I need some help to find out empirical relation for entrance length in channel flow in terms of Hartmann number and Reynolds number. As far as I know it is proportional to Ha/Re. I think the following paper has given some relation about this:
W. T. SNYDER. "Magnetohydrodynamic flow in the entrance region of a parallel-plate channel." AIAA Journal, Vol. 3.
Entrance length can be estimated as follows. First, the thickness of viscous boundary layer at the distance L from the entrance is d~(L*nu/U)^1/2 (as for Blasius profile). In the non-magnetic case, d has to spread over the whole channel height D, i.e., d~D, which leads to the classical result L~D*Re, where Re=UD/nu is Reynolds numbers. In MHD channel flow, d has to reach only the thickness of Hartmann layer ~D/Ha, which results in L~D*Re/Ha^2. In MHD duct/pipe flow, d has to reach the thickness of parallel layers ~D/Ha^1/2, which results in L~D*Re/Ha.
Thanks a lot. It would be great if i have exact empirical relation for entrance length. I am doing a numerical simulation on 90 degree bend for high Hartmann number .So i have to optimize my geometry to reduce computation time.
In that case you could just take a developed flow profile a few 'diameters' upstream from the bend and forget about the entrance length...
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