×

zbMATH — the first resource for mathematics

A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations. (English) Zbl 1427.76276
Summary: The coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular section are solved using combined finite volume method and spectral element technique, improved by means of Hermit interpolation. The transverse applied magnetic field may have an arbitrary orientation relative to the section of the pipe. The velocity and induced magnetic field are studied for various values of Hartmann number, wall conductivity and orientation of the applied magnetic field. Comparisons with the exact solution and also some other numerical methods are made in the special cases where the exact solution exists. The numerical results for these sample problems compare very well to analytical results.

MSC:
76W05 Magnetohydrodynamics and electrohydrodynamics
76M12 Finite volume methods applied to problems in fluid mechanics
76M22 Spectral methods applied to problems in fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alfvén, H., Existence of electromagnetic-hydrodynamic waves, Nature, 150, 405-406, (1942)
[2] Al Moatassime, H.; Esselaoui, D., A finite volume approach for unsteady viscoelastic fluid flows, Int. J. numer. meth. fluid, 39, 939-959, (2002) · Zbl 1101.76363
[3] Bailey, C.; Taylor, G.A.; Cross, M.; Chow, P., Discretisation procedures for multi-physics phenomena, J. comput. app. math., 103, 3-17, (1999) · Zbl 0962.76057
[4] Barrett, K.E., Duct flow with a transverse magnetic field at high Hartmann numbers, Int. J. numer. meth. eng., 50, 1893-1906, (2001) · Zbl 0998.76045
[5] Barth, T., Aspects of unstructured grids and finite volume solvers for the Euler and Navier-Stokes equations, 25th computational fluid dynamics lecture series, (1994), Von Karman Institute
[6] Bojarevics, V.; Sharamkin, V.I., MHD flows due to current spreading in an axisymmetric layer of finite thickness, Magnetohydrodynamics, 13, 172-177, (1977)
[7] Bojarevics, V., Magnetohydrodynamic interface waves and the distribution of heart caused by the dynamic interaction of currents in an aluminium electrolyte cell, Magnetohydrodynamics, 28, 360-367, (1992) · Zbl 0793.76108
[8] Bojarevics, V.; Pericleous, K., Magnetic levitation fluid dynamics, Magnetohydrodynamics, 37, 93-102, (2001)
[9] Bojarevics, V.; Pericleous, K., Comparison of MHD models for aluminium reduction cells, Light metals, 2, 347-352, (2006)
[10] Bozkaya, N.; Tezer-Sezgin, M., Time-domain BEM solution of convection-diffusion-type MHD equations, Int. J. numer. meth. fluids, 56, 1969-1991, (2008) · Zbl 1388.76191
[11] H. Branover, P. Gershon, MHD turbulence study, Ben-Gurion University, Rept. BGUN-RDA-100-76, 1976.
