The velocity profile of laminar MHD flows in circular conducting pipes

We present numerical simulations without modeling of an incompressible, laminar, unidirectional circular pipe flow of an electrically conducting fluid under the influence of a uniform transverse magnetic field. Our computations are performed using a finite-volume code that uses a charge-conserving formulation [called current-conservative formulation in references (Ni et al J Comput Phys 221(1):174-204, 2007, Ni et al J Comput Phys 227(1):205-228, 2007)]. Using high resolution unstructured meshes, we consider Hartmann numbers up to 3000 and various values of the wall conductance ratio c. In the limit $${c{\ll}{\rm Ha}^{-1}}$$ (insulating wall), our results are in excellent agreement with the so-called asymptotic solution (Shercliff J Fluid Mech 1:644-666, 1956). For higher values of the wall conductance ratio, a discrepancy with the asymptotic solution is observed and we exhibit regions of velocity overspeed in the Roberts layers. We characterise these overspeed regions as a function of the wall conductance ratio and the Hartmann number; a set of scaling laws is derived that is coherent with existing asymptotic analysis.