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

  • Oliver G. W. Cassells (a1), Tony Vo (a1), Alban Pothérat (a2) and Gregory J. Sheard (a1)


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 $0\leqslant \mathit{Ha}\leqslant 800$ 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 $\mathit{Ha}\approx 2$ 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 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.


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Airiau, C. & Castets, M. 2004 On the amplification of small disturbances in a channel flow with a normal magnetic field. Phys. Fluids 16 (8), 29913005.
Alemany, A., Moreau, R., Sulem, P. & Frisch, U. 1979 Influence of an external magnetic field on homogeneous MHD turbulence. J. Méc. 18 (2), 277313.
Baker, N. T., Pothérat, A., Davoust, L. & Debray, F. M. C. 2018 Inverse and direct energy cascades in three-dimensional magnetohydrodynamic turbulence at low magnetic Reynolds number. Phys. Rev. Lett. 120, 224502.
Barkley, D., Blackburn, H. & Sherwin, S. 2008 Direct optimal growth analysis for timesteppers. Intl J. Numer. Meth. Fluids 231, 120.
Biau, D. & Bottaro, A. 2004 Transient growth and minimal defects: two possible initial paths of transition to turbulence in plane shear flows. Phys. Fluids 16 (10), 35153529.
Biau, D., Soueid, H. & Bottaro, A. 2008 Transition to turbulence in duct flow. J. Fluid Mech. 596, 133142.
Blackburn, H. M., Sherwin, S. J. & Barkley, D. 2008 Convective instability and transient growth in steady and pulsatile stenotic flows. J. Fluid Mech. 607, 267277.
Böberg, L. & Brösa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. A 43 (8–9), 697726.
Cassells, O. G. W., Hussam, W. K. & Sheard, G. J. 2016 Heat transfer enhancement using rectangular vortex promoters in confined quasi-two-dimensional magnetohydrodynamic flows. Intl J. Heat Mass Transfer 93, 186199.
Dousset, V. & Pothérat, A. 2008 Numerical simulations of a cylinder wake under a strong axial magnetic field. Phys. Fluids 20, 017104.
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.
Gerard-Varet, D. 2002 Amplification of small perturbations in a Hartmann layer. Phys. Fluids 14 (4), 14581467.
Greenspan, H. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Hamid, A. H., Hussam, W. K., Pothérat, A. & Sheard, G. J. 2015 Spatial evolution of a quasi-two-dimensional Kármán vortex street subjected to a strong uniform magnetic field. Phys. Fluids 27 (5), 053602.
Hamid, A. H. A., Hussam, W. K. & Sheard, G. J. 2016 Combining an obstacle and electrically driven vortices to enhance heat transfer in a quasi-two-dimensional MHD duct flow. J. Fluid Mech. 792, 364396.
Hunt, J. C. R. & Stewartson, K. 1965 Magnetohydrodynamic flow in rectangular ducts. Part II. J. Fluid Mech. 23 (3), 563581.
Hussam, W. K. & Sheard, G. J. 2013 Heat transfer in a high Hartmann number MHD duct flow with a circular cylinder placed near the heated side-wall. Intl J. Heat Mass Transfer 67, 944954.
Hussam, W. K., Thompson, M. C. & Sheard, G. J. 2012a Enhancing heat transfer in a high Hartmann number magnetohydrodynamic channel flow via torsional oscillation of a cylindrical obstacle. Phys. Fluids 24, 113601.
Hussam, W. K., Thompson, M. C. & Sheard, G. J. 2012b Optimal transient disturbances behind a circular cylinder in a quasi-two-dimensional magnetohydrodynamic duct flow. Phys. Fluids 24, 024105.
Kanaris, N., Albets, X., Grigoriadis, D. & Kassinos, S. 2013 Three-dimensional numerical simulations of magnetohydrodynamic flow around a confined circular cylinder under low, moderate, and strong magnetic fields. Phys. Fluids 25, 074102.
Karniadakis, G. E., Israeli, M. & Orszag, S. A. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414443.
Klein, R. & Pothérat, A. 2010 Appearance of three dimensionality in wall-bounded MHD flows. Phys. Rev. Lett. 104 (3), 034502.
Krasnov, D., Zienicke, E., Zikanov, O., Boeck, T. & Thess, A. 2004 Numerical study of the instability of the Hartmann layer. J. Fluid Mech. 504, 183211.
Krasnov, D., Zikanov, O. & Boeck, T. 2012 Numerical study of magnetohydrodynamic duct flow at high Reynolds and Hartmann numbers. J. Fluid Mech. 704, 421.
Krasnov, D., Zikanov, O., Rossi, M. & Boeck, T. 2010 Optimal linear growth in magnetohydrodynamic duct flow. J. Fluid Mech. 653, 273299.
