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Current Density Conservative Scheme

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A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated meshIntroductionConservative formula of the Lorentz forceConsistent and conservative schemesConsistent scheme for calculation of current flux on cell faceConservative scheme for calculation of the Lorentz ForceBased on a conservative formula of the Lorentz forceBased on a conservative interpolation of current densityA non-conservative scheme for calculation of the Lorentz forceProjection method and detailed calculation procedures for MHDValidation of consistent and conservative schemesConservation of current densityShercliff ' s case on a rectangular mesh with core perturbationHunt ' s case on a triangular mesh and a prism meshFully 3D simulation - comparing with an experiment for circular pipeConclusionAcknowledgments blank ReferencesA current density conservative scheme for incompressibleMHD flows at a low magnetic Reynolds number.Part II: On an arbitrary collocated meshMing-Jiu Nia,c,*, Ramakanth Munipallib, Peter Huangb, Neil B. Morleya,Mohamed A. AbdouaaMAE Department, University of California at Los Angeles, CA 90095, USAbHyperComp Inc., Westlake Village, CA 91362, USAcPhysics Department, Graduate University of Chinese Academy of Sciences, Beijing 100049, ChinaReceived 19 October 2006; received in revised form 16 April 2007; accepted 22 July 2007Available online 8 August 2007AbstractA conservative formulation of the Lorentz force is given here for magnetohydrodynamic (MHD) flows at a low mag-netic Reynolds number with the current density calculated based on Ohm’s law and the electrical potential formula. Thisconservative formula shows that the total momentum contributed from the Lorentz force is conservative when the appliedmagnetic field is constant. For the case with a non-constant applied magnetic field, the Lorentz force has been divided intotwo parts: a strong globally conservative part and a weak locally conservative part.The conservative formula has been employed to develop a conservative scheme for the calculation of the Lorentz forceon an unstructured collocated mesh. Only the current density fluxes on the cell faces, which are calculated using a consis-tent scheme with good conservation, are needed for the calculation of the Lorentz force. Meanwhile, a conservative inter-polation technique is designed to get the current density at the cell center from the current density fluxes on the cell faces.This conservative interpolation can keep the current density at the cell center conservative, which can be used to calculatethe Lorentz force at the cell center with good accuracy. The Lorentz force calculated from the conservative current at thecell center is equivalent to the Lorentz force from the conservative formula when the applied magnetic field is constant,which can conserve the total momentum. We will further prove that the simple interpolation scheme used in the Part I[M.-J. Ni, R. Munipalli, N.B. Morley, P.Y. Huang, M. Abdou, A current density conservative scheme for MHD flowsat a low magnetic Reynolds number. Part I. On a rectangular collocated grid system, Journal of Computational Physics,in press, doi:10.1016/j.jcp.2007.07.025] of this series of papers is conservative on a rectangular grid and can keep the totalmomentum conservative in a rectangular grid. 2007 Elsevier Inc. All rights reserved.Keywords: Conservative formula of the Lorentz force; Consistent and conservative scheme; Projection method; MHD0021-9991/$ - see front matter  2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jcp.2007.07.023*Corresponding author. Address: Physics Department, Graduate University of Chinese Academy of Sciences, Beijing 100049, China.E-mail addresses: [email protected], [email protected] (M.-J. Ni).URL: http://www.fusion.ucla.edu (M.A. Abdou).Available online at www.sciencedirect.comJournal of Computational Physics 227 (2007) 205–228www.elsevier.com/locate/jcp1. IntroductionMagnetohydrodynamic (MHD) flows of electrically conducting liquids at high Hartmann numbers hasbeen a topic of great interest in the development of a fusion reactor blanket [1–4] . Two-dimensional MHDflows in a channel have been extensively studied by theoretical analysis and numerical simulation [5–8].The Hartmann number Ha is a measur e of the magnetic field strength for a given fluid in a duct of a givenscale. The thickness of the Hartmann layers on all walls normal to field scales with Ha1and is very thin;the side layers on all walls parallel to the magnetic field scale with Ha1/2and are much thicker than the Hart-mann layers at high Hartmann numbers. The developm ent of fusion reactors experiments with strong mag-netic fields leads to a growing interest in the study of 3D MHD phenomena. When inertial terms are small,the asymptotic method [9–11] that focuses on the main phenomena is a very efficient method to compute3D MHD flows. This efficient method is valid for high Hartmann numbers and high inter action parameters,but cannot always handle arbitrary complex geometries. When inertial terms are important, the direct simu-lation method accounting for all the physical effects is an important tool to study the 3D MHD phenomena.Using the direct simulation method, a very fine mesh is required to resolve the Hartmann layer and the sidelayer at high Hartmann numbers. For unsteady flows, the time step is proportional to the smallest grid sizewith an explicit update of the temporal term. The cost of direct simulation of 3D MHD flows at high Hart-mann number is high since it requires a fine mesh and therefore a small time step for uns teady flows. Three-dimensional numerical simulations of inertial flows are often limited to the steady regime and low Hartmannnumbers.For low magnetic Reynolds numbers, the electrical potential formula can be employed for MHD with goodaccuracy [12,13]. Consider the fully developed MHD flow in a rectangular channel, where flow velocity is inthe x-direction and magnetic field is applied in the y-direction. All derivatives with respect to x are zero exceptfor that of pressure, which is a constant in the flow and zero in solid wall regions. Current is computed fromOhm’s law asJ ¼ð0; jy; jzÞ¼r 0; ouoy; ouozþ u : ð1ÞIn the core of the flow away from a ny viscous or inertial boundary layers, the Lorentz force is balanced by thepressure gradient, as illustrated in [14], which leads


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