A generalized alternating-direction implicit scheme for incompressible magnetohydrodynamic viscous flows at low magnetic Reynolds number

Abstract This paper presents numerical simulations of incompressible fluid flows in the presence of a magnetic field at low magnetic Reynolds number. The equations governing the flow are the Navier–Stokes equations of fluid motion coupled with Maxwell’s equations of electromagnetics. The study of fluid flows under the influence of a magnetic field and with no free electric charges or electric fields is known as magnetohydrodynamics. The magnetohydrodynamics approximation is considered for the formulation of the non-dimensional problem and for the characterization of similarity parameters. A finite-difference technique is used to discretize the equations. In particular, an extension of the generalized Peaceman and Rachford alternating-direction implicit (ADI) scheme for simulating two-dimensional fluid flows is presented. The discretized conservation equations are solved in stream function–vorticity formulation. We compare the ADI and generalized ADI schemes, and show that the latter is more efficient in simulating low Reynolds number and magnetic Reynolds number problems. Numerical results demonstrating the applicability of this technique are also presented. The simulation of incompressible magnetohydrodynamic fluid flows is illustrated by numerical solution for two-dimensional cases.

[1]  A. I. Nesliturk,et al.  Finite element method solution of electrically driven magnetohydrodynamic flow , 2006 .

[2]  F. White.  Viscous Fluid Flow , 1974 .

[3]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[4]  W. Habashi,et al.  A finite element method for magnetohydrodynamics , 2001 .

[5]  Rolf Rannacher,et al.  Numerical methods for the Navier-Stokes equations , 1994 .

[6]  R. Glowinski,et al.  Incompressible Computational Fluid Dynamics: On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow , 1993 .

[7]  L. Quartapelle,et al.  A review of vorticity conditions in the numerical solution of the ζ–ψ equations , 1999 .

[8]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[9]  A. Acrivos,et al.  Steady flows in rectangular cavities , 1967, Journal of Fluid Mechanics.

[10]  W. Dai.  A Generalized Peaceman–Rachford ADI Scheme for Solving Two-Dimensional Parabolic Differential Equations , 1997 .

[11]  V. G. Ferreira,et al.  A NUMERICAL SCHEME FOR SOLVING CREEPING FLOWS , 2003 .

[12]  J. C. Simo,et al.  Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations☆ , 1996 .

[13]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[14]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[15]  O. Burggraf.  Analytical and numerical studies of the structure of steady separated flows , 1966, Journal of Fluid Mechanics.