Spectral Methods Based on the Least Dissipative Modes for Wall Bounded MHD Flows

We present a new approach for the Spectral Direct Numerical Simulation (DNS) of LowRm wall bounded magnetohydrodynamic (MHD) flows. The novelty is that instead of using bases like the usual Tchebychev polynomials, that are easy to implement but incur heavy computational costs in order to resolve the Hartmann boundary layers that arise along the walls, we use a basis made of elements that already incorporate flow structures such as anisotropic vortices and Hartmann layers. We show that such a basis can be obtained from the eigenvalue problem of the linear part of the governing equations with the problem’s boundary conditions. Since this basis is not always orthogonal, we develop a spectral method for non-orthogonal bases. We then demonstrate the efficiency of this method on the simple case of a laminar channel flow with periodic forcing. In particular, we show that this method eliminates the computational costs incurred by the Hartmann layer, and this for arbitrary high magnetic fields B. We then discuss the application of our method to nonlinear, turbulent flows for which the number of modes required to resolve the flow completely decreases strongly when B increases instead of increasing, as in current Tchebychev-based methods.