Magnetohydrodynamic Problem Research Articles

Gauge–Uzawa-based, highly efficient decoupled schemes for the diffuse interface model of two-phase magnetohydrodynamic

In this paper, we design two totally decoupled, linear and unconditionally energy stable schemes with first-order time accuracy for the multi-physics diffuse interface model of two-phase magnetohydrodynamic (MHD) problem. These schemes combine the semi-implicit stabilization method/invariant energy quadratization (IEQ) method for the phase-field equations, Gauge–Uzawa method for the MHD equations, with some subtle implicit–explicit treatments for nonlinear coupled terms. Then this strong nonlinear and coupled complex system is split into a series of small linear elliptic equations, which makes the calculation more efficient. Additionally, compared to the standard projection method, the given schemes can overcome the selection of the initial value about pressure p and the treatment of artificial boundary condition on p which can lead to boundary layers and lose accuracy. Finally, rigorous proof of unconditional energy stability and several simulations of numerical experiment are conducted to demonstrate the validity of the decoupled schemes.

Exact operational matrices for rational Bernstein polynomials and its application for solving MHD problems

In this paper, rational Bernstein polynomials on the semi‐infinite interval are adapted to solve a magnetohydrodynamic (MHD) problem. Also, the derivative, product, convert, and Galerkin exact operational matrices (EOMs) of these polynomials are produced. Using the spectral Galerkin method and the exact operational matrices (EOMs) of rational Bernstein polynomials, we solve the problem with high accuracy and speed, and we estimate an exponential convergence rate in this method. The problem is the flow of MHD micropolar over a moving plate with suction and injection boundary conditions. Comparing the results of the rational Bernstein Galerkin method with operational matrices and without operational matrices shows that the present method is faster than another method. Also, comparing the results of the rational Bernstein Galerkin method and rational Gegenbauer tau method shows that the present method is more accurate than another method.

Eigenvalue Problem for a Reduced Dynamo Model in Thick Astrophysical Discs

Magnetic fields of different astrophysical objects are generated by the dynamo mechanism. Dynamo is based on the alpha-effect and differential rotation, which are described using a system of parabolic equations. Their solution is an important problem in magnetohydrodynamics and mathematical physics. They can be solved assuming exponential growth of the solution, which leads to an eigenvalue problem for a differential operator connected with spatial coordinates. Here, we describe a system of equations connected with the generation of magnetic field in discs, which are associated with galaxies and binary systems. For an ideal case of an infinitely thin disc, the eigenvalue problem can be precisely solved. If we take into account the finite thickness of the disc, the problem becomes more difficult. The solution can be found using asymptotical methods based on perturbations of the eigenvalues. Here, we present two different models which describe field evolution for different cases. For the first, we find eigenvalues taking into account linear and quadratic terms for the perturbations in the eigenvalue problem. For the second, we find eigenvalues using only linear terms; this is quite sufficient. Results were verified through numerical modeling, and basic computational tests show proper correspondence between different methods.

FreeFEM++ based heat transfer analysis of an electrically induced magnetic flow within the framework of micropolar continuum

