The boundary elements method for magneto-hydrodynamic (MHD) channel flows at high Hartmann numbers

In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 10⁵${10}^5$ by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods.     

Fundamental solution for coupled magnetohydrodynamic flow equations

In this paper, a fundamental solution for the coupled convection–diffusion type equations is derived. The boundary element method (BEM) application then, is established with this fundamental solution, for solving the coupled equations of steady magnetohydrodynamic (MHD) duct flow in the presence of an external oblique magnetic field. Thus, it is possible to solve MHD duct flow problems with the most general form of wall conductivities and for large values of Hartmann number. The results for velocity and induced magnetic field is visualized in terms of graphics for values of Hartmann number M?300$M?300$.     

A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations

In this paper, the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular section are solved using combined finite volume method and spectral element technique, improved by means of Hermit interpolation. The transverse applied magnetic field may have an arbitrary orientation relative to the section of the pipe. The velocity and induced magnetic field are studied for various values of Hartmann number, wall conductivity and orientation of the applied magnetic field. Comparisons with the exact solution and also some other numerical methods are made in the special cases where the exact solution exists. The numerical results for these sample problems compare very well to analytical results.     

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

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

Numerical integration of logarithmic and nearly logarithmic singularity in BEMs

A new semi-analytical integration scheme is proposed for evaluation of logarithmically singular and/or nearly singular integrals occurring in 2D BEM formulations. Extensive numerical experiments are performed to study the accuracy by the proposed and other schemes of numerical integration. The accuracy of the numerical integration is almost exact even for curvilinear elements and the accuracy of numerical computation is determined by the accuracy of the approximation of the boundary density and geometry.     

薄体结构温度场的高阶边界元分析

分析了二维问题边界元法3节点二次单元的几何特征,区分和定义了源点相对高阶单元的Ⅰ型和Ⅱ型接近度.针对二维位势问题高阶边界元中奇异积分核,构造出具有相同Ⅱ型几乎奇异性的近似核函数,在几乎奇异积分单元上分离出积分核中主导的奇异函数部分.原积分核扣除其近似核函数后消除几乎奇异性,成为正则积分核函数,并采用常规Gauss数值方法计算该正则积分;对奇异核函数的积分推导出解析公式,从而建立了一种新的边界元法高阶单元几乎奇异积分半解析算法.应用该算法计算了二维薄体结构温度场算例,计算结果表明高阶单元半解析算法能充分发挥边界元法优势,显著提高计算精度.     

Local radial point interpolation method for the fully developed magnetohydrodynamic flow

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.     

Convergence,stability, and numerical solution of unsteady free convection magnetohydrodynamical flow between two slipping plates

In this study, the unsteady free convection magnetohydrodynamical flow of a viscous, incompressible, and electrically conducting fluid between two horizontally directed slipping plates is considered. The external magnetic filed is applied uniformly in the y-direction and the fluid is assumed to be of low conductivity so that the induced magnetic field is negligible. So the relevant variables, that is, the velocity and the temperature, depend only on one coordinate, the y-axis. The governing equations of velocity and temperature fields are obtained from the continuity, momentum, and energy equations. The boundary conditions for the velocity are taken in the most general form as Robins type which contain slipping parameter. Moreover, the upper plate is heated exponentially and the lower plate is adiabatic. Finite difference method (FDM) is used to simulate the numerical solutions of the problem in which the explicit forward difference in time variable t and central difference in space variable y is used. Hartmann number, Prandtl number, decay factor, and slipping parameter influences on the flow and temperature are shown graphically. It is seen that as the Hartmann number increases, the velocity magnitude drops, which is the well-known flattening tendency of the MHD flow. Also, the increase in decay factor causes an increase in both the velocity and temperature magnitudes at increasing time levels, but it does not change further close to the steady-state. Furthermore, the convergence and stability conditions of the considered scheme are obtained in terms of Hartmann number, Prandtl number, and the slip length.     

Numerical evaluation of various levels of singular integrals, arising in BEM and its application in hydrofoil analysis

The sound implementation of the boundary element method (BEM) is highly dependent on an accurate numerical integration of singular integrals. In this paper, a set of various types of singular domain integrals with three-dimensional boundary element discretization is evaluated based on a transformation integration technique. In the BEM, the integration domain (body surface) needs to be discretized into small elements. For each element, the integral I(x^p, x) is calculated on the domain dS. Several types of integrals IB_α and IC_α are numerically and analytically computed and compared with the relative error. The method is extended to evaluate singular integrals which arise in the solution of the three-dimensional Laplace’s equation. An example of the elliptic hydrofoil is performed to study the physical accuracy. The results obtained using both numerical and analytical methods are shown in good agreement with the experimental data.     

