Incompressible Magnetohydrodynamics Research Articles

Numerical solution for the electrically conducting hybrid nanofluid flow between two parallel rotating surfaces subject to thermal radiation

In the current analysis, the steady and incompressible magnetohydrodynamics hybrid nanofluid (hnf) flow across two spinning permeable surfaces is studied. The hybrid nanoliquid has been examined under the additional effects of heat source, magnetic field, and Arrhenius activation energy. The hnf is synthesized by the dispersion of silicon dioxide and molybdenum disulfide nanoparticles in ethylene glycol. The flow scenario has been communicated in the form of a system of nonlinear Partial Differential Equations (PDEs), which are degraded and dimensionless to a set of nonlinear ordinary differential equations through appropriate similarity replacement. For the solution of reduced first-order differential equations, a numerical technique is employed. The consequences of physical flow parameters on energy, mass, and velocity profiles are shown through figures. It has been observed that the radial velocity profile increases with the influence of the suction factor and decreases with the effect of surface stretching and the rotation factor. The heat transmission rate increases with the impact of the Reynolds number.

3D MHD analysis of prototypical manifold for liquid metal blankets

The water-cooled lead lithium (WCLL) and the dual-cooled lead lithium (DCLL) are two of the breeding blanket (BB) concepts that the EUROfusion consortium is pursuing in the framework of the development of the fusion reactor industrial demonstrator DEMO. Both involve the use of a liquid metal (LM) as working fluid, the lead-lithium eutectic alloy (PbLi), due to its excellent thermal properties and the possibility to serve as both the blanket coolant and tritium breeder and carrier. Unfortunately, due to the high electrical conductivity of LMs, their motion is influenced by the magnetic field used in the reactor to confine the plasma, generating a complex phenomenology which is studied by magnetohydrodynamics (MHD). In this work, a representative prototypical manifold of a BB bottom feeder is investigated for different configurations with the custom phiFoam solver, capable of simulating unsteady, incompressible and isothermal MHD flow. The aim of this study is to investigate which configuration minimizes the flow imbalance in the manifold for the WCLL or in the poloidal breeding zone channels for the DCLL. The distribution of the flow rate among the channels is strongly influenced by the position of the feeding pipe (FP) and by the development of the MHD internal layer near the expansion, which generates important jets close to the lower plate and the upper one, where the channels are attached. The channel aligned with the FP is the one carrying most of the flow, from 55% to 82%, while in the more distant one the flow is almost stagnant, carrying from 17% to 6% of the total flow rate. The total pressure loss is also estimated and its functional dependence on the manifold configuration is discussed.

An effective second-order scheme for the nonstationary incompressible magnetohydrodynamics equations

We are proposing the Crank-Nicolson/Adams-Bashforth (CN/AB) algorithm for the nonstationary magnetohydrodynamic (MHD) equations. For time discretization, the time derivative terms are approximated by a first-order Euler-backward scheme, an implicit second-order Crank-Nicolson scheme acting on linear terms, and an explicit Adams-Bashforth scheme dealing with nonlinear terms. We use the finite element method for spatial discretization. Theoretical studies show that the proposed algorithm is almost unconditionally stabilized with τ≤c. The optimal error estimate of the algorithm is obtained by a parabolic argument mathematical trick. Finally, the theoretical results are verified by numerical experiments.

Electromagnetic electron Kelvin–Helmholtz instability

On electron kinetic scales, ions and electrons decouple, and electron velocity shear on electron inertial length ∼de can trigger electromagnetic (EM) electron Kelvin–Helmholtz instability (EKHI). In this paper, we present an analytic study of EM EKHI in an inviscid collisionless plasma with a step-function electron shear flow. We show that in incompressible collisionless plasma, the ideal electron frozen-in condition E+ve×B/c=0 must be broken for the EM EKHI to occur. In a step-function electron shear flow, the ideal electron frozen-in condition is replaced by magnetic flux conservation, i.e., ∇×(E+ve×B/c)=0, resulting in a dispersion relation similar to that of the standard ideal and incompressible magnetohydrodynamics KHI. The magnetic field parallel to the electron streaming suppresses the EM EKHI due to magnetic tension. The threshold for the EM mode of the EKHI is (k·ΔUe)2>ne1+ne2ne1ne2[ne1(vAe1·k)2+ne2(vAe2·k)2], where vAe=B/(4πmene)1/2, ΔUe, and ne are the electron streaming velocity shear and densities, respectively. The growth rate of the EM mode is γem∼Ωce, which is the electron gyro-frequency.

An iterative method for the thermally coupled incompressible magnetohydrodynamics equations at high parameter

We design an iterative method based on linearization approach for the thermally coupled stationary incompressible magnetohydrodynamics equations at high physical parameters. The iterative method includes solving the considered equations by the linearization approach and applying the error correction strategy to reduce the iterative error arising from the linearization approach for solving the nonlinear equations. The iterative method not only can improve computational capability for solving the considered equations at high physical parameters but also can save computational time and iterative step. Moreover, we prove the stability of the iterative method and derive the iterative error estimates. Finally, some numerical tests are implemented to illustrate the efficiency of the presented method.

