Numerical simulations of quasi-static magnetohydrodynamics using ...

Numerical simulations of quasi-static magnetohydrodynamics using an unstructured finite volume solver: development and applications Stijn Vantieghem

Universit´ e Libre de Bruxelles

e en vue de esent´ ese pr´ Th` l’obtention du grade de Docteur en Sciences sous la direction du Prof. B. Knaepen. Janvier 2011

© 2011 Stijn Vantieghem All Rights Reserved

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Abstract In this dissertation, we are concerned with the flow of electrically conducting liquids in an externally imposed magnetic field. Such flows are governed by the equations of quasi-static magnetohydrodynamics (MHD), and are commonly encountered in applications of practical interest. Therefore, there is strong interest 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 presentation of the state-of-the-art numerical machinery which has been used and implemented to solve the (incompressible) quasi-static MHD equations. More precisely, we have contributed to the development of a parallel unstructured finite-volume flow solver. The discussion on these methods is accompanied by a numerical analysis which holds also for unstructured grids. In chapter 3, we verify 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 to study wall-bounded MHD flows in various configurations. The first geometry considered is the laminar flow in a circular pipe of infinite extent at high Hartmann number. We have investigated the sensitivity of the numerical results on the mesh topology and the numerical discretization scheme. Furthermore, our simulation results allow to characterize extensively the flow in pipes with well-conducting walls. This is achieved by an analysis of the scaling of the most relevant flow features with the non-dimensional parameters governing the flow. The subject of chapter 5 is the flow in a toroidal duct of square crosssection. A study of the laminar regime confirms existing asymptotic analysis for the shear layers. We have also provided simulations of turbulent flows in order to assess the effect of an externally imposed magnetic field on the state of the boundary layers. S Finally, in chapter 6, we investigated the inertialess MHD flow in a -bend and a backward elbow in a strong magnetic field. We explain how we can generate a mesh which can properly resolve all the shear layer at an affordable computational cost. Results of the present numerical method are compared against asymptotic core flow approximations.

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Acknowledgements Did you ever hear of Loukas Karrer? Probably you didn’t. He doesn’t even have an entry in Wikipedia. Nevertheless, he was an exceptional person who deserves to be paid tribute to until eternity. Much like the people mentioned below. I want to acknowledge my advisor, prof. Bernard Knaepen, for unwavering mentoring support during my first steps as a junior researcher. I have enormously appreciated his pedagogical qualities and his ability to offer a pertinent analysis and a systematic solution strategy whenever I got stuck. I also acknowledge his constructive criticism during the redaction of this dissertation. Finally, I thank him for giving me the confidence and freedom to pursue my own research ideas during the final years of this thesis, and for the relaxed way in which he heads his team. I want to extend a big thanks to prof. Daniele Carati for his efficient management of the Statistical Physics and Plasma team. If he had been on the rudder of the Titanic, it surely wouldn’t have sunk. I furthermore want to thank him for discreet encouragement, support and confidence, and for his conviviality, 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 scientific merits, and his complete lack of conceit, and offer my acknowledgements for many fruitful discussions, and for giving me a warm welcome at CORIA during my visit in the beginning of 2009. I sincerely hope that we can continue our collaboration in the near future. A special thanks is also due to dr. Chiara Mistrangelo and dr. Leo B¨ uhler, for enjoyable and fruitful discussions on the implementation of the unstructured quasi-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. Léon Brenig and prof. Gérard Degrez for accepting to be member of this jury. I would also like to thank the colleagues of the Statistical Physics and Plasma group who made work so enjoyable. In the first place, I think of Axelle vii

Viré, my office mate for four years. I have rarely met someone with whom I could collaborate so well. I would like to thank her for her kindness, and for always lending me a listening ear. I hope we will be able to stay in touch so that I’ll get the chance to taste the tiramisu she still owes me. I am also grateful to Bogdan Teaca for challenging my points of view in many scientific and nonscientific discussions, for our trip to Chicago, and simply for being a wonderful guy. I would like to salute Thomas Lessinnes for his cordiality and enthusiasm for taking on mathematical problems. A Flemish guy praising a real Carolo; who would have ever imagined that? I am thankful to Maxime Kinet for his kindness, and for patiently answering my computer-related questions. Special thanks also to Xavier Albets-Chico for intensive discussions and really good times. Furthermore, I would like to thank explicitly the other colleagues (in no particular order of importance): Sara Moradi, Chichi Lalescu, Paolo Burattini, Carlos Cartes, Michel Marc-Albrecht, Pierre Morel, Michael Leconte, Alejandro Banon-Navarro, Benjamin Cassart, Sotirios Kakarantzas, Chiara Toniola, Ioannis Sarris and Oleg Shyskin. I am indebted to Fabienne De Neyn and Marie-France Rogge for helping me out in my struggle with the bureaucratic merry-go-round, and above all, for being kind and caring office neighbors. The latter holds also for the colleagues of the Theoretical Non-Linear Optics group. Finally, thanks to Axel Coussement and Matthew Peavy for administrating our cluster. Outside the professional sphere, I am especially grateful to five friends for their rock-solid and longstanding friendship. A big thanks to Clara Verhelst for mental support and highly enjoyable train trips to hometown Kortrijk, to Frederik De Roo for exploring together Europe by bicycle and for never-ending discussions on Dawkins and Dostoevsky. Thanks also to Maarten Vanhee, for hospitality, coffee and cookies, and to Dieter Vanneste and Tim Bekaert for many road trips and not-so-lazy-sunday-afternoon jogging tours. All of you are cordially invited to Z¨ urich. Finally, there are my most faithful supporters along the road of study and life. I want to thank my family for believing in me and investing in me, and I dedicate this dissertation to them.

Stijn Vantieghem January 2011

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Contents Abstract

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Acknowledgements

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Introduction

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1 Introductory aspects of MHD 1.1 The fundamentals of hydrodynamics . . . . . . . 1.1.1 The continuum hypothesis . . . . . . . . . 1.1.2 Conservation of mass . . . . . . . . . . . . 1.1.3 Conservation of momentum . . . . . . . . 1.1.4 Conservation of energy . . . . . . . . . . . 1.1.5 Summary . . . . . . . . . . . . . . . . . . 1.2 Magnetohydrodynamics . . . . . . . . . . . . . . 1.2.1 Classical electromagnetism . . . . . . . . 1.2.2 The induction equation . . . . . . . . . . 1.2.3 Summary . . . . . . . . . . . . . . . . . . 1.3 The quasi-static approximation . . . . . . . . . . 1.3.1 Simplified equations for Rm ≪ 1 . . . . . 1.3.2 Phenomenology of the quasi-static regime 1.3.3 Examples of quasi-static MHD flows . . .

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1 1 1 2 3 5 6 6 7 9 9 11 11 15 21

2 Numerical framework 2.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Basic principle of the finite-volume method . . . . . 2.1.2 Construction of the control volumes . . . . . . . . . 2.1.3 Volume averages . . . . . . . . . . . . . . . . . . . . 2.1.4 The gradient operator . . . . . . . . . . . . . . . . . 2.1.5 The divergence operator . . . . . . . . . . . . . . . . 2.1.6 The Laplacian operator . . . . . . . . . . . . . . . . 2.1.7 Matrix representation of the discretization operators

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Time advancement . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Fractional-step methods . . . . . . . . . . . . . . . . . . 2.2.2 Time integration schemes for the momentum equation . 2.2.3 Kinetic energy conservation and the pressure term . . . 2.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . The quasi-static MHD equations . . . . . . . . . . . . . . . . . 2.3.1 Discretization of the Lorentz force . . . . . . . . . . . . 2.3.2 Boundary conditions for the potential . . . . . . . . . . 2.3.3 Coupling between the momentum and potential equations Solution techniques for systems of linear equations . . . . . . . 2.4.1 Jacobi iteration . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Algebraic multigrid methods . . . . . . . . . . . . . . . 2.4.3 Krylov subspace methods . . . . . . . . . . . . . . . . .

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3 Verification and validation 3.1 Taylor-Green vortex . . . . . . . . . . . . . . . . . . . . 3.2 Turbulent channel flow . . . . . . . . . . . . . . . . . . . 3.2.1 Physical background . . . . . . . . . . . . . . . . 3.2.2 Computational details . . . . . . . . . . . . . . . 3.2.3 Numerical results and discussion . . . . . . . . . 3.3 Two-dimensional MHD flows at high Hartmann number 3.3.1 Laminar MHD flow in a straight duct . . . . . . 3.3.2 Laminar MHD flow in a plane sudden expansion

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67 67 71 72 74 75 77 78 81

4 Laminar pipe flow 4.1 Introduction . . . . . . . . . . . . . . . 4.2 Phenomenology . . . . . . . . . . . . . 4.3 Computational details and grid study 4.3.1 Computational details . . . . . 4.3.2 Grid study . . . . . . . . . . . 4.4 Results and discussion . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . .

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5 Flow in a toroidal square duct 5.1 Motivation . . . . . . . . . . 5.2 Mathematical formulation . . 5.3 Laminar flow . . . . . . . . . 5.3.1 Computational set-up 5.3.2 Results and discussion 5.4 Turbulent flow . . . . . . . . 5.4.1 Computational set-up 5.4.2 Results . . . . . . . . 5.5 Conclusions . . . . . . . . . .

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6 Laminar flow in a right-angle bend 6.1 Introduction . .S. . . . . . . . . . . . . . . . . . . . . 6.2 MHD flow in a -bend . . . . . . . . . . . . . . . . . 6.2.1 Problem definition and computational set-up 6.2.2 Results and discussion . . . . . . . . . . . . . 6.3 MHD flow in a backward elbow . . . . . . . . . . . . 6.3.1 Problem definition and computational set-up 6.3.2 Results and discussion . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . .

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7 Conclusions and perspectives

123 124 126 126 130 136 136 141 143 145

A Elements of vector calculus A.1 Definition of the ∇ operators A.2 Integral theorems . . . . . . . A.3 Vector identities . . . . . . . A.4 Helmholtz’s decomposition . .

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147 147 148 148 149

B Elaborations on linear system solvers 151 B.1 A spectral analysis of the ω-Jacobi method for a Poisson equation151 B.2 Listing of the BiCGstab algorithm . . . . . . . . . . . . . . . . 153 B.3 Listing of the BiCGStab(2)-algorithm . . . . . . . . . . . . . . 154 C Asymptotic solutions at high Hartmann number 157 C.1 Asymptotic theory for circular pipes . . . . . . . . . . . . . . . 157 C.2 Free MHD shear layers near geometrical discontinuities . . . . . 160 S C.3 Asymptotic analysis of the core regions in a -bend . . . . . . 164

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Introduction The scope of this dissertation is at the crossroads between two important areas of research: magnetohydrodynamics (MHD) on the one hand, and computational fluid dynamics (CFD) on the other hand. Magnetohydrodynamics is the branch of physics which studies the interaction between the flow of electrically conducting fluids and electromagnetic fields. This coupling is due to three fundamental physical phenomena. According to Faraday’s and Ohm’s law, the presence of a magnetic field will induce an electric current in a moving conductor. Secondly, this current distribution is at the origin of an induced magnetic field. Finally, the interaction between the 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 can think of astrophysical or geophysical processes, like the spontaneous generation of the Earth’s magnetic field by the motion of the liquid iron core of the Earth, but MHD flows can also be encountered in applications of more practical interest. The common feature of almost all these industrial and laboratory flows is that the coupling between the flow and the magnetic field is virtually one-way, i.e. the magnetic field strongly affects the flow through the generation of a body force, but the flow does not act significantly upon the magnetic field. This regime is known under the name quasi-static magnetohydrodynamics. Sometimes, the term liquid metal MHD is also used to refer to this regime. Historically, the first application of MHD concerned flow measurement techniques. Pioneering work in this context was performed by Faraday, who attempted in 1832 in vain to estimate the flow rate of the river Thames by measuring the electric potential difference across the river, induced by the Earth’s magnetic field. Notwithstanding his failure, Faraday’s induction principle is still at the basis of most of the electromagnetic flow meters available on the market. It was only in the 1960’s that magnetic fields began to be used as tools for flow control and generation. In continuous casting processes, a better quality of the product is achieved by applying a (static) magnetic field during 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 other hand, rotating magnetic fields can be used to enhance mixing by inducing nonintrusively a stirring motion in a liquid metal or electrolyte. Such an approach is preferred in configurations where the use of a mechanical mixer is impractical, e.g. in high-temperature or corrosion-aggressive environments. Finally, rotating magnetic fields are also applied in heating or levitation processes. A recent review of the applications of MHD in materials processing can be found in [Dav99]. Apart from those applications where a magnetic field is imposed on purpose, there are situations in which there is an ambient magnetic field without specific intent for the flow. A notable example are so-called blankets for future thermonuclear fusion devices [B¨ 07]. Their role is to absorb the energy released in the fusion reaction, and transfer it subsequently to a power plant. Moreover, they should provide a shield against the neutron irradiation. Liquid metal alloys, like lead-lithium, are primary candidate coolant materials, mainly because of their operability at high temperatures. However, the flow of these media will be heavily affected by the presence of intense magnetic fields (up to 5-10 tesla) required to confine the plasma in the tokamak reactor. All the applications described above drive an ongoing scientific effort which aims at improving our current understanding of MHD. The mathematical framework describing the physics of MHD flows results from a combination of the laws of electromagnetism and the theory of fluid mechanics. The partial differential equations governing hydrodynamic and (quasi-static) MHD flow have a non-linear nature. Hence, for broad ranges of flow parameters, the flow may be in a turbulent state, i.e. exhibit a seemingly random chaotic behavior containing a large range of spatial and temporal scales. As such, solving exactly the flow equations, by means of analytical methods, is only possible for a 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 particular, we can distinguish between three approaches: physical experiments, analytical solutions of approximate equations and numerical simulations. Due to the fast increase in computational power and the development of parallel computing, numerical simulations have become a powerful predictive tool, and are a useful complementary alternative to experiments and approximate theories. The advantages of numerical experiments above physical ones are numerous. They are relatively cheap to perform, they give access to the complete flow field, they allow to study systematically the influence of geometrical parameters and they don’t require all kinds of precautions related to the handling of possibly dangerous working fluids. This holds a fortiori for quasi-static MHD flows, since most liquid metals are opaque; moreover, the generation of intense magnetic fields is very power-consuming, and thus expensive. The xiv

main drawback of a numerical approach is that the parameter ranges which are encountered in practical applications can not be reached (yet) with sufficient confidence. The term CFD refers, in the most general sense, to the study of fluid mechanics by means of numerical simulations, as well as to the development and analysis of numerical techniques used to solve the complete set of equations governing these flows. One of the earliest CFD calculations was performed by Richardson in 1916. To provide numerical weather predictions, he divided physical space in grid cells and applied a finite-difference technique. For his prediction of the globe’s weather over a period of 8 hours, he needed six weeks of computation time. Moreover, his attempt failed, presumably because of a lack of accuracy of the then available (mechanical) calculators. In order to produce 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 approximately 2000 grid points [Ric22]. The introduction of the digital computer brought new perspectives to the possibilities of computational fluid dynamics. However, for a very long time, numerical simulations were only affordable in the context of military research projects. During the 1960’s, many of the numerical techniques which are still used today were developed at Los Alamos National Laboratory [Har04], like the vorticity-streamfunction formulation or the k − ǫ turbulence model. It was also there that the first digital simulation of an unsteady von Karman vortex street was performed. None of these breakthroughs of the early days of CFD would have been possible without significant progress in the field of numerical analysis and numerical algorithms. Milestone achievements in these fields were, among others, the famous stability analysis of Courant, Friedrich and Lewy for the advancement of the advection equation [CFL28], and the introduction of von Neumann’s stability analysis method [vNR50]. The first commercial use of CFD codes should be situated in the second half of the 1970’s and early 1980’s, and concerned primarily the aircraft industry. These codes were based on finite-difference or finite-volume formulations. Computational resources were still very limited at that time. For instance, the Cray-X-MP, a state-of-the-art supercomputer in 1980, disposed of a total memory of 32 megabyte and had a peak performance of 400 MFLOPS. However, the range of treatable problems expanded rapidly because of the fast increase in computational power; codes based on unstructured meshes emerged in the 1990’s. In many branches of engineering, as well in industry and in academia, these numerical tools are now widely accepted as useful instruments for flow prediction. However, the systematic application of complex, unstructured CFD codes has 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 development of a numerical solver for the magnetohydrodynamic equations in the quasi-static limit, and used this solver to study a number of cases of theoretical and practical interest. The outline of this work is as follows. First, we derive the partial differential equations governing quasi-static MHD from first principles, and give a brief summary of the phenomena characterizing the quasi-static regime. In the second chapter, we introduce and discuss extensively the numerical techniques that have been implemented in the finite volume solver YALES2 [Mou10]; this is a versatile code, mainly developed at CORIA, for various types of flow problems. Several test cases, which were performed in order to validate the implementation of the numerical methods, are presented in the third chapter. We have then investigated various configurations with this code: a laminar MHD flow in a circular pipe in an intense magnetic field (chapter 4), the hydrodynamic and MHD flow in a toroidal duct of square cross-section (chapter 5), and the inertialess MHD flow in a right-angle bend in a strong magnetic field (chapter 6). In the seventh and last chapter, we summarize the main conclusions of this flow. Some chapters in this work are based on the following publications: • Chapter 4: – S. Vantieghem, X. Albets-Chico and B. Knaepen, “The velocity profile of laminar MHD flows in circular conducting pipes”, Theoretical and Computational Fluid Dynamics 23(6), (2009) 525. • Chapter 5: – S. Vantieghem and B. Knaepen, “Direct numerical simulation of quasi-static magnetohydrodynamic annular duct flow” Proceedings of the Fifth European Conference on Computational Fluid Dynamics, Lisbon, 14-17 June 2010, in press (2010). – S. Vantieghem and B. Knaepen, “Numerical simulation of magnetohydrodynamic flow in a toroidal duct of square cross-section”, submitted to International Journal of Heat and Fluid Flow (2010).

