ∼ N 1/2 put forward by Sommeria & Moreau (N is the interaction parameter) and, finally, the nature of the three-dimensionality involved is further clarified by distinguishing weak and strong three-dimensionalities previously introduced by Klein & Pothérat (Phys. Rev. Lett., vol. 104 (3), 2010, 034502). It is found that both types vanish only asymptotically in the limit N → ∞. This provides evidence that because of the no-slip walls, (i) the transition between quasi-two-dimensional and three-dimensional turbulence does not result from a global instability of the flow, unlike in domains with non-dissipative boundaries (Boeck et al. Phys. Rev. Lett., vol. 101, 2008, 244501), and (ii) it does not occur simultaneously at all scales.

Inspired by the experiment from Moresco and Alboussiere ͓J. Fluid Mech. 504, 167 ͑2004͔͒, we study the stability of a flow of liquid metal in a rectangular, electrically insulating duct with a steady homogeneous magnetic field perpendicular to two of the walls. In this configuration, the Lorentz force tends to eliminate the velocity variations in the direction of the magnetic field. This leads to a quasi-two-dimensional base flow with Hartmann boundary layers near the walls perpendicular to the magnetic field, and so-called Shercliff layers in the vicinity of the walls parallel to the field. Also, the Lorentz force tends to strongly oppose the growth of perturbations with a dependence along the magnetic field direction. On these grounds, we represent the flow using the model from Sommeria and Moreau ͓J. Fluid Mech. 118, 507 ͑1982͔͒, which essentially consists of two-dimensional ͑2D͒ motion equations with a linear friction term accounting for the effect of the Hartmann layers. The simplicity of this quasi-2D model makes it possible to study the stability and transient growth of quasi-two-dimensional perturbations over an extensive range of nondimensional parameters and reach the limit of high magnetic fields. In this asymptotic case, the Reynolds number based on the Shercliff layer thickness Re/ H 1/2 becomes the only relevant parameter. Tollmien-Schlichting waves are the most linearly unstable mode as for the Poiseuille flow, but for H տ 42, a second unstable mode, symmetric about the duct axis, appears with a lower growth rate. We find that these layers are linearly unstable for Re/ H 1/2 տ 48350 and energetically stable for Re/ H 1/2 Շ 65.32. Between these two bounds, some nonmodal quasi-two-dimensional perturbations undergo some significant transient growth ͑between two and seven times more than in the case of a purely 2D Poiseuille flow, and for much more subcritical values of Re͒. In the limit of a high magnetic field, the maximum gain G max associated with this transient growth is found to vary as G max ϳ͑Re/ Re c ͒ 2/3 and occur at time t G max ϳ͑Re/ Re c ͒ 1/3 for streamwise wavenumbers of the same order of magnitude as the critical wavenumber for the linear stability.

We characterize experimentally how three dimensionality appears in wall-bounded magnetohydrodynamic flows. Our analysis of the breakdown of a square array of vortices in a cubic container singles out two mechanisms: first, a form of three dimensionality we call weak appears through differential rotation in individual 2D vortices. Second, strong three dimensionality characterized by vortex disruption leads on the one hand to a remarkable vortex array that is both steady and 3D, and, on the other hand, to scale-selective breakdown of two dimensionality in chaotic flows. Most importantly, these phenomena are entirely driven by inertia, so they are relevant to other flows with a tendency to two dimensionality, such as rotating, or stratified flows in geophysics and astrophysics.