[12] Cai, Z., On the finite volume element method, Numer. math., 58, 713-735, (1991) · Zbl 0731.65093
[13] Cai, Z.; Mandel, J.; McCormick, S., The finite volume element method for diffusion equations on general triangulations, SIAM J. numer. anal., 28, 2, 392-402, (1991) · Zbl 0729.65086
[14] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag New York · Zbl 0658.76001
[15] Chainais-Hillairet, C., Second-order finite-volume schemes for a non-linear hyperbolic equation: error estimate, Math. method. appl. sci., 23, 467-490, (2000) · Zbl 0958.65096
[16] Chan, C.T.; Anastasiou, K., Solution of incompressible flows with or without a free surface using the finite volume method on unstructured triangular meshes, Int. J. numer. meth. fluids, 29, 35-57, (1999) · Zbl 0936.76037
[17] Chang, C.C.; Lundgren, T.S., Duct flow in magnetohydrodynamics, Zamp, 12, 100-114, (1961) · Zbl 0115.21904
[18] Chatzipantelidis, P., A finite volume method based on the Crouzeix-Raviart element for elliptic PDE’s in two dimensions, Numer. math., 82, 409-432, (1999) · Zbl 0942.65131
[19] Chatzipantelidis, P.; Lazarov, R.D.; Thomée, V., Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer. meth. partial differential eq., 20, 650-674, (2004) · Zbl 1067.65092
[20] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043
[21] Davidson, J.; Thess, A.; Davidson, P.A., Magnetohydrodynamics, (2002), Springer Vienna · Zbl 1030.00041
[22] Dehghan, M.; Mirzaei, D., Meshless local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl. numer. math., 59, 1043-1058, (2009) · Zbl 1159.76034
[23] Dehghan, M.; Mirzaei, D., Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes, Comput. phys. commun., 180, 1458-1466, (2009)
[24] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. simulation, 71, 16-30, (2006) · Zbl 1089.65085
[25] Dehghan, M., On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. methods partial differential eq., 21, 24-40, (2005) · Zbl 1059.65072
[26] Demendy, Z.; Nagy, T.; Hungary, M.-E., A new algorithm for solution of equations of MHD channel flows at moderate Hartmann numbers, Acta mech., 123, 135-149, (1997) · Zbl 0902.76058
[27] Dragos, L., Magneto-fluid dynamics, (1975), Abacus Press England
[28] Erduran, K.S.; Kutija, V.; Hewett, C.J.M., Performance of finite volume solutions to the shallow water equations with shock-capturing schemes, Int. J. numer. meth. fluids, 40, 1237-1273, (2002) · Zbl 1047.76059
[29] Ewing, R.E.; Lazarov, R.D.; Lin, Y., Finite volume element approximations of nonlocal in time one-dimensional flows in porous media, Computing, 64, 157-182, (2000) · Zbl 0969.76052
[30] Ewing, R.E.; Lin, T.; Lin, Y., On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. numer. anal., 39, 6, 1865-1888, (2002) · Zbl 1036.65084
[31] Eymard, R.; Gallouët, T.; Herbin, R., Finite volume methods, (), 713-1020 · Zbl 0981.65095
[32] Fuchs, F.G.; Mishra, S.; Risebro, N.H., Splitting based finite volume schemes for ideal MHD equations, J. comput. phys., 228, 641-660, (2009) · Zbl 1259.76021
[33] Gardner, L.R.T.; Gardner, G.A., A two-dimensional bi-cubic B-spline finite element used in a study of MHD duct flow, Comput. methods appl. mech. engrg., 124, 365-375, (1995) · Zbl 0844.65073
[34] Gold, R.R., Magnetohydrodynamic pipe flow. part 1, J. fluid mech., 13, 505-512, (1962) · Zbl 0117.43301
[35] Guermond, J.L.; Laguerre, R.; Léorat, J.; Nore, C., An interior penalty Galerkin method for the MHD equations in heterogeneous domains, J. comput. phys., 221, 349-369, (2007) · Zbl 1108.76040
