Transition and turbulence in MHD at very strong magnetic fields - PowerPoint PPT Presentation

Transition and turbulence in MHD at very strong magnetic fields Prasanta Dey, Yurong Zhao, Oleg Zikanov University of Michigan – Dearborn, USA Dmitry Krasnov, Thomas Boeck, Andre Thess Ilmenau University of Technology, Germany Yaroslav Listratov, Valentin Sviridov Moscow Power Engineering Institute, Russia Support: US National Science Foundation, German Science Foundation, Russian Ministry of Education and Science

F J B Lorentz Force b ( ) Electric current J E u B Induced B magnetic field Flow u Imposed Magnetic field

MHD Approximation Assumption: Instantaneous propagation of electromagnetic radiation, L / τ <<c. τ , L – typical time and length scales Neglect: Displacement currents, ϑ u in Ohm’s law, ϑ E in electromagnetic force 1 B 2 0 j u B B t 0 j E u B F j B

MHD flows Planetary dynamo Sunspots Dipole magnetic field Azimuthal magnetic field Mantle Nature 2002 Solid inner Liquid outer core core Magnetic confinement fusion www.iter.org

Metallurgical applications • Control of nozzle jet in continuous steel casting • Crystal growth • Primary aluminum production in Hall-Héroult cells • Induction heating and stirring • Vacuum arc remelting • Magnetic valves • Electromagnetic pumps • Electromagnetic flow meters • …

Fusion enabling technology Cooling/breeding blankets and divertors for TOKAMAK reactors

Flows of liquid metals (Li, Li-Pb, FLiBe) in strong magnetic fields

Magnetic Reynolds number 1 B 2 u B B Re t σ – Electrical conductivity m μ 0 – Magnetic permeability Re m =UL/ η =UL σμ 0 of vacuum σ [ Ω -1 m -1 ] η [m 2 s -1 ] Liquid Sea water 5 6.3x10 6 Al (750 C) 4x10 6 0.2 0.7x10 6 Steel (1500 C) 1 6x10 6 Na (400 C) 0.13 10 6 Hg (20 C) 0.8 3.3x10 6 GaInSn (20 C) 0.25

Magnetic Reynolds number Dipole magnetic field Azimuthal magnetic field Mantle Solid inner Liquid outer core core 10 2 10 8 0.01 - 1 Re m

MHD flows at low Re m (quasi-static approximation) 2 1 Ha u 2 ( ) u u u j e p b Re Re t u 0 j u e b 2 u b UL Re , Ha BL

Effect of magnetic field on flow structures (far from walls) 1 u d Joule dissipation: 2 dV dV dV 2 dt 1 2 J dV

Effect of magnetic field on flow structures (far from walls) 1 u d Joule dissipation: 2 dV dV dV 2 dt 1 2 MHD flows are found in laminar or J dV transitional state more often than ordinary hydrodynamic flows

Effect of magnetic field on flow structures (far from walls) 1 u d Joule dissipation: 2 dV dV dV 2 dt 1 2 MHD flows are found in laminar or J dV transitional state more often than ordinary hydrodynamic flows With magnet etic c Without t field magneti etic c field Anisotropy of flow structures: B

Anisotropy of gradients at low Re m 2 2 B u 1 [ ] F F u 2 z 2 2 2 k B B ˆ ˆ B ˆ ˆ ˆ B =0 2 [ ] z cos F F u u u 2 k ˆ 2 1 2 2 ( ) | ( , ) | cos k B u k t 3D Isotropic Instability of 2D Magnetic 3D Anisotropic structures, Field Nonlinear Interaction Quasi-2D

I. Archetypal MHD flow – duct with insulating walls in a uniform transverse magnetic field

Flow structure: Flat core and MHD boundary layers δ ~Ha -1/2 – Sidewall layer δ ~Ha -1 Hartmann layer velocity current Ha=10 Ha=50

Question Ha Transition between laminar and turbulent flow regimes Re R=Re/Ha=200 – 250 or 350 – 400 ?

Numerical method – Direct Numerical Simulations Finite difference solver (Krasnov et al, Comp. Fluids 2011) Time advancing: Adams-Bashforth/BWD 2nd order explicit with projection method for pressure/incompressibility Grid arrangement: Structured collocated grid with staggered fluxes Viscous term: 2nd-order finite differences Non-linear term: 2nd-order, divergent form, highly conservative operator (Morinishi et.al. 1998) MHD term: 2nd-order, charge-conserving scheme for potential eq. and Lorentz force (Ni et.al. 2007) Poisson solver: Fourier expansion + 2D direct solver Parallelization: Hybrid parallelization: MPI + OpenMP

Parameters, Grid, Boundary conditions • Re=10 5 (in terms of mean velocity and half- width) • Ha=0, 100, 200, 300, 350, 400 • Domain size: 4 π x2x2 • Numerical resolution: 2048x768x768 • Nearly Chebyshev-Gauss-Lobatto wall clustering • Electrically insulating walls • Periodic inlet/exit

Instantaneous streamwise velocity Ha=0 Ha=100 B

Instantaneous streamwise velocity Ha=200 Ha=300 B Laminar flow at Ha=400

Turbulence in sidewall layers Ha=350 2 nd eigenvalue of S ik S kj + Ω ik Ω k j (Jeong, Hussein, JFM 1995)

Summary of results Ha N U cl Re τ ,y Re τ ,z 0 0 1.1768 4253 4253 100 0.1 1.2304 3462 5269 200 0.4 1.3011 2487 5099 300 0.9 1.1466 1993 5865 350 1.225 1.1177 1901 6255 400 1.6 1.0465 1543 6512 (laminar) Time-averaged in fully developed flow