High order asymptotic preserving finite difference WENO schemes with constrained transport for MHD equations in all sonic Mach numbers

In this paper, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number ε ranging from 0 to O(1). High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially non-oscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit.

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

AbstractIn this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in and ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.

A regularity result for the incompressible inviscid magnetohydrodynamics equations in the Arbitrary Lagrangian-Eulerian coordinates

Consider herein the incompressible inviscid magnetohydrodynamics equations in a moving domain with free surface boundary conditions. Without smallness assumption in the fluid volume, a priori estimates for solutions of this model are established when the initial data in H2.5+δ (δ>0) upon the Arbitrary Lagrangian-Eulerian coordinates under the boundary condition p+12|B|2=0. Indeed, this is achieved by reformulating the system into a new formulation with the Arbitrary Lagrangian-Eulerian variables, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the curl and divergence estimates.

The virtual element method for a 2D incompressible MHD system

We present a novel discretization for the two-dimensional incompressible Magnetohydrodynamics (MHD) system coupling an electromagnetic model and a fluid flow model. Our approach follows the framework of the Virtual Element Method and offers two main advantages. The method can be implemented on unstructured meshes making it highly versatile and capable of handling a broad set of problems involving interfaces, free-boundaries, or adaptive refinements of the mesh. The second advantage concerns the divergence of the magnetic flux field and the fluid velocity. Our approach guarantees that the numerical approximation of the magnetic flux field and the fluid velocity are divergence free if their initial states are divergence free. Importantly, the divergence-free condition for the fluid velocity is satisfied in a pointwise sense. We include a theoretical proof of the condition on the magnetic flux field, energy estimates and a well-posedness study. Numerical testing confirms robustness of the method and its convergence properties on a variety of meshes.

Turbulence, Intermittency, and Cross-Scale Energy Transfer in an Interplanetary Coronal Mass Ejection

Solar wind measurements carried out by NASA’s Wind spacecraft before, during, and after the passing of an interplanetary coronal mass ejection (ICME), detected on 12 – 14 September 2014, have been used in order to examine several properties of magnetohydrodynamic (MHD) turbulence. Spectral indices and flatness scaling exponents of magnetic field, velocity, and proton density measurements were obtained, and provided a standard description of the characteristics of turbulence within different sub-regions of the ICME and its surroundings. This analysis was followed by the validation of the third-order moment scaling law for isotropic, incompressible MHD turbulence in the same sub-regions, which confirmed the fully developed nature of turbulence in the ICME plasma. The energy transfer rate was also estimated in each ICME sub-region and in the surrounding solar wind. An exceptionally high value was found within the ICME sheath, accompanied by enhanced intermittency, possibly related to the powerful energy injection associated with the arrival of the ICME.

FERMI: Fusion Energy Reactor Models Integrator

The Fusion Energy Reactor Models Integrator (FERMI) is an integrated simulation environment under development for the coupled simulation of the plasma, first wall, and blanket of fusion reactor designs. The FERMI goals are to shorten the overall design cycle while guaranteeing unprecedented accuracy, thus integrating fusion design activities, facilitating an optimal reactor design, and reducing development risks. These goals are achieved by coupling single-physics solvers into a multiphysics simulation environment (FERMI). The Integrated Plasma Simulator (IPS)–FASt TRANsport (IPS-FASTRAN) simulation framework is used for the following: plasma physics, MCNP/Shift codes for neutron and photon transport, OpenFoam for computational fluid dynamics and magnetohydrodynamics (MHD), HyPerComp Incompressible MHD solver for Arbitrary Geometry (HIMAG) for dual-coolant lead-lithium (DCLL) blankets, and DIABLO for structural mechanics simulations. These codes are coupled using the open-source library named precise Code Interaction Coupling Environment (preCICE). FERMI’s features are tested with the analysis of the liquid immersion blanket (LIB) [proposed in the Affordable Robust Compact (ARC)–class tokamak design], the DCLL blanket [proposed in the Fusion Pilot Plant (FPP) design], and other benchmark cases. The calculated figures of merit are the tritium breeding ratio, material activation, displacements per atom, shutdown dose rate, heat deposited in the vacuum vessel and blanket, temperature hot spots, and displacements caused by swelling and creep. A critical technical problem is multiphysics code coupling, which is tackled here, and the first three-dimensional (3D) simulations of the DCLL-FPP and LIB-ARC blankets are presented. To the authors’ knowledge, FERMI represents the first effort to perform 3D simulations of nuclear fusion first wall and blankets in a fully coupled multiphysics manner.