[36] Han, J.; Tang, H., An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics, J. comput. phys., 220, 791-812, (2007) · Zbl 1235.76084
[37] Hartmann, J.; Hg-Dynamics, I., Theory of the laminar flow of an electrically conducting liquid in a homogeneous magnetic field, K. dan. vidensk. selsk. mat.-fys. medd., 15, 1-27, (1937)
[38] Harris, R.E.; Wang, Z.J., High-order adaptive quadrature-free spectral volume method on unstructured grids, Computers and fluids, 38, 2006-2025, (2009) · Zbl 1242.76219
[39] Holroyd, E.T.; Hunt, J.C.R., A review of MHD flows in ducts with changing cross section areas and non-uniform magnetic fields, Euromech colloquium, 70, 16-19, (1976)
[40] Holroyd, R.J., An experimental study of the effects of wall conductivity, non-uniform magnetic field and variable-area ducts on liquid metal flow at high Hartmann number, part 1: ducts with non-conducting walls, J. fluid mech., 93, 609-630, (1979)
[41] Holroyd, R.J., MHD flow in a rectangular duct with pairs of conducting and non-conducting walls in the presence of a non-uniform magnetic field, J. fluid mech., 96, 335-353, (1980)
[42] Hua, T.Q.; Walker, J.S., Three-dimensional MHD flow in insulating circular ducts in non-uniform transverse magnetic fields, Int. J. engrg. sci., 27, 1079-1091, (1989) · Zbl 0693.76058
[43] Hughes, W.F.; Young, F.J., The electromagnetohydrodynamics of fluid, (1966), John Wiley and Sons New York
[44] Hughes, W.F.; McNAB, I.R.; Branover, H.; Lykoudis, P.S.; Yakhot, A., A quasi one dimensional analysis of an electromagnetic pump including end effects, Liquid-metal flows and magnetohydrodynamics, Progress in astronautics and aeronautics, 84, 287-312, (1983)
[45] Hughes, M.; Pericleous, K.A.; Cross, M., The CFD analysis of simple parabolic and elliptic MHD flows, Appl. math. modelling, 18, 150-155, (1994) · Zbl 0798.76051
[46] Hughes, M.; Pericleous, K.A.; Cross, M., The numerical modelling of DC electromagnetic pump and brake flow, Appl. math. model., 19, 713-723, (1995) · Zbl 0856.76093
[47] Huang, J.G.; Xi, S.T., On the finite volume element method for general self-adjoint elliptic problems, SIAM J. numer. anal., 35, 1762-1774, (1998) · Zbl 0913.65097
[48] Hunt, J.C.R., Magnetohydrodynamic flow in rectangular ducts, J. fluid mech., 21, 577-590, (1965) · Zbl 0125.18401
[49] Kao, A.; Djambazov, G.; Pericleous, K.; Voller, V., Thermoelectric MHD in dendritic solidification, Magnetohydrodynamics, 45, 305-316, (2009)
[50] Kechkar, N.; Silvester, D., Analysis of locally stabilized mixed finite element methods for the Stokes problem, Math. comput., 58, 1-10, (1992) · Zbl 0738.76040
[51] LeVeque, R.J., Finite volume methods for hyperbolic problems, (2002), University Press Cambridge · Zbl 1010.65040
[52] Lopes Verardi, S.L.; Machado, J.M.; Shiyou, Y., The application of interpolating MLS approximations to the analysis of MHD flows, Source finite elements in analysis and design, 39, 1173-1187, (2003)
[53] Lukáčová-Medvid’ová, M.; Vlk, Z., Well-balanced finite volume evolution Galerkin methods for the shallow water equations with source terms, Int. J. numer. meth. fluids, 47, 1165-1171, (2005) · Zbl 1064.76073
[54] Meir, A.J., Finite element analysis of magnetohydrodynamic pipe flow, Appl. math. comput., 57, 177-196, (1993) · Zbl 0778.76053
[55] Meir, A.J.; Schmidt, P.G., Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary, SIAM J. numer. anal., 36, 1304-1332, (1999) · Zbl 0948.76091
[56] Mignone, A., A simple and accurate Riemann solver for isothermal MHD, J. comput. phys., 225, 1427-1441, (2007) · Zbl 1343.76034