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Chapter 1

Introductory aspects of magnetohydrodynamics “Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.” Leonhard Euler In this first chapter, we will give a broad introduction to the field of magnetohydrodynamics (MHD). This is the branch of physics which studies the interaction between the flow of electrically conducting fluids and electromagnetic fields. As a starting point, we will develop the equations of conservation of mass, momentum and energy for an ordinary hydrodynamic flow. In a second step, we will detail the mutual interaction between a flow and an electromagnetic field. Under some conditions, this coupling may be virtually one-way, i.e. the velocity field only weakly influences the magnetic field. This regime, which is known under the name of quasi-static magnetohydrodynamics, is the main scope of this work. Its governing equations, together with a couple applications of industrial interest, will be presented in the third and final section of this chapter.

1.1 1.1.1

The fundamentals of hydrodynamics The continuum hypothesis

When we want to model the physics of fluid flows, the chosen description of the medium will depend on the length and time scale of the relevant physical 1

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CHAPTER 1. INTRODUCTORY ASPECTS OF MHD

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

1.1.2

Conservation of mass

To derive the equation of mass conservation, we consider an arbitrary lump of fluid which occupies a volume Ω in space. If we follow the lump throughout its 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 can express this as: Z d ρ(x, t) dV = 0 (1.1) dt Ω The size or shape of the integration domain may change in time, and this should be taken into account when bringing the time derivative within the integral. It can be shown that this leads to: Z d ρ(x, t) + ρ∇ · u dV = 0 (1.2) dt Ω Furthermore, the positions x of the fluid elements still depend on time. The total time derivative can be decomposed into an expression involving only partial derivatives by applying the chain rule: ∂ρ ∂x ∂ρ d ρ(x, t) = + · ∇ρ = + u · ∇ρ dt ∂t ∂t ∂t

(1.3)

We eventually obtain: Z Ω

∂ρ + ∇ · (ρu) dV = 0 ∂t 2

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1.1. THE FUNDAMENTALS OF HYDRODYNAMICS

3

Since this expression holds for any volume Ω, we can leave aside the volume integration; this yields a local relationship: ∂ρ + ∇ · (ρu) = 0 ∂t

(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 lump of fluid does not change with time. Such flows are called incompressible, and obey: Z d dV = 0 (1.6) dt Ω 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 sound in these media is typically far above the characteristic velocity scales of the applications of interest in this work. Therefore, we will always assume that the flow under consideration is incompressible.

1.1.3

Conservation of momentum

Newton’s second law states that the rate of change of momentum of a body equals the net force exerted on that body. We can again consider an arbitrary lump of fluid, and write its rate of change of momentum as: Z d ρu dV = F (1.8) dt Ω Following the same approach as in the previous section, we develop the lefthand side of this equation: Z Z d d (ρu) + (ρu)∇ · u dV ρu dV = dt Ω dt Ω Z d d (1.9) ρ u + u ρ + (ρu)∇ · u dV = dt dt Ω We can use (1.3) and (1.5) to show that the second and third term on the right-hand side of the result above cancel each other. Further development of the total time derivative of the first term leads to: Z Z ∂u d ρ + ρu · ∇u dV (1.10) ρu dV = dt Ω ∂t Ω 3

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We can furthermore write the net force F as the volume integral of a force density f . Following the approach of [Bat67], we will distinguish between two classes of forces contributing to f . On the one hand, we have volume or body forces fb , and surface forces fs on the other hand. The former stem from longrange interactions, like buoyancy, gravity or electromagnetic forces. The latter have a molecular origin, and are negligible unless there is direct mechanical contact between the interacting fluid elements. If we consider now a lump of fluid, all interior contributions of this force will cancel, since any elementary force exerted by a fluid element on its surroundings, is accompanied by an opposite reaction force of the surrounding on the element. It is thus worthwhile to write the resulting total surface force on the lump as the surface integral of a stress tensor τ : I Z fs dV = τ · dS (1.11) ∂Ω

Ω

Using Gauss’s divergence theorem, we obtain: ∂u + u · ∇u = −∇ · τ + fb ρ ∂t

(1.12)

As for now, we will not yet specify the form of the body forces, and concentrate on the structure of the tensor τ . It can be shown that angular momentum conservation 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 hold any 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 constrained by assuming that the stress tensor is an isotropic and linear function of the velocity gradient tensor. A fluid with such properties is called Newtonian. In its most general form, τ ′ can be written as: 2 T τ ′ = η ∇u + (∇u) − (∇ · u) 1 (1.13) 3 The quantity η is called the dynamic viscosity. In incompressible flows, the last term of the right-hand side in the expression above is zero (see (1.7)) and the previous constitutive relationship simplifies to: (1.14) τ ′ = η ∇u + (∇u)T 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: ∂u ρ + u · ∇u = −∇p + ρν∇2 u + fb (1.15) ∂t It is important to note that the idea of (mechanical) pressure is not to be confused with the one we know from thermodynamics. The latter one is a variable that is reserved for the description of equilibrium states. Both notions are only equivalent under hydrostatic equilibrium. To elucidate the role of the pressure in incompressible fluid dynamics, we take the divergence of equation (1.15) and rearrange the terms as follows: ∂ 2 (1.16) − ν∇ ∇ · u = −∇2 p − ρ∇ · (u · ∇u) + ∇ · fb ρ ∂t We now consider this as an equation for the unknown ∇ · u, with initial and boundary condition ∇ · u = 0. The solution of this equation is ∇ · u = 0 everywhere 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 incompressibility constraint due to the non-linear term u · ∇u or the body force term. We can interpret the pressure thus as a kind of Lagrange multiplier needed to satisfy the 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 this quantity can be obtained by taking the scalar product between the velocity and the Navier-stokes equation (1.15). For an incompressible flow, we find, after a few manipulations: 1 2 ∂ 1 2 = −∇ · u ρu − ∇ · (pu) ρu ∂t 2 2 +ρν∇ · (u · ∇u) − ρν||∇u||2

(1.17)

In this expression, P Pwe have introduced the norm of the velocity gradient tensor as ||∇u||2 = i j (∂i uj )2 . The first two terms on the right-hand side are in divergence form. If we integrate this equation over a given volume, we find that the total work caused by the non-linear and pressure term reduces to a boundary integral. We call these terms energy-conserving. The last two terms in the above equation concern viscous effects. Only the first one is energyconserving. The last term is always negative and represents a loss of kinetic energy; the effect of the viscous interaction is to dissipate kinetic energy into heat. 5

6

1.1.5


Summary

The following set of partial differential equations completely determines the incompressible motion of a fluid: ∇·u

∂u + u · ∇u ∂t

= 0

(1.18)

p + ν∇2 u + ρ−1 fb = −∇ ρ

(1.19)

This set of equations is not yet completely closed. We should still provide suitable boundary and initial conditions. Of particular interest is the boundary condition for a stationary rigid wall. In this work, we adopt the convention to denote the outward-pointing normal on the wall as n. Since the wall is impermeable, we have: u · n = un = 0

(1.20)

For the velocity components tangential to the wall, we assume that a viscous fluid ‘sticks’ to the rigid wall. Hence: u − u n n = uτ = 0

(1.21)

and both conditions together yield a homogeneous Dirichlet condition for the velocity at a rigid, stationary wall. Such a boundary condition is called a no-slip condition. Much confusion exists about the boundary conditions for the pressure. They should be such that they reconcile the incompressibility constraint and the velocity boundary conditions [McC89]. It is, generally spoken, not possible to define generic a priori conditions for p.

1.2

Magnetohydrodynamics

In this section, we will treat the coupling between fluid dynamics and electrodynamics. This coupling is twofold. On the one hand, electromagnetic body forces will enter into the momentum balance. On the other hand, the motion of a conducting medium may give rise to a complex dynamics of the electromagnetic field. We will first review classical electromagnetics, and introduce a slightly simplified version of it, leaving aside some complications which only matter 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 an evolution equation of the magnetic field, the so-called induction equation. We will derive it and analyse it in terms of a few non-dimensional parameters. 6

1.2. MAGNETOHYDRODYNAMICS

1.2.1

7

Classical electromagnetism

Classical electromagnetism [Jac99], in its most concise form, consists of the combination of Maxwell’s equations and an expression for the force on a charged particle 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/or magnetic 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 an electric current density J. We define these as the quantities which obey the following relationship for any arbitrary volume Ω: Z X ρe dV = q (1.23) ZΩ X J dV = qi ui (1.24) Ω

where the summation extends over all the particles i with charge qi and velocity ui within Ω. The Lorentz force density fL can then be written as: fL = ρe E + J × B

(1.25)

Just like mass, charge is a conserved quantity. The introduction of the quantities above allows us to express an electrical analogue of the mass conservation equation. This charge conservation equation reads: ∂ρe +∇·J= 0 ∂t

(1.26)

Maxwell’s equations on the other hand allow to compute the electric and magnetic fields for a specified charge and current distribution. We will only consider fluids which are neither dielectric nor diamagnetic; these equations take the following form: ∇×E ∇·E ∇·B ∇×B

∂B ∂t = ǫ−1 0 ρe = 0 ∂E = µ0 J + ǫ 0 ∂t = −

(1.27) (1.28) (1.29) (1.30)

In these expressions, ǫ0 and µ0 denote, respectively, the vacuum electric permittivity and magnetic permeability. By combining equation (1.28) with the 7

8


divergence of (1.30), and using (A.16), we immediately recover the law of charge conservation (1.26). The set of equations (1.27)-(1.30) defines eight constraints for ten unknowns (in three dimensions). Moreover, those constraints are not independent; taking the divergence of (1.27), together with (A.15), gives ∂t (∇ · B) = 0, which, together with an appropriate initial condition, reduces to (1.29). To close the system of equations, we still have to supply a constitutive relation which links the electric field to the current density. For stationary isotropic conducting media, and for low-frequency electromagnetic fields, there is empirical evidence that the current density is proportional to the electric field, with proportionality constant σ, 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’s law Lorentz invariant. We will however assume that the speed of the medium is small with respect to the speed of light, so that we can use an approximate version of Lorentz’s transformation laws: J = σ (E + u × B)

(1.32)

If we insert this expression into the charge conservation equation (1.26), and use (1.28), we obtain: σ ∂ρe + ρe = −σ∇ · (u × B) ∂t ǫ0

(1.33)

The left-hand side of this equation represents an exponential decay on a charge relaxation time of τC = ǫ0 σ −1 . Typical values of this time scale are of the order of 10−18 s. Since the flow phenomena in which we are interested, are characterized by much larger time scales, we may disregard the term ∂t ρe . It means 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)) = ǫ0 U B0 L−1 , where U , B0 and L are respectively a speed, magnetic field intensity and length scale which characterize the flow under consideration. Furthermore, Ohm’s law tells us that the electric field strength scales as O (E) = σ −1 J. Here, J is a typical magnitude of the current density. All this allows us to estimate the relative importance of the ‘electric’ and ‘magnetic’ part of the Lorentz force: O(ρe E) = ǫ

U J U U B0 = τC JB0 = τC O (J × B) L σ L L 8

(1.36)

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 term ρe E can be neglected with respect to the term J × B. We can, up to a good approximation, 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 the term ǫ0 µ0 ∂t E can be neglected in Ampère’s law (1.30). This term becomes important only if the characteristic velocity of the flow approach the speed of light c = (µ0 ǫ0 )−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 eliminate the induced electric field: ∂B = −∇ × E = −∇ × σ −1 J + ∇ × (u × B) ∂t

(1.39)

Furthermore, we use the pre-Maxwell version of Faraday’s law (1.38) to eliminate the current. The resulting expression becomes: ∂B 1 = − ∇ × ∇ × B + ∇ × (u × B) ∂t µσ

(1.40)

Using (A.17), (A.19) and the solenoidal character of both the velocity and magnetic field, we finally obtain: 1 2 ∂B + u · ∇B = B · ∇u + ∇ B ∂t µσ

1.2.3

(1.41)

Summary

We have now all the elements that are required to describe the incompressible flow of a viscous, conducting liquid in a magnetic field. The governing equations of incompressible MHD are given by a combination of the incompressibility constraint (1.18), the Navier-Stokes equations (1.15) and the induction equation for the magnetic field (1.41). This latter equation does only guarantee the 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

10


complete description of an MHD flow, but only as an initial condition: ∇·u ∂u + u · ∇u ∂t ∂B + u · ∇B ∂t ∇·B

=

0

(1.42)

=

−ρ−1 ∇p + ν∇2 u + ρ−1 J × B + ρ−1 f

(1.43)

=

B · ∇u +

=

0

1 2 ∇ B µσ

(1.44) (1.45)

We can write these equations under a non-dimensional form by the following substitutions: u → U u, B → B0 B, ∇ → L−1 ∇, t → LU −1 t, J → σU B0 J, p → ρU 2 p. We obtain: ∇·u ∂u + u · ∇u ∂t ∂B + u · ∇B ∂t ∇·B

= 0

(1.46)

= −∇p + Re−1 ∇2 u + N J × B

(1.47)

−1 2 = B · ∇u + Rm ∇ B

(1.48)

= 0

(1.49)

We see that, in a given geometry, we can characterize MHD flows by merely three dimensionless groups: the Reynolds number Re, the interaction parameter (or Stuart number ) N , and the magnetic Reynolds number Rm . The definition and physical meaning of these parameters is discussed below. The Reynolds number is a non-dimensional estimate of the ratio between convective and viscous forces in the Navier-Stokes equation. An order-ofmagnitude estimate yields: Re =

U 2 L−1 UL O (u · ∇u) = = O (ν∇2 u) νL−2 U ν

(1.50)

This is the only parameter in the hydrodynamic Navier-Stokes equations. If the Reynolds number is small, small-scale fluctuations can not overcome the dissipative action of the viscous forces, and will quickly be damped. This results in a homogenized flow in which only slow variations of the velocity field are possible. At large Reynolds number however, small-scale fluctuations can persist and grow due to the increasing impact of the (non-linear) convective term. This will give rise to a seemingly random behavior that is characterized by a large range of spatial temporal scales, a state known as turbulence. The analogous ratio between convective and diffusive terms in the magnetic induction equation is known under the name magnetic Reynolds number, and is defined as: O(u · ∇B) U L−1 B Rm = = µσ = µσU L (1.51) (µσ)−1 ∇2 B L−2 B 10

1.3. THE QUASI-STATIC APPROXIMATION

11

The phenomenology of MHD systems will depend strongly on the value of Rm , as it is the only parameter governing the induction equation. When Rm is small, magnetic field fluctuations relax quickly; velocity inhomogeneities, which are at the origin of these fluctuations, hardly affect the magnetic field. Such flows have a dissipative nature; the kinetic energy of the fluid is transformed into heat due to Joulean dissipation. Small values of the magnetic Reynolds number are typical for manmade flows, like the ones encountered in laboratory and industrial processes. We will analyse the case of Rm ≪ 1 in more detail in the following section. High values of Rm on the other hand are typical for large-scale terrestrial or astrophysical flows. One may think of the motion of the liquid core of the earth (Rm ≈ 104 − 105 ) or of astrophysical processes like sun 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 electromagnetic and inertial forces. Its definition reads: N=

σB02 L ρU

(1.52)

It is also instructive to interpret this parameter as the ratio between two time scales. On the one hand, there is the Joule damping time, τJ , which is the typical time needed by the Lorentz force to damp a vortex. It is given by: τJ = ρ/σB02 . On the other hand, we have the eddy-turnover time τe which is the 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=

τe τJ

(1.53)

At last, we will introduce a fourth parameter known as the Hartmann number, which is a hybrid of Re and N . r √ σ (1.54) M = B0 L = N Re ρν 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 negligible.

1.3 1.3.1

The quasi-static approximation Simplified equations for Rm ≪ 1

In the previous section, we found that the induction equation contains only one non-dimensional group: the magnetic Reynolds number Rm . For most 11

12


terrestrial, laboratory and industrial flows, this parameter is typically small compared to one. The assumption that Rm is vanishing, allows for a substantial simplification of the MHD equations [Rob67a, Dav01]. Consider therefore equation (1.41), in which we decompose the magnetic field in a uniform, stationary, externally imposed part Bext and a fluctuating part b: ∂b 1 2 + u · ∇b = Bext · ∇u + b · ∇u + ∇ b ∂t µσ

(1.55)

An order-of-magnitude estimate of the convective and diffusive terms gives: O (b · ∇u) U bL−1 O (u · ∇b) = µσU L = Rm = = O ((µσ)−1 ∇2 b) O ((µσ)−1 ∇2 b) (µσ)−1 bL−2

(1.56)

where b is a typical scale of the fluctuating magnetic field. We can thus neglect the second term on both the right- and left-hand side of (1.55). This leaves us with: ∂b 1 2 = Bext · ∇u + ∇ b (1.57) ∂t µσ This is a diffusion equation with a source term due to gradients in the velocity field. We can now pursue our analysis by considering the time scales τ associated with both terms on the right-hand side. The ratio between these is: τ (µσ)−1 ∇2 b L2 µσ = Rm (1.58) = τ (Bext · ∇u) LU −1 This means that the fast response of the diffusion term makes the magnetic field fluctuations adapt quasi-instantaneously to (slower) variations due to the flow. As such, we end up with the following static equation for the induced magnetic field: 1 2 ∇ b=0 (1.59) Bext · ∇u + µσ An order-of-magnitude estimate of this equation leads to the following result: O(b) = µσU L = Rm O(Bext )

(1.60)

We find thus that, in the limit of vanishing Rm , the induced magnetic field is negligible with respect to the externally imposed one. We can use this last result to formulate the quasi-static approximation in a different, but equivalent way. To this end, we use the pre-Maxwellian form of Ampère’s equation (1.38) to eliminate b from (1.59): 1 ∇ × u × Bext − ∇ × j = 0 σ 12

(1.61)


13

In this expression, the symbol j denotes the current density induced by the flow. Using Helmholtz’s decomposition (A.22), we may ‘uncurl’ this equation; this results in: 1 (1.62) u × Bext − j = ∇φ σ 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 is caused by an external electric field Eext = −∇φext . Together with (1.62), we obtain. J = σ −∇ φ + φext + u × Bext (1.63)

Comparison with Ohm’s law (1.32) teaches us that the electric field E in the quasi-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-conservation law (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 up with a Poisson equation for the electrical potential: ∇2 φ = ∇ · u × Bext (1.65) Upon the rescaling φ → φU LB0 , the full non-dimensional system of quasi-static MHD equations becomes: ∇·u = ∂u + u · ∇u = ∂t ∇2 φ =

0

(1.66)

−∇p + Re−1 ∇2 u + N (−∇φ + u × Bext ) × Bext(1.67) ∇ · u × Bext (1.68)

From now on, we will drop the superscript ext, and denote the external magnetic field as B. This variable is now however to be considered as an imposed parameter, and not as an unknown of the system (1.66-1.68). By using a scalar potential instead of the magnetic field fluctuations, we reduce the ‘electromagnetic’ unknowns from three to one. Furthermore, the magnetic field is infinitely extended, even if the electric currents are localized in space; the induction equation should thus in principle be also solved outside the domain of interest. Therefore, for many situations of practical interest electric boundary conditions 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 determined until suitable boundary conditions for the electric potential have been defined. In general, one should include the wall into the solution domain of the 13

14


σ Γ

tw σw

n

Γ’

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 conditions at the fluid-wall interface. These state that the normal component of the current across the interface and the tangential component of the electric field along the interface should remain constant. Furthermore, we assume that there is no contact resistance between the fluid and the wall, so that the potential is continuous across the interface. If we associate the subscript w with wall variables (see figure 1.1) and take into account Ohm’s law, these conditions read: ∂φ ∂φw = σ σw (1.69) ∂n Γ ∂n Γ (∇τ φw )|Γ = (∇τ φ)|Γ (1.70) φw |Γ

=

φ|Γ

(1.71)

When, however, the wall thickness is small compared to the fluid domain, an approximate boundary condition can be derived [Wal81], which allows us to restrict the solution of equation (1.68) to the fluid domain. To this end, we start from the charge conservation equation in the wall, which can be written under the following form: ∂jn,w = − (∇τ · jw,τ ) ∂n

(1.72)

Here, we have split the current and the nabla operator in a component normal and a component tangential to the wall, i.e: jn,w = n · jw , jw,τ = j − jw,n n and ∂n = n · ∇, ∇τ = ∇ − n∂n . We now integrate this expression with the assumption that the potential does not vary up to the leading order of approximation in the wall. The underlying physical idea is that wall currents 14


15

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 in the insulating domain outside the fluid-wall system. Taking into account the aforementioned jump conditions, we eventually obtain: ∂φ σw tw = ∇τ · ∇τ φ (1.74) ∂n σ or in non-dimensional form: ∂φ = ∇τ · (c∇τ φ) ∂n

(1.75)

where the wall conductance ratio c is defined as: c=

σw tw σL

(1.76)

We now consider two limiting cases of this thin-wall condition. If the wall is perfectly conducting, σw and c tend to infinity. Wall currents should however remain finite, and this can only be achieved if the tangential electric field in the wall vanishes. In other words, the potential along a perfectly conducting wall is constant, and the thin-wall condition reduces to a familiar Dirichlet condition for the potential: φ=C (1.77) where C is an arbitrary integration constant. When the boundary contains different 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 penetrate from the fluid into the wall. We see indeed that, for c = 0, the thin-wall condition simplifies to a homogeneous Neumann condition for the electric potential: ∂φ =0 (1.78) ∂n

1.3.2

Phenomenology of the quasi-static regime

In this subsection, we will give an overview of the most relevant phenomena emerging in conducting flows subjected to a static and uniform magnetic field. In a first stage, we will disregard boundary effects and concentrate on homogeneous flows. These are flows whose statistical properties are invariant under translation. We will thereafter study the effect of boundaries in the context of straight laminar channel and duct flow. Although not completely generic, it will allow us to introduce the most common boundary layer types. 15

16


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 interaction parameter N = 10. Snapshots of the kinetic energy contours after 2.85 (left), 10.2 (center) and 26.8 (right) Joule damping time units τJ . Figure taken 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 magnetic field. The most prominent effect is the emergence of anisotropy, i.e. a loss of statistical invariance with respect to rotation. More specifically, all variations along magnetic field lines tend to be suppressed, while inhomogeneities in directions perpendicular to the field are hardly affected. We can explain this property by a heuristic argument developed by Davidson [Dav97]. We consider therefore the evolution of the total energy and field-aligned angular momentum of a quasi-static MHD flow. For the sake of simplicity, we will neglect viscous effects; this high Reynolds number approximation implies that we are considering flows which are highly turbulent. The energy balance can be obtained by taking the scalar product between the velocity u and equation (1.67). After some mathematical manipulation, we obtain: Z

Ω

∂t

1 ρ u2 2

1 2 u p + ρu · dS = dV + 2 ∂Ω I Z 1 2 − φJ · dS J dV − σ Ω ∂Ω I

(1.79)

If we make abstraction from boundary terms, we see that the total kinetic energy of the flow is a monotonically decreasing function of time. The mechanism driving this loss of kinetic energy is Joulean dissipation. Newton’s laws on the other hand tell us that the rate of change of global angular momentum of a body equals the net torque that is exerted on that 16


17

body. Here, the only force contributing to this torque is the Lorentz force. Z Z d ρx × u dV = x × (J × B) dV (1.80) dt Ω Ω After some tedious algebra, we can rewrite this as: I Z Z d ... · dS σ ((x × u) × B) × B dV + ρx × u dV = dt Ω ∂Ω Ω

(1.81)

Again disregarding boundary effects, the final result shows us that the total angular 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 maintained, and that the flow cannot come to rest. The only possible way to satisfy both constraints is that the flow organizes itself in such a way that the Joule dissipation, and thus the electrical currents tend to zero. Thus, after some initial 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 nature of a boundary layer in a laminar channel and square duct flow. We will take x as the flow direction, so that all derivatives with respect to x vanish, with exception from the pressure gradient needed to drive the flow. The magnetic field is defined as B = B0 1y . 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 all z-dependencies, except for the potential, since we cannot exclude the presence of a spanwise-orientated induced electric field. We have thus u = u(y)1x , with u and φ obeying the following set of equations: ∂p ∂φ ∂2u − + ρν 2 + σB0 − B0 u = 0 (1.84) ∂x ∂y ∂z ∂2φ = 0 (1.85) ∂y 2 17

18


Figure 1.3: Channel (a) and duct (b) flow geometry. The Hartmann walls have a 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 =0 ∂y 2

(1.86)

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

= Az + B 1 ∂p cosh(M y) = − M −2 + σB0 A 1− ρν ∂x cosh(M )

(1.87) (1.88)

Here, A and B are integration constants, and M is the Hartmann number as defined in (1.54). The choice of B is arbitrary, but A is determined by the fact that the total current in the combined fluid-wall domain should integrate to zero. The velocity profile is then: 1 cosh(M y) 1+c 1 ∂p u= 1− (1.89) M cM + tanh(M ) ρν ∂x cosh(M ) We recall that c stands for the wall-conductance ratio, defined in (1.76). In figure 1.4, we show the shape of the velocity profile for different values of the Hartmann number. It consists of an extended, flat core, and thin, exponential boundary layers, which are called Hartmann layers. As M increases, their thickness decreases as M −1 . This illustrates again that, far enough away from boundaries, the main effect of the magnetic field is to suppress variations along magnetic field lines. 18


19

1.5

u/U b

1

0.5

0 −1

−0.5

M =0 M = 10 M = 100 0

y

0.5

1

Figure 1.4: Sketch of the velocity profile of a laminar MHD channel flow for several values of the Hartmann number. The normalization velocity Ub is defined as the average velocity.

The scaling of the velocity magnitude with the Hartmann number can be understood by the following arguments. In a channel with conducting walls, φ is zero, and the magnitude of the Lorentz force density is σuB02 . In the high Hartmann number regime, this force dominates the core flow; hence, the core velocity should scale as u ∝ −∂x p(σB02 )−1 ∝ −∂x pM −2 (ρν)−1 . If, on the other hand, the walls are perfectly insulating, a spanwise potential gradient will be induced which counteracts the effect of the term u × B. The effect of the Lorentz force density is now to brake the core flow, and to accelerate the bounday 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 momentum balance between y = −1 and y = 1. We find: ! Z 1 2 Z 1 ∂u ∂u ∂ u ∂p dy = ρν (1.90) dy = ρν − 2 ∂y y=1 ∂y y=−1 −1 ∂y −1 ∂x We see that in this case, the pressure gradient has to compensate for viscous 19

20


Figure 1.5: Laminar MHD flow in a straight duct with perfectly insulating walls: velocity along the duct centerline parallel (left) and perpendicular (center) 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 steep profile of the boundary layers, the velocity magnitude is an order-of-magnitude M 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 additional pair of walls located at z = ±1. These walls are called side walls and their respective boundary layers side layers. Analytical solutions are now only possible for a few specific combinations of the wall conductivity c, and we will limit ourselves 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 velocity profiles for three different combinations of c. In figure 1.5, all the walls are perfect insulators. A solution for this problem was provided by [She53]. The interaction between the flow and the magnetic field drives a current in z-direction, which brakes the flow. The current lines can however not enter into the insulating side walls, so that a potential difference is induced, which makes the current lines bend and close through the Hartmann layers. In the side layers, the current is almost parallel to the magnetic field so that the Lorentz force is weaker there. It turns out that these layers have a typical 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 case driven by the same pressure gradient. In the second case (figure 1.6), all walls are perfect conductors. The induced current closes its loops preferably through the walls because of their lower resistance. The current lines enter the walls perpendicularly, but are slightly deviated in the side layers; hence the component of the current perpendicular to the magnetic field in these zones is somewhat smaller than in the core. This gives rise to small overspeed zones above a certain threshold in the Hartmann 20


21

number. The ratio between the amplitude of the velocity in the side layers and the core scales as O(M 0 ). The core velocity itself scales as M −2 ∂x p.

