Comparison study on the numerical stability of dual reciprocity boundary element method for the MHD slip flow problem

Abstract

In this paper, unsteady Magnetohydrodynamics (MHD) flow in a cavity with insulated and slipping walls is studied. The nonlinear coupled MHD equations containing the velocity and the induced magnetic field are solved by the dual reciprocity boundary element method (DRBEM). The time derivative is discretized by the explicit Euler or the central difference methods. The grid independence is tested. The numerical stability is analyzed through the spectral radius of the coefficient matrix and the performances of the two time discretization methods are compared. In the studied ranges, the proper choices of relaxation parameter and time increment are found. The influences of Hartmann number (Ha), Reynolds number (Re), magnetic Reynolds number (Rm), and the slip length on the numerical stability are investigated. The flow behaviors in the steady-state and transient time levels are presented. It is found that relaxation parameter and time step close to one guarantee the numerical stability in the studied range. Increasing Re alters the stability condition more than Rm increment. DRBEM with the central difference method is the appropriate numerical technique considering the computational cost and the stability for solving MHD slip flow problems. The flow elongation time is postponed for large Rm and Re.