Magnetohydrodynamic Problem Research Articles

Error estimate of a fully discrete defect correction finite element method for unsteady incompressible Magnetohydrodynamics equations

ABSTRACTIn this study, a fully discrete defect correction finite element method for the unsteady incompressible Magnetohydrodynamics (MHD) equations, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. It is a continuous work of our formal paper [Math Method Appl Sci. 2017. DOI:10.1002/mma.4296]. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear MHD equation is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we introduce the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. Some numerical results [see Math Method Appl Sci. 2017. DOI:10.1002/mma.4296] show that this method is highly efficient for the unsteady incompressible MHD problems.

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L 2 and H 1 fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.

Time-decay solutions of the initial-boundary value problem of rotating magnetohydrodynamic fluids

We have investigated an initial-boundary problem for the perturbation equations of rotating, incompressible, and viscous magnetohydrodynamic (MHD) fluids with zero resistivity in a horizontally periodic domain. The velocity of the fluid in the domain is non-slip on both upper and lower flat boundaries. We switch the analysis of the initial-boundary problem from Euler coordinates to Lagrangian coordinates under proper initial data, and get a so-called transformed MHD problem. Then, we exploit the two-tiers energy method. We deduce the time-decay estimates for the transformed MHD problem which, together with a local well-posedness result, implies that there exists a unique time-decay solution to the transformed MHD problem. By an inverse transformation of coordinates, we also obtain the existence of a unique time-decay solution to the original initial-boundary problem with proper initial data.

Nonlinear thermal instability in the magnetohydrodynamics problem without heat conductivity

We investigate the nonlinear thermal instability of the magnetohydrodynamic problem for a full compressible viscous fluid with zero resistivity and zero heat conductivity in the presence of a uniform gravitational force in a bounded domain Ω∈R3. We establish that under some instability conditions, the equilibrium-state is linearly unstable by constructing a suitable energy functional and employing the modified variational method. Then, on the basis of the constructed linearly unstable solutions and the local well-posedness of classical solutions to the original nonlinear problem, we reconstruct the initial data of linearly unstable solutions to be the one of the original nonlinear problem and obtain an appropriate energy estimate of Gronwall-type. Finally, we establish that the equilibrium-state is nonlinearly unstable in view of the energy estimate by a bootstrap method.

CFD simulation of the magnetohydrodynamic flow inside the WCLL breeding blanket module

The interaction between the molten metal and the plasma-containing magnetic field in the breeding blanket causes the onset of a magnetohydrodynamic (MHD) flow. To properly design the blanket, it is important to quantify how and how much the flow features are modified compared with an ordinary hydrodynamic flow. This paper aims to characterize the evolution of the fluid inside one of the proposed concepts for DEMO, the Water-Cooled Lithium Lead (WCLL), focusing on the central cell of the equatorial outboard module. A preliminary validation was required to gauge the capability of ANSYS CFX to deal with MHD problems. The buoyant and pressure-driven fully developed laminar flows in a square duct were selected as benchmarks. Numerical results were compared with theoretical solutions and an excellent agreement was found. The channel analysis was realized on a simplified version of the latest available design geometry, developed by ENEA, for M≤1000. The simulation highlighted the formation of high velocity jets close to the baffle and the onset of an asymmetrical potential distribution.

Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

This article presents a local and parallel finite element method for the stationary incompressible magnetohydrodynamics problem. The key idea of this algorithm comes from the two‐grid discretization technique. Specifically, we solve the nonlinear system on a global coarse mesh, and then solve a series of linear problems on several subdomains in parallel. Furthermore, local a priori estimates are obtained on a general shape regular grid. The efficiency of the algorithm is also illustrated by some numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1513–1539, 2017

Almost sure existence of global weak solutions for incompressible MHD equations in negative-order Sobolev space

In this paper we consider the Cauchy problem of the incompressible Magnetohydrodynamics (MHD) equations in the periodic space domain TN, where N≥2. After a suitable randomization to the initial data, we obtain the almost sure existence of global weak solutions with the initial data in Hs(TN), s∈(−1,0). Specially, the global weak solution is unique when N=2.

