Incompressible Magnetohydrodynamics Research Articles

Some Exacts Solutions of Two Dimensional Unsteady Incompressible Magnetohydrodynamic Flow of Viscous Fluid by Ansatz Method

Some Exacts Solutions of Two Dimensional Unsteady Incompressible Magnetohydrodynamic Flow of Viscous Fluid by Ansatz Method

Global well-posedness for the 3D inhomogeneous incompressible magnetohydrodynamics system with axisymmetric data

We focus on the Cauchy problem of 3D inhomogeneous incompressible magnetohydrodynamics (MHD for short) system with axisymmetric initial data. It is shown that this MHD system has a unique global solution provided that ‖1r(1ρ0−1)‖L∞ is sufficiently small. Finally, we establish the L2 decay rate for any given globally smooth solution of the MHD system by using the Schonbek's Fourier splitting method.

A stabilized finite volume method based on the rotational pressure correction projection for the time-dependent incompressible MHD equations

This paper presents an efficient numerical scheme for solving the time-dependent incompressible Magnetohydrodynamics (MHD) equations based on the rotational pressure correction projection method within the finite volume framework. Utilizing the lowest equal-order mixed finite element pair (P1−P1−P1) to approximate the velocity, magnetic and pressure fields, our numerical scheme satisfies the discrete inf-sup condition by the pressure projection stabilization. To tackle the coupling inherent in the time-dependent incompressible MHD equations,the rotational pressure correction projection method is introduced to split the original problem into several linear subproblems. The unconditional stability of numerical schemes are provided,optimal error estimates in both L2 and H1-norms of numerical solutions are also presented.Finally, some numerical results are given to verify the established theoretical findings and show the performances of the considered numerical schemes.

Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform L∞ estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in L∞ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.

Accuracy and scalability of incompressible inductionless MHD codes applied to fusion technologies

Abstract It is well-known that magnetohydrodynamics (MHD) dominates the dynamic of the liquid metal flows inside the breeding blankets (BB) of future nuclear fusion plants by magnetic confinement. MHD is a multiphysics phenomenon involving both electromagnetism and incompressible fluid mechanics. From the computational point of view, the simulation of MHD flows in fusion relevant conditions entails a significant challenge. Indeed, due to the shape of the induced electrical currents inside the bulk of the fluid, high spatial resolutions are needed to capture the large gradients found in boundary layers and 3D effects. Besides, solving the equations accurately typically requires very small time steps for the transient algorithms. Over the past few decades, some parallel MHD codes have been developed with success to simulate complex flows in increasingly realistic geometries. Among them, the MHD tools of commercial CFD platforms have attracted attention due to their relatively soft learning curve. Most of these codes are based on the so called ϕ-formulation which, by applying the divergence free condition of the current density to the Ohms law, reduces the electromagnetic part of the problem to a single Poisson equation. As a downside, the approach segregates the fluid and electromagnetic problem. In practice, this establishes important limits to the mesh element size, to the mesh quality and to the time-step needed to obtain accurate and stable solutions that maintains charge conservation at a discrete level. In this work, these limits are explored for the commercial platform ANSYS-Fluent using a test geometry under different conditions. As an alternative, a new code based on Finite Element Methods (FEM) is introduced as well. This open-source code, called GridapMHD (https://github.com/gridapapps/GridapMHD.jl), aims at solving the full set of MHD equations using a monolithic approach. GridapMHD is still in early stages of development but it has already shown promising results.

Non-universality and dissipative anomaly in compressible magnetohydrodynamic turbulence

We systematically study the dissipative anomaly in compressible magnetohydrodynamic (MHD) turbulence using direct numerical simulations, and show that the total dissipation remains finite as viscosity diminishes. The dimensionless dissipation rate $\mathcal {C}_{\varepsilon }$ fits well with the model $\mathcal {C}_{\varepsilon } = \mathcal {C}_{\varepsilon,\infty } + \mathcal {D}/R_L^-$ for all levels of flow compressibility considered here, where $R_L^-$ is the generalized large-scale Reynolds number. The asymptotic value $\mathcal {C}_{\varepsilon,\infty }$ describes the total energy transfer flux, and decreases with increase of the flow compressibility, indicating non-universality of the dimensionless dissipation rate in compressible MHD turbulence. After introducing an empirically modified dissipation rate, the data from compressible cases collapse to a form similar to the incompressible MHD case depending only on the modified Reynolds number.

The Incompressible Magnetohydrodynamic Energy Cascade Rate Upstream of Mars: Effects of the Total Energy and the Cross-helicity on Solar Wind Turbulence

Solar wind turbulence is a dynamical phenomenon that evolves with heliocentric distance. Orbiting Mars since 2014 September, Mars Atmosphere and Volatile EvolutioN offers a unique opportunity to explore some of its main properties beyond ∼1.38 au. Here, we analyze solar wind turbulence upstream of Mars' bow shock, utilizing more than 5 years of magnetic field and plasma measurements. This analysis is based on two complementary methodologies: (1) the computation of magnetohydrodynamic (MHD) invariants characterizing incompressible fluctuations; (2) the estimation of the incompressible energy cascade rate at MHD scales (i.e., 〈ε T 〉MHD). Our results show the solar wind incompressible fluctuations are primarily in a magnetically dominated regime, with the component traveling away from the Sun having a higher median pseudoenergy. Moreover, turbulent fluctuations have a total energy per mass of up to ∼ 300 km2 s−2, a range smaller than reported at 1 au. For these conditions, we determine the probability distribution function of 〈ε T 〉MHD ranges mainly between ∼ −1 × 10−16 and ∼1 × 10−16 J m−3 s −1, with a median equal to −1.8 × 10−18 J m−3 s −1, suggesting back transfer of energy. Our results also suggest that ∣〈ε T 〉MHD∣ is correlated with the total energy per mass of fluctuations and that the median of 〈ε T 〉MHD does not vary significantly with the cross-helicity. We find, however, that the medians of the inward and outward pseudoenergy cascade rates vary with the solar wind cross-helicity. Finally, we discuss these results and their implications for future studies that can provide further insight into the factors affecting the solar wind energy transfer rate.

Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions

We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.

Error analysis of a fully discrete projection method for Cahn–Hilliard Inductionless MHD problems

This article investigates the fully discrete finite element approximation and error analysis for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equations, which are nonlinearly coupled through convection, stresses, and Lorentz forces. To address this highly nonlinear and multi-physics system, we propose a fully discrete energy stable scheme with the finite element projection method for spatial discretization, in which the velocity and pressure are decoupled. The temporal discretization is a combination of the first-order Euler semi-implicit scheme and a convex splitting energy strategy. We show that the proposed scheme is mass-conservative, charge-conservative and unconditional energy stable. The error estimates for the phase variable, chemical potential, velocity, pressure, current density and electric potential are rigorously established. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

On Liouville type results for the stationary MHD in R3

This paper is concerned with the Liouville type theorems for the steady incompressible magnetohydrodynamics (MHD) equations. We establish that the solution to the steady MHD equations is identically zero under the integrability assumptions on (v, b). We show that, in particular, a combination of a strong integrability condition on the velocity of a fluid and a weak integrability condition on the magnetic field gives a sufficient condition on the Liouville type theorems. Furthermore, we show that the combination of the growth condition of the potential for the fluid velocity and the integrability condition for the magnetic field leads to the triviality of the solution.

Existence of Local Solutions to a Free Boundary Problem for Incompressible Viscous Magnetohydrodynamics

We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to H2+α,1+α/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{2+\\alpha ,1+\\alpha /2}$$\\end{document}, α>5/8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >5/8$$\\end{document}.

Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations

Abstract In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $\beta =0$ and RPC-SAV-CN when $\beta \in (0,\frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-\alpha _{1}\varDelta t\nu \varDelta (\widetilde {\textbf {u}}^{n+1}-{\textbf {u}}^{n})$ and $\alpha _{2}\varDelta t\sigma ^{-1}\mbox {curl}\mbox {curl} (\textbf {H}^{n+1}-\textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.

Finite element iterative algorithm based on Anderson acceleration technique for incompressible MHD equations

In this paper, an Anderson’s acceleration algorithm for the finite element Oseen iterative scheme of the steady incompressible magnetohydrodynamics (MHD) problem is designed and analyzed. The Anderson acceleration algorithm based on solving an optimization problem in each iteration receives great attention to accelerate linear and nonlinear iterations. We prove that the proposed algorithm has higher convergence rate than that of the Oseen iterative method for the steady incompressible MHD, and give the accelerating convergence theory under different acceleration depths m. Numerical experiments including the singular solution and benchmark problems are carried out to verify the correctness and effectiveness of the algorithm. The tests show the marked improvement in the high physical parameters (including the hydrodynamic and magnetic Reynolds numbers, and the coupling number) when the Oseen iteration fails to converge.

