numerical simulations of quasi-static magnetohydrodynamics

Numerical simulations of

quasi-static magnetohydrodynamics

using an unstructured

finite volume solver:

development and applications

Stijn Vantieghem

Universite Libre de Bruxelles

These presente en vue de

l’obtention du grade de

Docteur en Sciences

sous la direction du

Prof. B. Knaepen.

Janvier 2011

Abstract

In this dissertation, we are concerned with the flow of electrically conductingliquids in an externally imposed magnetic field. Such flows are governed by theequations of quasi-static magnetohydrodynamics (MHD), and are commonlyencountered in applications of practical interest. Therefore, there is strong in-terest in numerical tools which can simulate these flows in complex geometries.

The first part of this thesis (chapters 2 and 3) is devoted to the presen-tation of the state-of-the-art numerical machinery which has been used andimplemented to solve the (incompressible) quasi-static MHD equations. Moreprecisely, we have contributed to the development of a parallel unstructuredfinite-volume flow solver. The discussion on these methods is accompanied bya numerical analysis which holds also for unstructured grids. In chapter 3, weverify our implementation through the simulation of a number of test cases,with a special emphasis on flows in strong magnetic fields.

In the second part of this thesis (chapters 4-6), we have used this solver tostudy wall-bounded MHD flows in various configurations. The first geometryconsidered is the laminar flow in a circular pipe of infinite extent at high Hart-mann number. We have investigated the sensitivity of the numerical resultson the mesh topology and the numerical discretization scheme. Furthermore,our simulation results allow to characterize extensively the flow in pipes withwell-conducting walls. This is achieved by an analysis of the scaling of the mostrelevant flow features with the non-dimensional parameters governing the flow.

The subject of chapter 5 is the flow in a toroidal duct of square cross-section. A study of the laminar regime confirms existing asymptotic analysisfor the shear layers. We have also provided simulations of turbulent flows inorder to assess the effect of an externally imposed magnetic field on the stateof the boundary layers.

Finally, in chapter 6, we investigated the inertialess MHD flow in a⋃

-bendand a backward elbow in a strong magnetic field. We explain how we cangenerate a mesh which can properly resolve all the shear layer at an affordablecomputational cost. Results of the present numerical method are comparedagainst asymptotic core flow approximations.

v

Acknowledgements

Did you ever hear of Loukas Karrer? Probably you didn’t. He doesn’t evenhave an entry in Wikipedia. Nevertheless, he was an exceptional person whodeserves to be paid tribute to until eternity. Much like the people mentionedbelow.

I want to acknowledge my advisor, prof. Bernard Knaepen, for unwaveringmentoring support during my first steps as a junior researcher. I have enor-mously appreciated his pedagogical qualities and his ability to offer a pertinentanalysis and a systematic solution strategy whenever I got stuck. I also ac-knowledge his constructive criticism during the redaction of this dissertation.Finally, I thank him for giving me the confidence and freedom to pursue myown research ideas during the final years of this thesis, and for the relaxed wayin which he heads his team.

I want to extend a big thanks to prof. Daniele Carati for his efficientmanagement of the Statistical Physics and Plasma team. If he had been onthe rudder of the Titanic, it surely wouldn’t have sunk. I furthermore wantto thank him for discreet encouragement, support and confidence, and for hisconviviality, which radiates onto the whole team.

I have been fortunate enough to have the opportunity to collaborate with dr.Vincent Moureau, my ‘finite-volume godfather’. I esteem him for his scientificmerits, and his complete lack of conceit, and offer my acknowledgements formany fruitful discussions, and for giving me a warm welcome at CORIA duringmy visit in the beginning of 2009. I sincerely hope that we can continue ourcollaboration in the near future.

A special thanks is also due to dr. Chiara Mistrangelo and dr. Leo Buhler,for enjoyable and fruitful discussions on the implementation of the unstructuredquasi-static module. Thanks also for providing me with many FZKA reports,and for hospitality during the MHD workshops of 2007 and 2010.

I am grateful to prof. Leon Brenig and prof. Gerard Degrez for acceptingto be member of this jury.

I would also like to thank the colleagues of the Statistical Physics andPlasma group who made work so enjoyable. In the first place, I think of Axelle

vii

Vire, my office mate for four years. I have rarely met someone with whom Icould collaborate so well. I would like to thank her for her kindness, and foralways lending me a listening ear. I hope we will be able to stay in touch so thatI’ll get the chance to taste the tiramisu she still owes me. I am also gratefulto Bogdan Teaca for challenging my points of view in many scientific and non-scientific discussions, for our trip to Chicago, and simply for being a wonderfulguy. I would like to salute Thomas Lessinnes for his cordiality and enthusiasmfor taking on mathematical problems. A Flemish guy praising a real Carolo;who would have ever imagined that? I am thankful to Maxime Kinet for hiskindness, and for patiently answering my computer-related questions. Specialthanks also to Xavier Albets-Chico for intensive discussions and really goodtimes.

Furthermore, I would like to thank explicitly the other colleagues (in noparticular order of importance): Sara Moradi, Chichi Lalescu, Paolo Burattini,Carlos Cartes, Michel Marc-Albrecht, Pierre Morel, Michael Leconte, Alejan-dro Banon-Navarro, Benjamin Cassart, Sotirios Kakarantzas, Chiara Toniola,Ioannis Sarris and Oleg Shyskin. I am indebted to Fabienne De Neyn andMarie-France Rogge for helping me out in my struggle with the bureaucraticmerry-go-round, and above all, for being kind and caring office neighbors. Thelatter holds also for the colleagues of the Theoretical Non-Linear Optics group.Finally, thanks to Axel Coussement and Matthew Peavy for administrating ourcluster.

Outside the professional sphere, I am especially grateful to five friends fortheir rock-solid and longstanding friendship. A big thanks to Clara Verhelstfor mental support and highly enjoyable train trips to hometown Kortrijk, toFrederik De Roo for exploring together Europe by bicycle and for never-endingdiscussions on Dawkins and Dostoevsky. Thanks also to Maarten Vanhee, forhospitality, coffee and cookies, and to Dieter Vanneste and Tim Bekaert formany road trips and not-so-lazy-sunday-afternoon jogging tours. All of youare cordially invited to Zurich.

Finally, there are my most faithful supporters along the road of study andlife. I want to thank my family for believing in me and investing in me, and Idedicate this dissertation to them.

Stijn VantieghemJanuary 2011

viii

Contents

Abstract v

Acknowledgements vii

Introduction xiii

1 Introductory aspects of MHD 1

1.1 The fundamentals of hydrodynamics . . . . . . . . . . . . . . . 11.1.1 The continuum hypothesis . . . . . . . . . . . . . . . . . 11.1.2 Conservation of mass . . . . . . . . . . . . . . . . . . . . 21.1.3 Conservation of momentum . . . . . . . . . . . . . . . . 31.1.4 Conservation of energy . . . . . . . . . . . . . . . . . . . 51.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Classical electromagnetism . . . . . . . . . . . . . . . . 71.2.2 The induction equation . . . . . . . . . . . . . . . . . . 91.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 The quasi-static approximation . . . . . . . . . . . . . . . . . . 111.3.1 Simplified equations for Rm ≪ 1 . . . . . . . . . . . . . 111.3.2 Phenomenology of the quasi-static regime . . . . . . . . 151.3.3 Examples of quasi-static MHD flows . . . . . . . . . . . 21

2 Numerical framework 25

2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . 262.1.1 Basic principle of the finite-volume method . . . . . . . 262.1.2 Construction of the control volumes . . . . . . . . . . . 272.1.3 Volume averages . . . . . . . . . . . . . . . . . . . . . . 312.1.4 The gradient operator . . . . . . . . . . . . . . . . . . . 332.1.5 The divergence operator . . . . . . . . . . . . . . . . . . 352.1.6 The Laplacian operator . . . . . . . . . . . . . . . . . . 362.1.7 Matrix representation of the discretization operators . . 40

ix

2.2 Time advancement . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.1 Fractional-step methods . . . . . . . . . . . . . . . . . . 432.2.2 Time integration schemes for the momentum equation . 472.2.3 Kinetic energy conservation and the pressure term . . . 502.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . 51

2.3 The quasi-static MHD equations . . . . . . . . . . . . . . . . . 532.3.1 Discretization of the Lorentz force . . . . . . . . . . . . 532.3.2 Boundary conditions for the potential . . . . . . . . . . 562.3.3 Coupling between the momentum and potential equations 57

2.4 Solution techniques for systems of linear equations . . . . . . . 582.4.1 Jacobi iteration . . . . . . . . . . . . . . . . . . . . . . . 592.4.2 Algebraic multigrid methods . . . . . . . . . . . . . . . 612.4.3 Krylov subspace methods . . . . . . . . . . . . . . . . . 64

3 Verification and validation 67

3.1 Taylor-Green vortex . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Turbulent channel flow . . . . . . . . . . . . . . . . . . . . . . . 71

3.2.1 Physical background . . . . . . . . . . . . . . . . . . . . 723.2.2 Computational details . . . . . . . . . . . . . . . . . . . 743.2.3 Numerical results and discussion . . . . . . . . . . . . . 75

3.3 Two-dimensional MHD flows at high Hartmann number . . . . 773.3.1 Laminar MHD flow in a straight duct . . . . . . . . . . 783.3.2 Laminar MHD flow in a plane sudden expansion . . . . 81

4 Laminar pipe flow 85

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.3 Computational details and grid study . . . . . . . . . . . . . . 89

4.3.1 Computational details . . . . . . . . . . . . . . . . . . . 894.3.2 Grid study . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 964.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 Flow in a toroidal square duct 105

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . 1075.3 Laminar flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3.1 Computational set-up . . . . . . . . . . . . . . . . . . . 1095.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . 109

5.4 Turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4.1 Computational set-up . . . . . . . . . . . . . . . . . . . 1165.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

x

6 Laminar flow in a right-angle bend 123

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2 MHD flow in a

⋃

-bend . . . . . . . . . . . . . . . . . . . . . . . 1266.2.1 Problem definition and computational set-up . . . . . . 1266.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . 130

6.3 MHD flow in a backward elbow . . . . . . . . . . . . . . . . . . 1366.3.1 Problem definition and computational set-up . . . . . . 1366.3.2 Results and discussion . . . . . . . . . . . . . . . . . . . 141

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7 Conclusions and perspectives 145

A Elements of vector calculus 147

A.1 Definition of the ∇ operators . . . . . . . . . . . . . . . . . . . 147A.2 Integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 148A.3 Vector identities . . . . . . . . . . . . . . . . . . . . . . . . . . 148A.4 Helmholtz’s decomposition . . . . . . . . . . . . . . . . . . . . . 149

B Elaborations on linear system solvers 151

B.1 A spectral analysis of the ω-Jacobi method for a Poisson equation151B.2 Listing of the BiCGstab algorithm . . . . . . . . . . . . . . . . 153B.3 Listing of the BiCGStab(2)-algorithm . . . . . . . . . . . . . . 154

C Asymptotic solutions at high Hartmann number 157

C.1 Asymptotic theory for circular pipes . . . . . . . . . . . . . . . 157C.2 Free MHD shear layers near geometrical discontinuities . . . . . 160C.3 Asymptotic analysis of the core regions in a

⋃

-bend . . . . . . 164

xi

Introduction

The scope of this dissertation is at the crossroads between two important areasof research: magnetohydrodynamics (MHD) on the one hand, and computa-tional fluid dynamics (CFD) on the other hand.

Magnetohydrodynamics is the branch of physics which studies the inter-action between the flow of electrically conducting fluids and electromagneticfields. This coupling is due to three fundamental physical phenomena. Accord-ing to Faraday’s and Ohm’s law, the presence of a magnetic field will inducean electric current in a moving conductor. Secondly, this current distributionis at the origin of an induced magnetic field. Finally, the interaction betweenthe resulting magnetic field and current distribution will cause a body force,which affects the momentum balance of the conducting medium.

Magnetohydrodynamics applies to a large variety of phenomena. One canthink of astrophysical or geophysical processes, like the spontaneous genera-tion of the Earth’s magnetic field by the motion of the liquid iron core of theEarth, but MHD flows can also be encountered in applications of more practi-cal interest. The common feature of almost all these industrial and laboratoryflows is that the coupling between the flow and the magnetic field is virtuallyone-way, i.e. the magnetic field strongly affects the flow through the genera-tion of a body force, but the flow does not act significantly upon the magneticfield. This regime is known under the name quasi-static magnetohydrodynam-ics. Sometimes, the term liquid metal MHD is also used to refer to this regime.

Historically, the first application of MHD concerned flow measurement tech-niques. Pioneering work in this context was performed by Faraday, who at-tempted in 1832 in vain to estimate the flow rate of the river Thames by mea-suring the electric potential difference across the river, induced by the Earth’smagnetic field. Notwithstanding his failure, Faraday’s induction principle isstill at the basis of most of the electromagnetic flow meters available on themarket. It was only in the 1960’s that magnetic fields began to be used astools for flow control and generation. In continuous casting processes, a betterquality of the product is achieved by applying a (static) magnetic field dur-ing the solidification process; it damps perturbations of the melt flow which

xiii

may be caused by natural convection or the inflow of the melt. On the otherhand, rotating magnetic fields can be used to enhance mixing by inducing non-intrusively a stirring motion in a liquid metal or electrolyte. Such an approachis preferred in configurations where the use of a mechanical mixer is imprac-tical, e.g. in high-temperature or corrosion-aggressive environments. Finally,rotating magnetic fields are also applied in heating or levitation processes. Arecent review of the applications of MHD in materials processing can be foundin [Dav99].

Apart from those applications where a magnetic field is imposed on pur-pose, there are situations in which there is an ambient magnetic field withoutspecific intent for the flow. A notable example are so-called blankets for fu-ture thermonuclear fusion devices [B07]. Their role is to absorb the energyreleased in the fusion reaction, and transfer it subsequently to a power plant.Moreover, they should provide a shield against the neutron irradiation. Liquidmetal alloys, like lead-lithium, are primary candidate coolant materials, mainlybecause of their operability at high temperatures. However, the flow of thesemedia will be heavily affected by the presence of intense magnetic fields (up to5-10 tesla) required to confine the plasma in the tokamak reactor.

All the applications described above drive an ongoing scientific effort whichaims at improving our current understanding of MHD. The mathematicalframework describing the physics of MHD flows results from a combinationof the laws of electromagnetism and the theory of fluid mechanics. The partialdifferential equations governing hydrodynamic and (quasi-static) MHD flowhave a non-linear nature. Hence, for broad ranges of flow parameters, the flowmay be in a turbulent state, i.e. exhibit a seemingly random chaotic behaviorcontaining a large range of spatial and temporal scales. As such, solving ex-actly the flow equations, by means of analytical methods, is only possible fora limited class of almost trivial geometries in the laminar (i.e steady) regime,and is pointless for turbulent flows.

To study fluid mechanics in general, and magnetohydrodynamics in par-ticular, we can distinguish between three approaches: physical experiments,analytical solutions of approximate equations and numerical simulations. Dueto the fast increase in computational power and the development of paral-lel computing, numerical simulations have become a powerful predictive tool,and are a useful complementary alternative to experiments and approximatetheories. The advantages of numerical experiments above physical ones arenumerous. They are relatively cheap to perform, they give access to the com-plete flow field, they allow to study systematically the influence of geometricalparameters and they don’t require all kinds of precautions related to the han-dling of possibly dangerous working fluids. This holds a fortiori for quasi-staticMHD flows, since most liquid metals are opaque; moreover, the generationof intense magnetic fields is very power-consuming, and thus expensive. The

xiv

main drawback of a numerical approach is that the parameter ranges which areencountered in practical applications can not be reached (yet) with sufficientconfidence.

The term CFD refers, in the most general sense, to the study of fluid me-chanics by means of numerical simulations, as well as to the development andanalysis of numerical techniques used to solve the complete set of equationsgoverning these flows. One of the earliest CFD calculations was performedby Richardson in 1916. To provide numerical weather predictions, he dividedphysical space in grid cells and applied a finite-difference technique. For hisprediction of the globe’s weather over a period of 8 hours, he needed six weeksof computation time. Moreover, his attempt failed, presumably because of alack of accuracy of the then available (mechanical) calculators. In order to pro-duce real-time predictions, he proposed the instauration of ‘forecast factories’,where continuously 64 000 people would, armed with a mechanical calculator,perform a part of the flow computation, on a grid consisting of approximately2000 grid points [Ric22].

The introduction of the digital computer brought new perspectives to thepossibilities of computational fluid dynamics. However, for a very long time,numerical simulations were only affordable in the context of military researchprojects. During the 1960’s, many of the numerical techniques which are stillused today were developed at Los Alamos National Laboratory [Har04], likethe vorticity-streamfunction formulation or the k− ǫ turbulence model. It wasalso there that the first digital simulation of an unsteady von Karman vortexstreet was performed. None of these breakthroughs of the early days of CFDwould have been possible without significant progress in the field of numericalanalysis and numerical algorithms. Milestone achievements in these fields were,among others, the famous stability analysis of Courant, Friedrich and Lewy forthe advancement of the advection equation [CFL28], and the introduction ofvon Neumann’s stability analysis method [vNR50].

The first commercial use of CFD codes should be situated in the secondhalf of the 1970’s and early 1980’s, and concerned primarily the aircraft indus-try. These codes were based on finite-difference or finite-volume formulations.Computational resources were still very limited at that time. For instance, theCray-X-MP, a state-of-the-art supercomputer in 1980, disposed of a total mem-ory of 32 megabyte and had a peak performance of 400 MFLOPS. However,the range of treatable problems expanded rapidly because of the fast increasein computational power; codes based on unstructured meshes emerged in the1990’s.

In many branches of engineering, as well in industry and in academia, thesenumerical tools are now widely accepted as useful instruments for flow predic-tion. However, the systematic application of complex, unstructured CFD codeshas not yet significantly trickled down into the quasi-static MHD community,

xv

and this brings us to the aim of this dissertation; we have contributed to the de-velopment of a numerical solver for the magnetohydrodynamic equations in thequasi-static limit, and used this solver to study a number of cases of theoreticaland practical interest.

The outline of this work is as follows. First, we derive the partial differen-tial equations governing quasi-static MHD from first principles, and give a briefsummary of the phenomena characterizing the quasi-static regime. In the sec-ond chapter, we introduce and discuss extensively the numerical techniques thathave been implemented in the finite volume solver YALES2 [Mou10]; this is aversatile code, mainly developed at CORIA, for various types of flow problems.Several test cases, which were performed in order to validate the implementa-tion of the numerical methods, are presented in the third chapter. We havethen investigated various configurations with this code: a laminar MHD flowin a circular pipe in an intense magnetic field (chapter 4), the hydrodynamicand MHD flow in a toroidal duct of square cross-section (chapter 5), and theinertialess MHD flow in a right-angle bend in a strong magnetic field (chapter6). In the seventh and last chapter, we summarize the main conclusions of thisflow.

Some chapters in this work are based on the following publications:

• Chapter 4:

– S. Vantieghem, X. Albets-Chico and B. Knaepen, “The velocity pro-file of laminar MHD flows in circular conducting pipes”, Theoreticaland Computational Fluid Dynamics 23(6), (2009) 525.

• Chapter 5:

– S. Vantieghem and B. Knaepen, “Direct numerical simulation ofquasi-static magnetohydrodynamic annular duct flow” Proceedingsof the Fifth European Conference on Computational Fluid Dynam-ics, Lisbon, 14-17 June 2010, in press (2010).

– S. Vantieghem and B. Knaepen, “Numerical simulation of magneto-hydrodynamic flow in a toroidal duct of square cross-section”, sub-mitted to International Journal of Heat and Fluid Flow (2010).

xvi

Chapter 1

Introductory aspects ofmagnetohydrodynamics

“Although to penetrate into the intimate mysteries of nature andthence to learn the true causes of phenomena is not allowed to us,nevertheless it can happen that a certain fictive hypothesis may suf-fice for explaining many phenomena.”Leonhard Euler

In this first chapter, we will give a broad introduction to the field of mag-netohydrodynamics (MHD). This is the branch of physics which studies theinteraction between the flow of electrically conducting fluids and electromag-netic fields. As a starting point, we will develop the equations of conservation ofmass, momentum and energy for an ordinary hydrodynamic flow. In a secondstep, we will detail the mutual interaction between a flow and an electromag-netic field. Under some conditions, this coupling may be virtually one-way, i.e.the velocity field only weakly influences the magnetic field. This regime, whichis known under the name of quasi-static magnetohydrodynamics, is the mainscope of this work. Its governing equations, together with a couple applica-tions of industrial interest, will be presented in the third and final section ofthis chapter.

1.1 The fundamentals of hydrodynamics

1.1.1 The continuum hypothesis

When we want to model the physics of fluid flows, the chosen description ofthe medium will depend on the length and time scale of the relevant physical

1

2 CHAPTER 1. INTRODUCTORY ASPECTS OF MHD

phenomena. If we are only interested in the macroscopic behavior, as is thecase in this work, we will use a fluid dynamic formulation to model the flow. Itsmain assumption, the so-called continuum hypothesis, states that a flow canbe completely characterized by means of continuous functions of the spatialcoordinates x, and of time t, like the mass density ρ(x, t) and the velocityu(x, t), etc. This is of course an idealization, since it is known that matteris built up out of discrete atoms or molecules at the microscopic level; thecontinuous functions should then be seen as the average over a volume whichis small with respect to the spatial variations of the flow, but large comparedto the distance between the individual particles. This description is extremelyaccurate as long as the length and time scales of the phenomena of our interestare macroscopic, and are thus much larger then their microscopic counterparts.If, on the other hand, microscopic effects are important, we should recourse toother formulations, like e.g. kinetic theory, which are based on the principlesof statistical physics.

1.1.2 Conservation of mass

To derive the equation of mass conservation, we consider an arbitrary lump offluid which occupies a volume Ω in space. If we follow the lump throughoutits motion, it is assumed that it will always consist of the same fluid elements;its mass will remain constant in time if no mass sources are present. We canexpress this as:

d

dt

∫

Ω

ρ(x, t) dV = 0 (1.1)

The size or shape of the integration domain may change in time, and this shouldbe taken into account when bringing the time derivative within the integral. Itcan be shown that this leads to:

∫

Ω

(

d

dtρ(x, t) + ρ∇ · u

)

dV = 0 (1.2)

Furthermore, the positions x of the fluid elements still depend on time. The to-tal time derivative can be decomposed into an expression involving only partialderivatives by applying the chain rule:

d

dtρ(x, t) =

∂ρ

∂t+∂x

∂t· ∇ρ =

∂ρ

∂t+ u · ∇ρ (1.3)

We eventually obtain:

∫

Ω

(

∂ρ

∂t+∇ · (ρu)

)

dV = 0 (1.4)

2

1.1. THE FUNDAMENTALS OF HYDRODYNAMICS 3

Since this expression holds for any volume Ω, we can leave aside the volumeintegration; this yields a local relationship:

∂ρ

∂t+∇ · (ρu) = 0 (1.5)

This result is called the mass conservation equation or continuity equation.If the flow is characterized by a velocity which is small with respect to the

speed of sound of the medium, we may assume that the volume of a lumpof fluid does not change with time. Such flows are called incompressible, andobey:

d

dt

∫

Ω

dV = 0 (1.6)

This is equivalent to a solenoidal constraint on the velocity:

∇ · u = 0 (1.7)

In this work, we will only be concerned with liquids. The speed of soundin these media is typically far above the characteristic velocity scales of theapplications of interest in this work. Therefore, we will always assume that theflow under consideration is incompressible.

1.1.3 Conservation of momentum

Newton’s second law states that the rate of change of momentum of a bodyequals the net force exerted on that body. We can again consider an arbitrarylump of fluid, and write its rate of change of momentum as:

d

dt

∫

Ω

ρu dV = F (1.8)

Following the same approach as in the previous section, we develop the left-hand side of this equation:

d

dt

∫

Ω

ρu dV =

∫

Ω

(

d

dt(ρu) + (ρu)∇ · u

)

dV

=

∫

Ω

(

ρd

dtu+ u

d

dtρ+ (ρu)∇ · u

)

dV (1.9)

We can use (1.3) and (1.5) to show that the second and third term on theright-hand side of the result above cancel each other. Further development ofthe total time derivative of the first term leads to:

d

dt

∫

Ω

ρu dV =

∫

Ω

(

ρ∂u

∂t+ ρu · ∇u

)

dV (1.10)

3


We can furthermore write the net force F as the volume integral of a forcedensity f . Following the approach of [Bat67], we will distinguish between twoclasses of forces contributing to f . On the one hand, we have volume or bodyforces fb, and surface forces fs on the other hand. The former stem from long-range interactions, like buoyancy, gravity or electromagnetic forces. The latterhave a molecular origin, and are negligible unless there is direct mechanicalcontact between the interacting fluid elements. If we consider now a lump offluid, all interior contributions of this force will cancel, since any elementaryforce exerted by a fluid element on its surroundings, is accompanied by anopposite reaction force of the surrounding on the element. It is thus worthwhileto write the resulting total surface force on the lump as the surface integral ofa stress tensor τ :

∫

Ω

fs dV =

∮

∂Ω

τ · dS (1.11)

Using Gauss’s divergence theorem, we obtain:

ρ

(

∂u

∂t+ u · ∇u

)

= −∇ · τ + fb (1.12)

As for now, we will not yet specify the form of the body forces, and concen-trate on the structure of the tensor τ . It can be shown that angular momentumconservation requires τ to be symmetric. Furthermore, for a fluid in rest, this

tensor is diagonal and isotropic, i.e the stress tensor can be written as τ = −p1,with p the thermodynamic pressure and 1 the unit tensor. This does not holdany more for a fluid in motion. However, we can still formally decompose the

stress tensor into a multiple of the unity tensor and a remainder: τ = −p1+τ ′.Here we define p as the mean normal stress, and term this quantity mechanical

pressure. It follows then that τ ′ is a deviatoric tensor, i.e. its trace is zero.

Since τ ′ is related to a kind of internal friction mechanism, it can not directly

depend on the values of the velocity itself. The structure of τ ′ is further con-strained by assuming that the stress tensor is an isotropic and linear function ofthe velocity gradient tensor. A fluid with such properties is called Newtonian.

In its most general form, τ ′ can be written as:

τ ′ = η

(

∇u+ (∇u)T − 2

3(∇ · u) 1

)

(1.13)

The quantity η is called the dynamic viscosity. In incompressible flows, the lastterm of the right-hand side in the expression above is zero (see (1.7)) and theprevious constitutive relationship simplifies to:

τ ′ = η(

∇u+ (∇u)T)

(1.14)

4

1.1. THE FUNDAMENTALS OF HYDRODYNAMICS 5

We will furthermore define the kinematic viscosity ν as ν = η/ρ.Eventually, we obtain the following form for the momentum equation, better

known as the notorious Navier-Stokes equations :

ρ

(

∂u

∂t+ u · ∇u

)

= −∇p+ ρν∇2u+ fb (1.15)

It is important to note that the idea of (mechanical) pressure is not to beconfused with the one we know from thermodynamics. The latter one is avariable that is reserved for the description of equilibrium states. Both notionsare only equivalent under hydrostatic equilibrium. To elucidate the role of thepressure in incompressible fluid dynamics, we take the divergence of equation(1.15) and rearrange the terms as follows:

ρ

(

∂

∂t− ν∇2

)

∇ · u = −∇2p− ρ∇ · (u · ∇u) +∇ · fb (1.16)

We now consider this as an equation for the unknown ∇ · u, with initial andboundary condition ∇ · u = 0. The solution of this equation is ∇ · u = 0everywhere in the domain if, and only if, the right-hand side of (1.16) is zero.The role of the pressure is thus to cancel the deviations of the incompressibilityconstraint due to the non-linear term u · ∇u or the body force term. We caninterpret the pressure thus as a kind of Lagrange multiplier needed to satisfythe incompressibility constraint of the velocity field.

1.1.4 Conservation of energy

The local kinetic energy density is defined as (ρu2)/2. An equation for thisquantity can be obtained by taking the scalar product between the velocityand the Navier-stokes equation (1.15). For an incompressible flow, we find,after a few manipulations:

∂

∂t

(

1

2ρu2

)

= −∇ ·(

u1

2ρu2

)

−∇ · (pu)

+ρν∇ · (u · ∇u)− ρν||∇u||2 (1.17)

In this expression, we have introduced the norm of the velocity gradient tensoras ||∇u||2 =

∑

i

∑

j(∂iuj)2. The first two terms on the right-hand side are in

divergence form. If we integrate this equation over a given volume, we findthat the total work caused by the non-linear and pressure term reduces to aboundary integral. We call these terms energy-conserving. The last two termsin the above equation concern viscous effects. Only the first one is energy-conserving. The last term is always negative and represents a loss of kineticenergy; the effect of the viscous interaction is to dissipate kinetic energy intoheat.

5


1.1.5 Summary

The following set of partial differential equations completely determines theincompressible motion of a fluid:

∇ · u = 0 (1.18)

∂u

∂t+ u · ∇u = −∇

(

p

ρ

)

+ ν∇2u+ ρ−1fb (1.19)

This set of equations is not yet completely closed. We should still providesuitable boundary and initial conditions. Of particular interest is the boundarycondition for a stationary rigid wall. In this work, we adopt the conventionto denote the outward-pointing normal on the wall as n. Since the wall isimpermeable, we have:

u · n = un = 0 (1.20)

For the velocity components tangential to the wall, we assume that a viscousfluid ‘sticks’ to the rigid wall. Hence:

u− unn = uτ = 0 (1.21)

and both conditions together yield a homogeneous Dirichlet condition for thevelocity at a rigid, stationary wall. Such a boundary condition is called a no-slipcondition.

Much confusion exists about the boundary conditions for the pressure. Theyshould be such that they reconcile the incompressibility constraint and thevelocity boundary conditions [McC89]. It is, generally spoken, not possible todefine generic a priori conditions for p.

1.2 Magnetohydrodynamics

In this section, we will treat the coupling between fluid dynamics and electro-dynamics. This coupling is twofold. On the one hand, electromagnetic bodyforces will enter into the momentum balance. On the other hand, the motionof a conducting medium may give rise to a complex dynamics of the electro-magnetic field. We will first review classical electromagnetics, and introducea slightly simplified version of it, leaving aside some complications which onlymatter for media which are moving with a speed close to the speed of light.These constituting laws of electromagnetism can then be combined to yield anevolution equation of the magnetic field, the so-called induction equation. Wewill derive it and analyse it in terms of a few non-dimensional parameters.

6

1.2. MAGNETOHYDRODYNAMICS 7

1.2.1 Classical electromagnetism

Classical electromagnetism [Jac99], in its most concise form, consists of thecombination ofMaxwell’s equations and an expression for the force on a chargedparticle or medium, the Lorentz force. This latter law states that a certain mass,carrying a charge q and moving with a velocity u in an electric field E and/ormagnetic field B, will undergo the following force FL:

FL = q (E+ u×B) (1.22)

For continuous media, it is convenient to introduce a charge density ρe and anelectric current density J. We define these as the quantities which obey thefollowing relationship for any arbitrary volume Ω:

∫

Ω

ρe dV =∑

q (1.23)

∫

Ω

JdV =∑

qiui (1.24)

where the summation extends over all the particles i with charge qi and velocityui within Ω. The Lorentz force density fL can then be written as:

fL = ρeE+ J×B (1.25)

Just like mass, charge is a conserved quantity. The introduction of the quanti-ties above allows us to express an electrical analogue of the mass conservationequation. This charge conservation equation reads:

∂ρe∂t

+∇ · J = 0 (1.26)

Maxwell’s equations on the other hand allow to compute the electric and mag-netic fields for a specified charge and current distribution. We will only con-sider fluids which are neither dielectric nor diamagnetic; these equations takethe following form:

∇×E = −∂B∂t

(1.27)

∇ · E = ǫ−10 ρe (1.28)

∇ ·B = 0 (1.29)

∇×B = µ0

(

J+ ǫ0∂E

∂t

)

(1.30)

In these expressions, ǫ0 and µ0 denote, respectively, the vacuum electric per-mittivity and magnetic permeability. By combining equation (1.28) with the

7


divergence of (1.30), and using (A.16), we immediately recover the law of chargeconservation (1.26).The set of equations (1.27)-(1.30) defines eight constraints for ten unknowns (inthree dimensions). Moreover, those constraints are not independent; taking thedivergence of (1.27), together with (A.15), gives ∂t (∇ ·B) = 0, which, togetherwith an appropriate initial condition, reduces to (1.29). To close the system ofequations, we still have to supply a constitutive relation which links the electricfield to the current density. For stationary isotropic conducting media, and forlow-frequency electromagnetic fields, there is empirical evidence that the cur-rent density is proportional to the electric field, with proportionality constantσ, termed electric conductivity. This is known as Ohm’s law :

J = σE (1.31)

When the conductor is moving, we have to adapt this expression to keep Ohm’slaw Lorentz invariant. We will however assume that the speed of the mediumis small with respect to the speed of light, so that we can use an approximateversion of Lorentz’s transformation laws:

J = σ (E+ u×B) (1.32)

If we insert this expression into the charge conservation equation (1.26), anduse (1.28), we obtain:

∂ρe∂t

+σ

ǫ0ρe = −σ∇ · (u×B) (1.33)

The left-hand side of this equation represents an exponential decay on a chargerelaxation time of τC = ǫ0σ

−1. Typical values of this time scale are of theorder of 10−18 s. Since the flow phenomena in which we are interested, arecharacterized by much larger time scales, we may disregard the term ∂tρe. Itmeans that the charge conservation equation reduces to:

∇ · J = 0 (1.34)

and that we are left with the pseudo-static equation:

ρe = −ǫ0∇ · (u×B) (1.35)

An order-of-magnitude analysis for the charge density learns us that O(ρe) =O(ǫ0∇ · (u×B)) = ǫ0UB0L

−1, where U , B0 and L are respectively a speed,magnetic field intensity and length scale which characterize the flow underconsideration. Furthermore, Ohm’s law tells us that the electric field strengthscales as O (E) = σ−1J . Here, J is a typical magnitude of the current den-sity. All this allows us to estimate the relative importance of the ‘electric’ and‘magnetic’ part of the Lorentz force:

O(ρeE) = ǫU

LB0

J

σ= τC

U

LJB0 = τC

U

LO (J×B) (1.36)

8

1.2. MAGNETOHYDRODYNAMICS 9

As previously mentioned, the mechanical time scale associated with the flow,L/U , is much larger than the charge relaxation time τC . As such, the termρeE can be neglected with respect to the term J × B. We can, up to a goodapproximation, restrain the Lorentz body force to the following term:

fL = J×B (1.37)

The previous estimates and considerations can also be used to show that theterm ǫ0µ0∂tE can be neglected in Ampere’s law (1.30). This term becomesimportant only if the characteristic velocity of the flow approach the speed oflight c = (µ0ǫ0)

−1/2. For much lower velocities however, we may approximate(1.30) as:

∇×B = µJ (1.38)

1.2.2 The induction equation

If we substitute Ohm’s law (1.32) into Faraday’s law (1.27), we can eliminatethe induced electric field:

∂B

∂t= −∇×E = −∇×

(

σ−1J)

+∇× (u×B) (1.39)

Furthermore, we use the pre-Maxwell version of Faraday’s law (1.38) to elimi-nate the current. The resulting expression becomes:

∂B

∂t= − 1

µσ∇×∇×B+∇× (u×B) (1.40)

Using (A.17), (A.19) and the solenoidal character of both the velocity andmagnetic field, we finally obtain:

∂B

∂t+ u · ∇B = B · ∇u+

1

µσ∇2B (1.41)

1.2.3 Summary

We have now all the elements that are required to describe the incompressibleflow of a viscous, conducting liquid in a magnetic field. The governing equa-tions of incompressible MHD are given by a combination of the incompress-ibility constraint (1.18), the Navier-Stokes equations (1.15) and the inductionequation for the magnetic field (1.41). This latter equation does only guaranteethe divergence-free character of the magnetic field, if B is initially solenoidal.Therefore, Gauss’s law for the magnetic field (1.29) is thus required also for a

9


complete description of an MHD flow, but only as an initial condition:

∇ · u = 0 (1.42)

∂u

∂t+ u · ∇u = −ρ−1∇p+ ν∇2u+ ρ−1J×B+ ρ−1f (1.43)

∂B

∂t+ u · ∇B = B · ∇u+

1

µσ∇2B (1.44)

∇ ·B = 0 (1.45)

We can write these equations under a non-dimensional form by the followingsubstitutions: u → Uu, B → B0B, ∇ → L−1∇, t → LU−1t, J → σUB0J,p→ ρU2p. We obtain:

∇ · u = 0 (1.46)

∂u

∂t+ u · ∇u = −∇p+Re−1∇2u+NJ×B (1.47)

∂B

∂t+ u · ∇B = B · ∇u+R−1

m ∇2B (1.48)

∇ ·B = 0 (1.49)

We see that, in a given geometry, we can characterize MHD flows by merelythree dimensionless groups: the Reynolds number Re, the interaction parameter(or Stuart number) N , and the magnetic Reynolds number Rm. The definitionand physical meaning of these parameters is discussed below.

The Reynolds number is a non-dimensional estimate of the ratio betweenconvective and viscous forces in the Navier-Stokes equation. An order-of-magnitude estimate yields:

Re =O (u · ∇u)

O (ν∇2u)=U2L−1

νL−2U=UL

ν(1.50)

This is the only parameter in the hydrodynamic Navier-Stokes equations. Ifthe Reynolds number is small, small-scale fluctuations can not overcome thedissipative action of the viscous forces, and will quickly be damped. Thisresults in a homogenized flow in which only slow variations of the velocity fieldare possible. At large Reynolds number however, small-scale fluctuations canpersist and grow due to the increasing impact of the (non-linear) convectiveterm. This will give rise to a seemingly random behavior that is characterizedby a large range of spatial temporal scales, a state known as turbulence.

The analogous ratio between convective and diffusive terms in the magneticinduction equation is known under the name magnetic Reynolds number, andis defined as:

Rm =O(u · ∇B)

(µσ)−1∇2B= µσ

UL−1B

L−2B= µσUL (1.51)

10

1.3. THE QUASI-STATIC APPROXIMATION 11

The phenomenology of MHD systems will depend strongly on the value ofRm, as it is the only parameter governing the induction equation. When Rm issmall, magnetic field fluctuations relax quickly; velocity inhomogeneities, whichare at the origin of these fluctuations, hardly affect the magnetic field. Suchflows have a dissipative nature; the kinetic energy of the fluid is transformedinto heat due to Joulean dissipation. Small values of the magnetic Reynoldsnumber are typical for manmade flows, like the ones encountered in laboratoryand industrial processes. We will analyse the case of Rm ≪ 1 in more detailin the following section. High values of Rm on the other hand are typical forlarge-scale terrestrial or astrophysical flows. One may think of the motion ofthe liquid core of the earth (Rm ≈ 104 − 105) or of astrophysical processes likesun spots (Rm ≈ 108). Such flows are such that the magnetic field is ‘frozen’into the fluid, and can exhibit wave-like behavior.

The interaction parameter is a measure for the ratio between electromag-netic and inertial forces. Its definition reads:

N =σB2

0L

ρU(1.52)

It is also instructive to interpret this parameter as the ratio between two timescales. On the one hand, there is the Joule damping time, τJ , which is thetypical time needed by the Lorentz force to damp a vortex. It is given by:τJ = ρ/σB2

0 . On the other hand, we have the eddy-turnover time τe which isthe time scale on which a vortex moves over a typical length scale L: τe = L/U ,and we can write the interaction parameter as:

N =τeτJ

(1.53)

At last, we will introduce a fourth parameter known as the Hartmann num-ber, which is a hybrid of Re and N .

M = B0L

√

σ

ρν=

√N Re (1.54)

The square of M measures electromagnetic forces with respect to viscous ones.It will be mainly useful in laminar cases in which the convective term is negli-gible.

1.3 The quasi-static approximation

1.3.1 Simplified equations for Rm≪ 1

In the previous section, we found that the induction equation contains onlyone non-dimensional group: the magnetic Reynolds number Rm. For most

11


terrestrial, laboratory and industrial flows, this parameter is typically smallcompared to one. The assumption that Rm is vanishing, allows for a substan-tial simplification of the MHD equations [Rob67a, Dav01]. Consider thereforeequation (1.41), in which we decompose the magnetic field in a uniform, sta-tionary, externally imposed part Bext and a fluctuating part b:

∂b

∂t+ u · ∇b = Bext · ∇u+ b · ∇u+

1

µσ∇2b (1.55)

An order-of-magnitude estimate of the convective and diffusive terms gives:

O (u · ∇b)

O ((µσ)−1∇2b)=

O (b · ∇u)

O ((µσ)−1∇2b)=

UbL−1

(µσ)−1bL−2= µσUL = Rm (1.56)

where b is a typical scale of the fluctuating magnetic field. We can thus neglectthe second term on both the right- and left-hand side of (1.55). This leaves uswith:

∂b

∂t= Bext · ∇u+

1

µσ∇2b (1.57)

This is a diffusion equation with a source term due to gradients in the veloc-ity field. We can now pursue our analysis by considering the time scales τassociated with both terms on the right-hand side. The ratio between these is:

τ(

(µσ)−1∇2b)

τ (Bext · ∇u)=L2µσ

LU−1= Rm (1.58)

This means that the fast response of the diffusion term makes the magneticfield fluctuations adapt quasi-instantaneously to (slower) variations due to theflow. As such, we end up with the following static equation for the inducedmagnetic field:

Bext · ∇u+1

µσ∇2b = 0 (1.59)

An order-of-magnitude estimate of this equation leads to the following result:

O(b)

O(Bext)= µσUL = Rm (1.60)

We find thus that, in the limit of vanishing Rm, the induced magnetic field isnegligible with respect to the externally imposed one.

We can use this last result to formulate the quasi-static approximation in adifferent, but equivalent way. To this end, we use the pre-Maxwellian form ofAmpere’s equation (1.38) to eliminate b from (1.59):

∇×(

u×Bext)

− 1

σ∇× j = 0 (1.61)

12


In this expression, the symbol j denotes the current density induced by theflow. Using Helmholtz’s decomposition (A.22), we may ‘uncurl’ this equation;this results in:

u×Bext − 1

σj = ∇φ (1.62)

Moreover, the total current density J can be written as a sum of its ‘induced’part j and an externally imposed current source Jext: J = Jext + j. Jext iscaused by an external electric field Eext = −∇φext. Together with (1.62), weobtain.

J = σ(

−∇(

φ+ φext)

+ u×Bext)

(1.63)

Comparison with Ohm’s law (1.32) teaches us that the electric field E in thequasi-static limit can be derived from a scalar potential φ+φext: E = −∇(φ+φext). If we replace now φ+φext by φ, we can express the charge-conservationlaw (1.34) as:

∇ ·(

σ(

−∇φ+ u×Bext))

= 0 (1.64)

If σ is a constant, which we will assume this throughout this work, we end upwith a Poisson equation for the electrical potential:

∇2φ = ∇ ·(

u×Bext)

(1.65)

Upon the rescaling φ→ φULB0, the full non-dimensional system of quasi-staticMHD equations becomes:

∇ · u = 0 (1.66)

∂u

∂t+ u · ∇u = −∇p+Re−1∇2u+N(−∇φ+ u×Bext)×Bext(1.67)

∇2φ = ∇ ·(

u×Bext)

(1.68)

From now on, we will drop the superscript ext, and denote the external mag-netic field as B. This variable is now however to be considered as an imposedparameter, and not as an unknown of the system (1.66-1.68). By using ascalar potential instead of the magnetic field fluctuations, we reduce the ‘elec-tromagnetic’ unknowns from three to one. Furthermore, the magnetic field isinfinitely extended, even if the electric currents are localized in space; the in-duction equation should thus in principle be also solved outside the domain ofinterest. Therefore, for many situations of practical interest electric boundaryconditions at the wall are more easily expressed in terms of the potential.

Boundary conditions for the electric potential

The solutions to the set of equations (1.66)-(1.68) are not completely deter-mined until suitable boundary conditions for the electric potential have beendefined. In general, one should include the wall into the solution domain of the

13


σ

σw

Γ

Γ’σw

tw

n

Figure 1.1: Sketch of a section of a wall portion: Γ is the fluid-wall interface,Γ’ the exterior wall, tw the (uniform) wall thickness.

Poisson equation (1.68) and solve this equation with appropriate jump condi-tions at the fluid-wall interface. These state that the normal component of thecurrent across the interface and the tangential component of the electric fieldalong the interface should remain constant. Furthermore, we assume that thereis no contact resistance between the fluid and the wall, so that the potentialis continuous across the interface. If we associate the subscript w with wallvariables (see figure 1.1) and take into account Ohm’s law, these conditionsread:

σw

(

∂φw∂n

)∣

∣

∣

∣

Γ

= σ∂φ

∂n

∣

∣

∣

∣

Γ

(1.69)

(∇τφw)|Γ = (∇τφ)|Γ (1.70)

φw |Γ = φ|Γ (1.71)

When, however, the wall thickness is small compared to the fluid domain, anapproximate boundary condition can be derived [Wal81], which allows us torestrict the solution of equation (1.68) to the fluid domain. To this end, westart from the charge conservation equation in the wall, which can be writtenunder the following form:

∂jn,w∂n

= − (∇τ · jw,τ ) (1.72)

Here, we have split the current and the nabla operator in a component normaland a component tangential to the wall, i.e: jn,w = n · jw , jw,τ = j − jw,nn

and ∂n = n · ∇, ∇τ = ∇ − n∂n. We now integrate this expression withthe assumption that the potential does not vary up to the leading order ofapproximation in the wall. The underlying physical idea is that wall currents

14


discharge tangentially in a quasi-two-dimensional way.

jn,w|Γ′ − jn,w|Γ = −tw∇τ · jτ,w (1.73)

The first term on the left hand side is zero since there are no currents inthe insulating domain outside the fluid-wall system. Taking into account theaforementioned jump conditions, we eventually obtain:

∂φ

∂n= ∇τ ·

(

σwtwσ

∇τφ

)

(1.74)

or in non-dimensional form:

∂φ

∂n= ∇τ · (c∇τφ) (1.75)

where the wall conductance ratio c is defined as:

c =σwtwσL

(1.76)

We now consider two limiting cases of this thin-wall condition. If the wall isperfectly conducting, σw and c tend to infinity. Wall currents should howeverremain finite, and this can only be achieved if the tangential electric field in thewall vanishes. In other words, the potential along a perfectly conducting wall isconstant, and the thin-wall condition reduces to a familiar Dirichlet conditionfor the potential:

φ = C (1.77)

where C is an arbitrary integration constant. When the boundary contains dif-ferent perfectly conducting wall portions which are not electrically connected,one should prescribe the potential difference between those portions.

If, on the other hand, the wall is perfectly insulating, currents cannot pen-etrate from the fluid into the wall. We see indeed that, for c = 0, the thin-wallcondition simplifies to a homogeneous Neumann condition for the electric po-tential:

∂φ

∂n= 0 (1.78)

1.3.2 Phenomenology of the quasi-static regime

In this subsection, we will give an overview of the most relevant phenomenaemerging in conducting flows subjected to a static and uniform magnetic field.In a first stage, we will disregard boundary effects and concentrate on homo-geneous flows. These are flows whose statistical properties are invariant undertranslation. We will thereafter study the effect of boundaries in the context ofstraight laminar channel and duct flow. Although not completely generic, itwill allow us to introduce the most common boundary layer types.

15


Figure 1.2: Evolution towards a quasi-2D state of a periodic box of decaying,initially isotropic, homogeneous turbulence in a uniform magnetic field at in-teraction parameter N = 10. Snapshots of the kinetic energy contours after2.85 (left), 10.2 (center) and 26.8 (right) Joule damping time units τJ . Figuretaken from [KM04].

Homogeneous flows

In figure 1.2, we illustrate how a periodic box of decaying, initially isotropic,homogeneous turbulence evolves after the application of a uniform magneticfield. The most prominent effect is the emergence of anisotropy, i.e. a loss ofstatistical invariance with respect to rotation. More specifically, all variationsalong magnetic field lines tend to be suppressed, while inhomogeneities in di-rections perpendicular to the field are hardly affected. We can explain thisproperty by a heuristic argument developed by Davidson [Dav97]. We considertherefore the evolution of the total energy and field-aligned angular momentumof a quasi-static MHD flow. For the sake of simplicity, we will neglect viscouseffects; this high Reynolds number approximation implies that we are consid-ering flows which are highly turbulent. The energy balance can be obtained bytaking the scalar product between the velocity u and equation (1.67). Aftersome mathematical manipulation, we obtain:

∫

Ω

∂t

(

ρ1

2u2

)

dV +

∮

∂Ω

u

(

p+1

2ρu2

)

· dS =

− 1

σ

∫

Ω

J2 dV −∮

∂Ω

φJ · dS (1.79)

If we make abstraction from boundary terms, we see that the total kinetic en-ergy of the flow is a monotonically decreasing function of time. The mechanismdriving this loss of kinetic energy is Joulean dissipation.

Newton’s laws on the other hand tell us that the rate of change of globalangular momentum of a body equals the net torque that is exerted on that

16


body. Here, the only force contributing to this torque is the Lorentz force.

d

dt

∫

Ω

ρx× u dV =

∫

Ω

x× (J×B) dV (1.80)

After some tedious algebra, we can rewrite this as:

d

dt

∫

Ω

ρx× u dV =

∫

Ω

σ ((x× u)×B)×B dV +

∮

∂Ω

... · dS (1.81)

Again disregarding boundary effects, the final result shows us that the totalangular momentum component along the magnetic field is a conserved quantity.

The combination of equations (1.80) and (1.81) now presents a paradox.The first equation prescribes that kinetic energy is destroyed without cease,while the second one states that a certain amount of motion should be main-tained, and that the flow cannot come to rest. The only possible way to satisfyboth constraints is that the flow organizes itself in such a way that the Jouledissipation, and thus the electrical currents tend to zero. Thus, after someinitial time, we should reach a state in which:

u×B ≈ ∇φ (1.82)

If we now take the curl of this expression, then the right-hand side vanishes,and we finally obtain:

B · ∇u ≈ 0 (1.83)

This explains why the flow reaches a so-called quasi-two-dimensional state (i.e.three non-zero components of the velocity depending only on two coordinates),with no variations along magnetic field lines, as shown in figure 1.2.

Boundary layers

We now investigate how the presence of a magnetic field influences the natureof a boundary layer in a laminar channel and square duct flow. We will takex as the flow direction, so that all derivatives with respect to x vanish, withexception from the pressure gradient needed to drive the flow. The magneticfield is defined as B = B01y. We now consider a channel (see figure 1.3(a))with its walls, located at y = ±1, perpendicular to the magnetic field lines.Furthermore, the flow is homogeneous in z-direction, and we can leave aside allz-dependencies, except for the potential, since we cannot exclude the presenceof a spanwise-orientated induced electric field. We have thus u = u(y)1x, withu and φ obeying the following set of equations:

− ∂p

∂x+ ρν

∂2u

∂y2+ σB0

(

∂φ

∂z−B0u

)

= 0 (1.84)

∂2φ

∂y2= 0 (1.85)

17


Figure 1.3: Channel (a) and duct (b) flow geometry. The Hartmann walls havea dark grey shading, the side walls are coloured in lighter grey.

It will be instructive to involve explicitly the wall domain into the calculation.The velocity in this domain is 0, and the potential obeys a Laplace equation:

∂2φ

∂y2= 0 (1.86)

The boundary condition for the velocity at y = ±1 is u = 0. Furthermore, weimpose that the potential is continuous across the interface, just like the wall-tangential components of the potential gradient and the normal component ofthe electric current density. All this leads to the following solution for φ and u:

φ = Az +B (1.87)

u =1

ρν

(

− ∂p

∂xM−2 + σB0A

)(

1− cosh(My)

cosh(M)

)

(1.88)

Here, A and B are integration constants, and M is the Hartmann number asdefined in (1.54). The choice of B is arbitrary, but A is determined by the factthat the total current in the combined fluid-wall domain should integrate tozero. The velocity profile is then:

u =1

M

1 + c

cM + tanh(M)

1

ρν

∂p

∂x

(

1− cosh(My)

cosh(M)

)

(1.89)

We recall that c stands for the wall-conductance ratio, defined in (1.76). Infigure 1.4, we show the shape of the velocity profile for different values of theHartmann number. It consists of an extended, flat core, and thin, exponentialboundary layers, which are called Hartmann layers. As M increases, theirthickness decreases as M−1. This illustrates again that, far enough away fromboundaries, the main effect of the magnetic field is to suppress variations alongmagnetic field lines.

18


−1 −0.5 0 0.5 10

0.5

1

1.5

y

u/U

b

M = 0M = 10M = 100

Figure 1.4: Sketch of the velocity profile of a laminar MHD channel flow forseveral values of the Hartmann number. The normalization velocity Ub isdefined as the average velocity.

The scaling of the velocity magnitude with the Hartmann number can beunderstood by the following arguments. In a channel with conducting walls,φ is zero, and the magnitude of the Lorentz force density is σuB2

0 . In thehigh Hartmann number regime, this force dominates the core flow; hence, thecore velocity should scale as u ∝ −∂xp(σB2

0)−1 ∝ −∂xpM−2(ρν)−1. If, on the

other hand, the walls are perfectly insulating, a spanwise potential gradientwill be induced which counteracts the effect of the term u ×B. The effect ofthe Lorentz force density is now to brake the core flow, and to accelerate thebounday layers. The integral of the J and fL between y = −1 and y = 1 is zero.To find the proper scaling, we integrate the remaining terms in the momentumbalance between y = −1 and y = 1. We find:

∫ 1

−1

∂p

∂xdy = ρν

∫ 1

−1

∂2u

∂y2dy = ρν

(

∂u

∂y

∣

∣

∣

∣

y=1

− ∂u

∂y

∣

∣

∣

∣

y=−1

)

(1.90)

We see that in this case, the pressure gradient has to compensate for viscous

19


Figure 1.5: Laminar MHD flow in a straight duct with perfectly insulatingwalls: velocity along the duct centerline parallel (left) and perpendicular (cen-ter) to the magnetic field for different values of the Hartmann number M .Sketch of the current streamlines (right).

momentum losses at the boundaries. Since these are large due to the steepprofile of the boundary layers, the velocity magnitude is an order-of-magnitudeM smaller then in the Poiseuille flow driven by the same pressure gradient.

For a square duct, like the one shown in figure 1.3(b), we have an additionalpair of walls located at z = ±1. These walls are called side walls and theirrespective boundary layers side layers. Analytical solutions are now only possi-ble for a few specific combinations of the wall conductivity c, and we will limitourselves to a rather qualitative discussion of the most representative cases.In figures 1.5 - 1.7, we show a sketch of the current stream lines and velocityprofiles for three different combinations of c.

In figure 1.5, all the walls are perfect insulators. A solution for this problemwas provided by [She53]. The interaction between the flow and the magneticfield drives a current in z-direction, which brakes the flow. The current lines canhowever not enter into the insulating side walls, so that a potential difference isinduced, which makes the current lines bend and close through the Hartmannlayers. In the side layers, the current is almost parallel to the magnetic fieldso that the Lorentz force is weaker there. It turns out that these layers have atypical thickness of O(M−1/2). Similarly to the channel with insulating walls,the velocity magnitude is an order M smaller then in the hydrodynamic casedriven by the same pressure gradient.

In the second case (figure 1.6), all walls are perfect conductors. The inducedcurrent closes its loops preferably through the walls because of their lowerresistance. The current lines enter the walls perpendicularly, but are slightlydeviated in the side layers; hence the component of the current perpendicularto the magnetic field in these zones is somewhat smaller than in the core. Thisgives rise to small overspeed zones above a certain threshold in the Hartmann

20


number. The ratio between the amplitude of the velocity in the side layers andthe core scales as O(M0). The core velocity itself scales as M−2∂xp.

Figure 1.6: Laminar MHD flow in a straight duct with perfectly conductingwalls: velocity along the duct centerline parallel (left) and perpendicular (cen-ter) to the magnetic field for different values of the Hartmann number M .Sketch of the current streamlines (right).

Finally, figure 1.7 sketches the behavior in a duct with perfectly conductingHartmann walls and perfectly insulating side walls. This case was first studiedby Hunt [Hun65]. Compared to the insulating case, the magnitude of thecurrents can be larger, since the current lines can form closed loops by enteringinto the Hartmann walls, which provide a path of much lower resistivity. Inthe side layers, the current flows parallel to the field towards the perfectlyconducting Hartmann walls. This means that the Lorentz force is vanishing inthese regions, and that the velocity is much higher than in the core. The ratiobetween the amplitude of these jets and the amplitude of the core flow scales asO(M). Since their thickness scales as M−1/2, we find that, at high Hartmannnumber, the mass flow rate carried by the core is negligible with respect tothe one in the side layers. We also note that the flow in the core may becomereversed if the Hartmann number is larger then 89.

1.3.3 Examples of quasi-static MHD flows

Liquid metal flows in fusion blankets Much effort is put in the realizationof thermonuclear fusion as a reliable and sustainable energy source. The reac-tion between a deuterium and tritium core can only take place if both reactantsare completely ionized. These plasmas are characterized by very high conduc-tivities and densities. Intense magnetic fields are used to confine these plasmaswithin the reaction vessel. While conventional MHD does not accurately de-scribe plasma-related effects, it comes into play in the context of thermonuclearfusion when the flow of liquid metals in the so-called blankets is considered.

21


Figure 1.7: Laminar MHD flow in a straight duct with perfectly conductingside walls and perfectly insulating Hartmann walls: velocity along the ductcenterline parallel (left) and perpendicular (center) to the magnetic field fordifferent values of the Hartmann number M . Sketch of the current streamlines(right).

These blankets are multifunctional: in the first place, they should absorb theneutron flux and convert the kinetic energy of the neutrons into heat, which canthen be used to drive a classical turbine process. Liquid metals are candidatecoolant liquids since they can be operated at high temperature and have highthermal conductivities [B07]. The second function of the blankets is to protectthe magnetic field coils from intense, damaging neutron radiation. In moreadvanced blanket designs (called self-cooling blankets), the liquid coolants alsohave to provide the tritium needed for the fusion reaction. This is possibleif the coolant contains lithium. Liquids that are considered are pure lithiumor eutectic lead-lithium alloy. The combination of their material properties atoperating conditions (750 K) and the ambient conditions in a fusion environ-ment (B0=10 T, L = 0.05 m, U = 0.5 m/s) are such that the non-dimensionalparameters typically have the following order of magnitude: M = 104 − 105,N = 103−105, Re = 104−105. Given the high values ofM , the velocity profilein these components often takes the form of laminar, inviscid, inertialess coreflows, surrounded by different types of boundary layers due to the presence ofsolid walls and geometrical discontinuities. On the other hand, the wall con-ductance ratio of some blanket components is such that strong jets occur in theside layers, and these may exhibit (quasi two-dimensional) turbulent behavior.

The main challenge for this type of application is that we have to computethe flow in fairly complex elements like expansions, bends, manifolds, helicalvanes, etc (see e.g. figure 1.8). The high value of the Hartmann numbermakes the flow morphology highly anisotropic, so that the problem is badlyconditioned for conventional computational fluid dynamic. On the other hand,the strength of the interaction parameter is such that turbulent behavior is

22


Figure 1.8: Left: Thermonuclear fusion device ITER (source:http://www.iter.org). Right: illustration of one of the possible blan-ket concepts, in casu the Dual Coolant Lead Lithium (DCLL) concept, withindication of the coolant flow direction (courtesy by C. Mistrangelo).

not a major issue here. Furthermore, many other complex effects still have tobe incorporated and studied. Among others, we mention: buoyant behavior,multi-channel effects, free-surface flows, evaporation phenomena, etc.

Classical and Lorentz force velocimetry The classical way of measuringflow rates with magnetic fields is a direct application of Faraday’s law. When,e.g. a straight channel flow like the one shown in figure 1.3(a), is subjectedto a wall-normal magnetic field, a potential difference is induced between thechannel walls. If the channel walls are good conductors, this potential differ-ence is directly proportional to the flow rate [HS65]. However this technique isdifficult to apply in high-temperature industrial melt flows (aluminium, glass),because the measuring electrodes suffer from corrosion under ambient condi-tions. Recently, a new technique, termed Lorentz force velocimetry [TVK06]was introduced, which has the benefit of being a non-contact technique. Itsprinciple is the following: when a flow is exposed to a permanent magnet, themagnet will exert a force on the flow. By virtue of the action-reaction princi-ple, the magnet will undergo an equal but opposite drag force. The flow ratecan then be inferred from the measurement of the force acting on the magnetsystem.

23


Magnetic damping in metallurgical processes We have already illus-trated that a magnetic field can suppress the motion of a liquid metal flowunder quasi-static conditions. The mechanism responsible for this is Jouledissipation. This property is often exploited in metallurgical processes likecontinuous steel casting or Czochralski growth of semiconductors. This motionis induced by the inflow of liquid metal in the mould or by natural convection.It is in general undesired since it creates inhomogeneities in the chemical com-position of the flow, which may result in a degradation of the metallurgicalstructure of the solid.

24

Chapter 2

Numerical framework

“Life can only be understood backwards; but it must be lived for-wards.”Soren Kierkegaard

In the previous chapter, we have described the partial differential equationswhich govern the physics of quasi-static magnetohydrodynamic flow. Due totheir non-linear nature, exact solutions, obtained with analytical methods, arenot readily available, with the exception of a few problems in simple geometriesand with trivial initial conditions. If we want to make predictions for flows ofpractical interest, we have to recourse to numerical methods, i.e approximate aset of partial differential equations by a system of algebraic equations. In otherwords, we must approximate continuous functions of space and time with afinite amount of information.

In certain types of methods, like the spectral or finite-element method,this is achieved by writing the variables (velocity, pressure, electric potential,etc.) as a linear combination of a finite number of basic functions. In finite-difference methods on the other hand, the variables are only represented ina finite number of points. We will adopt a third approach here, which iscalled the finite-volume method (FVM), and which is by far the most popularmethod in the field of computational fluid dynamics, with pioneering studiesperformed by McDonald [McD71] and MacCormack and Paullay [MP72] forthe case of the two-dimensional Euler equations. Shortly afterwards, Rizzi andInouye [RI73] applied a FVM approach to solve the Navier-Stokes equations(including viscous effects). The basic building block of FVM is a small controlvolume (CV), in which the solution domain is divided. The discretizationprocedure is then based on the integral of the conservation equations over thisCV. In the first section, on spatial discretization, we will introduce the FVMmethod in more detail, and we will show how we can construct approximations

25

26 CHAPTER 2. NUMERICAL FRAMEWORK

to various spatial differential operators.

This will allow us to transform the Navier-Stokes equations into a systemof ordinary differential equations with respect to time. The algorithms used toadvance this system in time are the subject of the second section. The majorissue here will be how to maintain (discrete) conservation of mass, since thetime derivative does not appear explicitly in the mass conservation equation.In these first two sections, we will leave out of consideration all aspects relatedto the numerical computation of the Lorentz force. This subject will be treatedin the third section of this chapter.

The combined space-time-discretization procedure will eventually lead to asystem of algebraic equations. In the fourth and last section, we will give anoverview of the different iterative methods which we have used to solve suchsystems.

The methods presented in this chapter have been implemented in an earlyversion of the unstructured parallel finite-volume code YALES2 (Yet AnotherLES Solver) [Mou10]. This is a versatile numerical solver for a broad range ofmulti-physics flow problems (combustion, magnetohydrodynamics, multi-phaseflows, etc.). It originates from, and is maintained at CORIA. In the context ofthe present work, the following contributions to this code were provided:

• A consistent formulation of a compact stencil for the Laplacian operatoron unstructured grids (see subsection 2.1.6).

• The implementation of implicit time-advancement methods for the mo-mentum equation (subsection 2.2.2).

• The implementation of a convective boundary condition for outflow bound-aries (subsection 2.2.4).

• The development of a module for the quasi-static MHD equations; thisincludes all aspects treated in section 2.3.

• The implementation of a Krylov subspace method and the coupling ofthe code with an algebraic multigrid solver for the solution of the Poissonequations for pressure and potential (see subsections 2.4.3-2.4.2).

2.1 Spatial discretization

2.1.1 Basic principle of the finite-volume method

To introduce the finite-volume method, we consider a polyhedral control volumeΩ, bounded by a number of faces fa, and integrate the momentum equation

26

2.1. SPATIAL DISCRETIZATION 27

over this control volume:∫

Ω

∂u

∂tdV =

∫

Ω

(

−∇ · (uu)− 1

ρ∇p+ ν∇2u+

1

ρfb

)

dV

=

∮

∂Ω

(

−uu− p

ρ1 + ν∇u

)

· dS+

∫

Ω

1

ρfb dV

=∑

fa

∫

Sfa

(

−uu− p

ρ1 + ν∇u

)

· dS+

∫

Ω

1

ρfb dV (2.1)

The surface integral term represents a transport of momentum towards neigh-bouring control volumes, and illustrates the conservative nature of the Navier-Stokes equations: the surface force terms do not create or annihilate momen-tum, they only redistribute it.

In FVM, we approximate equation (2.1) for a large set of adjacent controlvolumes. This means that we express the volume and surface integrals in (2.1)as a well-chosen combination of the values of the variables at specific locationswithin the control volume and its neighbours. The main advantage of thisapproach is that the discretized surface forces inherently conserve (discrete)momentum. If we sum the expression over a set of adjacent CV’s, each of theflux terms will appear twice, but with a different sign, so that these contribu-tions cancel each other; the remaining contributions to the net rate-of-changeof the global momentum are the fluxes through the exterior boundaries.

2.1.2 Construction of the control volumes

A mesh can be defined as a set of non-overlapping polygons (in two dimen-sions) or polyhedra (in three dimensions) which fill completely a well-defineddomain in space. We will refer to these constituting bricks as elements, to theirvertices as nodes and to their mutual interfaces as element faces. Moreover,we will restrict ourselves to the two types of two-dimensional and four typesof three-dimensional elements shown in figure 2.1. We can distinguish betweenstructured and unstructured meshes. The former can be mapped on simpledata structures. A subgroup of structured meshes are the cartesian meshes.In this case, the grid points are located along orthogonal grid lines. Despitethe huge increase in spatial resolution that has been reached over the past fewdecades, the generation of an appropriate mesh remains a critical issue for thesimulation of fluid flow in complex geometries. Badly designed meshes cangive rise to large numerical errors, which may completely destroy the physicalcontent of the simulation. We shall later specify which criteria should be takencare of.

Within one CV, we have some freedom on the arrangement of the vari-ables. In staggered meshes, the velocities are stored at the CV faces, and theother quantities (pressure, electric potential, transported scalar quantities) are

27


defined at the CV nodes. This arrangement is attractive because it does notlead to spurious pressure oscillations, known as the checkerboard problem (thiswill be explained in more detail in section 2.1.6). Furthermore, staggered ar-rangements do not require ad-hoc boundary conditions for the pressure, andcan simultaneously conserve mass, momentum and kinetic energy for an invis-cid flow [HW65]. However, for three-dimensional simulations and unstructuredmeshes, a staggered definition of the variables becomes very complex, since oneneeds separate CV’s for node and face defined quantities [MCM04]. Therefore,a collocated arrangement is chosen here.

There exist two common approaches to construct the CV’s once a mesh isgiven. In the cell-center-based methods, the elements themselves are the basicbuilding blocks for which the integral version of the conservation laws is solved.In the vertex-based approach, which was adopted in this work, a dual set ofcontrol volumes is created, which are centered around the vertices of the ele-ments. In [HMI06], it was found that the more accurate results for a turbulentchannel flow were obtained with a vertex-based formulation. However, thissingle study does not allow us to draw a conclusion on the superiority of thevertex-based approach for other geometries.

Figure 2.2 illustrates how the dual volumes, which we will call nodal vol-umes, are built up out of a number of simplexes, i.e. triangles (called subtri-angle and denoted subtri) in two, and tetrahedra (called subtetrahaedron anddenoted subtet) in three dimensions. For a two-dimensional grid, each triangleis obtained by connecting the node under consideration, the barycenter of theelement and the midpoint of an edge connected to this node. Similarly, inthree dimensions, the basic tetrahedra are defined by four vertices: the node,the barycenter of the element, the midpoint of an edge connected to the node,and the barycenter of an element face to which the edge belongs. Furthermore,the CV’s of boundary nodes are closed by attributing portions of the boundaryelement faces to their respective nodes. In two dimensions, the boundary seg-ment is just split in two; for three-dimensional meshes, we divide the boundarytriangles or quadrilaterals in subtris like explained before, and associate a sub-tri with a boundary node if the node is one of its vertices. The total volume ofthe nodal CV’s can be computed as the sum of the volume of its subtris/subtetsst. Let Sst be the normal vector on the subtri face opposing the node and xst

the vector connecting the node and the midpoint of the edge used to define st(see figure 2.2). Then we have:

Vnode =1

ndim

∑

st

xst · Sst (2.2)

Furthermore, the surface of the CV is formed by the union of all subtri/subtetfaces opposing the node. Since this surface is a closed, the subtri/subtet face

28


Figure 2.1: Constituting elements of a mesh: triangle (a), tetrahedron (b),prism (c), quadrilateral (d), pyramid (e), hexahedron (f).

normals satisfy:∑

st

Sst = 0 (2.3)

As mentioned before, we can express the different conservative terms in theNavier-Stokes equations as a surface integral over the surface of the CV. Thisrequires that we dispose of (approximated) values of the variables at the surfacelocation. These are obtained by a well-chosen interpolation of the values at thenodes. It will appear useful to assign distinct portions of the CV surface topairs of nodes, and we denote such a surface patch associated with a pair a pairface (in contrast to element faces). Hence, the surface integral for a certain CVcan be written as a sum over all the pairs to which belongs the CV node.

