R EYNOLDSNUMBER :S PECTRALPROPERTIESANDTRANSITIONMECHANISMINASQUAREDUCT MHDTURBULENCEATLOWMAGNETIC

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MHD TURBULENCE AT LOW MAGNETIC

R EYNOLDS NUMBER : S PECTRAL

PROPERTIES AND TRANSITION MECHANISM IN A SQUARE DUCT

Maxime Kinet

UNIVERSITE´ LIBRE DE BRUXELLES

THESE PR` ESENT´ EE EN VUE DE´

LOBTENTION DU GRADE DE

DOCTEUR ENSCIENCES SOUS LA DIRECTION DU

PROF. B. KNAEPEN.

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c 2009 Maxime Kinet All Rights Reserved

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Contents

Abstract v

Introduction vii

1 Fundamental principles 1

1.1 Governing equations . . . 1

1.1.1 Fluid dynamics . . . 1

1.1.2 Magnetohydrodynamics . . . 5

1.1.3 Dimensionless form and governing parameters . . . 9

1.1.4 Dynamics at low magnetic Reynolds number . . . 13

1.1.5 Boundary conditions . . . 16

1.1.6 Kinetic energy . . . 18

1.2 Turbulence . . . 20

1.2.1 The nature of turbulence . . . 20

1.2.2 Kolmogorov’s theory and the energy cascade . . . 21

1.2.3 Magnetohydrodynamic turbulence . . . 24

1.3 Numerical simulation of turbulent flows . . . 28

1.3.1 General approach of numerical methods . . . 29

1.3.2 Spatial discretization methods . . . 31

I Homogeneous Magnetohydrodynamic Turbulence 33

2 Homogeneous flows 35 2.1 The formalism of homogeneous turbulence . . . 36

2.1.1 Fourier representation of a velocity field . . . 36

2.1.2 Energy and the energy spectra . . . 38

2.2 MHD turbulence in Fourier space . . . 41

2.3 Numerical experiment . . . 44

2.3.1 Description of the algorithm . . . 44

2.3.2 Description of the simulations . . . 46

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3 Spectral Properties 51

3.1 Introduction . . . 51

3.2 Spectral energetics . . . 53

3.2.1 Kinetic energy and dissipation spectra . . . 53

3.2.2 Nonlinear transfer . . . 58

3.3 Velocity anisotropy . . . 64

3.3.1 One point statistics . . . 64

3.3.2 Anisotropy of velocity spectra . . . 66

3.4 Conclusions . . . 73

4 Passive scalar transport 75 4.1 Introduction . . . 75

4.2 Numerical details . . . 79

4.3 Results and discussion . . . 81

4.3.1 Scalar variance decay . . . 81

4.3.2 Scalar anisotropy . . . 82

4.3.3 Scalar variance distribution in Fourier space . . . 86

4.4 Conclusions . . . 91

II Wall bounded Magnetohydrodynamic Flow 93

5 Numerical simulation of laminar duct flow 95 5.1 Introduction . . . 95

5.2 Basic phenomena . . . 96

5.2.1 Hartmann layers . . . 97

5.2.2 Side layers . . . 98

5.3 Numerical method . . . 102

5.3.1 Spatial discretization . . . 102

5.3.2 Time advancement . . . 107

5.4 Basic test cases . . . 109

6 Instabilities in MHD duct flows 113 6.1 Introduction . . . 113

6.2 Description of the simulations . . . 117

6.3 Results. . . 119

6.3.1 Departure from laminar regime . . . 119

6.3.2 Instability in the form of TW vortices . . . 123

6.3.3 Partial jet detachment . . . 129

6.3.4 A Glimpse at turbulent regime . . . 136

6.3.5 Transition process . . . 137 ii

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6.4 Conclusions . . . 142

7 Conclusion 145 A Vector calculus identities 147 A.1 Index notation . . . 147

A.2 Vector identities . . . 148

A.3 Integral theorems . . . 148

B Properties of liquid metals 151 C Stability of the Numerical Scheme 153 C.1 Theoretical background . . . 153

C.2 Testcases . . . 157

C.2.1 Channel Flow . . . 157

C.2.2 Duct Flow. . . 158

C.3 Comments . . . 159

Bibliography 170

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Abstract

Magnetohydrodynamics describes the motions of an electrically conducting fluid under the influence of magnetic fields. Such flows are encountered in a large variety of applications, from steel industry to heat exchangers of nuclear fusion reactors.

Here we are concerned with situations where the magnetic field is relatively strong and the flow manifests turbulent motions. The interaction of the fluid with the electromagnetic field is still insufficiently understood and efficient predicting methods are lacking. Our goal is to provide more insight on this problem by making heavy use of numerical methods. In this work, two different classes of problem are investigated.

First we consider that the turbulent character of the fluid is well developed and that solid boundaries are sufficiently far away to be completely neglected. The main effects of a strong magnetic field in that case are to damp the motion and to homogenize the flow along its direction, leading to a quasi two dimensional state.

Using numerical simulations we have studied the dynamics of the flow in Fourier space and in particular the non linear energy transfers between turbulent eddies.

Further we investigated the scale-by-scale anisotropy and compared various meth- ods to address this quantity. Finally, the evolution of a passive scalar embedded in the flow was analyzed and it turned out that the characteristic anisotropy of the velocity field is reflected in the distribution of the scalar quantity.

In the second problem, the flow in a duct of square cross section subject to a transverse magnetic field has been considered. Here, unlike in the previous situ- ation, the magnetic field has globally a destabilizing effect on the flow, because of the strong inhomogeneities it produces. For instance, high velocity regions de- velop along the walls that are parallel to the magnetic field. There, we are mostly interested in the possible development of persistent time-dependent fluctuations.

It is observed that the transition between laminar and turbulent regimes occurs through at least two distinct bifurcations. The first one takes place at moder- ate Reynolds number and is characterized by highly organized fluctuations. The second is encountered at higher Reynolds number and presents very strong and localized disturbances.

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Introduction

There is a great number of industrial and technological settings where electrically conducting fluids can be encountered. Liquid metals, such as aluminium, mercury or steel are good examples of such fluids. The interaction of the moving fluid with a magnetic field gives rise to a rich variety of phenomena that may be exploited in several ways.

For instance, the application of an electric field causes an electric current to circulate through the fluid. If, at the same time, a magnetic field is applied perpen- dicularly to the current, then a force starts to act on the fluid. This is the working principle of an electromagnetic pump. Such devices offer the great advantage that they can be used even with corrosive, chemically reactive or very hot fluids. Fur- thermore they do not have any moving mechanical part which makes them very robust.

On the other hand, if a magnetic field is applied perpendicularly to the direc- tion of motion of a fluid, an electric voltage appears between the walls of the pipe.

