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

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quasi-static magnetohydrodynamics using an unstructured

finite volume solver:

development and applications

Stijn Vantieghem

Universit´e Libre de Bruxelles

Thèse présenté en vue de l’obtention du grade de Docteur en Sciences sous la direction du Prof. B. Knaepen.

Janvier 2011

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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 presen- tation 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 Hart- mann 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 cross- section. 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.

Finally, in chapter 6, we investigated the inertialess MHD flow in aS -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 enor- mously 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. L´eon Brenig and prof. G´erard 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

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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 non- scientific 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, Alejan- dro 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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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. Accord- ing 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 at- tempted in 1832 in vain to estimate the flow rate of the river Thames by mea- suring the electric potential difference across the river, induced by the Earth’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

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intrusively a stirring motion in a liquid metal or electrolyte. Such an approach is preferred in configurations where the use of a mechanical mixer is imprac- tical, e.g. in high-temperature or corrosion-aggressive environments. Finally, rotating magnetic fields are also applied in heating or levitation processes. 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 pur- pose, 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 ex- actly 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 han- dling 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

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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 pro- duce real-time predictions, he proposed the instauration of ‘forecast factories’, where continuously 64 000 people would, armed with a mechanical calculator, perform a part of the flow computation, on a grid consisting of 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 thek−ǫturbulence 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 mem- ory of 32 megabyte and had a peak performance of 400 MFLOPS. However, the range of treatable problems expanded rapidly because of the fast 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,

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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 Dynam- ics, 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”, sub- mitted to International Journal of Heat and Fluid Flow (2010).

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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 suf- fice 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 The fundamentals of hydrodynamics

1.1.1 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

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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:

d dt

Z

Ω

ρ(x, t) dV = 0 (1.1)

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

Ω

d

dtρ(x, t) +ρ∇ ·u

dV = 0 (1.2)

Furthermore, the positionsxof 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:

d

dtρ(x, t) = ∂ρ

∂t +∂x

∂t · ∇ρ= ∂ρ

∂t +u· ∇ρ (1.3)

We eventually obtain:

Z

Ω

∂ρ

∂t +∇ ·(ρu)

dV = 0 (1.4)

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

∂ρ

∂t +∇ ·(ρu) = 0 (1.5)

This result is called themass conservation equation orcontinuity 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:

d dt

Z

Ω

dV = 0 (1.6)

This is equivalent to a solenoidal constraint on the velocity:

∇ ·u= 0 (1.7)

In this work, we will only be concerned with liquids. The speed of 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:

d dt

Z

Ω

ρudV =F (1.8)

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

d dt

Z

Ω

ρudV = Z

Ω

d

dt(ρu) + (ρu)∇ ·u

dV

= Z

Ω

ρd

dtu+ud

dtρ+ (ρu)∇ ·u

dV (1.9)

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:

d dt

Z

Ω

ρudV = Z

Ω

ρ∂u

∂t +ρu· ∇u

dV (1.10)

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We can furthermore write the net forceFas the volume integral of a force densityf. Following the approach of [Bat67], we will distinguish between two classes of forces contributing tof. On the one hand, we have volume or body forcesf_b, and surface forcesf_son the other hand. The former stem from long- range 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:

Z

Ω

f_sdV = I

∂Ω

τ·dS (1.11)

Using Gauss’s divergence theorem, we obtain:

ρ ∂u

∂t +u· ∇u

=−∇ ·τ+f_b (1.12)

As for now, we will not yet specify the form of the body forces, and concentrate on the structure of the tensorτ. It can be shown that angular momentum conservation requiresτ to be symmetric. Furthermore, for a fluid in rest, this tensor is diagonal and isotropic, i.e the stress tensor can be written asτ =−p1, withpthe thermodynamicpressure 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 definepas the mean normal stress, and term this quantitymechanical pressure. It follows then that τ^′ is a deviatoric tensor, i.e. its trace is zero.

Sinceτ^′ is related to a kind of internal friction mechanism, it can not directly depend on the values of the velocity itself. The structure ofτ^′ is further con- strained by assuming that the stress tensor is an isotropic and linear function of the velocity gradient tensor. A fluid with such properties is calledNewtonian.

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

τ^′ =η

∇u+ (∇u)^T−2

3(∇ ·u) 1

(1.13) The quantityηis called thedynamic 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:

τ^′ =η

∇u+ (∇u)^T

(1.14)

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We will furthermore define thekinematic viscosity ν asν =η/ρ.

Eventually, we obtain the following form for the momentum equation, better known as the notoriousNavier-Stokes equations:

ρ ∂u

∂t +u· ∇u

=−∇p+ρν∇²u+f_b (1.15) 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:

ρ ∂

∂t−ν∇²

∇ ·u=−∇²p−ρ∇ ·(u· ∇u) +∇ ·f_b (1.16) 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 termu· ∇uor 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:

∂

∂t 1

2ρu²

= −∇ ·

u1 2ρu²

− ∇ ·(pu)

+ρν∇ ·(u· ∇u)−ρν||∇u||² (1.17) In this expression, we have introduced the norm of the velocity gradient tensor as ||∇u||² =P

i

P

j(∂iuj)². The first two terms on the right-hand side are in divergence form. If we integrate this equation over a given volume, we find that the total work caused by the non-linear and pressure term reduces to a boundary integral. We call these termsenergy-conserving. The last two terms in the above equation concern viscous effects. Only the first one is energy- conserving. 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.

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1.1.5 Summary

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

∇ ·u = 0 (1.18)

∂u

∂t +u· ∇u = −∇

p ρ

+ν∇²u+ρ⁻¹f_b (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−unn=u_τ= 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 ano-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 forp.

1.2 Magnetohydrodynamics

In this section, we will treat the coupling between fluid dynamics and electro- dynamics. This coupling is twofold. On the one hand, electromagnetic 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-calledinduction equation. We will derive it and analyse it in terms of a few non-dimensional parameters.

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1.2.1 Classical electromagnetism

Classical electromagnetism [Jac99], in its most concise form, consists of the combination ofMaxwell’s equationsand an expression for the force on a charged particle or medium, theLorentz force. This latter law states that a certain mass, carrying a chargeqand moving with a velocityuin an electric fieldEand/or magnetic fieldB, will undergo the following forceF_L:

FL=q(E+u×B) (1.22)

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

Z

Ω

ρedV = X

q (1.23)

Z

Ω

JdV = X

qiu_i (1.24)

where the summation extends over all the particlesiwith chargeqiand velocity u_i within Ω. The Lorentz force densityf_L can then be written as:

f_L=ρeE+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

∂t +∇ ·J= 0 (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 = −∂B

∂t (1.27)

∇ ·E = ǫ⁻¹₀ ρe (1.28)

∇ ·B = 0 (1.29)

∇ ×B = µ0

J+ǫ0

∂E

∂t

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

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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, termedelectric conductivity. This is known asOhm’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:

∂ρe

∂t + σ ǫ0

ρe=−σ∇ ·(u×B) (1.33) The left-hand side of this equation represents an exponential decay on acharge relaxation time of τC = ǫ0σ⁻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 ∂tρe. 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 thatO(ρe) = O(ǫ0∇ ·(u×B)) =ǫ0U B0L⁻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⁻¹J. 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(ρeE) =ǫU LB0J

σ =τCU

LJB0=τCU

LO(J×B) (1.36)

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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 ρeE 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:

f_L=J×B (1.37)

The previous estimates and considerations can also be used to show that the term ǫ0µ0∂tE can be neglected in Amp`ere’s law (1.30). This term becomes important only if the characteristic velocity of the flow approach the speed of lightc= (µ0ǫ0)^−1/2. For much lower velocities however, we may approximate (1.30) as:

∇ ×B=µJ (1.38)

1.2.2 The induction equation

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

∂B

∂t =−∇ ×E=−∇ × σ⁻¹J

+∇ ×(u×B) (1.39) Furthermore, we use the pre-Maxwell version of Faraday’s law (1.38) to eliminate the current. The resulting expression becomes:

∂B

∂t =− 1

µσ∇ × ∇ ×B+∇ ×(u×B) (1.40) Using (A.17), (A.19) and the solenoidal character of both the velocity and magnetic field, we finally obtain:

∂B

∂t +u· ∇B=B· ∇u+ 1

µσ∇²B (1.41)

1.2.3 Summary

We have now all the elements that are required to describe the 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, ifB is initially solenoidal.