We have the choice between two approaches for the definition of the pairs. Insparse stencils, we consider only those pairs which are connected by a physicalelement edge, whereas in dense stencils, we also construct pairs which are onlyvirtually connected, like pairs connected by diagonals. The attribution of thedifferent surface portions to the different pairs is illustrated in figures 2.3 - 2.4.

In the sparse approach, the CV surface of a subtri (subtet) is assigned to thepair, whose midpoint was used to define the subtri (subtet) under consideration.Consider for instance the surface patch Sst shown in figure 2.2(c). This surfacepatch is assigned to the node pair consisting of the hidden and purple nodedepicted in figure 2.4(e), since the midpoint of the edge between these nodes isa vertex of the triangle which defines Sst. In three dimensions (figures 2.3(b)

29


Figure 2.2: Construction of the control volumes: (a) Subtri of a node pairin a triangular element, (b) Control volume of a node in a mixed triangu-lar/quadrilateral mesh, (c) Subtri of an edge/face cobmination in a hexahedron,(d) Exterior triangles of all subtetrahedra of a hexahedral element.

and 2.4(c) and (e)), each pair face can be represented as a quadrilateral surface,which is the sum of two triangles; compare for instance figures 2.2(d) and 2.4(e).

We now explain how the pair face normals in the dense formulation can becomputed:

• In triangular and tetrahedral elements, there are no diagonals, and theattribution of CV surface patches to node pairs is the same for bothstencils, and is shown in figure 2.3.

• Within one quadrilateral element, each node is physically connected totwo other nodes. Hence, there are two pair faces in the sparse formulation(the red and blue line in figure 2.4(a)). In the dense approach, eachof these surface patches is split in two, and one half of each normal is

30


Figure 2.3: Illustration of the pair face for a node in a triangular (a) andtetrahedral element (b).

assigned to the diagonal pair involving the node under consideration; theresulting normal for the diagonal pair is shown in green in figure 2.4(b).The other halves remain associated with the ‘physical’ pairs.

• In prisms, the external triangular surfaces of the original subtets are splitin two if one of their edges is located in a quadrilateral element face,and one of the halves is assigned to the diagonal pair of the quadrilateralelement surface under consideration. The same is true for pyramids; theseare not shown here, because such elements have not been used further inthis work.

• For hexahaedra, we should consider space diagonals as well as diagonalswhich belong to element faces. Each of the exterior surface trianglesshown in figure 2.2 (d) is bisected, so that we obtain four surface patchesfor each ‘sparse’ pair face (see figure 2.4(e)). These are attributed asfollows: one to the given physical pair, one to the space diagonal involvingthe given node, and one to the diagonals of the faces to which the givenedge belongs.

We will use the following notation. The pair face normal to a pair of nodes i, jis Si,j . The fraction of this normal within the element el will be denoted Si,j|el.Furthermore, the set of all nodes j connected to i through a pair is π(i); πel(i)is the subset of this collection restricted to the nodes of the element el.

2.1.3 Volume averages

In equation (2.1), we need the average of the unsteady and body force termover the CV volume. For an arbitrary function g, we can write the volume

31


Figure 2.4: Illustration of sparse (a,c,e) and dense (b,d,f) discretization stencilsfor three types of elements. The CV surface patch is associated with a nodepair involving a node and the node of the same color.

32


integral as a Taylor expansion in the vicinity of the CV node i:∫

Ω

g dV =

∫

Ω

(

g|i + (∇g)|i · (x− xi) +O(∆2))

dV (2.4)

Here, ∆ is a length scale which is representative for the size of the CV. Weapproximate the average of g over the CV by its nodal value, i.e.:

∫

Ω

g dV ≈ Vi g|i (2.5)

For a generic mesh, this expression is clearly only first-order accurate. From(2.4), we learn that this approximation becomes second-order accurate if theCV center of mass is located at the node position. This is only true if themesh is locally uniform. We can significantly reduce the discretization errorby avoiding large size jumps between adjacent elements. If the physics requireus to locally refine the grid, this refinement should be obtained by a smoothdecrease in grid size towards the region of interest. Note also that, in boundaryCV’s, the node is on the surface. For such CV’s, this approximation can neverbe second-order accurate.

2.1.4 The gradient operator

The discretization stencil for the gradient of a function, evaluated at a node i,can be derived using Gauss’s theorem (A.10):

∫

Ωi

∇g dV =

∮

∂Ωi

g dS (2.6)

We can expand the left-hand side in a Taylor series around the location of theCV node, and retain the first-order approximation like in the previous section.For nodes not on the boundary, the surface integral can further be developedas a sum over the element pair faces (i, j) which contain the CV node i underconsideration:

∮

∂Ωi

g dS =∑

j∈π(i)

∫

Si,j

g dS

≈∑

j∈π(i)

g|(i,j)Si,j

≈∑

j∈π(i)

g|i + g|j2

Si,j (2.7)

Here, the vectors Si,j represent the normal vectors of the surface patches of theCV of node i, attributed to the pair (i, j). By construction, Si,j = −Sj,i

33


If the node is on the boundary, the surface integral in the right-hand sideof (2.6) can be completed by adding a similar summation over the boundaryfaces to the right-hand side.

The approximation∫

Si,jg dS = g|(i,j)Si,j is first-order accurate since the

midpoint (i, j) of the edge between nodes i and j is not necessarily situated atthe barycenter of the surface Si,j . See e.g. the pair (P,N) in figure 2.5(a). Wecan again argue that this first-order discretization error will disappear if themesh is locally uniform. Moreover, for quadrilateral and hexahedral grids, thesurface integral approximation is exactly second-order accurate when a densestencil is used. Consider therefore the two-dimensional situation sketched infigure 2.5(b), where we show how the surface patches of the (shaded) nodalvolume of node P are distributed among node pairs for a dense stencil. Thecolor of the surface patch of a pair (P,X) is the same as the color of node X .Furthermore, we (locally) adopt the convention that superscripts de indicatesquantities in the dense formulation. The notation SX→Y denotes the normalvector on the surface patch limited by the points X and Y . We now startfrom the expression for the discrete gradient (2.7) in the dense formulation andreformulate it as follows:

∑

j∈π(P )

∫

SP,j

g dS ≈ g|(P,W )Sde(P,W ) + g|(P,NW )S

de(P,NW ) + g|(P,N)S

de(P,N)

+g|(P,NE)Sde(P,NE) + g|(P,E)S

de(P,E) + ...

=1

2

(

g|(P,W ) + g|(P,NW )

)

S(P,W )→(P,NW )

+1

2

(

g|(P,NW ) + g|(P,N)

)

S(P,NW )→(P,N)

+1

2(g|P,N + g|P,NE)S(P,N)→(P,NE)

+1

2(g|P,NE + g|PE

)S(P,NE)→(P,E) + ...

Each term in the last expression can be interpreted as the approximation to asurface integral, in which the integrand has been evaluated at the barycenter(midpoint) of the surface patch. This technique is known as the trapezoid rule,and is second-order accurate. The last approximation in (2.7) is also second-order accurate, since the control volumes have been constructed such that paircenters (i, j) are always located at the midpoints between i and j. As such,we have proven that (2.7) is indeed second-order accurate for quadrilateralelements in the dense formulation.

We will denote the discretization stencil for a spatial differential operatorby a calligraphic letter. For instance, for the gradient operator, we have the

34


Figure 2.5: Illustration of the sparse (a) and dense (b) discretization stencilsfor the gradient operator.

stencil G:

G (g) |i =1

Vi

∑

j∈π(i)

g|i + g|j2

Si,j (2.8)

2.1.5 The divergence operator

To obtain a stencil for the discrete divergence of a vector u, we can adopt thesame approach as we did for the gradient. The stencil reads:

D (u) |i =1

Vi

∑

j∈π(i)

u|i + u|j2

· Si,j (2.9)

The convective term in the incompressible Navier-Stokes equations is ∇ · (uu).In section 2.2, we will show that it will be necessary to define an advectedand advecting velocity. The advected velocity u is a vectorial quantity whichis defined at the CV nodes; the advecting velocity U on the other hand isdefined at the pair face (i, j), and represents the mass flux through the pair facesurface (U is thus a scalar quantity despite its name). The discrete convectivedivergence C in the Navier-Stokes equations may then be computed with thefollowing stencil:

C (u) |i =1

Vi

∑

j

u|i + u|j2

U |(i,j) (2.10)

Both velocities are of course not completely independent. More details on theirmutual interplay will be given in section 2.2.

35


2.1.6 The Laplacian operator

The Laplacian operator appears twice in the quasi-static MHD equations. Oncein the viscous term in the momentum equation, and once in the Poisson equa-tion for the electric potential. In the next section, we will see that the timemarching algorithm also requires us to solve a Poisson equation for the pres-sure. The same discretization scheme for the Laplacian operator will be usedfor the three cases.

The most straightforward way to compute the Laplacian of a function gwould be to apply successively the gradient and divergence operator. Thisleads however to a non-compact stencil, which may be a source of non-physicalbehaviour when solving a Poisson equation. We can illustrate this for a one-dimensional grid with equidistant spacing ∆:

∂2g

∂x2

∣

∣

∣

∣

i

=1

2∆

(

∂g

∂x

∣

∣

∣

∣

i+1

− ∂g

∂x

∣

∣

∣

∣

i−1

)

+O(∆2)

The first-order derivatives can be discretized similarly:

± ∂g

∂x

∣

∣

∣

∣

i±1

=1

2∆(g|i±2 − g|i) +O(∆2)

Eventually, we obtain:

∂2g

∂x2

∣

∣

∣

∣

i

=1

4∆2(g|i+2 + g|i−2 − 2g|i) +O(∆2)

The major problem of this stencil come from the fact that it decouples odd-and-even numbered nodes. As such, it cannot accurately represent the Laplacianof the highest frequency components of g which can be resolved on the mesh.Indeed, the discrete Laplacian of a function taking values +1 at odd-numberedand -1 at even-numbered nodes (see figure 2.6), is 0. Inversely, we can addthese high-frequency components in an arbitrary amount to the solution of aPoisson equation for a given right-hand side; this gives rise to non-physicalsolutions. In mathematically more precise terms, these fluctuations can arisebecause the stencil above has a non-trivial kernel. This phenomenon is knownunder the name checkerboard problem.

Besides that, this discretization stencil is also troublesome when it comes toaspects of implementation. The data structures available (pairs, elements etc.)only allow to connect a node with its nearest neighbours. Since this schemeinvolves nodes that are further away, the computation of the Laplacian shouldbe performed in two steps, and is computationally more expensive.

We can circumvent these problems by constructing a scheme on a compactstencil. To this end, we start again from Gauss’s theorem, applied to the nodal

36


Figure 2.6: High-frequency function belonging to the null space of the nn-compact Laplacian discretization stencil.

volume Ωi of a given node i:∫

Ωi

∇2g dV =

∮

∂Ωi

∇g · dS =∑

j∈π(i)

∫

Si,j

∇g · dS

≈∑

j∈π(i)

(∇g) |(i,j) · Si,j (2.11)

We should now find a compact discretization of the projection of ∇g along theface normal at the CV faces. If the face normal is parallel to the edge connectingits two adjacent nodes, a second-order accurate estimate of the scalar product(∇g)|(i,j) · Si,j is easily found:

(∇g)|(i,j) · Si,j =g|j − g|i|xj − xi|

||Si,j ||+O(∆2) (2.12)

This expression is symmetric under exchange i j. For a one-dimensionalequidistant grid, we obtain:

∂2g

∂2x

∣

∣

∣

∣

i

≈ 1

∆2(g|i+1 + g|i−1 − 2g|i) +O(∆2) (2.13)

If we apply this stencil to the function shown in figure 2.6, we find that thediscrete Laplacian of g at node i takes the value −4∆−2g|i. This illustratesthat these high-frequency components do not belong to the null space of thisstencil.

On unstructured meshes however, a CV face normal is in general not parallelto the edge between its adjacent nodes (see l.h.s. of figure 2.7). This meansthat we have to approximate all the components of the gradient at the CV face.Several approaches have been proposed in the past. Zwart [Zwa99] for instance,suggested to decompose the gradient in a part parallel and perpendicular tothe pair edge xj − xi, i.e.:

Si,j = S||i,j + S⊥

i,j =

(

Si,j · (xj − xi)

|xj − xi|2)

(xj − xi) + S⊥i,j (2.14)

37


Figure 2.7: Illustration of the geometrical concepts used in various discretiza-tion stencils for the Laplacian. Left: Decomposition of the pair normal Si,j

of the pair (i, j) in a component parallel and perpendicular to the pair edge.Illustration for a sparse discretization stencil. Right Element surface patchesassociated to a certain node. The sum of the normal vectors of these patchesyield Sel.

Furthermore, the components of the gradient perpendicular to the pair edgeare obtained through interpolation between the values of the gradient at bothpair nodes:

Gp,Zwart(g)|(i,j) · Si,j =g|j − g|i|xj − xi|2

(xj − xi) · S||i,j

+1

2(G(g)|i + G(g)|j) · S⊥

i,j (2.15)

This approach removes the odd-even decoupling and is consistent, i.e. thediscretization error will tend to zero as the size of the CV’s shrinks. However,it is still not compact; this means that the computation of the Laplacian ofa given field φ at a given node i involves the values of φ at nodes which arenot directly connected to i. Therefore, this stencil is less suited for large-scalecomputations on unstructured meshes.

Ham et al. [HMI06] define the approximation of the gradient at a CV subtetface as the solution of the following 3-by-3 system:

g|j − g|i = Gp,Ham(g)|(i,j) · (xj − xi)g|fa − g|i,j = Gp,Ham(g)|(i,j)

(

xfa − x|(i,j))

g|el − g|i,j = Gp,Ham(g)|(i,j)(

xel − x(i,j)

)

(2.16)

The indices el and fa represent an averaging over all the nodes of the element

38


and face under consideration. The information used in this stencil is restrictedto one element, so that the obtained scheme is compact. The use of this methodis however restricted to sparse discretization schemes. For dense schemes, thesystem (2.16) can become singular for the non-physical edges. On a regularhexahedral mesh for instance, the first and third equation of this system arethe same for the pair associated with a space diagonal.

In this work, we propose an approach which combines ideas of both of theprevious stencils. We start again from a decomposition of the gradient in acomponent parallel and perpendicular to the edge. For the parallel projection,we can use the first term of (2.15). For the second term in this expression,we use a gradient approximation which is based on the application of Gauss’stheorem to the element el to which the pair under consideration belongs. Assuch, all nodes involved in the stencil are directly connected to both pair nodes;this scheme is thus also compact. We have:

∫

Ωel

∇g dV =∑

ef

∫

Sef

g dSef ≈∑

ef

1

Nefn

(

∑

k

g|k)

Sef (2.17)

Here, we have approximated the surface integral over an element face ef withfirst-order accuracy by taking a simple average over the nodes k of ef . We cannow associate an element surface patch Sel

k with each node k of the element:

Selk =

∑

ef

∑

efn

Sef

Nefn(2.18)

From figures 2.4 and 2.7 (r.h.s.), it is clear that the surface represented bySelk and the sum over all pairs of the node k within the element form a closed

surface. The above expression allows us recast the discrete, element-basedgradient as:

Ge(g)|el =1

Vel

∑

k

g|kSelk (2.19)

Eventually, we obtain the following result for the projection of the elementpair-face gradient along Si,j :

Gp(g)|(i,j) · Si,j =g|j − g|i|xj − xi|2

(xj − xi) · S||i,j +

∑

el

Ge(g)|el · S⊥i,j|el (2.20)

We now want to write the discretization stencil of the Laplacian of g at a nodei as a sum over the element pairs involving i. Therefore, we introduce the

notation w||i,j = (xj − xi) · S||

i,j/||xj − xi||2. Note that w||i,j is invariant under

39


the exchange i j. Using (2.19), (2.20), we can proceed as follows:

L(g)|i =1

Vi

∑

j∈π(i)

Gp(g)|i,j · Sij

=1

Vi

∑

j∈π(i)

w||i,j (g|j − g|i) +

∑

el

∑

j∈πel(i)

Ge(g)|el · S⊥i,j|el

=1

Vi

∑

j∈π(i)

w||i,j (g|j − g|i)

+∑

el

∑

j∈πel(i)

∑

k∈πel(i)

(g|k − g|i)Selk · S⊥

i,j|el

Vel

(2.21)

We now furthermore define w⊥i,k =

∑

el

∑

j∈πel(i)Selk ·S⊥

i,j|el/Vel. Upon exchange

of the dummy indices j and k, and introduction of wi,j = w||i,j + w⊥

i,j , we findeventually:

L(g)|i =1

V i

∑

j∈π(i)

wi,j (g|j − g|i) (2.22)

The factors w⊥i,j and thus the stencil L are asymmetric in i, j, and thus the

property wi,j = wj,i does not necessarily hold any more. The major drawbackof this is that we will not be able to take advantage of a number of fast Poissonsolvers specially intended for symmetric systems. Nevertheless, this skewnesscorrection is inevitable if one wants to avoid O(1) errors with a compact sten-cil on a random, unstructured mesh; this was also noted in [SGN08]. Theformulation of the skewness correction term discussed above has been develpedin collaboration with V. Moureau. Up to our knowledge, it has never beenpresented before.

2.1.7 Matrix representation of the discretization opera-tors

We can represent the action of the spatial discretization operators on a givenfield g as a matrix-vector product. The elements of this vector, which we willdenote G, are the values of g at every grid node. For later considerations,we now discuss briefly the form and spectral properties of these matrices. Wewill assume that the mesh is periodic, so that we don’t have to cope withcomplicating boundary terms.

The matrix form of the volume averaging operator is a diagonal matrix V,whose diagonal entries vii are the nodal volumes Vnode,i. Furthermore, the

40


convective divergence operator C can be represented as a sparse matrix C; theoff-diagonal entries on positions (i, j) are zero if the nodes i and j do not forma pair. The non-zero off-diagonal elements read:

cij =U |(i,j)2Vnode,i

(2.23)

By definition, U |(i,j) = −U |(j,i). In the following subsection, we will present atime-advancement algorithm for the Navier-Stokes which ensures incompress-ibility at the level of the convective velocities. This means that, for every nodei, the following sum over all its pairs will be satisfied:

∑

j∈π(i) Ui,j = 0. If thisproperty holds, the diagonal entries cii of C are zero. We can furthermore writeC as V−1C, where C is a skew-symmetric matrix, whose eigenvalues are purelyimaginary. Similarly, the matrix G has a zero diagonal and its off-diagonalentries are gi,j = Si,j/Vi.

The matrix L respresenting the Laplacian operator L has the same sparsitypattern as C. Referring to (2.22), we can write the non-diagonal elements lijrepresenting a pair (i, j) as:

lij =wi,j

Vi(2.24)

The diagonal elements of L are:

lii =1

Vi

∑

j∈π(i)

−wi,j (2.25)

Just like for the convective operator, we can introduce a matrix L = V−1L.The spectral properties of C and L are of utmost importance in the context

of (spatially discrete) kinetic energy conservation and decay, and for the sta-bility properties of the time integration schemes discussed in the next section.In most canonical textbooks [FP02, Hir07], the analysis of numerical schemesis limited to the case of structured, equidistant meshes. In this work, we havedeveloped our own, more generic approach, which also applies to unstructuredmeshes. To this end, we first introduce the Gershgorin circle theorem [Ger31],which allows to bound a priori the spectrum of an arbitrary matrix M, thuswithout computing explicitly the eigenvalues. This theorem reads as follows:

Gershgorin circle theorem. Let M = [mij ] be a n× n complex matrix, andlet Ri be the sum of the moduli of the off-diagonal elements in the i-th row,i.e. Ri =

∑

j,j 6=i |mij |. Then each eigenvalue of M lies in the union of circles|z −mii| ≤ Ri for i = 1, 2, ..., n.

Application of this theorem to the matrix L learns us that the real partof all its eigenvalues is not larger than lii +

∑

j 6=i |lij |. Since, by definition

41


lii = −∑j,j 6=i lij , we find that the real part of the eigenvalues of L satisfies:

ℜ(λ(L)) ≤∑

j,j 6=i

−lij + |lij | (2.26)

The expression in the right-hand side of this inequality is always larger than orequal to zero. It is only zero if all the lij have the same sign. These elements

are the sum of a main contribution w||i,j and a skewness correction term w⊥

i,j .The main contribution is always positive because the normal vector Si,j andthe edge vector xj − xi always point in the same direction. It is less obviouswhether the skewness correction is positive or negative; however, if the mesh isnot too distorted, this term will be small with respect to the main contribution,and it is unlikely that it may change the sign of lij .

We can use these matrix representations to write the spatially discretizedNavier-Stokes equations as follows:

∂U

∂t= −CU+ νLU−GP (2.27)

The total discrete kinetic energy can be computed as:

K =1

2UT

VU =∑

i

1

2ui · uiVi (2.28)

and the discrete equivalent of (1.17) is:

∂K

∂t= −UT

VCU+ νUTVLU+UT

VGP (2.29)

Since C = VC is skew-symmetric, the first-term on the right-hand side ofthis expression vanishes. The discretized convection term will thus mimic theenergy-conserving property of its space-continuous counterpart. Moreover, ifL is a negative semi-definite matrix, the second term of (2.29) will indeedrepresent a dissipation process. We can use this as a criterion to judge whethera mesh is acceptable or not. More specifically, and taking into account theabove considerations, we will always require that all the off-diagonal elementsof the discrete laplacian operator matrix L have the same sign, so that all itseigenvalues are located in the left-half of the complex plane. At last, the terminvolving the pressure in (2.29) is not in a quadratic form, and its analysis isthus more intricate. We will discuss it in more detail in subsection 2.2.3.

2.2 Time advancement

In the previous section we have shown that the discretization of the spatialdifferential operators transforms the Navier-Stokes equations in a system of

42

2.2. TIME ADVANCEMENT 43

coupled ordinary differential equations with respect to time, in which the pres-sure and the velocity at the nodes are the unknowns. For a given node i, thisdifferential equation is:

∂u|i∂t

= −C(u)|i + νL(u)|i − G(p)|i + ρ−1fb|i (2.30)

In this section, we will explain how we can compute these fields at an instanttn+1 = tn +∆t, given the fields at a prior time tn.

The major difficulty related to the advancement of these discrete Navier-Stokes equations is that the mass-conservation equation does not contain anexplicit time-derivative if the flow is incompressible. The incompressibility con-straint acts rather as a kinematic constraint on the velocity field, and couplesimplicitly pressure and velocity. Indeed, as we saw in the previous chapter,the pressure can be considered as an auxilary variable needed to maintain theincompressibility constraint. Fractional-step methods are without any doubtthe most widespread technique to decouple the computation of the pressurefrom the advancement of the momentum equation. The advantage of such anapproach is that the decoupled systems for p and u can be solved at a lowerexpense.

We will show how this splitting can be achieved in the first part of this sec-tion. The second subsection is devoted to the different time integration schemeswhich we can use to advance the decoupled equations for the velocity. A majorpoint of concern here is the stability of a scheme, i.e. does it prevent thatnumerical solutions become unbounded. A third aspect which we will addressis whether the presented methods mimic the energy-conserving properties oftheir time-continuous counterparts. In the last subsection, we will explain howthe boundary conditions can be enforced.

2.2.1 Fractional-step methods

In fractional step methods, we approximate the time-evolution equations basedon a decomposition of the operators it contains. More specifically, we willisolate the pressure gradient from the other terms in the momentum equation,and use it for the projection of the velocity field onto a solenoidal field. Thiscan be interpreted as a block-factorization of the operators in continuous time[Per93, LOK01]. The original formulation is due to Chorin [Cho68], and reads:

1. Compute the intermediate velocity u⋆ from the given velocity un by in-tegration of the Navier-Stokes equations without the pressure term.

u⋆|i − un|i∆t

= −C(u)|i + νL(u)|i + ρ−1fb|i (2.31)

43


In the right-hand side of this equation, we have not specified the depen-dency of u on un and u⋆. This aspect is proper to each time integrationscheme. We will deal with various approaches in subsection 2.2.2.

2. The ‘new’ velocity un+1 is related to the intermediate one u⋆ throughthe following relationship:

un+1|i − u⋆|i∆t

= −G(pn+1/2)|i (2.32)

Taking the divergence of this expression, and imposing the incompress-ibility constraint on un+1, yields a Poisson equation for the pressure. Aswe have shown previously, the successive application of the nodal opera-tors D and G leads to a stencil where the equations for odd and even nodesare decoupled. The compact operator L is based on the approximationof the (pressure) gradient at the CV faces. Therefore, we introduce theintermediate convecting velocity U⋆|(i,j):

U⋆|(i,j) =u⋆|i + u⋆|j

2· Si,j (2.33)

The pressure gradient at the faces can now be used to relate the ‘new’convecting velocity Un+1|(i,j) to U⋆|(i,j):

Un+1|(i,j) − U⋆|(i,j)∆t

= −Gp(pn+1/2)|(i,j) (2.34)

We now want to impose mass conservation at the level of the convectingvelocities, i.e., we want Un+1|(i,j) to satisfy:

∀i :∑

j∈π(i)

Un+1|(i,j) = 0 (2.35)

Combination of (2.33), (2.34) and (2.35) now leads to a Poisson equationfor the pressure at the CV nodes based on the compact operator L, andcan be expressed as

L(pn+1/2)|i =1

∆tD(u⋆) =

1

∆t

∑

j∈π(i)

U⋆|(i,j) (2.36)

3. Eventually, both the nodal and velocity are corrected with the gradientof the new pressure:

Un+1|(i,j) = U⋆|(i,j) −∆tGp(pn+1/2)|(i,j) (2.37)

un+1|i = u⋆|i −∆tG(pn+1/2)|i (2.38)

44


The pressure gradient correction for the convecting velocites U |(i,j) isconsistent with the Laplacian operator used to solve the Poisson equation;they will thus satisify the divergence-free condition (2.35) up to machineaccuracy. This is however not the case for the convected velocities un+1.Hence, D(un+1) will not be exactly zero.

The global time accuracy of this method is determined by three aspects.The splitting procedure itself is second-order accurate. Apart from that, theaccuracy depends also on the time integration schemes used in the first stepof the algorithm. These will be discussed in the following subsection. Finally,there is the accuracy of the boundary conditions for the pressure and the inter-mediate velocity. This is a source of ambiguity for several reasons. First of all,there are no generic boundary conditions for the pressure in the time continu-ous Navier-Stokes equations. Furthermore, the function p in the fractional-stepmethod is not exactly equivalent to the physical pressure, and contains in factcorrection terms due to the time-discretization in the first step of the algorithm.A standard choice of boundary conditions for the fractional-step could be:

u⋆|∂Ω = un+1|∂Ω (2.39)

∂pn+1/2

∂n

∣

∣

∣

∣

∂Ω

= 0 (2.40)

With this formulation, we can however not guarantee simultaneously that thetangential components of the pressure gradient are also zero. As such, theDirichlet boundary condition on the tangential boundary velocity are not re-spected.

The inconsistency is inherent to the decomposition, and the conditions(2.39)-(2.40) give rise to errors of first order in the time step. This was high-lighted by Kim and Moin [KM85]. For their analysis, they considered u⋆ as theapproximation to a fictitious velocity u†(tn+1) where the continuous functionu† satisfies:

∂u†

∂t= −u† · ∇u† + ν∇2u† + ρ−1fb (2.41)

with initial condition u†(tn) = un. We now develop u† in a Taylor series:

u†(tn+1) = un +∆t∂u†

∂t

∣

∣

∣

∣

n

+O(∆t)2

= un +∆t(

−u · ∇u+ ν∇2u+ ρ−1fb)

|n +O(∆t2) (2.42)

An expression for the ‘new’ velocity un+1 = u(tn+1) can be found in a similarway, by using the momentum equation (1.19) in stead of (2.41).

un+1 = un +∆t∂u

∂t

∣

∣

∣

∣

n

+O(∆t2)

= un +∆t(

−u · ∇u+ ν∇2u+ ρ−1fb −∇p)

|n +O(∆t2) (2.43)

45


Subtracting (2.43) from (2.42) yields:

u†(tn+1)− un+1 = ∆t∇pn +O(∆t2) (2.44)

This shows us that boundary condition (2.39) for u⋆ ≈ u†(tn+1) is only firstorder accurate. Furthermore, we have also:

∇pn = ∇pn−1/2 +∆t

2

∂∇p∂t

∣

∣

∣

∣

n−1/2

+O(∆t2) = ∇pn−1/2 +O(∆t) (2.45)

Together with (2.44), we find that the following conditions are second-orderaccurate:

u⋆|∂Ω = un+1|∂Ω +∆t∇pn−1/2 (2.46)

∂pn+1/2

∂n

∣

∣

∣

∣

∂Ω

= 0 (2.47)

This leads to a slightly modified version of the fractional-step algorithm:

1. Compute the intermediate velocity u from the ‘old’ velocity un, includingthe ‘old’ value of the pressure pn−1/2 for the pressure gradient term:

u|i − un|i∆t

= −C(u)|i + νL(u)|i − G(pn−1/2)|i + ρ−1fb|i (2.48)

with boundary conditions u|∂Ω = un+1|∂Ω2. Remove the old pressure contribution from u and require that the new

pressure pn+1/2 projects the new convecting velocities Un+1 on a solenoidalfield. This leads to a Poisson equation for the pressure, as previously dis-cussed:

u⋆|i = u|i +∆tG(pn−1/2)|i (2.49)

L(pn+1/2)|i =1

∆t

∑

j∈π(i)

U⋆|(i,j) (2.50)

The first of these equation shows us that the boundary values of u⋆ satisfythe second-order time-accurate boundary conditions (2.46) -(2.47).

3. Correct the convecting and convected velocities with the gradient of thenew pressure pn+1/2:

Un+1|(i,j) = U⋆|i,j −∆tGp(pn+1/2)|(i,j) (2.51)

un+1|i = u⋆|i −∆tG(pn+1/2)|i (2.52)

Compared to the original algorithm, this formulation requires one extra step,and is thus only slightly more computationally expensive. It is this version ofthe fractional step method which we have implemented in the code YALES2.

46


2.2.2 Time integration schemes for the momentum equa-tion

We can represent the first step of the fractional-step method as the followingintegral:

u− un

∆t=

1

∆t

∫ tn+1

tn

(

−C(u) + νL(u) + ρ−1fb)

dt− G(pn−1/2) (2.53)

There are two main families of integration schemes to approximate the integralin the right-hand side. In explicit methods, we only use values at tn and priorinstants for the time-discretization of this integral. Implicit methods on theother hand also involve information at tn+1. This latter approach comes at thecost of solving a large system of equations, and is thus computationally moreexpensive. However, explicit schemes are less stable, i.e. truncation errors areamplified and may eventually become unbounded if the time step ∆t is toolarge. The more general definition of stability of a numerical scheme which wewill use here, states that the solution should remain bounded at every tn asn→ ∞. In particular, we will require that the kinetic energy within the systemdoes not grow infinitely.In this work, we will restrain ourselves to an introductory discussion on thestability properties of various time integration schemes. To this end, we makethe following simplifying assumptions. First of all, we will neglect all boundaryeffects, so that our analysis will only hold for infinitely extended or periodicmeshes. Furthermore, we will deploy mathematical tools, which are strictlyonly valid for linear problems. Therefore, we will pursue the discussion on thestability properties of the various integration schemes in the context of theconvection-diffusion equation.

∂u

∂t= −u · ∇u+ ν∇2u (2.54)

or in discrete space:∂U

∂t= (−C+ νL)U = AU (2.55)

The effects of the pressure and Lorentz force term will be taken into accountin (sub)section 2.2.3 and 2.3. This also implies that we consider the convectingvelocities as a ‘given’ field, which is independent of the unknown convectedvelocities. We recall that we have assumed that the matrix L has only negativeeigenvalues. From this, and from the skew-symmetric character of C, it followsthat A = −C+ νL is negative semi-definite.After discretization in time of (2.55), we obtain a system of difference equations,which we can denote as:

U = AUn (2.56)

47


The matrix A depends on ∆t and A. A sufficient and necessary condition forthe stability of a scheme is that the discrete kinetic energy is conserved ordiminished after each time step. This is guaranteed if the eigenspectrum ofV−1/2AV1/2 occurs within the unit circle. Note that this matrix product hasthe same eigenvalues as A.

Explicit methods

The starting point for the derivation of explicit schemes, is the Taylor expansionu in the vicinity of un

u = un +∆t∂u

∂t

∣

∣

∣

∣

n

+∆t2

2

∂2u

∂t2

∣

∣

∣

∣

n

+O(∆t3) (2.57)

The corresponding expression in discrete space is:

U = Un +∆t(AUn) +∆t2

2(A2Un) +O(∆t3) (2.58)

If we truncate this series after the second term on the right-hand side, we obtainthe explicit Euler method ; this scheme is thus only first order accurate in time,and reads:

U = (I+∆tA)Un +O(∆t2) (2.59)

This scheme will be stable if the eigenvalues of V1/2(I + ∆t(−C + νL))V−1/2

have a norm smaller then one. However, using the previous considerations onthe nature of C and C, it can be easily shown that, in the inviscid limit, allthe eigenvalues λk of this matrix are of the form λk = 1 ± αki, where αk isbounded by:

αk ≤∑

j,j 6=k

∣

∣U |(k,j)∣

∣

∆t

2Vk= CFLk (2.60)

Since |λk| ≥ 1, the first-order explicit Euler method combined with a center-difference like spatial discretization stencil can thus never be stable for inviscidproblems. In order to obtain a stable explicit scheme, one has to use upwindstencils for the convective problem. This type of formulations only guaranteesstability under the condition that Courant-Friedrichs-Lewy number CFL =maxk CFLk is smaller then one.For purely viscous cases on the other hand, we find that the eigenvalues ofV1/2(I+∆tνL)V−1/2 are bounded by:

1− 2ν∆t∑

j,j 6=k

lkj < |αk| < 1 (2.61)

The left-hand side of this inequality should not become smaller then -1. Fromthis, it follows that an explicit Euler scheme is stable for a diffusion equation

48


under the following condition:

∀k : FOk =∑

j,j 6=k

lkjν∆t < 1 (2.62)

The non-dimensional number FO = maxk FOk is termed the Fourier num-ber. We can write this condition in a more explicit way for meshes where theskewness correction term is vanishing, e.g. on regular meshes:

FOk =∑

j,j 6=k

ν∆t

Vk|xj − xk|2Sk,j · (xk − xj) (2.63)

If we want to use higher-order methods, we should retain more terms in theTaylor series (2.57). For a second-order accurate time integration scheme e.g.,we can proceed as follows:

U = Un +∆t∂

∂t

(

Un +∆t

2

∂u

∂t

∣

∣

∣

∣

n)

+O(∆t3)

= Un +∆t∂

∂t

(

I+∆t

2A

)

Un +O(∆t)3

=

(

I+∆tA

(

I+∆t

2A

))

Un +O(∆t)3 (2.64)

This approach can easily be extended to higher orders of accuracy. One canshow that the stability properties of these time integration schemes are sim-ilar to the ones of the explicit Euler scheme, i.e. unconditionally unstablefor the convective term when combined with a central-difference like spatialdiscretization, and a limitation on ∆t determined by the Fourier number forviscous cases.

Implicit methods

In implicit methods, the velocity used to compute the right-hand side of (2.55)is expressed as a blend of un and u. In its most general form, we have:

U = Un +∆tA(

γU+ (1− γ)Un)

(2.65)

If we choose the blending parameter γ = 0.5, we obtain the Crank-Nicholsonscheme. This scheme has two major advantages. First of all, it is second orderaccurate provided that the coefficients of A are also second-order accurate. Tomaintain the accuracy, we should thus find a suitable interpolation which yieldsthe convective velocities at the mid-time-step. An implicit treatment of theseterms would result in a non-linear system for the momentum balance equation.

49


Having in mind the complexity of non-linear equation solvers, we will linearise

the problem, and compute the values of U |n+1/2(i,j) explicitly with a second-order

Adams-Bashforth method, i.e.:

Un+1/2|(i,j) =3

2Un|(i,j) −

1

2Un−1|(i,j) +O(∆t2) (2.66)

We will refer to this procedure as a semi-implicit treatment of the convectiveterm.The second avantage of the Crank-Nicholson scheme concerns its stability prop-erties. We can write (2.65) in a slightly different version:

U =

(

I− ∆t

2A

)−1(

I+∆t

2A

)

Un (2.67)

Since the eigenvalues λ of A are located in the left-half of the complex plane,we find that he eigenvalues λCN of the matrix product in (2.67) satisfy:

|λCN | =∣

∣

∣

∣

∣

1 + ∆t2 λ

1− ∆t2 λ

∣

∣

∣

∣

∣

≤ 1 (2.68)

The Crank-Nicholson scheme is thus unconditionally stable. Moreover, in thecase of inviscid flow, we obtain the limit |λCN | = 1. This means that it preservesthe kinetic-energy conserving property of the convective term if combined witha center-difference-like discretization stencil.Another popular implicit time advancement scheme is the so-called implicitEuler method. In this case, the value of the blending parameter γ = 1. Assuch, it is only first-order accurate, regardless of the time interpolation chosenfor U |(i,j). We can again express the eigenvalues λIE of the iteration matrix(I−∆tA)−1 as function of λ:

|λIE | =∣

∣

∣

∣

1

1−∆tλ

∣

∣

∣

∣

< 1 (2.69)

Just like the Crank-Nicholson scheme, the implicit Euler scheme is uncondi-tionally stable. In the case of purely imaginary λ however, the system will notconserve its kinetic energy; the eigenvalues λIE of the time-discrete system cannever be on the unit circle. This erroneous behaviour will become particularlyimportant if the imaginary part of λ has a large amplitude. This means thathigh spatial frequency components will be more affected by numerical dissipa-tion.

2.2.3 Kinetic energy conservation and the pressure term

In the continuous, incompressible Navier-Stokes equations, the pressure gradi-ent and the convective term both conserve kinetic energy. The issue of kinetic

50


energy conservation of the pressure term on collocated meshes was profoundlyinvestigated by Ham et al. [HI04]. As a starting point for a summary of theiranalysis, we write the last term of (2.29) in its nodal form:

UTVGP =

∑

i

Viui · G(p)|i

=∑

i

ui ·∑

j,j∈π(i)

1

2pjSi,j

=∑

i

pi∑

j,j∈π(i)

1

2uj · Si,j

=∑

i

piViD(u)|i (2.70)

As we have seen, the incompressibility constraint is not satisfied at the level ofthe convected velocities u, so that this discrete term does not vanish. Ham etal. [HI04] further analysed this erroneous non-conservation of kinetic energyand found that it scales as follows:

UTVGP ∝ −||∇4p||(∆t)∆2 (2.71)

In this expression, ∆ is a length-scale representing the size of the control vol-ume. As this term is slightly dissipative, it does not affect the stability criteriaof the semi-implicit time advancement schemes.

2.2.4 Boundary conditions

To conclude this section, we should still discuss how to impose the boundaryconditions for the intermediate velocity u. For solid walls, we can impose theDirichlet condition u|∂Ω = 0. At an inflow boundary, a velocity field u is tobe prescribed. This however not the case at an outflow boundary, since theNavier-Stokes equations are parabolic in time. In general, little is known aboutthe flow at an outlet, and as a primary rule of thumb, this kind of boundariesshould be located far enough from the domain of interest. For steady flows, wecan use a Neumann condition ∂nu = 0; for time-dependent flows however, thiscondition may cause an upstream propagation of errors. A popular alternativeis the so-called convective boundary condition [PMR90]:

∂u

∂t+ Uconv

∂u

∂n= 0 (2.72)

The quantitiy Uconv is a loosely defined characteristic velocity of the flow. Ingeneral, one takes the ratio between the mass flow rate and the outlet area.

51


This condition owes its popularity due to the fact that it is non-reflecting. Thevelocity profile at the interior of the domain is just convected in downstreamdirection at a velocity Uconv. This approach will give rise to non-physicalbehaviour at the outlet boundary condition if the non-linear term is not thedominant term in the momentum equation, i.e. if Re is small, or, for quasi-static MHD flows, if the interaction paramterN is large. In practice, for a givenfield un, we solve (2.72) to obtain the intermediate outlet velocity u|∂Ωout

, andimpose these values when solving the momentum equation. (2.72) is solvedwith a first-order explicit Euler method and an upwind formulation for thespatial discretization. As an illustration, we write this explicitly for the pointP in the cartesian two-dimensional mesh shown in figure 2.8:

un+1|P = un|P −∆tUconvn · G(un)|P= un|P − ∆tUconv

||xP − xW || (un|P − un|W ) (2.73)

This approach is only stable if the coefficient of u|P in this expression has anorm smaller then one. We see that this is fulfilled if ∆tUconv/||x|P−x|W || < 2.For a generic mesh, we find the following constraint:

∀i∈∂Ωout:∆t

2Vi

∑

j∈π(i)

|Ui,j |+ Uconv||Sbndi ||

< 1 (2.74)

A last issue is that a discrete Poisson equation for the pressure with Neumannboundary conditions on all the boundaries is only well-posed if the sum ofthe elements of its right-hand side is zero. In our case, this right-hand side isthe discrete divergence D(u⋆). All interior contributions disappear because ofthe anti-symmetric character of the stencil D. The remaining boundary termsshould satisfy the following condition

∑

i∈∂Ω

u⋆|i · Sbndi = 0 (2.75)

For solid walls, the velocity flux through the boundary faces is by definitionzero. However, the convective boundary condition (2.72) does not guaranteethat the mass flux leaving the domain is the same as the mass flux prescribedby the Dirichlet condition at the inlet boundary. As such, if there are oneor multiple inlet or/outlet boundaries, the condition (2.75) is not necessarilyfulfilled any more. To remedy to this, a uniform velocity is added to the outletvelocities such that (2.75) holds.

52

2.3. THE QUASI-STATIC MHD EQUATIONS 53

Figure 2.8: Sketch of a cartesian mesh close to an outflow boundary.

2.3 The quasi-static MHD equations

In this section, we discuss several aspects related to the computation of theLorentz force density fL. First, we will present two algorithms to obtain fLfor a given velocity field u. In the second subsection, we will explain how theboundary conditions for the Poisson equation for the potential (1.65) can beenforced. Finally, we will show how we can achieve the coupling between themomentum equation and the Poisson equation for the potential.

2.3.1 Discretization of the Lorentz force

The classical way to compute the current and Lorentz force density is similarto the pressure correction step in the fractional-step algorithm. Somehow,this should not surprise us, since there is a striking similarity between currentdensity and velocity on the one hand, and pressure and potential on the otherhand. Indeed, the latter scalar fields can both be considered as Lagrangemultipliers needed to satisfy the laws of mass (∇ · u = 0) and charge (∇ · J= 0) conservation. The classical algorithm to compute the Lorentz force thenreads as follows:

1. Take the vectorial product u×B at the CV nodes i and use these valuesto compute its discrete divergence D(u×B)|i.

2. Solve the discrete Poisson equation for the values of the potential at theCV nodes: L(φ)|i = D(u×B)|i with appropriate boundary conditions.

3. Compute the potential gradient G(φ) at the CV nodes, and the nodalcurrent density J as J|i = σ(−G(φ)|i + u|i ×B|i).

4. Obtain the Lorentz force density as: fL|i = J|i ×B|i.

53


Probably the most challenging aspect of the numerical computation ofquasi-static MHD flows in a strong magnetic field is the resolution of verythin shear layers. We recall from the first chapter that the presence of solidwalls leads to the emergence of side layers (non-dimensional thickness M−1/2)or Hartmann layers (non-dimensional thickness M−1). In earlier simulationsof high Hartmann number MHD flows, this issue was sometimes avoided byusing a wall model for the Hartmann layer [MGMB00, Wid03]. However, inthis work, we aim at a full solution of the complete set of quasi-static MHDequations. If we want to capture these layers properly, we need at least 3 or 4grid points within the layer under consideration. To meet this requirement atan acceptable computational cost, we have to use non-uniform meshes, whichare coarser in the core regions, and refined towards the shear layers.

However, many authors which have applied the above algorithm to com-pute laminar high Hartmann number flows on non-uniform grids, came to theconclusion that it was almost impossible to reach an accurate prediction for thepressure drop given the flow rate or vice versa [Leb99], certainly in domainswith insulating walls. The deficiency of the algorithm can be attributed to acombination of two factors. The first one of these is that the computation ofJ is a poorly conditioned operation in regions of the flow where it tends tozero. The terms from which it is computed, i.e. G(φ) and u×B are in generalmuch larger. It is commonly known that such a sum of two terms which arenearly equal in magnitude, but have an opposite sign, is very sensitive to smallnumerical errors. Therefore, this procedure will lead to erroneous contributionsto the Lorentz force.

The second factor contributing to the poor accuracy of the Lorentz forcedensity originates from the fact that the discretization procedure above doesnot mimic the conservative nature of fL if the magnetic field is uniform. Thisproperty can be elucidated by the following manipulations:

J×B = −J · ∇(B× x) = −∇ · (J(B× x)) (2.76)

In the last step, we have relied on the solenoidal character of the current densityfield. Likewise, we can alwo write J in conservative form:

J = ∇ · (Jx) (2.77)

Integration of (2.76) over the flow domain shows us that there is no net con-tribution of the ‘bulk’ of the domain to the global Lorentz force acting on thefluid; it only depends on the boundary conditions:

∫

Ω

J×B dV = −∮

∂Ω

(B× x)J · dS (2.78)

Indeed, as we have shown in the first chapter, the pressure drop in a ductwith insulating walls is solely due to viscous effects, and the Lorentz force itself

54


does not contribute to the pressure drop. However, since the discretizationprocedure described above is not conservative, erroneous contributions to fLat every CV, will not necessarily cancel out, and may add up to an important,non-physical bulk Lorentz force term.

To remediate to these deficiencies, we should compute the discrete Lorentzforce term in such a way that its conservative character is preserved. This wasoriginally recognized in the numerical study of accretion disks [Kle98], wherethe discretization of the Coriolis force term in a rotating frame of referenceleads to the same kind of problems. This lead Ni et al. [NMH+07, NMM+07]to introduce the following algorithm to compute the Lorentz force density:

1. Take the vectorial product u×B at the CV nodes i and use these valuesto compute its discrete divergence D(u×B)|i.

2. Solve the discrete Poisson equation for the values of the potential at theCV nodes: L(φ)|i = D(u×B)|i with appropriate boundary conditions.

3. Reconstruct the current density flux J |(i,j) at the CV face between i andj in a consistent way, i.e. compute the potential gradients in such a waythat

∑

j∈π(i) J |(i,j) is smaller than machine accuracy or the convergencethreshold of the linear solvers of the Poisson equation. To this end, thesefluxes should be computed as:

J |(i,j) = σ

(

−G(φ)|(i,j) +u|i + u|j

2×B

)

(2.79)

4. Compute the nodal current and Lorentz force using the conservative ex-pressions (2.77) and (2.76), i.e.:

J|i =1

Vi

∑

j∈π(i)

J |(i,j)x|(i,j) (2.80)

fL|i =1

Vi

∑

j∈π(i)

J |(i,j)(

B× x|(i,j))

(2.81)

The position coordinate x|(i,j) of the center of the node pair (i, j), is foundby interpolation of the positions of nodes i and j. This may however beproblematic in domains with periodic boundary conditions because ofthe jump in the position coordinate, but can easily be circumvented bylocating the origin of the coordinate system at the CV node:

J|i =1

Vi

∑

j∈π(i)

J |(i,j)(

x|(i,j) − xi

)

(2.82)

and likewise for fL.

55


Figure 2.9: Geometrical concepts for the implementation of Shercliff’s thin wallcondition. Boundary face normal Sbnd, and its contour ∂Sbnd.

2.3.2 Boundary conditions for the potential

One of the principal differences between the Poisson equations for pressure andpotential is the issue of the boundary conditions. For φ, we should considerthree different types of conditions.

The first one is the Neumann condition ∂nφ = 0 for electrically insulatingwalls. Its implementation is trivial. No current flux is passing through theboundaries, and no additional term is needed to complete the integral

∮

∂Ω∇φ ·dS over the surface of the boundary CV’s.

For perfectly conducting walls, the values of φ at the boundary nodes areprescribed, and are not unknowns. We can leave these nodes thus out of con-sideration when solving the system, and just impose the values φ.

Finally, for a discretization of Shercliff’s thin wall-condition (1.75) ∂nφ =cw∇2

τφ, we can apply the two-dimensional version of Gauss’s theorem to theboundary surface patch Sbnd of a boundary CV. The flux through Sbnd whichcompletes

∮

∂Ωi∇φ · dS can be developed as (see figure 2.9):

∫

Sbnd

∇φ · dS =

∫

Sbnd

∂φ

∂ndS =

∫

Sbnd

cw∇2τφdS =

∮

∂Sbnd

cw∇τφ · dn (2.83)

In the last step, we have applied Gausss divergence theorem (A.9) in the two-dimensional plane of the boundary face under consideration; the integrationin the rightmost expression concerns thus a surface integration over the one-dimensional surface consisting of the edges defining the boundary face. Thediscretization of this integral can then be performed in a way that is similar

56


to the procedure discussed in section 2.1.6. As far as we know, this is thefirst time that the thin-wall condition (1.75) has been implemented in a non-structured code. In earlier implementations [Mis05, NMMA06], the wall regionwas included in the solution domain for the Poisson potential equation.

2.3.3 Coupling between the momentum and potential equa-tions

Compared to the hydrodynamic Navier-Stokes equations, the quasi-static MHDequations have one extra term in the momentum equation, and an additionalPoisson equation for the unknown scalar potential φ. Moreover, the unknownsu and φ are tightly coupled. An implicit treatment of the Lorenz force wouldlead to a complicated algorithm in which the computation of the potentialinterferes with the pressure-velocity coupling and this would require a cumber-some nesting of iteration loops. Therefore, an explicit treatment of the Lorentzforce term is preferred.

However, as we have seen, the use of explicit time integration schemesleads to stringent conditions on the time step. We now evaluate how theincorporation of a body force term of the form ρ−1σ(u × B) × B affects thestability of the time advancement of the momentum equation. If we use aCrank-Nicholson algorithm for the other terms, and consider the componentsof the velocity field U⊥ perpendicular to the magnetic field B = B01B, wehave:

U⊥ −Un⊥ =

∆t

2A (U⋆

⊥ +Un⊥)−∆tρ−1σB2

0Un⊥ (2.84)

Following the same argumentation as for the Crank-Nicholson scheme in thehydrodynamic case, we find that linear stability is guaranteed if:

∣

∣

∣

∣

∣

1−∆tρ−1σB20 + ∆t

2 λi

1− ∆t2 λi

∣

∣

∣

∣

∣

< 1 (2.85)

with λi the eigenvalues of A, which are assumed to be located in the left-halfof the complex plane. As such, we have λi = Ai cos θi with π/2 < θi < 3π/2.This allows to cast (2.85) under the following form:

(

∆tσB2

0

ρ− 2

)(

σB20

ρ− 2Ai cos θi

)

< 0 (2.86)

The second factor is always positive for the given range of θi, so that we even-tually obtain the following restriction for the time step:

∆t <2ρ

σB20

(2.87)

57


The maximum allowed time step is thus proportional to the Joule dampingtime and scales as N−1 or M−2. The same result was obtained earlier byKinet [Kin09], who investigated the stability of the explicit treatment of theLorentz force for an inviscid, one-dimensional flow on an equidistant grid. Thepresent analysis extends this result to viscous flows on arbitrary meshes.

To evaluate the stringency of this condition, we compare it to the stabilitycriterion which would follow from an explicit treatment of the viscous term atthe same value of M . To this end, we can express (2.87) as ∆t < 2M−2L2/ν.In quasi-static MHD flows, the Hartmann layers are the zones which requirethe smallest resolution. If we want to capture them properly, we need more orless 4 points over a distance of approximately L/M ; the grid spacing in theselayers is thus L/4M , and the corresponding Fourier number will be of theorder of ν∆t(16M2/L2). An explicit treatment of the viscous terms requiresthat this number is smaller than one, and this leads to the following constrainton the time step: ∆t < M−2L2/(16ν). The constraint on the viscous term isthus considerably more restrictive then the limit on ∆t which follows from anexplicit treatment of fL.

Using an explicit, first-order time-accurate integration scheme for fL impliesthat the global temporal accuracy is lowered by one order. We can invoke thefollowing arguments to argue that this is not necessarily a major issue. If themagnetic field is weak, the interaction parameter N is low, and the time scale ofinertial effects is smaller than the Joule damping time, which characterizes elec-tromagnetic effects. To perform physically meaningful simulations, ∆t shouldbe chosen such that it is not larger then the inertial time scale. From this, itfollows that the variations of fL will be small on the scale of ∆t, as will be thetime discretization errors associated with this term. Quasi-static MHD flowsin a strong magnetic field on the other hand, tend to be steady. We are mostoften not interested in the exact temporal evolution from an initial conditioninto this steady state, so that the accuracy of the time integration scheme is ofminor importance.

2.4 Solution techniques for systems of linear equa-

tions

“I recommend this method to you for imitation. You will hardlyever again eliminate directly, at least not when you have more thentwo unknowns. The indirect [iterative] procedure can be done whilehalf asleep, or while thinking about other things.”Carl Friedrich Gauss, Letter to C.L Gerling, 1823

The solution algorithm for the quasi-static MHD equations as presented in theprevious sections, contains several steps in which a linear system of equations

58

2.4. SOLUTION TECHNIQUES FOR SYSTEMS OF LINEAR EQUATIONS 59

has to be solved. On the one hand, we have the Poisson equations for pressureand potential, on the other hand we have to solve a system in which the inter-mediate velocity u is the unknown if we apply a semi-implicit time stepping.The size of all these systems is the number of CV’s in the mesh, and the ma-trices associated with them are sparse, i.e. they contain only a small fractionof non-zero elements. Unfortunately, directly inverting a large matrix is com-putationally very expensive. Moreover, the inverse of sparse matrices are notnecessarily sparse themselves, which implies that a huge amount of memorymay be needed to store them. Therefore, we will rather use iterative methodsto solve these systems. The idea behind this type of methods is to start froma well-chosen, initial guess for the solution vector, and to construct a series ofimproved approximations until a certain convergence criterion is met. In thiswork, we have used a number of different approaches to solve linear systems.We will denote the system as:

MY = Z (2.88)

where an approximation of the solution Y has to be found for a given right-handside Z and system matrix M.

2.4.1 Jacobi iteration

As an ansatz for the Jacobi method [FP02, Hir07, VM07], we write the systemmatrix M as the sum of its diagonal part D and a matrix containing the off-diagonal entries N = M − D. We now tend to generate a series of improvedapproximations (denoted by a superscript) for the solution of Y (k) as follows:

Y (k) = D−1(Z − NY (k−1)) = D

−1(Z −MY (k−1)) + Y (k−1) (2.89)

The consecutive residuals R(k) are defined as:

R(k) = Z −MY (k) (2.90)

and satisfy the following iteration formula:

R(k) = (I−MD−1)R(k−1) (2.91)

A necessary and sufficient condition for the convergence of this method, isthat the norm of R is diminished after each iteration step; this requires thatthe norm of all the eigenvalues of the term within brackets in the previousexpression, is smaller then one. Application of the Gershgorin circle theoremtells us that this is the case if the system matrix M is diagonally dominant,i.e. if for every row i of the system matrix M, the matrix entries mij obey thefollowing condition:

|mii| ≥∑

i6=j

|mij | (2.92)

59


The computational work required to reach a certain convergence criterion scalesas O(n2 logn) where n is the size of the system.

We now investigate the condition of diagonal dominance in more detail forthe equations for which we applied the Jacobi method, i.e. the system associ-ated with the (semi-)implicit time advancement of the velocity. Most referencetextbooks [FP02, ?] on CFD do not analyse the iterative Jacobi method for thecase of a generic unstructured mesh; as in the preceding sections, we deliverhere our own, sui generis analysis. The matrix A contains contributions of thetransient, convective and viscous term. If we leave for a moment aside thenon-linear term, then we obtain the following expression for respectively thediagonal and non-diagonal entries of M = I− γν∆tL:

mii = 1 +∑

j,j 6=i

γν∆t

Vilij (2.93)

mij = −γν∆tVi

lij (2.94)

We recall that γ is the blending factor, which determines the weight of Un

and U in the implicit time-advancement of the momentum equation. Sincethe coefficients lij are assumed to be positive, the matrix M will always bediagonally dominant. This means that the Jacobi method will converge forany value of ∆t and γ.

We now consider the case of inviscid flow. The system matrix can be writtenas M = I+ γ∆tC, and its entries are:

mii = 1 +γ∆t

2Vi

∑

j∈π(i)

U |(i,j) = 1 (2.95)

mij =γ∆t

2ViU |(i,j) (2.96)

Here, we have made use of the discrete incompressibility constraint satisfiedby the convecting velocities:

∑

j,j∈π(i) U |(i,j) = 0. We see that, for inviscidflows, stability of the Jacobi method is only guaranteed as long as the followingcondition is respected, i.e.:

γ∆t

2Vi

∑

j,j 6=i

∣

∣U |(i,j)∣

∣ = γCFLi ≤ 1 (2.97)

This is similar to the condition that governs the stability of explicit time ad-vancement methods in combination with upwind discretization schemes. Thedrawback of using an explicit/upwind schemes however, is that it is only first-order accurate in space. The second-order spatial discretization error will gen-erate some artificial viscosity, and therefore, such approaches are slightly dis-sipative.

60


One can generalize the Jacobi method by using a relaxation technique;this means that we generate a blend of the previous and updated solution byintroducing a relaxation factor ω:

Y (k+1) = ωY (k) + (1− ω)D−1(Z − NY (k)) (2.98)

Hence, the residual satisfies the iteration formula:

R(k) = (I− (1− ω)MD−1)R(k−1) (2.99)

We will refer to this technique as the ω-Jacobi method. We can now tuneω, so that so that stability can be reached for higher values of ∆t, or fasterconvergence is obtained. We have not directly used this method. However,we will refer to it when introducing multigrid methods, discussed in the nextsubsection,

2.4.2 Algebraic multigrid methods

It has been observed that the convergence rate of the ω-Jacobi method exhibitsthe tendency to stall after a few iterations if the grid size is small with respectto the extent of the solution domain, even if the underrelaxation ω is carefullychosen. A spectral analysis of this phenomenon reveals that the iteration (2.98)quickly damps the short-wavelength components (i.e. wavelengths which are ofthe order of a few multiples of the mesh size) of the residual vector, but hardlyaffects the large-scale components (i.e. components with wavelengths of theorder of the domain size); in Appendix B.1, we offer an elaborative spectralanalysis of the ω-Jacobi method for the Poisson equation on a one-dimensionalequidistant grid. Multigrid methods tend to accelerate the convergence rate byperforming iterations on meshes of different sizes in order to obtain an optimalreduction of all spectral components of the residual vector.

The basic principle of this type of methods is the so-called two-grid cycle[TOS01, FP02, VM07] for solving linear systems on strongly refined meshes,and is illustrated in figure 2.10. It consists of several substeps; the first one iscalled smoothing and consists of performing a few ω-Jacobi iterations on thegiven (fine) mesh; as such, the short wavelength components of the residual arestrongly suppressed (figure 2.10(b)), and what remains is a residual which vir-tually only contains long wavelength contributions; subsequently, the solutionand residual vector are transferred to a coarser mesh, which only includes someof the nodes of the finer mesh (figure 2.10(c)). This step is called restrictionor coarsening. Furthermore, we also define a restricted system matrix, whichis an approximation of the original matrix on the coarser mesh. We can nowfurther reduce the residual by carrying out a few ω-Jacobi iterations on thecoarse mesh, a step called coarse grid correction. The remaining long wave-length components of the residual will decay faster on this mesh because the

61


grid size is larger. Once this has been achieved (figure 2.10(d)), we transfer thesolution and residual back to the original (finer) grid (figure 2.10(e)). Valuesfor nodes which do not exist on the coarse level, can be obtained through inter-polation between the coarse grid values. This cycle can be repeated a numberof times until a satisfactory reduction of the residual has been reached.

In multigrid methods, the concept of a two-grid cycle is now extended tomultiple grid levels. Each of these levels can be associated with a certainwavelength range. Furthermore, one can develop a multitude of cycling strate-gies, which each are specified by a suitable sequence and nesting of smoothing,restriction and interpolation operations. Because of this efficient interplay be-tween restriction, interpolation and smoothing, the computational effort scalesonly as O(n log n) (where n is the size of the system), in stead of O(n2 logn)for the classical ω-Jacobi method.

The remaining question is now how to generate a set of grid levels and the(restricted) system matrix associated with each individual level. On structuredmeshes, this can be done in a rather straightforward way; all mesh points arecharacterized by a unique index (i, j, k) and we can define a series of meshlevels by putting restrictions on the values of i, j and k. Moreover, since thesystem matrix is a representation of a discretization stencil of some differentialoperator, we can obtain restricted matrices by applying the same stencil on thecoarser mesh.

For unstructured meshes, we can use algebraic multigrid methods. Thegeneration of coarser grid level does not follow from geometrical considerations,but is solely based on the values of the matrix entries. Various proceduresexist to select the coarse points, but a description of these methods is notwithin the scope of this work. Once the coarse grid points are chosen, onecan compute the corresponding restriction and interpolation operators, andthe restricted matrices. Unfortunately most of the selection procedures arenot very well suited for parallel computation. Existing coarsening procedurestend to give poor results for points which are shared by different computationalunits. Hence, one should only apply these methods when the number of gridpoints per processor is sufficiently high.

Given these restrictions, multigrid methods tend to be very efficient forelliptic equations like the Poisson equation. Moreover, in our case, the matricesof these equations are the same at every time step. The coarsening, restrictionand interpolation operators should thus be computed only once during the set-up phase of the simulation; this leads to significant gain in CPU time, butcomes at the cost of storing a sparse matrix. This furthermore explains alsowhy we did not use multigrid methods to solve systems associated with thesemi-implicit time advancement of the velocity (which are different at everyiteration step).

More specifically, we have applied the parallel algorithmBoomerAMG [HY02]

62


−1 −0.5 0 0.5 1−0.5

0

0.5

x

R

−1 −0.5 0 0.5 1−0.5

0

0.5

x

R

−1 −0.5 0 0.5 1−0.5

0

0.5

x

R

−1 −0.5 0 0.5 1−0.5

0

0.5

x

R

−1 −0.5 0 0.5 1−0.5

0

0.5

x

R

(a)

(b)

(c)

(d)

(e)

Figure 2.10: Illustration of the substeps of a two-grid cycle for a one-dimensional Poisson equation on an equidistant grid. Residual R as functionof the grid node position x on fine (2) and coarse (∗) grid nodes. Fine gridresidual before (a) and after (b) initial smoothing, restricted residual before (c)and after (d) the coarse grid correction, and final interpolated residual (e).

63


which is part the library hypre. This is a library for solving sparse, linear sys-tems on massively parallel computers. The connection between the hypre sub-routines and our own subroutines has been established by using an intermediatelibrary, named PETSc [BBE+08].

2.4.3 Krylov subspace methods

A third class of algorithms are the so-called Krylov projection methods [vdV03].To introduce these, we start from the Cayley-Hamilton theorem:

Cayley-Hamilton theorem. Let M be a n× n-matrix, and let pM (λ) be itscharacteristic polynomial, i.e. pM (λ) = det(M − λI) =

∑nj=0 cjλ

j . Then M

satisfies its own characteristic equation, i.e.∑n

j=0 cjMj = 0.

A direct consequence of this theorem is that the inverse of M (if M issingular, c0 = 0), can be written as a linear combination of powers of M.

M−1 = − 1

c0

n∑

j=1

cjMj−1 (2.100)

Consider now the system (2.88), and an initial guess Y (0). By defining Y =Y − Y (0), we can rewrite it as MY = R0. With (2.100), we find that we cancast the exact solution of this system as:

Y = M−1R(0) = − 1

c0

n∑

j=1

cjMj−1R(0) (2.101)

In other terms, Y belongs to the vector space Kn spanned by the vectors

R(0),MR(0),M2R(0), ...,Mn−1R(0)

. Kn is called the Krylov subspace of ordern. Iterative Krylov subspace methods for linear systems now tend to constructapproximate solutions to Y in Krylov subspaces of lower order.

The best-known among the Krylov subspace methods, is the conjugate gra-dient method, which is limited to symmetric and positive-definite matrices M.For this type of matrices, solving the system MY = Z can be interpreted asfinding the minimum of ψ = 1

2YTMY − Y TZ. At each iteration step k, we

construct Y (k) such that ψ(

Y (k))

is minimal with respect to all vectors withinthe subspace Kk. In the most attractive implementation of this algorithm,one constructs a set of conjugate basis vectors Si for the Krylov subspace. Bythe term conjugate, we mean that the basis vectors Si satisfy an orthogonalityrelation of the form ST

i MSj = 0 if i 6= j. The iteration cycle then reads:

Y (i+1) = Y (i) + αiSi (2.102)

R(i+1) = R(i) − αiMSi (2.103)

Si+1 = R(i+1) + βi−1Si (2.104)

64


The coefficients αi and βi are given by:

αi =R(i)TR(i)

STi MSi

(2.105)

βi−1 =R(i)TR(i)

R(i−1)TR(i−1)(2.106)

The main advantage of this implementation is that additional basis vectors Si

can be computed from short recurrence relations; at each iteration step we onlyneed to know the previous basis vector and residual, and can ‘forget’ all theolder ones. Therefore, this implementation requires little memory.

If the system matrix M is non-symmetric, as is the case in this work, wecan consider the symmetrized system:

(

0 M

MT 0

)(

YY

)

=

(

Z

Z

)

(2.107)

where we have some additional freedom in the choice of the vector Z. Ap-plication of the conjugate gradient algorithm to this system, is called the Bi-Conjugate Gradient method (or BiCG method).

It has however been observed that this method exhibits an irregular con-vergence. We have applied two slightly different, but canonical approaches inthis work, which converge more smoothly and are more stable, and are termedthe BiCGStab and BiCGStab(2)-method. We will only sketch briefly the prin-ciples on which it is based, since a formal derivation is quite tedious and is notreally within the scope of this work; a detailed presentation can be found in[vdV03]. First, note that it can be inferred from (2.102-2.104) that the residualR(k) after k iterations of the conjugate gradient belongs to the Krylov subspaceKk(M, R0). Formally, we can express this as:

R(k) = Qk(M)R(0) (2.108)

where Qk is a polynomial of order k − 1. In the BiCG method, the samepolynomialsQk are used to be compute R(k) = Qk

(

MT)

R(0). In the BiCGStab

method, we construct R(k) differently. More specifically, we have:

R(k) =

k−1∏

j=0

(1− ωjM)

Qk (M) R(0) (2.109)

The coefficients ωk are chosen such that they minimize R(k) with respect toωk; it can be shown that this leads to the algorithm presented in appendixB.2. Like the conjugate gradient and BiCG method, the BiCGstab method

65


can be expressed in terms of short recurrence relations, and is thus also notvery demanding in terms of memory. Compared to BiCG however, it offers theadvantages of converging more smoothly.

The BiCGstab(2) method is a variant on the BiCGStab method, whichmakes a distinction between odd and even iteration steps. For odd-numberediterations k = 1, 3, 5, ..., we have:

R(k) = Qk (M) R(0) (2.110)

whereas for even-numbered iteration steps k = 0, 2, 4, ..., the residual is con-structed as:

R(k) =

k∏

j=0,2,...

(1− ωj1M− ωj2M2)

Qk (M) R(0) (2.111)

The resulting algorithm is given in Appendix B.3. It can be shown thatBiCGStab(2) is superior to BiCGStab in those cases where the skew-symmetricpart of M dominates its symmetric part. If on the other hand, M is approxi-mately symmetric, both methods exhibit similar convergence speeds.

The convergence speed of Krylov subspace methods in particular, and iter-ative methods in general, is strongly dependent on the condition number κ ofthe system matrix M, defined as the ratio between the amplitude of the largestand smallest eigenvalue of M. If κ is large, iterative methods will generallyconverge slower towards the solution of the system. This issue can be partiallyavoided by using a preconditioner. This is a matrix K, which is an approxima-tion to M

−1, and which is easy to store and to compute. Hence, the matrixK−1M will have a much lower condition number, and, as such, we can solvethe system K−1MY = K−1Z, which is equivalent to, but converges faster thanthe original system MY = Z. In this work, we have combined the BiCGstab orBiCGStab(2) with a Jacobi preconditioner; this means that we choose K = D,the diagonal of M.

66

Chapter 3

Verification and validation

“The formula ‘two plus two equals five’ is not without its attrac-tions.”Fyodor Dostoevsky

In this chapter, we discuss simulation runs of analytical solutions of the Navier-Stokes equations or well-documented test cases and compare them with the-oretical results or reference data available in the literature. Our goal is toverify and assess the implementation of the numerical methods presented inthe previous chapter, rather then benchmarking the performance of our solveragainst state-of-the-art simulation databases. Two criteria are of importancehere. First of all, does the numerical method generate a solution which approx-imates well the physical content? Secondly, does the solution converge towardsthe reference result as the resolution is increased?

3.1 Taylor-Green vortex

In two dimensions, the Taylor-Green vortex is an exact, non-trivial solutionof the Navier-Stokes equations in a square periodic box. If the side of thisrectangle is 2, the velocity and pressure field read:

ux = − cos(πx) sin(πy) exp(−2π2νt) (3.1)

uy = sin(πx) cos(πy) exp(−2π2νt) (3.2)

p =1

4(cos(2πx) + cos(2πy)) exp(−2π2νt) (3.3)

The total kinetic energy E of the flow is:

E = exp(−4π2νt) (3.4)

67

68 CHAPTER 3. VERIFICATION AND VALIDATION

The three-dimensional Taylor-Green vortex on the other hand describes thetransition of an initially two-dimensional vortex into a turbulent state. Theinitial condition is:

ux = − cos(πx) sin(πy) sin(πz) (3.5)

uy = sin(πx) cos(πy) sin(πz) (3.6)

uz = 0 (3.7)

The first series of tests concern simulations of an inviscid vortex, performedon a cartesian grid consisting of 64 equidistantially spaced grid points in eachdirection. The time step ∆t is 0.01; the CFL number corresponding to theinitial condition is 0.32. In figure 3.1(a), we show the total kinetic energyof a two-dimensional vortex as a function of time for three different time-integration schemes. We see that the kinetic energy becomes unbounded whenan explicit Euler method is used. This time integration scheme, combinedwith a center-difference-like discretization stencil, is indeed unstable for inviscidflows. Remarkably, we observe that the energy initially decreases. This maybe explained as follows: two terms affect the energy balance. On the one hand,we have the erroneous dissipation of the pressure term because of the non-solenoidal character of the convected velocity u. On the other hand, there isthe growing instability due to the explicit time-discretization of the non-linearterm. However, this last effect is initially very small and it takes a certain timebefore the unstable growth of energy of the non-linear term dominates thedissipation of the pressure term. The results furthermore confirm that boththe Crank-Nicholson and the implicit Euler scheme provide stable solutions,although they suffer from a slight dissipation due to aforementioned erroneousdissipation of the pressure term. Rather surprisingly, we find that, in twodimensions, both schemes perform equally well, notwithstanding the dissipativenature of the implicit Euler algorithm. This can be attributed to the fact thatFourier modes with low spatial frequencies, like the Taylor-Green vortex, arethe least affected by numerical dissipation.

In figure 3.1(b), we display the kinetic energy as function of time for thecase of a 3D Taylor-Green vortex. As expected, the first-order explicit schemeis again unstable. During a first stage of about one time unit, we see that theCrank-Nicholson algorithm almost perfectly conserves kinetic energy, while theimplicit Euler scheme is slightly dissipative. This is again conform with theproperties of these time integration methods discussed in the previous chapter.However, after this initial time interval, we see a sharp, non-physical decreasein kinetic energy for both implicit schemes. The reason for this is that theerroneous energy loss due to the pressure term increases with time. Indeed,this dissipation is proportional to ||∇4p|| and becomes more important as theflow develops and smaller spatial scales are excited. To illustrate this, we show

68

3.1. TAYLOR-GREEN VORTEX 69

0 2 4 6 8 100.8

0.9

1

1.1

1.2

t

E(co

mp)

0 2 4 6 8 100.8

0.9

1

1.1

1.2

t

a b

Figure 3.1: Kinetic energy E (solid line) and compensated kinetic energy Ecomp

(dashed line) evolution of an inviscid Taylor-Green vortex in two (a) and three(b) dimensions. Results for an explicit Euler (red), Crank-Nicholson (green)and implicit Euler (blue) time integration scheme compared to the exact re-sult (black). Compensated energy for the Crank-Nicholson and implicit Eulerscheme (dashed line).

also curves of Ecomp(t):

Ecomp(tn) = E(tn) +n∑

k=1

∆t

(

∑

i

Viuk|i · G

(

pk−1/2)∣

∣

∣

i

)

(3.8)

Ecomp(t) is thus the kinetic energy with a compensation term for the pressureerror. We find that Ecomp(t) remains close to one in the case of a Crank-Nicholson method; hence, the loss of energy is mainly due to the pressureterm. With an implicit Euler scheme however, we see that the pressure termis not the only term responsible for the loss in kinetic energy. An implicitEuler time-discretization of the non-linear term is not energy-conserving andthe numerical dissipation tends to be larger for smaller spatial scales.

Figure 3.2 shows the kinetic energy evolution for a viscous, decaying two-dimensional vortex. The value of ν was fixed at 0.01. We consider two-differenttime steps. In the case ∆t = 0.01, the Fourier number is approximately 0.41,

69


0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t

E

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

t

Figure 3.2: Kinetic energy (E) evolution of a two-dimensional viscous Taylor-Green viscous for ∆t = 0.01 (left) and ∆t = 0.04 (right). Results for anexplicit Euler (red), Crank-Nicholson (green) and implicit Euler (blue) schemecompared to the exact result (black).

and we obtain stable solutions with explicit as well as with implicit schemes.All curves virtually collapse. However, if we increase ∆t to 0.04, the Fouriernumber becomes larger then one, and the explicit scheme becomes unstable.Both implicit methods exhibit again a quasi-perfect agreement with the exactsolution.

In a last set of numerical experiments, we consider a two-dimensional in-viscid Taylor-Green vortex on a skewed mesh. To this end, we transform acartesian grid of equidistantly spaced grid points (x, y) into a non-cartesiangrid (x′, y′) according to the following transformation:

x′ = x+ 0.2 sin(πx) (3.9)

y′ = y + 0.2 sin(πy) (3.10)

The grid, and contours of constant kinetic energy for the exact solution (3.1-3.3) are shown in figure 3.3(a)-(b). In figure 3.3(c), we display these contoursfor a simulation performed with a Crank-Nicholson scheme. The computationleading to figure 3.3(d) was obtained by omitting the skewness correction term

70

3.2. TURBULENT CHANNEL FLOW 71

Figure 3.3: Inviscid Taylor-Green vortex on a skewed mesh. Mesh lines (a),contours of constant kinetic energy for the exact solution (b), and for the nu-merical simulation obtained with a Crank-Nicholson time advancement schemeand a center-difference-like spatial discretization stencil with (c) and without(d) skewness correction term for the Laplacian operator.

in the Laplacian operator used to solve the Poisson equation of the pressure.We see that we cannot correctly preserve the morphology of the flow withoutthis term.

With this set of simple test cases, we have illustrated the stability propertiesof various time integration schemes. We have also emphasized the necessity ofincorporating a skewness correction term in the discretization stencil of theLaplacian operator.

3.2 Turbulent channel flow

One of the most generic examples of a turbulent shear flow is the one betweentwo parallel plates of infinite extent (see figure 3.4). The flow is driven by auniform pressure gradient along one of the infinitely extended directions (saythe x-direction). As infinite domains are not very well suited for computational

71


Figure 3.4: Sketch of the geometry used for the study of a turbulent channelflow.

purposes, we can replace it by a periodic domain, provided that its extent ismuch larger then the correlation length of the flow in the streamwise (i.e. x)and spanwise (i.e the other direction of infinite extent, say z) directions. Thefirst numerical simulations of turbulent channel flow were performed by Kim. etal. [KMM87]. The friction Reynolds number Reτ (to be defined later) in theircase was 180, and they used a pseudo-spectral method, which is much fasterand more accurate then conventional finite-difference/finite-volume methods.Nowadays, values of Reτ up to 590 have been reached on grids containing morethan 37 million points [MKM99, dAJ03].

3.2.1 Physical background

Turbulent flows have, up to a certain degree, a random nature. Therefore, allrelevant physics should be expressed in terms of statistically averaged quanti-ties. In a fully developed channel flow, all variables, except the pressure, arestatistically invariant in the streamwise and spanwise directions, and in time.The mean of a given function g is defined as:

g(y) =

∫ Tavg

0dt∫ Lz

0dz∫ Lx

0dx g(x, y, z, t)

∫ Tavg

0dt∫ Lz

0dz∫ Lx

0dx

(3.11)

Here, Lx and Lz are the sizes of the box in the periodic directions. Tavg is atime which is large with respect to the characteristic time scales of the flow.In casu, Tavg should be mich larger than the eddy-turnover time τe. We canfurthermore decompose g in a mean and a fluctuating part:

g(x, y, z, t) = g(y) + g′(x, y, z, t) (3.12)

72


From symmetry arguments, we can deduce that the average spanwise velocityis zero. The averaged incompressibility constraint reduces to:

∂uy∂y

= 0 (3.13)

From this, we find that uy is also zero.Averaging the momentum equations, leads, together with the results above, to:

ρ∂

∂y

(

u′xu′y − ν

∂ux∂y

)

= − ∂p

∂x(3.14)

ρ∂u′yu

′y

∂y= −∂p

∂y(3.15)

We can define pw(x) = p + ρu′yu′y, which is, according to the last equation,

independent of y. We furthermore introduce the total shear stress τ (y) asthe sum of the Reynolds stress component −u′xu′y and the viscous shear stressν∂yux. With these definitions, we can recast the average streamwise momentumequation as:

ρ∂τ

∂y= −∂pw

∂x(3.16)

The left-hand side of this equation is function of y only, and the right-hand sidesolely function of x; both derivatives are thus constant. Taking into accountthe asymmetry of τ in y, we eventually obtain:

τ = τ |y=1y (3.17)

In the inner, near-wall region, the viscous stress dominates τ . We can introducean appropriate velocity scale uτ , termed friction velocity, and length scale y+,called wall unit :

u2τ = ρ−1τ |y=1 = ν∂ux∂y

∣

∣

∣

∣

y=1

(3.18)

u+ =u

uτ(3.19)

y+ = ±uτ (L∓ y)

νy ≷ 0 (3.20)

The friction Reynolds number is then defined as Reτ = uτL/ν. By now fur-ther deploying the techniques of dimensional analysis and asymptotic matching[Dav97, Pop00] between the inner and outer regions, one can derive that ux(y)can, up to good approximation, be matched to the following profiles:

ux =u2τνy y+ > 5 (3.21)

ux =uτA

log(uτy

ν

)

+B y+ ? 60, |(y − L)/L| > 0.15 (3.22)

73


In (3.22), known as the log-law, A and B are independent of y, but depend onthe friction Reynolds number. Furthermore, (3.22) is only relevant if the pointat y+ ≈ 60− 70 is located closer to the wall then the point |(y−L)/L| ≈ 0.15.From the definitions of Reτ and y+, it follows that a log law region can onlyexist for Reτ ? 900.

3.2.2 Computational details

A number of simulations of a channel flow at a Reτ = 180 were performed, andcompared to pseudo-spectral results of del Alamo and Jimenez [dAJ03]. Thecorresponding Reynolds number based on the bulk velocity is Re ≈ 3200. Thevalue of the friction Reynolds number is obtained by the following combinationof parameters: ν = 1/180 and fb = 1x. The simulation domain consisted ofa rectangular box of size Lx = 4π, Ly = 2, Lz = 2π. No-slip conditions wereimposed at y = ±1 and periodic boundary conditions were applied in x and zdirection. In [KMM87], it was shown that these streamwise and spanwise do-main lengths are much larger then the respective correlation lengths of the flowunder consideration. We will discuss two sets of results, which were obtainedrespectively at a resolution of 64 and 128 points in each direction. Grid pointswere distributed equidistantly along the periodic directions. The grid was how-ever stretched away from the walls to capture the near-wall region properly.Corresponding to the pseudo-spectral simulations, the points in wall-normaldirection are located at Chebyshev positions y′, which can be computed froman initial equidistant distribution y as follows:

y′ = sin(π

2y)

(3.23)

In the case of a minimum resolution of 64 points in wall-normal direction, thefirst point next to the wall is located at y+=0.216, and we have 3 points withinthe viscous sublayer.

Since the flow at Reτ = 180 is hardly out of the transitional regime, theturbulent regime will not necessarily spontaneously emerge if we start froma zero initial condition. Therefore, we impose an initial condition which is asuperposition of a Poiseuille profile and a large-scale random velocity field. Asemi-implicit Crank-Nicholson time integration scheme was used to advancethe flow in time. The time step ∆t was 6 · 10−4 for both resolution. Thelinear system from the implicit velocity advancement was solved with a Jacobiiteration. For these values of ∆t, the CFL number is about 0.13 (643 points)and 0.26 (1283 points). The pressure Poisson equation was solved with analgebraic multigrid method. We needed around 60000 time steps to obtaina statistical steady regime. For a resolution of 1283 and a simulation on 16CPU’s, we could perform approximately 6000 iterations per day. Statistical

74


10−1

100

101

102

0

2

4

6

8

10

12

14

16

18

20

y+

u+ x

Figure 3.5: Velocity profile of the mean streamwise velocity u+x in a turbulentchannel flow at Reτ = 180. Results of a reference pseudo-spectral method(solid line), and a finite-volume method with respectively 643 (∗) and 1283 (•)grid points. The near-wall profile (3.21) is depicted as a dashed line.

results were obtained by averaging over the periodic directions and over 10000steps in time. This corresponds to almost 100 eddy turn-over times.

3.2.3 Numerical results and discussion

We first investigate the influence of the discretization stencil. Therefore, weshow in figures 3.5-3.6 the averaged streamwise velocity for two different res-olutions together with the pseudo-spectral reference data. In figures 3.7-3.8,we show the total stress, and the individual contributions of the Reynolds andthe shear stress. We see that we obtain satisfactory results with the sparsestencil for both resolutions (figure 3.5). If we zoom in to the near-wall region,we can clearly distinguish the linear near-wall behaviour (3.21) for y+ < 4 Wefind also a good agreement for the viscous and turbulent stress (see figure 3.7).The total stress is virtually a linear function of the wall-normal coordinate y.Surprisingly enough, we even don’t find accurate velocity and stress profileswith the dense stencil (figures 3.6 and 3.8). This discrepancy may be due tothe large anisotropy of the elements close to the walls, although we haven’t

75


10−1

100

101

102

0

2

4

6

8

10

12

14

16

18

20

y+

u+ x

Figure 3.6: Velocity profile of the mean streamwise velocity u+x in a turbulentchannel flow at Reτ = 180. Results of a reference pseudo-spectral method (solidline), and a finite-volume method, using a dense spatial discretization stencil,with respectively 643 (∗) and 1283 (•) grid points. The near-wall profile (3.21)is depicted as a dashed line.

investigated this phenomenon profoundly. Nevertheless, we see that increasingresolution significantly reduces the mismatch with the pseudo-spectral results.

We now investigate the higher-order statistics for the sparse stencil 1. In thevelocity and stress profiles, a difference between the 643 and 1283 simulationswas hardly discernible. The effect of increasing the resolution becomes howeverclear when we consider the fluctuating velocity components. These quantitiesare representative for the level of turbulence within the channel. They are, e.g.for the streamwise component, defined as:

urmsx =

√

u2x − ux2 (3.24)

At a resolution of 643, we observe a large discrepancy between the present andthe pseudo-spectral data. With 1283 grid points however, the finite-volumeand the reference results show a much better agreement.

1The dense stencil is left out of consideration because it did not yield accurate predictions

76

3.3. TWO-DIMENSIONAL MHD FLOWS AT HIGH HARTMANN NUMBER 77

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y+

She

ar s

tres

s

Figure 3.7: Shear stress profiles for a turbulent channel flow computed with asparse discretization stencil: Reynolds stress (green), viscous stress (blue) andtotal shear stress (red). Results for a resolution of 643 (∗) and 1283 (•) gridpoints compared to pseudo-spectral reference data (solid line).

This study of turbulent channel flow showed us that satisfactory predictionsare only obtained with a sparse discretization scheme. However, we found thatall our results tend to the reference data as the spatial resolution is increased.This gives support to our claim that all the methods presented in chapter 2have been correctly implemented.

3.3 Two-dimensional MHD flows at high Hart-mann number

In this section, we validate the aspects of the solver related to the computationof the Lorentz force. Since we have a particular interest in the simulation offusion-relevant flows, we will focus on the high Hartmann number regime. Thefirst series of tests concern laminar MHD flow in a straight duct of infiniteextent, as presented in the first chapter. For the second test case, we consider

for the lower-order statistics.

77


0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

y+

She

ar s

tres

s

Figure 3.8: Shear stress profiles for a turbulent channel flow computed with adense discretization stencil: Reynolds stress (green), viscous stress (blue) andtotal shear stres (red). Results for a resolution of 643 (∗) and 1283 (•) gridpoints compared to pseudo-spectral reference data (solid line).

a more intricate geometry: a plane sudden expansion.

3.3.1 Laminar MHD flow in a straight duct

As we have shown in chapter 1, wall-bounded laminar MHD flows are char-acterized by shear layers whose thickness becomes smaller as the Hartmannnumber increases. The resolution of these viscous layer makes the simulationof such flows at high Hartmann number a challenging task. We consider a flowat M = 1000 in the case of a perfectly insulating duct and a duct with con-ducting Hartmann walls and insulating side walls. Since we are only interestedin the laminar behaviour, we can use a devoluted solution algorithm for unidi-rectional flow. In this algorithm, we have only to solve one velocity componentand we can omit the non-linear term. Furthermore, there is no need to solve aPoisson equation for the non-linear part of the pressure.

The velocity profiles result from a simulation on a mesh of 64 × 64 grid

78


0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

y+

urm

sx

,y,z

Figure 3.9: Mean fluctuating streamwise (urmsx ,red), wall-normal (urms

y ,blue)and spanwise (urms

z ,green) velocity components in a turbulent channel flow atReτ = 180. Results of a reference pseudo-spectral method (solid line), and afinite-volume method with respectively 643 (∗) and 1283 (•) grid points.

points. The grid was refined towards the walls to obtain a good resolution ofthe shear layers. The grid spacing in these layers was uniform in wall-normaldirection. The Hartmann layer consists of 3 or 5 points over a distance ofL/M . The side layers contain 20 points. The grid was stretched in the core byusing a tangent-hyperbolic-like transformation of an initially equidistant grid.The stretching parameters were furthermore chosen such that we obtained asmooth transition between the grid spacing in the core and in the shear layers.To avoid severe restrictions on the time step, the simulations were performedwith a Crank-Nicholson algorithm. In figure 3.10, we observe that a current-conservative formulation for the Lorentz force leads indeed to results whichare significantly more accurate then when we use the gradient discretizationstencil G to compute the potential gradient at the nodes. Furthermore, wesee that an increase in resolution of the Hartmann layers slightly improves theresults in the case of insulating walls, but does not affect the solution whenthese walls are perfectly conducting. The reason for this is that the Hartmannlayers play a much more critical role in the insulating case, since all the currentlines pass through this region, and the pressure drop is solely caused by viscous

79


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

z

u

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

z

Figure 3.10: Velocity profile of the MHD flow in a square duct at M = 1000along the centerline perpendicular to the magnetic field. Results for a perfectlyinsulating duct (left) and a duct with insulating side walls and conductingHartmann walls (right). Comparison between the analytic solution (solid line),a classical (blue) and a current-conservative (green) formulation of the Lorentzforce, and between a mesh with 3 (•) and 5 (∗) grid points in the Hartmannlayer.

momentum losses at the boundaries.

We show also results for a duct with an uniform wall conductance ratioc = 0.1 and c = 1.0 in figure 3.11; these were computed with a current-conservative formulation for the Lorentz force. In this case, the exact solutioncan not be expressed in terms of analytical functions; we will compare our re-sults to asymptotic approximations. In asymptotic methods, one assumes theexistence of an inviscid core, surrounded by thin shear layers. The solutionsprovided by this technique tend to become more accurate as the Hartmannnumber increases. To obtain the numerical results, we used the same grid asdescribed above. We find an excellent agreement between both sets of results;this validates our implementation of Shercliff’s thin wall condition (1.75).

80


0 0.5 10

1

2

3

4

5

6

z

ux

0 0.5 10

1

2

3

4

5

6

z

ux

0 0.5 10

0.2

0.4

0.6

0.8

1

z

φ

0 0.5 10

0.2

0.4

0.6

0.8

1

z

φ

Figure 3.11: Velocity (above) and potential (below) profile of the MHD flowin a square duct with a uniform wall conductivity c at M = 1000 along thecenterline perpendicular to the magnetic field. Numerical (∗) and asymptotic(solid line) results for c=0.1 (left) and c=1 (right).

3.3.2 Laminar MHD flow in a plane sudden expansion

We investigate the laminar MHD flow in a symmetric, plane sudden expan-sion with expansion ratio 4. All physical relevant variables are invariant inz-direction, and the magnetic field B = B01y is uniform and perpendicularto the expansion’s axis. All the walls are perfect conductors. The distancebetween the walls in the large duct is 2Ly. This situation is sketched in figure3.12(a); the origin of the coordinate system is located at the expansion, onthe symmetry axis of the configuration. This flow was studied previously byMistrangelo [Mis05] for the parameters Re = 1000 and M = 1000, and we nowinvestigate the same values of Re and M . The length scale used to define thesenon-dimensional number is Ly.

Far away from the expansion, the flow should be fully developed. At the

81


Figure 3.12: Sketch of the geometry for the MHD flow in a plane suddenexpansion (a). Detail of the mesh close to the corner x = 0, y = 0.25Ly for themesh with a growth factor of 1.5 (b) and 1.12 (c).

inflow boundary, located at a distance 3Ly from the expansion, we impose aHartmann channel profile (1.89) with M = 250. The outflow boundary is sit-uated at a distance 9Ly from the expansion. A convective outlet conditionwas chosen for this boundary. For this two-dimensional geometry, the Poissonequation for the potential reduces to ∇2φ = 0. We can apply Neumann bound-ary conditions for φ at the inflow and outflow boundary, since we assume fullydeveloped conditions at these locations. The solution of the Poisson equationis then φ = C, with C an arbitrary integration constant.

The geometrical discontinuity will give rise to an internal shear layer charac-terized by intense jets. According to the theory of Hunt and Leibovich [HL67],the flow should however relax to its fully developed profile over a distancewhich scales as O(N−1/3Ly) = O(0.1Ly) (this scaling law holds in the param-eter range M3/2 ≫ N ; see Appendix C.2).

The simulation grid is two-dimensional and consists of rectangular elementswhich are refined towards the expansion and the Hartmann layer. Close tothe walls, the distance between the grid-points in the wall-normal directionis ∆y = LyM

−1/5. In the neighbourhood of the expansion, the spacing instreamwise direction ∆x is LyN

−1/3/25 = 0.04Ly. These paramaters are thesame as in [Mis05]. The grid spacing grows exponentially if we move away fromthe refined region, until a maximum grid size for ∆x of 0.06Ly and for ∆y of

82


0.04Ly is reached. Two values of the mesh growth factor have been investigated.In the first case, it was 1.5, as in [Mis05] (see figure 3.12(b)) in the second one,we have chosen a more conservative value of 1.12 (figure 3.12(c)). In principle,the latter choice should yield more accurate results, since the mesh is (locally)more uniform. It turns however out that the results on both grids are virtuallyidentical. The complete grid consists of about 34000 (growth factor 1.5) and53000 (growth factor 1.12) grid nodes. The disadvantage of this approach isthat it leads to a useless clustering of points close to the axes y = ±Ly/4.

For the spatial discretization, a sparse stencil is used, and the equationsare advanced in time with a Crank-Nicholson method with a constant timestep. The Fourier number based on this time step is more or less 0.25 and themaximum CFL during the simulation was approximately 0.65. A fully devel-oped state is reached after about 25000 iterations. In figure (3.13), we compareour simulation results with the ones of Mistrangelo [Mis05]. We observe anexcellent agreement between both data sets, except for the profile of ux alongthe axis x = 0, close to the corners at y = ±0.25Ly. The present simulationspredicts peak streamwise velocities at x = 0 which are about 10 percent largerthan in the reference results. We don’t have a satisfactory explanation forthis disagreement; an in-depth analysis of this discrepancy would require moreinformation on the implementation details of the code used in [Mis05].

83


−2 −1 0 1 2 3 40.5

1

1.5

2

2.5

3

3.5

4

4.5

x

ux

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

y

ux

Figure 3.13: Velocity profile of the streamwise velocity ux along the centerlineof a plane sudden expansion (above) and at the expansion (below). Comparisonbetween present results (solid line) and reference data of Mistrangelo [Mis05](∗). Above: results along the axis y = 0. Below: results along the axes x = 0(green), x = 0.0625Ly (blue) and x = 1.25Ly (red).

84

Chapter 4

Laminar pipe flow

“There is no subject so old that something new cannot be said aboutit.”Fyodor Dostoevsky

This chapter is devoted to the numerical study of the laminar, incompressibleflow of a conducting fluid in a circular pipe under a uniform magnetic field, i.e.the MHD variant of Poiseuille pipe flow. Although one of the most elementaryand common geometries, the case of circular pipe flow in a uniform magneticfield has drawn little attention within the quasi-static MHD community, pre-sumably because of its apparent simplicity. However, in this chapter, we willshow some features of the velocity profile at high Hartmann number which werenever revealed prior to this work.

In the first section of this chapter, we pose the problem mathematically andbriefly sketch the historic achievements made. This is followed by an overviewof the most relevant physical phenomena appearing in such a configuration.In the third section, we compare several meshing strategies and discretizationstencils. Finally, we provide and discuss extensively the obtained numericaldatabase.

4.1 Introduction

We consider the incompressible, unidirectional flow u = u(x, y)1z of an electri-cally conducting fluid through a circular pipe of radius R and of infinite extentin the axial direction (see figure 4.1). An external magnetic field B = B01y isimposed, and the flow is driven by a uniform pressure gradient in the axial di-rection, ∇p = −f1z. To obtain non-dimensional equations, we rescale the vari-ables as follows: u → Uu, B → B01y, x → Rx,∇ → R−1∇, t → ρ(σB2

0)−1t,

85

86 CHAPTER 4. LAMINAR PIPE FLOW

Figure 4.1: Sketch of the geometry used for the study of the laminar flow in astraight circular pipe of infinite extent.

J → σUB0J, φ→ UB0Rφ and p→ σURB20p with U = f(σB2

0)−1. The quasi-

static MHD equations for such a flow become linear since there is no convectivetransport of momentum for a unidirectional flow:

∂u

∂t= M−2∇2u+ J× 1y + 1z = 0 (4.1)

J = −∇φ+ u× 1y (4.2)

∇2φ = −∂u∂x

(4.3)

The length scale used to define the Hartmann number in this case is the pipe’sradius R. It will also be convenient to introduce the radial coordinate r =√

x2 + y2 and the polar angle θ, with the convention that θ = 0 along themagnetic field direction. The (non-dimensional) boundary conditions are then:

u|r=1 = 0 (4.4)

∂φ

∂r

∣

∣

∣

∣

r=1

= c∂2φ

∂θ2

∣

∣

∣

∣

r=1

(4.5)

We see that such a flow can be described by merely two parameters: the Hart-mann number M and the wall conductance ratio c.

The problem of pipe flow under a magnetic field goes back to the very firstinvestigations of MHD phenomena. In their pioneering experimental work,Hartmann and Lazarus [Har37, HL37] studied the variation of the pressure dropwith the magnetic field intensity in the case of a mercury flow in a circular pipe.They found that the pressure drop first decreases with increasing magnetic fieldstrength, but then starts to grow if a certain threshold has been reached. Thefirst effect is due to a decrease of turbulent shear stress, the second one is theresult of the braking effect of the Lorentz force.

86

4.2. PHENOMENOLOGY 87

In the 1960’s, analytical solutions were found for pipes under a uniformmagnetic field with perfectly insulating [Gol62] as well as perfectly conductingwalls [IKM67]. A later analytical, and little known work by Samad [Sam81]considered the case of a pipe with finite wall thickness and observed overspeedregions in the velocity profile of MHD pipe flow. However, all these solutions areunder the form of infinite series expansions involving modified Bessel functions,which makes them difficult to evaluate, certainly when the Hartmann numberis high. This era saw also the birth of various approximate solutions, based onasymptotic methods [She56, CL61, She62]. The results obtained through thesemethods tend to become more accurate as the Hartmann number increases,but it was also pointed out that these approximate solutions, break down inthe region where the wall is parallel to the magnetic field.

At that point in history, the problem of the laminar flow under a uniformmagnetic field was considered to be more or less solved. Attention was thendirected to more challenging flows in complex geometries, magnetic field con-figurations or in turbulent regimes. This shift in focus was also favoured bythe exponential increase in computational power and the development of CFDtools. Nevertheless, the simulation of high Hartmann number flows with finite-difference or finite-volume codes remained to suffer from large numerical errors,as discussed in section 2.3. Among the few numerical studies of laminar MHDpipe flow in a uniform magnetic field, we cite the work of Barrett [Bar01], whoused a finite element method; the Hartmann number range reached in his workwas limited to M = 100. Of further interest for the present discussion is oneof the so-called MHD benchmark problems. This benchmark is related to anexperimental study of a liquid metal flow in a circular pipe in the fringing re-gion of a magnetic field [RPD85]. Several authors have attempted to reproducenumerically the experimental data set, both with asymptotic methods [MR03]and by means of direct simulations [ACRKK10]. The velocity profile which hasto be specified at the inlet boundary in these simulations, is exactly the oneinvestigated in this chapter. In this work, we now revisit the basic problem ofcircular pipe flow under the influence of a uniform magnetic field by means ofhigh-resolution numerical simulations.

4.2 Phenomenology

In figure 4.2, the basic features of the MHD pipe flow are shown; these can beinferred from the asymptotic analysis of MHD pipe flow, which is presentedin Appendix C.1. Similarly to the case of rectangular duct flow, discussedin the first chapter, three distinct regions can be considered in the limit ofhigh Hartmann number: the core of the flow, the Hartmann layer(s), and theRoberts layers, which are the counterpart of the side layers in rectangular ductflow.

87


Figure 4.2: Electrical current paths in the MHD pipe flow for a perfectly insu-lating pipe (left) and a perfectly conducting pipe (right).

In the core region, viscous effects are negligible, and the velocity profile isapproximately uniform due to the equalizing effect of the Lorentz force. Themotion of the fluid drives a uniform current, perpendicular to the flow andmagnetic field directions. Since current lines must form closed paths in thecombined wall-pipe domain, a potential is induced which changes the directionof the currents in the viscous boundary layers and drives them along the walland/or makes them enter into the wall, depending on the wall conductanceratio. For pipes with well-conducting walls, the effect of the potential gradienton the core current distribution is negligible. In insulating domains on the otherhand, the potential gradient almost completely cancels the induced electricfield u × B in the core region; the resulting current magnitude in this zoneis J = O(M−1)u × B (this is shown in Appendix C.1), and there is a netpotential difference between the end points of the center line perpendicular tothe magnetic field, which is proportional to the flow rate.

In the vicinity of the pipe wall, we have Hartmann layers, characterized bya typical exponential drop-off. Their thickness varies along the perimeter ofthe duct, and is given by:

δHL(θ) = (M cos θ)−1 (4.6)

However, the previous expression becomes singular in those regions of the pipe’scross section that are adjacent to the wall, and where the normal vector to

88

4.3. COMPUTATIONAL DETAILS AND GRID STUDY 89

the wall is perpendicular to the magnetic field, i.e those locations where thepolar coordinate θ satisfies 2θ = ±π. The effect of this singularity in theasymptotic approximation was profoundly investigated by Roberts [Rob67b];hence, these regions are now known as the Roberts layers. He found that theselayers extend over a distance O(M−2/3) along radial direction and O(M−1/3)in polar direction. An a priori prediction of the velocity profile in the Robertslayers is very difficult since it depends in a subtle way on the exact directionof the electric currents in these viscous boundary layers.

In the traditional asymptotic approximation of MHD pipe flow, the Robertslayers are not taken into account: the velocity profiles in the core and Hartmannlayers are matched and the core velocity is determined by taking into accountthe pipe’s electrical boundary condition. The resulting expressions for the corevelocity and the velocity profile along and perpendicular to the magnetic fielddirection in terms of dimensionless variables is [CL61]:

uc =(1 + c)M

1 +Mc(4.7)

u(x = 0, y) = uc(1− exp(M(|y| − 1)) (4.8)

uc(y = 0, x) =(1 + c)M

√1− x2

1 + cM√1− x2

(4.9)

4.3 Computational details and grid study

In this section, we first summarize the main simulation parameters. Sincethe accuracy of the results depends strongly on the combination of the meshdesign and the discretization stencil, we have devoted a second, more elaboratesubsection to a grid study.

4.3.1 Computational details

As we assume that the flow is laminar and unidirectional, we can use the‘simplified’ version of the solver, detailed in section 3.3, in which we solve onlyfor one component of the velocity and omit the non-linear term. As such, thereis no need to solve an equation for the pressure. We choose u = 0, φ = 0 asinitial condition, and use a Crank-Nicholson time-stepping for the viscous term.We have tested other initial conditions and backward Euler time integrationschemes; these alternatives lead to almost identical results.

4.3.2 Grid study

We now investigate the influence of the mesh on the simulation results. Through-out this subsection, the value of the Hartmann number is M = 100, and we

89


consider the two limiting values of c, i.e. c = 0 and c = ∞. Guiding principlesthat should be taken into account when generating a mesh for a high Hartmannnumber flow are the following:

• The current density is quasi-uniform in the core. Moreover, the velocityprofile does not vary fiercly in this region. As such, we can presume thatthe mesh spacing can be quite coarse if we are far enough away from theviscous boundary layers.

• Hartmann layers are characterized by an exponential drop-off in radialdirection for the velocity. They also carry large current densities if cM ≪1. As such, we want at least 4 points on a distance of (M cos θ)−1 inradial direction close to the wall. If we want to limit the total numberof grid points in the mesh, these near-wall elements have to be stronglyanisotropic.

Description of the mesh topology

In a first series of tests, we compare solutions between three types of gridswhich contain both of the features specified above (see figure 4.3). All threemeshes have 128 nodes along the boundary, and the grid spacing in the wall-normal direction is M−1/5 = 0.002. The first mesh is the popular, so-calledO-grid, which consists of a core of rectangular elements, surrounded by fourblocks of polar grids which are stretched away from the boundaries. As canbe seen, this type of mesh has several disadvantages. By definition, the totalnumber of grid points on the sides of the square and on the perimeter of thecircle are the same. This implies that a coarse mesh in the core can only beobtained by limiting severely the number of points on the boundary. Moreover,the distance between the square and the boundary is not the same along thediagonals and axes of the square. However, this distance has to be bridged witha same number of grid points so that this unavoidably leads to severe jumpsizes between adjacent cells.

For the second and third designs, we attempt to remedy the shortcomings ofthe O-grid. The stretched polar grid is now limited to the surroundings of theboundary layers. It is matched with a completely unstructured quadrilateralbut isotropic core, and the transition between the polar and the unstructuredparts is smooth. We will term this type of mesh Q-grid. The third type of meshis similar to the second, but the unstructured quadrilateral core is replace bya triangular one; we will refer to it as T-grid.

Results

In figures 4.4-4.5, simulation results for the three types of meshes are depicted,respectively for the cases c = 0 and c = ∞. Furthermore, we compare between

90


Figure 4.3: Grids used for the grid sensitivity study of section 4.3: O-grid with11649 grid nodes (a), Q-grid with 10609 nodes (b) and T-grid with 12041 nodes(c).

two discretization stencils. We will not comment yet the physical meaning ofthese results, but we assume that all variables should be smooth functions.

For the case of a perfectly insulating pipe (c = 0), we see that the profileof u along the x-axis is smooth for every grid/stencil combination. The resultsobtained on the T-grid with a dense scheme are however different from all theother ones, and do not follow the square-root-like profile of the asymptoticapproximation (4.9). The current distribution for this combination exhibits anon-physical behavior, with a magnitude far beyond the bounds of the figureand a severe jump at the interface between the boundary layer region andthe triangular core. This phenomenon is clearly due to the grid topology.Furthermore, we see that the profile of Jx along the x-axis contains some spikesfor all the stencil/grid combinations. For the O-grid, we have only one spike,which is, not surprisingly, located at the interface between the square and polargrid. At higher Hartmann number, this may lead to a non-smooth velocityprofile at this location. Finally, we also observe that the magnitude of the

91


0 0.5 10

5

10

x

u

0 0.4 0.8−1.05

−1

−0.95

x

Jx

0 0.5 10

5

10

x

u

0 0.4 0.8−1.05

−1

−0.95

x

Jx

0 0.5 10

5

10

x

u

0 0.4 0.8

−1

−1.05

−0.95

x

Jx

Figure 4.4: Profiles of the velocity u and current density component Jx alongthe x-axis in a pipe with perfectly insulating walls. Results on grid ‘O’ (top),‘Q’ (center), and ‘T’ (bottom) for a sparse (black solid) and dense (red dashed)discretization stencil.

spikes is significantly smaller when a dense discretization stencil is used.

For the case c = ∞ (see figure 4.5), we obtain smooth profiles for both thevelocity and current density as long as dense stencils are combined with fullyquadrilateral meshes. The use of sparse stencils on these quadrilateral meshesleads to high-frequency numerical noise; in the case of an O-grid, this noise isagain concentrated at the boundary between the central square and the polargrid regions. For the T-grid, we find a similar tendency as for the insulatingcase. A completely non-physical behavior results from the combination of adense stencil and a T-grid, while the use of a sparse stencil on this grid typeleads to a velocity and current density profile which is similar to the onesobtained on quadrilateral meshes with dense stencils. Note however that thereis a slight jump in the derivative of u at the interface between the triangularcore and the quadrilateral boundary layer region (marked by an arrow in figure4.5).

92


0 0.4 0.8−1.05

−1

−0.95

x

Jx

0 0.5 18

10

12

x

u

0 0.4 0.8−1.05

−1

−0.95

x

Jx

0 0.5 18

10

12

x

u

0 0.5 18

10

12

x

u

0 0.4 0.8−1.05

−1

−0.95

x

Jx

Figure 4.5: Profiles of the velocity u and current density component Jx alongthe x-axis in a pipe with perfectly conducting walls. Results on grid ‘O’ (top),‘Q’ (center), and ‘T’ (bottom) for a sparse (black solid) and dense (red dashed)discretization stencil.

Discussion

In section 2.3, we have argued that the computation of J in the core of aperfectly insulating duct is a badly conditioned operation. For the presentcase, the magnitude of Jx is about M = 100 times smaller than the one of theterms ∇φ and u×B. The relative variations of Jx are of the order a few tenthsof a percent if a dense stencil is used on the O- and Q-grid or a sparse stencilon a T-grid. We may conclude that these grid/stencil combinations lead toresults which are ‘satisfactorily’ accurate.

93


0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

x

u/u

c

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

x0 0.5 1

0

0.2

0.4

0.6

0.8

1

1.2

x

Figure 4.6: Velocity profile along the x-axis in a pipe with perfectly conductingwalls at M = 100 for three meshes of type ‘Q’ with respectively 128 (left), 512(center) and 2048 (right) grid nodes along the wall. Comparison between acombination of a sparse discretization stencil and a non-conservative formula-tion for J (blue), a sparse discretization stencil and a conservative formulationfor J (red) and a dense discretization stencil and a conservative formulation forJ (green).

On the other hand, the computation of J in the core of a perfectly con-ducting pipe is a well conditioned operation (Jx ≈ uB; |∂xφ| ≪ uB), and theviscous term should have a smoothing effect. Rather surprisingly, we find thatthe velocity and current density profiles for c = ∞ on Q-and O-grids sufferfrom severe numerical noise if a sparse stencil is used. We will now analysethese numerical problems more profoundly for the case of a Q-grid.

Analysis for the case c = ∞ on a Q-grid

As a first step, we perform a grid refinement study on Q-grids with respectively128, 512 and 2048 points along the boundary. In the most refined mesh, noboundary layer stretching is needed since the grid spacing in polar directionhas become smaller then the Hartmann layer thickness. In figure 4.6, we show

94


Figure 4.7: Velocity isocontours (red) for the MHD flow in a perfectly conduct-ing pipe, resulting from the use of a sparse discretization stencil on a Q-gridwith 128 nodes on the boundary.

plots of the velocity along the x-axis. We find that the noise is damped asthe grid size becomes smaller, and that all formulations eventually convergetowards the same solution. This provides evidence that the noise in the caseof a sparse discretization stencil is not caused by an implementation error, andthat the results obtained with a dense stencil on coarse meshes are already veryaccurate.

Furthermore, in figure 4.7, we show core velocity isocontours, resulting froma computation on a given Q-grid with 128 grid nodes on the wall, displayed aswell. The discretization stencil used was a sparse one. Numerical noise occursat those locations where the isolines are closely packed. We see that this is inthe vicinity of nodes marked by a black dot. These are ‘singular nodes’, in thesense that they are shared by three or five elements, and not by four like the‘regular’ nodes. The control volumes of these nodes are more irregular; we canthus presume that a more accurate discretization scheme is required to get ridof the cumbersome numerical behavior at these locations.

In a last set of experiments, we compare two discretization stencils for thereconstruction of the current density at the nodes. The Laplacian operator,the right-hand side of the Poisson equation for the potential and the currentdensity fluxes J |(i,j) at the CV faces are obtained with a sparse approach. Thefirst stencil for J is the classical sparse one, i.e.:

J|i =1

Vi

∑

j∈π(i)

J |(i,j)x|(i,j) (4.10)

For the second stencil, we use a dense-like approach for the position coordinate

95


0 0.5 18

10

12

14

x

u

0 0.4 0.8−1.05

−1

−0.95

x

Jx

Figure 4.8: Profile of the velocity (left) and current density (right) in a perfectlyconducting pipe along the x-axis on a Q-grid with 128 nodes on the boundary.Comparison between stencils (4.10) (black,solid) and (4.11) (red,dashed) forthe reconstruction of J at the nodes. All other terms have been discretizedwith a sparse stencil.

in the computation of the divergence :

J|i =1

Vi

∑

j∈π(i)

J |(i,j)x|(i,j) + x|elem

2(4.11)

Here, xelem is the barycenter of the element. Figure 4.8 shows that this second,more accurate approach does not suffer from any numerical noise. This test thusprovides an illustration that the poor numerical behavior can be attributed tothe low accuracy of the surface integral (4.10) which defines the nodal currentdensity in the sparse approach at irregularly shaped control volumes.

Summary

From the grid study, we found that overall, the most accurate results for bothvalues of c are obtained with a dense stencil on a Q- or O-grid, or with a sparsestencil on a triangular grid. Rather surprisingly, the most severe numericaldeficiencies on Q-grids were encountered in the case c = ∞. Further analysis ofthis phenomenon revealed that it was related to the presence of ‘singular nodes’,and to the current reconstruction scheme (4.10) in the sparse formulation. Thevelocity profiles presented in the next section are computed with a dense stencilon a Q-grid.

4.4 Results and discussion

In figure 4.9, we show numerical and asymptotical solutions for a flow at M =2000, for both a perfectly insulating and conducting pipe. We consider cuts of

96

4.4. RESULTS AND DISCUSSION 97

10−4

10−2

100

0

0.2

0.4

0.6

0.8

1

1 − yu/u

c

10−4

10−2

100

0

0.2

0.4

0.6

0.8

1

1 − y

u/u

c

0 0.5 10

0.2

0.4

0.6

0.8

1

x

u/u

c

0 0.5 10

0.2

0.4

0.6

0.8

1

x

u/u

c

Figure 4.9: Normalized velocity profiles for the laminar flow at M = 2000 in apipe with perfectly insulating (top) and perfectly conducting (bottom) walls.Cut along the axes perpendicular (left) and parallel (right) to the magneticfield. Comparison between the asymptotic approximation (solid line) and thenumerical solution (∗)

the velocity profile along the two main axes. For the insulating case (c = 0),we observe a quasi-perfect agreement between our numerical method and theasymptotic solution. In the conducting case (c = ∞), the agreement is excellentagain along the direction of the magnetic field, whereas there is a significantdiscrepancy in the results along the x-axis: small zones of overspeed appearin the numerical solution, whereas the asymptotic approximation predicts aflat profile. Simulation results on refined grids confirm the existence of thisoverspeed zone. The behavior observed in the conducting case, should in acertain sense not surprise us. The velocity profile of the analogous case ofMHD flow in a perfectly conducting square duct is also characterized by the

97


0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

u/u

c

0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

c = 0.001

Figure 4.10: Emergence of the overspeed zones with increasing c, illustratedfor a flow at M = 2000. Cuts along the pipe’s centerline perpendicular tothe magnetic field. Asymptotic approximations (solid line), and numericalsimulations (×).

presence of such small overspeed zones [Hun65].

In figure 4.10, we focus on the region between x = 0.8 and x = 1.0 for aflow at the same Hartmann number (M = 2000), but now we consider fourdifferent values of the wall conductance c. The figure shows that the overspeedzones appear gradually near the wall, with at first the emergence of a plateau,followed by the formation of small side bumps with velocities below the corevelocity, which grow eventually into zones of overspeed.

We choose to define the wall conductance parameter ccrit which marks theemergence of overspeed zones as the minimum c for which the velocity profilehas a local maximum at positions different from the pipe axis. This yields acurve in the (M, c)-plane that is displayed in figure 4.11. The results can besummarized as follows:

• The lowest Hartmann number for which we observe a velocity overspeed,is M = 12. Overspeed regions were observed earlier in this Hartmannnumber range (M = 18) by Samad [Sam81].

• The curve goes through a local minimum atM = 35 and a local maximumat M = 41. For lower values of the Hartmann number, the curve has a

98


101

102

103

104

10−3

10−2

10−1

100

M

c crit

M −2/3

Figure 4.11: Limiting values ccrit (∗) for the emergence of overspeed regions asa function of the Hartmann number and fitted power-law like behavior (−−−).

steep descent; at higher Hartmann numbers, ccrit follows a power-law likebehavior.

• We can fit the results for M ≥ 250 with the relationship ccrit ∝ M−2/3.We can explain this scaling law as follows. When the wall has a finitewall conductance, current loops can close both by entering the wall orby changing direction, in the fluid. We can interpret this as a system oftwo parallel resistors. For the Roberts layers, the fraction of the currentthat enters into the wall will depend on the conductance of the wall andof the layers. If we take into account Roberts’ estimate for the thicknessof these shear layers, we find that both conductances become equal ifc = O(M−2/3). If c is smaller, the current lines will preferably returnthrough the Roberts layers, as depicted in the left-hand side of figure4.2. In the opposite case, the situation in the right-hand side of thisfigure will be more accurate, and small overspeed zones may appear as aconsequence of subtle current distributions in the Roberts layers.

We define the relative amplitude α of the overspeed regions as the ratio betweenthe local maximum and core velocity. Figure 4.12 displays the value of α as a

99


101

102

103

1

1.02

1.04

1.06

1.08

1.1

1.12

M

α

Figure 4.12: Relative amplitude of the overspeed zones as a function of M fordifferent values of c. Curves for c = 0.1 (2), c = 1.0 (•) and c = ∞ (∗).

function of the Hartmann number for different values of the wall conductivity.For c = ∞ and c = 1.0 and in the limit of M → ∞, α clearly converges to avalue slightly above 11 %. For c = 0.1, the values of the Hartmann numberconsidered are not high enough to observe the same asymptote. However, itseems reasonable to conjecture that α scales as O(M0), provided that c andM are sufficiently high.

To find the proper scaling for the width of the overspeed regions along thex-axis, we introduce the coordinate ξ = 1−x and investigate how the position ofthe velocity maximum ξmax = maxξ u scales with the Hartmann number. Fromthe data in the left-hand side of figure 4.13, we conclude that, for M between250 and 3000, ξmax scales asM−2/3 (for c = 0.1, this scaling is probably reachedonly at the highest values of the Hartmann numbers considered). It makes senseto interpret ξmax as a measure of the radial extent of the Roberts layers. Inthe asymptotic analysis of the case c = 0 [Rob67a], the same O(M−2/3) scalinglaw was found for the radial length of these layers.

In the same work, Roberts also derived for c = 0 that these regions areresponsible for a contribution of O(M−7/3) to the asymptotic expression of theflow rate. In the right-hand side of figure 4.13, we show the relative flow rate

100


102

103

104

10−2

10−1

M

ξ

102

103

104

10−3

10−2

10−1

100

M

∆Q

(%)

M−2/3

M−1

Figure 4.13: Scaling laws in the conducting pipe: Position of the velocitymaximum ξmax(l.h.s.) and flow rate deficit ∆Q (r.h.s.) as function of theHartmann number for three different values of the wall conductance. Curvesfor c = 0.1 (2), c = 1.0 (•) and c = ∞ (∗).

deficit ∆Q as a function of the Hartmann number. This quantity is defined as:

∆Q =Qnumerical −Qasymptotic

Qasymptotic(4.12)

The asymptotic flow rate Qasymptotic does not contain Hartmann or side layer

101


correction terms, and is given by [CL61]:

Qasymptotic = 4uc(1 + cM)

π

4cM− 1

(cM)2+

π

2(cM)3

− 2

(cM)3

arctanh(√

cM−1cM+1

)

√

(cM)2 − 1

(4.13)

Figure 4.13 (right-hand-side) shows us that the combined Hartmann and sidelayer correction term scales as M−1. It is however not straightforward to con-sider the contribution of the side jets alone. Nevertheless, figure 4.14 gives anindication of how small the side layer correction term might be; the numericalprofile along the x-axis is partially below and partially above the asymptoticprofile. For the values of the Hartmann number and coductance ratio consid-ered, these positive and negative corrections to the flow rate always cancel eachother nearly perfectly; hence, it is very likely that the overspeed regions inducea flow rate correction that is much smaller than one would intuitively expect.

4.5 Conclusions

In this chapter, we have investigated in detail the laminar MHD flow in astraight, circular pipe up to M = 3000 with the numerical methods presentedand validated in chapter 2. We found that the results obtained are very sensitiveto the spatial discretization stencil and grid topology. It appears that thehighest accuracy is reached with a sparse stencil on a triangular mesh, or adense stencil on a quadrilateral mesh. The simulations results presented inthis chapter have been computed with the latter combination. They show thatoverspeed zones emerge at sufficiently high values of the Hartmann numberand the wall conductance ratio. These overspeed regions do not disappear asM → ∞, but are of the order of O(M0) compared to the core velocity. Thelateral extent of the overspeed zones scales as O(M−2/3), in agreement withthe asymptotic theory.

102

4.5. CONCLUSIONS 103

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

u/u

c

××× ×××

× ××

××

Figure 4.14: Contribution of the overspeed regions to the flow rate for a per-fectly conducting pipe at M = 400. Positive (×) and negative (2) correctionsof the numerical solution (solid line) to the asymptotic theory (dashed line).

103


104

Chapter 5

Flow in a toroidal squareduct

“When I meet God, I am going to ask him two questions: Whyrelativity? And why turbulence? I really believe he will have ananswer for the first.”Horace Lamb

In this chapter, we investigate the incompressible hydrodynamic and magneto-hydrodynamic flow in a toroidal duct of square cross-section. We will motivatethe choice for this particular geometry in the first section. Subsequently, we willdefine the problem mathematically. In the third and fourth section, we presentthe computational set-up and discuss the results for flows in the laminar andturbulent regime respectively.

5.1 Motivation

In the first chapter, we have shown that two different types of boundary lay-ers occur in MHD straight duct flow, submitted to a uniform magnetic field:Hartmann layers, with a non-dimensional thickness of O(M−1) at walls perpen-dicular to the field and side layers at walls parallel to the field, whose thicknessscales as O(M−1/2). In the past years, there has been considerable interestin the role of these shear layers in the transition process towards turbulencein MHD duct and channel flows. Krasnov et al. [KZZ+04] performed a com-putational study of the instability of the Hartmann layer in a channel flow,and found that the parameter which governs the transition, is the ratio be-tween the Reynolds number and the Hartmann number. The critical value ofRe/M lies between 350 and 400. Moresco and Alboussiere on the other hand,

105

106 CHAPTER 5. FLOW IN A TOROIDAL SQUARE DUCT

performed friction factor measurements in an electrically driven toroidal ductof square cross section at high Hartmann (M ≈ 260 − 1700) and Reynoldsnumber (Re ≈ 105 − 106). Since the major part of the friction occurs in theHartmann layer for high Hartmann number flows, they conjectured that a sud-den change in the behavior of the friction factor is related to a transition inthe Hartmann layer. Their measurements showed that this transition occursat Re/M ≈ 380, regardless of the exact value of the Hartmann number. Alinear stability analysis of Lingwood and Alboussiere [LA99] yielded a criticalvalue of Re/M ≈ 50000. This large discrepancy indicates that the transitionis triggered by nonlinear effects.

The experimental method used in [MA04] did however not allow to study thebehavior of the side layers. A computational study of Boeck et al. [BKRZ09]on the other hand, showed that the nuclei of instability in MHD straight ductflow are located in the side layers. The common feature in all these studiesis that the Hartmann walls are insulating. Other authors have consideredthe instability in duct flows with conducting Hartmann walls [RP89, KKM09].These flows however are characterized by strong side wall jets, and undergoa completely different transition; they are not directly relevant to the presentwork.

It is clear that many aspects of the experiment performed by Moresco andAlboussiere are not yet completely understood. Currently, it is unfortunatelynot possible to access the whole paramater range covered in the experimentwith numerical simulations based on an unstructured finite-volume formulation.Moreover, even the laminar behavior of MHD toroidal duct flow is relativelyunexplored. To the best of our knowledge, only two studies of the laminar flowin such a configuration have been undertaken. The first one was performed byBaylis and Hunt [BH71]. They used an asymptotic approach and showed thatthe inertial term is negligible when the aspect ratio between the duct side andthe average radius of the duct is small compared to M/Re. One decade later,Tabeling and Chabrerie [TC81] performed a more detailed analysis, in whichthey considered the curvature as a small parameter, and used a perturbationseries approach to study the flow in the limit of high Hartmann number and lowcurvature. This allowed them to compute the secondary flow profile in the shearlayers for sufficiently low values of the curvature and the Reynolds number andhigh values of the Hartmann number. They predicted that streamwise-orientedvortices would occur in the parallel layers whose exact shape depends on theelectric boundary conditions. Unfortunately, they did not investigate thesevortices for the case with insulating Hartmann walls and conducting side walls,the configuration used in [MA04]. Furthermore, they found that the secondaryflow has a strong radially inward component along the Hartmann layers.

In this chapter, we discuss numerical simulation results of purely hydrody-namic and MHD flow in a toroidal duct of square cross-section for the config-

106

5.2. MATHEMATICAL FORMULATION 107

Figure 5.1: Section of a toroidal duct of square cross-section with the magneticfield B along the axis of the torus.

uration used in [MA04]; we investigate two different regimes in the parameterspace (Re,M). In the first set of simulations, we focus on the laminar behaviorat relatively low values of Re and high values of M . Such a study results in amore complete characterization of the laminar base flow of the aforementionedexperiment. For the second series of numerical experiments, we consider moremoderate values of M (between 0 and 30), and significantly higher values ofRe; under these conditions, the flow is (partially) in a turbulent state. Thisyields some insight on the possibility of the coexistence of different flow regimesfor the Hartmann and side layers.

5.2 Mathematical formulation

We consider the incompressible flow, characterized by a velocity field u, of afluid in a square annular duct with mean radius R and length 2L (see figure 5.1).The axis of the torus is along the y-direction, and the uniform magnetic fieldB = B01y is parallel to this axis. It will be instructive to use cylindrical coor-dinates, with the radial coordinate r defined as r =

√x2 + z2. The azimuthal

coordinate will be denoted by θ, and is defined as θ = ±acos(x/√x2 + z2) for

x R 0.The Hartmann walls, located at y = ±L are electrically insulating, and the

side walls are perfect conductors. The flow is driven by imposing a potentialdifference V between the side walls. We have thus:

u = 0,∂φ

∂n= 0 at y = ±L (5.1)

u = 0, φ = ±1

2V at r = R± L (5.2)

107


By imposing a potential difference between the side walls, a radial current isinjected in the fluid. The Lorentz force resulting from the interaction betweenthis current and the magnetic field, provides the necessary forcing of the flow,and thus, there is no need to specify an external forcing to drive the flow. Theseboundary conditions are inspired by, but nevertheless slightly different from theones in the experiment performed by Moresco and Alboussiere [MA04]. In thepresent formulation, the imposed potential difference between the side wallsdoes not fluctuate in time. This was however not the case in the experiment,where the total current between the walls was kept constant. If u × B isfluctuating in time, J = −σ(∇φ+u×B) can only remain constant if∇φ exhibitsthe same fluctuations as u×B. In the laminar regime, both formulations areof course strictly equivalent.

We can now use the linearity of the Laplacian operator to split the potentialin two parts: φ = φ1 + φ2. Here, φ1 is a solution of the non-homogeneousPoisson equation for the potential with Neumann conditions on the Hartmannwalls and homogeneous Dirichlet conditions φ1 = 0 on the side walls. φ2, onthe other hand, satisfies the Laplace equation ∇2φ2 = 0, also with Neumannconditions at the Hartmann walls, but now with non-homogeneous Dirichletconditions φ2 = ±V/2 at r = R±L. The external forcing density fext can thenbe defined as the forcing density due to the gradient of φ2:

φ2 =V

log(

R+LR−L

) ln

(

r√R2 − L2

)

(5.3)

fext = −σ∇φ2 ×B = fext1θ =V B

r ln(

R+LR−L

)1θ (5.4)

This flow can be characterized by three non-dimensional parameters. We haveused L as a length scale to define the Hartmann number (1.54) and Reynoldsnumber (1.50); the velocity scale in Re is given by the average streamwisevelocity in the toroidal duct. Apart from Re and M , an extra parameter isneeded to specify the geometry. We will denote it by Ar, and it is defined asthe ratio between the half duct side L and the radius of the torus:

Ar =L

R(5.5)

5.3 Laminar flow

To characterize the laminar regime, we investigate all combinations of the pa-rameter values M = 25, 100, 400 and Re = 0, Re ≈ 100, 800. The value of Aris 1/9, as in [MA04]. For these values of Re,M , it is assumed that the flowis steady and independent of the azimuthal coordinate θ. This assumption is

108

5.3. LAMINAR FLOW 109

based on the asymptotic estimate of Baylis and Hunt [BH71], stating that theinertial term is negligible under the following condition:

Ar2Re2

M4≪ 1 (5.6)

We have tested this assumption, by simulating the flow in a complete torus forthe parameter combination which yields the highest value of Ar2 × (Re2/M4)i.e. M = 25 and Re = 800. The number of points in the mesh for thisverification step was 512 in the streamwise (azimuthal) direction, and 100 inthe y- and r-direction. The initial condition was a random, chaotic velocityfield of high amplitude. We found that the flow eventually reaches a state whichis independent of time or the azimuthal coordinate. The results for Re = 0 areobtained by omitting the non-linear term in the momentum budget; the non-dimensional equations (1.66-1.68) are then independent of the velocity scaleU ; hence, the results do not depend on the velocity or forcing scale used toperform the simulations.

5.3.1 Computational set-up

The validation of the assumption of laminarity allows us to simulate only asmall section of the torus (see figure 5.2). As such, we can use a small gridspacing along axial and radial direction while maintaining an acceptable com-putational cost. For all simulations, the grids contain 192 × 192 × 16 pointsin r-, y- and θ- direction respectively. The size of the grid in the streamwisedirection is chosen such that the grid spacings in the radial and streamwisedirection were approximately equal in the core.

The grid consists of quadrilateral elements and is structured. the nodeslie along lines of constant values of r, y, and θ. Hence, the grid elements aretrapezoidal. The grid is symmetric with respect to the planes y = 0 and r = R,and appropriately refined in the Hartmann and side layer regions (respectivelyy- and z-direction in figure 5.2). The grid spacing in the Hartmann layer isone fifth of the typical Hartmann layer thickness L/M ; in the side layer, wehave 10 points over a distance L/M−1/2. A tangent-hyperbolic-like stretchingfunction is used to obtain a smooth transition between the coarser grid in thecore and finer grid in the shear layers.

5.3.2 Results and discussion

Mean flow

Figure 5.3 shows the non-dimensionalized product of the radial coordinate andstreamwise velocity profile along the centerline of the duct in the radial di-rection, i.e. uθr/(U0R). In the previous expression, the reference centerline

109


Figure 5.2: Grid lines of the mesh used to simulate the laminar flow atM = 400in a cross-section along the plane y = 0 (a) and a quarter cross-section alongthe plane x = 0 (z = r < R and y > 0) (b). Detail of the mesh in the planex = 0 near the corner between the Hartmann and side wall (c).

velocity U0 is defined as:

U0 =V

B0R log(

R+LR−L

) (5.7)

The results are plotted together with the asymptotic solution of Khalzov etal. [KSI10], which reads:

uθr

uθ(r = R)R= 1− r

I1(2LH)

(

1

R− LI1 (H(R+ L− r))

+1

R+ LI1 (H(r − (R − L)))

)

(5.8)

110


4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

r

uθr/U

0R

4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

r4 4.5 5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

r

Figure 5.3: Streamwise velocity profiles for different Reynolds and Hartmannnumbers along the radial centerline of the duct. Computational resultsM = 25(left), M = 100 (center),M = 400 (right) and Re = 0 (green), Re ≈ 100 (blue),Re ≈ 800 (red), together with the asymptotic solution (5.8) (black,dashed).

In this expression, the symbol I1 stands for the modified Bessel function ofthe first kind of order 1, and the parameter H is defined as H =

√2M/L.

As expected, we see that the velocity in the core becomes more and moreuniform as M increases. The results also illustrate that the velocity scales, upto the leading order in M , as fextM

−2 ∝ V/B. This is in agreement with theasymptotic analysis of [HS65].

When the Hartmann number is high, the only relevant force in the core isthe Lorentz force jrB0, and the effect of the curvature is a simple decrease of thestreamwise velocity in the radial direction, a reflection of the 1/r behavior of theexternal forcing as discussed in the previous section. This is clear from figure5.3, where we have a uniform profile of u(r, y = 0)r along the radial centerlinein the core, with minor overspeed zones in the side layers; these zones existalso in the limit of R → ∞, i.e. in a straight, square duct. The asymptotictheory does not predict the existence of such overspeed zones, neither in thecase of a straight duct nor in the case of a curved duct. At lower Hartmannnumber (M = 25), inertia starts to play a role. A useful parameter in this

111


context is the interaction parameter N , an estimate of the ratio between theLorentz force and inertia, defined in (1.52) as N = M2/Re. This parameterbecomes of order one for M = 25 and Re ≈ 800. In that case, the streamwisevelocity is pushed towards the exterior wall. The same tendency was observedin [KSI10], where the flow in a toroidal duct of non-square cross-section wasinvestigated. At higher Hartmann number, the profiles for different values ofRe are virtually identical.

Secondary flow

Since ∂θuθ = 0 in the laminar case, the vector field us = uy1y + ur1r is aproper, two-dimensional incompressible velocity field, and is termed secondaryflow. In figure 5.4(a), we display the magnitude of the secondary flow field inthe complete cross-section of the torus, rescaled by U0, i.e:

Us(r, y) =

√

u2y(r, y) + u2r(r, y)

U0(5.9)

Figure 5.4(b) shows the streamlines of the secondary flow. These can be com-puted as the isolines of a streamfunction ψ, which satisfies:

1

r∇ψ × 1θ = us (5.10)

Component-wise this reads:

ur = −1

r

∂ψ

∂y(5.11)

uy =1

r

∂ψ

∂r(5.12)

It follows that:∂2ψ

∂r2+∂2ψ

∂y2=

∂

∂r(ruy)−

∂

∂y(rur) (5.13)

If we now define u′s = uy1r − ur1y, we can write (5.13) as:

∇2ψ = ∇ · (ru′s) (5.14)

in which r and y are to be considered as if they were cartesian variables. Inorder to compute ψ, we can proceed as follows: we compute the right-handside of (5.14), and solve subsequently the Poisson equation (5.14) on a two-dimensional mesh which is an exact copy of the cut along planes θ = cst ofthe 3D mesh which was used to perform the simulations, like the one shown infigure 5.2(b).

112


Figure 5.4: Secondary flow profile: magnitude Us (a) and streamlines (b) forRe ≈ 100, and for different values of the Hartmann number. M = 25 (top),M = 100 (center), M = 400 (bottom).

We only show results in the upper half plane, since Us and ψ are respectivelysymmetric and antisymmetric around the axis y = 0. First, from figure 5.4(a),we note that the overall magnitude of the secondary flow strongly decreaseswith increasing M . We observe that Us is the most intense in the Hartmannlayer. AsM grows, the relative magnitude of Us in the side layers (compared toits value in the core or Hartmann layers) becomes more pronounced; moreover,we see that Us is larger in the inner side layer then in the outer one.

For M = 25, we see that we have one vortex cell, rotating in counterclock-wise direction, with a stagnation point (A in figure 5.4(b)) towards the innerwall. As the Hartmann number increases, the main vortex starts to developtwo subvortices (B in figure 5.4(b)) of unequal strength; the one at the innerside wall has a higher velocity than the one at the outer side wall. These sub-vortices exhibit a bump (C in figure 5.4(b)) in the side layers: they have a large

113


Figure 5.5: Detail of the secondary flow profile for Re ≈ 100 and M = 400.Magnitude of the secondary flow Us at the inner (a) and outer side wall (b);arrows indicate the location of the the stagnation points of the side wall vortices.Streamlines at the inner (c) and outer side wall (d).

component in y-direction and then suddenly turn into the opposite direction.This bump reveals itself in the profile of Us as a valley between two hills (Cin figure 5.4(a)). At last, for M = 100 and M = 400, we see the emergenceof side wall vortices, which are rotating in the clockwise direction (D in figure5.4(b)), as predicted by Tabeling and Chabrerie [TC81]. Due to their smallspatial extent in the case M = 400, it is difficult to discern them in figure 5.4.Therefore, we plot the details of Us and ψ in the side wall regions in figure5.5, in which the radial coordinate has been rescaled to units of the side layerthickness: s± =

√M(R ± L ∓ r); the location of the side wall vortices are

indicated by arrows in figure 5.5(a) and (b). These side wall vortices are muchweaker than the main secondary flow vortex.

Comparison with the asymptotic solution

In figure 5.6, the radial velocity is plotted along the direction of the magneticfield at the center line r = R. We only show results in the vicinity of theHartmann layer. The coordinate η is a rescaling of y with the Hartmann

114


0 5 10 15−0.1

0

0.1

0.2

0.3

0.4

η

Ura

d

0 5 10 15−0.1

0

0.1

0.2

0.3

0.4

η0 5 10 15

−0.1

0

0.1

0.2

0.3

0.4

η

Figure 5.6: Radial velocity profiles in the Hartmann layer: Comparison betweenthe asymptotic solution (green) and the numerical results for different valuesof the Reynolds and Hartmann number: M = 25 (left), M = 100 (center),M = 400 (right), Re ≈ 100 (blue) and Re ≈ 800 (red).

number and the duct length:

η =M(

1 +y

L

)

(5.15)

The velocity along this center line (r = R) is rescaled as:

Urad(η) = −M2 νR

L2U−20 ur(η, r = R) (5.16)

and can be compared to its asymptotic counterpart in the Hartmann layer[TC81]:

Urad,as(η) =1

6[6(η − 1/3) exp(−η) + 2 exp−2η]

+5

6M exp(−η)O(M−3) (5.17)

The simulation results only converge very slowly towards the asymptotic oneswith increasing M . Even at M = 400, the difference between both expressionsfor the maximum value of Urad is still around six percent, and this is much largerthan one would expect from a leading-error term ofO(M3). On the other hand,we see that the results for both values of Re almost collapse. The discrepancy

115


Figure 5.7: Non-physical behavior on strongly anisotropic meshes.

between the asymptotic and numerical results might be explained by the factthat expression (5.17) represents only a first-order term in an expansion in Ar,and is only exact in the limit of Ar → 0. In the present case, Ar takes the valueof 1/9, so that higher-order terms due to curvature-induced non-linearities, maystill have a considerable effect. We finally note that Urad has the same order-of-magnitude for all combinations of Re and M . Given (5.17), it illustratesthat ur scales as U2

0M−2, as found in [TC81].

5.4 Turbulent flow

In the section, we investigate the nature of the turbulent regime for purely hy-drodynamic as well as magnetohydrodynamic flow. To the best of our knowl-edge, no profound study of this hydrodynamic case has been performed untilnow. We consider three different values of the Hartmann number: M = 0, 10and 30. In each of the simulations, Re is between 3900 and 4000, and Ar =1/18. The value of Ar is smaller than in the study of the laminar regime,because it was observed that non-physical numerical behavior emerges if theanisotropy of the grid elements at the inner side wall becomes to large. Thisis illustrated in figure 5.7, where we show a color plot of the radial velocity urin the plane y = 0. In this case, the ratio between the grid size in streamwiseand radial direction is approximately 6 at the inner wall. Close to this wall, weobserve an oscillation in ur, whose wavelength appears to be exactly the gridsize in streamwise direction.

5.4.1 Computational set-up

We do not simulate a complete torus, but only a fraction of it, which coversan angle of π/4, and impose periodic boundary conditions at θ = ±π/8. This

116

5.4. TURBULENT FLOW 117

implies that the streamwise extent of the simulation domain is about seventimes the distance between the duct walls. In earlier studies of turbulent hy-drodynamic and MHD shear flows in periodic domains [Gav92, BKZ07], it wasfound that such an extent is much longer then the correlation length of theflow in the streamwise direction.

For the hydrodynamic case (M = 0), the flow is forced by imposing astreamwise mean pressure gradient, which is constant in time. It can be writtenas r−1∂θP , with ∂θP a constant. Note that this last case is not physical: the

integral∫ 2π

θ=0 ∂θP dθ should be strictly zero, since the pressure has a uniquevalue at every point in space. On the other hand, the Navier-Stokes equationscontain a pressure gradient term, but do not depend directly on the value of thepressure itself. Since we only simulate a section of the torus, this mathematicalartefact is of no direct relevance for our computations.

The structured, quadrilateral mesh consists of 256 nodes in streamwise di-rection, and 100 × 100 nodes in the cross-section of the duct, where a stretchingwas applied. This is a rather small resolution for the simulation of a flow atRe = 4000. We assume however that even a slightly underresolved simulationcan reveal some interesting features. In the hydrodynamic case, the stretch-ing is such that the mesh contained at least three points within a distance ofone (turbulent) viscous sublaye thickness δBL of the walls. According to theanalysis in section 3.2, we can write δBL = ν/uτ , where we define the frictionvelocity uτ :

u2τ =−1

2LR

∂P

∂θ(5.18)

This formula is based on the expression for uτ in a straight duct flow driven bya uniform pressure gradient [Gav92]. To obtain (5.18), we have replaced thisuniform pressure gradient by the value of the 1/r-like forcing at the center ofthe curved duct. Strictly spoken, this last formula thus only holds in the limitR → ∞, but it should also provide a reasonable estimate of the viscous sublayerthickness in the present case, given the rather small value of the curvaturethat is considered. The corresponding fritcion Reynolds number Reτ is 165(M = 0). The same mesh is used for the caseM = 10 (Reτ =224). ForM = 30(Reτ =266) however, a less severe stretching is applied in the Hartmann layers,since we presume that these layers should be stabilized due to the presence ofthe magnetic field.

The mesh topology detailed here is similar to the one discussed in the sectionon turbulent channel flow (section 3.2); in both cases, we have a structuredquadrilateral grid with severely anisotropic elements close to the wall. We haveseen that, for the turbulent flow in a straight channel, more accurate resultswere reached when using a sparse discretization stencil. Therefore, a sparsestencil is also used here. As initial condition, we impose a random velocity fieldof large amplitude. The momentum equation is advanced in time with a Crank-Nicholson method, and the time step is such that the CFL number is 0.4. The

117


linear system resulting from this implict time stepping, is solved with a Jacobimethod. The Poisson equations for pressure and potential are solved with analgebraic multigrid method. Statistics are obtained by an averaging over thestreamwise direction, and over approximately 720 (M = 0, 220000 iterations),420 (M = 10, 140000 iterations) and 240 (M = 30, 80000 iterations) convectivetime scales L/U . The relative variations of the bulk Reynolds number Re inthis steady regime are less then 2 percent. The computations have been carriedout on 32 CPU’s. As such, we could perform approximately 5000 iterations perday for M = 10 and M = 30 and 8000 iterations per day for M = 0.

5.4.2 Results

Mean velocity and secondary flow

In figure 5.8(a), we compare the main streamwise profile uθ for different valuesof the Hartmann number; this quantity has been normalized by its averageover the cross-section, Uθ. In the case M = 0, we see that the centrifugal forceshifts the velocity maximum towards the outer wall. For smaller values of r,the velocity maximum is not located on the symmetry axis y = 0, but closeto the upper and lower wall. We find a similar profile for the case M = 10.The most striking tendency is that variations along the magnetic field directiontend to disappear in the case M = 30. Streamlines of the mean secondary floware shown in figure 5.8(b). These were computed with the same procedureas outlined in section 5.3. We have again two counterrotating vortex cells,whose stagnation points shift towards the outer wall as the Hartmann numberincreases. In the case M = 10 this vortex pair is accompanied by a muchweaker, counterrotating pair, located close to the outer side wall.

Streamwise velocity fluctuation

In figure 5.9(a), we show isolines of the fluctuating streamwise velocity urmsθ

(like defined in (3.24)). We recall that urmsθ can be considered as a measure

for the turbulence intensity. In the hydrodynamic case (M = 0), we see thatall shear layers are turbulent and that urms

θ takes a smaller, although non-negligible value in the core. Furthermore, there is an asymmetry between theinner and outer side wall. The turbulence is more intense at the outer side layer,in spite of the fact that the external force is the weakest in this region. Thisis in agreement with previous studies of curved channel flow [MM87, NK04].As the Hartmann number increases, urms

θ is strongly suppressed in the core,the inner side layer and the Hartmann layers, whereas a significant level ofturbulence persists in the outer side layer. This illustrates that unstable sidelayers may exist in an otherwise stable flow.

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5.4. TURBULENT FLOW 119

Figure 5.8: Isolines of the normalized average streamwise velocity (a) andstreamlines of the average secondary flow (b). Results forM = 0 (left), M = 10(middle) and M = 30 (right).

Instantaneous velocity magnitude

We display a snapshot of the isocontours of the velocity magnitude in thecenter plane perpendicular to the magnetic field and in a plane perpendicularto the streamwise direction in figure 5.9(b). This figure provides an additionalillustration of the flow feautures described above, i.e. an asymmetry in the levelof turbulence between the inner and outer side layer, and an uniformizationalong the magnetic field direction and suppression of turbulence in the innerside layer and Hartmann layers as M increases.

Coherent vortices

The previous observations are confirmed by a visualization of coherent vor-tices in the flow. According to Jeong and Hussain [JH95], such a vortex canbe defined as a connected region in space with two negative eigenvalues ofCik = SijSjk +ΩijΩjk, with Sij and Ωij being respectively the symmetric andanti-symmetric part of the velocity gradient tensor ∂iuj . In figure 5.10, weshow the regions in the flow for which the second largest eigenvalue of Cik issmaller then -0.08; this means that the weakest vortex cores (with a secondeigenvalue between -0.08 and 0.0) are left away to not overload the figure. In

119


the hydrodynamic case, we see that these structures are distributed over thecomplete flow domain, with a higher density of vortices in the vicinity of theouter wall. When a magnetic field is imposed, all these structures tend to besuppressed with exception of the ones close the outer wall. Contrary to otherstudies of MHD shear flows [KKM09], no elongation of these structures alongthe magnetic field direction is found. Probably the value of the interactionparameter N ≈ 0.2 is too small for this to be observed.

5.5 Conclusions

In this work, we presented an analysis of the quasi-static MHD flow in a toroidalduct of square cross-section by means of numerical simulations. In the case oflaminar flow, we saw that an increase of the external magnetic field leads to adrastic change of the main and secondary flows; this secondary flow consists oftwo counter-rotating vortex cells, but also exhibits side layer vortices at highHartmann number. Our results confirmed and extended the earlier asymptoticanalysis of [TC81]. In the study of the turbulent regime, we found that thestreamline curvature leads to a higher level of turbulence at the outer sidewall than at the inner one. We also showed that application of a magneticfield results in a uniformization of the flow along magnetic field lines and asuppression of turbulence. At M = 30, the core, the Hartmann layers andthe inner side layer are completely stabilized, whereas a significant level ofturbulence persists in the outer side layer. This implies that unstable sidelayers can coexist with stable Hartmann layers. Finally, a study of coherentstructures in the flow yielded an additional illustration of the main features ofthe turbulent regime.

120


Figure 5.9: Turbulent (MHD) flow in a toroidal duct of square cross-section.Statistics of the fluctuating streamwise velocity urms

θ (a) and isocontours ofthe instantaneous velocity magnitude (b) for different values of the Hartmannnumber: M = 0 (above), M = 10 (center), M = 30 (below).

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Figure 5.10: Coherent structures in the flow. Isosurfaces of the second largesteigenvalue λ2 = −0.08 of the tensor Cik: M = 0 (above), M = 10 (center),M = 30 (below). The grey plane indicates the bottom wall of the torus.

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Chapter 6

Laminar flow in aright-angle bend

“Still around the corner may wait,A new road, or a secret gate.”J. R. R. Tolkien

The subject of this sixth chapter is the MHD flow in a right-angle bend be-tween two ducts of square cross-section, with a strong, uniform magnetic fieldin the plane of the axes of the ducts. The interest in this particular geome-try originates from the design of fusion blankets. However, MHD right-anglebend flow is also an interesting subject in its own right, because it contains allthe characteristic features of fully three-dimensional MHD flows in their mostclearly manifested form.

First, we briefly discuss the relevance of this geometry for fusion blanketdesign, and review some of the historical achievements made in the understand-ing of the physics of MHD bend flows. We then present simulation details andresults for two different geometries and orientations of the magnetic field: aso-called

⋃

-bend, a system of two bends, in which the magnetic field is alignedwith the axis of the central duct, and a backward elbow, in which the magneticfield is perpendicular to the line connecting the two corners of the junction.For both studies, we consider the inertialess limit, i.e the regime in which theconvective term in the quasi-static MHD equations is negligible with respectto the other terms.

123

124 CHAPTER 6. LAMINAR FLOW IN A RIGHT-ANGLE BEND

6.1 Introduction

As discussed in the first chapter, the main role of fusion blanket devices isto absorb and transfer the energy generated in the fusion reaction. Therefore,high velocities are desired in those components which face the tokamak plasma.However, in the presence of a magnetic field, the flow will be affected by astrong body force, whose strength is proportional to U⊥M

α, where U⊥ is atypical velocity scale of the velocity components perpendicular to the magneticfield direction, and α is an exponent which depends on the conductivity of thewalls. At high values of M , the electromagnetic contribution to the pressuredrop required to drive a given mass flow rate completely dominates viscous andinertial ones. It has been suggested to minimize the pressure drop in blanketsby aligning the flow with the magnetic field in the plasma-facing ducts, and byfeeding these ducts by larger (i.e. with lower velocities) ones, in which the flowis perpendicular to the magnetic field over a short distance [Mal88].

The flow in such configurations is characterized by a free, internal shearlayer, which emanates from the geometrical discontinuity at the junction be-tween the ducts, and which is parallel to the magnetic field direction. Thistype of layers is named after Ludford [Lud61], who was the first to identifythem in the MHD flow around obstacles; the generic theory of Ludford layerswas established by Hunt and Leibovich [HL67], and is extensively discussed inAppendix C.2.

The introduction of the blanket design described above gave rise to a largenumber of investigations of the flow pattern and, specifically, the internal shearlayer in the context of right-angle bend flow. Here, we only discuss the mostelaborative ones among these studies.

Molokov and Buhler [MB94] provided a core flow approximation of the flowin a

⋃

-bend, in which the magnetic field is aligned with the axis of the centralduct of a system of two right-angle bends (see figure 6.1(a)). In this approxima-tion, the core flow is governed by a simple balance between a pressure gradientand the Lorentz body force. In the shear layers, viscosity is taken into account,and estimates for the flow rate carried by these layers are based on dimensionalarguments and an appropriate matching with the core solution. The authorsalso found that the flow is very sensitive to certain parameters like the wallconductance ratio or the aspect ratio of the duct aligned with the magneticfield. They also predicted that, for certain parameter combinations, the flowpattern exhibits a rather ‘exotic’ behaviour, like e.g. helical and vortical flowstructures in the central duct, which may enhance the heat transfer.

The situation in which the magnetic field is aligned with none of both ductsof the junction, is called an elbow geometry. A core flow approximation for theflow in such a configuration was provided by Moon and Walker [MW90], andMoon, Hua and Walker [MHW91]. For an angle of 45 degrees between themagnetic field and the downstream axis (see figure 6.1(b)), they found, among

124

6.1. INTRODUCTION 125

Figure 6.1: Conceptual sketch of the geometries studied in previous investi-gations of MHD bend flow: [MB94] (a), [MHW91] (b) and [SBBM96] (c) and(d).

other things, that a large portion of the mass flow rate is carried by the internalshear layer for modestly conducting walls. This fraction further increases as thewall conductance ratio is decreased. Furthermore, it was also found that thepressure drop due to the presence of the internal shear layer is small comparedto the MHD pressure drop of the fully developed duct flow.

Stieglitz et al. [SBBM96] experimentally investigated the flow in a right-angle bend for two orientations of the magnetic field (figures 6.1(c)-(d)), andfor the parameter range M ≈ 2000 − 8000, N ≈ 1000 − 50000. They per-formed measurements of the side wall potential and the pressure drop, and

125


were the first to retrieve undoubtedly the exact scaling law of the viscous andinertial contribution to the pressure drop. In the case of an elbow geometry,they reported considerable discrepancies between asymptotically predicted andexperimentally measured potential gradients in the downstream duct. Further-more, these authors also investigated the instability of the Ludford layer for theelbow by recording the time signal of the electric potential at various locations;they found that the layer becomes unstable if N < 10300.

Only a few attempts have been made to simulate numerically the laminarMHD flow in a right-angle bend. With limited computational resources, Aitovet al. [AKT78] could reach a Hartmann number of 30 with a finite-differencemesh of 32× 16× 16 grid nodes. A similar method was used by Kunugi et al.[KTA91] in the early 1990’s to obtain the steady state velocity profile in anelbow at M = 100 and N = 200, with a uniform wall conductivity c = 0.01.In this case, the mesh contained 44× 44× 16 grid points. They observed thatthe strongest velocities occur near the inner corner of the configuration.

The results discussed in this chapter are part of a larger (ongoing) investiga-tion of the effect of inertia on MHD flows in right-angle bends for fusion-relevantparameters. As a first part of this project, we wanted to compare the presentnumerical methods against the core flow approximation. To this end, we con-sider an inertialess flow (Re = 0) and a high but finite value of the Hartmannnumber (M = 2000). Results for the limit of vanishing Reynolds number areobtained by dropping the convective term in the momentum equation.

6.2 MHD flow in a⋃

-bend

6.2.1 Problem definition and computational set-up

We consider a system of two right-angle bends (see figure 6.2), like the onestudied by Molokov and Buhler [MB94]. The uniform magnetic field B = B01y

is aligned with the axis of the central duct. The three ducts have a square cross-section of side 2L, and the distance between the outer Hartmann walls is 24L.The origin of our coordinate system is located in the plane of the axes of theducts, on the outer corner between the entrance and central duct (see figure6.2(a)). The geometrical discontinuity at x = 2L will give rise to two Ludfordlayers, whose location is shown in figure 6.2(d). Since Re = 0, their extent alongthe x-direction is, according to appendix C.2, O(M−1/2L). In principle, theentrance and exit ducts (with their axes along x-direction) are of semi-infiniteextent. However, far enough from the junction, the fully developed duct flowprofile should be recovered. We impose in-and outflow conditions at a distance3L from the junctions. This distance is large compared to the thickness of theLudford layer. All the walls have a uniform wall conductivity of c = 0.1; this isthe value of c for which [MB94] offers the most elaborative results. The length

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6.2. MHD FLOW IN A⋃

-BEND 127

Figure 6.2: MHD flow in a⋃

-bend. Detailed geometry of the flow domain (a),location of the Hartmann layer (b), side layer (c) and Ludford layer (d) regions.

scale used to define the Hartmann number is the half duct side L.In this chapter, we will deal with non-dimensional quantities, which have

been obtained by the following rescaling: u → Uu, B → B01y, x → Lx,∇ →L−1∇, t → ρ(σB2

0)−1t, J → σUB0J, φ → UB0Lφ and p → σULB2

0p. The ve-locity scale U is defined as the mean streamwise velocity. The non-dimensionalequations governing the fully developed inertialess regime are:

M−2∇2u+ J×B−∇p = 0 (6.1)

J = −∇φ+ u×B (6.2)

∇2φ = ∇ · (u×B) (6.3)

A cartesian, non-uniform grid is used to perform the simulations (see figures

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Name Type Location Grid Spacing Growth FactorA1 Hartmann y = 2 M−1/5 1.07A2 Hartmann y = 2 M−1/5 1.12B Ludford/Side x = 2 M−1/2/25 1.08C Hartmann y = 0 M−1/5 1.12

D Side x = 0 M−1/2/25 1.08E Side z = −1 M−1/2/8 1.08

F Side z = 1 M−1/2/8 1.08

Table 6.1: Summary of the mesh spacing and growth factor for the shear layersas shown in figure 6.3-6.4(a).

6.3-6.4(a)), with appropriate grid refinements in the vicinity of all the shearlayers. These refinements are discussed here, shown in figures 6.3 and 6.4, andsummarized in table 6.1. In the Hartmann layers, we have 5 points in wall-normal direction and a uniform grid spacing of M−1/5. Further away fromthe wall, the grid spacing grows exponentially until a maximum size of 0.035is reached. The growth factor between adjacent cells is 1.12 (‘A2’ and ‘C’ infigure 6.3 and table 6.1) towards the axes of the entrance and exit ducts, and1.07 (‘A1’) in the central duct. The wall-normal grid spacing at the planesz = ±1 (‘E’) is M−1/2/8. Around the planes x = 0 (‘D’) and x = 2 (‘B’),the distance between the nodes in the direction of shear is M−1/2/25. Thereason for this smaller resolution is that we want to limit the anisotropy ofthe cells at the inner corner (cfr. figure 6.3(b)). Away from these planes, thespacing increases exponentially with a growth factor of 1.08, up to a maximumgrid spacing of 0.035. These specifications result in a grid which consists ofapproximately 18 200 000 nodes. The only drawback of this approach is that itleads to a clustering of points in the core of the central duct close to the innerHartmann wall planes at y = 2 and y = 22.

As mentioned before, we assume that the velocity profile is fully developedat the inflow boundary. The velocity profile which we impose at this locationis computed with the devoluted solver for fully-developed duct flow discussedin section 3.3. To this end, we use a two-dimensional mesh which is an exactcopy of the inlet boundary mesh (figure 6.4(a)) used to perform the three-dimensional simulations. For the parameters c = 0.1 and M = 2000, we have avelocity profile which is characterized by strong side layer jets, whose velocitiesare O(M1/2) larger than the core velocities (see figure 6.4(b)).

At the outflow boundary, we apply the convective boundary condition(2.72). This is somehow strange, because convective terms are neglected in therest of the domain (since Re = 0). However, the only effect of this approach isto convect the velocity profile from the interior of the solution domain across

128


-BEND 129

Figure 6.3: Mesh design for a⋃

-bend. Lower half of a along the x − y-plane(a) (the mesh is symmetric around the plane y = 12), detail close to the innercorner (b) and the outer corner (c). Details of the spacing and growth factor ofthe grid in the shear layers (denoted by a capital letter) can be found in table6.1.

the outflow boundary at a uniform streamwise velocity, and does not affect theflow in upstream direction. Hence, this formulation does not suffer from anyconceptual difficulty. Note that, when a steady state is reached (∂t ≡ 0), thecondition ∂tu+ Uconv∂nu = 0 is equivalent to a Neumann condition.

The spatial differential operators have been discretized with a sparse stencil,since it was observed in chapter 3 that the use of this stencil for the simulationof high Hartmann number flows on cartesian meshes leads to very accurateresults.

129


Figure 6.4: Boundary mesh at the inlet boundary (a). Details of the spacingand growth factor of the grid in the shear layers (denoted by a capital letter)can be found in table 6.1. Fully developed inlet velocity profile (b).

To obtain the steady solution, we start from an initial conditon u = 0,φ = 0 and advance the discrete quasi-static MHD equations in (pseudo)-timewith a Crank-Nicholson method until the pressure, potential and velocity havereached a steady state. The time step is ∆t = 0.2; the Fourier number for thistime step is of the order of 100. We use an algebraic multigrid method to solvethe Poisson equation for the pressure, and a BiCGStab(2)-solver for the poten-tial equation. The momentum equation is also solved with the BiCGStab(2)-algorithm. Convergence was decided if the relative variation of u and φ wasnot larger than 10−5 over an interval of 5000 time steps. 55000 iteration stepswere needed to reach convergence. These computations were carried out on192 CPU’s and required more or less 100000 CPU hours.


We start our discussion by inspecting the current density streamlines in thecore of the entrance duct. In figure 6.5, we show the projection of these linesonto the plane y = 1. Far enough away from the junction, the flow should reacha fully developed regime, which is independent of x; the current paths for thisstate occur in y-z-planes. As such, figure 6.5 gives us an indication of thedeviation from the fully developed regime. We find that the x-component of J

130


-BEND 131

Figure 6.5: Projection on the plane y = 1 of the streamlines of the currentdensity J in the core of the entrance duct; the streamlines are colored by thevalue of the pressure p.

is negligible close to the inlet boundary at x = 5. However, as we approach thejunction, the lines become more and more bent, and the resulting Lorentz forcefL,z = jxBy drives the flow towards the side walls at z = ±1. This deviationfrom the two-dimensional state was also observed in [MB94]. We will brieflycome back to this effect later.

These streamlines have furthermore been colored by the value of the pres-sure p. According to asymptotic analysis (see appendix C.3), the core pressureprovides a streamfunction for the streamlines of J⊥ = Jx1x+Jz1z; isolines of pare thus streamlines of J⊥. We observe indeed that the color of the streamlines(i.e. the value of p) hardly changes along their path. This confirms that, forM = 2000, we recover the main features of the asymptotic core flow solutionin the entrance duct.

In figure 6.6, we show a plot of the velocity component uy(x, y, z = 0) alongx for different values of y. We see that the magnitude of uy along the axis of theentrance duct (i.e. y = 1), upstream from the junction (i.e. for x > 2) remainsclose to zero up to a short distance from the junction. The internal shear layerat x = 2 layer takes the form of a jet, which becomes very sharp as we approachthe inner corner (i.e close to y = 2). Since uy > 0 in this internal layer, massis transferred in the positive y-direction. For y > 2, the internal shear layerbecomes a side layer, which we will term inner side layer. As we move awayfrom the junction, we observe that, in the inner side layer, uy rapidly decreases

131


in magnitude and that its profile becomes a bit broader. Moreover, at y = 3.6and y = 6, uy is negative in the vicinity of x = 2. Close to the wall x = 0, wehave another side layer, called outer side layer, which also takes the form of astrong jet. We observe that the magnitude of uy increases with y up to y = 3.6.For larger values of y, the maximum value of uy slightly decreases. Just likefor the inner side layer, we find that the jet becomes broader with increasing y.Both side layers are separated by the core of the central duct, and uy in thisregion is vanishing. This is in agreement with the core flow approximation (seeAppendix C.3). The flow in the core of the central duct is thus approximatelyhorizontal.

Furthermore, side layers also occur along the walls z = ±1. We show theevolution of their profiles in the planes x = 1 with increasing y in figure 6.7.It appears that the side jets become thicker as we move away from y = 0. Thebroadening of the side layers can be explained as follows. Far downstream fromthe junctions, we should recover a fully developed flow. For the central duct(with the magnetic field aligned with the flow direction), the fully developedflow is the laminar hydrodynamic (Poiseuille-like) duct flow. As we move awayfrom the junction, the side jets should thus fade out. According to asymptotictheory however, the developed regime is only reached at a distance O(M).Finally, we also note that the amplitude of the jets first grows until y = 3.6,but then decreases as we approach the plane y = 12.

A color plot of uy(x, y = 6, z) is depicted in figure 6.8(a). While not com-pletely generic, it illustrates the most relevant features of uy in planes y = cst(for y > 2). Some of these features could already be inferred from figures 6.6and 6.7, like the vanishing of uy in the core, and the presence of jets of largeamplitude close to x = 0 and z = ±1. Moreover, we see that the highest veloc-ities occur in the corner regions. Remarkably, the magnitude of uy close to thecorners of the side wall x = 2 is many times larger than its value at the centerof this side layer.

In the core flow solution, uy is exactly zero in the core of the central duct.Hence, the secondary flow u⊥ = ux1x+uy1y in this region can be derived froma stream function. It appears that the electric potential φ is a stream functionfor u⊥, i.e. u⊥ = ∇× φ1y (see Appendix C.3). This feature has been verifiedby tracing a couple of streamlines originating from seed points along the axisx = 1.6, y = 6. In figure 6.8(b), we show the projection of the streamlines onthe core of the plane y = 6, i.e. between x = 0.25 and x = 1.75, and z = −0.75and z = 0.75. The streamlines are coloured according to the value of φ alongthe line. We do not observe notable color variations along streamlines. Thismeans that these are, up to a good approximation, isolines of φ, and that φ isindeed a streamfunction for u ≈ u⊥ in the core of the central duct. Moreover,we note that the main tendency of the core secondary flow is to transfer massfrom the inner side layer to the outer one and the ones at z = ±1.

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-BEND 133

0 0.5 1 1.5 2 2.5 3−1

0

1

2

3

4

5

6

7

8

9

x

uy

y = 1y = 2y = 3.6y = 6y = 12

Figure 6.6: Velocity component uy(x, y, z = 0) along x for different values ofy.

Once the quasi-horizontal flow in the core of the central duct enters one ofthe side layers, it becomes fully three-dimensional, and this results in a complexmotion. This is illustrated in figure 6.8(c), where we trace the streamlinesoriginating from three different seed points in the entrance duct. The symmetryof the flow allows us to show only one quarter of the configuration (i.e. forz > 0 and y < 12). The first of these points (red streamline) is located onthe axis of the duct. From symmetry considerations, we find that uz is zeroin the plane z = 0. Therefore, this streamline is constrained to this plane.Close to the junction, it enters the internal shear layer and follows over ashort distance the direction of the magnetic field. Then, it bends and flowsquasi-horizontally in x-direction towards the outer side layer. The seed of thesecond (blue) streamline is situated in the side layer close to the wall z = 1.At the junction, it makes a large bend, but remains close to the plane z = 1.Subsequently, it enters the inner side layer (close to the wall x = 2), from whereit is transferred in horizontal direction towards the outer side layer, and joins

133


−1 −0.5 0 0.5 1−1

0

1

2

3

4

5

6

7

z

uy

y = 1

y = 2

y = 3.6

y = 6

y = 12

Figure 6.7: Velocity component uy(x = 1, y, z) along z for different values of y.

the red streamline. Finally, the green streamline originates from the seed withcoordinates (5, 1, 0.5). Close to the junction, it enters the shear layer and issimultaneously driven towards the wall z = 1; this was already inferred fromthe bending of the current streamlines in figure 6.5. Once it has arrived inthe central duct, it follows a complex motion, in which it hops back and forthbetween the side layers at the inner wall and the wall z = 1.

Finally, a more quantitative comparison with the core flow approximationis shown in figure 6.9, where we plot the relative fraction of the mass flow rateQ carried by the boundary layers of the central duct against the coordinatey. Q has been normalized such that the total mass flow rate in the centralduct equals 2 for y > 2. As mentioned before, the core mass flow rate inthe asymptotic limit is exactly zero. However, for the present simulations, thestreamwise core flow velocity is not exactly zero. Moreover, the thickness ofthe side layers is not very well defined, and does vary with y. This implies thatwe have to use a slightly different definition for Q. The values of Q which resultfrom our simulations (solid line in figure 6.9) have been obtained by integrating

134


-BEND 135

Figure 6.8: Color plot of uy in the plane y = 6 (a). Velocity streamlinespassing trough the axis y = 6, x = 1.6 projected on the plane y = 6, andcolored according to the value of the electric potential φ along the streamline(b). Three-dimensional velocity streamlines in the lower half of the

⋃

-bendoriginating from three different seed points.

uy over triangular domains, which are sketched in the upper left corner of figure6.9. More specifically, the non-normalized flow rates for the side layers at x = 0

135


(green line), x = 1 (blue line) and z = ±1 (red line) are computed as:

Qx=0 =

∫ 1

x=0

∫ z=−x+1

z=−1+x

uy dz dx (6.4)

Qx=2 =

∫ 2

x=1

∫ z=x−1

z=−x+1

uy dz dx (6.5)

Qz=±1 = 2

∫ L

z=0

∫ x=1+z

x=1−z

uy dxdz (6.6)

In figure 6.9, we see that both sets of results agree well for the outer sidelayer. On the other hand, the present simulations do not predict reversed masstransfer in the inner side layer, in contrast to the asymptotic study. It followsthat the results for the side layers at the walls z = ±1 also exhibit severediscrepancies.

This poor agreement may be partially explained by the influence of thecorner regions, where the side layers at z = ±1 overlap with those at x = 0, x =2. In the asymptotic study, these overlap regions do not contribute to the massflow rate up to the leading order, since the velocity in the corners is of thesame order as in the non-overlapping zones, whereas they extend spatially overa region which is O

(

M−1/2)

times smaller. However, from figure 6.8(a) e.g., wemay deduce the corner regions carry a significant fraction of the mass flow rate

included in Qx=2. On the other hand, the integral∫ 2

x=1 uy(x, y = 6, z = 0) dx(the integrand is shown in figure 6.6) is negative. This indicates that themass transfer in the inner side layer is indeed reversed far away from the wallsz = ±1, but that this feature is overcome by the contribution of the corners.The ‘contamination’ of Qx=2 and Qz=±1 by corner effects is however difficultto quantify because the extent of the corner regions is not very well defined.

6.3 MHD flow in a backward elbow

6.3.1 Problem definition and computational set-up

We consider a right-angle bend between two semi-infinite ducts of square cross-section (see figure 6.10(a)); the side of these ducts is 2L. The origin of thecoordinate system is located on the outer corner and in the plane of symmetrywith respect to the z-direction, and the ducts extend along the positive x-axis (upstream) and y-axis (downstream). A strong uniform magnetic fieldis imposed along one of the diagonals between the axes of the ducts: B =B0/

√2(1y − 1x). All the walls have again a wall conductance ratio c = 0.1.

This configuration is exactly the same as the one investigated in [MHW91].The variables have been rescaled in the same way as in the previous section,and the equations which govern the flow are given by (6.1-6.3). The theory

136

6.3. MHD FLOW IN A BACKWARD ELBOW 137

Figure 6.9: Distribution of the mass flow rate between the side layers in theduct parallel to the magnetic field. Comparison between the present numericalresults (solid) and the core flow approximation of [MB94] (dashed). Massflow rate fraction carried by the outer wall (green), inner wall (blue), and sidewall (red) boundary layers. The integration domains of equations (6.4-6.6) aresketched in the upper left corner.

of MHD shear layers near geometrical discontinuities predicts that, for theseparameters, a Ludford layer of thickness O(M−1/2) will emerge close to theinner corner of the system, which extends along the magnetic field direction.The inflow and outflow boundaries of the simulation domain are imposed in theplanes x = 7 and y = 7. We assume that fully developed duct flow conditionswill apply in these planes, since they are located on a distance from the Ludfordlayer which is large with respect to its thickness.

One of the major problems related to the numerical simulation of this con-figuration is the design of an appropriate mesh. The internal shear layer ex-tends in both axial directions over a distance 4. Properly resolving the Ludfordlayer with a cartesian mesh would imply that a grid spacing should be smallerthan M−1/2 in the region limited by the planes y = 0, y = 4, x = 0 andx = 4. Therefore, we have to use unstructured meshes if we want to providesimulations of the high Hartmann number regime at reasonable computational

137


Figure 6.10: MHD flow in a backward elbow. Detailed geometry (a), andlocation of the Hartmann layer (b), side layer (c) and Ludford layer (d) regions.

expense. On the other hand, the wall-normal spacing in the Hartmann layersshould be of the order of M−1. This layer extends over a distance of multiplelength units in the directions tangential to the wall. Far away from the cor-ners and side walls, all the variables are quasi-uniform along these directions.Hence, meshes consisting of strongly anisotropic elements are preferred in thesezones. We now explain how this is achieved.

These considerations imply that we have to tailor the mesh such that a

138


Name 3D element Min.Size Max. Size Growth FactorA Hexa M−1/5 0.03 1.06B Hexa M−1/5 0.03 or ‘C’ or ’D’ 1.12C Prism M−1/5 M−1/2/8 1.08D Prism ‘C’ 0.03 1.08

E Hexa M−1/2/8 0.03 1.06F Hexa M−1/2/8 0.03 1.08

Table 6.2: Summary of the mesh spacing and growth factor for the shear layersas shown in figure 6.11.

structured anisotropic mesh in the Hartmann layers is matched with an un-structured, isotropic one in the core and Ludford layers regions. We first con-sider the plane of the axes of the duct (see figure 6.11(a)-(b),(d)-(e)). At theinner corner, two Hartmann layers meet, and the grid spacing at this point isM−1/5 (see figure 6.11(e)) in both x- and y-direction. The grid spacing betweenthe wall nodes grows exponentially away from this corner until a maximum of0.03 is reached (‘A’ in figure 6.11(d)). Simultaneously, the extent of the struc-tured quadrilateral region also increases in wall-normal direction, so that thelast quadrilateral point is approximately isotropic (‘B’ in figure 6.11(b)-(d)).This approach is necessary if we want to obtain a smooth matching betweenthe structured boundary layer and the unstructured core.

Furthermore, a refined, unstructured triangular layer emanates from theinner corner along the magnetic field direction; the size of the elements growswith a factor 1.08 away from the inner corner until a maximum size ofM−1/2/8(‘C’ in figure 6.11(d)). If we move away from this axis, the grid size increasesfurther up to a maximum of 0.03 (‘D’ in figure 6.11(d)). We prefer triangularelements above quadrilateral ones because they are more easily generated bythe preprocessor. Moreover, in section 4.3, we have seen that a mesh whichconsists of a triangular core and a structured quadrilateral boundary layeryields reliable numerical results for MHD pipe flow with well-conducting walls,when combined with a sparse discretization stencil.

Finally, for the outer Hartmann walls, we use a similar matching procedureas for the inner ones. We see that the structured boundary layer is tailored suchthat a smooth transition with the refined unstructured Ludford layer region isobtained (‘E’ in figure 6.11(b)).

To generate the three-dimensional mesh, we simply extrude the two-dimen-sional one. Hence, the mesh elements are hexahaedra and rectangular prisms.The mesh spacing in z-direction is such that it allows a proper resolution ofthe side layers, with six points over a distance M−1/2 next to the wall (‘F’ infigure 6.11(c)). Adopting the above strategy results in a mesh which contains

139


Figure 6.11: Grid topology for the simulation of a backward elbow: Cut alongthe x − y-plane (a). Detail of (a) close to x = 4, y = 0 (b). Structuredquadrilateral mesh at the inlet boundary (c). Detail of (a) close to the innercorner at x = y = 2 (d). Capital letters indicate a stretched grid region.

between 15 and 16 million grid nodes, and consists of hexahedral elements inthe structured zones of the mesh, and prisms in the unstructured regions. Theinformation on the stretched regions is summarized in table 6.2. A cartesianmesh with the same resolution in the shear layers would contain at least 10times more grid points.

The other simulation details are similar to those discussed in the previous

140


section. The computations were performed on 160 CPU’s and approximately32000 iterations were needed to reach convergence. This required about 60000CPU hours.


In figure 6.12, we show profiles of ux(x, y, z = 0) and uy(x, y, z = 0) along thex-axis for different values of y. Note that, because of symmetry uz(x, y, z = 0)is zero, so that the flow in the plane z = 0 remains restricted to it. The axisy = 1 is the axis of the entrance duct, and is intersected by the internal shearlayer at x = 3. Far upstream from this internal layer, the flow adopts the fullydeveloped state of a straight duct; it is independent of x, and uy = 0. The valueof the core velocity is approximately ux ≈ −0.21. This value is uniform over thecross-section of the duct, with exception of the side layers and Hartmann layerswhich are asymptotically thin. Therefore, we can deduce that the core carriesapproximately 21 percent of the mass flow rate. In the Ludford layer, ux anduy have the same order of magnitude, which is much larger than the core valueof ux. Furthermore, the magnitude of ux downstream from the shear layer, isconsiderably smaller than upstream. This suggests that a fraction of the massflow rate carried by the core upstream from the Ludford layer is passed to it.