This voltage is proportional to the flow rate which can therefore be easily mea- sured. The concept of electromagnetic flowmeter was already known by Faraday who, in the nineteenth century, tried to calculate the flow rate of the Thames river by measuring the voltage induced by its motion through earth’s magnetic field.

Despite this pioneering experiment, it was only in the early 1960’s that mag- netic fields started to be considered as a potentially useful tool for metallurgy.

Static fields are now commonly used to control the flow motions and to suppress the undesirable turbulent fluctuations. For instance, in steel casting, the fluid must be as quiet as possible during its solidification in order to offer a high quality product. Another example is provided by the growth of semi-conductors crystals, where a very high level of purity can be achieved by electromagnetic control. On the contrary, oscillating magnetic fields can be used to stir the molten metal while it solidifies and hence reach a high level of homogeneity. The objective is to avoid the intrusion of an agitating device which, regarding the high temperature of liq- uid metals might be impractical. Other applications of rotating magnetic fields include heating and levitation. A recent review of the applications of magnetohy- drodynamics in materials processing is presented in Ref. [24].

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Aside from those applications where electromagnetic fields are used to pro- duce a desired effect, there are situations where the magnetic field is present for some unrelated reason. The electromagnetic interaction is then a constraint that must be taken into account in the design process. This is the case of aluminium re- duction cells. There large electric currents are driven through a solution of cryolite and alumina to produce aluminium. The efficiency of the process is strongly re- duced by instabilities created at the interface between the alumina and the molten aluminium. These instabilities finds their origin in the interaction of the electric current with the ambient magnetic field. The same kind of constrain occur in the so-called blankets of the future nuclear fusion reactors. Fusion energy has been recognized as a promising source for more than half a century and even if it is commonly admitted that industrial power plants will not be achievable within the next century, research is very active in this area. Therefore, the conceptual design of practical reactors is already well defined. Blankets are fundamental compo- nents for energy conversion, heat removal and tritium production. They consist in a network of pipes in which a liquid metal (lithium or a lead-lithium alloy) is circulated. They are located right behind the first wall of the plasma chamber and must fill several functions. First of all, they collect the fast neutrons emitted by the fusion reactions and convert their energy into heat. Secondly, they shield the superconductor magnets from any damaging radiation. Finally, they produce tritium, which is one of the fuel component that does not exist in nature. This latter operation is achieved through a nuclear reaction between lithium and a fast neutron. If the liquid metal serves as the cooling fluid, as well as the breeder mate- rial, it must flow through the blanket fast enough to ensure that the temperature of the structure remains within acceptable bounds. In addition, because the blankets are placed between the plasma and the magnetic coils, they undergo the presence of the confining magnetic field, whose intensity can be as large as5−10teslas.

Strong electromagnetic effects result and therefore, the thermohydraulic proper- ties of such a heat exchanger cannot be predicted by classical hydrodynamics.

All of theses problems belong to the branch of physics called magnetohy- drodynamics (MHD), which describes the coupling between fluid mechanics and electromagnetism1. Research on liquid metal MHD has received an increasing attention in the last decades, motivated by the numerous industrial applications related to it. For long, the investigation has been based on experimental observa- tions. With respect to hydrodynamics, experiments on MHD flows rise a certain number of issues especially regarding the handling of liquid metals and of very intense magnetic fields. As a consequence, experiments are usually very expen-

1It should be mentioned that, besides flows of liquid metals,MHDapplies to a large variety of subjects among which are solar physics, astrophysics, plasma physics, geo-dynamo theory, etc..

However, in this work, we shall only be concerned with liquid metals flows such as those occurring in the applications discussed.

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sive. In parallel, methods based on analytical developments have also been widely applied. Although they provide very useful information, their scope is often fairly limited due to the intrinsic complexity of fluid flows.

Recently, methods based on the numerical resolution of the mathematical equations have been developed. This technique, also known as computational fluid mechanics (CFD), has been improving continuously. The main reason for this is the fast evolution of high performance supercomputers. The advantages of computational methods with respect to experimental or analytical investigations are numerous : they provide accurate and detailed information about the flow variables, they can take into account the non-linearity of the equations, and the financial cost remains relatively affordable. The main drawback is that they are unable to reproduce exactly the complicated flows that are encountered in practi- cal situations. Therefore they are limited to rather simple cases. In the scientific community, and in engineering in general, they are now widely accepted as a very efficient prediction tool that will play a predominant role in the future.

The main purpose of this work is to investigate, using CFD methods, some aspects of liquid metal flows under intense magnetic fields. We will adopt a fun- damental approach which does not restrict the conclusions to one particular appli- cation. For this reason, only relatively simple problems, for which almost exact resolution methods can be applied, are considered. Two main types of problem have been investigated. In the first one, we consider that the fluid is in a turbulent state, and we explore the fundamental influence of a magnetic field on the tur- bulence of the fluid. In order to simplify the problem, we neglect the interaction of the fluid with any solid boundary and consider that the fluid is enclosed in an infinitely extended domain. There, the characteristics of the flow are statistically invariant by translation and we consider that the turbulence is homogeneous over the space domain. The second problem we have explored concerns the flow in a square duct of infinite length. In that case, the influence of the walls is critical and we are interested in the conditions of occurrence of turbulence.

The outline of this dissertation is as follows. In chapter1we introduce the nec- essary mathematical background to solve a problem of magnetohydrodynamics.

In chapter 2, we introduce the notion of homogeneous (isotropic or anisotropic) turbulence and we briefly discuss the elementary concepts that apply to magne- tohydrodynamic turbulence. Chapters 3 and 4 present studies conducted in the field of homogeneous turbulence. In chapter5 we begin the second part of this work. General features of magnetohydrodynamics in ducts are described. We also present the code of numerical resolution that was used for the purpose of simula- tions of duct flows. Chapter6is related to the development of unsteady flows in a square duct. Finally, chapter7summarizes the main results that were obtained and draws the conclusions.

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Several chapters of this manuscript are based on articles written in collabora- tion with other authors :

− Chapter 3 :

◦ P. Burattini, M. Kinet, D. Carati and B. Knaepen, Kinetic energy repar- tition in MHD turbulence, Advances in Turbulence XI, Proceedings of the 11th EUROMECH European Turbulence Conference, June 25-28, 2007, Porto, Portugal.

◦ P. Burattini, M. Kinet, D. Carati and B. Knaepen, Spectral energetics of quasi-static MHD turbulence, Physica D: Nonlinear Phenomena, 237(14-17), 2062-2066, 2008.

◦ P. Burattini, M. Kinet, D. Carati and B. Knaepen, Anisotropy of veloc- ity spectra in quasi-static magnetohydrodynamic turbulence, Physics of Fluids, 20(6):0651100, 2008.

