FEM-BEM Coupling for the MHD Pipe Flow in an Exterior Region

ABSTRACT

In this study, the magnetohydrodynamic
(MHD) flow through a circular pipe under the influence of a transverse mag- netic field when the outside medium is also
electrically conducting is solved numerically by using FEM-BEM coupling
approach. The coupled partial differential equations defined for the interior
medium are transformed into homogenous modified Helmholtz equations. For the
exterior medium on an infinite region, the
Laplace equation is considered for the exterior magnetic field. Unknowns in the
equations are also related with the corresponding Dirichlet and Neumann type
coupled boundary conditions. Unknown values of the magnetic field on the
boundary and for the exterior region are obtained by using BEM, and the unknown
velocity and magnetic field inside the pipe are obtained by using SUPG type
stabilized FEM. Computations are carried for very high values of magnetic
Reynolds numbers *Rm*_{1}, Reynolds number *Re* and magnetic pressure *Rh* of the fluid. The results show that using stabilized method enables us to get stable and accurate numerical
approximations consistent with the physical configuration of the problem over
rough mesh which also results a cheap computational cost.

Cite this paper

S. Han Aydin, "FEM-BEM Coupling for the MHD Pipe Flow in an Exterior Region,"*Open Journal of Fluid Dynamics*, Vol. 3 No. 3, 2013, pp. 184-190. doi: 10.4236/ojfd.2013.33023.

S. Han Aydin, "FEM-BEM Coupling for the MHD Pipe Flow in an Exterior Region,"

References

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[2] J. A. Shercliff, “Steady Motion of Conducting Fluid in a Pipe under Transverse Magnetic Fields,” Journal of Fluid Mechanics, Vol. 1, No. 6, 1956, pp. 644-666. doi:10.1017/S0022112056000421

[3] A. Carabineanu, A. Dinu and I. Oprea, “The Application of the Boundary Element Method to the Magnetohydrodynamic Duct Flow,” Zeitschrift für Angewandte Mathe matik und Physik, Vol. 46, No. 6, 1995, pp. 971-981. doi:10.1007/BF00917881

[4] T. W. H. Sheu and R. K. Lin, “Development of a Convection-Diffusion-Reaction Magnetohydrodynamic Solver on Nonstaggared Grids,” International Journal for Numerical Methods in Fluids, Vol. 45, No. 11, 2004, pp. 1209- 1233. doi:10.1002/fld.738

[5] K. E. Barrett, “Duct Flow with a Transverse Magnetic Field at High Hartmann Numbers,” International Journal for Numerical Methods in Engineering, Vol. 50, 2001, pp. 1893-1906. doi:10.1002/nme.101

[6] H. W. Liu and S. P. Zhu, “The Dual Reciprocity Boundary Element Method for Magnetohydrodynamic Channel Flows,” ANZIAM Journal, Vol. 44, 2002, pp. 305-322. doi:10.1017/S1446181100013961

[7] M. Tezer-Sezgin and S. H. Aydn, “Dual Reciprocity Boundary Element Method for Magnetohydrodynamic Flow Using Radial Basis Functions,” International Journal of Computational Fluid Dynamics, Vol. 16, No. 1, 2002, pp. 49-63. doi:10.1080/10618560290004026

[8] M. Tezer-Sezgin, “Boundary Element Method Solution of MHD Flow in Rectangular Duct,” International Journal for Numerical Methods in Fluids, Vol. 18, No. 10, 1994, pp. 937-952. doi:10.1002/fld.1650181004

[9] M. Dehghan and D. Mirzai, “Meshless Local Boundary Integral Equation (LBIE) Method for the Unsteady Magnetohydrodynamic (MHD) Flow in Rectangular and Circular Pipes,” Computer Physics Communications, Vol. 180, No. 9, 2009, pp. 1458-1466. doi:10.1016/j.cpc.2009.03.007

[10] F. Paris and J. Canas, “Boundary Element Method, Fundamentals and Applications,” Oxford University Press Inc., New York, 1997, pp. 64-78.