[57] Mitchell, W.R., Optimal multilevel iterative methods for adaptive grids, SIAM J. sci. stat. comp., 13, 146-167, (1992) · Zbl 0746.65087
[58] Neslitürk, A.I.; Tezer-Sezgin, M., The finite element method for MHD flow at high Hartmann numbers, Comput. methods appl. mech. engrg., 194, 1201-1224, (2005) · Zbl 1091.76036
[59] Neslitürk, A.I.; Tezer-Sezgin, M., Finite element method solution of electrically driven magnetohydrodynamic flow, J. comput. appl. math., 192, 339-352, (2006) · Zbl 1155.76357
[60] 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 1: on a rectangular collocated grid system, J. comput. phys., 227, 174-204, (2007) · Zbl 1280.76045
[61] Oliveira, P.J., On the numerical implementation of nonlinear viscoelastic models in a finite-volume method, Numer. heat transfer, part B, 40, 283-301, (2001)
[62] Patera, A.T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. comput. phys., 54, 468-488, (1984) · Zbl 0535.76035
[63] Pozrikidis, C., Introduction to finite and spectral element methods using Matlab, (2005), Chapman and Hall/CRC · Zbl 1078.65109
[64] Pericleous, K.A.; Bojarevics, V., Pseudo-spectral solutions for fluid flow and heat transfer in electro-metallurgical applications, Progr. comput. fluid dynam., 7, 118-127, (2007) · Zbl 1388.76229
[65] Pericleous, K.A.; Hughes, M.; Cross, M., The CFD analysis of simple parabolic and elliptic MHD flows, Appl. math. modelling, 18, 150-155, (1994) · Zbl 0798.76051
[66] Ramos, J.I.; Winowich, N.S., Magnetohydrodynamic channel flow study, Phys. fluids, 29, 992-997, (1986) · Zbl 0593.76114
[67] Ramos, J.I.; Winowich, N.S., Finite difference and finite element methods for MHD channel flows, Int. J. numer. meth. fluids, 11, 907-934, (1990) · Zbl 0704.76066
[68] Ravindran, S.S., Linear feedback control and approximation for a system governed by unsteady MHD equations, Comput. methods appl. mech. engrg., 198, 524-541, (2008) · Zbl 1228.76194
[69] Reynolds, D.R.; Samtaney, R.; Woodward, C.S., A fully implicit numerical method for single-fluid resistive magnetohydrodynamics, J. comput. phys., 219, 144-162, (2006) · Zbl 1103.76036
[70] Salah, N.B.; Soulaimani, A.; Habashi, W.G., A finite element method for magnetohydrodynamics, Comput. methods appl. mech. engrg., 190, 5867-5892, (2001) · Zbl 1044.76030
[71] Seungsoo, L.; Dulikravich, G.S., Magnetohydrodynamic steady flow computation in three dimensions, Int. J. numer. meth. fluids, 13, 917-936, (1991) · Zbl 0741.76049
[72] Shercliff, J.A., Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. camb. phil. soc., 49, 136-144, (1953) · Zbl 0050.19404
[73] Sheu, T.W.H.; Lin, R.K., Development of a convection-diffusion-reaction magnetohydrodynamic solver on nonstaggered grids, Int. J. numer. meth. fluids, 45, 1209-1233, (2004) · Zbl 1060.76619
[74] Singh, B.; Lal, J., MHD axial flow in a triangular pipe under transverse magnetic field, Indian J. pure appl. math., 9, 101-115, (1978) · Zbl 0383.76094
[75] Singh, B.; Lal, J., MHD axial flow in a triangular pipe under transverse magnetic field parallel to a side of the triangle, Ind. J. tech., 17, 184-189, (1979) · Zbl 0413.76094
[76] Singh, B.; Lal, J., FEM in MHD channel flow problems, Int. J. numer. meth. eng., 18, 1104-1111, (1982) · Zbl 0489.76119
[77] Singh, B.; Lal, J., Heat transfer for MHD flow through a rectangular pipe with discontinuity in wall temperatures, J. heat moss transfer, 25, 1523-1529, (1982) · Zbl 0517.76120
[78] Singh, B.; Lal, J., FEM for unsteady MHD flow through pipes with arbitrary wall conductivity, Int. J. numer. meth. fluids, 4, 291-302, (1984) · Zbl 0547.76119