− Chapter 4 :

◦ M. Kinet, P. Burattini, D. Carati and B. Knaepen, Anisotropy of trans- port properties in magnetohydrodynamic turbulent flows, Proceeding of the conference “Turbulence and Interactions”, May 29-June 02, 2006, Porquerolles, France.

◦ M. Kinet, P. Burattini, D. Carati and B. Knaepen, Decay of passive scalar fluctuations in homogeneous magnetohydrodynamic turbulence, Physics of Fluids, 20(7):075105, 2008.

− Chapter 6 :

◦ M. Kinet, S. Molokov and B. Knaepen, Instabilities and transition in ducts with electrically conducting walls, submitted to Physical Review Letters.

◦ B. Knaepen, M. Kinet and S. Molokov, Instabilities of side-layer jets in magnetohydrodynamic duct flows, Proceeding of the 6thInternational Conference on Electromagnetic Processing of Materials, October 19- 23, 2009, Dresden, Germany.

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Chapter 1

Fundamental principles

1.1 Governing equations

Magnetohydrodynamics (MHD for short) is mainly concerned with the dynamics of fluids that are good conductors of electricity and the interaction of their motions with a magnetic field. This discipline results from the coupling of hydrodynamics, which examines the motions of a fluid under the action of forces, with the theory of electromagnetism, characterizing the behavior of electromagnetic fields.

In this section, we derive the mathematical equations that govern magneto- hydrodynamics. It is shown that the interaction of the induced currents with the magnetic field gives rise to an additional volumetric force that may strongly affect the flow and bring completely new phenomena to the problem of fluid dynamics.

Furthermore, since we are mostly interested in the flows of liquid metals, we in- troduce an approximation, valid when such fluids are considered, which greatly simplifies the equations.

1.1.1 Fluid dynamics

In a first step, we leave the magnetic field aside and concentrate on the fluid part of the problem. Fluid mechanics is the science devoted to the study of fluid motions from a macroscopic point of view. This means that, although a liquid is made of microscopic molecules, it is treated as a continuous medium. Accordingly, when speaking about an infinitely small element of volume in the fluid, we mean that it is “physically” small, i.e. smaller than the spatial variations of the flow, but at the same time much larger than the interatomic distances. Therefore, expressions such as “motion of a fluid particle” is to be understood as the motion of a volume element containing many particles but considered as a single point, and not the motion of an individual particle within the fluid.

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

The state of a fluid is completely determined by the distribution of the fluid velocity, u(x, t), and two thermodynamic quantities, the pressure, P (x, t), and the density,ρ(x, t), which are all in principle functions of space coordinates,x= (x1, x2, x3) = (x, y, z), and time,t.

Conservation of mass

The law of conservation of matter can be expressed by considering some volume of fluid V, and establishing that the mass can only vary by flowing out of the domain. The total mass of the fluid is equal to R

ρdV and the fraction flowing through an infinitesimal element of the bounding surface per unit time is given byρu· dS. Here, dS is a vector whose magnitude is equal to the surface of the element, and pointing along the outward normal direction to the surface. The total mass of fluid flowing out of the surface ofV per unit time is equal toH

ρu· dS, and the variation of the mass insideV is

∂t Z

ρdV =− I

ρu· dS (1.1)

which, thanks to GAUSS’s theorem1, can be rewritten as

⇒∂

∂t Z

ρdV =− Z

∇·(ρu) dV

Z ∂ρ

∂t +∇·(ρu)

dV = 0

Since this must hold for every volume, the integrand in the latter must vanish :

∂ρ

∂t +∇·(ρu) = 0 (1.2)

or, in index notation2

∂ρ

∂t + ∂

∂xi

(ρui) = 0 (1.3)

This is known as the continuity equation.

Conservation of momentum

The equation of motion can be derived by applying NEWTON’s law to an infinites- imal fluid element. If the volume of the element is δV, its mass is ρδV and the

1See AppendixA, (A.13).

2EINSTEIN’s convention of implicit summation over repeated indices is assumed everywhere in the manuscript, unless explicitly mentioned.

2

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1.1. GOVERNING EQUATIONS total force acting on it is the sum of the pressure and viscous friction on its sur- face on one hand, and the sum of any additional volumetric force on the other.

Newton’s law writes (ρδV)Du

Dt = I

δS

(−P) dS+ I

δS

τ · dS+FbδV. (1.4) We shall analyse each term of this balance separately.

− The left hand side is the acceleration of the fluid element. The derivative Du/Dt is therefore the rate of change of the velocity of the fluid element as it moves in space, and not the rate of change of the velocity at a fixed point. An expression ofD(·)/Dtfor a given variable,f, can be obtained by expressing that a change inf due to small variations inx, y, z andtis given by

δf = ∂f

∂tδt+∂f

∂xδx+ ∂f

∂yδy+∂f

∂zδz.

Since we are interested in the changes in f following a fluid particle, we haveδx =uxδt, etc. Therefore we may write,

Df Dt = ∂f

∂t +∂f

∂xux+ ∂f

∂yuy+∂f

∂zuz

= ∂f

∂t + (u·∇)f = ∂f

∂t +ui

∂f

∂xi

, (1.5)

where ∂t is the rate of change offevaluated at a fixed location in space.

− The first term of the right hand side of (1.4) corresponds to the net pressure force acting on the fluid element. From Gauss’s theorem, it can be rewritten as follows

I

δS

(−P) dS= Z

δV

(−∇P) dV ∼=−(∇P)δV (1.6)

− The second term on the right hand side, whereτ =τij is the viscous stress tensor, represents the effects of internal friction on the motion of the fluid.

It can be transformed as : I

δS

τ · dS= I

δS

τjidSj

= Z

δV

∂τji

∂xj

dV ∼= ∂τji

∂xj

δV (1.7)

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

To proceed further, we need a constitutive relation betweenτij and the ve- locity field. When the velocity gradients are not too large, it can be assumed that the viscous stress tensor depends linearly on the velocity gradients as follows [59]

τij =ξ ∂ui

∂xj

+ ∂uj

∂xi −2 3δij

∂uk

∂xk

+ζδij

∂uk

∂xk

(1.8) where ξ and ζ are named the dynamic viscosity and the second viscosity respectively, and are independent of the velocity field. δij is the Kronecker delta (A.3).

− Finally,Fb, includes every other body force that may occur in the flow under consideration : gravity, buoyancy, Lorentz force, external stirring, etc.

Substituting (1.5), (1.6), (1.7) and (1.8) in (1.4), we obtain the famous NAVIER- STOKESequations

ρ ∂u

∂t + (u·∇)u

=−∇P +ξ∇2u+

ζ+1 3ξ

∇ (∇·u) +Fb (1.9) which, concurrently with mass conservation (1.2) and a state equation relatingP andρcompletely describes the motion of the fluid.

