SIMULATION OF LIQUID METAL MHD FLOWS IN COMPLEX GEOMETRIES

Vinayak Eswaran Department of Mechanical Engineering Indian Institute of Technology Kanpur (with V.Naveen , R. Paniharan, Profs M.K.Verma and K.Muralidhar)

Main features of a CFD software being developed at IITK (2004- ) The aim is to develop a general-purpose CFD code which will  allow the numerical solutions of a wide variety of problems with flow and heat-transfer, chemical reaction, combustion, turbulence, and many other specialized applications,  run on a parallel cluster,  allow used-defined modules,  and be capable of enhancement and a life of 20 years

Brief Overview of Solver Features. 

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A multi-block Finite-volume solver for nonorthogonal hexahedral grids, that can read grids and write solutions in CGNS format. Solves Navier Stokes, Continuity, Temperature and Species transport equations, for constant density and variable density flows . Solves 2-D, 2-D axi-symmetric and 3-D problems in complex geometries. Time Stepping Schemes: First order (Implicit) and Second order (Crank Nicolson) schemes.

Physics incorporated in the Solver       

Conduction in solids Laminar and turbulent (6 models) forced, natural and mixed convection Conjugate Heat transfer with laminar and turbulent flows Melting and solidification problems Variable density laminar and turbulent flows Reactive laminar and turbulent flows with variable density Combustion with fast chemistry

Physics…       

Flow of ionised gasses in electric fields Flow and contaminant transport through ground-water and porous media Thermal radiation in enclosures Combined flow and thermal radiation in participative media Homogeneous equilibrium model for two-phase flow Two-fluid model for gas-liquid two-phase flow (with turbulence model) Two-fluid model for particle-gas two-phase flow (with turbulence model)

A typical application :Temperature Distribution in a Heat Exchanger : This is a conjugate heat transfer problem. Here the velocity and temperature distribution through out the Heat Exchanger vessel is to be simulated numerically using turbulence models, and the temperature distribution in the vessel walls is to be found.

NUMERICAL SIMULATION OF LIQUID METAL MHD FLOWS WITH IMPOSED AND INDUCED MAGNETIC FIELD 

Magneto Hydro Dynamics deals with flows of electrically conducting and non-magnetic fluids which are subjected to a magnetic field.

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Typical industrial applications of liquid metal MHD flows include electromagnetic flow meter, conduction pump, liquid metal blankets in fusion reactor etc.

Governing Equations (for induction-less approach) Derived from Maxwell’s equations and Navier-Stokes and continuity equation

Governing Equations (including induced magnetic field) 

In addition we also solve the induction equation:

where

is the magnetic diffusivity.

Important Parameters 

Hartmann number:

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Interaction parameter or Stuart number:

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Magnetic Reynolds number:

Solution methodology  Structured, non-uniform, non-orthogonal and

collocated grid. 

FVM Discretization.

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Semi-coupled method.

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BTCS with blending of QUICK/upwind for convection terms.

Results

Hartmann Flow 

The steady flow of liquid metal between infinitely broad parallel plates in presence of magnetic field. Analytical solution given by Chang et. al. (1961)

where c is the wall conductance ratio

Insulating walls

Variation of velocity for Rem=10

Perfectly conducting walls

Variation of velocity for Rem=1

Arbitrary conducting walls

Variation of velocity for Rem=10

Buoyancy driven convection in a rectangular cavity in presence of magnetic field (induction-less):

Schematic of the problem

Stream lines of the flow

a) Ha = 0; b) Ha = 5; c) Ha = 50; d) Ha = 100

Comparison of horizontal velocity with the analytical solution given by Garandet et. al. (1992)

Variation of normalized vertical velocity along the horizontal direction at the mid length of the cavity

Isotherms

Isotherms giving the temperature distribution in the cavity a) Ha = 0; b) Ha = 5; c) Ha = 50; d) Ha = 100

Variation of applied magnetic field along the length of the channel

Variation of axial velocity for Ha=10 and Rem=1

Flow in 3D channel (Induction-less approach)

Ha=0 Insulated walls

Ha=5

Contd.

Ha=10

Ha=20

Induction less approach (M-profile)

M-Profile for Ha=50

M-Profile for Ha=600

Conclusions  An algorithm is developed for solving velocity and 

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magnetic field to simulate MHD flows. The present algorithm works well for capturing flow and magnetic fields accurately for several cases of channel flows. The induction less algorithm works well for 3-D geometries and captures the familiar M-profile.

Scope for the future work  Code should be thoroughly tested on complex

geometries. 

In case of high Hartmann numbers grids need to be finer especially near the wall due to Hartmann layer and side layer. This can be eliminated by core flow approximation.

THANK YOU

Boundary conditions for Φ 

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The boundary conditions at the inlet and the exit are chosen such that no electric currents leave or enter the domain.

for insulating walls

for perfectly conducting walls

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Boundary conditions for Arbitrary conducting walls:

Boundary conditions for B The boundary conditions for the total magnetic field : 

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At the inlet a homogeneous Dirichlet condition is specified and at outlet homogeneous Neumann condition is given. for perfectly conducting walls

for insulating walls depending on the direction in which magnetic field is applied or

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