Figure 1.6: Laminar MHD flow in a straight duct with perfectly conducting walls: velocity along the duct centerline parallel (left) and perpendicular (center) 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 conducting Hartmann walls and perfectly insulating side walls. This case was first studied by Hunt [Hun65]. Compared to the insulating case, the magnitude of the currents can be larger, since the current lines can form closed loops by entering into the Hartmann walls, which provide a path of much lower resistivity. In the side layers, the current flows parallel to the field towards the perfectly conducting Hartmann walls. This means that the Lorentz force is vanishing in these regions, and that the velocity is much higher than in the core. The ratio between the amplitude of these jets and the amplitude of the core flow scales as O(M ). Since their thickness scales as M −1/2 , we find that, at high Hartmann number, the mass flow rate carried by the core is negligible with respect to the one in the side layers. We also note that the flow in the core may become reversed 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 realization of thermonuclear fusion as a reliable and sustainable energy source. The reaction between a deuterium and tritium core can only take place if both reactants are completely ionized. These plasmas are characterized by very high conductivities and densities. Intense magnetic fields are used to confine these plasmas within the reaction vessel. While conventional MHD does not accurately describe plasma-related effects, it comes into play in the context of thermonuclear fusion when the flow of liquid metals in the so-called blankets is considered. 21

22


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

These blankets are multifunctional: in the first place, they should absorb the neutron flux and convert the kinetic energy of the neutrons into heat, which can then be used to drive a classical turbine process. Liquid metals are candidate coolant liquids since they can be operated at high temperature and have high thermal conductivities [B¨ 07]. The second function of the blankets is to protect the magnetic field coils from intense, damaging neutron radiation. In more advanced blanket designs (called self-cooling blankets), the liquid coolants also have to provide the tritium needed for the fusion reaction. This is possible if the coolant contains lithium. Liquids that are considered are pure lithium or eutectic lead-lithium alloy. The combination of their material properties at operating conditions (750 K) and the ambient conditions in a fusion environment (B0 =10 T, L = 0.05 m, U = 0.5 m/s) are such that the non-dimensional parameters typically have the following order of magnitude: M = 104 − 105 , N = 103 − 105 , Re = 104 − 105 . Given the high values of M , the velocity profile in these components often takes the form of laminar, inviscid, inertialess core flows, surrounded by different types of boundary layers due to the presence of solid walls and geometrical discontinuities. On the other hand, the wall conductance ratio of some blanket components is such that strong jets occur in the side layers, and these may exhibit (quasi two-dimensional) turbulent behavior. The main challenge for this type of application is that we have to compute the flow in fairly complex elements like expansions, bends, manifolds, helical vanes, etc (see e.g. figure 1.8). The high value of the Hartmann number makes the flow morphology highly anisotropic, so that the problem is badly conditioned for conventional computational fluid dynamic. On the other hand, the strength of the interaction parameter is such that turbulent behavior is 22


23

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

not a major issue here. Furthermore, many other complex effects still have to be 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 measuring flow 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 subjected to a wall-normal magnetic field, a potential difference is induced between the channel walls. If the channel walls are good conductors, this potential difference is directly proportional to the flow rate [HS65]. However this technique is difficult to apply in high-temperature industrial melt flows (aluminium, glass), because the measuring electrodes suffer from corrosion under ambient conditions. Recently, a new technique, termed Lorentz force velocimetry [TVK06] was introduced, which has the benefit of being a non-contact technique. Its principle is the following: when a flow is exposed to a permanent magnet, the magnet will exert a force on the flow. By virtue of the action-reaction principle, the magnet will undergo an equal but opposite drag force. The flow rate can then be inferred from the measurement of the force acting on the magnet system. 23

24


Magnetic damping in metallurgical processes We have already illustrated that a magnetic field can suppress the motion of a liquid metal flow under quasi-static conditions. The mechanism responsible for this is Joule dissipation. This property is often exploited in metallurgical processes like continuous steel casting or Czochralski growth of semiconductors. This motion is 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 composition of the flow, which may result in a degradation of the metallurgical structure of the solid.

24

Chapter 2

Numerical framework “Life can only be understood backwards; but it must be lived forwards.” S¨ oren Kierkegaard In the previous chapter, we have described the partial differential equations which govern the physics of quasi-static magnetohydrodynamic flow. Due to their non-linear nature, exact solutions, obtained with analytical methods, are not readily available, with the exception of a few problems in simple geometries and with trivial initial conditions. If we want to make predictions for flows of practical interest, we have to recourse to numerical methods, i.e approximate a set of partial differential equations by a system of algebraic equations. In other words, we must approximate continuous functions of space and time with a finite 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 finitedifference methods on the other hand, the variables are only represented in a finite number of points. We will adopt a third approach here, which is called the finite-volume method (FVM), and which is by far the most popular method in the field of computational fluid dynamics, with pioneering studies performed by McDonald [McD71] and MacCormack and Paullay [MP72] for the case of the two-dimensional Euler equations. Shortly afterwards, Rizzi and Inouye [RI73] applied a FVM approach to solve the Navier-Stokes equations (including viscous effects). The basic building block of FVM is a small control volume (CV), in which the solution domain is divided. The discretization procedure is then based on the integral of the conservation equations over this CV. In the first section, on spatial discretization, we will introduce the FVM method 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 system of ordinary differential equations with respect to time. The algorithms used to advance this system in time are the subject of the second section. The major issue here will be how to maintain (discrete) conservation of mass, since the time derivative does not appear explicitly in the mass conservation equation. In these first two sections, we will leave out of consideration all aspects related to the numerical computation of the Lorentz force. This subject will be treated in the third section of this chapter. The combined space-time-discretization procedure will eventually lead to a system of algebraic equations. In the fourth and last section, we will give an overview of the different iterative methods which we have used to solve such systems. The methods presented in this chapter have been implemented in an early version of the unstructured parallel finite-volume code YALES2 (Yet Another LES Solver) [Mou10]. This is a versatile numerical solver for a broad range of multi-physics flow problems (combustion, magnetohydrodynamics, multi-phase flows, etc.). It originates from, and is maintained at CORIA. In the context of the present work, the following contributions to this code were provided: • A consistent formulation of a compact stencil for the Laplacian operator on unstructured grids (see subsection 2.1.6). • The implementation of implicit time-advancement methods for the momentum equation (subsection 2.2.2). • The implementation of a convective boundary condition for outflow boundaries (subsection 2.2.4). • The development of a module for the quasi-static MHD equations; this includes all aspects treated in section 2.3. • The implementation of a Krylov subspace method and the coupling of the code with an algebraic multigrid solver for the solution of the Poisson equations for pressure and potential (see subsections 2.4.3-2.4.2).

2.1 2.1.1

Spatial discretization Basic principle of the finite-volume method

To introduce the finite-volume method, we consider a polyhedral control volume Ω, bounded by a number of faces f a, and integrate the momentum equation 26

2.1. SPATIAL DISCRETIZATION

27

over this control volume: Z Z 1 ∂u 1 −∇ · (uu) − ∇p + ν∇2 u + fb dV dV = ρ ρ Ω ∂t Ω I Z p 1 fb dV = −uu − 1 + ν∇u · dS + ρ ρ ∂Ω Ω Z XZ p 1 = −uu − 1 + ν∇u · dS + fb dV ρ ρ Sf a Ω

(2.1)

fa

The surface integral term represents a transport of momentum towards neighbouring control volumes, and illustrates the conservative nature of the NavierStokes equations: the surface force terms do not create or annihilate momentum, they only redistribute it. In FVM, we approximate equation (2.1) for a large set of adjacent control volumes. 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 locations within the control volume and its neighbours. The main advantage of this approach is that the discretized surface forces inherently conserve (discrete) momentum. If we sum the expression over a set of adjacent CV’s, each of the flux terms will appear twice, but with a different sign, so that these contributions cancel each other; the remaining contributions to the net rate-of-change of 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 dimensions) or polyhedra (in three dimensions) which fill completely a well-defined domain in space. We will refer to these constituting bricks as elements, to their vertices as nodes and to their mutual interfaces as element faces. Moreover, we will restrict ourselves to the two types of two-dimensional and four types of three-dimensional elements shown in figure 2.1. We can distinguish between structured and unstructured meshes. The former can be mapped on simple data structures. A subgroup of structured meshes are the cartesian meshes. In this case, the grid points are located along orthogonal grid lines. Despite the huge increase in spatial resolution that has been reached over the past few decades, the generation of an appropriate mesh remains a critical issue for the simulation of fluid flow in complex geometries. Badly designed meshes can give rise to large numerical errors, which may completely destroy the physical content of the simulation. We shall later specify which criteria should be taken care of. Within one CV, we have some freedom on the arrangement of the variables. In staggered meshes, the velocities are stored at the CV faces, and the other quantities (pressure, electric potential, transported scalar quantities) are 27

28


defined at the CV nodes. This arrangement is attractive because it does not lead to spurious pressure oscillations, known as the checkerboard problem (this will be explained in more detail in section 2.1.6). Furthermore, staggered arrangements do not require ad-hoc boundary conditions for the pressure, and can simultaneously conserve mass, momentum and kinetic energy for an inviscid flow [HW65]. However, for three-dimensional simulations and unstructured meshes, a staggered definition of the variables becomes very complex, since one needs 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 is given. In the cell-center-based methods, the elements themselves are the basic building 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 of control volumes is created, which are centered around the vertices of the elements. In [HMI06], it was found that the more accurate results for a turbulent channel flow were obtained with a vertex-based formulation. However, this single study does not allow us to draw a conclusion on the superiority of the vertex-based approach for other geometries. Figure 2.2 illustrates how the dual volumes, which we will call nodal volumes, are built up out of a number of simplexes, i.e. triangles (called subtriangle and denoted subtri) in two, and tetrahedra (called subtetrahaedron and denoted subtet ) in three dimensions. For a two-dimensional grid, each triangle is obtained by connecting the node under consideration, the barycenter of the element and the midpoint of an edge connected to this node. Similarly, in three 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 boundary element faces to their respective nodes. In two dimensions, the boundary segment is just split in two; for three-dimensional meshes, we divide the boundary triangles or quadrilaterals in subtris like explained before, and associate a subtri with a boundary node if the node is one of its vertices. The total volume of the nodal CV’s can be computed as the sum of the volume of its subtris/subtets st. 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

X st

xst · Sst

(2.2)

Furthermore, the surface of the CV is formed by the union of all subtri/subtet faces opposing the node. Since this surface is a closed, the subtri/subtet face 28


29

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

normals satisfy: X

Sst = 0

(2.3)

st

As mentioned before, we can express the different conservative terms in the Navier-Stokes equations as a surface integral over the surface of the CV. This requires that we dispose of (approximated) values of the variables at the surface location. These are obtained by a well-chosen interpolation of the values at the nodes. It will appear useful to assign distinct portions of the CV surface to pairs of nodes, and we denote such a surface patch associated with a pair a pair face (in contrast to element faces). Hence, the surface integral for a certain CV can 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. In sparse stencils, we consider only those pairs which are connected by a physical element edge, whereas in dense stencils, we also construct pairs which are only virtually connected, like pairs connected by diagonals. The attribution of the different 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 the pair, 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 surface patch is assigned to the node pair consisting of the hidden and purple node depicted in figure 2.4(e), since the midpoint of the edge between these nodes is a vertex of the triangle which defines Sst . In three dimensions (figures 2.3(b) 29

30


Figure 2.2: Construction of the control volumes: (a) Subtri of a node pair in a triangular element, (b) Control volume of a node in a mixed triangular/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 be computed: • In triangular and tetrahedral elements, there are no diagonals, and the attribution of CV surface patches to node pairs is the same for both stencils, and is shown in figure 2.3. • Within one quadrilateral element, each node is physically connected to two 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, each of these surface patches is split in two, and one half of each normal is 30


31

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

assigned to the diagonal pair involving the node under consideration; the resulting 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 split in 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 quadrilateral element surface under consideration. The same is true for pyramids; these are not shown here, because such elements have not been used further in this work. • For hexahaedra, we should consider space diagonals as well as diagonals which belong to element faces. Each of the exterior surface triangles shown in figure 2.2 (d) is bisected, so that we obtain four surface patches for each ‘sparse’ pair face (see figure 2.4(e)). These are attributed as follows: one to the given physical pair, one to the space diagonal involving the given node, and one to the diagonals of the faces to which the given edge belongs. We will use the following notation. The pair face normal to a pair of nodes i, j is 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 term over the CV volume. For an arbitrary function g, we can write the volume 31

32


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

32


33

integral as a Taylor expansion in the vicinity of the CV node i: Z Z g|i + (∇g)|i · (x − xi ) + O(∆2 ) dV g dV = Ω

(2.4)

Ω

Here, ∆ is a length scale which is representative for the size of the CV. We approximate the average of g over the CV by its nodal value, i.e.: Z g dV ≈ Vi g|i (2.5) Ω

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 the CV center of mass is located at the node position. This is only true if the mesh is locally uniform. We can significantly reduce the discretization error by avoiding large size jumps between adjacent elements. If the physics require us to locally refine the grid, this refinement should be obtained by a smooth decrease in grid size towards the region of interest. Note also that, in boundary CV’s, the node is on the surface. For such CV’s, this approximation can never be 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): I Z g dS (2.6) ∇g dV = ∂Ωi

Ωi

We can expand the left-hand side in a Taylor series around the location of the CV node, and retain the first-order approximation like in the previous section. For nodes not on the boundary, the surface integral can further be developed as a sum over the element pair faces (i, j) which contain the CV node i under consideration: I X Z g dS = g dS ∂Ωi

j∈π(i)

≈ ≈

X

Si,j

g|(i,j) Si,j

j∈π(i)

X g|i + g|j Si,j 2

(2.7)

j∈π(i)

Here, the vectors Si,j represent the normal vectors of the surface patches of the CV of node i, attributed to the pair (i, j). By construction, Si,j = −Sj,i 33

34


If the node is on the boundary, the surface integral in the right-hand side of (2.6) can be completed by adding a similar summation over the boundary faces to the right-hand side. R The approximation Si,j g 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 at the barycenter of the surface Si,j . See e.g. the pair (P, N ) in figure 2.5(a). We can again argue that this first-order discretization error will disappear if the mesh is locally uniform. Moreover, for quadrilateral and hexahedral grids, the surface integral approximation is exactly second-order accurate when a dense stencil is used. Consider therefore the two-dimensional situation sketched in figure 2.5(b), where we show how the surface patches of the (shaded) nodal volume of node P are distributed among node pairs for a dense stencil. The color 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 indicates quantities in the dense formulation. The notation SX→Y denotes the normal vector on the surface patch limited by the points X and Y . We now start from the expression for the discrete gradient (2.7) in the dense formulation and reformulate it as follows: X Z

j∈π(P )

SP,j

g dS

≈

=

de de g|(P,W ) Sde (P,W ) + g|(P,N W ) S(P,N W ) + g|(P,N ) S(P,N ) de +g|(P,N E)Sde (P,N E) + g|(P,E) S(P,E) + ... 1 g|(P,W ) + g|(P,N W ) S(P,W )→(P,N W ) 2 1 + g|(P,N W ) + g|(P,N ) S(P,N W )→(P,N ) 2 1 + (g|P,N + g|P,N E )S(P,N )→(P,N E) 2 1 + (g|P,N E + g|PE )S(P,N E)→(P,E) + ... 2

Each term in the last expression can be interpreted as the approximation to a surface 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 secondorder accurate, since the control volumes have been constructed such that pair centers (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 quadrilateral elements in the dense formulation. We will denote the discretization stencil for a spatial differential operator by a calligraphic letter. For instance, for the gradient operator, we have the 34


35

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

stencil G: G (g) |i =

2.1.5

1 X g|i + g|j Si,j Vi 2

(2.8)

j∈π(i)

The divergence operator

To obtain a stencil for the discrete divergence of a vector u, we can adopt the same approach as we did for the gradient. The stencil reads: D (u) |i =

1 X u|i + u|j · Si,j Vi 2

(2.9)

j∈π(i)

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 advected and advecting velocity. The advected velocity u is a vectorial quantity which is defined at the CV nodes; the advecting velocity U on the other hand is defined at the pair face (i, j), and represents the mass flux through the pair face surface (U is thus a scalar quantity despite its name). The discrete convective divergence C in the Navier-Stokes equations may then be computed with the following stencil: 1 X u|i + u|j C (u) |i = U |(i,j) (2.10) Vi j 2 Both velocities are of course not completely independent. More details on their mutual interplay will be given in section 2.2. 35

36

2.1.6


The Laplacian operator

The Laplacian operator appears twice in the quasi-static MHD equations. Once in the viscous term in the momentum equation, and once in the Poisson equation for the electric potential. In the next section, we will see that the time marching algorithm also requires us to solve a Poisson equation for the pressure. The same discretization scheme for the Laplacian operator will be used for the three cases. The most straightforward way to compute the Laplacian of a function g would be to apply successively the gradient and divergence operator. This leads however to a non-compact stencil, which may be a source of non-physical behaviour when solving a Poisson equation. We can illustrate this for a onedimensional grid with equidistant spacing ∆: ! ∂g 1 ∂g ∂ 2 g = + O(∆2 ) − ∂x2 2∆ ∂x ∂x i

i+1

i−1

The first-order derivatives can be discretized similarly: ∂g 1 (g|i±2 − g|i ) + O(∆2 ) ± = ∂x i±1 2∆