Incompressible flows

In many cases the density of the fluid can be assumed invariable, i.e. constant throughout the volume of the fluid and throughout its motion. This means that there are no appreciable compressions or expansions of the fluid. This is the case of incompressible flows and the continuity equation (1.2) reduces to

∇·u = 0 or ∂ui

∂xi = 0 (1.10)

This relation is also sometimes referred to as the divergence-free or solenoidal condition on the vector field. Navier-Stokes equations (1.9) become considerably simpler in this case :

∂u

∂t + (u·∇)u=−∇p+ν∇2u+fb (1.11) whereP has been replaced by the kinematic pressurep=P/ρ,ξby the kinematic viscosityν = ξ/ρand Fb by fb = Fb/ρ. We draw the attention to the fact that 4

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1.1. GOVERNING EQUATIONS pressure is no longer an independent variable. Indeed, taking the divergence of (1.11), and taking into account (1.10) results in the following Poisson equation

2p=−∇·[(u·∇)u] +∇·fb. (1.12) Hence, the divergence-free condition onu is sufficient to unambiguously deter- mine the pressure and a thermodynamic state equation is not needed anymore.

Incompressibility can reasonably be assumed when the fluid velocity is small as compared to the speed of sound.

1.1.2 Magnetohydrodynamics

In the following, we will describe the coupling between a fluid that conducts elec- tricity and a magnetic field. Practically, such a magnetic field is produced by electric current sources which may be either external to the field (in which case we talk about “applied” magnetic field), or induced by the motions of the fluid itself. Before entering the mathematical details, it is interesting to describe the physical phenomena that take place. Although this is somewhat artificial, it can be separated into three different steps :

(i) As the conducting fluid moves across the magnetic field lines, an electro- motive force develops in accordance to FARADAY’s law of induction. As a consequence, electrical currents appear in the fluid.

(ii) Those electrical currents give birth to an induced magnetic field, according to AMPERE` ’s law.

(iii) The total magnetic field (applied plus induced) interacts with the electrical currents to produce a LORENTZ force, whose action is generally to oppose the motion.

This highlights the fact that, if a magnetic field can modify the motion of a con- ducting fluid, the reciprocal is also true : a conducting fluid can affect a magnetic field.

Force acting on a fluid element

Let us first consider the force experienced by a moving particle in an electromag- netic field. It is given by the sum of Coulomb and Lorentz forces

FL =qe(E+u×B), (1.13)

where qe is the electric charge of the particle, u its velocity and, E and B are the electric and magnetic fields, respectively. If this equation is summed over an

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CHAPTER 1. FUNDAMENTAL PRINCIPLES elementary volume,P

qe becomes the charge density, ρe, and qeu becomes the current density,J. Hence the bulk force per unit of volume on a small element of the fluid is

FLeE+J×B. (1.14)

However, in most practical situations involving liquid metals, we can neglect the first term of this relationship. Indeed, at the macroscopic level, we can reasonably assume that charge neutrality is ensured. In other words, this means thatρe is so small thatρeE≪J×B. This hypothesis is called quasi-neutrality and does not, however, implies that the electrostatic field completely vanishes. Consequently, (1.14) simplifies into:

FL =J×B. (1.15)

This definition of the Lorentz force is not sufficient to determine the motions of the flow. As mentioned previously, the electric currents depend on the magnetic field, which itself depends on the velocity field. We need some extra equations to close the system and those are provided by MAXWELL’s theory of electromag- netism.

Maxwell’s equations

We refer to commonly known textbooks (like [58,31]) for the physical origin and meaning of those equations.

In a medium of permittivityǫ and permeabilityµ, any electromagnetic field obeys the following equations :

∇·E= ρe

ǫ (GAUSS’s law) (1.16a)

∇·B= 0 (Solenoidal constraint onB) (1.16b)

∇×E=−∂B

∂t (FARADAY’s law) (1.16c)

∇×B=µ

J+ǫ∂E

∂t

. (AMPERE` ’s law) (1.16d)

These equations must be supplemented by OHM’s law, which states that the current density is proportional to the electric field. If we assume that the medium is linear and isotropic, it is written, in the frame of the conductor :

J=σE.

When the medium is in motion, the electric field has to be modified through the 6

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1.1. GOVERNING EQUATIONS approximate Lorentz transformation3E→E+u×Band Ohm’s law then reads

J=σ(E+u×B). (1.17)

Amp`ere’s law (1.16d), specifies the magnetic field generated by a given dis- tribution of current. The second term of the right hand side is the displacement current, and is often referred to as the correction of Maxwell. Taking the diver- gence of (1.16d), and noting that∇·∇×(·) = 0yields

∇·J=−ǫ∂∇·E

∂t =−∂ρe

∂t

where Gauss’s law (1.16a), has been used for the last equality. This relationship is nothing more than the expression of charge conservation, which establishes that the rate at which electric charge is changing in a small control volume is equal to the density of charge flowing out of the volume per unit time. Consequently, Maxwell’s correction was introduced to ensure that electrical charge remains con- served. However in MHD this correction can reasonably be neglected as we show now. Using Ohm’s law the displacement current can be evaluated as

ǫ∂E

∂t ∼ ǫ σ

∂J

∂t ∼τe

∂J

∂t.

Here, τe = σ/ǫis called the charge relaxation time. It corresponds to the time for a charge perturbation to be redistributed in the medium at rest. In most MHD applications, it is of order10−18and therefore, we can assume thatτetJ≪J.

Note that this assumption is equivalent to neglecting events that happen at a speed of the order of the speed of light. Indeed, the assumption of negligible displacement currents implies that

µǫ∂E

∂t ≪∇×B.

Now, we can perform a dimensional analysis of this relationship. Thanks to (1.16c), E is of the order ℓ2B0v−1, ℓ, v and B0 being typical length, velocity and magnetic field intensity of the particular problem considered. Therefore, we must have

1

c2 ℓ τ−2B0 ≪ℓ−1B0

where c = (µǫ)−1/2 is the speed of light. Finally, the typical velocityv = ℓ/τ must satisfy

v c

2

≪1

3More precisely, we apply a development of the Lorentz transformation correct at the first order in vc2

, wherecis the speed of light andvthe speed of the frame of reference.

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

for the hypothesis of negligible displacement currents to hold.

Consequently, Amp`ere’s law can be written as

∇×B =µJ

and taking the divergence of the latter yields the modified equation of charge conservation :

∇·J = 0. (1.18)

We can now summarize the equations needed to describe the electrodynamic part of the problem :

∇·B= 0 (1.19a)

∇×B=µJ (1.19b)

∇×E=−∂B

∂t (1.19c)

J=σ(E+u×B) (1.19d)

FL=J×B (1.19e)

Induction equation

SubstitutingEin Faraday’s law (1.19c) by its value given by Ohm’s law, and using (1.19b) yields :

∂B

∂t =−∇×E

=−∇× J

σ −u×B

=∇×(u×B)− 1

µσ∇×∇×B.