An analytical solution to the heat transfer problem in thick-walled hunt flow

The flow of a liquid metal in a rectangular duct, subject to a strong transverse magnetic field is of interest in a number of applications. An important application of such flows is in the context of coolants in fusion reactors, where heat is transferred to a lead-lithium eutectic. It is vital, therefore, that the heat transfer mechanisms are understood. Forced convection heat transfer is strongly dependent on the flow profile. In the hydrodynamic case, Nusselt numbers and the like, have long been well characterised in duct geometries. In the case of liquid metals in strong magnetic fields (magnetohydrodynamics), the flow profiles are very different and one can expect a concomitant effect on convective heat transfer. For fully developed laminar flows, the magnetohydrodynamic problem can be characterised in terms of two coupled partial differential equations. The problem of heat transfer for perfectly electrically insulating boundaries (Shercliff case) has been studied previously (Bluck et al., 2015). In this paper, we demonstrate corresponding analytical solutions for the case of conducting hartmann walls of arbitrary thickness. The flow is very different from the Shercliff case, exhibiting jets near the side walls and core flow suppression which have profound effects on heat transfer.

Finite Element Solution for MHD Flow of Nanofluids with Heat and Mass Transfer through a Porous Media with Thermal Radiation, Viscous Dissipation and Chemical Reaction Effects

AbstractIn this paper, the problem of magnetohydrodynamics (MHD) boundary layer flow of nanofluid with heat and mass transfer through a porous media in the presence of thermal radiation, viscous dissipation and chemical reaction is studied. Three types of nanofluids, namely Copper (Cu)-water, Alumina (Al2O3)-water and Titanium Oxide (TiO2)-water are considered. The governing set of partial differential equations of the problem is reduced into the coupled nonlinear system of ordinary differential equations (ODEs) by means of similarity transformations. Finite element solution of the resulting system of nonlinear differential equations is obtained using continuous Galerkin-Petrov discretization together with the well-known shooting technique. The obtained results are validated using MATLAB “bvp4c” function and with the existing results in the literature. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and the impact of various physical parameters such as the magnetic parameterM, solid volume fraction of nanoparticles 𝜙 and type of nanofluid on the flow is discussed. The results obtained in this study confirm the idea that the finite element method (FEM) is a powerful mathematical technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

Preconditioners for Mixed Finite Element Discretizations of Incompressible MHD Equations

We consider preconditioning techniques for a mixed finite element discretization of an incompressible magnetohydrodynamics (MHD) problem. Upon discretization and linearization, a 4-by-4 nonsymmetric block-structured linear system needs to be (repeatedly) solved. One of the principal challenges is the presence of a skew-symmetric term that couples the fluid velocity with the magnetic field. We propose a preconditioner that exploits the block structure of the underlying linear system, utilizing and combining effective solvers for the mixed Maxwell and the Navier--Stokes subproblems. We perform a spectral analysis for an ideal version of the preconditioner, and develop and test a practical version of it. Large-scale numerical results for linear systems of dimensions up to $10^7$ in two and three dimensions validate the effectiveness of our approach.

On the performance of exponential integrators for problems in magnetohydrodynamics

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EPIRK-type are used to simulate magnetic reconnection and the Kevin–Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD.

Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers

This paper presents an anisotropic adaptive finite element method (FEM) to solve the governing equations of steady magnetohydrodynamic (MHD) duct flow. A residual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu estimates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.

Error analysis of a fractional-step method for magnetohydrodynamics equations

This paper focuses on a fractional-step finite element method for the magnetohydrodynamics problems in three-dimensional bounded domains. It is shown that the proposed fractional-step scheme allows for a discrete energy identity. A rigorous error analysis is presented. We derive the temporal and spatial error estimates of O(Δt+h) for the velocity and the magnetic field in the discrete space l2(H1)∩l∞(L2) under the constraint Δt≥Ch.

A reconstructed discontinuous Galerkin method for magnetohydrodynamics on arbitrary grids

A reconstructed discontinuous Galerkin (rDG) method, designed not only to enhance the accuracy of DG methods but also to ensure the nonlinear stability of the rDG method, is developed for solving the Magnetohydrodynamics (MHD) equations on arbitrary grids. In this rDG(P1P2) method, a quadratic polynomial solution (P2) is first obtained using a Hermite Weighted Essentially Non-oscillatory (WENO) reconstruction from the underlying linear polynomial (P1) discontinuous Galerkin solution to ensure linear stability of the rDG method and to improves efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the first derivatives in the DG formulation, the stencils used in reconstruction involve only Von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the nonlinear stability of the rDG method. The HLLD Riemann solver introduced in the literature for one-dimensional MHD problems is adopted in normal direction to compute numerical fluxes. The divergence free constraint is satisfied using the Locally Divergence Free (LDF) approach. The developed rDG method is used to compute a variety of 2D and 3D MHD problems on arbitrary grids to demonstrate its accuracy, robustness, and non-oscillatory property. Our numerical experiments indicate that the rDG(P1P2) method is able to capture shock waves sharply essentially without any spurious oscillations, and achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method.

First Studies for the Development of Computational Tools for the Design of Liquid Metal Electromagnetic Pumps

Abstract Liquid alloy systems have a high degree of thermal conductivity, far superior to ordinary nonmetallic liquids and inherent high densities and electrical conductivities. This results in the use of these materials for specific heat conducting and dissipation applications for the nuclear and space sectors. Uniquely, they can be used to conduct heat and electricity between nonmetallic and metallic surfaces. The motion of liquid metals in strong magnetic fields generally induces electric currents, which, while interacting with the magnetic field, produce electromagnetic forces. Electromagnetic pumps exploit the fact that liquid metals are conducting fluids capable of carrying currents, which is a source of electromagnetic fields useful for pumping and diagnostics. The coupling between the electromagnetics and thermo-fluid mechanical phenomena and the determination of its geometry and electrical configuration, gives rise to complex engineering magnetohydrodynamics problems. The development of tools to model, characterize, design, and build liquid metal thermo-magnetic systems for space, nuclear, and industrial applications are of primordial importance and represent a cross-cutting technology that can provide unique design and development capabilities as well as a better understanding of the physics behind the magneto-hydrodynamics of liquid metals. First studies for the development of computational tools for the design of liquid metal electromagnetic pumps are discussed.

Analytical Model of an Asymmetric Sunspot with a Steady Plasma Flow in its Penumbra

A new exact analytical solution to the stationary problem of ideal magnetohydrodynamics is derived for an unipolar asymmetric sunspot immersed in a realistic solar atmosphere. The radial and vertical profiles of pressure, plasma density, and temperature in the visible layers of the sunspot are calculated. The reduction in plasma density in the magnetic funnel of the sunspot, corresponding to the Wilson depression, is also obtained. The magnetic structure of the sunspot is given analytically in a realistic way: a part of the magnetic flux of the sunspot approaches the surrounding photosphere at the outer edge of the penumbra. The magnetic field of the sunspot is not assumed to be axially symmetric. For the first time, the angular dependence of the physical variables in this model allows us to simulate not only a deviation from the circular shape of the sunspot, but also a fine filamentary structure of the sunspot penumbra. The Alfven Mach number (the ratio of the plasma speed to the Alfven speed) is zero at the center of the sunspot and rises slowly toward the periphery of the sunspot; this corresponds to the structure of the Evershed flow in the penumbra. The Evershed flow in our model is mainly concentrated in dark penumbral filaments, as is observed.

Lie group symmetry method for MHD double-diffusive convection from a permeable vertical stretching/shrinking sheet

In this paper, the problem of the steady magnetohydrodynamics (MHD) double-diffusive convection of a viscous and electrically conducting fluid from a permeable vertical stretching/shrinking sheet with the presence of buoyancy force is investigated numerically. The effect of the MHD on the flow, heat transfer and species concentration has been also discussed. The partial differential equations are reduced to ordinary (similarity) equations using the method of the Lie group symmetry. Numerical solutions of the resulting nonlinear boundary value problem are carried out using the function bvp4c from Matlab for different values of the governing parameters. It is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the stretching/shrinking, buoyancy, suction and magnetic parameters. It is shown that the reported results are in excellent agreement with those from the open literature for a forced convection flow when the concentration gradient is absent and the stretching sheet is impermeable. In order to establish which of upper and lower solutions are stable and which are not, a stability analysis has been performed. Thus, it is found that the upper branch solution is stable and, therefore, physically realizable in practice, while the lower branch solution is not stable and, thus, physically not realizable. The effects of the governing parameters on the reduced skin friction coefficient, reduced Nusselt number and reduced Sherwood number are shown in tables and figures. It results in that the governing parameters affect considerably the flow, heat transfer and concentration characteristics. In our opinion, the results are new and original.

Magnetohydrodynamics Mixed Convection in a Lid-Driven Cavity Having a Corrugated Bottom Wall and Filled With a Non-Newtonian Power-Law Fluid Under the Influence of an Inclined Magnetic Field

In this study, the problem of magnetohydrodynamics (MHD) mixed convection of lid-driven cavity with a triangular-wave shaped corrugated bottom wall filled with a non-Newtonian power-law fluid is numerically studied. The bottom corrugated wall of the cavity is heated and the top moving wall is kept at a constant lower temperature while the vertical walls of the enclosure are considered to be adiabatic. The governing equations are solved by the Galerkin weighted residual finite element formulation. The influence of the Richardson number (between 0.01 and 100), Hartmann number (between 0 and 50), inclination angle of the magnetic field (between 0 deg and 90 deg), and the power-law index (between 0.6 and 1.4) on the fluid flow and heat transfer characteristics are numerically investigated. It is observed that the effects of free convection are more pronounced for a shear-thinning fluid and the buoyancy force is weaker for the dilatant fluid flow compared to that of the Newtonian fluid. The averaged heat transfer decreases with increasing values of the Richardson number and enhancement is more effective for a shear-thickening fluid. At the highest value of the Hartmann number, the averaged heat transfer is the lowest for a pseudoplastic fluid. As the inclination angle of the magnetic field increases, the averaged Nusselt number generally enhances.

Iterative penalty finite element methods for the steady incompressible magnetohydrodynamic problem

In the paper we consider the one-level and two-level iterative penalty finite element methods for the steady incompressible magnetohydrodynamic problem based on the iteration of pressure with a factor of penalty parameter. Firstly, the $$\mathbf {H}^1$$ and $$\mathbf {L}^2$$ error estimates of numerical solutions of one-level iterative penalty finite element method are provided. Secondly, the stability and convergence of two-level iterative penalty finite element method are analyzed. Finally, some numerical results are provided to verify the effectiveness of the developed numerical schemes.

Nodal-based finite element methods with local projection stabilization for linearized incompressible magnetohydrodynamics

We consider the linearized resistive magnetohydrodynamics (MHD) model in incompressible media and its numerical solution. Conforming nodal-based finite element approximations of the MHD model both for inf-sup stable and equal order finite elements with respect to velocity and pressure are considered. As opposed to a residual-based stabilization method by Badia et al. (2013), we consider a local projection stabilization for the numerical solution. A detailed stability and error analysis for the arising discrete problem is given. Some numerical experiments like Hartmann’s MHD problem and singular solutions are examined.