Abstract
We present numerical simulations without modeling of an incompressible, laminar, unidirectional circular pipe flow of an electrically conducting fluid under the influence of a uniform transverse magnetic field. Our computations are performed using a finite-volume code that uses a charge-conserving formulation [called current-conservative formulation in references (Ni et al J Comput Phys 221(1):174–204, 2007, Ni et al J Comput Phys 227(1):205–228, 2007)]. Using high resolution unstructured meshes, we consider Hartmann numbers up to 3000 and various values of the wall conductance ratio c. In the limit \({c{\ll}{\rm Ha}^{-1}}\) (insulating wall), our results are in excellent agreement with the so-called asymptotic solution (Shercliff J Fluid Mech 1:644–666, 1956). For higher values of the wall conductance ratio, a discrepancy with the asymptotic solution is observed and we exhibit regions of velocity overspeed in the Roberts layers. We characterise these overspeed regions as a function of the wall conductance ratio and the Hartmann number; a set of scaling laws is derived that is coherent with existing asymptotic analysis.
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Hartmann J.: Hg-dynamics I. Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. K. Dan. Vidensk. Selsk. Mat. Fys. Medd. XV(6), 1–28 (1937)
Hartmann J., Lazarus F.: Hg-dynamics II. Experimental investigations on the flow of mercury in a homogeneous magnetic field. K. Dan. Vidensk. Selsk. Mat. Fys. Medd. XV(7), 1–45 (1937)
Gold R.R.: Magnetohydrodynamic pipe flow. Part 1. J. Fluid Mech. 13, 505–512 (1962)
Ihara S., Kiyohiro T., Matsushima A.: The flow of conducting fluids in circular pipes with finite conductivity under uniform transverse magnetic fields. J. Appl. Mech. 34(1), 29–36 (1967)
Samad S.: The flow of conducting fluids through circular pipes having finite conductivity and finite thickness under uniform transverse magnetic fields. Int. J. Eng. Sci. 19, 1221–1232 (1981)
Shercliff A.: The flow of conducting fluids in circular pipes under transverse magnetic fields. J. Fluid Mech. 1, 644–666 (1956)
Shercliff A.: Magnetohydrodynamic pipe flow. Part 2. High Hartmann number. J. Fluid Mech. 13, 513–518 (1962)
Chang C., Lundgren S.: Duct flow in magnetohydrodynamics. ZAMP XII, 100–114 (1961)
Ni M.-J., Munipalli R., Morley N.B., Huang P., Abdou M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic reynolds number. Part I: on rectangular collocated grid system. J. Comput. Phys. 221(1), 174–204 (2007)
Ni M.-J., Munipalli R., Huang P., Morley N.B., Abdou M.A.: 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. 227(1), 205–228 (2007)
Mistrangelo, C.: Three-dimensional MHD flow in sudden expansions. PhD Thesis, Fakültat für Maschinenbau, Universität Karlsruhe (2005)
Roberts P.H.: An Introduction to Magnetohydrodynamics. American Elsevier Publishing Company, Inc., New York (1967)
Walker J.S.: Magnetohydrodynamic flows in rectangular ducts with thin conducting walls. Journal de Mécanique 20(1), 79–112 (1981)
Gnatyuk V.V., Paramonova T.: Effect of the conductivity of the walls on the velocity profile in a circular tube. Magnetohydrodynamics 7(1), 145–147 (1971)
Roberts P.H.: Singularities of hartmann layers. Proc. R. Soc. A 300, 94–107 (1967)
Hunt J.C.R.: Magnetohydrodynamic flow in rectangular ducts. J. Fluid Mech. 21(4), 577–590 (1965)
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Communicated by O. Zikanov
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Vantieghem, S., Albets-Chico, X. & Knaepen, B. The velocity profile of laminar MHD flows in circular conducting pipes. Theor. Comput. Fluid Dyn. 23, 525–533 (2009). https://doi.org/10.1007/s00162-009-0163-0
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DOI: https://doi.org/10.1007/s00162-009-0163-0