Optimal transient disturbances behind a circular cylinder in a quasi-two-dimensional magnetohydrodynamic duct flow

The transient response of optimal linear perturbations of liquid metal flow under a strong axial magnetic field in an electrically insulated rectangular duct is considered. The focus is on the subcritical regime, below the onset of von Karman vortex shedding, to determine the role of optimal disturbances in developing wake instabilities. In this configuration, the flow is quasi-two-dimensional and can be solved over a two-dimensional domain. Parameter ranges considered include Reynolds numbers 50 <= Re less than or similar to 2100, modified Hartmann numbers 50 = Ha(star) less than or similar to 500, and blockage ratios 0.1 <= beta <= 0.4. In some instances, the optimal disturbances are found to generate energy growth of greater than four orders of magnitude. Variation in the wake recirculation length in the steady flow regime is determined as a function of Reynolds number, Hartman number, and blockage ratio, and a universal expression is proposed. For all beta, the energy amplification of the disturbances is found to decrease significantly with increasing Hartmann number and the peak growth shifts towards smaller times. The optimal initial disturbances are consistently located in the vicinity of the boundary layer separation from the cylinder, and the structure of these disturbances is consistent for all Hartmann numbers and blockage ratios considered in this study. The time evolution of the optimal perturbations is presented, and is shown to correspond to sinuous oscillations of the shear layer downstream of the wake recirculation. The critical Reynolds number for the onset of growth at different Hartmann numbers and blockage ratios is determined. It is found that it increases rapidly with increasing Hartmann number and blockage ratio. For all beta, the peak energy amplification grows exponentially with Re at low and high Hartmann numbers. Direct numerical simulation in which the inflow is perturbed by a random white noise confirms the predictions arising from the transient growth analysis: that is, the perturbation excites and feeds energy into the global mode. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3686809]