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Exponential compact ADI method for a coupled system of convection-diffusion equations arising from the 2D unsteady magnetohydrodynamic (MHD) flows
Abstract
In this paper, an exponential high-order compact alternating direction implicit (EHOC-ADI) difference method is developed for solving the coupled equations representing the unsteady incompressible, viscous magnetohydrodynamic (MHD) flow through a straight pipe of rectangular section. The method, in which the Crank-Nicolson scheme is used for the time discretization and an original EHOC difference scheme established for the steady 1D coupled system of convection-diffusion equations is used for the spatial discretization, is second order accurate in time and fourth-order accurate in space and requires only a regular five-point 2D stencil similar to that in the standard second-order methods and the three-point stencil for each 1D operator. A distinguishing desirable property of the proposed method is combining the computational efficiency of the lower order methods with superior accuracy inherent in high order approximations. The unconditional stable character of the method is verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are carried out to illustrate and assess the performance and the accuracy of the method proposed currently. Computational results of the MHD flow in the 2D square-channel problems are presented for Hartmann numbers ranging from 0 to 10⁶ and compared with the exact solutions and those obtained using other available methods in the literatures.
... for t > 0. Equations (2)-(3) are associated with the following conditions as given in Eq. (4). The space vector x ¼ ðx; yÞ; M is the magnetic parameter, ...
... x; y ð Þ 2 @X; t > 0: (4) The parameters and variables given in (5) are key to converting the nondimensionalized system of the model given in (7)- (9), in which the bar refers to the dimensional quantities, l is a characteristic length of the flow section of the duct, and M is the Hartmann number 4 ...
... While the top wall of the rectangular duct is subject to a small boundary perturbation e, Fendo glu et al. 9 studied the timedependent MHD flow of an incompressible, viscous, and electrically conductive liquid. Wu et al. 4 researched the development of an exponential high-order compact alternating direction implicit (EHOC-ADI) difference scheme to numerically seek the behavior of coupled equations presenting the time-dependent MHD flow model. The nonlocal characteristics of unsteady coupled convection-diffusion systems that emerge in MHD flows were documented by Hamid et al. 10 Boundary element techniques for MHD duct flows and convectiondiffusion models are among the important contributions made by the mathematics community. ...
A difficult matter for researchers is to develop numerical methods to analyze the behavior of nonlinear dynamical systems that arise in mathematical physics due to their nonlinearity and multi-dimensional nature. In this affection, a novel and effective coupling of spectral and one-step approaches is suggested and used to inspect the robust solutions to the mathematical problems. This proposed method is grounded on the one-step sixth-order Runge–Kutta (RK) method and the shifted Vieta–Fibonacci (sVF) polynomial-based spectral approach. First, use the sixth-order one-step technique for the temporal approximation, then utilize the sVF polynomials to estimate the spatial variables. The sVF polynomial-based new operational matrices are used to calculate the derivative terms of the mathematical models. Related theorems are presented in the paper to provide a mathematical validation of the technique. The suggested semi-spectral techniques transform the investigated nonlinear models into a set of linear algebraic equations, making them simpler to solve. We conduct theoretical and numerical research into the stability, convergence, and error-bound analysis to support the computational algorithm's mathematical formulation. Several mathematical models, including the Hartmann numbers ranging from 0 to 10⁶, are taken into consideration to demonstrate the precision and efficacy of the proposed computational technique. A thorough comparison analysis illustrates the recommended computational approach's validity, correctness, and dependability. The suggested construction of sixth-order in the temporal direction and exponential order around 3.5545e0.1515M in the spatial direction is determined to be exceptional in handling nonlinear problems and examining the exact solutions.
... The authors considered the flows for 0 ≤ Ha ≤ 500. Wu et al. [27] used an EHOC alternating direction implicit (EHOC-ADI) difference method for solving unsteady MHD flows through straight pipes of rectangular sections for a wide range of Ha values up to Ha = 10 7 . Evcin et al. [28,29] considered MHD flows by employing mixed finite element formulations in the optimal control framework. ...
... The unsteady, incompressible, and laminar flow of an electrically conducting and viscous fluid occupying a long channel of a rectangular cross-section under the influence of a constant and uniform magnetic field imposed along x 1 -axis in the following dimensionless form can be described as follows [10,15,27]: ...
... where V (x 1 , x 2 , t) and B (x 1 , x 2 , t) represent the velocity and magnetic fields. The set of suitable dimensionless variables and appropriate scaling can be found in [27]. The flow is assumed to be driven by a constant axial pressure gradient in the x 3 -direction, and the induced magnetic field and velocity vector are parallel to the x 3 -axis. ...
The concern of this manuscript is stabilized finite element simulations of two-dimensional incompressible and viscous magnetohydrodynamic (MHD) duct flows governed by a coupled system of partial differential equations of the convection-diffusion type. In such flows, the high values of the Hartmann numbers (Ha) indicate that the convection process dominates the flow field. As is typical trouble encountered in convection-dominated flow simulations, classical discretization methods fail to function properly for MHD flows at high Hartmann numbers, yielding several numerical instabilities. In this study, the core computational tool we employ in order to overcome such challenges is the streamline-upwind/Petrov-Galerkin (SUPG) formulation. Beyond that, since the SUPG-stabilized formulation requires additional treatment where solutions experience rapid changes, we also enhance the SUPG-stabilized finite element formulation with a shock-capturing operator for achieving better solution profiles around sharp layers. The formulations are derived for both steady-state and time-dependent models. The proposed method, the so-called SUPG-YZβ formulation, techniques used, and in-house-developed finite element solvers are evaluated on a comprehensive set of numerical experiments. The test computations reveal that the SUPG-YZβ formulation yields quite good solution profiles near strong gradients without any significant numerical instabilities, even for quite challenging values of Ha, whereas the SUPG usually fails alone. Furthermore, by using only linear elements on relatively coarser meshes, the proposed formulation achieves this without the need for any adaptive mesh strategy, in contrast to most previously reported studies.
... In the literature, finding the analytical solutions of the coupled diffusion-convection-wave equations has gained considerable attention because of its widespread application as a mathematical formulation for the dynamics of long-range interactions between more than one element or species in physical processes like convection-dominated transport media problems. Some important classes of coupled diffusion-convection equations include the coupled Burgers equations, the hydrodynamic model for semiconductor device simulation, magneto-hydrodynamics, shallow water equations, and the Navier-stokes equations [72][73][74]. ...
... where the unknown functions δ sj (t), s = 1, 2, j = 1, 2, 3, 4, can be determined by following the same procedure of the invariant subspace method as used in Example 3.1. Note that the obtained analytical solution (3.26) is more general than the derived solution in [73]. ...
This paper discusses how to generalize the invariant subspace method for deriving the analytical solutions of the multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo fractional-order derivative for the first time. Specifically, we describe how to systematically find different invariant product linear spaces with various dimensions for the considered system. Also, we observe that the obtained invariant product linear spaces help to reduce the multi-component [Formula: see text]-dimensional coupled NTFPDEs into a system of fractional-order ODEs, which can then be solved using the well-known analytical methods. More precisely, we illustrate the effectiveness and importance of this developed method for obtaining a long list of invariant product linear spaces for the multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations. In addition, we have shown how to find different kinds of generalized separable analytical solutions for a multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations along with the initial and boundary conditions using invariant product linear spaces obtained. Finally, we provide appropriate graphical representations of some of the derived generalized separable analytical solutions with various fractional-order values.
... (1.1b) (u(x, y, t), v(x, y, t)) = (ϕ 1 (x, y, t), ϕ 2 (x, y, t)) , (x, y) ∈ ∂Ω, as, linear or nonlinear reaction-diffusions equations [27][28][29], delay parabolic equations [30][31][32], convection-diffusions equations [33,34], Schrödinger equations with or without fractional derivatives [35][36][37]. Also, there have been a lot of researches on numerical solutions of multi-dimensional linear and nonlinear hyperbolic equations by compact ADI methods with the convergent rates of O(Δt 2 + h 4 x + h 4 y ), such as, linear or nonlinear telegraph equations [38,39], generalized sine-Gordon equations [40]. ...