In this article, we aim to analyze electrically induced magnetic incompressible flow and heat transfer through a channel within the framework of micropolar continuum. Finite element method is employed to compute the numerical solution of the governing dynamics formulated in the form of PDEs along with associated boundary conditions on the channel. The weak form is presented and variational problem is implemented in FreeFem++ language. To validate, the correct implementation of the problem and obtained numerical results, a reduced micropolar magnetohydrodynamic model is taken into consideration. The analytical solution is presented. Computed numerical results are compared with the exact solution in case of reduced micropolar model and a good agreement is achieved. An excellent convergence of the numerical solution to the exact solution is shown through Tables where L 2 -norm and H 1 -errors are computed at refined meshes. Effect of different physical parameters such as magnetic Reynolds number ( R m ), Hartmann number (Ha), micorpolar constants (m and l 1 2 ), micropolar coupling number (N), Prandtl number (Pr) and Brinkmann number (Br) are studied and discussed in detail. The development of thermal, translational and micro-rotational velocity profiles over the cross-sectional domain of the channel is shown for varying values of these material parameters. Some interesting and new findings in this investigation are presented and discussed. It is observed that the maximum temperature in the medium shifts from the center of the domain toward the boundaries if the micropolar coupling constant N is increased. Maximum micro-rotations are always found near the channel’s boundary where it is evidenced that the particle’s micro-rotations have a direct proportionality with the micropolar constant m. Moreover, the counter rotations of the continuum particles are evidenced at the channel’s boundary. Furthermore, the micorpolar coupling constant l 1 2 is found to resist the micro motions of the particles in the channel. The successful implementation of one and higher dimensional magnetohydrodynamic model in FreeFEM++ in general and within the context of micorpolar continuum in particular is delineated. This shows that it provides an efficient platform to simulate magnetohydrodynamic problems in higher order continuum which has applications in designing magneto-rheological airbags, aircraft take-off gear, mechanical heart valves and cooling systems.

Gas kinetic schemes for solving the magnetohydrodynamic equations with pressure anisotropy

In many astrophysical plasmas, the Coulomb collision is insufficient to maintain an isotropic temperature, and the system is driven to the anisotropic regime. In this case, magnetohydrodynamic (MHD) models with anisotropic pressure are needed to describe such a plasma system. To solve the anisotropic MHD equation numerically, we develop a robust Gas-Kinetic flux scheme for non-linear MHD flows. Using anisotropic velocity distribution functions, the numerical flux functions are derived for updating the macroscopic plasma variables. The schemes are suitable for finite-volume solvers which utilize a conservative form of the mass, momentum and total energy equations, and can be easily applied to multi-fluid problems and extended to more generalized double polytropic plasma systems. Test results show that the numerical scheme is very robust and performs well for both linear wave and non-linear MHD problems.

Thermal conductivity effects on mixed convection flow of electrically conducting fluid along vertical magnetized plate embedded in porous medium with convective boundary condition

The temperature dependent thermal conductivity, surface heat flux and magnetohydrodynamic effects on electromagnetic fluid across the magnetically vertical surface placed in porous material have been performed numerically. To reduce excessive heating, the aligned magnetic field acts like a coating material to insulate the heat which is a very important mechanism in modern technologies. To solve this problem, the primary goal of the current analysis is to magnetize the surface. The developed partial differential equations of the present magnetic mechanism are changed in nonlinear coupled ordinary differential equations with defined stream functions and similarity variables for smooth algorithm and integration. The changed ODEs are again converted in similar form for numerical outcomes by applying Keller Box approach. The numerical outcomes are deduced in graphs and tabular form with the help of MATLAB program. The distinct parameters like thermal-conductivity number ξ, Biot number Bi, and porous number Ω in the flow model are affected by physical features such skin friction, heat transfer, and magnetic flux for velocity profile, magnetic field profile, and temperature profile together with their slopes. After obtaining the velocity graph, magnetic-field graph, and temperature graph, their gradients are numerically examined across the magnetically vertical surface. The current thermal and magneto hydrodynamics problem is very important in the fields of MRI resonance sequences, artificial heart wolves, internal heart cavities, and nanoburning technologies.

Local well-posedness of the free-boundary problem in compressible resistive magnetohydrodynamics

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.

Two‐level methods based on the Arrow–Hurwicz iteration for the steady incompressible magnetohydrodynamic system

AbstractWe consider the two‐level methods based on Arrow–Hurwicz (A‐H) iteration for solving the stationary incompressible magnetohydrodynamics problem. The methods carry out some A‐H iterations on the coarse mesh, improve the obtained solutions by using one‐step scheme on the fine mesh. Such methods enjoy some advantages including that the number of A‐H iterations can be flexibly and controllably selected, and they are valid for high hydrodynamic Reynolds number. The convergence analyses are established and various numerical experiments are performed to validate and illustrate the stability, accuracy, and efficiency of the new methods.

Error Analysis of Fully Discrete Scheme for the Cahn–Hilliard–Magneto-Hydrodynamics Problem

In this paper we analyze a fully discrete scheme for a general Cahn–Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stable and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence.

A Fully Discrete Decoupled Finite Element Method for the Thermally Coupled Incompressible Magnetohydrodynamic Problem

A Fully Discrete Decoupled Finite Element Method for the Thermally Coupled Incompressible Magnetohydrodynamic Problem

MARPLE: программное обеспечение для мультифизического моделирования в задачах сплошных сред

The research code MARPLE was originally created to model high-speed dynamic processes caused by the action of high-intensity energy fluxes on matter. At present, it is a universal tool able to solve various continuum mechanics problems. The implemented physical models are the following: single-fluid two-temperature MHD model of plasma dynamics, including electron-ion energy exchange and generalized Ohm's law; model of electrical and thermal conductivity taking into account the anisotropy in the magnetic field; radiative heat transfer: models pertinent to optically thin as well as optically thick media: techniques for taking into account radiative cooling losses, spectral multigroup diffusion transfer, laser radiation propagation etc.; model of multicomponent flow. Calculations are performed using wide-range equations of state, transport and optical data. The MARPLE code utilises modern computational technologies based on block structured and unstructured meshes consisting of tetrahedral, hexahedral, prismatic elements and their combinations. The solvers implement conservation laws using high-resolution techniques. We apply the physical splitting to solve the governing system. The object-oriented approach to software design is used, as well as methods of object and generic programming (C++ implementation language). Design of computational domains is provided by means of integrated SALOME open source CAD-CAE system. Marple works as MPI application for modern HPC systems. The paper presents examples of problems in plasma dynamics, magnetohydrodynamics, astrophysics, and solid thermomechanics solved by means of the MARPLE code.

Double multiple-relaxation-time model of lattice-Boltzmann magnetohydrodynamics at low magnetic Reynolds numbers

We develop an improved lattice-Boltzmann numerical scheme to solve magnetohydrodynamic (MHD) equations in the regime of low magnetic Reynolds numbers, grounded on the central-moment (CM) and multi-relaxation-time (MRT) collision models. The simulation of the magnetic induction equation within the lattice-Boltzmann approach to MHD has been usually devised along the lines of the simplest phenomenological description—the single relaxation time (SRT) model to solve the complete induction equation. In order to deal with well-known stability difficulties of the SRT framework for larger magnetic relaxation time scales, we introduce, alternatively, a MRT technique for the solution of the magnetic induction equation, which proves to be efficient in extending the domain of applicability of the lattice-Boltzmann method to MHD problems. We also put forward a novel and practical boundary condition method to cope with the subtleties of magnetic Boltzmann-like distributions on curved boundaries. As supporting applications, we discuss the performance of the CM–MRT algorithm to describe the complex dynamics of the 3D Orszag–Tang vortex problem and open issues related to transient flow regimes in MHD pipe flows, subject to uniform and non-uniform magnetic fields.

Zero surface tension limit of the free-boundary problem in incompressible magnetohydrodynamics* *XG was supported in part by NSFC Grant 12031006. CL was supported in part by the RGC Grant of Hong Kong CUHK-24304621 and CUHK Direct Grant for Research No. 4053457.

We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh–Taylor sign condition holds initially. This result is a continuation of the authors’ previous works (Gu et al 2020 arXiv:2105.00596; Gu and Wang 2019 J. Math. Pures Appl. 128 1–41; Luo and Zhang 2021 SIAM J. Math. Anal. 53 2595–630). Our proof is based on the combination of the techniques developed in our previous works (Gu et al 2020 arXiv:2105.00596; Gu and Wang 2019 J. Math. Pures Appl. 128 1–41; Luo and Zhang 2021 SIAM J. Math. Anal. 53 2595–630), Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.

