Local radial point interpolation method for the fully developed magnetohydrodynamic flow

doi:10.1016/j.amc.2010.11.004

Applied Mathematics and Computation

Volume 217, Issue 9, 1 January 2011, Pages 4529-4539

https://doi.org/10.1016/j.amc.2010.11.004 Get rights and content

Abstract

In this paper, a local radial point interpolation method (LRPIM) is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight duct of rectangular section with arbitrary wall conductivity and orientation of applied magnetic field. Local weak forms are developed using weighted residual method locally for the governing equations of fully developed MHD flow. The shape functions from LRPIM possess the delta function property. Therefore, essential boundary conditions can be applied as easily as that in the finite-element method. The implementation procedure of LRPIM method is node based, and it doesn’t need any “mesh” or “element”. Computations have been carried out for different Hartmann numbers, wall conductivities and orientations of applied magnetic field.

Introduction

The MHD flow has been applied in more broad fields than ever such as the cooling system with liquid metals for nuclear fission or fusion reactors, electromagnetic pumps, MHD generators. It is very important to devise effective numerical methods to obtain the approximate solutions of the magnetohydrodynamic (MHD) flow problem since the exact solution can be achieved only in some special cases [1].

The methods for solving the MHD problems are mostly based on mesh, such as the finite difference method (FDM) [2], [3] and the finite-element method (FEM) [4], [5], [6], [7], [8], [9]. The common characteristic for these methods is that the quality of mesh has a great impact on the computed results. However, sometimes the high-quality meshes are hard to be generated for the irregular shape of computed area and the procedure of generating mesh consumes much time and labor. Then the meshless methods, which don’t need any mesh production, become more and more attractive nowadays.

Many meshless methods have been proposed and employed in several fields in recent years, such as reproducing kernel particle method (RKPM) [10], the partition of unity finite-element method (PUFEM) [11], Hp-cloud method [12], natural element method (NEM) [13], Mesh-free radial basis function method [14], [15], the meshless local weak–strong (MLWS) method [16], local boundary integral equation (LBIE) method [17], [18]. Some of them have been applied in MHD problems. Smoothed particle hydrodynamics method (SPH) was first introduced in 1977 by Lucy [19], Gingold [20] in the context of astrophysical phenomena. SPH has been used in MHD Poiseuille flow by Jiang [21]. Børve applied SPH to analyze multidimensional MHD shock [22]. The meshless local Petrov–Galerkin (MLPG) method was first proposed by Atluri and Zhu [23], and later discussed deeply in Atluri and shen [24], [25]. Dehghan analyzed unsteady MHD flow through a duct with arbitrary wall conductivity with MLPG [26]. Belytschko et al. [27] proposed the element-free Galerkin (EFG) method and applied it to elasticity and heat conduction problems. Verardi applied the element-free Galerkin method [28] to study the fully developed MHD duct flows. Dehghan employed the LBIE method [18] to deal with the unsteady magnetohydrodynamic flow in rectangular and circular pipes.

The local radial point interpolation method (LRPIM) is proposed by Liu [29]. In LRPIM, the point interpolation with the radial basis functions is used to construct shape functions with the delta function property. The LRPIM has been applied for two-dimensional static analyses [29], free vibration analyses of two-dimensional solids [30], numerical simulation of two-dimensional sine–Gordon equation [31] and dissipation process of excess pore water pressure [32]. Very good results have been obtained.

LRPIM for two-dimensional fully developed MHD flow analyses are presented in this paper. Local weak forms are developed using weighted residual method locally for the MHD governing equations. The essential boundary conditions can be easily implemented in MHD analyses because the shape functions possess the delta function property. A number of numerical solutions of MHD analyses are presented and compared with the exact solutions [1] for the flow in an insulated duct. Furthermore, numerical solutions are also obtained for arbitrary wall conductivity and oblique applied magnetic field.

The organization of the rest of this paper is as follows: In Section 2 the basic equations are presented. In Section 3 a brief discussion of point interpolation using radial basis functions is outlined. In Section 4 a local weak form of LRPIM for the fully developed MHD flow is presented. In Section 5 the discretized equations of MHD flow are obtained. The numerical results are reported in Section 6. Finally some concluding remarks are given in Section 7.

Section snippets

Transverse magnetic field

The geometry of the duct is shown in Fig. 1, where the external applied magnetic field B₀ is along the x-axis, the y-axis is perpendicular to it and lying in a section, the z-axis is along the axial direction in which the flow is taking place for Newtonian fluid of constant physical properties driven by a constant pressure gradient.

