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

Abstract: 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 $0 \leq Ha \leq 800$ 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 $R_H$, and Reynolds number built upon the Shercliff layer thickness $R_S$, respectively. Transition between regimes respectively occurs at $Ha \approx 2$ and no lower than $R_H \approx 33.\dot{3}$. Notably for the high Hartmann number regime, quasi-two-dimensional magnetohydrodynamic models are shown to be an excellent predictor 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.