Effects of wall perturbations on the stabilized FEM solution of steady MHD flow in a duct

Year 2024, Volume: 4 Issue: 5-Special Issue: ICAME'24, 45 - 63, 31.12.2024

Münevver Tezer-sezgin , Selçuk Han Aydın

https://doi.org/10.53391/mmnsa.1524369

Abstract

In this study, the effects of curved boundary perturbations on the solution of steady magnetohydrodynamic (MHD) duct flow are investigated. Hartmann (upper and bottom) walls are perturbly curved and perfectly conducting while the side walls are insulated and plane. The velocity of the flow and induced magnetic field are obtained numerically by solving the steady MHD flow coupled equations using the finite element method (FEM with Streamline Upwind Petrov Galerkin (SUPG)) stabilization to inhibit instabilities in the flow. The results are obtained for Hartmann number ($Ha$) values up to $500$, for several definitions of the curved upper and bottom walls, and for several values of perturbation parameters of the curved walls ($0 \le \epsilon_u , \epsilon_b \le 0.3$). The velocity and the induced magnetic field sensitivity to the curved wall shapes are visualized in terms of equivelocity and current lines. It is found that the flow and the induced magnetic field are affected by the curved boundary shapes especially near those boundaries and also, to some extent, in the whole duct. It is also observed that increasing the Hartman number pushes the flow near the upper boundary even if both upper and bottom walls are perturbed since the external magnetic field applies vertically. The increase in the perturbation parameter of the curved upper boundary forces the flow to move through this wall and the induced magnetic field reaches its maximum value near the maximum points of the perturbed curve. Further, an increase in ${\rm Ha}$ delays the effect of the curved boundaries and gives rise to flattened flow with side layers and stagnant fluid at the central part of the duct overwhelming the effects of curved boundaries.

Keywords

Steady MHD duct flow, stabilized FEM, perturbed boundary

References

• [1] 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), 1-28, (1937).
• [2] Shercliff, J.A. Steady motion of conducting fluids in pipes under transverse magnetic fields. In Proceedings, Mathematical Proceedings of the Cambridge Philosophical Society pp. 136-144. Cambridge University Press, (1953, January).
• [3] Singh, B. and Lal, J. MHD axial flow in a triangular pipe under transverse magnetic field. Indian Journal of Pure and Applied Mathematics, 18, 101-115, (1978).
• [4] Singh B. and Jia L. MHD axial flow in a triangular pipe under transverse magnetic field parallel to a side of the triangle. Indian Journal of Pure and Applied Mathematics, 9(2), 101-115, (1978).
• [5] Sheu, T.W.H. and Lin, R.K. Development of a convection–diffusion-reaction magnetohydrodynamic solver on non-staggered grids. International Journal for Numerical Methods in Fluids, 45(11), 1209-1233, (2004).
• [6] Singh, B. and Lal, J. FEM in MHD channel flow problems. International Journal for Numerical Methods in Engineering, 18, 1104-1111, (1982).
• [7] Singh, B. and Lal, J. Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity. International Journal for Numerical Methods in Fluids, 4(3), 291-302, (1984).
• [8] Tezer-Sezgin, M. and Koksal, S. Finite element method for solving MHD flow in a rectangular duct. International Journal for Numerical Methods in Engineering, 28(2), 445-459, (1989).
• [9] Tezer-Sezgin, M. Boundary element method solution of MHD flow in a rectangular duct. International Journal for Numerical Methods in Fluids, 18(10), 937-952, (1994).
• [10] Tezer-Sezgin, M. and Bozkaya, C. Boundary element method solution of magnetohydrodynamic flow in a rectangular duct with conducting walls parallel to applied magnetic field. Computational Mechanics, 41, 769-775, (2008).
• [11] Tezer-Sezgin, M. and Han Aydin, S. Dual reciprocity boundary element method for magnetohydrodynamic flow using radial basis functions. International Journal of Computational Fluid Dynamics, 16(1), 49-63, (2002).
• [12] Tezer-Sezgin, M. Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method. Computers & Fluids, 33(4), 533-547, (2004).
• [13] Tezer-Sezgin, M. and Bozkaya, C. Boundary Element Method for Magnetohydrodynamic Flow: 2D MHD Duct Flow Problems, (Vol. 14). Springer Nature: Switzerland, (2024).
• [14] Nesliturk, A.I. and Tezer-Sezgin, M. The finite element method for MHD flow at high Hartmann numbers. Computer Methods in Applied Mechanics and Engineering, 194(9-11), 1201-1224, (2005).
• [15] Nesliturk, A.I. and Tezer-Sezgin, M. Finite element method solution of electrically driven magnetohydrodynamic flow. Journal of Computational and Applied Mathematics, 192(2), 339-352, (2006).
• [16] Mahabaleshwar, U.S. Pazanin, I. Radulovic, M. and Suárez Grau. F.J. Effects of small boundary perturbation on the MHD duct flow. Theoretical and Applied Mechanics, 44(1), 83-101, (2017).
• [17] Aydin, C. and Tezer-Sezgin, M. DRBEM solution of the Cauchy MHD duct flow with a slipping perturbed boundary. Engineering Analysis with Boundary Elements, 93, 94-104, (2018).
• [18] Fendoglu, H., Bozkaya, C. and Tezer-Sezgin, M. MHD flow in a rectangular duct with a perturbed boundary. Computers & Mathematics with Applications, 77(2), 374-388, (2019).
• [19] Yang, L., Mao, J. and Xiong, B. Numerical simulation of liquid metal MHD flows in a conducting rectangular duct with triangular strips. Fusion Engineering and Design, 163, 112152, (2021).
• [20] Okechi, N.F., Asghari S. and Charreh, D. Magnetohydrodynamic flow through a wavy curved channel. AIP Advances, 10(3), 035114, (2020).
• [21] Marušic–Paloka, E., Pazanin, I. and Radulovic, M. MHD flow through a perturbed channel filled with a porous medium. Bulletin of the Malaysian Mathematical Sciences Society, 45, 2441- 2471, (2022).
• [22] Prasanna Jeyanthi, M. and Ganesh, S. Numerical solution of magnetohydrodynamic flow through duct with perturbated boundary using RBF-FD method. International Journal of Ambient Energy, 45(1), 2276130, (2024).
• [23] Tezer-Sezgin, M. and Han Aydin, S. The Stabilized FEM Solution of the MHD Flow in a Rectangular Duct with Perturbed Boundary. CMES 2018 (presentation), 3rd International Conference on Computational Mathematics and Engineering Sciences, 04-06 May 2018, Girne, Cyprus.
• [24] Tezer-Sezgin, M. and Han Aydın, S. Steady MHD Flow in a Duct with Non-Rectangular Cross-Section Using the Stabilized FEM Solution. ICAME 24, The 3rd International Conference on Applied Mathematics in Engineering, 26-28 June 2024, Balikesir, Türkiye.
• [25] Brooks, A.N. and Hughes. T.J.R. 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, 32(1-3), 199-259, (1982).
• [26] Müller, U. and Bühler, L. Magnetofluiddynamics in Channels and Containers. Springer-Verlag: Berlin, (2001).
• [27] Moreau, R.J. Magnetohydrodynamics, Fluid Mechanics and its applications, Kluwer Academic Publisher, (1990).
• [28] Branover, G.G and Tsinober, A. B. Magnetohydrodynamic of incompressible media, Moscow: Nauka, (1970).