[79] Singh, B.; Lal, J.; Agarwal, P.K., Finite element method for unsteady MHD channel flow with arbitrary wall conductivity and orientation of applied magnetic field, Indian J. pure appl. math., 16, 11, 1390-1398, (1985) · Zbl 0593.76116
[80] Slone, A.K.; Bailey, C.; Cross, M., Dynamic solid mechanics using finite volume methods, Appl. math. model., 27, 69-87, (2003) · Zbl 1135.74319
[81] Smith, P., Some asymptotic extremum principles for magnetohydrodynamic pipe flow, Appl. sci. res., 24, 452-466, (1971) · Zbl 0235.76053
[82] Sterl, A., Numerical simulation of liquid-metal MHD flows in rectangular ducts, J. fluid mech., 216, 161-191, (1990) · Zbl 0698.76123
[83] Suwon, C.; Sang, H.H., The magnetic field and performance calculations for an electromagnetic pump of a liquid metal, J. phys. D: appl. phys., 31, 2754-2759, (1998)
[84] Takhar, H.S.; Singh, A.K.; Nath, G., Unsteady MHD flow and heat transfer on a rotating disk in an ambient fluid, Int. J. therm. sci., 41, 147-155, (2002)
[85] Tezer-Sezgin, M., Magnetohydrodynamic flow in a rectangular duct, Int. J. numer. meth. fluids, 7, 697-718, (2005) · Zbl 0639.76112
[86] Tezer-Sezgin, M.; Han Aydın, S., Dual reciprocity boundary element method for magnetohydrodynamic flow using radial basis functions, Int. J. comput. fluid dyn., 16, 1, 88-92, (2002) · Zbl 1082.76580
[87] Tezer-Sezgin, M.; Köksal, S., Finite element method for solving MHD flow in a rectangular duct, Int. J. numer. meth. eng., 28, 445-459, (1989) · Zbl 0669.76140
[88] Tikhonov, A.N.; Samarskii, A.A., Homogeneous difference schemes, USSR comput. math. math. phys., 1, 5-67, (1962) · Zbl 0131.34102
[89] Tikhonov, A.N.; Samarskii, A.A., Homogeneous difference schemes on nonuniform nets, USSR comput. math. math. phys., 2, 927-953, (1963) · Zbl 0128.36702
[90] Versteeg, H.; Malalasekra, W., An introduction to computational fluid dynamics: the finite volume method, (2007), Prentice Hall
[91] Walker, J.S.; Ludford, G.S.S., MHD flow in insulating circular expansions with strong transverse magnetic fields, Int. J. eng. sci., 12, 1045-1061, (1974) · Zbl 0285.76046
[92] Wan, Q.; Wan, H.; Zhou, C.; Wu, Y., Simulating the hydraulic characteristics of the lower yellow river by the finite-volume technique, Hydrol. process., 16, 2767-2779, (2002)
[93] Winowich, N.S.; Hughes, W.F.; Ramos, J.I., Numerical simulation of electromagnetic pump flow, Numer. methods laminar turbulent flow, 5, 1228-1240, (1987)
[94] Xia, G.; Lin, C.L., An unstructured finite volume approach for structural dynamics in response to fluid motions, Comput. struct., 86, 684-701, (2008)
[95] Xiong, Z.; Chen, Y., Finite volume element method with interpolated coefficients for two-point boundary value problem of semilinear differential equations, Comput. methods appl. mech. engrg., 196, 3798-3804, (2007) · Zbl 1173.65329
[96] Yagawa, G.; Masuda, M., Finite element analysis of magnetohydrodynamics and its application to lithium blanket design of fusion reactor, Nucl. eng. design, 71, 121-136, (1982)
[97] Yee, H.C.; Sjögreen, B., Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems, J. comput. phys., 225, 910-934, (2007) · Zbl 1343.76053
[98] Zhang, M.; Yu, S.T. John; Lin, S.C. Henry; Chang, S.C.; Blankson, I., Solving the MHD equations by the space-time conservation element and solution element method, J. comput. phys., 214, 599-617, (2006) · Zbl 1136.76399
[99] Zang, Z.P.; Teng, B.; Bai, W.; Cheng, L., A finite volume solution of wave forces on submarine pipelines, Ocean eng., 34, 1955-1964, (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.