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Simulation of MHD Flows using the Lattice Boltzmann Method

Simulation of MHD Flows using the Lattice Boltzmann Method. Kannan N. Premnath & Martin J. Pattison MetaHeuristics LLC Santa Barbara, CA 93105. Phase II SBIR – DOE Grant No. DE-FG02-03ER83715. Main Topics. Lattice Boltzmann Models for MHD.

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Simulation of MHD Flows using the Lattice Boltzmann Method

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  1. Simulation of MHD Flows using the Lattice Boltzmann Method Kannan N. Premnath & Martin J. Pattison MetaHeuristics LLC Santa Barbara, CA 93105 Phase II SBIR – DOE Grant No. DE-FG02-03ER83715

  2. Main Topics • Lattice Boltzmann Models for MHD • New Lattice Boltzmann Model for high Hartmann number MHD • Simulation results in 2D and 3D • Summary and Conclusions

  3. Magnetohydrodynamic (MHD) Equations Fluid dynamical equations Lorentz force Magnetic induction equation Dimensionless numbers Reynolds number Stuart number = magnetic force /inertial force Hartmann number = magnetic force /viscous force Magnetic Reynolds number

  4. Lattice Boltzmann Model for MHD I. Hydrodynamics Macroscopic fields 3D moments of distribution function Coincident lattices Scalar distribution function for hydrodynamics streaming collision Equilibrium distribution functions functions of macro fields

  5. Lattice Boltzmann Model for MHD (cont…) II. Magnetic induction Vector distribution function for magnetic induction 3D streaming collision Equilibrium distribution functions Coincident lattices functions of macro fields Macroscopic fields moments of distribution function

  6. Interior layer Wall Ghost layer Lattice Boltzmann Model for MHD (cont…) Transport coefficients (Diffusivities) Magnetic Prandtl number Boundary conditions Special cases Bounce-back Insulating Specular reflection Conducting General Case Extrapolation method (proposed) Electromagnetic domain extending outside fluid flow domain Fluid boundary Post-collision Electromagnetic boundary Post-collision

  7. Advantages of LBM for MHD flows • Avoids time-consuming solution of Poisson-type pressure equation All information obtained locally • Naturally amenable for implementation on parallel computers Efficient calculationprocedure for handling large problems • Well suited to MHD flows in complex geometries • Other advantages • Complete field formulation • Calculated fields solenoidal to machine round-off error • Current density as higher moment of the distribution function (no finite differencing)

  8. Comparison of LBM with Projection Method Sample problem: Flow through rectangular duct Different grid sizes For 20 timesteps same machine same problem:

  9. MHD Results in 2D Orszag -Tang vortex Time evolution Vorticity Current density Results comparable to other sources

  10. y x MHD Results in 2D - II Hartmann Flow Induced magnetic field profiles Velocity profiles

  11. MHD Results in 2D -III MHD Lid-driven Cavity Fluid flow Domain: 128128 Electromagnetic domain: 162162 y Induced magnetic fields set to zero on the electromagnetic boundary x Streamlines: Re = 100, Ha = 15.2, Rem = 100

  12. MHD Results in 2D -III MHD Lid-driven Cavity u-velocity profiles v-velocity profiles

  13. A new LB model for high Ha MHD flows - I • Adjustable magnetic Prandtl numbers (Prm) for liquid metals • High Hartmann number (Ha) flows require the resolution of various thin viscous boundary or shear layers • Hartmann layers • Side layers • Ludford free shear layers from sharp bends • Standard LB MHD model restricted to uniform lattice grids • Standard LB MHD model uses a single relaxation time (SRT), which restricts stability for a given resolution and variations in Prm • A new Multiple Relaxation Time (MRT) Interpolation Supplemented Lattice Boltzmann Model (ISLBM) developed for non-uniform or stretched grids with improved stability

  14. A new LB model for high Ha MHD flows - II Scalar distribution function for hydrodynamics MRT Model Forcing term representing Lorentz force streaming collision Components of the MRT matrix Vector distribution function for magnetic induction Interpolation step Non-uniform Grid Second order Interpolation of distribution functions

  15. A new LB model for high Ha MHD flows - III Hartmann Flow Non-uniform grid with simple step-changes in grid resolutions Velocity profile (Ha = 700) Boundary layer stretching transformations (e.g. Roberts transformation) can be used to further increase Ha