[11] A. J. Meir, “Finite Element Analysis of Magnetohydrodynamic Pipe Flow,” Applied Mathematics and Computation, Vol. 57, No. 2-3, 1993, pp. 177-196. doi:10.1016/0096-3003(93)90145-5

[12] M. Tezer-Sezgin and S. Koksal, “FEM for Solving MHD Flow in a Rectangular Duct,” International Journal for Numerical Methods in Engineering, Vol. 28, No. 2, 1989, pp. 445-459. doi:10.1002/nme.1620280213

[13] N. B. Salah, A. Soulaimani, W. G. Habash and M. Fortin, “A Conservative Stabilized Finite Element Method for the Magneto-Hydrodynamic Equations,” International Journal for Numerical Methods in Fluids, Vol. 29, No. 5, 1999, pp. 535-554. doi:10.1002/(SICI)1097-0363(19990315)29:5<535::AID-FLD799>3.0.CO;2-D

[14] N. B. Salah, A. Soulaimani and W. G. Habashi, “A Finite Element Method for Magnetohydrodynamics,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 43-44, 2001, pp. 5867-5892. doi:10.1016/S0045-7825(01)00196-7

[15] A. I. Nesliturk and M. Tezer-Sezgin, “The Finite Element Method for MHD Flow at High Hartmann Numbers”, Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 9-11, 2005, pp. 1201-1224. doi:10.1016/j.cma.2004.06.035

[16] J. F. Gerbeau, “A Stabilized Finite Element Method for the Incompressible Magnetohydrodynamic Equations,” Numerische Mathematik, Vol. 87, No. 1, 2000, pp. 83-111. doi:10.1007/s002110000193

[17] R. Codina and N. H. Silva, “Stabilized Finite Element Approximation of the Stationary Magnetohydrodynamics Equations,” Computational Mechanics, Vol. 38, No. 4-5, 2006, pp. 344-355. doi:10.1007/s00466-006-0037-x

[18] S. H. Aydin, A. I. Nesliturk and M. Tezer-Sezgin, “Two-Level Finite Element Method with a Stabilizing Subgrid for the Incompressible MHD Equations,” International Journal for Numerical Methods in Fluids, Vol. 62, No. 2, 2010, pp. 188-210.

[19] A. N. Brook and T. J. R. Hughes, “Streamline Upwind/ Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations,” Computer Methods in Applied Mechanics and Engineering, Vol. 32, No. 1-3, 1982, pp. 199-259. doi:10.1016/0045-7825(82)90071-8?

[20] A. Carabineanu and E. Lungu, “Pseudospectral Method for MHD Pipe Flow,” International Journal for Numerical Methods in Engineering, Vol. 68, No. 2, 2006, pp. 173-191. doi:10.1002/nme.1706

[21] M. Tezer-Sezgin and S. H. Aydn, “BEM Solution of MHD Flow in a Pipe Coupled with Magnetic Induction of Exterior Region,” Computing, Vol. 95, No. 1, 2013, pp. 751-770. doi:10.1007/s00607-012-0270-4?

[22] J. N. Reddy, “An Introduction to the Finite Element Method,” McGraw-Hill, New York, 1993.

[1] L. Dragos, “Magnetofluid Dynamics,” Abacus Press, Preston, 1975, pp. 92-99.

[2] J. A. Shercliff, “Steady Motion of Conducting Fluid in a Pipe under Transverse Magnetic Fields,” Journal of Fluid Mechanics, Vol. 1, No. 6, 1956, pp. 644-666. doi:10.1017/S0022112056000421

[3] A. Carabineanu, A. Dinu and I. Oprea, “The Application of the Boundary Element Method to the Magnetohydrodynamic Duct Flow,” Zeitschrift für Angewandte Mathe matik und Physik, Vol. 46, No. 6, 1995, pp. 971-981. doi:10.1007/BF00917881

[4] T. W. H. Sheu and R. K. Lin, “Development of a Convection-Diffusion-Reaction Magnetohydrodynamic Solver on Nonstaggared Grids,” International Journal for Numerical Methods in Fluids, Vol. 45, No. 11, 2004, pp. 1209- 1233. doi:10.1002/fld.738

[5] K. E. Barrett, “Duct Flow with a Transverse Magnetic Field at High Hartmann Numbers,” International Journal for Numerical Methods in Engineering, Vol. 50, 2001, pp. 1893-1906. doi:10.1002/nme.101

[6] H. W. Liu and S. P. Zhu, “The Dual Reciprocity Boundary Element Method for Magnetohydrodynamic Channel Flows,” ANZIAM Journal, Vol. 44, 2002, pp. 305-322. doi:10.1017/S1446181100013961

