Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Laminar fully developed flows in a square duct and a pipe. Streamwise velocity distributions are shown for ducts at Ha = 10 (a), 100 (b) and 1000 (d) and for pipe at Ha = 50. (e) Distributions of electric currents j are shown for duct at Ha = 100 (c) and pipe at Ha = 50 (f). Direction of magnetic field is as indicated by arrows. The solutions are obtained numerically using the finite-difference model [14]. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Results of DNS [43] of duct flow at Re = Instantaneous distributions of streamwise velocity are shown for flows at Ha = 100 (a) and Ha = 300 (b). The magnetic field is in the vertical direction. The contour levels are the same in both plots ranging from 0 (near the walls) to 1.25 (at the center of the duct). Copyright by the Cambridge University Press (2012). Reproduced with permission. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Results of DNS [43] of duct flow at Re = Friction coefficients at Hartmann (CfHa) and Shercliff (CfSh) walls are shown as functions of Ha. The total friction coefficient (14) is λ=CfHa+CfSh. Results obtained for laminar flow solutions at the same Re and Ha are shown for comparison. Copyright by the Cambridge University Press (2012). Reproduced with permission. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Optimal modes of transient growth in the MHD duct flow. The flow is in the positive x-direction and the orientation of the imposed magnetic field is indicated by arrows. The optimal modes are visualized by iso-surfaces of streamwise velocity perturbations at the time of maximum amplification. Three cases of duct flow at Re = 5000 and Ha = 50 are shown: square duct (left), duct of aspect ratio β = 1/9 under spanwise field (middle), and duct of aspect ratio β = 9 (right, only a half of the flow domain is shown). The optimal perturbations in a duct look similar to superpositions of symmetric modes with opposite oblique angles observed in Ref. [80] in the channel flow with spanwise magnetic field. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Intermittency cycle in the channel flow at Re = 5333 with strong spanwise magnetic field at Ha = 80 [92]. The bottom row visualizes stages of the cycle using isosurfaces of streamwise velocity perturbations with respect to the mean flow (from left to right): exponentially growing 2D Tollmien-Schlichting modes in the beginning of the cycle, short burst into 3D turbulence, decay phase with streamwise streaks, crossover from streamwise streaks back to 2D T-S modes. The isolevels are adjusted to instantaneous maxima and minima of perturbation amplitude and differ by many orders of magnitude at different stages of the flow evolution. The top row shows the time evolution of the wall friction coefficient scaled by the coefficient of the laminar base (Poiseuille) flow. For comparison, friction coefficient and typical flow structures are shown for the 2D spanwise-independent state, to which the flow converges at Ha ≥ 160, and turbulent flow at a lower Hartmann number Ha = 40. Copyright (2008) by the American Physical Society. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Friction coefficient (14) in flows in (a) pipe at Re = 3500 and (b) square duct at Re = Results of DNS in periodic inlet-exit domains [44] and of DNS of spatially evolving flows [97] are shown in comparison with the experimental data [5]. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Characteristics of optimal transient amplification modes in the channel flow with spanwise magnetic field [80]. Results for Re = 5000 and varying Ha are shown. (a), the wavenumber vector determining the orientation of the optimal mode in the wall-parallel plane. k x and k y are, respectively, streamwise and spanwise wavenumbers. (b), the maximum energy amplification E(tmax)/E(t=0) for global optimal modes (M tot ) and purely streamwise optimal modes (M stream ). The horizontal lines indicate the no-amplification level and the level of amplification of purely spanwise (k y = 0) Orr modes insensitive to the magnetic field. Copyright by the Cambridge University Press (2008). Reproduced with permission. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Transition to turbulence in Hartmann channel flow via transient growth and breakdown of streamwise streaks [79]. Instantaneous distributions of streamwise velocity ux(x=const,y,z,t) at R = 500 (Re = 10,000 and Ha = 20) are shown at principal stages of evolution: growth of streamwise streaks (a), maximum amplification of kinetic energy of the streaks and formation of strong inflection points (b), streak breakdown (c) and, finally, turbulent flow (d). Optimal perturbations in the form of streamwise vortices, residing in the region of Hartmann walls, are imposed on the laminar profile (prior to stage (a)). The arrows indicate the direction of the applied magnetic field. Copyright by the American Institute of Physics (2005). Reproduced with permission. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Flow regimes with isolated turbulent spots in pipe (a) and duct (b)–(e) flows computed at Re = 5000 [44]. The flow states are visualized using turbulent kinetic energy of transverse velocity components. The isosurface corresponding to 2% of the maximum energy in the domain is shown. (a), turbulent puffs in pipe at Ha = 22. (b) and (c), double- and single-sided puffs in duct at Ha = 25. (d) and (e), extended turbulent zones in duct at Ha = 22 illustrating cases of double- and single-sided patterns. The total length of the computational domain is 80 pipe radii in (a) and 32π duct half-widths in (b)–(e). Copyright (2013) by the American Physical Society. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Patterned turbulence regimes realized in preliminary simulations of duct flow at Re = 10 5 and Ha = 400 (a), Ha = 450 (b), and Ha = 500 (c). The isosurfaces of turbulent kinetic energy of transverse velocity components corresponding to 2% of the maximum energy in the domain are shown. Figure Legend:

Date of download: 11/12/2016 Copyright © ASME. All rights reserved. From: Laminar-Turbulent Transition in Magnetohydrodynamic Duct, Pipe, and Channel Flows Appl. Mech. Rev. 2014;66(3): doi: / Possible states of MHD tube flows with transverse magnetic field Figure Legend: