Stream function-velocity-magnetic induction compact difference method for the 2D steady incompressible full magnetohydrodynamic equations

doi:10.1016/j.cpc.2017.05.007

Computer Physics Communications

Volume 219, October 2017, Pages 45-69

https://doi.org/10.1016/j.cpc.2017.05.007 Get rights and content

Abstract

In this paper, an effective and accurate numerical model that involves a suggested mathematical formulation, viz., the stream functions ( $ψ$ and $A$ )-velocity-magnetic induction formulation and a fourth-order compact difference algorithm is proposed for solving the two-dimensional (2D) steady incompressible full magnetohydrodynamic (MHD) flow equations. The stream functions-velocity-magnetic induction formulation of the 2D incompressible full MHD equations is able to circumvent the difficulty of handling the pressure variable in the primitive variable formulation or determining the vorticity values on the boundary in the stream function-vorticity formulation, and also ensure the divergence-free constraint condition of the magnetic field inherently.

A test problem with the analytical solution, the well-studied lid-driven cavity problem in viscous fluid flow and the lid-driven MHD flow in a square cavity are performed to assess and verify the accuracy and the behavior of the method proposed currently. Numerical results for the present method are compared with the analytical solution and the other high-order accurate results. It is shown that the proposed stream function-velocity-magnetic induction compact difference method not only has the excellent performances in computational accuracy and efficiency, but also matches well with the divergence-free constraint of the magnetic field. Moreover, the benchmark solutions for the lid-driven cavity MHD flow in the presence of the aligned and transverse magnetic field for Reynolds number ( $R e$ ) up to 5000 are provided for the wide range of magnetic Reynolds number ( $R e_{m}$ ) from 0.01 to 100 and Hartmann number ( $H a$ ) up to 4000.

Introduction

Viscous incompressible laminar flow of electrically conducting fluids in the presence of magnetic field has been one of the major interesting research subjects due to its wide engineering applications in the cooling system with liquid metals for fusion reactors, electromagnetic pumps, etc. The incompressible full MHD governing equations involve the coupling of the Navier–Stokes equationsrepresenting incompressible viscous fluid flows with Maxwell’s equations of electromagnetics through Ohm’s law, so that analytical solutions are available only in some special conditions. In the general situation, the incompressible MHD governing equations can only be numerically solved. The main difficulties for solving the incompressible full MHD governing equations numerically come from the nonlinear properties of the equations, the additional terms with the existence of Lorentz’s force and the divergence-free constraint satisfied for both the velocity and the magnetic fields, i.e., $\nabla \cdot \vec{V} = 0$ and $\nabla \cdot \vec{B} = 0$ [1], [2]. Hence, establishing and developing accurate and effective numerical methods and algorithms suitable for the incompressible full MHD flow equations become more significant.

In the last two decades, many scholars have been devoted to developing various numerical models for the 2D and 3D incompressible full MHD equations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. According to the incompressible Navier–Stokes equations in the different variable formulations, the mathematical formulations of the solutions of the 2D incompressible full MHD equations may be categorized into three types, viz. the primitive variable formulation, the vorticity formulation and the pure stream function or stream function-velocity formulation. For the primitive variable formulation of the incompressible full MHD equations, Liu and Wang [13] developed a second-order accurate finite difference method, which combined the MAC scheme for the Navier–Stokes equation and Yee’s scheme for the Maxwell equations. However, the authors did not take into account the divergence-free constraint of the magnetic field in the numerical model. In [12], Aydın et al. developed a two-level finite element method with a stabilizing subgrid and tried to solve the lid-driven cavity MHD flow problems. Unfortunately, they also did not pay attention to the constraint of $\nabla \cdot \vec{B} = 0$ . In [2], Ben Salah et al. established a finite element method for the full MHD equations based on the ( $\vec{B}, q$ ) formulation, in which the gradient of a scalar variable $q$ were introduced in the magnetic induction equations and consequently, the divergence-free equation on the magnetic field was also included in the system of equations while overcoming the over-determination of the system.

The second type is the vorticity formulation of the incompressible full MHD governing equations. For the 2D incompressible Navier–Stokes equations, to avoid the difficulty of handling with the pressure variable, an alternative form is the vorticity formulation. The vorticity–velocity formulation is a popular vorticity formulation. According to [19], the vorticity–velocity formulation can also be classified into two basic categories considering the different formulations to update the velocity. Among them, solving the Poisson equations for velocity, which was proposed by Fasel [20], is more popular. Another popular vorticity formulation is the stream function-vorticity formulation, in which only two unknown variables involving stream function and vorticity require to be solved, whereas three unknown variables should be solved in the primitive variable formulation or the vorticity–velocity formulation. For the 2D incompressible full MHD equations, the vorticity formulation has also been considered. For example, the stream function-vorticity formulation can be extended to stream function-vorticity-current density formulation, and some finite element and finite difference methods have been developed [16], [17], [18]. In [1], Bozkaya et al. presented a stream function-vorticity-magnetic induction-current density formulation of the 2D incompressible full MHD equations, in which a magnetic induction-current density formulation for the Maxwell equations was suggested, which is the same as vorticity–velocity formulation of the Navier–Stokes equations in the form, while the Navier–Stokes equation was still expressed as the stream function-vorticity formulation.