In this article, a two-level linearized compact alternating direction implicit (ADI) scheme is proposed for solving two-dimensional (2D) nonlinear coupled wave equations (NCWEqs). In comparison with the existent compact ADI methods for NCWEqs, there are two obvious characters. (1) Numerical solutions at time level one, which have effect on the global stability and convergence of the numerical solutions, are not needed to be solved firstly by using another numerical scheme. (2) The computational cost is comparatively low and acceptable because the new compact ADI method is a linear scheme, thus avoiding Newton’s iterations. By using the energy analysis method, it is shown that numerical solutions converge to exact solutions with an order of 𝒪(Δt2+hx4+hy4) in H¹- and L∞-norms, and are uniquely solvable. Numerical examples are given to illustrate the exactness of the theoretical findings and the high-performance of our methods.
... The alternating direction implicit (ADI) method is preferred, because it can reduce high-dimensional problems to a series of independent one-dimensional problems, then improves the computational efficiency, see [10,34]. For fractional problems, many various ADI schemes had been proposed in previous works, more details can be found in [35,31,4,32,24,25]. For the current model (1.1)-(1.3), ...
A high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time in this paper. The L1 formula and trapezoidal PI rule on nonuniform meshes, which greatly improve the temporal accuracy compared to the method on uniform grids, are adopted to approximate the Caputo derivative and the Riemann-Liouville integral, respectively. With the help of a modified discrete fractional Grönwall inequality and some crucial skills, the stability and convergence of the proposed scheme are analyzed. Numerical results confirm the sharpness of the error analysis.
... Neumann boundary conditions were considered by Fu, Tian and Liu [58]. Previously, Tian and Ge [217] developed a fourth-order exponential scheme for solving 2D unsteady convection-diffusion problems, and used in unsteady magnetohydrodynamics flows by Wu, Peng and Tian [239]. Extension to a threedimensional case of the Tian-Dai's scheme [216] was done by Mohamed, Mohamed and Seddek [138]. ...
In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order
quadratures are used to achieve numerical results close to machine-accuracy
even with coarse grids.
As the discretization is essentially one-dimensional, getting the machine-accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists of a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.
The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics (MHD) flows. The time derivative is expressed by means of Caputo’s fractional derivative concept, while the model is solved via the full-spectral method (FSM) and the semi-spectral scheme (SSS). The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques. The SSS is developed by discretizing the time variable, and the space domain is collocated by using equal points. A detailed comparative analysis is made through graphs for various parameters and tables with existing literature. The contour graphs are made to show the behaviors of the velocity and magnetic fields. The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows, and the concept may be extended for variable order models arising in MHD flows.
A new implicit finite difference method with a compact correction term is established for solving unsteady convection-diffusion equations. The correction terms by connecting classical and compact finite difference formulas are proposed for improving the accuracies of numerical solutions. This new method with fourth order accuracy can be used for the numerical calculations of convection diffusion equation on both uniform and nonuniform grid systems. It can be extended not only from one-dimensional to multi-dimensional equations, but also from steady to unsteady convection-diffusion equations. Several convection-diffusion problems are stimulated for verifying the flexibility and high accuracy of this method.
In this article, an exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank–Nicolson scheme is used for the time discretization and an exponential fourth-order compact difference formula for the steady-state 1D convection–diffusion problem is used for the spatial discretization, is presented for the solution of the unsteady 2D convection–diffusion problems. The method is temporally second-order accurate and spatially fourth order accurate, which requires only a regular five-point 2D stencil similar to that in the standard second-order methods. The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of Karaa and Zhang and the spatial third-order compact scheme of Note and Tan.
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 105) 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 Petrov–Galerkin (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.
In this paper, an effective and accurate numerical model that involves a suggested mathematical formulation, viz., the stream functions (ψ and A)-velocity-magnetic induction formulation and a fourth-order compact difference algorithm is proposed for solving the two-dimensional (2D) steady incompressible full magnetohydrodynamic (MHD) flow equations. The stream functions-velocity-magnetic induction formulation of the 2D incompressible full MHD equations is able to circumvent the difficulty of handling the pressure variable in the primitive variable formulation or determining the vorticity values on the boundary in the stream function-vorticity formulation, and also ensure the divergence-free constraint condition of the magnetic field inherently.A test problem with the analytical solution, the well-studied lid-driven cavity problem in viscous fluid flow and the lid-driven MHD flow in a square cavity are performed to assess and verify the accuracy and the behavior of the method proposed currently. Numerical results for the present method are compared with the analytical solution and the other high-order accurate results. It is shown that the proposed stream function-velocity-magnetic induction compact difference method not only has the excellent performances in computational accuracy and efficiency, but also matches well with the divergence-free constraint of the magnetic field. Moreover, the benchmark solutions for the lid-driven cavity MHD flow in the presence of the aligned and transverse magnetic field for Reynolds number (Re) up to 5000 are provided for the wide range of magnetic Reynolds number (Rem) from 0.01 to 100 and Hartmann number (Ha) up to 4000.
In this paper, we consider the initial-boundary value problem for the three-dimensional incompressible magnetohydrodynamic equations. We first establish certain regularity results for the solution (u, p, H) under the assumption that u, H is an element of L-4(0, T;H-1 (Omega)(3)). Next, we consider the fully discrete first-order Euler semi-implicit scheme based on the mixed finite element method. Then we obtain the L-2-unconditional convergence of this scheme by using the negative norm technique.
In this paper, the exponential higher order compact (EHOC) finite difference schemes proposed by Tian and Dai (2007) [10] for solving the one and two dimensional steady convection diffusion equations with constant or variable convection coefficients are extended to the three dimensional case. The proposed EHOC scheme has the feature that it provides very accurate solution (exact in case of constant convection coefficient in 1D) of the homogeneous equation that is responsible for the fundamental singularity of the homogeneous solution while approximates the particular part of the solution by fourth order accuracy over the nineteen point compact stencil. The key properties of this scheme are its stability, accuracy and efficiency so that high gradients near the boundary layers can be effectively resolved even on coarse uniform meshes. To validate the present EHOC method, three test problems, mostly with boundary or internal layers are solved. Comparisons are made between numerical results for the present EHOC scheme and other available methods in the literature.
In this paper, Newton iteration and two-level finite element algorithm are combined for solving numerically the stationary incompressible magnetohydrodynamics (MHD) under a strong uniqueness condition. The method consists of solving the nonlinear MHD system by m Newton iterations on a coarse mesh with size H and then computing the Stokes and Maxwell problems on a fine mesh with size . The uniform stability and optimal error estimates of both Newton iterative method and two-level Newton iterative method are given. The error analysis shows that the two-level Newton iterative solution is of the same convergence order as the Newton iterative solution on a fine grid with . However, the two-level Newton iterative method for solving the stationary incompressible MHD equations is simpler and more efficient than Newton iterative one. Finally, the effectiveness of the two-level Newton iterative method is illustrated by several numerical investigations.
The steady MHD axial flow in an infinite pipe with equilateral triangular cross-section with transverse magnetic field parallel to a side of the triangle, has been solved numerically using the grid method. The variation of velocity and induced magnetic field along different lines in the cross-section for various Hartmann numbers has been calculated. The results obtained using grids of different sizes have been improved by Richardson's deferred approach to the limit. Flux and average velocity through a section have been obtained for various Hartmann numbers by developing a quadrature formula.
Finite element method is described for unsteady MHD flow in a straight pipe of arbitrary wall conductivity. The transverse applied magnetic field may have an arbitrary orientation relative to the section of the pipe. The velocity, induced magnetic field and the flux through a section have been worked out for a pipe of square section at various time levels for various values of Hartmann number, wall conductivity and orientation of the applied magnetic field. Comparison has been made with known values in special cases. It is found that increase in wall conductivity and/or the Hartmann number has a retarding effect on the flow rate. Also the flux is found to be maximum when the applied magnetic field is parallel to a diagonal.