A hybrid high-order scheme for the stationary, incompressible magnetohydrodynamics equations

Abstract We propose and analyse a hybrid high-order scheme for the stationary incompressible magnetohydrodynamics equations. The scheme has an arbitrary order of accuracy and is applicable on generic polyhedral meshes. For sources that are small enough, we prove error estimates in energy norm for the velocity and magnetic field, and $L^2$-norm for the pressure; these estimates are fully robust with respect to small faces, and of optimal order with respect to the mesh size. Using compactness techniques, we also prove that the scheme converges to a solution of the continuous problem, irrespective of the source being small or large. Finally, we illustrate our theoretical results through 3D numerical tests on tetrahedral and Voronoi mesh families.

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.

Stability and sharp decay for 3D incompressible MHD system with fractional horizontal dissipation and magnetic diffusion

This paper aims as the stability and large-time behavior of 3D incompressible magnetohydrodynamic (MHD) equations with fractional horizontal dissipation and magnetic diffusion. By using the energy methods, we obtain that if the initial data are small enough in $$H^3(\mathbb {R}^3)$$ , then this system possesses a global solution, and whose horizontal derivatives decay at least at the rate of $$(1+t)^{-\frac{1}{2}}$$ . Moreover, if we control the initial data further small in $$H^3(\mathbb {R}^3)\cap H_h^{-1}(\mathbb {R}^3)$$ , the sharp decay of this solution and its first-order derivatives is established.

Coronal Loop Heating by Nearly Incompressible Magnetohydrodynamic and Reduced Magnetohydrodynamic Turbulence Models

The transport of waves and turbulence beyond the photosphere is central to the coronal heating problem. Turbulence in the quiet solar corona has been modeled on the basis of the nearly incompressible magnetohydrodynamic (NI MHD) theory to describe the transport of low-frequency turbulence in open magnetic field regions. It describes the evolution of the coupled majority quasi-2D and minority slab component, driven by the magnetic carpet and advected by a subsonic, sub-Alfvénic flow from the lower corona. In this paper, we couple the NI MHD turbulence transport model with an MHD model of the solar corona to study the heating problem in a coronal loop. In a realistic benchmark coronal loop problem, we find that a loop can be heated to ∼1.5 million K by transport and dissipation of MHD turbulence described by the NI MHD model. We also find that the majority 2D component is as important as the minority slab component in the heating of the coronal loop. We compare our coupled MHD/NI MHD model results with a reduced MHD (RMHD) model. An important distinction between these models is that RMHD solves for small-scale velocity and magnetic field fluctuations and obtains the actual viscous/resistive dissipation associated with their evolution whereas NI MHD evolves scalar moments of the fluctuating velocity and magnetic fields and approximates dissipation using an MHD turbulence phenomenology. Despite the basic differences between the models, their simulation results match remarkably well, yielding almost identical heating rates inside the corona.

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.

New analysis of mixed FEMs for dynamical incompressible magnetohydrodynamics

New analysis of mixed FEMs for dynamical incompressible magnetohydrodynamics

Three interior penalty DG methods for stationary incompressible magnetohydrodynamics

In this paper, we propose and analyze three interior penalty DG methods for the stationary incompressible magnetohydrodynamics (MHD) equations. We use div-conforming Brezzi–Douglas–Marini (BDM) elements to discretize the velocity field and therefore the approximation of the velocity field is exactly divergence-free. The magnetic field is approximated by curl-conforming Nédélec elements. In the mixed formulation, the zero mean-valued constraint of the pressure is enforced via a Lagrange multiplier technique. The stability and convergence of the proposed methods are proved in a general Lipschitz domain. Finally, a series of numerical experiments are presented to verify the theoretical results.

A multilevel block preconditioner for the HDG trace system applied to incompressible resistive MHD

We present a scalable block preconditioning strategy for the trace system arising from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block for which a system algebraic multigrid (AMG) is used to approximate the inverse. The complexity of the MHD equations together with the algebraic nature of the statically condensed HDG trace system makes the choice of smoother in the system AMG part critical for the convergence and performance of the block preconditioner. Our numerical experiments show GMRES preconditioned by ILU(0) of overlap zero as a smoother inside system AMG performs best in terms of robustness, time per nonlinear iteration and memory requirements. With several transient test cases in 2D and 3D including the island coalescence problem at high Lundquist number we demonstrate the robustness and parallel scalability of the block preconditioner. Additionally for the upper block a preliminary study of an alternate nodal block system solver based on a multilevel approximate nested dissection is presented. On a 2D island coalescence problem the multilevel approximate nested dissection preconditioner shows better scalability with respect to mesh refinement than the system AMG, but is relatively less robust with respect to Lundquist number scaling. In the Appendix B, we rigorously show: (1) the uniqueness of the solution of the nonlinear MHD system for small time, (2) the convergence of the Picard iterations for each backward Euler step, and (3) the convergence of the entire time stepping procedure.

Analysis of a new mixed FEM for stationary incompressible magneto-hydrodynamics

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided.