Stabilized FEM solution of MHD flow over array of cubic domains

Year 2023, Volume: 72 Issue: 3, 839 - 856, 30.09.2023

Selçuk Han Aydın

https://doi.org/10.31801/cfsuasmas.1202192

Abstract

In this study, 3D magnetohydrodynamic (MHD) equations are considered in array of cubic domains having insulated external boundaries separated by conducting thin walls. In order to obtain stable solutions, stabilized version of the Galerkin finite element method is used as a numerical scheme. Different problem parameters and configurations are tested in order to visualize the accuracy and efficiency of the proposed algorithm. Obtained solutions are visualized as contour lines of 2D slices taken from the obtained 3D domain solutions.

Keywords

3D MHD Flow, stabilized FEM, array of cubic domains

References

• Hartmann, J., Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 15(6) (1937), 1–28.
• Shercliff, J.A., Steady motion of conducting fluid in a pipes under transverse magnetic fields, J. Fluid Mech., 1(6) (1956), 644–666. https://doi.org/10.1017/S0022112056000421
• Drago¸s, L., Magnetofluid Dynamics, Abacus Pres, 1975.
• Davidson, P.A., An Introduction to Magnetohydrodynamic, Cambridge Texts in Applied Mathematics, Vol. 1, Cambridge University Press, 2001. https://doi.org/10.1017/CBO9780511626333
• Carabineanu, A., Dinu, A., Oprea, I., The application of the boundary element method to the magnetohydrodynamic duct flow, The Journal of Applied Mathematics and Physics (ZAMP), 46 (1995), 971–981. https://doi.org/10.1007/BF00917881
• Meir, A.J., Finite element analysis of magnetohydrodynamic pipe flow, Applied Mathematics and Computation, 57 (1993), 177–196. https://doi.org/10.1016/0096-3003(93)90145-5
• Sheu, T.W.H., Lin, R.K., Development of a ranvection-diflusion-reaction magnetohydrodynamic solver on nonstaggared grids, International Journal for Numerical Methods in Fluids, 45 (2004), 1209–1233. https://doi.org/10.1002/fld.738
• Singh, B., Lal, J., Finite element method of MHD channel flow with arbitrary wall conductivity, Journal of Mathematical and Physical Sciences, 18 (1984), 501–516.
• Tezer-Sezgin, M., Han Aydin, S., Dual reciprocity boundary element method for magnetohydrodynamic flow using radial basis functions, International Journal of Computational Fluid Dynamics, 16(1) (2002), 49–63. https://doi.org/10.1080/10618560290004026
• Tezer-Sezgin, M., Bozkaya, C., Boundary-element method solution of magnetohydrodynamic flow in a rectangular duct with conducting walls parallel to applied magnetic field, Computational Mechanics, 41 (2008), 769–775. https://doi.org/10.1007/s00466-006-0139-5
• Tezer-Sezgin, M., Han Aydin, S., BEM solution of MHD flow in a pipe coupled with magnetic induction of exterior region, Computing, 95(1) (2013), 751–770. https://doi.org/10.1007/s00607-012-0270-4
• Carabineanu, A., Lungu, E., Pseudospectral method for MHD pipe flow, Int. J. Numer. Methods Eng., 68(2) (2006), 173–191. https://doi.org/10.1002/nme.1706
• Han Aydın, S., Tezer- Sezgin, M., DRBEM solution of MHD pipe flow in a conducting medium, J. Comput. Appl. Math., 259(B) (2014), 720–729. https://doi.org/10.1016/j.cam.2013.05.010
• Tezer-Sezgin, M., Han Aydın, S., FEM Solution of MHD Flow Equations Coupled on a Pipe Wall in a Conducting Medium, PAMIR, 2014.
• Cai, X., Qiang, H., Dong, S., Lu, J., Wang, D., Numerical simulations on the fully developedliquid-metal MHD flow at high Hartmann numbers in the rectangular duct, Advances in Intelligent Systems Research, 143 (2018), 68–71. https://doi.org/10.2991/ammsa-18.2018.14
• Dehghan, M., Mirzai, D., Meshless local boundary integral equation (LBIE) method for theunsteady magnetohydrodynamic(MHD) flow in rectangular and circular pipes, Computer Physics Communications, 180 (2009), 1458–66. https://doi.org/10.1016/j.cpc.2009.03.007
• Loukopoulos, V.C., Bourantas, G.C., Skouras, E.D., Nikiforidis, G.C., Localized meshless point collocation method for time-dependent magnetohydrodynamic flow through pipes under a variety of wall conductivity conditions, Computational Mechanics, 47(2) (2011), 137–159. https://doi.org/10.1007/s00466-010-0535-8
