From three-dimensional to quasi-two-dimensional: Transient growth in magnetohydrodynamic duct flows

This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of 5000 and 15 000 and an extensive range of Hartmann numbers 0 ≤ Ha ≤ 800 were investigated. Three broad regimes are identified in which optimal mode topology and non-modal growth mechanisms are distinct. These regimes, corresponding to low, moderate and high magnetic field strengths, are found to be governed by the independent parameters; Hartmann number, Reynolds number based on the Hartmann layer thickness RH and Reynolds number built upon the Shercliff layer thickness RS, respectively. Transition between regimes respectively occurs at Ha~2 and no lower than RH ~ 33: P3. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be excellent predictors of not only transient growth magnitudes, but also the fundamental growth mechanisms of linear disturbances. This paves the way for a precise analysis of transition to quasi-two-dimensional turbulence at much higher Hartmann numbers than is currently achievable.