FLUENT 6.1 Magnetohydrodynamics (MHD) Module Manual

FLUENT 6.1 Magnetohydrodynamics (MHD) Module Manual

February 2003

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Contents

1 Introduction

1-1

2 Technical Background

2-1

2.1

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2-1

2.2

Magnetic Induction Method . . . . . . . . . . . . . . . . . . . . . . . . .

2-2

2.2.1

Case 1: B~0 generated in non-conducting media. . . . . . . . . . .

2-3

2.2.2

Case 2: B~0 generated in conducting media. . . . . . . . . . . . .

2-3

Electric Potential Method . . . . . . . . . . . . . . . . . . . . . . . . . .

2-4

2.3

3 Implementation

3-1

3.1

Solving Induction Equations . . . . . . . . . . . . . . . . . . . . . . . . .

3-1

3.2

Solving Electric Potential Equation . . . . . . . . . . . . . . . . . . . . .

3-2

3.3

Calculation of MHD Variables . . . . . . . . . . . . . . . . . . . . . . . .

3-2

3.4

MHD Interaction with Fluid Flows . . . . . . . . . . . . . . . . . . . . .

3-2

3.5

MHD Interaction with Discrete Phase Model

. . . . . . . . . . . . . . .

3-2

3.6

General UDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3-3

4 Using the FLUENT MHD Model 4.1

4-1

MHD Module Installation . . . . . . . . . . . . . . . . . . . . . . . . . .

4-1

4.1.1

Installing the MHD Model on Unix/Linux Platforms . . . . . . .

4-1

4.1.2

Installing the MHD Model on NT/Windows Platforms . . . . . .

4-2

4.2

Getting Started With the MHD Model . . . . . . . . . . . . . . . . . . .

4-2

4.3

MHD Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-3

4.3.1

External Applied Field . . . . . . . . . . . . . . . . . . . . . . .

4-3

4.3.2

User-Defined Scalar Allocation . . . . . . . . . . . . . . . . . . .

4-3

4.3.3

Solving User-Defined Scalars . . . . . . . . . . . . . . . . . . . .

4-7

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i

CONTENTS

4.4

4.5

4.3.4

Allocating User-Defined Memory . . . . . . . . . . . . . . . . . .

4-8

4.3.5

Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4-8

4.3.6

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . .

4-8

4.3.7

MHD/DPM Interaction . . . . . . . . . . . . . . . . . . . . . . . 4-11

4.3.8

General UDF Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 4-14

Solving and Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . 4-14 4.4.1

MHD Model Initialization . . . . . . . . . . . . . . . . . . . . . . 4-14

4.4.2

Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15

4.4.3

Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-15

Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-18

A Guidelines For Using the FLUENT MHD Model

A-1

A.1 Installing the MHD Module . . . . . . . . . . . . . . . . . . . . . . . . . A-1 A.2 An Overview of Using the MHD Module . . . . . . . . . . . . . . . . . . A-2 B Definitions of the Magnetic Field

B-1

B.1 Magnetic Field Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . B-1 C External Magnetic Field Data Format C.1 Magnetic Field Data Format

ii

C-1

. . . . . . . . . . . . . . . . . . . . . . . . C-1

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Chapter 1.

Introduction

Magnetohydrodynamics (MHD) refers to the interaction between an applied electromagnetic field and a flowing, electrically-conductive fluid. The FLUENT MHD model allows you to analyze the behavior of electrically conducting fluid flow under the influence of constant (DC) or oscillating (AC) electromagnetic fields. The externally-imposed magnetic field may be generated either by selecting simple built-in functions or by importing a user-supplied data file. For multiphase flows, the MHD model is compatible with both the discrete phase model (DPM) and the volume-of-fluid (VOF) approach in FLUENT, including the effects of a discrete phase on the electrical conductivity of the mixture. This document describes the FLUENT MHD model. Chapter 2 provides theoretical background information. Chapter 3 summarizes the UDF-based software implementation. Instructions for getting started with the model are provided in Chapter 4. Appendix A provides an condensed overview on how to use the MHD model, while Appendix B contains defintions for the magnetic field, and Appendix C describes the external magnetic field data format.