Mean streamwise velocity B Ha=0 Ha=100 Ha=300 Ha=200

Log-layer? γ =z + dU + /dz + γ =y + dU + /dy +

Conclusions • Transitional flow regimes with turbulence restricted to sidewall layers in a wide range of Ha • Within sidewall layers, turbulence is small-scale and approaching isotropy near walls, but becomes large-scale, weak, and strongly anisotropic toward the center • Non-trivial transformation of mean flow profile in the spanwise direction: lin-log Krasnov et al, J. Fluid Mech. 2012, 704 , 421-446

Fully laminar Ha Fully turbulent Re

II. Mixed convection with strong transverse magnetic field Flow of Hg in a horizontal pipe with transverse magnetic field: Institute of High Temperatures RAS Pipe inner diameter: d=19 mm Walls: stainless steel 0.5 mm Length of working segment: 2m Heated length: 0.812 m (43d) Uniform magn. field: 0.5m (26d) Max. heat flux: q<55 kW/m 2 Max. magn. field: B<1T

Considered Case • Horizontal pipe • Perpendicular horizontal magnetic field • Heated lower half • Thermally insulated upper half Re d =10 4 Ha d =0, 100, 300, 500 Gr d =8.3x10 7 (q=35 kW/m 2 ) Pr=0.022

Experimental Temperature fluctuations: r=0.7R, bottom, x/d=40 data 4 2 0 Ha=0 -2 t, c -4 0 1 2 3 4 5 6 7 8 9 10 4 2 0 Ha=100 -2 t, c -4 0 1 2 3 4 5 6 7 8 9 10 4 2 Ha=300 0 -2 t, c -4 0 1 2 3 4 5 6 7 8 9 10

Experimental data Ha=300

Hypothesis Ha=100 Ha=300

Linear stability analysis: Base flow B Ha d =300 U x 0.34 1.14 0.30 1.06 0.27 0.98 0.23 0.91 0.20 0.83 0.16 0.76 0.13 0.68 0.09 0.61 0.05 0.53 0.02 0.45 -0.02 0.38 -0.05 0.30 -0.09 0.23 -0.12 0.15 B -0.16 0.08

Linear Stability Analysis: 2D (streamwise-uniform) mode Volume-averaged perturbations: E2d, E θ 2d – x-independent mode E3d, E θ 3d – mode of x-periodicity λ -1 -1 10 10 -2 -2 10 10 -3 -3 10 10 -4 -4 10 10 -5 -5 10 10 E 2D E 2D -6 -6 10 10 -7 -7 10 10 Ha=100 Ha=300 -8 -8 10 10 Ha=500 -9 -9 10 10 -10 -10 10 10 50 100 150 200 50 100 150 200 t t

Linear Stability Analysis: 2D (streamwise-uniform) mode Ha=100 Ha=300 u u 0.15 2.00E-02 0.12 1.61E-02 0.09 1.21E-02 0.06 8.21E-03 0.04 4.29E-03 0.01 3.57E-04 -0.02 -3.57E-03 -0.05 -7.50E-03 -0.08 -1.14E-02 -0.11 -1.54E-02 -0.14 -1.93E-02 -0.16 -2.32E-02 -0.19 -2.71E-02 g -0.22 B -3.11E-02 -0.25 -3.50E-02 Ha=500 Ha=100 u + 5.00E-03 0.34 3.79E-03 0.30 2.57E-03 0.27 1.36E-03 0.23 1.43E-04 0.20 -1.07E-03 0.16 -2.29E-03 0.13 -3.50E-03 0.09 -4.71E-03 0.05 -5.93E-03 0.02 -7.14E-03 -0.02 -8.36E-03 -0.05 -9.57E-03 -0.09 -1.08E-02 -0.12 -1.20E-02 -0.16

Linear Stability Analysis: 2D + 3D modes Example: Ha d =300, λ =1.0d -4 10 -6 10 E2d E 2d E3d -8 10 E 3d Energy -10 10 -12 10 Exponential growth -14 10 E3d E 3d exp(2 t) -16 10 0 20 40 60 80 100 t

Linear Stability Analysis Example: Ha d =300, λ =1.0d -0.0075 0.115 Point signals of -0.0077 velocity and temperature during 0.114 u x exponential growth -0.0079 0.113 -0.0081 0.112 50 60 70 80 90 100 t

Linear Stability Analysis t=100, horizontal cross-section Example: Ha d =300, λ =1.0d through pipe axis Temperature perturbations Vertical velocity perturbations 1 1 0.75 0.75 0.5 0.5 0.25 0.25 R R 0 0 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 X X Magnetic field Flow

Linear Stability Analysis Temperature and velocity Example: Ha d =300, λ =1.0d perturbations 1 0.75 t=100, vertical 0.5 cross-section through pipe axis 0.25 R 0 -0.25 -0.5 -0.75 -1 -1 -0.5 0 0.5 1 X

Linear Stability Analysis Ha d =300 Growth rate Phase velocity 0.25 1.2 0.2 c= /period 0.15 1.1 0.1 1 0.05 0 0.9 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 /d /d Fastest growing mode: λ =0.9d, period T=0.8 Dimensional frequency ~ 3.2 Hz (compare with 2-3 Hz in experiment)

Linear Stability Analysis Ha d =500 Phase Velocity Growth rate 0.25 1.2 0.2 c= /period 0.15 1.1 0.1 1 0.05 0 0.9 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 /d /d Fastest growing mode: λ =0.9d, period T=0.8 Growth rate ~ 10% higher than at Ha=300