A Crank–Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

Unconditional optimal first‐order error estimates of a full pressure segregation scheme for the magnetohydrodynamics equations

AbstractIn this article, a first‐order linear fully discrete pressure segregation scheme is studied for the time‐dependent incompressible magnetohydrodynamics (MHD) equations in three‐dimensional bounded domain. Based on an incremental pressure projection method, this scheme allows us to decouple the MHD system into two sub‐problems at each time step, one is the velocity‐magnetic field system, the other is the pressure system. Firstly, a coupled linear elliptic system is solved for the velocity and the magnetic field. Next, a Poisson‐Neumann problem is treated for the pressure. We analyze the temporal error and the spatial error, respectively, and derive the temporal‐spatial error estimates of for the velocity and the magnetic field in the discrete space and for the pressure in the discrete space without imposing constraints on the mesh width and the time step size . Finally, some numerical results are presented to confirm the theoretical predictions and demonstrate the efficiency of the method.

Global solutions of two‐dimensional inviscid and resistive magnetohydrodynamics system near an equilibrium

We investigate the global existence of classical solutions for two‐dimensional incompressible, inviscid, and resistive magnetohydrodynamics (MHD) system. Assuming that the initial magnetic field is close to an equilibrium state of constant magnetic vorticity, the global existence result could be obtained using a time‐weighted energy estimate.

On a relaxation approximation of the 2-D incompressible magnetohydrodynamic equations

We study the Cauchy problem of a 2-D hyperbolic relaxation system for the incompressible magnetohydrodynamic (MHD) equations, which is related with the scientific computational aspects. We prove that this relaxed problem possesses a global strong solution, provided that the relaxation parameter is small and the appropriate norm of the initial data is not very large. We also show that such global solution converges to the one of the original incompressible MHD equations, as the relaxation parameter goes to zero.

A divergence-free hybrid finite volume / finite element scheme for the incompressible MHD equations based on compatible finite element spaces with a posteriori limiting

We present a novel semi-implicit hybrid finite volume/finite element (FV/FE) method for the equations of viscous and resistive incompressible magnetohydrodynamics (MHD). The scheme preserves the divergence-free property of the magnetic field exactly on the discrete level, is second order accurate in space, and is stable in the limit of vanishing viscosity and resistivity. In particular, the MHD system is first split into a fluid and a magnetic subsystem. The former is then addressed via a semi-implicit hybrid FV/FE method. The latter belongs to a class of generalized advection-diffusion problems, for which we propose a novel semi-implicit scheme based on compatible finite element spaces and interpolation-contraction operators. In the case of vanishing resistivity, the new algorithm avoids the solution of a linear system at each time step, thanks to the framework of physical degrees of freedom, which allow to express the exterior derivative as the incidence matrix of an appropriate refinement of the original mesh. Regarding the timestepping, a local fully-discrete Lax-Wendroff-type evolution is proposed as an alternative to the classical semi-discrete method of lines. In order to suppress spurious oscillations in the presence of discontinuities or steep gradients in the magnetic field, a new simple but efficient a posteriori limiting procedure based on a nonlinear artificial resistivity is introduced. The novel methodology is validated with several test cases in two and three space dimensions using unstructured simplex meshes.

Kolmogorov’s dissipation number and determining wavenumber for dyadic models

We study some dyadic models for incompressible magnetohydrodynamics and Navier–Stokes equation. The existence of fixed point and stability of the fixed point are established. The scaling law of Kolmogorov’s dissipation wavenumber arises from heuristic analysis. In addition, a time-dependent determining wavenumber is shown to exist; moreover, the time average of the determining wavenumber is proved to be bounded above by Kolmogorov’s dissipation wavenumber. Additionally, based on the knowledge of the fixed point and stability of the fixed point, numerical simulations are performed to illustrate the energy spectrum in the inertial range below Kolmogorov’s dissipation wavenumber.

Well-posedness of the three-dimensional axisymmetric incompressible magnetohydrodynamics system in the exterior of a cylinder

In this paper, we investigate the three-dimensional axisymmetric incompressible magnetohydrodynamics (MHD) system. Under the slip boundary conditions for both the velocity and the magnetic field, we prove that the MHD system has a unique global solution in the exterior of a cylinder Π={x=(x1,x2,x3)∈R3||xh|>1,xh=(x1,x2)} under initial axisymmetric data.