Upwinding meshfree point collocation method for steady MHD flow with arbitrary orientation of applied magnetic field at high Hartmann numbers

Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers

“…When γ = π 2 , this problem is the Shercliff problem [47]. As a test example used efficiently in the literature [1,2,4,7,[17][18][19]21], this problem is also numerically solved by using the present EHOC scheme. Numerical experiments are performed for M = 10 2 and M = 10 4 at γ = π 4 , γ = π 3 , π 6 and π 2 on a uniform and non-uniform 21 × 21, 41 × 41 and 81 × 81 mesh size, respectively.…”

“…This is the so-called Shercliff problem [47], which is a frequently used test problem in the literature [1,2,4,7,[17][18][19]21]. Its exact solution is given by Shercliff [47]:…”

“…It is well known that the difficulty of the numerical solution of the governing equations for the MHD problems with high Ha, which is a measure of the magnetic field strength for a given fluid in a duct of a given scale, is similar to that of the solution of the convection-diffusion equation with convection-dominated [1,7,17,21]. The thickness of the Hartmann layers on all walls which perpendicular to the magnetic field scale with O(Ha −1 ) is very thin, and the secondary (side) layers on all walls parallel to the magnetic field scale with O(Ha − 1 2 ) which are much thicker than the Hartmann layers at high Ha [28].…”

“…The MHD flow problem may obtain analytical solution in some special conditions, but it can only be solved numerically in the general situation . In the past several decades, various numerical approximation methods, involving finite element method (FEM) [14][15][16][17][18][19][20][21][22][23], finite difference method (FDM) [7][8][9][10], boundary element method (BEM) [4][5][6], differential quadrature method (DQM) [24], meshfree (meshless) method (MM) [1][2][3], finite volume spectral element method (FVSEM) [25], have been suggested to solve the coupled equations in velocity and magnetic field for the steady magnetohydrodynamic (MHD) duct flow. However, most of them cannot solve the MHD problems with high Ha effectively.…”

An exponential compact difference scheme for solving 2D steady magnetohydrodynamic (MHD) duct flow problems

“…The MHD flow problem, only in some special conditions, has analytical solution, the general situation can only be solved numerically. On the other hand, it is well known that the difficulty of the solution of the governing equations for the MHD problems at high Hartmann numbers is similar to that of the solution of the convection-diffusion equation with convection-dominated [1,4,13,19], which leads to some special numerical difficulty. Hence, developing stable, accurate and effective numerical methods suitable for the MHD flow problems at high Hartmann numbers has become a hot research topic [3,4,13,14,19].…”