| Volume 8, Number 1, 2018, Pages 42-65 DOI：10.11948/2018.42 |
| Development of a high order discretization scheme for solving fully nonlinear magnetohydrodynamic equations |
| A. D. Abin Rejeesh,S. Udhayakumar,T. V. S. Sekhar,Rajagopalan Sivakumar |
| Keywords:High order compact scheme, Full-MHD equations, forced convective heat transfer, divided differences. |
| Abstract: |
| We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell''s equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of $64 \times 64$ is sufficient for weakly nonlinear problems and higher grids up to $512 \times 512$ are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for $Re=73$ which is in agreement with literature. When the magnetic Reynolds number $Rm<1$ separation length decreases linearly with strength of magnetic field on a log-log scale whereas if $Rm>1$, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order. |
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