Finite element method solution of electrically driven magnetohydrodynamic flow

The magnetohydrodynamic (MHD) flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes, placed one in each of the walls of the duct where the applied magnetic field is perpendicular. The flow is steady, laminar and the fluid is incompressible, viscous and electrically conducting. A stabilized finite element with the residual-free bubble (RFB) functions is used for solving the governing equations. The finite element method employing the RFB functions is capable of resolving high gradients near the layer regions without refining the mesh. Thus, it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the Hartmann number. Before employing the bubble functions in the global problem, we have to find them inside each element by means of a local problem. This is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem. Equivelocity and current lines are drawn to show the well-known behaviours of the MHD flow. Those are the boundary layer formation close to the insulated walls for increasing values of the Hartmann number and the layers emanating from the endpoints of the electrodes. The changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field.     

Evolution of the magnetic field in plasma in a neighborhood of zero points

We consider a mathematical model of magnetic reconnection in a neighborhood of singular points in the framework of 3D magnetohydrodynamics. In the case of linear dependence of the magnetic field and the velocity on coordinates, we present self-similar solutions of the system of MHD equations. For an arbitrary initial magnetic configuration, we state a mixed problem, which is solved numerically. We describe a semi-implicit algorithm for the solution of the difference problem and present the results of several numerical experiments.     

Numerical solution of a singular integral equation appearing in magnetohydrodynamics

The numerical solution for the velocity and induced magnetic field has been obtained for the MHD flow through a rectangular pipe with perfectly conducting electrodes. The problem reduces to the solution of a singular integral equation which has been solved numerically. It is found that as the Hartmann number is increased the velocity profile shows a flattening tendency and the flux through a section is reduced. Also as compared with the case of nonconducting walls the flux is found to be smaller. Graphs and tables are given for the solution of the integral equation and the velocity and induced magnetic field.

---

Zusammenfassung Für den MHD Fluß durch ein rechteckiges Rohr mit gut leitenden Elektroden wurde die numerische Lösung für die Geschwindigkeit und das induzierte Feld ermittelt. Das Problem ließ sich auf eine singuläre Integralgleichung zurückführen, die numerisch gelöst wurde. Es hat sich herausgestellt, daß wenn die Hartmann-Zahl größer wird, das Geschwindigkeitsprofil eine Tendenz zur Abflachung zeigt und der Fluß durch den Querschnitt zurückgeht. Im Vergleich mit dem Einsatz von nicht leitenden Wänden wurde ebenfalls ein geringerer Fluß festgestellt. Für die Lösung der Integralgleichung, die Geschwindigkeit und das magnetische induzierte Feld sind graphische Darstellungen und Tabellen angegeben.

     

Evaluation of regular and singular domain integrals with boundary-only discretization—theory and Fortran code

In this paper, a set of boundary integrals are derived based on a radial integration technique to accurately evaluate two dimensional (2D) and three dimensional (3D), regular and singular domain integrals. A self-contained Fortran code is listed and described for numerical implementation of these boundary integrals. The main feature of the theory is that only the boundary of the integration domain needs to be discretized into elements. This feature cannot only save considerable efforts in discretizing the integration domain into internal cells (as in the conventional method), but also make computational results for singular domain integrals more accurate since the integrals have been regularized. Some examples are provided to verify the correctness of the presented formulations and the included code.     

Thermally coupled,stationary, incompressible MHD flow; existence,uniqueness, and finite element approximation

This article addresses the questions of existence, uniqueness, and finite element approximation (including some computational aspects) of solutions to the equations of steady-state magnetohy-drodynamic (MHD) when buoyancy effects due to temperature differences in the flow cannot be neglected. We couple the MHD equations to the heat equation and employ the well-known Boussinesq approximation. We consider the equations posed on a bounded three-dimensional domain. The boundary conditions for the velocity are of Dirichlet type; the boundary conditions for the temperature are mixed (of Dirichlet type and of Neumann type); we also specify the normal component of the magnetic field and tangential component of the electric field on the boundary. We point out that these problems are relevant to many physical phenomena such as the cooling of nuclear reactors by electrically conducting fluids, continuous metal casting, crystal growth, and semi-conductor manufacture. © 1995 John Wiley & Sons, Inc.     