Furthermore, noting that4∇×∇×B =−∇2B, this simplifies into

∂B

∂t =∇×(u×B) +χ∇2B (1.20)

whereχ = (µσ)−1 is the magnetic diffusivity. This equation constitutes a trans- port equation for B since, ifu is prescribed, it entirely governs the evolution of B. Using the vectorial identity (A.12) and the solenoidal nature of both fields, we can rewrite that transport equation as

∂B

∂t + (u·∇)B= (B·∇)u+χ∇2B. (1.21)

4See AppendixA

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1.1. GOVERNING EQUATIONS This allows us to distinguish a term of advection,(u·∇)B, and a source term, (B·∇)u, which represents the intensification of magnetic field by mechanical stretching of the field lines.

Bringing Navier-Stokes and Maxwell together

With the induction equation, we can write down the full set of equations governing the motion of a conducting fluid in a magnetic field.

In Eq. (1.11), we assumed that the volumetric Lorentz force was embedded in the body force term,fb. With the expression (1.19e), we can separate it from other additional body forces (which we denote byf) : fb = (1/ρ)J×B+f. The equa- tions needed to solve magnetohydrodynamic problems can thus be summarized as:

∂u

∂t + (u·∇)u=−∇p+ν∇2u+ 1

ρJ×B+f (1.22)

∂B

∂t + (u·∇)B= (B·∇)u+χ∇2B (1.23)

∇·u= 0 (1.24)

∇·B= 0 (1.25)

We voluntarily kept the current density in the equations since it can be expressed either by (1.19b) or using Ohm’s law (1.19d), leading to two equivalent formula- tions of the MHD equations.

1.1.3 Dimensionless form and governing parameters

It is often useful to nondimensionalize the variables appearing in the governing equations. This is achieved by performing the following replacements

u −→ U u

∇ −→ L−1 ∇ t −→ LU−1 t f −→ U2L−1 f B −→ B0 B

J −→ σU B0 J p −→ σL B02 U p

whereU is a typical velocity (e.g., the maximum velocity, or some mean velocity), La typical length scale (e.g., the geometrical dimension of a body immersed in the flow or the distance between solid walls, or in general, a distance over which

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

the velocity undergoes a perceptible change of the order of U), and B0 a typi- cal induction (e.g., the intensity of the applied magnetic field) of the flow under consideration5. Substituting in the governing equations (1.22), (1.23), (1.24) and (1.25), we obtain, after a little algebra :

∂u

∂t + (u·∇)u=−N∇p+Re−12u+NJ×B+f (1.26)

∂B

∂t + (u·∇)B =Rem−12B+ (B·∇)u (1.27)

∇·u= 0 (1.28)

∇·B = 0. (1.29)

In those equations appear the dimensionless parameters Re, Rem and N which are called the Reynolds number, magnetic Reynolds number and interaction pa- rameter, respectively. The exact definition of these parameters is provided in the next sections together with some physical interpretation.

Reynolds number

The Reynolds number,Reis defined as the ratio Re= U L

ν . (1.30)

It plays a crucial role in turbulence and in fluid dynamics in general. On the basis of dimensional arguments, it can be interpreted as the ratio of the order of magnitude of inertial forces to the order of magnitude of viscous forces:

Re= Inertial forces

Viscous forces ≃ |(u·∇)u|

|ν∇2u| .

In the case of small Reynolds number, Re ≪ 1, the viscosity has a consider- able effect on the entire flow by smoothing out all existing small-scale inho- mogeneities; hence space variations of the fluid quantities must take place very smoothly in the case of small Reynolds numbers. In the case of large Reynolds numbers the dominant role in the flow is played by the inertia forces, the action of which leads to the transfer of energy from the large-scale components of motion to the small-scale components, and consequently to the formation of sharp local

5Note that we introduced a scaling with electromagnetic quantities for the pressure. Another possibility could have been

p −→ U2p This is only a matter of convenience.

10

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1.1. GOVERNING EQUATIONS irregularities; In many systems of practical interest,Recan reach values as large as105–108. Therefore, we shall be mainly concerned with high Reynolds number flows in the subsequent chapters.

Interaction parameter N is defined as

N = σ L B20

ρ U . (1.31)

It represents dimensionally the ratio of the Lorentz force to the inertia of the fluid

N = Electromagnetic forces

Inertial forces ≃ |J×B|/ρ

|(u·∇)u|.

WhenN ≪1, the non-linear terms dominate the Lorentz force and the magnetic field has only a weak dynamical effect on the flow. In contrast, when N ≫ 1 the Lorentz force is significantly larger than inertia and the effect of the magnetic field becomes significant. From a physical point of view, it is also instructive to interpretN as the ratio of time scales characterizing the electromagnetic damping on the one hand, and the inertial process on the other. If we consider an isolated eddy of sizeLand velocityU, the mean time over which it significantly moves is the eddy-turnover timeτu =L/U, while the mean time it takes the Lorentz force to damp its energy is the Joule time τJ = ρ/σB02. The interaction parameter is exactly the ratio of those two time scales

N = τu τJ

.

Hence, whenN ≫ 1, the Lorentz force modifies significantly the eddy while, in the meantime, inertia can only weakly alter it.

Hartmann number

The ratio of electromagnetic to viscous forces leads to another dimensionless group, the Hartmann number

Ha2 = Electromagnetic forces

Viscous forces ≃ |J×B|

|ν∇2u|, which can be expressed as

Ha=L B0

r σ ρν =√

N Re. (1.32)

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

Two parameters among Re, N and Ha are necessary to characterize MHD problems. Conventionally, we choose (Re, N) when inertia plays a significant role, such as in fully developed homogeneous turbulence; and(Re, Ha)when the viscous phenomena are of primary importance, as in the presence of boundary layers. In fusion blankets applications, the Hartmann number is likely to be as large as104, due to the intensity of the magnetic field [10].

Magnetic Reynolds number

Finally, the so-called magnetic Reynolds number is given by Rem = U L

χ . (1.33)

and is indicative of the relative strength of advection versus diffusion in the induc- tion equation,

Rem = Advection of magnetic field

Diffusion of magnetic field ≃ |(u·∇)B|

|χ∇2B| .

The two major areas of magnetohydrodynamics may be discriminated in terms of this single dimensionless number

(i) Rem ≪ 1: This is the domain of liquid-metal magnetohydrodynamics, at least in practical applications. In this situation, the magnetic diffusion com- pletely overcomes the intensification of magnetic field due to the motion of the fluid. The magnetic field inside the fluid is determined, at the leading or- der, by the field generated by external magnets. The additional field, which is induced by the fluid, is weak in comparison.

