Abstract
A benchmark solution is of great importance in validating algorithms and codes for magnetohydrodynamic (MHD) flows. Hunt and Shercliff’s solutions are usually employed as benchmarks for MHD flows in a duct with insulated walls or with thin conductive walls, in which wall effects on MHD are represented by the wall conductance ratio. With wall thickness resolved, it is stressed that the solution of Sloan and Smith’s and the solution of Butler’s can be used to check the error of the thin wall approximation condition used for Hunt’s solutions. It is noted that Tao and Ni’s solutions can be used as a benchmark for MHD flows in a duct with wall symmetrical or unsymmetrical, thick or thin. When the walls are symmetrical, Tao and Ni’s solutions are reduced to Sloan and Smith’s solution and Butler’s solution, respectively.
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References
Abdou A, Sze D, Wong C, et al. U.S. plans and strategy for ITER blanket testing. Fusion Sci Technol, 2005, 47: 475–487
Molokov S, Moreau R, Moffatt H K. Magnetohydrodynamics—Historical Evolution and Trends. New York: Springer, 2007
Kulikovskii A G. Slow steady flows of a conducting fluid at large Hartmann numbers. Fluid Dyn, 1968, 3: 1–5
Molokov S, Bühler L. Liquid metal flow in a U-bend in a strong uniform magnetic field. J Fluid Mech, 1994, 267: 325–352
Reed C B, Piclolglou B F, Hua T Q, et al. ALEX results—a comparison of measurements from a round and a rectangular duct with 3-D code predictions. In: the 12th Symposium on Fusion Engineering, Monkey, CA, October, 1987
Bühler L, Horanyi S. Experimental investigations of MHD flows in a sudden expansion. FZKA 7245, Forschungszentrum Karlsruhe, 2006
Ni M J, Li J F. A consistent and conservative scheme for incompressible MHD flows at low magnetic Reynolds number. Part III: On a staggered mesh. J Comput Phys, 2011, 231: 281–298
Schercliff J A. Steady motion of conducting fluids in pipes under transverse magnetic fields. Proc Cambe Philos Soc, 1953, 49: 126–144
Uflyand Y S. Flow stability of a conducting fluid in a rectangular channel in a transverse magnetic field. Soviet Phys Tech Phys, 1961, 5: 1194–1196
Chang C C, Lundgren T S. Duct flow in magnetohydrodynamics. Z Angew Math Phys, 1961, 12: 100–114
Hunt J C R. 1965 magnetohydrodynamic flow in rectangular ducts. J Fluid Mech, 1965, 21: 577–590
Sloan D M, Smith P. Magnetohydrodynamic flow in a rectangular pipe between conducting plates. Z Angew Math Mech, 1966, 46: 439–443
Tao Z, Ni M J. Exact solutions for MHD flows in a rectangular duct with unsymmetrical Hartmann walls. Submitted for review
Butler G F. A note on magnetohydrodynamic duct flow. Proc Camb Philos Soc, 1969, 66: 66–77
Tao Z, Ni M J. Return to Hunt’s case I of MHD flows in a rectangular duct with unsymmetrical walls. Submitted for review
Jaworski M A, Gray T K, Antonelli M, et al. Thermoelectric magnetohydrodynamic stirring of liquid metals. Phys Rev Lett, 2010, 104: 094503
Davidson P A. An Introduction to Magnetohydrodynamics. Cambridge: Cambridge University Press, 2001
Ni M J, Munipalli R, Morley N B, et al. A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: On a rectangular collocated mesh. J Comput Phys, 2007, 227: 187–204
Ni M J, Munipalli R, Huang P, et al. A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh. J Comput Phys, 2007, 227: 205–228
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Tao, Z., Ni, M. Benchmark solutions for MHD solver development. Sci. China Phys. Mech. Astron. 56, 378–382 (2013). https://doi.org/10.1007/s11433-013-4997-5
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DOI: https://doi.org/10.1007/s11433-013-4997-5