Simulation of magneto-hydrodynamic (MHD) flows: electric potential formulation

Transcription

1 Simulation of magneto-hdrodnamic (MHD) flows: electric potential formulation, Ola Widlund 5th OpenFOAM workshop Göteborg, June 22-24, 2010

2 Outline Motivations for studing MHD flows Wh a formulation with electric potential? available mhdfoam new solver mhdepotfoam Magneto-hdrodnamic (MHD) equations Issues for MHD flow simulations Mesh constraints Numerical algorithm MHD flow in electricall conducting ducts new solver conjugatemhdfoam MHD flows in ducts: solver validation Summar & Outlook

3 Liquid metal flows in fusion blankets Fusion plasma (D + T He + n + energ) Fusion blanket Radiation shielding Test blanket module (TM) reeding of tritium 6 Li + n He + T + energ Heat removal: conversion of nuclear kinetic energ into electric energ Magnetic coils Requirements can be accomplished with Li-containing liquids as breeder and coolant International Thermonuclear Experimental Reactor Magnetic confinement of plasma

4 Liquid metal flows in fusion blankets Fusion plasma (D + T He + n + energ) Fusion blanket Radiation shielding Test blanket module (TM) reeding of tritium 6 Li + n He + T + energ Heat removal: conversion of nuclear kinetic energ into electric energ Magnetic coils Requirements can be accomplished with Li-containing liquids as breeder and coolant Moving electricall conducting fluid magnetic field Liquid metal magneto-hdrodnamics (MHD)

5 Liquid metal flows in fusion blankets Other MHD applications: Development of measuring techniques in liquid metal flows, electromagnetic flow meters and pumps. Fusion blanket Radiation shielding Metallurgical technolog continuous casting (surface treatment, MHD liquid metal stirring), industrial processes Focus on fusion applications: High magnetic fields ( = 4 11T) ver thin MHD boundar laers Coupled phenomena Complex geometries Strict numerical issues and requirements Simulation is a challenging task! reeding of tritium 6 Li + n He + T + energ Heat removal: conversion of nuclear kinetic energ into electric energ Requirements can be accomplished with Li-containing liquids as breeder and coolant Moving electricall conducting fluid magnetic field Liquid metal magneto-hdrodnamics (MHD)

6 Need of electric potential formulation mhdfoam available solver Induction equation 1 2 t Re v v (transport eq. for ) Magnetic Renolds number u 0 L Re m 1/ m Magnetic diffusivit mhdfoam available solver mhdepotfoam new solver

7 Need of electric potential formulation mhdfoam available solver Induction equation 1 2 t Re v v (transport eq. for ) Magnetic Renolds number u 0 L Re m 1/ m Magnetic diffusivit mhdfoam available solver oundar conditions mhdepotfoam new solver Full developed MHD duct flow (induced magnetic field constant along duct axis) Local Dirichlet const 0 or Neumann / n 0 at Cs can be set Induced magnetic field serves as streamfunction for current in duct cross-section 3D MHD flow No local Cs can be defined induced field in external space has to be considered (j ext = 0 Ampère s law 2 0 define at, outside 0)

8 Need of electric potential formulation mhdfoam available solver Induction equation 1 2 t Re m v v (transport eq. for ) Magnetic Renolds number mhdepotfoam new solver u 0 L Re m 1/ Magnetic diffusivit MHD channel flow externall applied Initial boundar value problem: = f (v) is determined depending on the flow field for Re m << 1 (liquid metals, industrial applications) f (v, t) x E = f (t) t = 0 E = Inductionless Approximation (the magnetic field is not affected b the flow. The induced magnetic field can be neglected.)

9 Magnetohdrodnamic equations (Re m <<1) Conservation of Momentum Mass & Charge Ohm s law 1 N j v t Ha v v p v j v 0, j 0 v Lorent force Poisson eq. for 2 v Dimensionless groups Interaction parameter N Induced electric field L u 0 2 el.magn. force inertia force Fusion reactors N 10 5 Hartmann number 2 L Ha 2 2 el.magn. force viscous force Ha 10 4 Renolds number Re Ha 2 / N inertia force viscous force

10 MHD flow features and numerical issues MHD full developed flow Velocit distribution Hartmann wall ( ) Side laers s ~ Ha -1/2 Side wall (II ) x v s ~ Ha -1/2 u Core Ha ~ Ha -1 L Hartmann laer Ha ~ Ha -1

