Spectral methods based on the least dissipative modes for wall bounded MHD flows

We present a new approach for the Spectral Direct Numerical Simulation (DNS) of Low-Rm wall-bounded magnetohydrodynamic (MHD) flows. The novelty is that instead of using bases similar to the usual Chebyshev polynomials, which are easy to implement but incur heavy computational costs to resolve the Hartmann boundary layers that arise along the walls, we use a basis made of elements that already incorporate flow structures such as anisotropic vortices and Hartmann layers. We show that such a basis can be obtained from the eigenvalue problem of the linear part of the governing equations with the problem’s boundary conditions. Since this basis is not always orthogonal, we develop a spectral method for non-orthogonal bases. We then demonstrate the efficiency of this method on the simple case of a laminar channel flow with periodic forcing. In particular, we show that this method eliminates the computational costs incurred this Hartmann layer, and this for arbitrary high magnetic fields B. We then discuss the application of our method to nonlinear, turbulent flows for which the number of modes required to resolve the flow completely decreases strongly when B increases, instead of increasing as in the case of currently employed Chebyshev-based methods.     

Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery-Hamel problem

A heuristic technique is developed for a nonlinear magnetohydrodynamics(MHD) Jeffery-Hamel problem with the help of the feed-forward artificial neural network(ANN) optimized with the genetic algorithm(GA) and the sequential quadratic programming(SQP) method. The two-dimensional(2D) MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem(BVP) of ordinary differential equations(ODEs). The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the angles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.     

Heat transfer and entropy generation analysis of non-Newtonian fluid flow through vertical microchannel with convective boundary condition

The entropy generation and heat transfer characteristics of magnetohydrodynamic(MHD) third-grade fluid flow through a vertical porous microchannel with a convective boundary condition are analyzed. Entropy generation due to flow of MHD non-Newtonian third-grade fluid within a microchannel and temperature-dependent viscosity is studied using the entropy generation rate and Vogel's model. The equations describing flow and heat transport along with boundary conditions are first made dimensionless using proper non-dimensional transformations and then solved numerically via the finite element method(FEM). An appropriate comparison is made with the previously published results in the literature as a limiting case of the considered problem.The comparison confirms excellent agreement. The effects of the Grashof number, the Hartmann number, the Biot number, the exponential space-and thermal-dependent heat source(ESHS/THS) parameters, and the viscous dissipation parameter on the temperature and velocity are studied and presented graphically. The entropy generation and the Bejan number are also calculated. From the comprehensive parametric study, it is recognized that the production of entropy can be improved with convective heating and viscous dissipation aspects. It is also found that the ESHS aspect dominates the THS aspect.     

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.     

Numerical simulation of magnetohydrodynamic flow in a toroidal duct of square cross-section

We present numerical simulation results of the quasi-static magnetohydrodynamic (MHD) flow in a toroidal duct of square cross-section with insulating Hartmann walls and conducting side walls. Both laminar and turbulent flows are considered. In the case of steady flows, we present a comprehensive analysis of the secondary flow. It consists of two counter-rotating vortex cells, with additional side wall vortices emerging at sufficiently high Hartmann number. Our results agree well with existing asymptotic analysis. In the turbulent regime, we make a comparison between hydrodynamic and MHD flows. We find that the curvature induces an asymmetry between the inner and outer side of the duct, with higher turbulence intensities occurring at the outer side wall. The magnetic field is seen to stabilize the flow so that only the outer side layer remains unstable. These features are illustrated both by a study of statistically averaged quantities and by a visualization of (instantaneous) coherent vortices.     

SELF-ADAPTIVE STRATEGY FOR ONE-DIMENSIONAL FINITE ELEMENT METHOD BASED ON ELEMENT ENERGY PROJECTION METHOD

Based on the newly-developed element energy projection (EEP) method for computation of super-convergent results in one-dimensional finite element method (FEM), the task of self-adaptive FEM analysis was converted into the task of adaptive piecewise polynomial interpolation. As a result, a satisfactory FEM mesh can be obtained, and further FEM analysis on this mesh would immediately produce an FEM solution which usually satisfies the user specified error tolerance. Even though the error tolerance was not completely satisfied, one or two steps of further local refinements would be sufficient. This strategy was found to be very simple, rapid, cheap and efficient. Taking the elliptical ordinary differential equation of second order as the model problem, the fundamental idea, implementation strategy and detailed algorithm are described. Representative numerical examples are given to show the effectiveness and reliability of the proposed approach.     

Magnetohydrodynamic flow on a half-plane

We investigate the magnetohydrodynamic flow (MHD) on the upper, half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour.     

Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order

Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.     

A new approach on MHD natural convection boundary layer flow past a flat plate of finite dimensions

A new approach on MHD natural convection boundary layer flow from a finite flat plate of arbitrary inclination in a rotating environment, is presented. This problem plays a significant role on boundary layer flow control. It is shown that taking into account the pressure rise region at the leading edge of the plate leads to avoid separation and the back flow is reduced by the strong magnetic field. It is also shown that the frictional drag at the leading edge of the plate is reduced when the inclination angle α=π/4. In the case of isothermal flat plate, the bulk temperature becomes identical for any value of Gr (Grashof number) when the value of M ² (Hartmann number) and K ² (rotation parameter) are kept fixed.     

