Optimal growth and transition to turbulence in channel flow with spanwise magnetic field

Instability and transition to turbulence in a magnetohydrodynamic channel flow are studied numerically for the case of a uniform magnetic field imposed along the spanwise direction. Optimal perturbations and their maximum amplifications over finite time intervals are computed in the framework of the linear problem using an iterative scheme based on direct and adjoint governing equations. It is shown that, at sufficiently strong magnetic field, the maximum amplification is no longer provided by classical streamwise rolls, but rather by rolls oriented at an oblique angle to the basic flow direction. The angle grows with the Hartmann number Ha and reaches the limit corresponding to purely spanwise rolls at Ha between 50 and 100 depending on the Reynolds number. Direct numerical simulations are applied to investigate the transition to turbulence at a single subcritical Reynolds number Re = 5000 and various Hartmann numbers. The transition is caused by the transient growth and subsequent breakdown of optimal perturbations, which take the form of one or two symmetric optimal modes (streamwise, oblique or spanwise modes depending on Ha) with low-amplitude three-dimensional noise added at the moment of strongest energy amplification. A sufficiently strong magnetic field (Ha larger than approximately 30) is found to completely suppress the instability. At smaller Hartmann numbers, the transition is observed but it is modified in comparison with the pure hydrodynamic case.