On Convergence of Support Operator Method Schemes for Differential Rotational Operations on Tetrahedral Meshes Applied to Magnetohydrodynamic Problems

The problem of constructing and justifying the discrete algorithms of the support operator method for numerical modeling of differential repeated rotational operations of vector analysis (curlcurl) in application to problems of magnetohydrodynamics is considered. Difference schemes of the support operator method on the unstructured meshes do not approximate equations in the local sense. Therefore, it is necessary to prove the convergence of these schemes to the exact solution, which is possible after analyzing the error structure of their approximation. For this analysis, a decomposition of the space of mesh vector functions into an orthogonal direct sum of subspaces of potential and vortex fields is introduced. Generalized centroid-tensor metric representations of repeated operations of tensor analysis (div, grad, and curl) are constructed. Representations have flux-circulation properties that are integrally consistent on spatial meshes of irregular structure. On smooth solutions of the model magnetostatic problem on a tetrahedral mesh with the first order of accuracy in the rms sense, the convergence of the constructed difference schemes is proved. The algorithms constructed in this work can be used to solve physical problems with discontinuous magnetic viscosity, dielectric permittivity, or thermal resistance of the medium.

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

In this paper, we develop and analyze a fully discrete mixed finite element method for unsteady inductionless magnetohydrodynamics problem. An Euler semi-implicit scheme is proposed with a mixed variational formulation based on the variables (u,p,j,ϕ), in which the Navier–Stokes equations are approximated by stable finite elements and the current density is discretized by the divergence-conforming finite element. This scheme has the feature that the discrete current density keeps charge conservation property. It is shown that both the continuous problem and its fully discrete Euler semi-implicit scheme are well-posed. We prove unconditionally convergence and error estimates for velocity, pressure and current density. Finally, numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

Convergence analysis of a conservative finite element scheme for the thermally coupled incompressible inductionless MHD problem

In this paper, a stable mixed finite element method for unsteady inductionless magnetohydrodynamics (MHD) problem coupled heat equation is devised. We first propose a mixed variational formulation based on all the variables, in which the thermal equation and Navier-Stokes equations are approximated by mini finite element method and the current density is discretized by the divergence-conforming elements. The novel feature of this scheme is that the discrete current density keeps charge conservation property. It is shown that the fully discrete first order Euler semi-implicit scheme is well-posed and unconditionally stable. The existence and uniqueness of the weak solutions for continuous problem is established by a numerical version analysis. Furthermore, under the low regularity hypothesis for the exact solutions, we prove the optimal error estimates of the velocity, current density and electric potential. Finally, some numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

An MHD Marangoni Boundary Layer Flow and Heat Transfer with Mass Transpiration and Radiation: An Analytical Study

This examination is carried out on the two-dimensional magnetohydrodynamic problem for a steady incompressible flow over a porous medium. The Cu−Al2O3 nanoparticles are added to the water base fluid in order to improve thermal efficiency. The transverse magnetic field with strength B0 is applied. The governing equations formed for the defined flow form a system of partial differential equations that are then converted to a system of ordinary differential equations upon applying the suitable similarity transformations. On analytically solving the obtained system, the solutions for velocity profile and temperature distribution are obtained in terms of exponential and Gamma functions, respectively. In addition, the physical parameter of interest, the local Nusselt number, is obtained. The results are analyzed through plotting graphs, and the effect of different parameters is analyzed. Furthermore, we observe that the suction/injection parameter enhances the axial velocity. The porous and radiation parameters enhance the temperature distribution, and the suction/injection parameter suppresses the temperature distribution.

Solution of Steady Incompressible MHD Problems with Quasi-Least Square Method

A quasi-least-squares (QLS) mixed finite element method (MFE) based on the L2-inner product is utilized to solve an incompressible magnetohydrodynamic (MHD) model. These models are associated with the three unknown terms, i.e., fluid velocity, fluid pressure, and magnetic field. For the MHD-based models, common theories and algorithms for approximation of the solutions are not always applicable because of the choice of the functional spaces during the utilization of the weak formulation. It is well known that the spaces used for the approximation of the different unknowns, e.g., the spaces for the unknowns, cannot be chosen independently for the variational formulation, and may have to satisfy strict stability conditions such as the inf-sup, or Ladyzhenskaya–Babuska–Brezzi (LBB) condition. The dependency of the selection of the spaces for the unknowns are critical and always not applicable for some pair of unknowns. Because of this, the numerical or theoretical solutions must have to face some stability issue. The proposed scheme (L2-inner product) is introduced to circumvent this deficiency of the conditions (inf-sup or LBB) and obtained a well-posed solution theoretically. The model equations are nonlinear and highly coupled with the combination of Navier–Stokes and Maxwell relations. First, these nonlinear models are made linear around a specific state wherein the modified system represents an algebraic equation in a first-order symmetric form. Secondly, a direct iteration technique is applied to solve the nonlinearities and obtain a theoretical convergent rate for a general initial guess. Theoretical results show that only a single parameter with a single initial guess is sufficient to establish the well-posedness of the solution.

An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.

SCELT (symbolic computation aided eigenvalue and linear code for tokamaks): A full MHD eigenvalue code in toroidal geometry developed with the use of a symbolic computation technique

In this work, we report the construction of an eigenvalue computer code largely in C++ language, SCELT, by using the symbolic computation technique for the first time to solve the linearized single fluid magnetohydrodynamic (MHD) eigenvalue problem in toroidal geometry. A symbolic vector analysis module is developed to function the automatic derivation of the tedious linearized full MHD equations in the magnetic flux coordinate system. Furthermore, another module is developed to implement the automatic numerical discretization. These two modules dramatically reduce the human workload and obviate the possibility of a mistake during code development. The tools provide a means of constructing matrices from differential operations and can be used for (generalized) linear problems, such as source driven and eigenvalue problems. Demo uses of both the symbolic vector analysis module and automatic numerical discretization module, such as the Poisson equation and tokamak equilibrium equation, are presented to demonstrate their advantages and potential broad applications. The full MHD eigenvalue code developed with these two modules is verified by the internal kink mode and tearing mode tests. Program summaryProgram Title: SCELT (Symbolic Computation aided Eigenvalue and Linear code for Tokamaks)CPC Library link to program files:https://doi.org/10.17632/35h3xmc28k.1Licensing provisions: GNU General Public License 3.0Programming language: C++, MATLABExternal routines/libraries:• EIGEN (https://eigen.tuxfamily.org/index.php?title=Main_Page)Nature of problem: The eigenvalue full MHD model determines a wide range of physical phenomena arising from the basic destabilizing forces, which are of considerable both fundamental and practical importance for magnetic fusion plasmas. From the viewpoint of physics, solving the model is equivalent to finding the spectrum of the system, and for realistic tokamak geometry, it becomes possible only with the help of computer simulation. From the viewpoint of mathematics, it is actually a problem of solving a set of partial differential equations in a general curvilinear coordinate system.Solution method: Different from the conventional method, a symbolic vector analysis module using the symbolic computation technique is developed to conduct the automatic expansion of the full MHD equations in the magnetic flux coordinate system. In addition, an automatic numerical discretization module is developed to automatically implement the numerical discretization. Facilitated by the two modules, the coefficients of the matrices of the eigenvalue equation will be automatically generated and passed to the linear eigenvalue solver (eig function of MATLAB). The eigenvalues and eigenvectors are resolved by the solver.Additional comments including restrictions and unusual features: As originally motivated by the development of an eigenvalue full MHD code, a symbolic vector analysis module by using symbolic computation technique and a closely related automatic numerical discretization module are developed and directly applied to physics code development for the first time at least in the fusion field. Many (generalized) linear physics problems can benefit from the standalone or integrated application of these two modules. The tokamak equilibrium and eigenvalue MHD problems presented in the paper can be considered two important and practical examples.