Eventually, we obtain: ∂ 2 g 1 = (g|i+2 + g|i−2 − 2g|i ) + O(∆2 ) ∂x2 i 4∆2

The major problem of this stencil come from the fact that it decouples odd-andeven numbered nodes. As such, it cannot accurately represent the Laplacian of 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-numbered and -1 at even-numbered nodes (see figure 2.6), is 0. Inversely, we can add these high-frequency components in an arbitrary amount to the solution of a Poisson equation for a given right-hand side; this gives rise to non-physical solutions. In mathematically more precise terms, these fluctuations can arise because the stencil above has a non-trivial kernel. This phenomenon is known under the name checkerboard problem. Besides that, this discretization stencil is also troublesome when it comes to aspects of implementation. The data structures available (pairs, elements etc.) only allow to connect a node with its nearest neighbours. Since this scheme involves nodes that are further away, the computation of the Laplacian should be performed in two steps, and is computationally more expensive. We can circumvent these problems by constructing a scheme on a compact stencil. To this end, we start again from Gauss’s theorem, applied to the nodal 36


37

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

volume Ωi of a given node i: I Z ∇2 g dV = Ωi

∂Ωi

∇g · dS

=

X Z

j∈π(i)

≈

X

j∈π(i)

Si,j

∇g · dS

(∇g) |(i,j) · Si,j

(2.11)

We should now find a compact discretization of the projection of ∇g along the face normal at the CV faces. If the face normal is parallel to the edge connecting its 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 ||Si,j || + O(∆2 ) |xj − xi |

(2.12)

This expression is symmetric under exchange i ⇋ j. For a one-dimensional equidistant grid, we obtain: ∂ 2 g 1 ≈ (g|i+1 + g|i−1 − 2g|i ) + O(∆2 ) (2.13) ∂ 2 x i ∆2

If we apply this stencil to the function shown in figure 2.6, we find that the discrete Laplacian of g at node i takes the value −4∆−2 g|i . This illustrates that these high-frequency components do not belong to the null space of this stencil. On unstructured meshes however, a CV face normal is in general not parallel to the edge between its adjacent nodes (see l.h.s. of figure 2.7). This means that 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 to the pair edge xj − xi , i.e.: Si,j · (xj − xi ) || ⊥ (xj − xi ) + S⊥ (2.14) Si,j = Si,j + Si,j = i,j |xj − xi |2 37

38


Figure 2.7: Illustration of the geometrical concepts used in various discretization 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 patches associated to a certain node. The sum of the normal vectors of these patches yield Sel .

Furthermore, the components of the gradient perpendicular to the pair edge are obtained through interpolation between the values of the gradient at both pair nodes: Gp,Zwart (g)|(i,j) · Si,j

g|j − g|i || (xj − xi ) · Si,j |xj − xi |2 1 (G(g)|i + G(g)|j ) · S⊥ i,j 2

= +

(2.15)

This approach removes the odd-even decoupling and is consistent, i.e. the discretization 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 of a given field φ at a given node i involves the values of φ at nodes which are not directly connected to i. Therefore, this stencil is less suited for large-scale computations on unstructured meshes. Ham et al. [HMI06] define the approximation of the gradient at a CV subtet face as the solution of the following 3-by-3 system:  = Gp,Ham (g)|(i,j) · (xj − xi )  g|j − g|i g|f a − g|i,j = Gp,Ham (g)|(i,j) xf a − x|(i,j) (2.16)  g|el − g|i,j = Gp,Ham (g)|(i,j) xel − x(i,j)

The indices el and f a represent an averaging over all the nodes of the element 38


39

and face under consideration. The information used in this stencil is restricted to one element, so that the obtained scheme is compact. The use of this method is however restricted to sparse discretization schemes. For dense schemes, the system (2.16) can become singular for the non-physical edges. On a regular hexahedral mesh for instance, the first and third equation of this system are the same for the pair associated with a space diagonal. In this work, we propose an approach which combines ideas of both of the previous stencils. We start again from a decomposition of the gradient in a component 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’s theorem to the element el to which the pair under consideration belongs. As such, all nodes involved in the stencil are directly connected to both pair nodes; this scheme is thus also compact. We have: Z

Ωel

∇g dV =

XZ ef

Sef

g dSef ≈

X ef

1

X

Nef n

k

g|k

!

Sef

(2.17)

Here, we have approximated the surface integral over an element face ef with first-order accuracy by taking a simple average over the nodes k of ef . We can now associate an element surface patch Sel k with each node k of the element: Sel k =

X X Sef Nef n

(2.18)

ef ef n

From figures 2.4 and 2.7 (r.h.s.), it is clear that the surface represented by Sel k 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-based gradient as: 1 X Ge (g)|el = g|k Sel (2.19) k Vel k

Eventually, we obtain the following result for the projection of the element pair-face gradient along Si,j : Gp (g)|(i,j) · Si,j =

X g|j − g|i || (xj − xi ) · Si,j + Ge (g)|el · S⊥ i,j|el 2 |xj − xi |

(2.20)

el

We now want to write the discretization stencil of the Laplacian of g at a node i as a sum over the element pairs involving i. Therefore, we introduce the || || || notation wi,j = (xj − xi ) · Si,j /||xj − xi ||2 . Note that wi,j is invariant under 39

40


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

1 X Gp (g)|i,j · Sij Vi j∈π(i)   X X 1  X ||  Ge (g)|el · S⊥ wi,j (g|j − g|i ) + = i,j|el Vi el j∈πel (i) j∈π(i)  1  X || = wi,j (g|j − g|i ) Vi j∈π(i)  ⊥ X X X Sel k · Si,j|el  (2.21) (g|k − g|i ) + Vel =

el j∈πel (i) k∈πel (i)

⊥ We now furthermore define wi,k =

P P el

j∈πel (i)

⊥ Sel k ·Si,j|el /Vel . Upon exchange ||

⊥ , we find of the dummy indices j and k, and introduction of wi,j = wi,j + wi,j eventually: 1 X L(g)|i = wi,j (g|j − g|i ) (2.22) Vi j∈π(i)

⊥ The factors wi,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 drawback of this is that we will not be able to take advantage of a number of fast Poisson solvers specially intended for symmetric systems. Nevertheless, this skewness correction is inevitable if one wants to avoid O(1) errors with a compact stencil on a random, unstructured mesh; this was also noted in [SGN08]. The formulation of the skewness correction term discussed above has been develped in collaboration with V. Moureau. Up to our knowledge, it has never been presented before.

2.1.7

Matrix representation of the discretization operators

We can represent the action of the spatial discretization operators on a given field g as a matrix-vector product. The elements of this vector, which we will denote 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. We will assume that the mesh is periodic, so that we don’t have to cope with complicating 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


41

convective divergence operator C can be represented as a sparse matrix C; the off-diagonal entries on positions (i, j) are zero if the nodes i and j do not form a 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 a time-advancement algorithm for the Navier-Stokes which ensures incompressibility at the level of the convective velocities. This means P that, for every node i, the following sum over all its pairs will be satisfied: j∈π(i) Ui,j = 0. If this property holds, the diagonal entries cii of C are zero. We can furthermore write ˆ where C ˆ is a skew-symmetric matrix, whose eigenvalues are purely C as V−1 C, imaginary. Similarly, the matrix G has a zero diagonal and its off-diagonal entries are gi,j = Si,j /Vi . The matrix L respresenting the Laplacian operator L has the same sparsity pattern as C. Referring to (2.22), we can write the non-diagonal elements lij representing a pair (i, j) as: wi,j lij = (2.24) Vi The diagonal elements of L are: lii =

1 X −wi,j Vi

(2.25)

j∈π(i)

ˆ Just like for the convective operator, we can introduce a matrix L = V−1 L. 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 stability properties of the time integration schemes discussed in the next section. In most canonical textbooks [FP02, Hir07], the analysis of numerical schemes is limited to the case of structured, equidistant meshes. In this work, we have developed our own, more generic approach, which also applies to unstructured meshes. To this end, we first introduce the Gershgorin circle theorem [Ger31], which allows to bound a priori the spectrum of an arbitrary matrix M, thus without computing explicitly the eigenvalues. This theorem reads as follows: Gershgorin circle theorem. Let M = [mij ] be a n × n complex matrix, and let Ri be the P sum of the moduli of the off-diagonal elements in the i-th row, i.e. Ri = j,j6=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 matrixP L learns us that the real part of all its eigenvalues is not larger than lii + j6=i |lij |. Since, by definition 41

42

lii = −


P

j,j6=i lij ,

we find that the real part of the eigenvalues of L satisfies: X ℜ(λ(L)) ≤ −lij + |lij | (2.26) j,j6=i

The expression in the right-hand side of this inequality is always larger than or equal to zero. It is only zero if all the lij have the same sign. These elements || ⊥ . are the sum of a main contribution wi,j and a skewness correction term wi,j The main contribution is always positive because the normal vector Si,j and the edge vector xj − xi always point in the same direction. It is less obvious whether the skewness correction is positive or negative; however, if the mesh is not 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 discretized Navier-Stokes equations as follows: ∂U = −CU + νLU − GP ∂t The total discrete kinetic energy can be computed as: X1 1 ui · ui Vi K = UT VU = 2 2 i

(2.27)

(2.28)

and the discrete equivalent of (1.17) is: ∂K = −UT VCU + νUT VLU + UT VGP ∂t

(2.29)

ˆ = VC is skew-symmetric, the first-term on the right-hand side of Since C this expression vanishes. The discretized convection term will thus mimic the energy-conserving property of its space-continuous counterpart. Moreover, if L is a negative semi-definite matrix, the second term of (2.29) will indeed represent a dissipation process. We can use this as a criterion to judge whether a mesh is acceptable or not. More specifically, and taking into account the above considerations, we will always require that all the off-diagonal elements of the discrete laplacian operator matrix L have the same sign, so that all its eigenvalues are located in the left-half of the complex plane. At last, the term involving the pressure in (2.29) is not in a quadratic form, and its analysis is thus 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 spatial differential 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 pressure and the velocity at the nodes are the unknowns. For a given node i, this differential equation is: ∂u|i = −C(u)|i + νL(u)|i − G(p)|i + ρ−1 fb |i ∂t

(2.30)

In this section, we will explain how we can compute these fields at an instant tn+1 = tn + ∆t, given the fields at a prior time tn . The major difficulty related to the advancement of these discrete NavierStokes equations is that the mass-conservation equation does not contain an explicit time-derivative if the flow is incompressible. The incompressibility constraint acts rather as a kinematic constraint on the velocity field, and couples implicitly pressure and velocity. Indeed, as we saw in the previous chapter, the pressure can be considered as an auxilary variable needed to maintain the incompressibility constraint. Fractional-step methods are without any doubt the most widespread technique to decouple the computation of the pressure from the advancement of the momentum equation. The advantage of such an approach is that the decoupled systems for p and u can be solved at a lower expense. We will show how this splitting can be achieved in the first part of this section. The second subsection is devoted to the different time integration schemes which we can use to advance the decoupled equations for the velocity. A major point of concern here is the stability of a scheme, i.e. does it prevent that numerical solutions become unbounded. A third aspect which we will address is whether the presented methods mimic the energy-conserving properties of their time-continuous counterparts. In the last subsection, we will explain how the boundary conditions can be enforced.

2.2.1

Fractional-step methods

In fractional step methods, we approximate the time-evolution equations based on a decomposition of the operators it contains. More specifically, we will isolate 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. This can 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 integration of the Navier-Stokes equations without the pressure term. u ⋆ | i − u n |i = −C(u)|i + νL(u)|i + ρ−1 fb |i ∆t 43

(2.31)

44


In the right-hand side of this equation, we have not specified the dependency of u on un and u⋆ . This aspect is proper to each time integration scheme. We will deal with various approaches in subsection 2.2.2. 2. The ‘new’ velocity un+1 is related to the intermediate one u⋆ through the following relationship: un+1 |i − u⋆ |i = −G(pn+1/2 )|i ∆t

(2.32)

Taking the divergence of this expression, and imposing the incompressibility constraint on un+1 , yields a Poisson equation for the pressure. As we have shown previously, the successive application of the nodal operators D and G leads to a stencil where the equations for odd and even nodes are decoupled. The compact operator L is based on the approximation of the (pressure) gradient at the CV faces. Therefore, we introduce the intermediate convecting velocity U ⋆ |(i,j) : U ⋆ |(i,j) =

u ⋆ |i + u ⋆ |j · Si,j 2

(2.33)

The pressure gradient at the faces can now be used to relate the ‘new’ convecting velocity U n+1 |(i,j) to U ⋆ |(i,j) : U n+1 |(i,j) − U ⋆ |(i,j) = −Gp (pn+1/2 )|(i,j) ∆t

(2.34)

We now want to impose mass conservation at the level of the convecting velocities, i.e., we want U n+1 |(i,j) to satisfy: ∀i :

X

j∈π(i)

U n+1 |(i,j) = 0

(2.35)

Combination of (2.33), (2.34) and (2.35) now leads to a Poisson equation for the pressure at the CV nodes based on the compact operator L, and can be expressed as L(pn+1/2 )|i =

1 X ⋆ 1 U |(i,j) D(u⋆ ) = ∆t ∆t

(2.36)

j∈π(i)

3. Eventually, both the nodal and velocity are corrected with the gradient of the new pressure: U n+1 |(i,j) u

n+1

|i

= U ⋆ |(i,j) − ∆tGp (pn+1/2 )|(i,j) ⋆

= u |i − ∆tG(p 44

n+1/2

)|i

(2.37) (2.38)


45

The pressure gradient correction for the convecting velocites U |(i,j) is consistent with the Laplacian operator used to solve the Poisson equation; they will thus satisify the divergence-free condition (2.35) up to machine accuracy. 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, the accuracy depends also on the time integration schemes used in the first step of the algorithm. These will be discussed in the following subsection. Finally, there is the accuracy of the boundary conditions for the pressure and the intermediate 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 continuous Navier-Stokes equations. Furthermore, the function p in the fractional-step method is not exactly equivalent to the physical pressure, and contains in fact correction 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 ∂pn+1/2 ∂n

=

un+1 |∂Ω

(2.39)

=

0

(2.40)

∂Ω

With this formulation, we can however not guarantee simultaneously that the tangential components of the pressure gradient are also zero. As such, the Dirichlet boundary condition on the tangential boundary velocity are not respected. 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 highlighted by Kim and Moin [KM85]. For their analysis, they considered u⋆ as the approximation to a fictitious velocity u† (tn+1 ) where the continuous function u† satisfies: ∂u† = −u† · ∇u† + ν∇2 u† + ρ−1 fb (2.41) ∂t with initial condition u† (tn ) = un . We now develop u† in a Taylor series: n ∂u† † n u (tn+1 ) = u + ∆t + O(∆t)2 ∂t = un + ∆t −u · ∇u + ν∇2 u + ρ−1 fb |n + O(∆t2 ) (2.42)

An expression for the ‘new’ velocity un+1 = u(tn+1 ) can be found in a similar way, by using the momentum equation (1.19) in stead of (2.41). n ∂u n+1 n + O(∆t2 ) u = u + ∆t ∂t = un + ∆t −u · ∇u + ν∇2 u + ρ−1 fb − ∇p |n + O(∆t2 ) (2.43) 45

46


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 first order accurate. Furthermore, we have also: n−1/2 ∆t ∂∇p n n−1/2 + O(∆t2 ) = ∇pn−1/2 + O(∆t) (2.45) ∇p = ∇p + 2 ∂t

Together with (2.44), we find that the following conditions are second-order accurate: u⋆ |∂Ω ∂pn+1/2 ∂n

= un+1 |∂Ω + ∆t∇pn−1/2

(2.46)

= 0

(2.47)

∂Ω

This leads to a slightly modified version of the fractional-step algorithm: ˆ from the ‘old’ velocity un , including 1. Compute the intermediate velocity u n−1/2 the ‘old’ value of the pressure p for the pressure gradient term: ˆ |i − u n |i u = −C(u)|i + νL(u)|i − G(pn−1/2 )|i + ρ−1 fb |i ∆t

(2.48)

ˆ |∂Ω = un+1 |∂Ω with boundary conditions u 2. Remove the old pressure contribution from u ˆ and require that the new pressure pn+1/2 projects the new convecting velocities U n+1 on a solenoidal field. This leads to a Poisson equation for the pressure, as previously discussed: u ⋆ |i

=

L(pn+1/2 )|i

=

ˆ |i + ∆tG(pn−1/2 )|i u 1 X ⋆ U |(i,j) ∆t

(2.49) (2.50)

j∈π(i)

The first of these equation shows us that the boundary values of u⋆ satisfy the second-order time-accurate boundary conditions (2.46) -(2.47). 3. Correct the convecting and convected velocities with the gradient of the new pressure pn+1/2 : U n+1 |(i,j)

un+1 |i

= U ⋆ |i,j − ∆tGp (pn+1/2 )|(i,j)

= u⋆ |i − ∆tG(pn+1/2 )|i

(2.51) (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 of the fractional step method which we have implemented in the code YALES2. 46


2.2.2

47

Time integration schemes for the momentum equation

We can represent the first step of the fractional-step method as the following integral: Z tn+1 ˆ − un u 1 = (2.53) −C(u) + νL(u) + ρ−1 fb dt − G(pn−1/2 ) ∆t ∆t tn There are two main families of integration schemes to approximate the integral in the right-hand side. In explicit methods, we only use values at tn and prior instants for the time-discretization of this integral. Implicit methods on the other hand also involve information at tn+1 . This latter approach comes at the cost of solving a large system of equations, and is thus computationally more expensive. However, explicit schemes are less stable, i.e. truncation errors are amplified and may eventually become unbounded if the time step ∆t is too large. The more general definition of stability of a numerical scheme which we will use here, states that the solution should remain bounded at every tn as n → ∞. In particular, we will require that the kinetic energy within the system does not grow infinitely. In this work, we will restrain ourselves to an introductory discussion on the stability properties of various time integration schemes. To this end, we make the following simplifying assumptions. First of all, we will neglect all boundary effects, so that our analysis will only hold for infinitely extended or periodic meshes. Furthermore, we will deploy mathematical tools, which are strictly only valid for linear problems. Therefore, we will pursue the discussion on the stability properties of the various integration schemes in the context of the convection-diffusion equation. ∂u = −u · ∇u + ν∇2 u ∂t

(2.54)

or in discrete space: ∂U = (−C + νL)U = AU (2.55) ∂t The effects of the pressure and Lorentz force term will be taken into account in (sub)section 2.2.3 and 2.3. This also implies that we consider the convecting velocities as a ‘given’ field, which is independent of the unknown convected velocities. We recall that we have assumed that the matrix L has only negative ˆ it follows eigenvalues. From this, and from the skew-symmetric character of C, that 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: ˜ n ˆ = AU U (2.56) 47

48


˜ depends on ∆t and A. A sufficient and necessary condition for The matrix A the stability of a scheme is that the discrete kinetic energy is conserved or diminished after each time step. This is guaranteed if the eigenspectrum of ˜ 1/2 occurs within the unit circle. Note that this matrix product has V−1/2 AV ˜ the same eigenvalues as A. Explicit methods The starting point for the derivation of explicit schemes, is the Taylor expansion ˆ in the vicinity of un u n n ∆t2 ∂ 2 u ∂u ˆ = un + ∆t + + O(∆t3 ) (2.57) u ∂t 2 ∂t2 The corresponding expression in discrete space is:

2 ˆ = Un + ∆t(AUn ) + ∆t (A2 Un ) + O(∆t3 ) U 2

(2.58)

If we truncate this series after the second term on the right-hand side, we obtain the explicit Euler method ; this scheme is thus only first order accurate in time, and reads: ˆ = (I + ∆tA)Un + O(∆t2 ) U (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 on ˆ it can be easily shown that, in the inviscid limit, all the nature of C and C, the eigenvalues λk of this matrix are of the form λk = 1 ± αk i, where αk is bounded by: X U |(k,j) ∆t = CF Lk αk ≤ (2.60) 2Vk j,j6=k

Since |λk | ≥ 1, the first-order explicit Euler method combined with a centerdifference like spatial discretization stencil can thus never be stable for inviscid problems. In order to obtain a stable explicit scheme, one has to use upwind stencils for the convective problem. This type of formulations only guarantees stability under the condition that Courant-Friedrichs-Lewy number CF L = maxk CF Lk is smaller then one. For purely viscous cases on the other hand, we find that the eigenvalues of V1/2 (I + ∆tνL)V−1/2 are bounded by: X 1 − 2ν∆t lkj < |αk | < 1 (2.61) j,j6=k

The left-hand side of this inequality should not become smaller then -1. From this, it follows that an explicit Euler scheme is stable for a diffusion equation 48


49

under the following condition: ∀k : F Ok =

X

lkj ν∆t < 1

(2.62)

j,j6=k

The non-dimensional number F O = maxk F Ok is termed the Fourier number. We can write this condition in a more explicit way for meshes where the skewness correction term is vanishing, e.g. on regular meshes: F Ok =

X

j,j6=k

ν∆t Sk,j · (xk − xj ) Vk |xj − xk |2

(2.63)

If we want to use higher-order methods, we should retain more terms in the Taylor series (2.57). For a second-order accurate time integration scheme e.g., we can proceed as follows: n ˆ = Un + ∆t ∂ Un + ∆t ∂u + O(∆t3 ) U ∂t 2 ∂t ∂ ∆t = Un + ∆t I+ A Un + O(∆t)3 ∂t 2 ∆t A Un + O(∆t)3 (2.64) = I + ∆t A I + 2 This approach can easily be extended to higher orders of accuracy. One can show that the stability properties of these time integration schemes are similar to the ones of the explicit Euler scheme, i.e. unconditionally unstable for the convective term when combined with a central-difference like spatial discretization, and a limitation on ∆t determined by the Fourier number for viscous cases. Implicit methods In implicit methods, the velocity used to compute the right-hand side of (2.55) ˆ . In its most general form, we have: is expressed as a blend of un and u ˆ = Un + ∆tA γ U ˆ + (1 − γ)Un U (2.65)

If we choose the blending parameter γ = 0.5, we obtain the Crank-Nicholson scheme. This scheme has two major advantages. First of all, it is second order accurate provided that the coefficients of A are also second-order accurate. To maintain the accuracy, we should thus find a suitable interpolation which yields the convective velocities at the mid-time-step. An implicit treatment of these terms would result in a non-linear system for the momentum balance equation. 49

50


Having in mind the complexity of non-linear equation solvers, we will linearise n+1/2 the problem, and compute the values of U |(i,j) explicitly with a second-order Adams-Bashforth method, i.e.: U n+1/2 |(i,j) =

3 n 1 U |(i,j) − U n−1 |(i,j) + O(∆t2 ) 2 2

(2.66)

We will refer to this procedure as a semi-implicit treatment of the convective term. The second avantage of the Crank-Nicholson scheme concerns its stability properties. We can write (2.65) in a slightly different version: −1 ∆t ˆ = I − ∆t A I+ A Un (2.67) U 2 2 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: 1 + ∆t λ 2 (2.68) |λCN | = ≤1 1 − ∆t λ 2

The Crank-Nicholson scheme is thus unconditionally stable. Moreover, in the case of inviscid flow, we obtain the limit |λCN | = 1. This means that it preserves the kinetic-energy conserving property of the convective term if combined with a center-difference-like discretization stencil. Another popular implicit time advancement scheme is the so-called implicit Euler method. In this case, the value of the blending parameter γ = 1. As such, it is only first-order accurate, regardless of the time interpolation chosen for U |(i,j) . We can again express the eigenvalues λIE of the iteration matrix (I − ∆tA)−1 as function of λ: 1 2) remains close to zero up to a short distance from the junction. The internal shear layer at x = 2 layer takes the form of a jet, which becomes very sharp as we approach the inner corner (i.e close to y = 2). Since uy > 0 in this internal layer, mass is transferred in the positive y-direction. For y > 2, the internal shear layer becomes a side layer, which we will term inner side layer. As we move away from the junction, we observe that, in the inner side layer, uy rapidly decreases 131

132

CHAPTER 6. LAMINAR FLOW IN A RIGHT-ANGLE BEND

in magnitude and that its profile becomes a bit broader. Moreover, at y = 3.6 and y = 6, uy is negative in the vicinity of x = 2. Close to the wall x = 0, we have another side layer, called outer side layer, which also takes the form of a strong 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 like for 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 this region is vanishing. This is in agreement with the core flow approximation (see Appendix C.3). The flow in the core of the central duct is thus approximately horizontal. Furthermore, side layers also occur along the walls z = ±1. We show the evolution 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. The broadening of the side layers can be explained as follows. Far downstream from the junctions, we should recover a fully developed flow. For the central duct (with the magnetic field aligned with the flow direction), the fully developed flow is the laminar hydrodynamic (Poiseuille-like) duct flow. As we move away from the junction, the side jets should thus fade out. According to asymptotic theory 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 completely 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 6.6 and 6.7, like the vanishing of uy in the core, and the presence of jets of large amplitude close to x = 0 and z = ±1. Moreover, we see that the highest velocities occur in the corner regions. Remarkably, the magnitude of uy close to the corners of the side wall x = 2 is many times larger than its value at the center of this side layer. In the core flow solution, uy is exactly zero in the core of the central duct. Hence, the secondary flow u⊥ = ux 1x + uy 1y in this region can be derived from a stream function. It appears that the electric potential φ is a stream function for u⊥ , i.e. u⊥ = ∇ × φ1y (see Appendix C.3). This feature has been verified by tracing a couple of streamlines originating from seed points along the axis x = 1.6, y = 6. In figure 6.8(b), we show the projection of the streamlines on the core of the plane y = 6, i.e. between x = 0.25 and x = 1.75, and z = −0.75 and z = 0.75. The streamlines are coloured according to the value of φ along the line. We do not observe notable color variations along streamlines. This means that these are, up to a good approximation, isolines of φ, and that φ is indeed 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 mass from the inner side layer to the outer one and the ones at z = ±1. 132

6.2. MHD FLOW IN A

S

-BEND

133

9

y y y y y

8 7

= = = = =

1 2 3.6 6 12

6

uy

5 4 3 2 1 0 −1 0

0.5

1

1.5

x

2

2.5

3

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

Once the quasi-horizontal flow in the core of the central duct enters one of the side layers, it becomes fully three-dimensional, and this results in a complex motion. This is illustrated in figure 6.8(c), where we trace the streamlines originating from three different seed points in the entrance duct. The symmetry of the flow allows us to show only one quarter of the configuration (i.e. for z > 0 and y < 12). The first of these points (red streamline) is located on the axis of the duct. From symmetry considerations, we find that uz is zero in 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 a short distance the direction of the magnetic field. Then, it bends and flows quasi-horizontally in x-direction towards the outer side layer. The seed of the second (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 where it is transferred in horizontal direction towards the outer side layer, and joins 133

134


7 y=1 y=2

6

y = 3.6 y=6

5

y = 12

uy

4 3 2 1 0 −1 −1

−0.5

0

z

0.5

1

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 with coordinates (5, 1, 0.5). Close to the junction, it enters the shear layer and is simultaneously driven towards the wall z = 1; this was already inferred from the bending of the current streamlines in figure 6.5. Once it has arrived in the central duct, it follows a complex motion, in which it hops back and forth between the side layers at the inner wall and the wall z = 1. Finally, a more quantitative comparison with the core flow approximation is shown in figure 6.9, where we plot the relative fraction of the mass flow rate Q carried by the boundary layers of the central duct against the coordinate y. Q has been normalized such that the total mass flow rate in the central duct equals 2 for y > 2. As mentioned before, the core mass flow rate in the asymptotic limit is exactly zero. However, for the present simulations, the streamwise core flow velocity is not exactly zero. Moreover, the thickness of the side layers is not very well defined, and does vary with y. This implies that we have to use a slightly different definition for Q. The values of Q which result from our simulations (solid line in figure 6.9) have been obtained by integrating 134

6.2. MHD FLOW IN A

S

-BEND

135

Figure 6.8: Color plot of uy in the plane y = 6 (a). Velocity streamlines passing trough the axis y = 6, x = 1.6 projected on the plane y = 6, and colored according to the value of the electric potential φ along the streamline S (b). Three-dimensional velocity streamlines in the lower half of the -bend originating from three different seed points.

uy over triangular domains, which are sketched in the upper left corner of figure 6.9. More specifically, the non-normalized flow rates for the side layers at x = 0 135

136


(green line), x = 1 (blue line) and z = ±1 (red line) are computed as: Z 1 Z z=−x+1 Qx=0 = uy dz dx Qx=2 Qz=±1

= =

x=0 Z 2

2

z=−1+x Z z=x−1

uy dz dx

x=1 z=−x+1 Z L Z x=1+z z=0

uy dx dz

(6.4) (6.5) (6.6)

x=1−z

In figure 6.9, we see that both sets of results agree well for the outer side layer. On the other hand, the present simulations do not predict reversed mass transfer in the inner side layer, in contrast to the asymptotic study. It follows that the results for the side layers at the walls z = ±1 also exhibit severe discrepancies. This poor agreement may be partially explained by the influence of the corner 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 mass flow rate up to the leading order, since the velocity in the corners is of the same order as in the non-overlapping zones, whereas they extend spatially over a region which is O M −1/2 times smaller. However, from figure 6.8(a) e.g., we may deduce the corner regions carry a significant fraction of the mass flow rate R2 included in Qx=2 . On the other hand, the integral x=1 uy (x, y = 6, z = 0) dx (the integrand is shown in figure 6.6) is negative. This indicates that the mass transfer in the inner side layer is indeed reversed far away from the walls z = ±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 difficult to quantify because the extent of the corner regions is not very well defined.

6.3 6.3.1

MHD flow in a backward elbow Problem definition and computational set-up

We consider a right-angle bend between two semi-infinite ducts of square crosssection (see figure 6.10(a)); the side of these ducts is 2L. The origin of the coordinate system is located on the outer corner and in the plane of symmetry with respect to the z-direction, and the ducts extend along the positive xaxis (upstream) and y-axis (downstream). A strong uniform magnetic field is 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 the duct parallel to the magnetic field. Comparison between the present numerical results (solid) and the core flow approximation of [MB94] (dashed). Mass flow rate fraction carried by the outer wall (green), inner wall (blue), and side wall (red) boundary layers. The integration domains of equations (6.4-6.6) are sketched in the upper left corner.

of MHD shear layers near geometrical discontinuities predicts that, for these parameters, a Ludford layer of thickness O(M −1/2 ) will emerge close to the inner corner of the system, which extends along the magnetic field direction. The inflow and outflow boundaries of the simulation domain are imposed in the planes x = 7 and y = 7. We assume that fully developed duct flow conditions will apply in these planes, since they are located on a distance from the Ludford layer which is large with respect to its thickness. One of the major problems related to the numerical simulation of this configuration is the design of an appropriate mesh. The internal shear layer extends in both axial directions over a distance 4. Properly resolving the Ludford layer with a cartesian mesh would imply that a grid spacing should be smaller than M −1/2 in the region limited by the planes y = 0, y = 4, x = 0 and x = 4. Therefore, we have to use unstructured meshes if we want to provide simulations of the high Hartmann number regime at reasonable computational 137

138


Figure 6.10: MHD flow in a backward elbow. Detailed geometry (a), and location 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 layers should be of the order of M −1 . This layer extends over a distance of multiple length units in the directions tangential to the wall. Far away from the corners and side walls, all the variables are quasi-uniform along these directions. Hence, meshes consisting of strongly anisotropic elements are preferred in these zones. We now explain how this is achieved. These considerations imply that we have to tailor the mesh such that a 138


Name A B C D E F

3D element Hexa Hexa Prism Prism Hexa Hexa

Min.Size M −1 /5 M −1 /5 M −1 /5 ‘C’ M −1/2 /8 M −1/2 /8

Max. Size 0.03 0.03 or ‘C’ or ’D’ M −1/2 /8 0.03 0.03 0.03

139

Growth Factor 1.06 1.12 1.08 1.08 1.06 1.08

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

structured anisotropic mesh in the Hartmann layers is matched with an unstructured, isotropic one in the core and Ludford layers regions. We first consider the plane of the axes of the duct (see figure 6.11(a)-(b),(d)-(e)). At the inner corner, two Hartmann layers meet, and the grid spacing at this point is M −1 /5 (see figure 6.11(e)) in both x- and y-direction. The grid spacing between the wall nodes grows exponentially away from this corner until a maximum of 0.03 is reached (‘A’ in figure 6.11(d)). Simultaneously, the extent of the structured quadrilateral region also increases in wall-normal direction, so that the last quadrilateral point is approximately isotropic (‘B’ in figure 6.11(b)-(d)). This approach is necessary if we want to obtain a smooth matching between the structured boundary layer and the unstructured core. Furthermore, a refined, unstructured triangular layer emanates from the inner corner along the magnetic field direction; the size of the elements grows with a factor 1.08 away from the inner corner until a maximum size of M −1/2 /8 (‘C’ in figure 6.11(d)). If we move away from this axis, the grid size increases further up to a maximum of 0.03 (‘D’ in figure 6.11(d)). We prefer triangular elements above quadrilateral ones because they are more easily generated by the preprocessor. Moreover, in section 4.3, we have seen that a mesh which consists of a triangular core and a structured quadrilateral boundary layer yields 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 procedure as for the inner ones. We see that the structured boundary layer is tailored such that a smooth transition with the refined unstructured Ludford layer region is obtained (‘E’ in figure 6.11(b)). To generate the three-dimensional mesh, we simply extrude the two-dimensional one. Hence, the mesh elements are hexahaedra and rectangular prisms. The mesh spacing in z-direction is such that it allows a proper resolution of the side layers, with six points over a distance M −1/2 next to the wall (‘F’ in figure 6.11(c)). Adopting the above strategy results in a mesh which contains 139

140


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

between 15 and 16 million grid nodes, and consists of hexahedral elements in the structured zones of the mesh, and prisms in the unstructured regions. The information on the stretched regions is summarized in table 6.2. A cartesian mesh with the same resolution in the shear layers would contain at least 10 times more grid points. The other simulation details are similar to those discussed in the previous 140


141

section. The computations were performed on 160 CPU’s and approximately 32000 iterations were needed to reach convergence. This required about 60000 CPU hours.

6.3.2

Results and discussion

In figure 6.12, we show profiles of ux (x, y, z = 0) and uy (x, y, z = 0) along the x-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 axis y = 1 is the axis of the entrance duct, and is intersected by the internal shear layer at x = 3. Far upstream from this internal layer, the flow adopts the fully developed state of a straight duct; it is independent of x, and uy = 0. The value of the core velocity is approximately ux ≈ −0.21. This value is uniform over the cross-section of the duct, with exception of the side layers and Hartmann layers which are asymptotically thin. Therefore, we can deduce that the core carries approximately 21 percent of the mass flow rate. In the Ludford layer, ux and uy have the same order of magnitude, which is much larger than the core value of ux . Furthermore, the magnitude of ux downstream from the shear layer, is considerably smaller than upstream. This suggests that a fraction of the mass flow 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, except near the inner corner at x = 2. It appears the velocity magnitude in the plane z = 0 reaches its maximum close to this point, and the velocity profile takes the form of an intense and sharp jet. Since ux and uy are both opposite in sign and virtually equal in magnitude, it follows that the flow direction in this jet is paralel to the magnetic field. What we observe in fact, is that a significant fraction of the total mass flow is gathered in the Ludford layer, and is transferred from the entrance to exit duct through a small corridor near the inner corner. For y > 2, the Ludford layer will gradually release the mass flow rate it carries into the core of the exit duct. Hence, the velocity magnitude within 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 note that the layer has become broader compared to the plane y = 2 and that ux is vanishing outside the Ludford layer. Finally, at y = 5, the flow has reached again 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 significant fraction of the mass flow rate is transferred along the Ludford layer from the entrance to the exit duct. This behavior is also found in figure 6.13(a), where we trace streamlines originating from seed points in the plane y = 7; only one 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

142


0.6

2

y=1

1

ux , y

ux , y

0.3 0 −0.3

2

x

4

−2 0

6

0.6

y=3

0.5

1

1.5

2

1

1.5

2

x

y=5

0.3

ux , y

ux , y

0.3 0

0 −0.3

−0.3 −0.6 0

0 −1

−0.6 0

0.6

y=2

0.5

1

x

1.5

2

−0.6 0

0.5

x

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

z = ±1. The streamlines close to these side walls exhibit a small kink when approaching the Ludford layer, and migrate between both ducts along a path which is somewhat curved with respect to the magnetic field direction. We also see that a unidirectional flow is recovered at a very short distance from the Ludford layer. The topology of the streamlines agrees, at least in qualitative sense, well with the findings of [MHW91]. An inspection of the color plot of the velocity magnitude depicted in figure 6.13(b) gives a final illustration of the fact that the Ludford layer takes the form of a jet which becomes sharper and 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 lines uB = 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 this investigation, is not u = 0, but ∂n uτ = 0, where uτ is the velocity tangential to 142

6.4. CONCLUSIONS

143

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

the wall. This means that abstraction is made from the Hartmann layer or side layer 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 fact be interpreted as the velocity at the edge between the core and the Hartmann layer region. Therefore, we have compared them here with present values of uB which have been evaluated along x = M −1 . From figure 6.14, we see that the agreement between both sets of results is rather poor. In particular, we find that the present numerical simulations predict a smoother variation of the side layer velocity profile in the vicinity of the internal shear layer. It is not yet clear whether this is caused by the finite value of the Hartmann number, a false hypothesis in the core flow approximation or a 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 a strong magnetic field by means of high-resolution numerical simulations. For the backward elbow, we have demonstrated how we can reduce the number of 143

144


2

0

−0.1

1.5

φ

uB

−0.2 1

−0.3 0.5 −0.4

0 0

1

2

3

y

4

5

6

−0.5 0

7

1

2

3

y

4

5

6

7

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

required grid nodes by appropriately tailoring the mesh. We have also performed simulations of the configurations studied in [MB94] and [MHW91]. In both cases, we observe that the present numerical method yields velocity distributions which agree qualitatively rather well with the core flow approximation. However, when it comes to quantitative measures, we find considerable deviations between both sets of results. In the near future, we will investigate more profoundly the origin of these discrepancies. Moreover, we plan also to study the effect of inertia at fusion-relevant paramaters. One of the more interesting questions in this context is under which conditions the internal shear layer may become unstable.

144

Chapter 7

Conclusions and perspectives “We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time.” T. S. Eliot The contents of this dissertation can roughly be divided in two parts. In chapters 2 and 3, we dealt with the development of a parallel, unstructured finite-volume solver for the MHD equations in the quasi-static limit. We have provided state-of-the-art numerical technology for this solver, whether it concerns iterative methods to solve large, sparse linear systems, the computation of the Lorentz force, or the implementation of Shercliff’s thin-wall condition for the electric potential. This was accompanied by a (limited) numerical analysis, based on Gershgorin’s circle theorem, which holds also for unstructured meshes. Our implementations were furthermore successfully verified through the simulation of a number of test cases discussed in chapter 3. In the mean time, a newer and more advanced version of the code YALES2 has been developed, and we have recently ported our implementations to this more recent release. The solver developed was then used to study several configurations of interest in chapters 4-6. In chapter 4, we considered the laminar unidirectional flow in a circular pipe. We could reveal the existence of overspeed zones in wellconducting pipes at high Hartmann number, and characterize them in terms of scaling laws. These are consistent with previous results, like the scaling of the radial extent of the Roberts layers in insulating pipes. Although the implications 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 simulations are a valid complement next to theoretical considerations for the fundamental study of MHD flows. The work presented in chapter 5 concerned the MHD flow in a toroidal duct of square cross-section and both the laminar and turbulent regimes were investigated. The results for the laminar regime confirmed the asymptotic boundary layer analysis [TC81]. Moreover, the present approach allowed us also to study the secondary flow in the core regions. Although somewhat underresolved, the present simulations of the turbulent regime illustrated the main features of the flow. We observed that the curvature induces an asymmetry between the convex and concave side wall. The application of an external magnetic field leads to the stabilization of the core, the Hartmann layers and the concave side layers whereas a certain level of turbulence can persist in the convex side layer. Structured codes (in a cylindrical coordinate system) are probably more appropriate to study in more detail the original experiment of Moresco and Alboussière [MA04]. Finally, in chapter 6, we have investigated the inertialess S flow in a rightangle bend in a strong magnetic field. In the case of a -bend, we could recover qualitatively most of the aspects characterizing the asymptotic core flow solution. Important discrepancies between the present solution and the core flow solution were found for the flow rates of the side layers in the central S duct of the -bend. Also for the backward elbow, considerable disagreements were 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 the configurations. In particular, the sophistication and complexity of the mesh for the investigation of the backward elbow is unprecedented in the domain of (quasi-static) magnetohydrodynamics. The design of a reliable, unstructured computational mesh for high Hartmann number flows is an issue which has often been subtly avoided in previous investigations by focusing on those geometries which can be simulated on cartesian meshes. This issue will however become more predominant in the near future, and may confront us with a fundamental stumbling stone, which is due to the following inherent opposition. Unstructured finite volume methods tend to become more accurate as the control volumes are more isotropic, whereas quasi-static MHD flows in a strong magnetic field are severely anisotropic by nature. Therefore, the development and analysis of numerical techniques for unstructured formulations in the context of MHD flows will remain of critical importance. Moreover, consideration may 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, the divergence of F, the curl of F and the Laplacian of φ, respectively ∇φ, ∇ · F, ∇ × F and ∇2 φ = ∇ · ∇φ, are defined as follows. • In Cartesian coordinates x, y and z, the definitions read: ∇φ =

∂φ ∂φ ∂φ 1x + 1y + 1z ∂x ∂y ∂z

∇·F=

∂Fy ∂Fz ∂Fx + + ∂x ∂y ∂z

∂Fx ∂Fy ∂Fz ∂Fz − − 1x + 1y ∇×F = ∂y ∂z ∂z ∂x ∂Fy ∂Fx + 1z − ∂x ∂y

(A.1) (A.2)

∇2 φ =

∂2φ ∂2φ ∂2φ + 2 + 2 ∂x2 ∂y ∂z

(A.3)

(A.4)

• In cylindrical coordinates r, θ and y, we have: ∇φ = ∇·F=

∂φ 1 ∂φ ∂φ 1r + 1θ + 1y ∂r r ∂r ∂y

(A.5)

1 ∂Fθ ∂Fy 1 ∂ (rFr ) + + r ∂r r ∂θ ∂y

(A.6)

147

148

APPENDIX A. ELEMENTS OF VECTOR CALCULUS

∂Fr ∂Fθ ∂Fy 1 ∂Fy 1r + 1θ − − ∇×F = r ∂θ ∂y ∂y ∂r 1 ∂ 1 ∂Fr + 1y (rFθ ) − r ∂r r ∂θ ∂φ 1 ∂2φ ∂2φ 1 ∂ 2 r + 2 2 + 2 ∇ φ= r ∂r ∂r r ∂θ ∂y

A.2

(A.7) (A.8)

Integral theorems

Let Ω be an arbitrary closed volume, and ∂Ω its bounding surface. For any vector field F and any scalar function φ, the following identities hold: Z I ∇ · F dV = F · dS (A.9) Ω

Z

Z

Ω

∂Ω

Ω

∇φ dV =

∇ × F dV =

I

φ dS

(A.10)

F × dS

(A.11)

∂Ω

I

∂Ω

If Σ is an arbitrary surface, ∂Σ its boundary, and F and φ respectively any vector field and scalar function. Then, the following identities hold: Z I ∇ × F · dS = F · dl (A.12) Σ

∂Σ

Z

Σ

A.3

∇φ × dS =

I

φ dl

(A.13)

∂Σ

Vector identities

For any scalar field φ, and every vector field F and G, we have the following relationships: ∇ × ∇φ

= 0

(A.14)

∇ · (∇ × F) ∇ (∇ · F)

= 0 = ∇2 F + ∇ × (∇ × F) 1 (∇ × F) × F = (F · ∇) F − ∇ (F · F) 2 ∇ · (F × G) = G · (∇ × F) − F · (∇ × G)

(∇ × F) × G = (G · ∇) F − (F · ∇) G

∇ × (F × G) ∇ × (φF)

(A.15) (A.16) (A.17) (A.18) (A.19)

= F (∇ · G) − G (∇ · F) + (G · ∇) F − (F · ∇) G(A.20) = φ∇ × F + (∇φ) × F (A.21) 148

A.4. HELMHOLTZ’S DECOMPOSITION

A.4

149

Helmholtz’s decomposition

Any vector field A may be written as the sum of an irrotational and solenoidal field. The irrotational field may, in turn, be written in terms of a scalar potential φ, and the solenoidal field in terms of a vector potential F: A = −∇φ + ∇ × F

(A.22)

The two potentials are the solutions of: ∇2 φ =

∇2 F =

−∇ · A

−∇ × A

149

(A.23) (A.24)

150

APPENDIX A. ELEMENTS OF VECTOR CALCULUS

150

Appendix B

Elaborations on linear system solvers B.1

A spectral analysis of the ω-Jacobi method for a Poisson equation

We consider an one-dimensional equidistant mesh between the boundary points π x = 0 and x = π. The grid spacing is ∆ = N . The grid contains thus N + 1 points, located at positions xi = i∆ for i = 0, 1, 2, ..., N . We now want to solve a discrete Poisson equation with Neumann boundary conditions on this grid; the analysis is not fundamentally different for Dirichlet or periodic boundary conditions. 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 the form: Φk = cos(kxi ) k = 0, 1, 2, ..., N (B.2) We can easily proof this by injecting (B.2) in (B.1): cos(k(xi + ∆)) + cos(k(xi − ∆)) − 2 cos(kxi ) ∆2 4 k∆ cos(kxi ) (B.3) = − 2 sin2 ∆ 2 The eigenvalues of L (or L) are λk = − ∆42 sin2 k∆ 2 . We furthermore note that the diagonal of L is D = − ∆22 I. L(Φk )|i

=

151

152

APPENDIX B. ELABORATIONS ON LINEAR SYSTEM SOLVERS

To analyse the ω-Jacobi method, we first write the residual R(m) after m ω-Jacobi iterations as a linear combinations of eigenvectors. This can always be done since the N + 1 eigenvectors Φk form a complete set: R(m) =

N X

(m)

c k Φk

(B.4)

k=0

We recall the iteration formula (2.99) for R: R(m+1) = I − (1 − ω)MD−1 R(m)

The iteration matrix I − (1 − ω)MD eigenvalues µk read:

−1

(B.5)

has the same eigenvectors as L but its 2

µk = 1 − 2(1 − ω) sin

k∆ 2

(B.6)

(m)