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1-1

Introduction

1-2

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Chapter 2. 2.1

Technical Background

Introduction

The coupling between the fluid flow field and the magnetic field can be understood on the basis of two fundamental effects: the induction of electric current due to the movement of conducting material in a magnetic field, and the effect of Lorentz force as the result of electric current and magnetic field interaction. In general, the induced electric current and the Lorentz force tend to oppose the mechanisms that create them. Movements that lead to electromagnetic induction are therefore systematically braked by the resulting Lorentz force. Electric induction can also occur in the presence of a time-varying magnetic field. The effect is the stirring of fluid movement by the Lorentz force. Electromagnetic fields are described by Maxwell’s equations: ~ =0 ∇·B

(2.1-1)

~ ~ = − ∂B ∇×E ∂t

(2.1-2)

~ =q ∇·D

(2.1-3)

~ = ~ + ∇×H

~ ∂D ∂t

(2.1-4)

~ (Tesla) and E ~ (V/m) are the magnetic and electric fields, respectively, and H ~ where B ~ are the induction fields for the magnetic and electric fields, respectively. q (C/m3 ) and D is the electric charge density, and ~ (A/m2 ) is the electric current density vector. ~ and D ~ are defined as: The induction fields H

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~ = 1B ~ H µ

(2.1-5)

~ = εE ~ D

(2.1-6)

2-1

Technical Background

where µ and ε are the magnetic permeability and the electric permittivity, respectively. For sufficiently conducting media such as liquid metals, the electric charge density q and ~ the displacement current ∂∂tD are customarily neglected [1]. In studying the interaction between flow field and electromagnetic field, it is critical to know the current density ~ due to induction. Generally, two approaches may be used to evaluate the current density. One is through the solution of a magnetic induction equation; the other is through solving an electric potential equation.

2.2

Magnetic Induction Method

In the first approach, the magnetic induction equation is derived from Ohm’s law and Maxwell’s equation. The equation provides the coupling between the flow field and the magnetic field. In general, Ohm’s law that defines the current density is given by: ~ ~ = σ E

(2.2-1)

~ in a magnetic where σ is the electrical conductivity of the media. For fluid velocity field U ~ field B, Ohm’s law takes the form: ~ +U ~ × B) ~ ~ = σ(E

(2.2-2)

From Ohm’s law and Maxwell’s equation, the induction equation can be derived as: ~ ∂B ~ · ∇)B ~ = 1 ∇2 B ~ + (B ~ · ∇)U ~ + (U ∂t µσ

(2.2-3)

~ the current density ~ can be calculated using Ampere’s From the solved magnetic field B, relation as: ~ =

1 ~ ∇×B µ

(2.2-4)

~ in a MHD problem can be decomposed into the externally Generally, the magnetic field B imposed field B~0 and the induced field ~b due to fluid motion. Only the induced field ~b needs to be solved. From Maxwell’s equations, the imposed field B~0 satisfies the following equation: ∇2 B~0 − µσ 0

2-2

∂ B~0 =0 ∂t

(2.2-5)

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2.2 Magnetic Induction Method where σ 0 is the electrical conductivity of the media in which field B~0 is generated. Two cases need to be considered.

2.2.1

~0 generated in non-conducting media. Case 1: B

In this case the imposed field B~0 satisfies the following conditions: ∇ × B~0 = 0

(2.2-6)

∇2 B~0 = 0

(2.2-7)

~ = B~0 + ~b, the induction equation (Equation 2.2-3) can be written as: With B ~ ∂~b ~ · ∇)~b = 1 ∇2~b + ((B~0 + ~b) · ∇)U ~ − (U ~ · ∇)B~0 − ∂ B0 + (U ∂t µσ ∂t

(2.2-8)

The current density is given by: ~ =

2.2.2

1 ∇ × ~b µ

(2.2-9)

~0 generated in conducting media. Case 2: B

In this case the conditions given in Equations 2.2-6 and 2.2-7 are not true. Assuming that the electrical conductivity of the media in which field B~0 is generated is the same as that of the flow, i.e. σ 0 = σ, from Equations 2.2-3 and 2.2-5 the induction equation can be written as: ∂~b ~ · ∇)~b = 1 ∇2~b + ((B~0 + ~b) · ∇)U ~ − (U ~ · ∇)B~0 + (U ∂t µσ

(2.2-10)

and the current density is given by: ~ =

1 ∇ × (B~0 + ~b) µ

(2.2-11)

For the induction equation Equations 2.2-8 or 2.2-10, the boundary conditions for the induced field are given by: ~b = {bn

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bt1

bt2 }T = b~∗

(2.2-12)

2-3

Technical Background where the subscripts denote the normal and tangential components of the field and b~∗ is specified by the user. For an electrically insulating boundary, as jn = 0 at the boundary, from Ampere’s relation one has bt1 = bt2 = 0 at the boundary.

2.3

Electric Potential Method

The second approach for the current density is to solve the electric potential equation ~ can and calculate the current density using Ohm’s law. In general, the electric field E be expressed as: ~ ~ = −∇ϕ − ∂ A E ∂t

(2.3-1)

~ are the scalar potential and the vector potential, respectively. For a static where ϕ and A field and assuming ~b