• Salah, N.B., Soulaimani, A., Habashi, W.G., A finite element method for magnetohydrodynamics, Comput. Methods Appl. Mech. Engrg., 190 (2001) 5867–5892. https://doi.org/10.1016/S0045-7825(01)00196-7
• Dong, X., He, Y., Two-level Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics, J. Sci. Comput., 63 (2015), 426–451. https://doi.org/10.1007/s10915-014-9900-7
• Wang, L., Li, J. Huang, P., An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics based on the finite element method, International Communications in Heat and Mass Transfer, 98 (2018), 183–190. https://doi.org/10.1016/j.icheatmasstransfer.2018.02.019
• Xu, J., Feng, X., Su, H., Two-level Newton iterative method based on nonconforming finiteelement discretization for 2D/3D stationary MHD equations, Computers and Fluids, 238 (2022), 105372. https://doi.org/10.1016/j.compfluid.2022.105372
• Dong, X., He, Y., Zhang, Y., Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics, Comput. Methods Appl. Mech. Engrg., 276 (2014), 287–311. https://doi.org/10.1016/j.cma.2014.03.022
• Xu, J., Su, H., Li, Z., Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations, Numerical Algorithms. https://doi.org/10.1007/s11075-021-01224-4 (2021)
• Li, L., Zheng, W., A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D, Journal of Computational Physics, 351 (2017), 254–270. https://doi.org/10.1016/j.jcp.2017.09.025
• Zhang, G.D., He, X., Yang, X., A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations, Journal of Computational Physics, 448 (2022), 110752. https://doi.org/10.1016/j.jcp.2021.110752
• Skala, J., Baruffa, F., Buechner, J., Rampp, M., The 3D MHD Code GOEMHD3 for large-Reynolds-number astrophysical plasmas, Astron. Astrophys., 580 (2015), A48. https://doi.org/10.1051/0004-6361/201425274
• Sutevski, D., Smolentsev, S., Morley, N., Abdou, M., 3D numerical study of MHD flow in a rectangular duct with a flow channel insert, Fusion Science and Technology, 60(2) (2011), 513-517. https://doi.org/10.13182/FST11-A12433
• Huba, J.D., Lyon, J.G., A new 3D MHD algorithm: the distribution function method, J. Plasma Physics., 61(3) (1999), 391–405. https://doi.org/10.1017/S0022377899007503
• Barnes, D.C., Rousculp, C.L., Accurate, finite-volume methods for 3D MHD on unstructured Lagrangian meshes, Nuclear explosives code developers conference (NECDC), Las Vegas, NV (United States), October, 1998.
• Wu, J., Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395–413. https://doi.org/10.1007/s00332-002-0486-0
• Ni, L., Guo, Z., Zhou, Y., Some new regularity criteria for the 3D MHD equations, J. Math. Anal. Appl., 396 (2012), 108–118. https://doi.org/10.1016/j.jmaa.2012.05.076
• Zhang, Z., Ouyang, X., Zhong, D., Qiu, S., Remarks on the regularity criteria for the MHD equations in the multiplier spaces, Boundary Value Problems, (2013), 270. https://doi.org/10.1186/1687-2770-2013-270
• Jia, X., Zhou, Y., Regularity criteria for the 3D MHD equations involving partial components, Nonlinear Analysis, Real World Applications, 13 (2012), 410–418. https://doi.org/10.1016/j.nonrwa.2011.07.055
• Yea, Z., Zhang, Z., A remark on regularity criterion for the 3D Hall-MHD equations based on the vorticity, Applied Mathematics and Computation., 301 (2017), 70–77. https://doi.org/10.1016/j.amc.2016.12.011
• Caoa, C., Wu, J., Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263–2274. https://doi.org/10.1016/j.jde.2009.09.020
• Tassone, A., Gramiccia, L., Caruso, G., Three-dimensional MHD flow and heat transfer in a channel with internal obstacle, International Journal of Heat and Technology, 36(4) (2018), 1367-1377. https://doi.org/10.18280/ijht.360428
• Ud-Doula, A., Sundqvist, J., Owocki, S.P., Petit, V., Townsend, RHD First 3D MHD simulation of a massive-star magnetosphere with application to H alpha emission from theta(1) Ori C, Monthly Notices of the Royal Astronomical Society, 428(3) (2013), 2723-2730. https://doi.org/10.1093/mnras/sts246