After each iteration, every coefficient ck in (B.4) will be multiplied with a factor µk . To converge, |µk | should be smaller then one; this condition is satisfied 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 value of ω. This implies that the long-wavelength contributions (small k) the residual vector will be slowly damped, independently of ω; moreover, we also see that the convergence slows down as ∆ becomes smaller, i.e. if the number of grid points N increases. Note that a (discrete) Poisson equation with Neumann boundaries is only well-posed if the sum of the elements of the right-hand side is zero. If this condition is fulfilled, the inital coefficient c00 will also be zero. Therefore, the limit µk = 1 does not a formal problem for convergence. • High frequencies: If k → N , we find that µk → −1 + 2ω. We see that the classic Jacobi approach (ω = 0) hardly affects the high-frequency components. For k = N , we have µk = 1, and this means that the (0) classic Jacobi method can never converge if the coefficient cN of the initial residual is non-zero. However, for 0 < ω < 1, we can reduce the absolute value of µk , and obtain an efficient reduction of these short wavelength components. We may thus conclude that, for a proper choice of ω, application of the ω-Jacobi method to a one-dimensional Poisson equation efficiently damps the shortwavelength components of the residual, but hardly affects the long-wavelength contributions. This observation forms the ansatz for the development of multigrid techniques. 152

B.2. LISTING OF THE BICGSTAB ALGORITHM

B.2

153

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 threshold ǫ. Choose the parameters ρ, α and ω and vectors P and V initially as follows: ρ0 = α = ωq =1, P (0) = V (0) = 0. ˜ e.g.: R ˜ = R(0) Choose R, (q) while ||R || ≥ ǫ AND q ≤ qmax do q = q+1 ˜ T Rq−1 ρq = R ρ α β = ρq−1qωq−1 P (q) = Rq−1 + β(P (q−1) − ωq−1 V (q−1) ) V (q) = MP (q) ρ α = R˜ T Vq(q) T S = R(q−1) − αq V (q) T = MS T ωq = TT T TS Y (q) = Y (q−1) + αq P (q) + ωq S R(q) = S − ωq T end while

153

154

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 threshold ǫ. Choose the parameters ρ0 , α and ω2 and vector U initially as follows: ρ0 = ω2 = 1, α = 0, U = 0. ˜ e.g.: R ˜ = R(0) Choose R, (q) while ||R || ≥ ǫ AND q ≤ qmax do q = q+2 ρ0 = −ω2 ρ0 EVEN STEP: ˜ T R(q) , β = αρ1 , ρ0 = ρ1 ρ1 = R ρ U = R(q) − βU V = MU ˜ (0) , α = ρ0 γ = V TR γ R = R(q) − αV S = MR Y = Y (q) + αU ODD STEP: ˜ T S, β = αρ1 , ρ0 = ρ1 ρ1 = R ρ V = S − βV W = MV ˜ α = ρ0 γ = W T R, γ U = R − βU R = R − αV , S = S − αW T = MS Y = Y (q) + αU QUADRATIC-POLYNOMIAL-PART: ω1 = RT S, µ = S T S, ν = S T T , τ = T T T , ω2 = RT T 2 τ = τ − νµ −νω1 ω2 = µω2µτ ω1 −νω2 ω1 = µ Y (q+2) = Y + ω1 R + ω2 S + αU R(q+2) = R − ω1 S − ω2 T U = U − ω1 V − ω2 W end while

154

B.3. LISTING OF THE BICGSTAB(2)-ALGORITHM

155

155

156


156

Appendix C

Asymptotic solutions at high Hartmann number The term asymptotic theory refers to an approximate solution method for the fully developed (i.e. steady) profile of wall-bounded MHD flows at high Hartmann number and interaction parameter. In this formulation, the existence of different regions is assumed; the flow domain consists a core, surrounded by various shear layers. These layers may occur due to the presence of solid walls (e.g. Hartmann layers or side layers ), or geometrical discontinuous (e.g. Ludford layers). For each region, a simplified set of equations is derived, which can be solved analytically. These solutions are then matched to each other and the boundary conditions, such that a smooth profile is obtained. This type of methods is sometimes also called core flow approximation. Useful reference works 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 a straight pipe of radius R and of infinite extent along the axial direction, as sketched in section 4.1. The variables have been rescaled in the same way as in section 4.1, but with a different definition for U . Here, U = f M (σB02 )−1 . The asymptotic approximations to equations (4.1-4.3) for this case are solved more easily in terms of the induced magnetic field b. For fully developed unidirectional flows, b has only one component: b = b(x, y)1z . We now rescale b as −1 follows: b → µσU B0 Lb = Rem B0 b. The pre-Maxwellian form of Ampère’s law (1.38) reads: ∇×b=J 157

(C.1)

158

APPENDIX C. ASYMPTOTIC SOLUTIONS AT HIGH HARTMANN NUMBER

By substituting this in the streamwise component of the momentum budget, we find: 2 ∂ u ∂2u 1 ∂b (C.2) + 2 = −M −1 + ∂y M 2 ∂x2 ∂y Furthermore, we have the dimensionless form of the quasi-static induction equation (1.59): ∂u ∂2b ∂2b (C.3) + 2 + 2 =0 ∂y ∂x ∂y 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 ˆb = M −1 b in (C.2)-(C.3). We obtain: ∂ˆb 1 ∂2u ∂2u = −1 (C.4) + 2 + ∂y M ∂x2 ∂y ! ∂ 2ˆb ∂ 2ˆb 1 ∂u = 0 (C.5) + 2 + 2 ∂y M ∂x ∂y √ 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 magnetic induction as: 1 ∂b + b=0 (C.6) ∂n c 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 , ˆbc (for the core) and uH , ˆbH (for the Hartmann layers). The global solution can then be written as u = uc + uH and ˆb = ˆbc + ˆbH The core If M ≫ 1, we may neglect the viscous terms in the core; (C.4)-(C.5) takes the simplified form: ∂ˆbc ∂y ∂uc ∂y

=

−1

(C.7)

=

0

(C.8)

−y

(C.9)

The solution of these equations is: ˆbc

=

uc

=

uc (x)

(C.10)

Note that there is no integration constant in the solution for ˆbc . This follows from the constraint that ˆb 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 y into η = M (y − Y (x)). If we substitute this form in (C.4)–(C.5), we obtain: 1 ∂ˆbH + ∂η M

∂ 2 uH ∂ 2 uH M2 = −1 + ∂x2 ∂η 2 ! 2ˆ ∂u ˆH ∂ 2ˆbH 1 2 ∂ bH M M = 0 + + ∂η M ∂x2 ∂η 2

M

(C.11) (C.12)

If we retain only the leading-order terms in M , we find the set of ordinary differential equations: ∂uH ∂ˆbH + ∂η ∂η 2 ∂uH ∂ˆbH + ∂η ∂η 2

= 0

(C.13)

= 0

(C.14)

Solutions of (C.13)-(C.14) which satisfy the boundary condition uH = A exp(η)+ B and ˆbH = −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 , ˆbH → 0 for 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 = ˆb =

uc (1 − exp(η))

(C.15)

−y + uc exp(η)

(C.16)

The unknown core velocity uc follows from imposing the thin-wall condition (C.6). If we write n = nx 1x + ny 1y , (C.6) becomes: ! ∂ˆb ∂ˆb 1 ˆ + ny + b nx ∂x ∂y c

=0

(C.17)

y=Y (x)

We now assume that the first term on the left-hand side can be neglected. This assumption will be validated later. We obtain: 1 ny (−1 + M uc ) + (−Y + u) = 0 c 159

(C.18)

160


We now solve uc from this expression: uc (x) =

ny c + Y ny cM + 1

(C.19)

√ For Y (x) = ± 1 − x2 , we have n = x1x + Y (x)1y ., and we arrive at: √ (c + 1) 1 − x2 √ uc (x) = cM 1 − x2 + 1

(C.20)

This solution differs by a factor M from (4.7)-(4.9) since the velocity scale U has been defined differently in this appendix then in chapter 4. We now check under which conditions the assumption nx ∂xˆb ≪ ny ∂y ˆb fails. With the results above, we can compute √ both the right-hand side of the inequality explicitly at the wall: ny ∂y ˆb = ± 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 the wall. In this region, the asymptotic approximation breaks down and a more detailed analysis is required. From the previous results, we find for the core current: Jc = ∇ × (bc 1z ) = M −1 ∇ × (ˆbc 1z ) = M −1 1x

(C.21)

Discussion for limiting values of c We now compare the order-of-magnitude of Jc and uc for the two limiting values of c, i.e. c = 0 and c = ∞. For both values of c, the dimensionless core current Jc 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 almost completely cancels the electric field induced by the flow. For c = ∞ on the other hand, (C.20) yields uc = M −1 1x . Together with (C.21), we find Jc = uc × B. It follows that the potential gradient in the core is negligible up to the leading order in M .

C.2

Free MHD shear layers near geometrical discontinuities

In this section, we summarize the theory of free MHD (internal) shear layers which emanate from geometrical discontinuities in a strong uniform magnetic field B = B0 1y [HL67, MB01]. Consider therefore the two-dimensional situation sketched in figure C.1. We have a channel with a sharp corner in the bottom 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 geometrical discontinuity. The extent of the shear layer which emanates from it is shaded in light grey.

be specified by means of functions Yt (x) and Yb (x). The Poisson equation for the potential reads: ∇2 φ = ∇ · (u × B) = ∂z (ux B) = 0

(C.22)

If the boundary conditions for φ are homogeneous and uniform, the solution of this equation is φ = αz + β, with β an arbitrary integration constant, 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 ∂z φ1x = αB0 1x . We may absorb this contribution in the pressure gradient term. Hence, we may treat the momentum equations, without loss of generality, as if ∂z φ = 0. The (non-dimensional) quasi-static MHD equations for fully developed flow then read:

∂ 1 ux N ∂x ∂ 1 ux N ∂x

∂ux ∂uy + ∂x ∂y ∂ ux + uy ∂y ∂ uy + uy ∂y

= 0 ∂p + ∂x ∂p = − + ∂y = −

(C.23) 2 ∂ ∂2 1 ux − ux (C.24) + M 2 ∂x2 ∂y 2 2 ∂ ∂2 1 uy (C.25) + M 2 ∂x2 ∂y 2

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 the existence 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 assumptions break down for the obtained solution. 161

162


The equations governing the core flow are: ux,c

=

0 =

∂pc ∂x ∂pc − ∂y

−

(C.26) (C.27)

These expressions can be combined into: ∂ 2 pc ∂ux,c =− =0 ∂y ∂x∂y

(C.28)

It follows that ux,c is independent of y. Of course, this solution can not satisfy the kinematic boundary conditions u = 0 at y = Yt and y = Yb . Close to these walls, viscous Hartmann layers of non-dimensional thickness O(M −1 ) will emerge, and the tangential component of u will fall off exponentially to zero. Therefore, we may relax the condition u = 0, and require only that the wall-normal velocity un,c = n · uc is zero at the top and bottom wall. RY Mass conservation requires that Ybt ux dy = C, with C a constant which specifies the mass flow rate in the channel. For the sake of simplicity, we will assume that the reference velocity U0 used is chosen such that C = 1. Since the velocity in the Hartmann layers has the same order-of-magnitude as the core velocity and the spatial extent of these layers is O(M −1 ) smaller than the core, the Hartmann layers do not contribute to the mass flow rate up to the RY leading order in M . This implies that Ybt ux,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 conservation equation (C.23), in which we have substituted ux,c by (C.29). In the case of figure C.1, the kinematic boundary condition at y = Yt is uy,c (y = Yt ) = 0, and we find: Z Yt Z Yt ∂uy,c ∂ux,c uy,c = dy = − (C.30) ∂y ∂x y y The explicit solution for uy,c is: uy,c = (Yt − y)

∂ Yt − y ∂Yb (Yt − Yb )−1 = ∂x (Yt − Yb )2 ∂x

(C.31)

This solution is discontinuous at x = 0. For x < 0, uy,c = 0 since the bottom wall is horizontal. Downstream from x = 0 however, the flow has also 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 ∂x uy,c = O(N ). We now compute this term explicitly: 2 ! ∂uy,c ∂ 2 Yb 2 Yt − y ∂Yb (C.32) + = ∂x (Yt − Yb )2 ∂x2 Yt − Yb ∂x We see thus that the core flow solution breaks down if ∂x2 Yb = O(N ) or ∂x Yb = O(N 1/2 ). The sudden occurrence of a velocity component along the magnetic field direction gives rise to a local shear flow, which extends over a distance δ, whose dependence on M and N is to be specified. For further analysis, we introduce the 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 typical length 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 and convective term, we find: ∂uy ∂Ux + ∂ξ ∂y 1 ∂Ux N ∂ξ 1 1 ∂uy N δ ∂ξ

= 0

(C.34)

1 ∂p 1 1 ∂ 2 Ux − (1 + δUx ) − 2 δ ∂ξ M δ ∂ξ 2 ∂p 1 1 ∂ 2 uy = − + 2 2 ∂y M δ ∂ξ 2 = −

(C.35) (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 ∂p = −1 − δUx (ξ, y) δ ∂ξ

(C.37)

This result implies that the pressure drop across the shear layer is of the orderof-magnitude of δ. We can now use (C.34) and (C.37) to eliminate the pressure term from (C.36). This yields: 2 1 1 ∂ 4 uy 1 1 ∂ 3 uy 2 ∂ uy = − δ N δ ∂ξ 3 M 2 δ 2 ∂ξ 4 ∂y 2

(C.38)

This equation expresses a balance between an inertial, viscous, and electromagnetic term. Depending on the values of M and N , we can distinguish between several regimes: 163

164


• The viscous-electromagnetic balance holds if the inertial term in (C.38) is −2 2 negligible with respect to the other terms, i.e if O (M δ) = O δ ≫ O (N δ)−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 ≫ M 3/2 . • The condition O (N δ)−1 = O δ 2 ≫ (M δ)−2 governs the inertialelectromagnetic balance. This leads to the following scaling law for δ: δ ∼ N −1/3

(C.40)

Hence, this regime hold is if N ≪ M 3/2 . • 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 ≈ M 3/2 . • Theoretically, a fourth possibility would be the one of a viscous-inertial balance. It can however be shown that no solution of (C.38) without electromagnetic term exists which satisfies the condition that uy matches the core solution as η → ±∞. For the three possible regimes, one can easily show that the scaling law for δ is consistent with the assumption δ > O(N −1 ), δ > O(M −1 ).

C.3

Asymptotic analysis of the core regions in S a -bend

We consider the configuration discussed in section 6.2.1, i.e the inertialess flow S in a -bend in a strong magnetic field, i.e Re = 0 and M ≫ 1. Here, we summarize 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 neglected, and we have a simple balance between the pressure gradient and the Lorentz force 1 : −∇p + J × 1y = 0 (C.43) 1 Since 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

S

-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⊥ = Jx 1x + Jz 1z . This shows that the pressure is a streamfunction for J⊥ in x-z planes. We now concentrate on the core of the central duct. First, we note that the plane 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 of generality, we may choose C = 0. Using Ohm’s law (1.32), we can formulate (C.42) componentwise as: ∂p ∂φ + − ux ∂x ∂z ∂p − ∂y ∂p ∂φ − − − uz ∂z ∂x

−

=

0

(C.45)

=

0

(C.46)

=

0

(C.47)

From (C.46), together with the symmetry condition p(x, y = 12, z) = 0, it follows that p(x, y, z) is zero everywhere in the core of the central duct. As such, ∇p = 0 in this region, and the momentum balances (C.45) and (C.47) reduce to: ux uz

∂φ ∂z ∂φ = − ∂x

=

(C.48) (C.49)

If we substitute (C.48) and (C.49) in the incompressibility constraint ∇ · u = 0, we find: ∂uy =0 (C.50) ∂y Since it is assumed the wall-normal velocity does not vary up to the leading order in the Hartmann layers, uy satisfies the kinematic boundary condition uy = 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 asymptotic core flow solution in this region is thus a two-dimensional field u = u⊥ = ux 1x + uz 1z for which, according to (C.48)-(C.49), the electric potential φ provides a streamfunction, i.e. u⊥ = −∇ × φ1y .

165

166


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