The following non-dimensional variables and parameters are introduced as: $X = \frac{x}{a}, Y = \frac{y}{a}, B = - \frac{B_{z}}{(a^{2} / η) (\partial p / \partial z) μ_{0} \sqrt{η σ_{1}}}, V = - \frac{v_{z}}{(a^{2} / η) (\partial p / \partial z)}, M = B_{0} a \sqrt{\frac{σ_{1}}{η}}, θ = \frac{σ_{1} a}{σ_{2} h},$ where η, σ₁ are the

Point interpolation using radial basis functions

Consider a continuous function u(x) defined on a two-dimensional domain Ω with a set of suitably located nodes in it. An interpolation of u(x) in the neighborhood of a point x_q using radial basis functions (RBFs) and polynomial basis is written as: $u (x) = \sum_{i = 1}^{n} R_{i} (x) a_{i} + \sum_{j = 1}^{m} p_{j} (x) b_{j} = R^{T} (x) a + p^{T} (x) b,$ where R_i(x) is a radial basis function associated with node i, p_j(x) is a monomial in the space coordinates x^T = [x, y], n is the number of nodes in the neighborhood of x_q, m is the number of monomial basis

Local weak form of LRPIM for the fully developed MHD flow

A local weak form of Eqs. (1), (2) over a local sub-domain Ω_q can be obtained using the weighted residual method $\int_{Ω_{q}} (\nabla^{2} V + M \frac{\partial B}{\partial X}) w_{i} d Ω = - \int_{Ω_{q}} w_{i} d Ω,$ $\int_{Ω_{q}} (\nabla^{2} B + M \frac{\partial V}{\partial X}) w_{i} d Ω = 0,$ where w_i is the test function of node i.

The support sub-domain $Ω_{s}^{i}$ of a node x_i is a domain in which w(x_i) ≠ 0. These local sub-domains do not form a contiguous mesh globally; but these disjointed local sub-domains may overlap each other. An arbitrary shape of support domain can be used. A rectangular support domain is used in this paper for

Discretized equations

Consider N nodal points on the boundary and domain of the problem. Some of them are distributed across the global boundary Γ. The problem domain Ω is represented by properly scattered nodes. The point interpolation approximations (38), (39), (40), (41) are used to approximate the values of a point x_q.

Substituting Eqs. (38), (39), (40), (41) into the local weak form (34), (35) for all nodes and imposing natural boundary conditions lead to the following discrete system equations: $\sum_{j = 1}^{n} \int_{Ω_{q}} (\frac{\partial ϕ_{j}}{\partial X} \frac{\partial w_{i}}{\partial X})$

Numerical results

We have taken a long duct of square cross-section defined by ∣X∣ ⩽ 1 and ∣Y∣ ⩽ 1 for this problem. Velocity and induced magnetic field across the section have been evaluated and graphed for different values of M, θ and α. In this paper, the linear monomial basis is used as p^T(x) = [1, x, y]. The calculations are performed using N = 51 × 51, a_c = 4.0, q = 1.03.

Fig. 4, Fig. 5, Fig. 6 present the numerical solutions of velocity and magnetic field, when θ = ∞ for some selected points forM = 5, 30 and 50,

Conclusion

In this paper, a local radial point interpolation method (LRPIM) is applied to the fully developed MHD flow problem in a straight duct. The numerical solutions are compared with the Shercliff’s exact solutions under the condition that the duct walls are insulated for different Hartmann numbers. With different wall conductivities and different orientations of external applied magnetic fields, the velocity and induced magnetic fields are also presented. LRPIM is a truly meshless method, which

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Local radial point interpolation method for the fully developed magnetohydrodynamic flow

Abstract

Introduction

Section snippets

Transverse magnetic field

Point interpolation using radial basis functions

Local weak form of LRPIM for the fully developed MHD flow

Discretized equations

Numerical results

Conclusion

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

Math. Comput. Simulat.

J. Comput. Appl. Math.

Eng. Anal. Boundary Elem.

Comput. Phys. Commun.

Appl. Numer. Math.

J. Sound Vibr.

Comput. Phys. Commun.

Comput. Fluids

Comput. Math. Appl.

Appl. Math. Comput.

Steady motion of conducting fluids in pipes under transverse magnetic fields

Proc. Camb. Phil. Soc.

MHD axial flow in a triangular pipe under transverse magnetic field

Ind. J. Pure Appl. Math.

MHD axial flow in a triangular pipe under transverse magnetic field parallel to a side of the triangle

Ind. J. Tech.

Finite element method in MHD channel flow problems

Int. J. Numer. Meth. Eng.