Special Boundary Integral Equations for Approximate Solution of Potential Problems in Three-dimensional Regions with Slender Cavities of Circular Cross-section

A special boundary integral method is developed for solvingpotential problems in a general three-dimensional region withslender internal cavities of circular cross-section. The solutionon the boundary of each cavity is assumed to be axisymmetricso that the surface integrals on a cavity boundary may be reducedto contour integrals along the centre line of the cavity. Specialintegral equations are introduced to determine the solutionalong each cavity. Although the contour integrals in these newequations are never singular, they have a nearly singular characterwhich gives them computational advantages similar to traditionalboundary integral equations with out the danger of ill-conditioningcaused by the strongly contrasting length scales introducedby the slender cavities. For the special case of parallel toroidalcavities, the method gives results with accuracy comparableto previous perturbation methods. The numerical characteristicsof the new integral equations are explored by solving the problemsof two perpendicular, interlocking tori and two perpendicular,finite cylindrical cavities, both in unbounded regions. Theequations exhibit excellent numerical characteristics over abroad range of parameters.     

Numerical calculation of singular integrals appearing in three-dimensional potential flow problems

Two approximate methods for calculating singular integrals appearing in the numerical solution of three-dimensional potential flow problems are presented. The first method is a self-adaptive, fully numerical method based on special copy formulas of Gaussian quadrature rules. The singularity is treated by refining the partitions of the copy formula in the vicinity of the singular point. The second method is a semianalytic method based on asymptotic considerations. Under the small curvature hypothesis, asymptotic expansions are derived for the integrals that are involved in the calculation of the scalar potential, the velocity as well as the deformation field induced from curved quadrilateral surface elements. Compared to other methods, the proposed integration schemes, when applied to practical flow field calculations, require less computational effort.     

The boundary element solution of the magnetohydrodynamic flow in an infinite region

We consider the magnetohydrodynamic (MHD) flow which is laminar, steady and incompressible, of a viscous and electrically conducting fluid on the half plane (y≥0)$(y\ge 0)$. The boundary y=0$y=0$ is partly insulated and partly perfectly conducting. An external circuit is connected so that current enters the fluid at discontinuity points through external circuits and moves up on the plane. The flow is driven by the interaction of imposed electric currents and a uniform, transverse magnetic field applied perpendicular to the wall, y=0$y=0$. The MHD equations are coupled in terms of the velocity and the induced magnetic field. The boundary element method (BEM) is applied here by using a fundamental solution which enables treating the MHD equations in coupled form with general wall conditions. Constant elements are used for the discretization of the boundary y=0$y=0$ only since the boundary integral equation is restricted to this boundary due to the regularity conditions at infinity. The solution is presented for the values of the Hartmann number up to M=700$M=700$ in terms of equivelocity and induced magnetic field contours which show the well-known characteristics of the MHD flow. Also, the thickness of the parabolic boundary layer propagating in the field from the discontinuity points in the boundary conditions, is calculated.     

A level-set topological optimization method to analyze two-dimensional thermal problem using BEM

A level-set based topological optimization approach is proposed using boundary element method (BEM) to solve two-dimensional(2D) thermal problems. The objective function is considered as a function of temperature and thermal flux defined on boundaries with Dirichlet and Neumann boundary conditions. The topological sensitivity is derived combining BEM under the assumption of insulating topological boundaries generated during optimization. Smooth boundaries represented by the level-set function is updated using topological sensitivity with a regularization term. Numerical examples with different objective functions considering the real-world problems are presented to show the effectiveness of the proposed approach. The topological sensitivity, computational time and boundary smoothness are verified by comparing with finite difference method (FDM).     

A note on the ill-posedness of shear flow for the MHD boundary layer equations

For the two-dimensional Magnetohydrodynamics(MHD)boundary layer system,it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in the tangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by G′erard-Varet and Dormy(2010).This partially shows the necessity of the non-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.     

A new extension of boundary elements method for classical elliptic partial differential equations with constant coefficients

This article is devoted to an extension of boundary elements method (BEM) for solving elliptic partial differential equations of general type with constant coefficients. As the fundamental solution of these equations was not available in the literature, BEM was not able to handle them, directly. So the dual reciprocity method (DRM) has been applied to tackle these problems. In this work, a fundamental solution for these equations is obtained and a new formulation is derived to solve them. Besides, we show that the rate of convergence of the new scheme is quadratic when singular (boundary and domain) integrals are calculated, accurately. The new scheme is applicable on complex domains, without needing internal nodes, just same as conventional BEM. So the CPU time of the new scheme is much less than that of the DRM. Numerical examples presented in the article show ability and efficiency of the new scheme in solving two‐dimensional nonhomogenous elliptic boundary value problems, clearly. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2027–2042, 2015