Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

Abstract

In this article a numerical solution of the time dependent, coupled system equations of magnetohydrodynamics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approximation of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collocation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform magnetic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for several Hartmann numbers (up to 10 5 ) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form meshless methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local PetrovGalerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L 2 and L ∞ , the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems.

A DRBEM solution for MHD pipe flow in a conducting medium

“…Among them we can count the works by Sheu and Lin [9] with the finite difference method (FDM), Singh and Lal [10], Barrett [11] and Neslitürk and Tezer-Sezgin [12] with the FEM, and Liu and Zhu [13], Tezer-Sezgin and Aydin [14], and Tezer-Sezgin and Bozkaya [15] with the boundary element method (BEM). Some meshless methods have also been used for solving MHD flow equations in channels for arbitrary cross-section and arbitrary wall conductivities in [16,17]. A numerical solution based on the pseudospectral (PS) collocation method was proposed in [18], taking a conducting pipe wall which also has magnetic induction generated by the effect of the external magnetic field.…”

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

“…The boundary element method has been used extensively for solving MHD flow in ducts by Liu and Zhu, Hosseinzadeh et al, Tezer‐Sezgin and Bozkaya, and Tezer‐Sezgin and Han Aydın . Finite difference solution of MHD flow was given by Sheu and Lin, and some meshless method solutions are due to Dehghan and Mirzai, Loukopoulos et al, and Cai et al in channels for arbitrary cross section and arbitrary wall conductivities. Dehghan and Mohammadi and Dehghan and Salehi carried meshless methods based on radial basis functions as multiquadrics, inverse quadrics and Wendland function, and meshfree collocation method in strong form for essential boundary and meshless local Petrov‐Galerkin method in weak form for natural boundary, respectively.…”

BEM solution to magnetohydrodynamic flow in a semi‐infinite duct

“…The flow is driven by the current produced by a pressure gradient. Loukopoulos [18] and Bourantas et al [19] solved MHD duct flow with various wall conductivities and cross sections using localized meshless point collocation method and domain-type meshless method, respectively. The wall that connects the two semi-infinite walls is partly non-conducting and partly conducting (in the middle).…”

“…When we find V , B, @V =@n, and @B=@n everywhere on the boundary , one can obtain the values of V and B at any point (sufficiently distant) on the semi-infinite duct on the upper half plane by taking c A D 1 in Equation (18). After the insertion of boundary conditions (4), the resulting linear system of equations is solved for the unknowns B and @V =@n on the conducting boundary and @B=@n and @V =@n on the nonconducting boundary.…”

“…Among these, we can count the work by Dehghan and Mirzai [17] as meshless local Petrov-Galerkin and meshless local boundary integral equation methods, respectively, for solving unsteady MHD flow through pipes. Loukopoulos [18] and Bourantas et al [19] solved MHD duct flow with various wall conductivities and cross sections using localized meshless point collocation method and domain-type meshless method, respectively. A finite volume spectral element method was used by Shakeri and Dehghan [20], and a local radial point interpolation method was proposed, in the work of Cai et al [21], for solving MHD flow through pipes of rectangular cross section.…”