At y = 2, we see that both ux and uy are nearly zero everywhere, exceptnear the inner corner at x = 2. It appears the velocity magnitude in theplane z = 0 reaches its maximum close to this point, and the velocity profiletakes the form of an intense and sharp jet. Since ux and uy are both oppositein sign and virtually equal in magnitude, it follows that the flow direction inthis jet is paralel to the magnetic field. What we observe in fact, is that asignificant fraction of the total mass flow is gathered in the Ludford layer, andis transferred from the entrance to exit duct through a small corridor near theinner corner. For y > 2, the Ludford layer will gradually release the mass flowrate it carries into the core of the exit duct. Hence, the velocity magnitudewithin the layer should decrease with increasing y.

For y = 3, we find a velocity profile inside the Ludford layer (close to x = 1),which is similar to the one observed in the plane y = 1. Furthermore, we notethat the layer has become broader compared to the plane y = 2 and that uxis vanishing outside the Ludford layer. Finally, at y = 5, the flow has reachedagain the fully developed duct regime. It is virtually unidirectional along y,and we recover a Hartmann-like profile for uy with a core value of uy ≈ 0.21.

The discussion on the profiles of ux and uy above suggested that a significantfraction of the mass flow rate is transferred along the Ludford layer from theentrance to the exit duct. This behavior is also found in figure 6.13(a), wherewe trace streamlines originating from seed points in the plane y = 7; onlyone half of the configuration is shown. This figure furthermore reveals that,outside the plane z = 0, the flow is slightly pushed towards the side wall at

141


0 2 4 6−0.6

−0.3

0

0.3

0.6

x

ux

,y

0 0.5 1 1.5 2−2

−1

0

1

2

x

ux

,y

0 0.5 1 1.5 2−0.6

−0.3

0

0.3

0.6

x

ux

,y

0 0.5 1 1.5 2−0.6

−0.3

0

0.3

0.6

x

ux

,y

y = 1 y = 2

y = 5y = 3

Figure 6.12: Normalized profile of ux (solid) and uy (dashed) along x in theplane z = 0 for different values of y.

z = ±1. The streamlines close to these side walls exhibit a small kink whenapproaching the Ludford layer, and migrate between both ducts along a pathwhich is somewhat curved with respect to the magnetic field direction. Wealso see that a unidirectional flow is recovered at a very short distance fromthe Ludford layer. The topology of the streamlines agrees, at least in qualitativesense, well with the findings of [MHW91]. An inspection of the color plot ofthe velocity magnitude depicted in figure 6.13(b) gives a final illustration ofthe fact that the Ludford layer takes the form of a jet which becomes sharperand more intense as we approach the inner corner at x = y = 2.

The present results are compared with the asymptotic estimates of [MHW91]in figure 6.14. We consider the velocity component along magnetic field linesuB = 1/

√2(uy−ux) and the electric potential. In [MHW91], a solution is given

for the core flow; the kinematic boundary condition on solid boundaries in thisinvestigation, is not u = 0, but ∂nuτ = 0, where uτ is the velocity tangential to

142


Figure 6.13: Flow in a backward elbow: streamlines originating from seedpoints in the plane y = 1 (for z > 0)) (a). Color plot of the velocity magnitudein the plane z = 0 (b).

the wall. This means that abstraction is made from the Hartmann layer or sidelayer regions, and that the velocity on the boundaries takes a non-zero value.The asymptotic estimates of uB which are presented in [MHW91] should in factbe interpreted as the velocity at the edge between the core and the Hartmannlayer region. Therefore, we have compared them here with present values ofuB which have been evaluated along x =M−1.

From figure 6.14, we see that the agreement between both sets of results israther poor. In particular, we find that the present numerical simulations pre-dict a smoother variation of the side layer velocity profile in the vicinity of theinternal shear layer. It is not yet clear whether this is caused by the finite valueof the Hartmann number, a false hypothesis in the core flow approximation ora lack of numerical accuracy of the present simulations.

6.4 Conclusions

In this chapter, we have investigated the MHD flow in a right-angle bend in astrong magnetic field by means of high-resolution numerical simulations. Forthe backward elbow, we have demonstrated how we can reduce the number of

143


0 1 2 3 4 5 6 70

0.5

1

1.5

2

y

uB

0 1 2 3 4 5 6 7−0.5

−0.4

−0.3

−0.2

−0.1

0

y

φFigure 6.14: Comparison between the present numerical results (solid) and thecore flow approximation of [MHW91] (dashed). Velocity component along themagnetic field direction (left) against the y-coordinate close the outer Hart-mann wall x = 0, and in the plane z = 0.95 (blue) or z = 0.05 (red); electricpotential along the axis y = 0, z = 1 (right).

required grid nodes by appropriately tailoring the mesh. We have also per-formed simulations of the configurations studied in [MB94] and [MHW91]. Inboth cases, we observe that the present numerical method yields velocity distri-butions which agree qualitatively rather well with the core flow approximation.However, when it comes to quantitative measures, we find considerable devia-tions between both sets of results. In the near future, we will investigate moreprofoundly the origin of these discrepancies.

Moreover, we plan also to study the effect of inertia at fusion-relevant para-maters. One of the more interesting questions in this context is under whichconditions the internal shear layer may become unstable.

144

Chapter 7

Conclusions andperspectives

“We shall not cease from explorationAnd the end of all our exploringWill be to arrive where we startedAnd know the place for the first time.”T. S. Eliot

The contents of this dissertation can roughly be divided in two parts. Inchapters 2 and 3, we dealt with the development of a parallel, unstructuredfinite-volume solver for the MHD equations in the quasi-static limit. We haveprovided state-of-the-art numerical technology for this solver, whether it con-cerns iterative methods to solve large, sparse linear systems, the computationof the Lorentz force, or the implementation of Shercliff’s thin-wall conditionfor the electric potential. This was accompanied by a (limited) numerical anal-ysis, based on Gershgorin’s circle theorem, which holds also for unstructuredmeshes. Our implementations were furthermore successfully verified throughthe simulation of a number of test cases discussed in chapter 3. In the meantime, a newer and more advanced version of the code YALES2 has been de-veloped, and we have recently ported our implementations to this more recentrelease.

The solver developed was then used to study several configurations of inter-est in chapters 4-6. In chapter 4, we considered the laminar unidirectional flowin a circular pipe. We could reveal the existence of overspeed zones in well-conducting pipes at high Hartmann number, and characterize them in terms ofscaling laws. These are consistent with previous results, like the scaling of theradial extent of the Roberts layers in insulating pipes. Although the implica-tions of this feature for existing asymptotic theories are probably rather small,

145

146 CHAPTER 7. CONCLUSIONS AND PERSPECTIVES

this ‘discovery’ illustrates anyhow that high-resolution numerical simulationsare a valid complement next to theoretical considerations for the fundamentalstudy of MHD flows.

The work presented in chapter 5 concerned the MHD flow in a toroidal ductof square cross-section and both the laminar and turbulent regimes were inves-tigated. The results for the laminar regime confirmed the asymptotic boundarylayer analysis [TC81]. Moreover, the present approach allowed us also to studythe secondary flow in the core regions. Although somewhat underresolved, thepresent simulations of the turbulent regime illustrated the main features ofthe flow. We observed that the curvature induces an asymmetry between theconvex and concave side wall. The application of an external magnetic fieldleads to the stabilization of the core, the Hartmann layers and the concaveside layers whereas a certain level of turbulence can persist in the convex sidelayer. Structured codes (in a cylindrical coordinate system) are probably moreappropriate to study in more detail the original experiment of Moresco andAlboussiere [MA04].

Finally, in chapter 6, we have investigated the inertialess flow in a right-angle bend in a strong magnetic field. In the case of a

⋃

-bend, we couldrecover qualitatively most of the aspects characterizing the asymptotic coreflow solution. Important discrepancies between the present solution and thecore flow solution were found for the flow rates of the side layers in the centralduct of the

⋃

-bend. Also for the backward elbow, considerable disagreementswere found. In the near future, we will attempt to quantify the effect of inertia.

In chapters 4-6, we have systematically increased the complexity of theconfigurations. In particular, the sophistication and complexity of the meshfor the investigation of the backward elbow is unprecedented in the domain of(quasi-static) magnetohydrodynamics. The design of a reliable, unstructuredcomputational mesh for high Hartmann number flows is an issue which hasoften been subtly avoided in previous investigations by focusing on those ge-ometries which can be simulated on cartesian meshes. This issue will howeverbecome more predominant in the near future, and may confront us with a fun-damental stumbling stone, which is due to the following inherent opposition.Unstructured finite volume methods tend to become more accurate as the con-trol volumes are more isotropic, whereas quasi-static MHD flows in a strongmagnetic field are severely anisotropic by nature. Therefore, the developmentand analysis of numerical techniques for unstructured formulations in the con-text of MHD flows will remain of critical importance. Moreover, considerationmay also be given to alternative techniques like adaptive mesh refinement.

146

Appendix A

Elements of vector calculus

A.1 Definition of the ∇ operators

Given are a scalar function φ and a vector field F. The gradient of φ, thedivergence of F, the curl of F and the Laplacian of φ, respectively ∇φ, ∇ · F,∇× F and ∇2φ = ∇ · ∇φ, are defined as follows.

• In Cartesian coordinates x, y and z, the definitions read:

∇φ =∂φ

∂x1x +

∂φ

∂y1y +

∂φ

∂z1z (A.1)

∇ ·F =∂Fx

∂x+∂Fy

∂y+∂Fz

∂z(A.2)

∇× F =

(

∂Fz

∂y− ∂Fy

∂z

)

1x +

(

∂Fx

∂z− ∂Fz

∂x

)

1y

+

(

∂Fy

∂x− ∂Fx

∂y

)

1z (A.3)

∇2φ =∂2φ

∂x2+∂2φ

∂y2+∂2φ

∂z2(A.4)

• In cylindrical coordinates r, θ and y, we have:

∇φ =∂φ

∂r1r +

1

r

∂φ

∂r1θ +

∂φ

∂y1y (A.5)

∇ · F =1

r

∂

∂r(rFr) +

1

r

∂Fθ

∂θ+∂Fy

∂y(A.6)

147

148 APPENDIX A. ELEMENTS OF VECTOR CALCULUS

∇× F =

(

1

r

∂Fy

∂θ− ∂Fθ

∂y

)

1r +

(

∂Fr

∂y− ∂Fy

∂r

)

1θ

+

(

1

r

∂

∂r(rFθ)−

1

r

∂Fr

∂θ

)

1y (A.7)

∇2φ =1

r

∂

∂r

(

r∂φ

∂r

)

+1

r2∂2φ

∂θ2+∂2φ

∂y2(A.8)

A.2 Integral theorems

Let Ω be an arbitrary closed volume, and ∂Ω its bounding surface. For anyvector field F and any scalar function φ, the following identities hold:

∫

Ω

∇ ·F dV =

∮

∂Ω

F · dS (A.9)

∫

Ω

∇φdV =

∮

∂Ω

φdS (A.10)

∫

Ω

∇× F dV =

∮

∂Ω

F× dS (A.11)

If Σ is an arbitrary surface, ∂Σ its boundary, and F and φ respectively anyvector field and scalar function. Then, the following identities hold:

∫

Σ

∇× F · dS =

∮

∂Σ

F · dl (A.12)

∫

Σ

∇φ× dS =

∮

∂Σ

φdl (A.13)

A.3 Vector identities

For any scalar field φ, and every vector field F and G, we have the followingrelationships:

∇×∇φ = 0 (A.14)

∇ · (∇× F) = 0 (A.15)

∇ (∇ ·F) = ∇2F+∇× (∇× F) (A.16)

(∇× F)× F = (F · ∇)F− 1

2∇ (F ·F) (A.17)

∇ · (F×G) = G · (∇× F)− F · (∇×G) (A.18)

(∇× F)×G = (G · ∇)F− (F · ∇)G (A.19)

∇× (F×G) = F (∇ ·G)−G (∇ · F) + (G · ∇)F− (F · ∇)G(A.20)

∇× (φF) = φ∇× F+ (∇φ) × F (A.21)

148

A.4. HELMHOLTZ’S DECOMPOSITION 149

A.4 Helmholtz’s decomposition

Any vector field A may be written as the sum of an irrotational and solenoidalfield. The irrotational field may, in turn, be written in terms of a scalar poten-tial φ, and the solenoidal field in terms of a vector potential F:

A = −∇φ+∇× F (A.22)

The two potentials are the solutions of:

∇2φ = −∇ ·A (A.23)

∇2F = −∇×A (A.24)

149

150 APPENDIX A. ELEMENTS OF VECTOR CALCULUS

150

Appendix B

Elaborations on linearsystem solvers

B.1 A spectral analysis of the ω-Jacobi methodfor a Poisson equation

We consider an one-dimensional equidistant mesh between the boundary pointsx = 0 and x = π. The grid spacing is ∆ = π

N . The grid contains thus N + 1points, located at positions xi = i∆ for i = 0, 1, 2, ..., N . We now want to solvea discrete Poisson equation with Neumann boundary conditions on this grid;the analysis is not fundamentally different for Dirichlet or periodic boundaryconditions. The discrete Laplacian operator is given in (2.13):

L(φ)|i =φ|i+1 + φ|i−1 − 2φ|i

∆2(B.1)

The eigenvectors Φk of this operator, and of its associated matrix L, are of theform:

Φk = cos(kxi) k = 0, 1, 2, ..., N (B.2)

We can easily proof this by injecting (B.2) in (B.1):

L(Φk)|i =cos(k(xi +∆)) + cos(k(xi −∆))− 2 cos(kxi)

∆2

= − 4

∆2sin2

(

k∆

2

)

cos(kxi) (B.3)

The eigenvalues of L (or L) are λk = − 4∆2 sin

2(

k∆2

)

. We furthermore notethat the diagonal of L is D = − 2

∆2 I.

151

152 APPENDIX B. ELABORATIONS ON LINEAR SYSTEM SOLVERS

To analyse the ω-Jacobi method, we first write the residual R(m) after mω-Jacobi iterations as a linear combinations of eigenvectors. This can alwaysbe done since the N + 1 eigenvectors Φk form a complete set:

R(m) =

N∑

k=0

c(m)k Φk (B.4)

We recall the iteration formula (2.99) for R:

R(m+1) =(

I− (1− ω)MD−1)

R(m) (B.5)

The iteration matrix I− (1 − ω)MD−1 has the same eigenvectors as L but itseigenvalues µk read:

µk = 1− 2(1− ω) sin2(

k∆

2

)

(B.6)

After each iteration, every coefficient c(m)k in (B.4) will be multiplied with a

factor µk. To converge, |µk| should be smaller then one; this condition issatisfied for 0 < ω < 1.

We are now interested in the limit of µk for low (k → 0) and high (k → N)spatial frequency components:

• Low frequencies : If k → 0, we obtain µk → 1, independently of the valueof ω. This implies that the long-wavelength contributions (small k) theresidual vector will be slowly damped, independently of ω; moreover, wealso see that the convergence slows down as ∆ becomes smaller, i.e. ifthe number of grid points N increases. Note that a (discrete) Poissonequation with Neumann boundaries is only well-posed if the sum of theelements of the right-hand side is zero. If this condition is fulfilled, theinital coefficient c00 will also be zero. Therefore, the limit µk = 1 does nota formal problem for convergence.

• High frequencies : If k → N , we find that µk → −1 + 2ω. We see thatthe classic Jacobi approach (ω = 0) hardly affects the high-frequencycomponents. For k = N , we have µk = 1, and this means that the

classic Jacobi method can never converge if the coefficient c(0)N of the initial

residual is non-zero. However, for 0 < ω < 1, we can reduce the absolutevalue of µk, and obtain an efficient reduction of these short wavelengthcomponents.

We may thus conclude that, for a proper choice of ω, application of the ω-Jacobimethod to a one-dimensional Poisson equation efficiently damps the short-wavelength components of the residual, but hardly affects the long-wavelengthcontributions. This observation forms the ansatz for the development of multi-grid techniques.

152

B.2. LISTING OF THE BICGSTAB ALGORITHM 153

B.2 Listing of the BiCGstab algorithm

Algorithm 1 BiCGStab

Y (0) is an initial guess; R(0) = Z −MY (0)

Choose the maximum number of iterations qmax and a convergence thresholdǫ. Choose the parameters ρ, α and ω and vectors P and V initially as follows:ρ0 = α = ωq=1, P (0) = V (0) = 0.

Choose R, e.g.: R = R(0)

while ||R(q)|| ≥ ǫ AND q ≤ qmax do

q = q+1ρq = RTRq−1

β =ρqα

ρq−1ωq−1

P (q) = Rq−1 + β(P (q−1) − ωq−1V(q−1))

V (q) = MP (q)

α =ρq

RTV (q)T

S = R(q−1) − αqV(q)

T = MSωq = TT T

TT S

Y (q) = Y (q−1) + αqP(q) + ωqS

R(q) = S − ωqTend while

153


B.3 Listing of the BiCGStab(2)-algorithm

Algorithm 2 BiCGStab(2)

Y (0) is an initial guess; R(0) = Z −MY (0)

Choose the maximum number of iterations qmax and a convergence thresholdǫ. Choose the parameters ρ0, α and ω2 and vector U initially as follows:ρ0 = ω2 = 1, α = 0, U = 0.Choose R, e.g.: R = R(0)

while ||R(q)|| ≥ ǫ AND q ≤ qmax do

q = q+2ρ0 = −ω2ρ0EVEN STEP:ρ1 = RTR(q), β = αρ1

ρ , ρ0 = ρ1

U = R(q) − βUV = MUγ = V T R(0), α = ρ0

γ

R = R(q) − αVS = MRY = Y (q) + αUODD STEP:ρ1 = RTS, β = αρ1

ρ , ρ0 = ρ1V = S − βVW = MVγ =WT R, α = ρ0

γU = R− βUR = R− αV , S = S − αWT = MSY = Y (q) + αUQUADRATIC-POLYNOMIAL-PART:ω1 = RTS, µ = STS, ν = STT , τ = T TT , ω2 = RTT

τ = τ − ν2

µ

ω2 = µω2−νω1

µτ

ω1 = ω1−νω2

µ

Y (q+2) = Y + ω1R + ω2S + αUR(q+2) = R − ω1S − ω2TU = U − ω1V − ω2W

end while

154

B.3. LISTING OF THE BICGSTAB(2)-ALGORITHM 155

155


156

Appendix C

Asymptotic solutions athigh Hartmann number

The term asymptotic theory refers to an approximate solution method for thefully developed (i.e. steady) profile of wall-bounded MHD flows at high Hart-mann number and interaction parameter. In this formulation, the existenceof different regions is assumed; the flow domain consists a core, surroundedby various shear layers. These layers may occur due to the presence of solidwalls (e.g. Hartmann layers or side layers ), or geometrical discontinuous (e.g.Ludford layers). For each region, a simplified set of equations is derived, whichcan be solved analytically. These solutions are then matched to each other andthe boundary conditions, such that a smooth profile is obtained. This typeof methods is sometimes also called core flow approximation. Useful referenceworks on this subject are [MB01, Mor90]

C.1 Asymptotic theory for circular pipes

We consider the incompressible, unidirectional MHD flow u = u(x, y)1z in astraight pipe of radius R and of infinite extent along the axial direction, assketched in section 4.1. The variables have been rescaled in the same way as insection 4.1, but with a different definition for U . Here, U = fM(σB2

0)−1. The

asymptotic approximations to equations (4.1-4.3) for this case are solved moreeasily in terms of the induced magnetic field b. For fully developed unidirec-tional flows, b has only one component: b = b(x, y)1z. We now rescale b asfollows: b → µσUB0Lb = Re−1

m B0b. The pre-Maxwellian form of Ampere’slaw (1.38) reads:

∇× b = J (C.1)

157

158 APPENDIX C. ASYMPTOTIC SOLUTIONS AT HIGH HARTMANN NUMBER

By substituting this in the streamwise component of the momentum budget,we find:

∂b

∂y+

1

M2

(

∂2u

∂x2+∂2u

∂y2

)

= −M−1 (C.2)

Furthermore, we have the dimensionless form of the quasi-static induction equa-tion (1.59):

∂u

∂y+∂2b

∂x2+∂2b

∂y2= 0 (C.3)

Note that this equation can also be derived from taking the curl of Ohm’s law(1.32). It will be convenient to replace b by b = M−1b in (C.2)-(C.3). Weobtain:

∂b

∂y+

1

M

(

∂2u

∂x2+∂2u

∂y2

)

= −1 (C.4)

∂u

∂y+

1

M

(

∂2b

∂x2+∂2b

∂y2

)

= 0 (C.5)

The coordinates of the wall satisfy y = ±√1− x2 = ±Y (x). For fully developed

flow, we can express the thin-wall condition (1.75) in terms of the magneticinduction as:

∂b

∂n+

1

cb = 0 (C.6)

We now derive approximations to (C.4)-(C.5) for the core and Hartmann layer

regions. We will denote the solutions to these approximations as uc, bc (for the

core) and uH , bH (for the Hartmann layers). The global solution can then be

written as u = uc + uH and b = bc + bH

The core

If M ≫ 1, we may neglect the viscous terms in the core; (C.4)-(C.5) takes thesimplified form:

∂bc∂y

= −1 (C.7)

∂uc∂y

= 0 (C.8)

The solution of these equations is:

bc = −y (C.9)

uc = uc(x) (C.10)

Note that there is no integration constant in the solution for bc. This followsfrom the constraint that b should be antisymmetric in y.

158

C.1. ASYMPTOTIC THEORY FOR CIRCULAR PIPES 159

The Hartmann layers

To obtain the equations for the Hartmann layers, we rescale the coordinate yinto η =M(y − Y (x)). If we substitute this form in (C.4)–(C.5), we obtain:

M∂bH∂η

+1

M

(

M2 ∂2uH∂x2

+∂2uH∂η2

)

= −1 (C.11)

M∂uH∂η

+1

M

(

M2 ∂2bH∂x2

+∂2bH∂η2

)

= 0 (C.12)

If we retain only the leading-order terms in M , we find the set of ordinarydifferential equations:

∂bH∂η

+∂uH∂η2

= 0 (C.13)

∂uH∂η

+∂bH∂η2

= 0 (C.14)

Solutions of (C.13)-(C.14) which satisfy the boundary condition uH = A exp(η)+

B and bH = −A exp(η) +C, with A, B and C integration constants. B and C

are zero since we require that the solutions match the core solutions uH , bH → 0for η → −∞ (i.e. far away from the wall).

The global solution

The composite core-Hartmann layer solution which satisfies the constraint u =uc + uH = 0 at η = 0 is:

u = uc(1− exp(η)) (C.15)

b = −y + uc exp(η) (C.16)

The unknown core velocity uc follows from imposing the thin-wall condition(C.6). If we write n = nx1x + ny1y, (C.6) becomes:

(

nx∂b

∂x+ ny

∂b

∂y+

1

cb

)∣

∣

∣

∣

∣

y=Y (x)

= 0 (C.17)

We now assume that the first term on the left-hand side can be neglected. Thisassumption will be validated later. We obtain:

ny (−1 +Muc) +1

c(−Y + u) = 0 (C.18)

159


We now solve uc from this expression:

uc(x) =nyc+ Y

nycM + 1(C.19)

For Y (x) = ±√1− x2, we have n = x1x + Y (x)1y ., and we arrive at:

uc(x) =(c+ 1)

√1− x2

cM√1− x2 + 1

(C.20)

This solution differs by a factor M from (4.7)-(4.9) since the velocity scale Uhas been defined differently in this appendix then in chapter 4. We now checkunder which conditions the assumption nx∂xb ≪ ny∂y b fails. With the resultsabove, we can compute both the right-hand side of the inequality explicitly atthe wall: ny∂y b = ±

√1− x2(1+M). We see that this term tends to zero close

to the point (0, 1), i.e. close to where the magnetic field is nearly parallel to thewall. In this region, the asymptotic approximation breaks down and a moredetailed analysis is required.From the previous results, we find for the core current:

Jc = ∇× (bc1z) =M−1∇× (bc1z) =M−11x (C.21)

Discussion for limiting values of c

We now compare the order-of-magnitude of Jc and uc for the two limiting valuesof c, i.e. c = 0 and c = ∞. For both values of c, the dimensionless core currentJc scales as O(M−1). If the walls of the pipe are perfectly insulating (c = 0),we find that uc has order-of-magnitude O(1). Hence, Jc = O(M−1)uc × B,as stated in section 4.2. This implies that the core potential gradient almostcompletely cancels the electric field induced by the flow. For c = ∞ on the otherhand, (C.20) yields uc =M−11x. Together with (C.21), we find Jc = uc ×B.It follows that the potential gradient in the core is negligible up to the leadingorder in M .

C.2 Free MHD shear layers near geometricaldiscontinuities

In this section, we summarize the theory of free MHD (internal) shear layerswhich emanate from geometrical discontinuities in a strong uniform magneticfield B = B01y [HL67, MB01]. Consider therefore the two-dimensional situ-ation sketched in figure C.1. We have a channel with a sharp corner in thebottom wall, located at x = 0. The position of the top and bottom wall can

160

C.2. FREE MHD SHEAR LAYERS NEAR GEOMETRICAL DISCONTINUITIES 161

Figure C.1: Sketch of a two-dimensional MHD flow involving a geometricaldiscontinuity. The extent of the shear layer which emanates from it is shadedin light grey.

be specified by means of functions Yt(x) and Yb(x). The Poisson equation forthe potential reads:

∇2φ = ∇ · (u×B) = ∂z(uxB) = 0 (C.22)

If the boundary conditions for φ are homogeneous and uniform, the solu-tion of this equation is φ = αz + β, with β an arbitrary integration con-stant, and α a constant which depends on the conductivity of the walls. The(non-dimensional) Lorentz force density caused by this flow is: −∇φ × B =B0∂zφ1x = αB01x. We may absorb this contribution in the pressure gradientterm. Hence, we may treat the momentum equations, without loss of general-ity, as if ∂zφ = 0. The (non-dimensional) quasi-static MHD equations for fullydeveloped flow then read:

∂ux∂x

+∂uy∂y

= 0 (C.23)

1

N

(

ux∂

∂x+ uy

∂

∂y

)

ux = − ∂p

∂x+

1

M2

(

∂2

∂x2+

∂2

∂y2

)

ux − ux (C.24)

1

N

(

ux∂

∂x+ uy

∂

∂y

)

uy = −∂p∂y

+1

M2

(

∂2

∂x2+

∂2

∂y2

)

uy (C.25)

This set of equations can in general not be solved in analytical form. However,we can use an asymptotic approach to provide an analysis for M ≫ 1, N ≫1. The outline of our presentation is as follows. We now first postulate theexistence of core regions, in which the viscous and inertial terms in (C.24-C.25)can be neglected. This allows to solve a simplified set of momentum equations,and we will verify afterwards under which conditions our original assumptionsbreak down for the obtained solution.

161


The equations governing the core flow are:

ux,c = −∂pc∂x

(C.26)

0 = −∂pc∂y

(C.27)

These expressions can be combined into:

∂ux,c∂y

= − ∂2pc∂x∂y

= 0 (C.28)

It follows that ux,c is independent of y. Of course, this solution can not satisfythe kinematic boundary conditions u = 0 at y = Yt and y = Yb. Close tothese walls, viscous Hartmann layers of non-dimensional thickness O(M−1)will emerge, and the tangential component of u will fall off exponentially tozero. Therefore, we may relax the condition u = 0, and require only that thewall-normal velocity un,c = n · uc is zero at the top and bottom wall.

Mass conservation requires that∫ Yt

Ybux dy = C, with C a constant which

specifies the mass flow rate in the channel. For the sake of simplicity, we willassume that the reference velocity U0 used is chosen such that C = 1. Sincethe velocity in the Hartmann layers has the same order-of-magnitude as thecore velocity and the spatial extent of these layers is O(M−1) smaller than thecore, the Hartmann layers do not contribute to the mass flow rate up to the

leading order in M . This implies that∫ Yt

Ybux,c dy = 1, and, hence:

ux,c(x) =1

Yt(x)− Yb(x)(C.29)

We can now find an expression for uy,c by integrating the mass conservationequation (C.23), in which we have substituted ux,c by (C.29). In the case offigure C.1, the kinematic boundary condition at y = Yt is uy,c(y = Yt) = 0,and we find:

uy,c =

∫ Yt

y

∂uy,c∂y

dy = −∫ Yt

y

∂ux,c∂x

(C.30)

The explicit solution for uy,c is:

uy,c = (Yt − y)∂

∂x(Yt − Yb)

−1 =Yt − y

(Yt − Yb)2∂Yb∂x

(C.31)

This solution is discontinuous at x = 0. For x < 0, uy,c = 0 since thebottom wall is horizontal. Downstream from x = 0 however, the flow hasalso a component along magnetic field lines. Because of this discontinuity,the convective and viscous terms in the momentum equation (C.24) can not

162

C.2. FREE MHD SHEAR LAYERS NEAR GEOMETRICAL DISCONTINUITIES 163

be neglected close to x = 0. More specifically, the convective term in (C.24)becomes O(1) if ∂xuy,c = O(N). We now compute this term explicitly:

∂uy,c∂x

=Yt − y

(Yt − Yb)2

(

∂2Yb∂x2

+2

Yt − Yb

(

∂Yb∂x

)2)

(C.32)

We see thus that the core flow solution breaks down if ∂2xYb = O(N) or ∂xYb =O(N1/2).

The sudden occurrence of a velocity component along the magnetic fielddirection gives rise to a local shear flow, which extends over a distance δ, whosedependence on M and N is to be specified. For further analysis, we introducethe stretched coordinate ξ = x/δ. Moreover, since ux,c is continuous at x = 0,its variation across the shear layer is O(δ). Finally, we assume that the typicallength scale L used to define N and M is chosen such that ux,c(x = 0) = 1, i.e.L = Yt(x = 0)− Yb(x = 0). Then we have:

u = 1 + δUx(ξ, y) (C.33)

Here, Ux is a quantity wich is O(1). If this form is injected into (C.23-C.25),and if we retain only the leading-order contributions in δ in the viscous andconvective term, we find:

∂Ux

∂ξ+∂uy∂y

= 0 (C.34)

1

N

∂Ux

∂ξ= −1

δ

∂p

∂ξ− 1

M2

1

δ

∂2Ux

∂ξ2− (1 + δUx) (C.35)

1

N

1

δ

∂uy∂ξ

= −∂p∂y

+1

M2

1

δ2∂2uy∂ξ2

(C.36)

We now furthermore assume that δ > O(M−1) and δ > O(N−1). As such,(C.35) is up to the leading order in δ:

1

δ

∂p

∂ξ= −1− δUx(ξ, y) (C.37)

This result implies that the pressure drop across the shear layer is of the order-of-magnitude of δ. We can now use (C.34) and (C.37) to eliminate the pressureterm from (C.36). This yields:

1

N

1

δ

∂3uy∂ξ3

=1

M2

1

δ2∂4uy∂ξ4

− δ2∂2uy∂y2

(C.38)

This equation expresses a balance between an inertial, viscous, and electromag-netic term. Depending on the values of M and N , we can distinguish betweenseveral regimes:

163


• The viscous-electromagnetic balance holds if the inertial term in (C.38) isnegligible with respect to the other terms, i.e if O

(

(Mδ)−2)

= O(

δ2)

≫O(

(Nδ)−1)

. This implies that the shear layer thickness scales as:

δ ∼M−1/2 (C.39)

From this, we find that this regime holds if N ≫M3/2.

• The condition O(

(Nδ)−1)

= O(

δ2)

≫ (Mδ)−2 governs the inertial-electromagnetic balance. This leads to the following scaling law for δ:

δ ∼ N−1/3 (C.40)

Hence, this regime hold is if N ≪M3/2.

• The three terms in (C.38) have the same order-of-magnitude if :

δ ∼ N−1/3 ∼M−1/2 (C.41)

This viscous-inertial-electromagnetic balance requires that N ≈M3/2.

• Theoretically, a fourth possibility would be the one of a viscous-inertialbalance. It can however be shown that no solution of (C.38) withoutelectromagnetic term exists which satisfies the condition that uy matchesthe core solution as η → ±∞.

For the three possible regimes, one can easily show that the scaling law for δis consistent with the assumption δ > O(N−1), δ > O(M−1).

C.3 Asymptotic analysis of the core regions ina⋃

-bend

We consider the configuration discussed in section 6.2.1, i.e the inertialess flowin a

⋃

-bend in a strong magnetic field, i.e Re = 0 and M ≫ 1. Here, we sum-marize some elements of the core flow approximation, as presented in [MB94].The equations for fully developed flow read:

−∇p+M−2∇u+ J× 1y = 0 (C.42)

In the core regions, the viscous term in the momentum equation can be ne-glected, and we have a simple balance between the pressure gradient and theLorentz force 1:

−∇p+ J× 1y = 0 (C.43)

1Since we limit ourselves to core regions in this section, we will leave away the subscript

‘c’ for core variables.

164

C.3. ASYMPTOTIC ANALYSIS OF THE CORE REGIONS IN A⋃

-BEND 165

If we take the vector product of (C.43) with 1y, we find:

−∇× (p1y) = J⊥ (C.44)

where J⊥ is defined as J⊥ = Jx1x + Jz1z. This shows that the pressure is astreamfunction for J⊥ in x-z planes.

We now concentrate on the core of the central duct. First, we note that theplane y = 12 is a symmetry plane for the momentum equation (C.42). Hence,we find that p(x, y = 12, z) = C with C an arbitrary constant. Without loss ofgenerality, we may choose C = 0. Using Ohm’s law (1.32), we can formulate(C.42) componentwise as:

− ∂p

∂x+∂φ

∂z− ux = 0 (C.45)

−∂p∂y

= 0 (C.46)

−∂p∂z

− ∂φ

∂x− uz = 0 (C.47)

From (C.46), together with the symmetry condition p(x, y = 12, z) = 0, itfollows that p(x, y, z) is zero everywhere in the core of the central duct. Assuch, ∇p = 0 in this region, and the momentum balances (C.45) and (C.47)reduce to:

ux =∂φ

∂z(C.48)

uz = −∂φ∂x

(C.49)

If we substitute (C.48) and (C.49) in the incompressibility constraint ∇·u = 0,we find:

∂uy∂y

= 0 (C.50)

Since it is assumed the wall-normal velocity does not vary up to the leadingorder in the Hartmann layers, uy satisfies the kinematic boundary conditionuy = 0 at the Hartmann walls at y = 0 and y = 24. Integration of (C.50)results in uy = 0 everywhere in the core of the central duct. The asymptoticcore flow solution in this region is thus a two-dimensional field u = u⊥ =ux1x + uz1z for which, according to (C.48)-(C.49), the electric potential φprovides a streamfunction, i.e. u⊥ = −∇× φ1y.

165


166

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