Therefore, Gauss’s law for the magnetic field (1.29) is thus required also for a

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

∇ ·u = 0 (1.42)

∂u

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

∂B

∂t +u· ∇B = B· ∇u+ 1

µσ∇²B (1.44)

∇ ·B = 0 (1.45)

We can write these equations under a non-dimensional form by the following substitutions: u → Uu, B → B0B, ∇ → L⁻¹∇, t → LU⁻¹t, J → σU B0J, p→ρU²p. We obtain:

∇ ·u = 0 (1.46)

∂u

∂t +u· ∇u = −∇p+Re⁻¹∇²u+NJ×B (1.47)

∂B

∂t +u· ∇B = B· ∇u+R_m⁻¹∇²B (1.48)

∇ ·B = 0 (1.49)

We see that, in a given geometry, we can characterize MHD flows by merely three dimensionless groups: theReynolds numberRe, theinteraction parameter (orStuart number)N, and themagnetic 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-of- magnitude estimate yields:

Re=O(u· ∇u)

O(ν∇²u) = U²L⁻¹ νL⁻²U =U L

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 asturbulence.

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:

Rm= O(u· ∇B)

(µσ)⁻¹∇²B =µσU L⁻¹B

L⁻²B =µσU L (1.51)

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The phenomenology of MHD systems will depend strongly on the value of Rm, as it is the only parameter governing the induction equation. WhenRmis 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 ofRmon 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= σB₀²L

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 scaleL: 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 theHartmann number, which is a hybrid ofReand N.

M =B0L rσ

ρν =√

N Re (1.54)

The square ofM 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 The quasi-static approximation

1.3.1 Simplified equations for R

_m

≪ 1

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

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terrestrial, laboratory and industrial flows, this parameter is typically small compared to one. The assumption thatRmis vanishing, allows for a substan- tial 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 partB^ext and a fluctuating partb:

∂b

∂t +u· ∇b=B^ext· ∇u+b· ∇u+ 1

µσ∇²b (1.55) An order-of-magnitude estimate of the convective and diffusive terms gives:

O(u· ∇b)

O((µσ)⁻¹∇²b) = O(b· ∇u)

O((µσ)⁻¹∇²b)= U bL⁻¹

(µσ)⁻¹bL⁻² =µσU L=Rm (1.56) wherebis 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

∂t =B^ext· ∇u+ 1

µσ∇²b (1.57)

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∇²b

τ(B^ext· ∇u) = L²µσ

LU⁻¹ =Rm (1.58)

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:

B^ext· ∇u+ 1

µσ∇²b= 0 (1.59)

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

O(b)

O(B^ext) =µσU L=Rm (1.60) We find thus that, in the limit of vanishingRm, 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 Amp`ere’s equation (1.38) to eliminatebfrom (1.59):

∇ × u×B^ext

− 1

σ∇ ×j= 0 (1.61)

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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:

u×B^ext−1

σj=∇φ (1.62)

Moreover, the total current densityJcan be written as a sum of its ‘induced’

part j and an externally imposed current source Jêxt: J = Jêxt+j. Jêxt is caused by an external electric fieldEêxt=−∇φêxt. Together with (1.62), we obtain.

J=σ −∇ φ+φ^ext

+u×B^ext

(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 potentialφ+φêxt: E=−∇(φ+ φêxt). If we replace nowφ+φêxt byφ, we can express the charge-conservation law (1.34) as:

∇ · σ −∇φ+u×B^ext

= 0 (1.64)

Ifσis a constant, which we will assume this throughout this work, we end up with a Poisson equation for the electrical potential:

∇²φ=∇ · u×B^ext

(1.65) Upon the rescalingφ→φU LB0, the full non-dimensional system of quasi-static MHD equations becomes:

∇ ·u = 0 (1.66)