11 MHD flow features and numerical issues MHD full developed flow Velocit distribution Hartmann wall ( ) Side laers s ~ Ha -1/2 Side wall (II ) x v s ~ Ha -1/2 u Core Ha ~ Ha -1 L Hartmann laer Ha ~ Ha -1 MHD simulation issues: MESH (discretiation error) Suitable resolution of MHD boundar laers - Refinement in boundar laers - Smooth grid transition between various regions (core - laers) - Good cell aspect ratio has to be maintained increasing Ha (i.e. ) the total number of nodes becomes larger Walls of finite electric conductivit: strong current turns in the thin wall The corner region has to be properl resolved

12 MHD flow features and numerical issues MHD full developed flow Side wall (II ) Hartmann wall ( ) x v Side laers s ~ Ha -1/2 Governing equations v t j 1 2 v v p Re v N j v 0, j 0 v 2 v Hartmann laer Ha ~ Ha -1 L MHD simulation issues: Numerics Phsics (modeling error) Error in current densit j is amplified b N in mom. Eq. when used to calculate the Lorent force High accurac required to compute the current densit Charge conservation has to be ensured: in FVM balance of fluxes through cell faces Source term in Eq. comes from a part of the current densit and has to be given at cell faces Lorent force defined at cell center proper interpolation of j from cell face to center

13 MHD duct flow: electricall insulating walls Current streamlines = 1 Ha = 500 Current closes its path in boundar laers (Ls) Resolution of Ls critical for the accurac of the solution (velocit and pressure gradient) x = 1 Velocit profile Velocit distribution u u

14 MHD duct flow: walls of finite electric conductivit conjugateheatfoam solver (OF dev) conjugatemhdfoam new solver Momentum Eq. solved onl on fluid mesh and Eq. on both meshes (combined matrix for fluid-solid coupledfvscalarmatrix) Fluid j v v j n j w n w Solid / Wall j w w w w 0 w, w electric conductivit of fluid and wall Definition of the electric conductivit field Coupled boundar conditions for and InterF { tpe regioncouple;. } x fluid w solid InterW { tpe regioncouple;. }

15 MHD duct flow: walls of finite electric conductivit conjugateheatfoam solver (OF dev) conjugatemhdfoam new solver t w, w 0 Insulating side walls Conducting Hartmann walls v x core u Side laers s Transverse velocit profile Ha 3000 c c Ha S wtw L Hartmann laers Ha -aligned velocit profile core L core u Ha 3000 c c Ha S wtw L Ha ~ Ha -1

16 MHD duct flow: walls of finite electric conductivit conjugateheatfoam solver (OF dev) conjugatemhdfoam new solver Conducting Hartmann walls t w, w 0 Insulating side walls Hartmann laers Ha -aligned velocit profile core v x core L Side laers s u t w, w 0 core Current streamlines u Ha 3000 c c Ha S wtw L Ha ~ Ha -1 Side wall w = 0 x

17 Summar Explanation of need of electric potential formulation to simulate 3D MHD flows difficult Cs for 3D MHD flows in case of induction equation approach Description of magneto-hdrodnamic (MHD) equations Lorent force in mom. Eq., Ohm s law, Poisson Eq. Issues for MHD flow simulations: MESH: proper resolution of thin boundar laers, L ~ Ha n ALGORITHM-MODELING: accurac of j prediction, interpolation of j from cell face to center, charge conservation new solver mhdepotfoam Code validation: perfect agreement with analtical solutions up to Ha = 5000 Successful application to 3D MHD problems Channels can have walls of arbitrar electric conductivit new solver conjugatemhdfoam from conjugateheatfoam (OF 1.5-dev)

18 Outlook Optimiation of present solver version (speed, numerical scheme, grid sensitivit studies, mesh skewness ) Development of wall functions (boundar laer models) Implementation of thin wall condition jn n c t w Wall element with tw w c L Simulation of MHD flow in ducts with walls of finite electric conductivit 1 st approach based on conjugateheatfoam solver (OF 1.5-dev) conjugatemhdfoam solver 2 nd approach as in chtmultiregionfoam solver (OF 1.6, 1.6.x) mhdmultiregionfoam solver t w j t j n j t j t j t Cooperation Ola Widlund, A A, Corporate Research Center, Västerås, Sweden Vincent Dousset, Coventr Universit, UK Elisabet Mas de Les Valls, Technical Universit of Catalonia, arcelona, Spain