Landahl, M. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98 (2), 243251.
Lehoucq, R., Sorensen, D. & Yang, C.1998 ARPACK users’ guide. Society for Industrial and Applied Mathematics.
Leigh, M. A., Tsai, T. & Sheard, G. J. 2016 Probing horizontal convection instability via perturbation of the forcing boundary layer using a synthetic jet. Intl J. Therm. Sci. 110, 251260.
Moresco, P. & Alboussiere, T. 2004 Experimental study of the instability of the Hartmann layer. J. Fluid Mech. 504, 167181.
Mück, B., Günther, C., Müller, U. & Bühler, L. 2000 Three-dimensional MHD flows in rectangular ducts with internal obstacles. J. Fluid Mech. 418, 265295.
Müller, U. & Bühler, L. 2001 Magnetofluiddynamics in Channels and Containers, 1st edn. Springer.
Ng, Z. Y., Vo, T., Hussam, W. K. & Sheard, G. J. 2016 Two-dimensional wake dynamics behind cylinders with triangular cross-section under incidence angle variation. J. Fluids Struct. 63, 302324.
Ni, M.-J., Munipalli, R., Morley, N. B., Huang, P. & Abdou, M. A. 2007 A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I. On a rectangular collocated grid system. J. Comput. Phys. 227 (1), 174204.
Ó Náraigh, L. 2015 Global modes in nonlinear non-normal evolutionary models: exact solutions, perturbation theory, direct numerical simulation, and chaos. Physica D 309, 2036.
Orr, W. M. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II. A viscous liquid. Proceedings of the Royal Irish Academy, Section A: Mathematical and Physical Sciences 27, 69138.
Paret, J., Marteau, D., Paireau, O. & Tabeling, P. 1997 Are flows electromagnetically forced in thin stratified layers two dimensional? Phys. Fluids 9 (10), 31023104.
Pothérat, A. 2007 Quasi-two-dimensional perturbations in duct flows under transverse magnetic field. Phys. Fluids 19, 074104.
Pothérat, A. 2012 Three-dimensionality in quasi-two-dimensional flows: recirculations and Barrel effects. Europhys. Lett. 98 (6), 64003.
Pothérat, A. & Klein, R. 2014 Why, how and when MHD turbulence at low Rm becomes three-dimensional. J. Fluid Mech. 761, 168205.
Pothérat, A., Sommeria, J. & Moreau, R. 2000 An effective two-dimensional model for MHD flows with transverse magnetic field. J. Fluid Mech. 424, 75100.
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.
Reshotko, E. 2001 Transient growth: a factor in bypass transition. Phys. Fluids 13 (5), 10671075.
Sapardi, A. M., Hussam, W. K., Pothérat, A. & Sheard, G. J. 2017 Linear stability of confined flow around a 180-degree sharp bend. J. Fluid Mech. 822, 813847.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Sheard, G. J., Hussam, W. K. & Tsai, T. 2016 Linear stability and energetics of rotating radial horizontal convection. J. Fluid Mech. 795, 135.
Smolentsev, S. & Moreau, R. 2007 One-equation model for quasi-two-dimensional turbulent magnetohydrodynamic flows. Phys. Fluids 19 (7), 078101.
Smolentsev, S., Wong, C., Malang, S., Dagher, M. & Abdou, M. 2010 MHD considerations for the DCLL inboard blanket and access ducts. Fusion Engng Des. 85 (7), 10071011.
Sommeria, J. & Moreau, R. 1982 Why, how, and when, MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507518.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578584.
Tsai, T., Hussam, W. K., Fouras, A. & Sheard, G. J. 2016 The origin of instability in enclosed horizontally driven convection. Intl J. Heat Mass Transfer 94, 509515.
Vo, T., Montabone, L., Read, P. L. & Sheard, G. J. 2015 Non-axisymmetric flows in a differential-disk rotating system. J. Fluid Mech. 775, 349386.
Vo, T., Pothérat, A. & Sheard, G. J. 2017 Linear stability of horizontal, laminar fully developed, quasi-two-dimensional liquid metal duct flow under a transverse magnetic field and heated from below. Phys. Rev. Fluids 2 (3), 033902.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883900.
Zienicke, E. & Krasnov, D. 2005 Parametric study of streak breakdown mechanism in Hartmann flow. Phys. Fluids 17 (11), 114101.
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From three-dimensional to quasi-two-dimensional: transient growth in magnetohydrodynamic duct flows

  • Oliver G. W. Cassells (a1), Tony Vo (a1), Alban Pothérat (a2) and Gregory J. Sheard (a1)


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