∂u

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

∇²φ = ∇ · u×B^ext

(1.68) From now on, we will drop the superscriptext, and denote the external magnetic field asB. 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

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σ

σ w

Γ

Γ ’

σ w

t _w

n

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

Poisson equation (1.68) and solve this equation with appropriate jump 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:

σw

∂φw

∂n

_Γ

= σ ∂φ

∂n _Γ

(1.69) (∇^τφw)|^Γ = (∇^τφ)|^Γ (1.70)

φw|Γ = φ|Γ (1.71)

When, however, the wall thickness is small compared to the fluid domain, 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

∂n =−(∇τ·j_w,τ) (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·j_w , j_w,τ =j−jw,nn 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

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discharge tangentially in a quasi-two-dimensional way.

j_n,w|Γ^′−j_n,w|Γ =−tw∇τ·j_τ,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

∂n =∇^τ· σwtw

σ ∇^τφ

(1.74) or in non-dimensional form:

∂φ

∂n=∇^τ·(c∇^τφ) (1.75)

where thewall conductance ratio cis defined as:

c= σwtw

σL (1.76)

We now consider two limiting cases of this thin-wall condition. If the wall is perfectly conducting,σwand 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)

whereCis 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, forc= 0, the thin-wall condition simplifies to a homogeneous Neumann condition for the electric potential:

∂φ

∂n= 0 (1.78)

1.3.2 Phenomenology of the quasi-static regime

In this subsection, we will give an overview of the most relevant 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.

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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 unitsτJ. 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 di- rections 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

ρ1

2u²

dV + I

∂Ω

u

p+1 2ρu²

·dS=

−1 σ

Z

Ω

J²dV − I

∂Ω

φJ·dS (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

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

d dt

Z

Ω

ρx×udV = Z

Ω

x×(J×B) dV (1.80) After some tedious algebra, we can rewrite this as:

d dt

Z

Ω

ρx×udV = Z

Ω

σ((x×u)×B)×BdV + I

∂Ω

...·dS (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 main- tained, 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 xas 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 =B01_y. 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 inz-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 thusu=u(y)1x, with uandφobeying the following set of equations:

−∂p

∂x+ρν∂²u

∂y² +σB0

∂φ

∂z −B0u

= 0 (1.84)

∂²φ

∂y² = 0 (1.85)

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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

∂y² = 0 (1.86)

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

φ = Az+B (1.87)

u = 1 ρν

−∂p

∂xM⁻²+σB0A 1−cosh(M y) cosh(M)

(1.88) Here,A andB are integration constants, andM is the Hartmann number as defined in (1.54). The choice ofBis arbitrary, butAis determined by the fact that the total current in the combined fluid-wall domain should integrate to zero. The velocity profile is then:

u= 1 M

1 +c cM+ tanh(M)

1 ρν

∂p

∂x

1−cosh(M y) cosh(M)

(1.89) 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 asM⁻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.

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−10 −0.5 0 0.5 1 0.5

1 1.5

y u / U

b

M = 0 M = 10 M = 100

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 asu∝ −∂xp(σB₀²)⁻¹∝ −∂xpM⁻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 theJandf_Lbetweeny=−1 andy= 1 is zero.

To find the proper scaling, we integrate the remaining terms in the momentum balance betweeny=−1 andy= 1. We find:

Z 1

−1

∂p

∂xdy=ρν Z 1

−1

∂²u

∂y²dy=ρν ∂u

∂y

_y=1− ∂u

∂y _y=−1

!

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

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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 layersside layers. Analytical solutions are now only possible for a few specific combinations of the wall conductivityc, 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 ofc.

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 inz-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 ofO(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

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

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 asM^−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.

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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 numberM. 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 environ- ment (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 ofM, 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

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

quasi-static magnetohydrodynamics using an unstructured

finite volume solver:

development and applications

Stijn Vantieghem

Abstract

Acknowledgements

Contents

Introduction

Introductory aspects of magnetohydrodynamics

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 R

≪ 1

σ

σ w

Γ

Γ ’

σ w

t w

n

1.3.2 Phenomenology of the quasi-static regime

y u / U

1.3.3 Examples of quasi-static MHD flows

t _w