  16. 3D MHD Flows - I Developed MHD duct flow Induced magnetic field profile Velocity profile Hartmann walls – perfectly insulating, Side walls - perfectly insulating (Ha = 30)

  17. 3D MHD Flows - I Developed MHD duct flow Side wall jets Induced magnetic field profile Velocity profile Hartmann walls – conducting, Side walls - perfectly insulating (Ha = 30)

  18. z y x 3D MHD Flows - II 3D Developing MHD Duct Flow – Sterl problem hydrodynamic MHD effect Streamwise sharp gradient in the applied magnetic field Pressure Variation along streamwise direction (Ha = 44)

  19. 3D MHD Flows - II 3D Developing MHD Duct Flow – Sterl problem Velocity profile at the exit plane Induced magnetic field at the exit plane

  20. Summary and Conclusions • Lattice Boltzmann simulations for for 2D and 3D MHD performed • Simulations of MHD test problems in 2D and 3D show qualitative and quantitative agreement • A new multiple relaxation time (MRT) interpolation supplemented lattice Boltzmann model (ISLBM) for simulating high Ha liquid metal MHD flows

  21. Ongoing and Future Work Code Version 1 Capabilities Code Release - end of June, 2005 • MHD flows at intermediate Hartmann numbers • Multiphase flows • Heat transfer with non-uniform thermal conductivities • Complex geometries • Parallel processing using MPI • Pre-processor: Cart3D from NASA • Post-processor: FieldView Code will be implemented on a smaller cluster at MetaHeuristics and a larger cluster at National Center for Supercomputing Applications (NCSA)

  22. Ongoing and Future Work Code Version 2 Additional Capabilities • 3D MHD flows at high Hartmann numbers with multiple relaxation time (MRT) model • 3D complex geometries • Non-uniform grids • Turbulence modeling capability using Smagorinsky type large eddy simulation (LES) model Code Release - end of October, 2005

  23. Multiphase Flow Capabilities

  24. Example Problem - I: Drop Collisions Head-on collision resulting in reflexive separation Off-center collision resulting in stretching separation

  25. Example Problem - II: Drop subjected to Magnetic Field

  26. Example Problem - III: Rayleigh Instability and Satellite Droplet Formation Liquid Cylindrical Column perturbed by shorter wavelength surface disturbance Liquid Cylindrical Column perturbed by longer wavelength surface disturbance

  27. Supplementary Slides

  28. Pre-conditioning LBM for Accelerating Convergence to Steady State

  29. New Pre-conditioned LB MHD Model for Acceleration to Steady-State Evolution equations Pre-conditioning parameters: Equilibrium distribution functions Macroscopic Fields Transport coefficients

  30. Local Grid Refinement Technique for LB MHD model

  31. xc c {tc, c} f {tf, f} xf New Local Grid Refinement Schemes for LBM with Forcing Terms and SRT/MRT Models - I LBE with forcing term with single relaxation time (SRT) model where forcing term is given by Grid refinement factors Transformation Relations Here, tilde refers to post-collision value Similar transformation relations can be developed for the vector distribution function representing magnetic induction

  32. xc c {tc, c} f {tf, f} xf New Local Grid Refinement Schemes for LBM with Forcing Terms and SRT/MRT Models - II LBE with forcing term with multi relaxation time (MRT) model where forcing term is given by Grid refinement factors

  33. New Local Grid Refinement Schemes for LBM with Forcing Terms and SRT/MRT Models - III LBE with forcing term with multi relaxation time (MRT) model (cont…) Transformation Relations Here, tilde refers to post-collision value

  34. Curved Boundary Treatment for MHD Flows using LBM

  35. wall ff f w b New Curved Boundary Treatment - I Scalar LBE with forcing term Here, tilde refers to post-collision value Reconstructed distribution function from the solid side where

  36. New Curved Boundary Treatment - II Vector LBE Here, tilde refers to post-collision value Reconstructed distribution function from the solid side where is a free parameter

  37. Sub Grid Scale (SGS) Turbulence Modeling for MHD Flows using LBM

  38. Sub Grid Scale (SGS) Modeling of MHD Turbulent Flows For LES using LBM Evolution equation of “coarse-grained” LBE Total relaxation time Laminar kinematic viscosity Smagorinsky SGS eddy viscosity Effective Eddy viscosity due to magnetic field Magnetic damping factor (Shimomura, Phys. Fluids., 3: 3098 (1991)) Total kinematic “viscosity”

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