• Fernandez-Dalgo, P.G., Jarrin, O., Weak suitable solutions for 3D MHD equations for intermittent initial data, hal-02490130 (2020).
• Liu, F., Wang, Y.Z., Global solutions to three-dimensional generalized MHD equations with large initial data, Z. Angew. Math. Phys., 70(69) (2019). https://doi.org/10.1007/s00033-019-1113-3
• Bluck, M.J., Wolfandale, M.J., An analytical solution to electromagnetically coupled duct flow in MHD, Journal of Fluid Mechanics, 771 (2015), 595–623. https://doi.org/10.1017/jfm.2015.202
• Hunt, J.C.R., Stewartson, K., Magnetohydrodynamics flow in rectangular ducts. II., Journal of Fluid Mechanics, 23(3) (1965), 563–581. https://doi.org/10.1017/S0022112065001544
• Tezer-Sezgin, M., Aydin, S.H., FEM solution of MHD flow in an array of electromagnetically coupled rectangular ducts, Progress in Computational Fluid Dynamics, An International Journal., 20 (2020), 40–50. https://doi.org/10.1504/PCFD.2020.104706
• Aydin, S.H., 3-D MHD flow over array of cubic ducts, International Conference on Applied Mathematics in Engineering (ICAME 21), September 1-3, (2021), Balikesir, Turkey.
• Brooks, A.N., Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32 (1982), 199–2592. https://doi.org/10.1016/0045-7825(82)90071-8
• Salah, N.B., Soulaimani, A., Habashi, W.G., Fortin, M., A conservative stabilized finite element method for the magnet-hydrodynamic equations, Internation Journal for Numerical Methods in Fluids, 29 (1999), 535–554. https://doi.org/10.1002/(SICI)1097-0363(19990315)29:5<535::AID-FLD799>3.0.CO;2-D
• Shadid, J.N., Powlowski, R.P., Cyr, E.C., Tuminaro, R.S., Chacon, L., Weber, P.D., Scalable implicit incompressible resistive MHD with stabilized FE and fullycoupled Newton-Krylov-AMG, Comput. Methods Appl. Mech. Engrg., 304 (2016), 1–25. https://doi.org/10.1016/j.cma.2016.01.019
• Gerbeau, J.F., A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numerische Mathematik, 87 (2000), 83–111. https://doi.org/10.1007/s002110000193
• Nesliturk, A.I., Tezer-Sezgin, M., The finite element method for MHD flow at high Hartmann numbers, Comput. Methods Appl. Mech. Engrg., 194 (2005), 1201–1224. https://doi.org/10.1016/j.cma.2004.06.035
• Nesliturk, A.I., Tezer-Sezgin, M., Finite element method solution of electrically driven magnetohydrodynamic flow, Journal of Computational and Applied Mathematics, 192 (2006), 339–352. https://doi.org/10.1016/j.cam.2005.05.015
• Codina, R., Silva, N.H., Stabilized finite element approximation of the stationary magneto-hydrodynamics equations, Computational Mechanics, 38 (2006), 344–355. https://doi.org/10.1007/s00466-006-0037-x
• Aydin, S.H., Nesliturk, A.I., Tezer-Sezgin, M., Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations, International Journal for Numerical Methods in Fluids, 62(2) (2010), 188–210. https://doi.org/10.1002/fld.2019
• Marchandise, E., Remacle, J.F., A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows, Journal of Computational Physics, 219 (2006), 780–800. https://doi.org/10.1016/j.jcp.2006.04.015
• Nesliturk, A.I., Aydin, S.H., Tezer-Sezgin, M., Two-level finite element method with a stabilizing subgrid for the incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids, 58 (2007), 551–572. https://doi.org/10.1002/fld.1753
• Hachem, E., Rivaux, B., Kloczko, T., Digonnet, H., Coupez, T., Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, 229 (2010), 8643–8665. https://doi.org/10.1016/j.jcp.2010.07.030
• Wang, A., Zhao, X., Qin, P., Xie, D., An oseen two-level stabilized mixed finite-element method for the 2D/3D stationary Navier-Stokes equations, Abstract and Applied Analysis, 2012 (2012), 1–12. https://doi.org/10.1155/2012/520818
• Reddy, J.N., An Introduction to the Finite Element Method, 2nd ed., McGraw-Hill, New York, 1993.
• Muller, U., Buhler, L., Magnetofluiddynamics in Channels and Containers, Springer, 2001.