Abstract
We investigate aspects of lowmagneticReynoldsnumber flow between two parallel, perfectly insulating walls in the presence of an imposed magnetic field parallel to the bounding walls. We find a functional basis to describe the flow, well adapted to the problem of finding the attractor dimension and which is also used in subsequent direct numerical simulation of these flows. For given Reynolds and Hartmann numbers, we obtain an upper bound for the dimension of the attractor by means of known bounds on the nonlinear inertial term and this functional basis for the flow. Three distinct flow regimes emerge: a quasiisotropic threedimensional (3D) flow, a nonisotropic 3D flow, and a 2D flow. We find the transition curves between these regimes in the space parametrized by Hartmann number Ha and attractor dimension datt. We find how the attractor dimension scales as a function of Reynolds and Hartmann numbers (Re and Ha) in each regime. We also investigate the thickness of the boundary layer along the bounding wall and find that in all regimes this scales as 1/Re, independently of the value of Ha, unlike Hartmann boundary layers found when the field is normal to the channel. The structure of the set of least dissipative modes is indeed quite different between these two cases but the properties of turbulence far from the walls (smallest scales and number of degrees of freedom) are found to be very similar.
Original language  English 

Journal  Physical Review E 
Volume  91 
Issue number  053022 
DOIs  
Publication status  Published  27 May 2015 
Bibliographical note
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Alban Potherat
 Research Centre for Fluid and Complex Systems  Centre Director
Person: Teaching and Research