Secondary optimal energy growth and magnetic damping of turbulence in Hartmann channel flow

The transient amplification of secondary linear perturbations in Hartmann channel flow is investigated. Optimal linear growth on either antisymmetric or symmetric streaky base flow is calculated by iteratively solving the direct and adjoint governing equations. The result shows that there is still residual interaction between the top and bottom Hartmann layer at relatively large Hartmann number. Strong amplification of secondary perturbations due to inflectional instability of modulated Hartmann channel flow is observed when the primary perturbations have a sufficiently large amplitude. The characteristic streamwise wavelength of the secondary perturbation is finite and scales with the thickness of the Hartmann layer. For weak modulation of the basic flow by the streaks, the secondary perturbations are streamwise independent vortices that resemble the primary optimal perturbations. The influence of the magnetic field is examined by means of the perturbation energy budgets, and the Joule dissipation turns out to be weak compared with the viscous dissipation. The secondary instability is therefore similar to that of an asymptotic suction boundary layer. Only for the mean velocity profile the Lorentz force is decisive. The weak influence of the magnetic field on the dynamics within Hartmann layer is verified by additional direct numerical simulations where the Lorentz force is only taken into account in the mean streamwise momentum equation. The results are close to full simulations of turbulent Hartmann flow, and the differences reduce with growing Reynolds number R based on the Hartmann layer thickness.