[7] M. Tezer-Sezgin and S. H. Aydn, “Dual Reciprocity Boundary Element Method for Magnetohydrodynamic Flow Using Radial Basis Functions,” International Journal of Computational Fluid Dynamics, Vol. 16, No. 1, 2002, pp. 49-63. doi:10.1080/10618560290004026

[8] M. Tezer-Sezgin, “Boundary Element Method Solution of MHD Flow in Rectangular Duct,” International Journal for Numerical Methods in Fluids, Vol. 18, No. 10, 1994, pp. 937-952. doi:10.1002/fld.1650181004

[9] M. Dehghan and D. Mirzai, “Meshless Local Boundary Integral Equation (LBIE) Method for the Unsteady Magnetohydrodynamic (MHD) Flow in Rectangular and Circular Pipes,” Computer Physics Communications, Vol. 180, No. 9, 2009, pp. 1458-1466. doi:10.1016/j.cpc.2009.03.007

[10] F. Paris and J. Canas, “Boundary Element Method, Fundamentals and Applications,” Oxford University Press Inc., New York, 1997, pp. 64-78.

[11] A. J. Meir, “Finite Element Analysis of Magnetohydrodynamic Pipe Flow,” Applied Mathematics and Computation, Vol. 57, No. 2-3, 1993, pp. 177-196. doi:10.1016/0096-3003(93)90145-5

[12] M. Tezer-Sezgin and S. Koksal, “FEM for Solving MHD Flow in a Rectangular Duct,” International Journal for Numerical Methods in Engineering, Vol. 28, No. 2, 1989, pp. 445-459. doi:10.1002/nme.1620280213

[13] N. B. Salah, A. Soulaimani, W. G. Habash and M. Fortin, “A Conservative Stabilized Finite Element Method for the Magneto-Hydrodynamic Equations,” International Journal for Numerical Methods in Fluids, Vol. 29, No. 5, 1999, pp. 535-554. doi:10.1002/(SICI)1097-0363(19990315)29:5<535::AID-FLD799>3.0.CO;2-D

[14] N. B. Salah, A. Soulaimani and W. G. Habashi, “A Finite Element Method for Magnetohydrodynamics,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 43-44, 2001, pp. 5867-5892. doi:10.1016/S0045-7825(01)00196-7

[15] A. I. Nesliturk and M. Tezer-Sezgin, “The Finite Element Method for MHD Flow at High Hartmann Numbers”, Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 9-11, 2005, pp. 1201-1224. doi:10.1016/j.cma.2004.06.035

[16] J. F. Gerbeau, “A Stabilized Finite Element Method for the Incompressible Magnetohydrodynamic Equations,” Numerische Mathematik, Vol. 87, No. 1, 2000, pp. 83-111. doi:10.1007/s002110000193

[17] R. Codina and N. H. Silva, “Stabilized Finite Element Approximation of the Stationary Magnetohydrodynamics Equations,” Computational Mechanics, Vol. 38, No. 4-5, 2006, pp. 344-355. doi:10.1007/s00466-006-0037-x

[18] S. H. Aydin, A. I. Nesliturk and M. Tezer-Sezgin, “Two-Level Finite Element Method with a Stabilizing Subgrid for the Incompressible MHD Equations,” International Journal for Numerical Methods in Fluids, Vol. 62, No. 2, 2010, pp. 188-210.

[19] A. N. Brook and T. J. R. Hughes, “Streamline Upwind/ Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations,” Computer Methods in Applied Mechanics and Engineering, Vol. 32, No. 1-3, 1982, pp. 199-259. doi:10.1016/0045-7825(82)90071-8?

[20] A. Carabineanu and E. Lungu, “Pseudospectral Method for MHD Pipe Flow,” International Journal for Numerical Methods in Engineering, Vol. 68, No. 2, 2006, pp. 173-191. doi:10.1002/nme.1706

[21] M. Tezer-Sezgin and S. H. Aydn, “BEM Solution of MHD Flow in a Pipe Coupled with Magnetic Induction of Exterior Region,” Computing, Vol. 95, No. 1, 2013, pp. 751-770. doi:10.1007/s00607-012-0270-4?

[22] J. N. Reddy, “An Introduction to the Finite Element Method,” McGraw-Hill, New York, 1993.