倾斜非均匀磁场下导电方管磁流体管流的数值模拟研究

热核聚变反应堆液态金属包层应用中的一个重要问题是液态金属在导电管中流动和强磁场相互作用产生的额外的磁流体动力学压降.这种磁流体动力学压降远远大于普通水力学压降.美国阿贡国家实验室ALEX研究小组,对非均匀磁场下导电管中液态金属磁流体动力学效应进行了实验研究,其实验结果成为液态金属包层数值验证的标准模型之一.液态金属包层在应用中会受到不同方向的磁场作用,本文以ALEX的非均匀磁场下导电方管中液态金属管流实验中的一组参数为基础,保持哈特曼数、雷诺数和壁面电导率不变,采用三维直接数值模拟的方法,研究了外加磁场与侧壁之间的倾角对导电方管内液态金属流动的速度、电流和压降分布的影响.研究结果表明:沿流向相同横截面上的速度、电流以及压力分布均随磁场的倾斜而同向旋转.倾斜磁场均匀段,横截面上的高速区位于平行磁场方向的哈特曼层和平行层交叉位置,压力梯度随磁场倾角的增大先增大后减小.倾斜磁场递减段,在三维磁流体动力学效应作用下,横截面上的高速射流位置向垂直磁场方向偏移.磁场递减段的三维磁流体动力学压降随磁场倾角的增大而增大.随磁场倾斜,截面上的射流峰值逐渐减小,二次流增强,引发层流向湍流的转捩.      

Solution of magnetohydrodynamic flow in a rectangular duct by Chebyshev collocation method

In this study, matrix representation of the Chebyshev collocation method for partial differential equation has been represented and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Numerical solution of velocity and induced magnetic field is obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. The Chebyshev collocation method is used with a reasonable number of collocations points, which gives accurate numerical solutions of the MHD flow problem. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H≤1000. Copyright © 2010 John Wiley & Sons, Ltd.     

基于无网格Galerkin法求解矩形绝缘管道内的磁流体流动

无量纲磁流体流动控制方程中的哈特曼数较大将导致数值计算发散或误差过大。将无网格Galerkin法引入绝缘管道内的稳定磁流体流动计算中,针对磁流体控制方程中大哈特曼数导致计算误差大的情况,对无网格Galerkin法添加了稳定项。计算结果表明,同等条件下,添加了稳定项的无网格Galerkin法总体相对误差远小于标准无网格Galerkin法的结果,且可以计算哈特曼数最大达50000绝缘管道内的磁流体流动。     

The DRBEM solution of incompressible MHD flow equations

This paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier‐Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function‐vorticity‐magnetic induction‐current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson‐type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time‐dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward‐facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley & Sons, Ltd.     

时域自适应算法求解弹性地基薄板的动力问题

为更精确地求解弹性地基薄板的动力响应,发展了一种分段时域自适应算法,通过变量在离散时段内的展开,将时空耦合的初边值问题转化为一系列递推的基于有限元(FEM)的空间问题求解,通过自适应计算保持稳定的计算精度。数值算例表明:本文解与解析解相比最大相对误差不超过3.59%;当步长较大时四阶Runge-Kutta法和Newmark法均失效,本文所提算法仍可得到满意的计算结果。     

Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method

The element energy projection （EEP） method for computation of super- convergent resulting in a one-dimensional finite element method （FEM） is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems. The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton＇s method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a re- sult, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed. The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation （ODE） of second-order as the model problem, this paper describes the related fundamental idea, the imple- mentation strategy, and the computational algorithm. Representative numerical exam- ples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity

A finite element method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady MHD flow through a pipe having arbitrarily conducting walls. Pipes of rectangular, circular and triangular sections have been taken for illustration. Computations have been carried out for different Hartmann numbers and wall conductivity at various time levels. It is found that if the wall conductivity increases, the flux through a section decreases. The same is the effect of increasing the Hartmann number. It is also observed that the steady state is approached at a faster rate for larger Hartmann numbers or larger wall conductivity. Selected graphs are given showing the behaviour of velocity, induced magnetic field and flux across a section.     

Two‐level finite element method with a stabilizing subgrid for the incompressible MHD equations

We consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite‐dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual‐free bubble functions. To find the bubble part of the solution, a two‐level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate solutions obtained show the well‐known characteristics of the MHD flow. Copyright © 2009 John Wiley & Sons, Ltd.     

Fully developed magneto convection flow in a vertical rectangular duct

An analysis is performed to study the MHD free convection flow in a vertical rectangular duct for laminar and fully developed regime taking into consideration the effects of Ohmic heating and viscous dissipation. Numerical solutions are found using finite difference method of second-order accuracy. The effects of various physical parameters such as Hartmann number, aspect ratio, buoyancy parameter and circuit parameter are presented graphically. It is found that as Hartmann number, buoyancy parameter and aspect ratio increase, the upward and downward flow rates are increased for open circuit but decrease for short circuit.     

Direct simulation of MHD flows in dual-coolant liquid metal fusion blanket using a consistent and conservative scheme

Direct simulation of 3-D MHD (magnetohydrodynamics) flows in liquid metal fusion blanket with flow channel insert (FCI) has been conducted. Two kinds of pressure equilibrium slot (PES) in FCI, which are used to balance the pressure difference between the inside and outside of FCI, are considered with a slot in Hartmann wall or a slot in side wall, respectively. The velocity and pressure distribution of FCI made of SiC/SiC_f are numerically studied to illustrate the 3-D MHD flow effects, which clearly show that the flows in fusion blanket with FCI are typical three-dimensional issues and the assumption of 2-D fully developed flows is not the real physical problem of the MHD flows in dual-coolant liquid metal fusion blanket. The optimum opening location of PES has been analyzed based on the 3-D pressure and velocity distributions.