Can anyone please advise me on a 2D MHD Benchmark problem w.r.t availabilty in books/journals, etc.? Thanks in advance.

I solved some 2D and 3D MHD problems in my PhD using CFX.

Some details follow, please feel free to contact me if you want to know more.

Title "Coupled heat/species transfer and buoyancy-driven magnetohydrodynamic convection"

Author Duncan Maclean, Department of Engineering, University of Cambridge, December 2000.

Abstract:

This thesis presents a numerical and analytical study of the effects of a steady magnetic field on thermal and solutal buoyancy-driven convection in liquid metals and liquid semiconductors of electricity. Of particular interest is the coupling between the heat/species transfer and the magnetohydrodynamic equations when the magnetic field is strong. Numerical simulations with commercial CFD codes which are modified to account for the electromagnetic effects are used to validate all of the analysis. The work on solutal buoyancy-driven convection relates specifically to long-capillary methods for determining coefficients of molecular diffusion. The work on thermal buoyancy-driven convection is fundamental and may have applications where convective effects are dominant such as semiconductor crystal growth and the design of nuclear fusion reactors.

An effective diffusivity model is developed for mass transport by molecular diffusion and unsteady solute buoyancy-driven convection in a horizontal capillary by a method similar to that used for Taylor dispersion in forced flow. A 1D non-linear diffusion equation is obtained and an analytical asymptotic solution is found for dominant convective mass transport. A scaling analysis clarifies the boundary between dominant molecular diffusion and dominant convective mass transport. The use of a steady vertical magnetic field for liquid metals and semi-conductors is found to damp the convective mass transport by a factor of $H\!a^{-4}$ where $H\!a$ is the Hartmann number.

Thermal buoyancy-driven convection in a tall cavity is studied for cases where there is a strong magnetic field and heat transfer is predominantly convective, this requires respectively that $H\hspace{-0.05cm}a$ and $Ra/H\hspace{-0.05cm}a^{2}$ are large where $Ra$ is the Rayleigh number. A 3D analytical model is developed for conditions of uniform heat flux on two opposite side-walls with the magnetic field parallel to the imposed thermal gradient. In the asymptotic limits the core is stationary and thermally stratified with all the flow passing through thin boundary layers. Analytical solutions are obtained for velocity and temperature fields and a Nusselt number based on the temperature difference between the heated walls is found asymptotically to approach $\frac{1}{2}\left( Ra/H\hspace{-0.05cm}a^{2}\right) ^{2/5}$. An idealistic 2D model of the same cavity is considered but with uniform and different temperatures on the side-walls. The core is again thermally stratified and has only a horizontal component of velocity. Approximate analytical solutions are obtained for velocity and temperature and an average Nusselt number is found to be $\sqrt{LRa/H\hspace{-0.05cm}a^{2}H}$ where $H$ and $L$ are the height and width of the cavity respectively. Numerical results are presented for a 2D square cavity which is tilted at some angle and subject to a vertical magnetic field with uniform heat flux through two opposite side-walls. These results confirm asymptotic analytical solutions by Cowley (1996)\ for velocity, temperature and Nusselt number, the latter being a function only of the angle of tilt.

[prodriguez7]

Dear Duncan Maclean,

I am trying to find your published paper on "Coupled heat/species transfer and buoyancy-driven magnetohydrodynamic convection" but I have had no luck in finding it. I also had another question regarding MHD, where can I find tutorials specifically on MHD. Thanks

[alainou]

i am very interested to have a copy of your work thak u in advance

[alainou]

Originally Posted by Duncan Maclean
;15089

I solved some 2D and 3D MHD problems in my PhD using CFX.

Some details follow, please feel free to contact me if you want to know more.

Title "Coupled heat/species transfer and buoyancy-driven magnetohydrodynamic convection"

Author Duncan Maclean, Department of Engineering, University of Cambridge, December 2000.

Abstract:

This thesis presents a numerical and analytical study of the effects of a steady magnetic field on thermal and solutal buoyancy-driven convection in liquid metals and liquid semiconductors of electricity. Of particular interest is the coupling between the heat/species transfer and the magnetohydrodynamic equations when the magnetic field is strong. Numerical simulations with commercial CFD codes which are modified to account for the electromagnetic effects are used to validate all of the analysis. The work on solutal buoyancy-driven convection relates specifically to long-capillary methods for determining coefficients of molecular diffusion. The work on thermal buoyancy-driven convection is fundamental and may have applications where convective effects are dominant such as semiconductor crystal growth and the design of nuclear fusion reactors.

An effective diffusivity model is developed for mass transport by molecular diffusion and unsteady solute buoyancy-driven convection in a horizontal capillary by a method similar to that used for Taylor dispersion in forced flow. A 1D non-linear diffusion equation is obtained and an analytical asymptotic solution is found for dominant convective mass transport. A scaling analysis clarifies the boundary between dominant molecular diffusion and dominant convective mass transport. The use of a steady vertical magnetic field for liquid metals and semi-conductors is found to damp the convective mass transport by a factor of $H\!a^{-4}$ where $H\!a$ is the Hartmann number.

Thermal buoyancy-driven convection in a tall cavity is studied for cases where there is a strong magnetic field and heat transfer is predominantly convective, this requires respectively that $H\hspace{-0.05cm}a$ and $Ra/H\hspace{-0.05cm}a^{2}$ are large where $Ra$ is the Rayleigh number. A 3D analytical model is developed for conditions of uniform heat flux on two opposite side-walls with the magnetic field parallel to the imposed thermal gradient. In the asymptotic limits the core is stationary and thermally stratified with all the flow passing through thin boundary layers. Analytical solutions are obtained for velocity and temperature fields and a Nusselt number based on the temperature difference between the heated walls is found asymptotically to approach $\frac{1}{2}\left( Ra/H\hspace{-0.05cm}a^{2}\right) ^{2/5}$. An idealistic 2D model of the same cavity is considered but with uniform and different temperatures on the side-walls. The core is again thermally stratified and has only a horizontal component of velocity. Approximate analytical solutions are obtained for velocity and temperature and an average Nusselt number is found to be $\sqrt{LRa/H\hspace{-0.05cm}a^{2}H}$ where $H$ and $L$ are the height and width of the cavity respectively. Numerical results are presented for a 2D square cavity which is tilted at some angle and subject to a vertical magnetic field with uniform heat flux through two opposite side-walls. These results confirm asymptotic analytical solutions by Cowley (1996)\ for velocity, temperature and Nusselt number, the latter being a function only of the angle of tilt.

I am very interested to have electronic copy of your work, thank u