The third type pays attention to the pure stream function or stream function-velocity formulation of the incompressible Navier–Stokesequations, in which the vorticity is also eliminated by the relationship of the vorticity and stream function [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]. Comparing with the vorticity formulation, the main advantage of this formulation is that there is no need to establish the numerical boundary conditions for the vorticity. However, there are some challenges of the numerical method for the formulation because the governing equation becomes a fourth-order partial difference equation. In order to establish the compact scheme of this equation, an alternative approach is to treat the first derivatives of the stream function (velocities) as the unknown variables as well and becomes stream function-velocity formulation. In recent years, several second-order compact schemes [21], [23], [24], [26], [28], [29], [30] and fourth-order compact algorithms [22], [27] were proposed based on the stream function-velocity formulation to solve the 2D incompressible fluid flows. The simplified model-based MHD stream function-velocity formulation has been developed to solve the natural convection in a uniform magnetic field [31]. For the 2D incompressible MHD equations, Krzemiński et al. [32] firstly used this type of formulation. A method involving a biharmonic mathematical model with stream function $ψ$ , and the magnetic potential $A$ was used, which was called as ( $ψ, A$ ) formulation. However, their formulation was limited for the incompressible full MHD equations, which would be discussed in the next section. Recently, a second-order compact scheme has been developed in [33] to compare the differences between the simplified and the full models for the MHD flow past a circular cylinder.

The purpose of this paper is to develop an accurate and effective numerical algorithm for the 2D steady incompressible full MHD equations and to solve the 2D lid-driven cavity MHD flow. The stream functions ( $ψ$ and $A$ )-velocity-magnetic induction formulation of the 2D full MHD governing equations is first introduced, in which the governing equations of the magnetic field are similar to the stream function-velocity formulation of the incompressible Navier–Stokes equations. The mathematical model can enforce to satisfy the divergence-free constraint of the magnetic field and circumvent difficulties existing in the velocity–pressure or stream function-vorticity formulation of the Navier–Stokes equations. Then, noticing the similarity of the formulation, a fourth-order compact difference method based on the stream function-velocity formulation of the Navier–Stokes equation is extended to solve the stream function-velocity-magnetic induction formulation directly. Furthermore, we would provide the benchmark solutions of the lid-driven cavity MHD flow by the currently proposed fourth-order compact difference method because there is no consistent numerical solutions for this problem.

The remainder of this paper is organized as follows. The stream functions ( $ψ$ and $A$ )-velocity-magnetic induction formulation is introduced for the 2D incompressible full MHD equations in Section 2. In Section 3, the numerical methods involving a fourth-order accurate compact finite difference scheme and the solution of algebraic systems are proposed to solve the stream functions ( $ψ$ and $A$ )-velocity-magnetic induction formulation. Numerical experiments for three problems are performed to validate the accuracy and effectiveness of the newly established numerical methods in Section 4. In Section 5, the benchmark solutions for the lid-driven cavity flow in the presence of the aligned and transverse magnetic field are provided. At last, the whole work is summarized in Section 6.

Section snippets

Mathematical formulations

In the 2D incompressible fluid flow problems, the primitive variable (velocity–pressure) formulation, the vorticity (vorticity–velocity, vorticity-stream function) formulation and the stream function or stream function-velocity formulation of the Navier–Stokes equations are usually used. Those formulations can be extended to establish the different forms of formulations of the 2D incompressible full MHD equations. However, it is worthy noticing that only primitive variable formulation and

Numerical method

In this section, both a second- and a fourth-order compact schemes are established for the stream function-velocity-magnetic induction formulation of the MHD governing equations. For convenience, the same space mesh size $h$ is employed in both $x$ and $y$ directions and the grid points are labeled from 0 to $N$ both on the $x$ -axis and $y$ -axis.

The test problem with an analytic solution

To verify the accuracy of the present numerical methods, we consider a test problem on the unit square with the following analytic solution for the forced MHD flow problem. $ψ = e^{x + y} R e, u = e^{x + y} R e, v = - e^{x + y} R e A = {(x + y)}^{2}, B_{x} = 2 (x + y), B_{y} = - 2 (x + y)$ and thus the additional force term $f_{a}$ besides the magnetic terms in the stream function equation (32) is given by $f_{a} = 4 e^{x + y} .$

In our test, we choose $R e = 1$ , $H a = 5$ and $R e_{m} = 100$ and set the convergence criterion $ε = 2 \times 1 0^{- 14}$ . Table 1 shows $L^{2}$ errors of all the

Result and discussion

In this section, the newly proposed stream function-velocity-magnetic induction formulation-based fourth-order compact difference method is carried out to solve the lid-driven cavity MHD flow in the transverse and aligned magnetic fields at $0 \leq H a \leq 4000$ and $0.01 \leq R e_{m} \leq 100$ for the higher range of Reynolds number up to 5000.

Conclusions

In this work, an effective and accurate numerical model, which is highly accurate and satisfies divergence-free of magnetic field, has been proposed for solving the steady 2D incompressible full MHD equations. We first describe the stream function-velocity-magnetic induction formulation of the 2D incompressible full MHD equations. The main advantage of this type of formulation is that it can not only circumvent difficulties existing in the primitive variable or the stream function-vorticity

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 11372075, 91330112 and 11502054, China Postdoctoral Science Foundation under Grant 2014M550211 and Special Program for Applied Research on Super Computation of the NSFC-Guang dong Joint Fund (the second phase).

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Stream function-velocity-magnetic induction compact difference method for the 2D steady incompressible full magnetohydrodynamic equations

Abstract

Introduction

Section snippets

Mathematical formulations

Numerical method

The test problem with an analytic solution

Result and discussion

Conclusions

Acknowledgments

Comput. Methods Appl. Mech. Engrg.

Int. J. Non-Linear Mech.

Comput. Methods Appl. Mech. Engrg.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Appl. Math. Comput.

Appl. Numer. Math.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. Phys. Comm.

Comput. Math. Appl.

Int. J. Heat Mass Transfer

Appl. Math. Model.

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.

Comput. Methods Appl. Mech. Engrg.