MAGNETOHYDRODYNAMIC AXIAL FLOW IN A TRIANGULAR PIPE UNDER TRANSVERSE MAGNETIC FIELD
Three finite element iterative methods are designed and analyzed for solving 2D/3D stationary incompressible magnetohydrodynamics (MHD). By a new technique, strong uniqueness conditions for both Stokes type iterative method (Iterative method I) and Newton iterative method (Iterative method II) are obtained, which are weaker than the ones reported in open literature. Stability and optimal convergence rates for the above two methods are derived, where the Iterative method II has an exponential convergent part with respect to iterative step m. Moreover, Oseen type iterative method (Iterative method III) is unconditionally stable and convergent under a uniqueness condition. Finally, performance of the three proposed methods is investigated by numerical experiments.
In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods.
Magnetohydrodynamic flows in coupled rectangular channels are numerically investigated under an external, horizontally applied magnetic field. The flows are driven by constant pressure gradients in the channels, which are separated with a thin partly insulating and partly conducting barrier. A direct boundary element formulation is utilized to solve these two-dimensional steady, convection–diffusion type coupled partial differential equations in terms of velocity and induced magnetic fields. The resulting system of linear equations is solved by reordering the unknown vector due to the insertion of the coupled boundary conditions along the conducting partition of the barrier. This study aims to examine the consequence of high values of Hartmann number on the velocity and induced magnetic fields. Further, the alteration in flow behavior due to the variations in the length of the conducting partition along the thin barrier and in the value of ratio of the pressure gradients of two channels is also analyzed.
A spectral resolutioned exponential compact higher order scheme (SRECHOS) is developed to solve stationary convection–diffusion type of differential equations with constant convection and diffusion coefficients. The scheme is O(h6) for one-dimensional problems and produces a tri-diagonal system of equations that can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an O(h4+k4) accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit (ADI) procedure. The efficiency of the developed scheme is measured using wave number analysis. The analysis shows that the SRECHOS has a much better spectral resolution than any of the existing higher order schemes. Numerical examples are solved and better performance of the SRECHOS in terms of accuracy over the existing schemes in the literature is demonstrated.
In this Letter, a new element free Galerkin method, namely the two-level element free Galerkin method, is presented for solving the governing equations of steady magnetohydrodynamic duct flow. Because this element free Galerkin method makes use of the nodal point configurations which do not require a mesh, therefore it differs from FEM-like approaches by avoiding the need of meshing, a very demanding task for complicated geometry problems. Another distinguished feature of the proposed method is the resolving capability of high gradients near the layer regions without local or adaptive refinements. Numerical results indicate that no matter how large the Hartmann number is, this method has the ability to produce the satisfactory results for the velocity and the magnetic field simultaneously. That is to say, the presented method has some excellent properties, such as better stability and accuracy.
An exact solution for unsteady magnetohydrodynamic flow in a circular pipe (with a uniform magnetic field parallel to a diameter of the cross section) with insulating walls has been obtained. For comparison, graphs have been plotted for the magnetic field in the magnetohydrodynamic case and for velocity in the hydrodynamic and magnetohydrodynamic cases at different times.
This paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents. The equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel. When the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The layers on the walls parallel to the imposed field are of a novel character. An analogous degenerate solution for channels of any symmetrical shape is developed. The predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes.
The solution is obtained to the problem of the steady one-dimensional flow of an incompressible, viscous, electrically conducting fluid through a circular pipe in the presence of an applied (transverse) uniform magnetic field. A no-slip condition on the velocity is assumed at the non-conducting wall. The solution is exact and thus valid for all values of the Hartmann number. Excellent agreement exists between the present theoretical results and the experimental values obtained by Hartmann & Lazarus (1937) in the low to medium Hartmann number range. The high Hartmann number case is treated by Shercliff (1962) in the following paper.
Finite element methods can be used for determining the flow in a straight channel under a variety of wall conductivity conditions when a uniform magnetic field is imposed perpendicular to the flow direction. At high Hartmann numbers oscillatory solutions are found unless sufficient points are placed within the Hartmann layers. In some one- and two-dimensional problems it appears that it is adequate to place one or two points within the Hartmann layer to remove the oscillations. Central core values can sometimes be predicted with good accuracy even when the Hartmann layers are not resolved adequately. A nine-point Gauss point rule has been used to evaluate the stiffness and other matrices for the eight-node elements. Copyright © 2001 John Wiley & Sons, Ltd.
A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M (M ≤ 100). A special solution procedure has been used for the resulting block tridiagonal system of equations. Computations have been carried out for several values of Hartmann number (5 ≤ M ≤ 100) and wall conductivity. It is also found that, if the wall conductivity increases, the flux decreases. The same is the effect of increasing the Hartmann number. Selected graphs are given showing the behaviour of the velocity field and induced magnetic field.
The governing equations and the boundary conditions of steady incompressible magnetohydrodynamic (MHD) problems are adapted to the case of steady MHD flow in simply connected cross sections with arbitrary wall conductance. The problem is then solved numerically by means of the finite element method based on the weighted residual approach, and solutions for industrially relevant Hartmann numbers and non-symmetric kinetic fields are presented.
Methods based on exponential finite difference approximations of h4 accuracy are developed to solve one and two-dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one-dimensional case, the numerical scheme developed uses three points. For the two-dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri-diagonal systems. The methods are applied on a number of linear and non-linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non-uniform mesh is also illustrated. The h4 accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.
A numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) flow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic field are transformed first into the decoupled time-dependent convection–diffusion-type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step-by-step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M⩽50) at transient and the steady-state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic field and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady-state solution. Copyright © 2005 John Wiley & Sons, Ltd.
High-order compact finite difference schemes for two-dimensional convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h4) schemes conform to a 3 × 3 stencil. We show that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h2) centred differencing and commonly used O(h) and O(h3) upwinding schemes.
The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods. Computations have been carried out for several values of the Hartmann number (1 ⩽ M ⩽ 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.
In this paper, a meshfree point collocation method, with an upwinding scheme, is presented to obtain the numerical solutions of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through an insulated straight duct of rectangular section. The moving least-square (MLS) approximation is employed to construct the shape functions in conjunction with the framework of the point collocation method. Computations have been carried out for different applied magnetic field orientations and a wide range of values of Hartmann number from 5 to 106. As the adaptive upwinding local support domain is introduced in the meshless point collocation method, numerical results show that the method may compute MHD problems not only at low and moderate values but also at high values of the Hartmann number with high accuracy and good convergence.
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.
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.
A complete three-dimensional mathematical model has been developed governing the steady, laminar flow of an incompressible fluid subjected to a magnetic field and including internal heating due to the Joule effect, heat transfer due to conduction, and thermally induced buoyancy forces. The thermally induced buoyancy was accounted for via the Boussinesq approximation. The entire system of eight partial differential equations was solved by integrating intermittently a system of five fluid flow equations and a system of three magnetic field equations and transferring the information through source-like terms. An explicit Runge-Kutta time-stepping algorithm and a finite difference scheme with artificial compressibility were used in the general non-orthogonal curvilinear boundary-conforming co-ordinate system. Comparison of computational results and known analytical solutions in two and three dimensions demonstrates high accuracy and smooth monotone convergence of the iterative algorithm. Results of test cases with thermally induced buoyancy demonstrate the stabilizing effect of the magnetic field on the recirculating flows.
This paper presents a convection–diffusion-reaction (CDR) model for solving magnetic induction equations and incompressible Navier–Stokes equations. For purposes of increasing the prediction accuracy, the general solution to the one-dimensional constant-coefficient CDR equation is employed. For purposes of extending this discrete formulation to two-dimensional analysis, the alternating direction implicit solution algorithm is applied. Numerical tests that are amenable to analytic solutions were performed in order to validate the proposed scheme. Results show good agreement with the analytic solutions and high rate of convergence. Like many magnetohydrodynamic studies, the Hartmann–Poiseuille problem is considered as a benchmark test to validate the code. Copyright © 2004 John Wiley & Sons, Ltd.
A new analytic finite element method (AFEM) is proposed for solving the governing equations of steady magnetohydrodynamic (MHD) duct flows. By the AFEM code one is able to calculate the flow field, the induced magnetic field, and the first partial derivatives of these fields. The process of the code generation is rather lengthy and complicated, therefore, to save space, the actual formulation is presented only for rectangular ducts. A distinguished feature of the AFEM code is the resolving capability of the high gradients near the walls without use of local mesh refinement. Results of traditional FEM, AFEM and finite difference method (FDM) are compared with analytic results demonstrating the manifest superiority of the AFEM code. The programs for the AFEM codes are implemented in GAUSS using traditional computer arithmetic and work in the range of low and moderate Hartmann numbersM<1000.
The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with
one conducting and one insulating pair of opposite walls under an external magnetic field parallel to the conducting walls,
is investigated. The MHD equations are coupled in terms of velocity and magnetic field and cannot be decoupled with conducting
wall boundary conditions since then boundary conditions are coupled and involve an unknown function. The boundary element
method (BEM) is applied here by using a fundamental solution which enables to treat the MHD equations in coupled form with
the most general form of wall conductivities. Also, with this fundamental solution it is possible to obtain BEM solution for
values of Hartmann number (M) up to 300 which was not available before. The equivelocity and induced magnetic field contours which show the well-known
characteristics of MHD duct flow are presented for several values of M.
In this paper, the meshless method is introduced to
magnetohydrodynamics. A numerical scheme based on the element-free
Galerkin method is used to solve the laminar steady-state
two-dimensional fully developed magnetohydrodynamic flow in a
rectangular duct. Accurate and convergent solutions are achieved for low
to moderately high Hartmann numbers
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.
The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.
In this article a meshless local Petrov–Galerkin (MLPG) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular section having arbitrary conducting walls. Computations have been carried out for different Hartmann numbers and wall conductivity at various time levels. The method is based on the local weak form and the moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally numerical results are presented showing the behaviour of velocity and induced magnetic field across the section.
A class of high-order compact (HOC) exponential finite difference (FD) methods is proposed for solving one- and two-dimensional steady-state convection–diffusion problems. The newly proposed HOC exponential FD schemes have nonoscillation property and yield high accuracy approximation solution as well as are suitable for convection-dominated problems. The O(h4) compact exponential FD schemes developed for the one-dimensional (1D) problems produce diagonally dominant tri-diagonal system of equations which can be solved by applying the tridiagonal Thomas algorithm. For the two-dimensional (2D) problems, O(h4 + k4) compact exponential FD schemes are formulated on the nine-point 2D stencil and the line iterative approach with alternating direction implicit (ADI) procedure enables us to deal with diagonally dominant tridiagonal matrix equations which can be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. To validate the present HOC exponential FD methods, three linear and nonlinear problems, mostly with boundary or internal layers where sharp gradients may appear due to high Peclet or Reynolds numbers, are numerically solved. Comparisons are made between analytical solutions and numerical results for the currently proposed HOC exponential FD methods and some previously published HOC methods. The present HOC exponential FD methods produce excellent results for all test problems. It is shown that, besides including the excellent performances in computational accuracy, efficiency and stability, the present method has the advantage of better scale resolution. The method developed in this article is easy to implement and has been applied to obtain the numerical solutions of the lid driven cavity flow problem governed by the 2D incompressible Navier–Stokes equations using the stream function-vorticity formulation.
This paper presents a finite element method for the solution of 3D incompressible magnetohydrodynamic (MHD) flows. Two important issues are thoroughly addressed. First, appropriate formulations for the magnetic governing equations and the corresponding weak variational forms are discussed. The selected formulation is conservative in the sense that the local divergence-free condition of the magnetic field is accounted for in the variational sense. A Galerkin-least-squares variational formulation is used allowing equal-order approximations for all unknowns. In the second issue, a solution algorithm is developed for the solution of the coupled problem which is valid for both high and low magnetic Reynolds numbers. Several numerical benchmark tests are carried out to assess the stability and accuracy of the finite element method and to test the behavior of the solution algorithm.
In this paper we devise a stabilized least-squares finite element method using the residual-free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow. We convert the original system of second-order partial differential equations into a first-order system formulation by introducing two additional variables. Then the least-squares finite element method using C0 linear elements enriched with the residual-free bubble functions for all unknowns is applied to obtain approximations to the first-order system. The most advantageous features of this approach are that the resulting linear system is symmetric and positive definite, and it is capable of resolving high gradients near the layer regions without refining the mesh. Thus, this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the Hartmann number. Before incoorperating the bubble functions into the global problem, we apply the Galerkin least-squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements. Therefore, we indeed introduce a two-level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual-free bubble functions. Numerical results confirming theoretical findings are presented for several examples including the Shercliff problem.
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.
The polynomial based differential quadrature and the Fourier expansion based differential quadrature method are applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of a transverse external oblique magnetic field. Numerical solution for velocity and induced magnetic field is obtained for the steady-state, fully developed, incompressible flow of a conducting fluid inside of the duct. Equal and unequal grid point discretizations are both used in the domain and it is found that the polynomial based differential quadrature method with a reasonable number of unequally spaced grid points gives accurate numerical solution of the MHD flow problem. Some graphs are presented showing the behaviours of the velocity and the induced magnetic field for several values of Hartmann number, number of grid points and the direction of the applied magnetic field.
The element-free Galerkin method (EFGM) is a very attractive technique for solutions of partial differential equations, since it makes use of nodal point configurations which do not require a mesh. Therefore, it differs from FEM-like approaches by avoiding the need of meshing, a very demanding task for complicated geometry problems. However, the imposition of boundary conditions is not straightforward, since the EFGM is based on moving-least-squares (MLS) approximations which are not necessarily interpolants. This feature requires, for instance, the introduction of modified functionals with additional unknown parameters such as Lagrange multipliers, a serious drawback which leads to poor conditionings of the matrix equations. In this paper, an interpolatory formulation for MLS approximants is presented: it allows the direct introduction of boundary conditions, reducing the processing time and improving the condition numbers. The formulation is applied to the study of two-dimensional magnetohydrodynamic flow problems, and the computed results confirm the accuracy and correctness of the proposed formulation.
We conduct experimental study on numerical solution of the two dimensional convection–diffusion equation discretized by three finite difference schemes: the traditional central difference scheme, the standard upwind scheme and the fourth-order compact scheme. We study the computed accuracy achievable by each scheme, the algebraic properties of the coefficient matrices arising from different schemes and the performance of the Gauss–Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from these schemes.
A stabilized finite element method using the residual-free bubble functions (RFB) is proposed for solving the governing equations of steady magnetohydrodynamic duct flow. A distinguished feature of the RFB method is the resolving capability of high gradients near the layer regions without refining mesh. We show that the RFB method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the Hartmann number. Numerical results confirming theoretical findings are presented for several configurations of interest. The approximate solution obtained by the RFB method is also compared with the analytical solution of Shercliff’s problem.
In this paper, we extend a previous work on a compact scheme for the steady NavierStokes equations (Li, Tang, and Fornberg (1995), Int. J. Numer. Methods Fluids, 20, 11371151) to the unsteady case. By exploiting the coupling relation between the streamfunction and vorticity equations, the NavierStokes equa- tions are discretized in space within a 3_3 stencil such that a fourth order accuracy is achieved. The time derivatives are discretized in such a way as to maintain the compactness of the stencil. We explore several known time-stepping approaches including second-order BDF method, fourth-order BDF method and the CrankNicolson method. Numerical solutions are obtained for the driven cavity problem and are compared with solutions available in the litera- ture. For large values of the Reynolds number, it is found that high-order time discretizations outperform the low-order ones.
In this paper, we develop two new upwind difference schemes for solving a coupled system of convection–diffusion equations arising from the steady incompressible MHD duct flow problem with a transverse magnetic field at high Hartmann numbers. Such an MHD duct flow is convection-dominated and its solution may exhibit localized phenomena such as boundary layers, namely, narrow boundary regions where the solution changes rapidly. Most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy. In contrast, the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability, and they are capable of resolving high gradients near the layer regions without refining the grid. The accuracy of the first new upwind scheme is O(h+k) and the second one improves the accuracy to O(ε2(h+k)+ε(h2+k2)+(h3+k3)), where 0