(ii) Rem ≫ 1 : Here we are in the domain of “nearly infinite conductivity”, in which inductive effects dominate diffusion. This is usually the case in astrophysics where, because of the dimensions of the objects involved,Rem

is typically of the order104–1010. In this regime the magnetic field lines are almost frozen in the fluid. Moreover, the motions of the fluid may lead to the growth of a magnetic field whose origin is purely internal, a phenomenon known as the dynamo effect.

12

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1.1. GOVERNING EQUATIONS In the present work, we are mostly interested in the flows characterized by

Re≫1 (1.34)

Ha≫1 (1.35)

N ≥1 (1.36)

Rem ≪1. (1.37)

1.1.4 Dynamics at low magnetic Reynolds number

The assumption (1.37) usually holds for liquid metals at the laboratory scale. In- deed, for scientific experiments or industrial processes, χ ∼ 1m2/s,L ∼ 0.1m andU ∼0.001m/s →0.1 m/s . This gives6Rem ∼10−4→10−2[75].

When the magnetic Reynolds number is small, the governing equations can be substantially simplified [91,75,25], as we show now. The induction equation, in which we separate the externally applied part of the magnetic field,B0, suppos- edly constant and homogeneous, from its fluctuating part,Bcan be written

∂tB =−(u·∇)B+ ((B0+B)·∇)u+χ∇2B.

In the limit of smallRem, the advection terms involving the fluctuations of mag- netic field can be neglected with respect to diffusive terms. This can be justified by the following dimensional arguments

(u·∇)B ∼ U B

L , (B·∇)u ∼ U B

L , χ∇2B ∼ χB L2 ,

=⇒ (u·∇)B

χ∇2B = (B ·∇)u

χ∇2B ∼ U L χ ≪ 1

whereB is the typical order of magnitude of the fluctuations of magnetic field.

The induction equation reduces to

∂tB = (B0·∇)u+χ∇2B. (1.38) Additionally, we can assume that∂tB ≈0. This can be understood by analyzing the characteristic time scales of the right hand side of the latter equation. For the term (B0·∇)u, it is τu = L/U, while for the diffusive term, τχ = L2/χ; the ratio of those two terms isRem. Therefore, for vanishingRem, the diffusive time of the magnetic field is much smaller than the eddy turnover time, which means

6A table containing the main physical properties of some typical liquid metal can be found at AppendixB.

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

that the velocity evolves much slower thanB. As a consequence, the fluctuations of the magnetic field almost instantaneously adapt to the velocity field and can be specified by solving the following vectorial Poisson equation :

2B =−1

χ(B0·∇)u (1.39)

Note that assumingB ≪B0does not implyB ≈0, because then, there wouldn’t be any significant electrical currents, and hence no Lorentz force.

From this we derive the expression of the Lorentz force J×B = (B·∇)B−∇

B·B 2

∼= (B0·∇)B−∇(B0·B)

where (A.8) has been used, and the second order terms inB are neglected. The gradient term in the right hand side is called the magnetic pressure and can be included in the pressure. The system of equations to solve is thus :













∂u

∂t + (u·∇)u=−∇p˜+ν∇2u+ 1

µρ(B0·∇)B

2B =−1

χ(B0·∇)u

∇·u = 0

(1.40)

˜

pbeing the sum of the kinematic and magnetic pressurep˜=p+B0·B/2ρ. This simplified system is called the quasi-static approximation of MHD. The term inductionless is also sometimes used in the literature.

It should be mentioned that the quasi-static approximation is frequently formu- lated in a different, although strictly equivalent, way. In this approach, the current is computed from Ohm’s law (1.19d), rather than from Amp`ere’s law (1.19b). The starting point is to separate the electromagnetic vector field (E, J, B) into their mean values (E0, J0, B0) and perturbations (E,J, B) which occur due to the velocity field. These quantities are governed by

∇×E0 = 0, J0 =σE0,

∇×E =−∂B

∂t , J =σ(E+u×B0), 14

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1.1. GOVERNING EQUATIONS where we have neglectedu×Bin the last equation. Noting that Faraday’s equa- tion givesE ∼UB, the perturbation of the electric field can be neglected. Ohm’s law now becomes

J=J0+J =σ(E0+u×B0)

and sinceE0is irrotational, it can be written as the gradient of a potential,φ

J=σ(−∇φ+u×B0) (1.41)

while the leading order term in the Lorentz force is F=J×B0.

This is all we need to evaluate the Lorentz force in the momentum balance, since conservation of charge (1.18), provide a Poisson equation linkingφtouandB0 :

2φ =∇·(u×B0). (1.42)

From now on, we shall drop the subscript under B0 on understanding that B represents the imposed, steady, magnetic field. The simplified system of equations at hand is then











∂u

∂t + (u·∇)u=−∇p+ν∇2u+σ

ρ (−∇φ+u×B)×B

2φ=∇·(u×B)

∇·u= 0

(1.43)

The first formulation of quasi-static MHD equations (1.40) will be more con- venient when working in infinitely expanded space domain, where the Fourier transform allows the immediate resolution of the vectorial Poisson equation. When working in finite domains, such as channels, or ducts, the second formulation will be more suitable since only one Poisson equation needs to be solved, instead of three in the other formulation. Furthermore the magnetic field goes out of the fluid where Maxwell equations need to be solved.

The quasi-static approximation deserves some comments on its physical mean- ing. Two key elements are necessary for the quasi-static approximation to hold [91]:

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

(i) The first one assumes that the induction is dominated by diffusion phenom- ena. The magnetic field then behaves as it would in a motionless fluid, in the sense that any perturbation due to the motion of the fluid is strongly damped by diffusion, while it is not enhanced by advection. Therefore, fluctuations ofB are kept to a low level and do not alter the field produced by external coils significantly.

(ii) The second supposes that the characteristic time scale of diffusion of the magnetic field is vanishingly small, with respect to the time scale of fluid motions. This means that the magnetic field instantaneously adapts itself to the configuration of the fluid. We may therefore consider that, on the time scale of the motions of the fluid, the fluctuations of magnetic field are time-independent.

The main advantage of the approximation is that we can get rid of the induc- tion equation. As an aside, we should point out that solving exactly the full set of MHD equation for lowRem problems is a hard task, especially when numeri- cal methods are used. This comes, again, from the time scales of the phenomena that are very different. The evolution is discretely advanced in time by timesteps whose size is governed by the typical time scale of the fastest process to be mod- elled, in our case the diffusion of the induced magnetic field. This would result in an unaffordable increase of the total computation time. When using numeri- cal methods to investigate low magnetic Reynolds number MHD the quasi-static approximation is very helpful [19].

1.1.5 Boundary conditions

To be mathematically well-posed, MHD problems must be supplemented with appropriate initial and boundary conditions.

Velocity field

For time-dependent problems, the initial state of the velocity field has to be pro- vided along with conditions at the boundary of the physical domain,Γ. At a sta- tionary wall with outward normal unitn, we impose that the boundary conditions satisfied by the velocity are the impermeability condition

n·u= 0, and the no slip condition

u−n(n·u) = 0, which together yield

u = 0 onΓ. (1.44)

16

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1.1. GOVERNING EQUATIONS Electric current

Since the current density is a solenoidal vector field, we can show that the normal component must be continuous through a solid interface. Integrating the charge conservation equation over a cylindrical pillbox enclosing the interface, and re- ducing the volume of integration to zero yields :

0 = Z

Box

∇·JdV = I

ΓBox

J· dS= (J1−J2)·n.

At the interface between the fluid and a solid wall, the amount of current crossing the boundary will be determined by the conductivity of the external medium,σw. We can identify three main boundary conditions depending on the value ofσw. Insulating wall : In that case, σw = 0 and no current can penetrate the outer domain. Therefore, we have

J·n=Jn = 0 onΓ

This can be translated in a condition on φ by expressing Ohm’s law at the wall and taking into account the no-slip boundary condition :

Jn=−σ∂φ

∂n = 0→ ∂φ

∂n = 0 onΓ (1.45)

Perfectly conducting wall : For a perfectly conducting wall, withσw =∞, the potential has to be uniform in order to keep the electrical current to a finite value.

The wall potential can then be set to zero without loss of generality.

φ = 0 onΓ (1.46)

Conducting wall : If the walls have a finite conductivity,0 < σw < ∞, the current flowing in the wall has to be determined by a coupled electromagnetic problem in the outer domain. The boundary condition would then be given by the continuity condition on the current at the interface. However, if the wall is thin (in the sense that the thickness of the wall, tw, is much smaller than the characteristic dimension of the vessel) a simplified boundary condition can be derived [111]. Indeed, the local current entering the wall is discharged into the thin wall in a quasi two-dimensional way. To describe this phenomena we use the charge conservation equation in the wall in the form

∂jn

∂n =−∇τ ·jτ,

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

where the subscriptτ represents the projections on the plane tangential to the thin wall, such that j = jτ +jnn, or∇ = ∇τ +n∂n . Applying Ohm’s law in the wall, integrating it in the wall normal direction, and taking into account that the potential does not vary across the wall to the leading order of approximation, we finally find the relationship

−j·n=σ∂φ

∂n =∇τ ·(σwtwτφw) onΓ

wherenis the outward unit normal to the wall,φw is the electric potential at the fluid-wall interface, and tw the thickness of the conducting wall. If there is no contact resistance, we haveφw =φand we can write in dimensionless form

∂φ

∂n =∇τ·(c∇τφ) onΓ. (1.47)

cis called the wall conductance ratio, and is equal to

c= σwtw

σL .

1.1.6 Kinetic energy

The density of kinetic energy of a flow enclosed in a domain of volume V is defined as

e= 1 V

Z uiui

2 dV. (1.48)

The evolution of the kinetic energy with time can be established by multiply- ing the momentum equation (1.22) byui :

t

uiui

2 +uiujjui =−uiip+νuijjui+ 1

ρuiεijkJjBk+uifi

where εijk is the Levi-Civita symbol (see Appendix A). Integrating over the volume of fluid, using Gauss’s integral theorem (A.13), and taking into account 18

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1.1. GOVERNING EQUATIONS (1.10), (1.18) and (1.19d), this can be further developed as

∂t

Z uiui

2 dV

V =− I

uj

uiui

2 dSj

| {z V }

(1)=0

+ I

ujp dSj

| {z V }

(2)=0

+ 2ν I

uisij

dSj

| {z V }

(3)=0

−2ν Z

sijsij dV

| {z V}

(4)=ǫν

− 1 σρ

Z JiJi

dV

| {z V}

(5)=ǫJ

− 1 ρ

I

Jiφ dSi

| {z V}

(6)=ǫext

+ Z

uifi

dV

| {z V}

(7)=ǫf

.

The first three terms of the right hand side cancel out if the boundaries of the domain are stationary, because of the no-slip boundary condition on the velocity field. The term(4)is the viscous dissipation term, which involves the rate of strain tensor,sij = (∂jui+∂iuj)/2, while(5)is the Joule term. They are both strictly negative and therefore dissipate the kinetic energy. We can already conclude that the net effect ofB is to increase the dissipation of kinetic energy. The term(6) is a surface term as well. If the boundaries of the vessel are perfectly insulating, we haveJidSi = 0so that this term integrates to zero. If, on the other hand, the walls are perfectly conducting, the potential is zero at the wall and the integral cancels as well. If the walls have a finite conductivity, then there is an electric current that flows out of the fluid domain in the walls of the container. Since this current experiences Ohmic losses in the wall, this term corresponds to a flux of energy leaving the domain. The last term represents the power of any additional volumetric term, it may be negative, dissipating the energy just likeǫν andǫJ, or on the contrary positive, in which case it constitutes a injection of kinetic energy in the system. Finally, we end up with

te(t) =−ǫν −ǫJ −ǫextf. (1.49)

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

1.2 Turbulence

Let us now briefly outline one of the key issues of fluid dynamics, namely the turbulence of fluid flows. In a first stage, we set the MHD equations aside to focus on hydrodynamic turbulence.

1.2.1 The nature of turbulence

A flow is said to be turbulent whenever its velocity, pressure or temperature fluc- tuates in a disordered manner with extremely sharp and irregular space- and time- variations. Contrastingly, laminar flows are much smoother, well-behaved and

“predictable”. Unfortunately, this latter situation appears to be the exception rather than the rule, and it turns out that virtually all flows are turbulent, from those occurring in nature to those encountered in engineering applications. The main reason is that fluid motion is always inherently unstable and that emerg- ing instabilities can be suppressed only if the viscous dissipation is high enough.

More precisely turbulence develops when the inertial forces are predominant over the viscous ones, or when the Reynolds number is large. As a matter of fact, many fluids have a low viscosity, which result in very high Reynolds numbers. Conse- quently, any instability that may arise quickly develops and eventually generates chaotic motion, that is to say, turbulence. This chaotic behavior causes the flow to be completely unpredictable, despite the fact that the governing equations are perfectly deterministic. This is a consequence of the sensibility of the Navier- Stokes equations to initial and boundary conditions, i.e. that a small change to the initial conditions produces a large change to the subsequent motions. To illustrate the latter statement consider an experiment, for instance the flow around a cylin- der. We measure the velocity,u(x0, t)as a function of time at a given position, x0, in the wake of the cylinder. In general the time signal will be quite different from one realization of the experiment to another, although exactly similar, one.

Indeed, no matter how careful the experimentalist is, there will always be small dissimilarities in the conditions of the experiment, and those small differences will lead to totally different instantaneous measurements. This extreme sensitivity to initial conditions is now recognized has the signature of mathematical chaos. The complexity of turbulent flows implies that in practice there are relatively few situ- ations in which definite predictions can be made. Indeed, the Navier-Stokes equa- tions cannot in general be rigorously solved and it appears completely hopeless to search for a mathematical expression of the solution of an individual turbulent flow.

However, many decades of experiment have demonstrated that, although the velocity field appears to be chaotic, its statistical quantities (such as the mean ve- locity) are comparatively much more regular with space and time. This means 20

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1.2. TURBULENCE that it is possible to separate any velocity field into a mean part, which is a smooth and ordered function of position, and a fluctuating part, which incorporates the disordered nature ofu. It is now believed that any attempt at understanding tur- bulence must focus on the characterization of correlation functions and other sta- tistical moments of the velocity (or pressure, temperature) and of its fluctuations.

In the experimental example provided before, one should be rather interested for instance in

hhui(x0, t0)ii,hhui(x0, t0)uj(x1, t1)ii, . . .

where the double brackets denotes an averaging over several independent realiza- tions of the flow field. It seems natural then to seek dynamical equations for those statistical quantities from the Navier-Stokes equations, and actually, this turns out to be possible. However, this is not really a step forward. Indeed the system of equations hence obtained is not closed in the sense that, no matter how many ma- nipulations we perform there are always more statistical unknowns than equations relating them. This is known as the closure problem of turbulence.

One of the most enlightening picture of what is going on in a turbulent flow is provided by KOLMOGOROV’s theory of turbulence, which is briefly outlined in the next section.

1.2.2 Kolmogorov’s theory and the energy cascade

We consider a steady-on-average flow, characterized by a large Reynolds number.

The exact geometry of the problem is not important for now, we just keep in mind that the flow is highly turbulent. As mentioned earlier, it is suitable to decompose the velocity field into a mean, stationary part and a randomly fluctuating one :

u(x, t) =hhu(x, t)ii+u(x, t). (1.50) At any instant, the fluctuationsu may be regarded as a random ensemble of eddies. The size of the largest eddies is comparable to the characteristic length scale of the mean flow; but there are also many eddies that are much smaller.

Indeed fully developed turbulence contains a broad spectrum of eddy sizes.

It has been shown at section1.1.6that the rate at which energy is dissipated (in the absence of magnetic field) isǫ = 2νsijsij, where sij is the rate of strain tensor. Thus the dissipation of kinetic energy is particularly strong in regions of intense velocity gradients. This suggests that the mechanical energy in a turbulent flow must be dissipated in the smallest eddies. This observation, along with the fact that there exist a broad spectrum of eddy sizes led RICHARDSONto introduce the concept of energy cascade for high-Returbulence.

It can be described as follows. The big eddies, which are created by instabili- ties in the mean flow for instance, are break down under the action of inertia and

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CHAPTER 1. FUNDAMENTAL PRINCIPLES

rapidly evolve into smaller vortices. The latter are themselves unstable and give birth to even smaller structures. As a consequence, there is a continuous cascade of energy from the large scales of the motion to the small ones (Figure 1.1). It

Dissipation Injection

Energy Flux

Figure 1.1: Schematic representation of the energy cascade in a turbulent flow.

is important to note that in this process, viscosity plays absolutely no role. That is to say, sinceReis large, the large eddies do not experience significant viscous stresses. The whole process of energy transfer between two “generations” of ed- dies is governed by inertial forces. However, when the eddy size becomes so small thatRe (based on the size and velocity of the smallest eddies) is approximately unity, the cascade is interrupted. The viscous forces are no longer negligible and dissipation becomes important.

Later, Kolmogorov [56] extended the idea of an energy cascade to determine the small scales of turbulence. LetU and υ be the typical velocities associated with the largest and smallest eddies of the flow respectively. Also, letLandη be the length scales of the largest and smallest structures. The eddies break-up on a timescale of their turn-over time, that isτu =L/U, and so the rate at which energy is “cascading” from the large to the small eddies is of the orderΠ∼U2/(L/U) = U3/L. Since the flow is considered to be statistically steady, this must be exactly equal to the rate of dissipation of energy at the smallest scales, which isǫ∼νςijςij. ςij is the rate of strain tensor of the small eddies and scales therefore asςij ∼υ/η.

This establishes that the dissipation rate scales asǫ ∼ ν(υ/η)2. By equating the energy fluxΠin the cascade to the dissipation rateǫ, we have

U3/L∼ν υ22 . 22

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1.2. TURBULENCE On the other hand, the Reynolds number based on the size and velocity of the smallest scales must be of order unity

υη/ν ∼1.

The two last expressions may be combined to get

η∼LRe−3/4 or η∼ ν31/4

(1.51)

and υ ∼URe−1/4 or υ ∼(νǫ)1/4 (1.52)

where Reis based on the large scales eddies Re = UL/ν. The scales η and υ are called the Kolmogorov microscales of turbulence. The arguments leading to (1.51) and (1.52) may appear a little heuristic. Actually, the theory of Kolmogorov assumes that the small scales of turbulence are locally isotropic, unlike the very large scales which are influenced by the geometry of the problem, and therefore necessarily anisotropic. Hence, according to Kolmogorov, the small scales of tur- bulence are identical for jets, wakes, boundary layers, etc. The statistical behavior of the small scales is somehow universal and determined by the mean dissipation rate,ǫ, and the viscosity,ν, only. As a consequence of this hypothesis, it is natural to assume that when the range of fluctuations subject to this universal statistical regime extends to scales much greater thanη, there must exist a range of scales which are at the same time much larger thanηand much smaller than L. In this range, called the inertial range, viscosity plays no role and the statistics depend only on the energy flux in the cascade, ǫ. If the typical size of an eddy in the inertial range isℓ, its typical velocity is therefore of orderu∼(ǫℓ)3.

Although Kolmogorov’s theory has proven to be incomplete since then7, the estimations (1.51) and (1.52) turn out to conform remarkably well to the experi- mental data. Indeed, they represent some of the more useful results in turbulence theory!

One of the most interesting quantity in turbulence is the energy spectrum (see Chap. 2), E(k) which describes how the energy is distributed among eddies of different sizes. Usually it is expressed as a function of the wavenumber,k ∼ℓ−1, rather than the size of the eddies. The most striking result of Kolmogorov’s theory is the prediction that the energy spectrumE(k)behaves as [56]

E(k)∼ǫ2/3k−5/3 (1.53)

for scales belonging to the inertial subrange. This power law has been confirmed many times in experiments of turbulence.

7See for instance the discussion in [26], p377.

Figure

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